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{{Infobox Book
{{Infobox Book
|series       = Petroleum Engineering Handbook
|series = Petroleum Engineering Handbook
|editor-in-chief   = Larry W. Lake
|editor-in-chief = Larry W. Lake
|volume       = Volume I – General Engineering
|volume = Volume I – General Engineering
|vol editor = John R. Fanchi, Editor
|vol editor = John R. Fanchi, Editor
|date         = 2007
|date = 2007
|publisher   = Society of Petroleum Engineers
|publisher = Society of Petroleum Engineers
|image       = [[File:Vol1GECover.png|center|120px]]
|image = [[File:Vol1GECover.png|center|120px]]
|imagestyle   =  
|imagestyle =  
|chapter = Chapter 13 – Rock Properties
|chapter = Chapter 13 – Rock Properties
|ch author = M. Batzle, Colorado School of Mines, D.-H. Han, U. of Houston, and R. Hofmann, Colorado School of Mines
|ch author = M. Batzle, Colorado School of Mines, D.-H. Han, U. of Houston, and R. Hofmann, Colorado School of Mines
|ch author info =  
|ch author info =  
|page numbers   = 571-685
|page numbers = 571-685
|ISBN   = 978-1-55563-108-6
|ISBN = 978-1-55563-108-6
}}
}}<br/><br/>__TOC__
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== Introduction ==
== Introduction ==
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
<br>
<br/>Rock and fluid properties provide the common denominator around which we build the models, interpretations, and predictions of petroleum engineering, as well as geology and geophysics. We consider here the properties of sedimentary rocks, particularly those that make up hydrocarbon reservoirs. Usually, these consist of sandstones, limestones, and dolomites. We must be more inclusive, and consider rocks such as shales, evaporates, and diatomites because these provide the seals, bounding materials, or source rocks to our reservoirs. It is important to note that shales and claystones make up the most abundant rock type in the typical sedimentary column. Features such as seismic signature depend as much on the enclosing shale as on the reservoir sands.<br/><br/>In this chapter, we will tabulate important mineral and rock properties, and provide many of the mathematical models used to describe and predict properties. Much of this summary is drawn upon the extensive work and compellations already available. As examples, Clark<ref name="r1">_</ref> provides an extensive list of mineral and rock properties; Birch<ref name="r2">_</ref> presents tables of compressional velocities, and Gregory<ref name="r3">_</ref> gives a detailed overview of the use of rock property information in seismic interpretation. Castagna ''et al''.<ref name="r4">_</ref> focused on rock properties for use in amplitude versus offset analyses. Useful handbooks on this topic include Carmichael<ref name="r5">_</ref> and Lama and Vutukuri.<ref name="r6">_</ref><ref name="r7">_</ref> Probably the best reference covering a wide range of rock property formulas and models is Mavko ''et al''.<ref name="r8">_</ref> These references can be consulted for details not presented here.
Rock and fluid properties provide the common denominator around which we build the models, interpretations, and predictions of petroleum engineering, as well as geology and geophysics. We consider here the properties of sedimentary rocks, particularly those that make up hydrocarbon reservoirs. Usually, these consist of sandstones, limestones, and dolomites. We must be more inclusive, and consider rocks such as shales, evaporates, and diatomites because these provide the seals, bounding materials, or source rocks to our reservoirs. It is important to note that shales and claystones make up the most abundant rock type in the typical sedimentary column. Features such as seismic signature depend as much on the enclosing shale as on the reservoir sands.  
<br>
<br>
In this chapter, we will tabulate important mineral and rock properties, and provide many of the mathematical models used to describe and predict properties. Much of this summary is drawn upon the extensive work and compellations already available. As examples, Clark<ref name="r1" /> provides an extensive list of mineral and rock properties; Birch<ref name="r2" /> presents tables of compressional velocities, and Gregory<ref name="r3" /> gives a detailed overview of the use of rock property information in seismic interpretation. Castagna ''et al''.<ref name="r4" /> focused on rock properties for use in amplitude versus offset analyses. Useful handbooks on this topic include Carmichael<ref name="r5" /> and Lama and Vutukuri.<ref name="r6" /><ref name="r7" /> Probably the best reference covering a wide range of rock property formulas and models is Mavko ''et al''.<ref name="r8" /> These references can be consulted for details not presented here.  


=== Knowledge of Rock Properties Is Largely Empirical ===
=== Knowledge of Rock Properties Is Largely Empirical ===


Many theoretical models have been developed to predict or correlate specific physical properties of porous rock. Most theoretical models are built on simplified physical concepts: what are the properties of an ideal porous media. However, in comparison with real rocks, these models are always oversimplified (they must be, to be solvable). Most of these models are capable of "forward modeling" or predicting rock properties with one or more arbitrary parameters. However, as is typical in Earth science, our models cannot be inverted from measurements to predict uniquely real rock and pore-fluid properties. Many efforts have been made and will continue to be made to build porous rock models, but progress is very limited. Some of the most fundamental questions are still unanswered.  
Many theoretical models have been developed to predict or correlate specific physical properties of porous rock. Most theoretical models are built on simplified physical concepts: what are the properties of an ideal porous media. However, in comparison with real rocks, these models are always oversimplified (they must be, to be solvable). Most of these models are capable of "forward modeling" or predicting rock properties with one or more arbitrary parameters. However, as is typical in Earth science, our models cannot be inverted from measurements to predict uniquely real rock and pore-fluid properties. Many efforts have been made and will continue to be made to build porous rock models, but progress is very limited. Some of the most fundamental questions are still unanswered.<br/><br/>To establish the basic relationships between physical properties and rock parameters, laboratory investigations are made. Laboratory measurements of rock samples can provide controlled conditions and high data quality ("hard data"). These relationships can be extended to a larger scale, or can even be made scaleless. Typically, models and relations based on laboratory data are then applied to in-situ measurements to derive the parameters we actually need (say, permeability) from information we can actually collect (say, density and gamma ray radiation). The relative merits and problems associated with several rock and fluid measurement techniques are presented in '''Table 13.1'''.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
To establish the basic relationships between physical properties and rock parameters, laboratory investigations are made. Laboratory measurements of rock samples can provide controlled conditions and high data quality ("hard data"). These relationships can be extended to a larger scale, or can even be made scaleless. Typically, models and relations based on laboratory data are then applied to in-situ measurements to derive the parameters we actually need (say, permeability) from information we can actually collect (say, density and gamma ray radiation). The relative merits and problems associated with several rock and fluid measurement techniques are presented in '''Table 13.1'''.
<br>
<br>
<gallery widths=300px heights=200px>
File:Vol1 Page 572 Image 0001.png|'''Table 13.1'''
File:Vol1 Page 572 Image 0001.png|'''Table 13.1'''
</gallery>
</gallery><br/>Although many empirical relationships already have been established, when facing a frontier basin, new development areas, or untested portions of known formations, valid prediction of rock properties usually requires core data (including "sidewall" plugs). For many applications, standard trend data may not be adequate. A broad investigation is needed.
<br>
Although many empirical relationships already have been established, when facing a frontier basin, new development areas, or untested portions of known formations, valid prediction of rock properties usually requires core data (including "sidewall" plugs). For many applications, standard trend data may not be adequate. A broad investigation is needed.  


=== Philosophy for Rock Properties ===
=== Philosophy for Rock Properties ===


Many of the factors affecting rock properties are incompletely ascertained. For example, acoustic velocities can be affected by numerous parameters, many of which cannot be measured. In addressing a rock physics problem, the following aspects should be remembered:
Many of the factors affecting rock properties are incompletely ascertained. For example, acoustic velocities can be affected by numerous parameters, many of which cannot be measured. In addressing a rock physics problem, the following aspects should be remembered:
<br>
 
* There may be no exact solution.
*There may be no exact solution.
* Rock properties are controlled by rock parameters, and these physical correlations can be examined and recognized (although perhaps not understood).
*Rock properties are controlled by rock parameters, and these physical correlations can be examined and recognized (although perhaps not understood).
* Often nature gives us a break. At certain conditions, relationships between the rock properties and rock parameters can be simplified (such as Archie’s Law).
*Often nature gives us a break. At certain conditions, relationships between the rock properties and rock parameters can be simplified (such as Archie’s Law).
* We usually must settle on imperfect solutions with some uncertainty. Statistical trends or high and low bounds might be used to handle the uncertainty.
*We usually must settle on imperfect solutions with some uncertainty. Statistical trends or high and low bounds might be used to handle the uncertainty.
* Every measurement is, to some degree, wrong. The question is: Can we tolerate the errors and understand how they propagate through our analyses?  
*Every measurement is, to some degree, wrong. The question is: Can we tolerate the errors and understand how they propagate through our analyses?
 
We will begin this chapter with a suite of definitions and examples, then move on to data and models of individual properties. By necessity, we will be restricted in the material we can cover in a single chapter. As a result, we will not go into many details of rock fabrics and petrography. Also, with a few exceptions, the information provided here assumes that rocks are homogeneous and isotropic.
We will begin this chapter with a suite of definitions and examples, then move on to data and models of individual properties. By necessity, we will be restricted in the material we can cover in a single chapter. As a result, we will not go into many details of rock fabrics and petrography. Also, with a few exceptions, the information provided here assumes that rocks are homogeneous and isotropic.
<br>


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</div></div><div class="toccolours mw-collapsible mw-collapsed">
<div class="toccolours mw-collapsible mw-collapsed" >
== Rocks: Minerals Plus Pores ==
== Rocks: Minerals Plus Pores ==
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
<br>
<br/>Rocks are defined for our purposes as aggregates or mixtures of minerals plus pores. The three general rock types are classified as igneous, metamorphic, and sedimentary. Although hydrocarbon reservoirs have been found in all three rock types, we will consider here primarily sedimentary rocks, by far the most common rocks associated with hydrocarbons.<br/><br/>Minerals are defined as naturally occurring solids: They have a definite structure, composition, and suite of properties that are either fixed or vary systematically within a definite range. Although there are dozens of elements and hundreds of described minerals available in the Earth’s crust, the actual number that we must concern ourselves with for reservoir engineering purposes is remarkably small. Classification can be broken into silicates, carbonates, sulfates, sulfides, and oxides. In addition, "solid" organic mixtures such as coal or bitumen can be abundant. Common sedimentary silicates include quartz, feldspars, micas, zeolites, and clays. Carbonates usually consist of calcite and dolomite, although siderite may be present. Gypsum and anhydrite are the most common sulfates, with pyrite the typical sulfide. Oxides are usually materials such as magnetite and hematite. For most of our purposes, we can further restrict our attention to the subset of quartz, feldspars, clays, calcite, dolomite, and anhydrite. A working knowledge of six or so minerals fulfills most engineering needs.<br/><br/>Clays represent an entire family of minerals with widely differing properties. This situation is compounded by the fact that clays are among the most abundant minerals in the sedimentary section. Clays are also problematic because their properties can vary with the in-situ pressure, temperature, and chemical environment. These issues have led to an unfortunate bias against clays when measuring or describing rocks. A "clean" sand, for example, is one that has little or no clay. "Dirty" sandstones or limestones have significant amounts of clay. Clays and their influence on rock properties remain poorly understood and continue to be an area requiring intensive research.<br/><br/>The properties of primary engineering interest are often controlled more by the rock fabric than by the bulk composition. The "holes" are usually more important than the mineral frame. With the following few examples, we will see many of the most common sedimentary rock forms and textures. Numerous attempts have been made to extract rock properties from images of the rock and pore space.<ref name="r9">_</ref><ref name="r10">_</ref><ref name="r11">_</ref> These techniques often work well, but depend on the observation scale, representative nature of the image, and internal heterogeneity.<br/><br/>A thin section of clean sandstone is shown in '''Fig. 13.1'''. Under plane-polarized light, quartz grains appear white and pores are stained blue. This is a high-porosity, friable sample that has not undergone substantial consolidation. Silica cement can be seen coating the individual grains and bonding the largely unchanged, rounded quartz grains. Grain-to-grain stress is indicated by the fractures radiating from points of grain contact. Although these fractures have a relatively small volume, they have a disproportionately large influence on the mechanical properties, particularly the pressure dependence. With continued diagenesis, quartz grains typically would become intergrown, and large amounts of cement would develop, reducing the pore volume.<br/><br/><gallery widths="300px" heights="200px">
Rocks are defined for our purposes as aggregates or mixtures of minerals plus pores. The three general rock types are classified as igneous, metamorphic, and sedimentary. Although hydrocarbon reservoirs have been found in all three rock types, we will consider here primarily sedimentary rocks, by far the most common rocks associated with hydrocarbons.  
<br>
<br>
Minerals are defined as naturally occurring solids: They have a definite structure, composition, and suite of properties that are either fixed or vary systematically within a definite range. Although there are dozens of elements and hundreds of described minerals available in the Earth’s crust, the actual number that we must concern ourselves with for reservoir engineering purposes is remarkably small. Classification can be broken into silicates, carbonates, sulfates, sulfides, and oxides. In addition, "solid" organic mixtures such as coal or bitumen can be abundant. Common sedimentary silicates include quartz, feldspars, micas, zeolites, and clays. Carbonates usually consist of calcite and dolomite, although siderite may be present. Gypsum and anhydrite are the most common sulfates, with pyrite the typical sulfide. Oxides are usually materials such as magnetite and hematite. For most of our purposes, we can further restrict our attention to the subset of quartz, feldspars, clays, calcite, dolomite, and anhydrite. A working knowledge of six or so minerals fulfills most engineering needs.  
<br>
<br>
Clays represent an entire family of minerals with widely differing properties. This situation is compounded by the fact that clays are among the most abundant minerals in the sedimentary section. Clays are also problematic because their properties can vary with the in-situ pressure, temperature, and chemical environment. These issues have led to an unfortunate bias against clays when measuring or describing rocks. A "clean" sand, for example, is one that has little or no clay. "Dirty" sandstones or limestones have significant amounts of clay. Clays and their influence on rock properties remain poorly understood and continue to be an area requiring intensive research.  
<br>
<br>
The properties of primary engineering interest are often controlled more by the rock fabric than by the bulk composition. The "holes" are usually more important than the mineral frame. With the following few examples, we will see many of the most common sedimentary rock forms and textures. Numerous attempts have been made to extract rock properties from images of the rock and pore space.<ref name="r9" /><ref name="r10" /><ref name="r11" /> These techniques often work well, but depend on the observation scale, representative nature of the image, and internal heterogeneity.  
<br>
<br>
A thin section of clean sandstone is shown in '''Fig. 13.1'''. Under plane-polarized light, quartz grains appear white and pores are stained blue. This is a high-porosity, friable sample that has not undergone substantial consolidation. Silica cement can be seen coating the individual grains and bonding the largely unchanged, rounded quartz grains. Grain-to-grain stress is indicated by the fractures radiating from points of grain contact. Although these fractures have a relatively small volume, they have a disproportionately large influence on the mechanical properties, particularly the pressure dependence. With continued diagenesis, quartz grains typically would become intergrown, and large amounts of cement would develop, reducing the pore volume.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 574 Image 0001.png|'''Fig. 13.1 – Common sandstone textures include point contacts, cements, and microfractures. These microstructures determine the properties of the rock on a whole.'''
File:vol1 Page 574 Image 0001.png|'''Fig. 13.1 – Common sandstone textures include point contacts, cements, and microfractures. These microstructures determine the properties of the rock on a whole.'''
</gallery>
</gallery><br/>A Scanning Electron Microscope (SEM) image of another sandstone is seen in '''Fig. 13.2'''. A higher degree of compaction is indicated here by the intergrown, sutured contacts of the quartz grains (gray areas). A grain undergoing alteration (a) as well as some of the matrix quartz (b) contain isolated, ineffective porosity. Fractures are again present, particularly near point of grain contact. Many of these fractures, however, may be caused by stress relief as the sample was cored, or by the cutting and polishing. The most obvious features are the contorted and rotated mica grains (d). These micas were crushed due to compaction, and now host numerous sets of parallel fractures. Some diagenetic clays are also beginning to grow in the pore spaces and act as a cement.<br/><br/><gallery widths="300px" heights="200px">
<br/>
A Scanning Electron Microscope (SEM) image of another sandstone is seen in '''Fig. 13.2'''. A higher degree of compaction is indicated here by the intergrown, sutured contacts of the quartz grains (gray areas). A grain undergoing alteration (a) as well as some of the matrix quartz (b) contain isolated, ineffective porosity. Fractures are again present, particularly near point of grain contact. Many of these fractures, however, may be caused by stress relief as the sample was cored, or by the cutting and polishing. The most obvious features are the contorted and rotated mica grains (d). These micas were crushed due to compaction, and now host numerous sets of parallel fractures. Some diagenetic clays are also beginning to grow in the pore spaces and act as a cement.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 575 Image 0001.png|'''Fig. 13.2 – Scanning electron microscope (SEM) image of sandstone AT49 showing numerous compaction features. Some grains are either altering (a) or have internal, ineffective porosity (b). Fractures (c) cut numerous grains. Mica plates (d) are rotated and crushed, forming parallel sets of microfractures.'''
File:vol1 Page 575 Image 0001.png|'''Fig. 13.2 – Scanning electron microscope (SEM) image of sandstone AT49 showing numerous compaction features. Some grains are either altering (a) or have internal, ineffective porosity (b). Fractures (c) cut numerous grains. Mica plates (d) are rotated and crushed, forming parallel sets of microfractures.'''
</gallery>
</gallery><br/>A cementation "front" is visible in '''Fig. 13.3'''. Cements come in a wide variety of forms. Open pores are black in the SEM image. In this case, the lighter gray calcite has filled the pores in the lower portion of the image. Unlike the dispersed silica and clay cements seen in the previous figures, the calcite is deposited with an abrupt front. This kind of texture is common for carbonate cements in sands and is probably caused by the availability of crystal nucleation sites available to a slightly supersaturated pore fluid. We would obviously expect vastly different properties of the uncemented vs. cemented portions separated by only a few grain diameters. This rock is an example of the extreme heterogeneity that can frequently occur even within the same small geologic unit of the same formation.<br/><br/><gallery widths="300px" heights="200px">
<br/>
A cementation "front" is visible in '''Fig. 13.3'''. Cements come in a wide variety of forms. Open pores are black in the SEM image. In this case, the lighter gray calcite has filled the pores in the lower portion of the image. Unlike the dispersed silica and clay cements seen in the previous figures, the calcite is deposited with an abrupt front. This kind of texture is common for carbonate cements in sands and is probably caused by the availability of crystal nucleation sites available to a slightly supersaturated pore fluid. We would obviously expect vastly different properties of the uncemented vs. cemented portions separated by only a few grain diameters. This rock is an example of the extreme heterogeneity that can frequently occur even within the same small geologic unit of the same formation.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 576 Image 0001.png|'''Fig. 13.3 – SEM image of sandstone AT41 showing a progressing calcite cementation front.'''
File:vol1 Page 576 Image 0001.png|'''Fig. 13.3 – SEM image of sandstone AT41 showing a progressing calcite cementation front.'''
</gallery>
</gallery><br/>Carbonates can have extremely complex textures resulting form the mixture of fossils and matrix building the rock. In '''Fig. 13.4''', an optical image demonstrates the multitude of forms that can be present. Shell fragments appear as crescent shapes in cross section. Much of the material between fragments can be filled with carbonate mud, reducing the porosity substantially. In this sample, bulk porosity is dominated by the larger disconnected vugs. Such vugs can occur as parts of fossils or as a result of chemical dissolution after deposition. Here, a coating of crystals has grown on the vug surfaces. Because of the wide range of sizes, shapes, and compositions that can occur in carbonate rocks, they are often difficult to characterize with core or even log sampling.<br/><br/><gallery widths="300px" heights="200px">
<br/>
Carbonates can have extremely complex textures resulting form the mixture of fossils and matrix building the rock. In '''Fig. 13.4''', an optical image demonstrates the multitude of forms that can be present. Shell fragments appear as crescent shapes in cross section. Much of the material between fragments can be filled with carbonate mud, reducing the porosity substantially. In this sample, bulk porosity is dominated by the larger disconnected vugs. Such vugs can occur as parts of fossils or as a result of chemical dissolution after deposition. Here, a coating of crystals has grown on the vug surfaces. Because of the wide range of sizes, shapes, and compositions that can occur in carbonate rocks, they are often difficult to characterize with core or even log sampling.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 577 Image 0001.png|'''Fig. 13.4 – Thin-section image of carbonate textures; in this limestone, numerous curved shell fragments [e.g., at (a)] are packed together and bound by a fine-grained lime mud. Calcite crystals (b) are growing into the pore spaces.'''
File:vol1 Page 577 Image 0001.png|'''Fig. 13.4 – Thin-section image of carbonate textures; in this limestone, numerous curved shell fragments [e.g., at (a)] are packed together and bound by a fine-grained lime mud. Calcite crystals (b) are growing into the pore spaces.'''
</gallery>
</gallery><br/>Dolomites are usually formed by recrystallization of original aragonite or calcite crystals in sediments. Magnesium in the pore fluids replace some of the calcium, forming a Mg-Ca carbonate structure. Because of the greater density of dolomite, this transformation can include a porosity increase. Sometimes, the replacement can be subtle, and original sedimentary structures and fossil forms can be preserved. Often, however, the recrystalliztion largely destroys the original rock fabric and rhombohedral dolomite crystals appear, as at (a) in '''Fig. 13.5'''. The other intergrown dolomite crystals form porosity that is polygonal. In this sample, many of the pores are coated (b) with pyrobitumin, a complex organic material similar to coal. This pyrobitumen is sometimes incased within dolomite crystals. In this case, it will lower the apparent grain density and strength of the rock.<br/><br/><gallery widths="300px" heights="200px">
<br/>
Dolomites are usually formed by recrystallization of original aragonite or calcite crystals in sediments. Magnesium in the pore fluids replace some of the calcium, forming a Mg-Ca carbonate structure. Because of the greater density of dolomite, this transformation can include a porosity increase. Sometimes, the replacement can be subtle, and original sedimentary structures and fossil forms can be preserved. Often, however, the recrystalliztion largely destroys the original rock fabric and rhombohedral dolomite crystals appear, as at (a) in '''Fig. 13.5'''. The other intergrown dolomite crystals form porosity that is polygonal. In this sample, many of the pores are coated (b) with pyrobitumin, a complex organic material similar to coal. This pyrobitumen is sometimes incased within dolomite crystals. In this case, it will lower the apparent grain density and strength of the rock.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 578 Image 0001.png|'''Fig. 13.5 – Thin-section image of a dolomite. Dolomite rhombohedra (a) are common. Within the pore space and between grains are black layers of phyrobitumen (b and c).'''
File:vol1 Page 578 Image 0001.png|'''Fig. 13.5 – Thin-section image of a dolomite. Dolomite rhombohedra (a) are common. Within the pore space and between grains are black layers of phyrobitumen (b and c).'''
</gallery>
</gallery><br/>As mentioned, clays are among the most abundant minerals. These minerals can influence or control physical properties to a major degree. In addition, many clays are sensitive to the environment and will change properties and forms under different conditions. An example of such "sensitive" clay fabrics is shown in '''Fig. 13.6'''. Note that the scale is much finer here than in previous figures. In '''Fig. 13.6a''', chlorite originally coats the quartz grains. On top of the chlorite, a smectite coating was developed. This core sample was allowed to dry, and the smectite collapsed, forming long slender columns in the pore space. Resaturating the rock with distilled water allowed the smectite coating to expand and fill the pore space ('''Fig. 13.6b'''). The closed pores will obviously have different fluid-flow characteristics. In this case, we cannot assume the mineral in is a passive, inert solid. This rock will change properties according to pore fluid chemistry.<br/><br/><gallery widths="300px" heights="200px">
<br/>
As mentioned, clays are among the most abundant minerals. These minerals can influence or control physical properties to a major degree. In addition, many clays are sensitive to the environment and will change properties and forms under different conditions. An example of such "sensitive" clay fabrics is shown in '''Fig. 13.6'''. Note that the scale is much finer here than in previous figures. In '''Fig. 13.6a''', chlorite originally coats the quartz grains. On top of the chlorite, a smectite coating was developed. This core sample was allowed to dry, and the smectite collapsed, forming long slender columns in the pore space. Resaturating the rock with distilled water allowed the smectite coating to expand and fill the pore space ('''Fig. 13.6b'''). The closed pores will obviously have different fluid-flow characteristics. In this case, we cannot assume the mineral in is a passive, inert solid. This rock will change properties according to pore fluid chemistry.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 579 Image 0001.png|'''Fig. 13.6 – SEM images of a smectite-rich sandstone. When dry (a), the smectites have collapsed. After saturation with distilled water, the smectites have expanded (b) to plug the pore space.'''
File:vol1 Page 579 Image 0001.png|'''Fig. 13.6 – SEM images of a smectite-rich sandstone. When dry (a), the smectites have collapsed. After saturation with distilled water, the smectites have expanded (b) to plug the pore space.'''
</gallery>
</gallery><br/>The most common sedimentary rock types are shales and silts. In '''Fig. 13.7''', white quartz grains float in the surrounding clay matrix. Black organic material in thin layers indicates the horizontal bedding. As a result, this rock has properties that vary strongly with direction and are thus anisotropic. This material could serve as both a source rock and reservoir seal. This sample demonstrates how a mudstone or shale could have a complex composition. Although clays typically make up a large portion of fine-grained rocks, terms such as "clay" and "shale" are not synonymous.<br/><br/><gallery widths="300px" heights="200px">
<br/>
The most common sedimentary rock types are shales and silts. In '''Fig. 13.7''', white quartz grains float in the surrounding clay matrix. Black organic material in thin layers indicates the horizontal bedding. As a result, this rock has properties that vary strongly with direction and are thus anisotropic. This material could serve as both a source rock and reservoir seal. This sample demonstrates how a mudstone or shale could have a complex composition. Although clays typically make up a large portion of fine-grained rocks, terms such as "clay" and "shale" are not synonymous.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 580 Image 0001.png|'''Fig. 13.7 – Thin-section image of an organic-rich siltstone. Quartz grains (white) are surrounded by silt and clay (brown). Thin organic layers (black) are aligned to give the rock a strong anisotropy.'''
File:vol1 Page 580 Image 0001.png|'''Fig. 13.7 – Thin-section image of an organic-rich siltstone. Quartz grains (white) are surrounded by silt and clay (brown). Thin organic layers (black) are aligned to give the rock a strong anisotropy.'''
</gallery>
</gallery><br/>Most sedimentary rocks have porosities under 0.50 (fractional). This is easy to understand, particularly with coarser clastic sediments, in which open grain packings that can support a matrix framework have maximum porosities around 0.45. Exceptions to this and other generalizations can occur, and an example is shown in '''Fig. 13.8'''. This globigerina "ooze" is composed largely of the small shells or tests of organisms. The matrix mud fills the region between tests, but interiors remain empty. In addition, the tests themselves are porous. As a result, porosities can be as high as 0.8. Despite these huge porosities, because of the isolated nature of the pores, permeability can be in the microdarcy range. A similar situation often occurs in shallow clay-rich sediments where the open clay plate structure results in initial very high porosities. In the remainder of this chapter, however, these types of sediments will be considered exceptional and will not be included in our analyses.<br/><br/><gallery widths="300px" heights="200px">
<br/>
Most sedimentary rocks have porosities under 0.50 (fractional). This is easy to understand, particularly with coarser clastic sediments, in which open grain packings that can support a matrix framework have maximum porosities around 0.45. Exceptions to this and other generalizations can occur, and an example is shown in '''Fig. 13.8'''. This globigerina "ooze" is composed largely of the small shells or tests of organisms. The matrix mud fills the region between tests, but interiors remain empty. In addition, the tests themselves are porous. As a result, porosities can be as high as 0.8. Despite these huge porosities, because of the isolated nature of the pores, permeability can be in the microdarcy range. A similar situation often occurs in shallow clay-rich sediments where the open clay plate structure results in initial very high porosities. In the remainder of this chapter, however, these types of sediments will be considered exceptional and will not be included in our analyses.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 581 Image 0001.png|'''Fig. 13.8 – Thin-section image of fossil-rich globigerina ooze. Because of the porosity of the individual fossil tests, the total rock porosity reaches 80%. Much of this is microporosity. Despite the high porosity and soft nature of this rock, a fracture has formed across the sample.'''
File:vol1 Page 581 Image 0001.png|'''Fig. 13.8 – Thin-section image of fossil-rich globigerina ooze. Because of the porosity of the individual fossil tests, the total rock porosity reaches 80%. Much of this is microporosity. Despite the high porosity and soft nature of this rock, a fracture has formed across the sample.'''
</gallery>
</gallery><br/>The rock images shown in these several figures are meant to convey a feel for the types of textures common in sedimentary rocks, and that influence physical properties. We will refer back to these images later in the chapter. These few images can in no way be considered a complete description of rock textures. For a more thorough treatment, the reader should consult one of the standard petrography texts or pertinent papers.<ref name="r12">_</ref><ref name="r13">_</ref><ref name="r14">_</ref>
<br/>
</div></div><div class="toccolours mw-collapsible mw-collapsed">
The rock images shown in these several figures are meant to convey a feel for the types of textures common in sedimentary rocks, and that influence physical properties. We will refer back to these images later in the chapter. These few images can in no way be considered a complete description of rock textures. For a more thorough treatment, the reader should consult one of the standard petrography texts or pertinent papers.<ref name="r12" /><ref name="r13" /><ref name="r14" />  
<br>
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" >
== Density and Porosity ==
== Density and Porosity ==
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
=== Basics and Definitions ===
=== Basics and Definitions ===


Density is defined as the mass per volume of a substance.
Density is defined as the mass per volume of a substance.<br/><br/>[[File:Vol1 page 0578 eq 001.png|RTENOTITLE]]....................(13.1)<br/><br/>typically with units of g/cm<sup>3</sup> or kg/m<sup>3</sup>. Other units that might be encountered are lbm/gallon or lbm/ft<sup>3</sup> (see '''Table 13.2''').<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
[[File:Vol1 page 0578 eq 001.png]]....................(13.1)
<br>
<br>
typically with units of g/cm<sup>3</sup> or kg/m<sup>3</sup>. Other units that might be encountered are lbm/gallon or lbm/ft<sup>3</sup> (see '''Table 13.2''').
<br>
<br>
<gallery widths=300px heights=200px>
File:Vol1 Page 581 Image 0002.png|'''Table 13.2'''
File:Vol1 Page 581 Image 0002.png|'''Table 13.2'''
</gallery>
</gallery><br/>For simple, completely homogeneous (single-phase) material, this definition of density is straightforward. However, Earth materials involved in petroleum engineering are mixtures of several phases, both solids (minerals) and fluids. Rocks, in particular, are porous, and porosity is intimately related to density. For rocks, porosity (''Φ'') is defined as the nonsolid or pore-volume fraction.<br/><br/>[[File:Vol1 page 0578 eq 002.png|RTENOTITLE]]....................(13.2)<br/><br/>Porosity is a volume ratio and thus dimensionless, and is usually reported as a fraction or percent. To avoid confusion, particularly when variable or changing porosities are involved, it is often reported in porosity units (1 PU = 1%).<br/><br/>Several volume definitions are required to describe porosity:<br/><br/>[[File:Vol1 page 0578 eq 003.png|RTENOTITLE]]<br/>[[File:Vol1 page 0579 eq 001.png|RTENOTITLE]]....................(13.3)<br/><br/>From these we can define the various kinds of porosity encountered:<br/><br/>[[File:Vol1 page 0580 eq 001.png|RTENOTITLE]]....................(13.4)<br/><br/>'''Fig. 13.9''' shows the appearance of these types of porosity in a sandstone.<br/><br/><gallery widths="300px" heights="200px">
<br>
For simple, completely homogeneous (single-phase) material, this definition of density is straightforward. However, Earth materials involved in petroleum engineering are mixtures of several phases, both solids (minerals) and fluids. Rocks, in particular, are porous, and porosity is intimately related to density. For rocks, porosity (''Φ'') is defined as the nonsolid or pore-volume fraction.
<br>
<br>
[[File:Vol1 page 0578 eq 002.png]]....................(13.2)
<br>
<br>
Porosity is a volume ratio and thus dimensionless, and is usually reported as a fraction or percent. To avoid confusion, particularly when variable or changing porosities are involved, it is often reported in porosity units (1 PU = 1%).  
<br>
<br>
Several volume definitions are required to describe porosity:
<br>
<br>
[[File:Vol1 page 0578 eq 003.png]]<br>
[[File:Vol1 page 0579 eq 001.png]]....................(13.3)
<br>
<br>
From these we can define the various kinds of porosity encountered:
<br>
<br>
[[File:Vol1 page 0580 eq 001.png]]....................(13.4)
<br>
<br>
'''Fig. 13.9''' shows the appearance of these types of porosity in a sandstone.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 582 Image 0001.png|'''Fig. 13.9 – Calculated density vs. porosity for sandstone, limestone, and dolomite.'''
File:vol1 Page 582 Image 0001.png|'''Fig. 13.9 – Calculated density vs. porosity for sandstone, limestone, and dolomite.'''
</gallery>
</gallery><br/>Similarly, the definitions of the standard densities associated with rocks then follows:<br/><br/>[[File:Vol1 page 0580 eq 002.png|RTENOTITLE]]....................(13.5)<br/><br/>where ''M''<sub>''s''</sub>, ''M''<sub>''d''</sub>, ''M''<sub>''sat''</sub>, ''M''<sub>''b''</sub>, and ''M''<sub>''fl''</sub> are the mass of the solid, dry rock, saturated rock, buoyant rock, and fluid, respectively.
<br/>
Similarly, the definitions of the standard densities associated with rocks then follows:
<br>
<br>
[[File:Vol1 page 0580 eq 002.png]]....................(13.5)
<br>
<br>
where ''M''<sub>''s''</sub>, ''M''<sub>''d''</sub>, ''M''<sub>''sat''</sub>, ''M''<sub>''b''</sub>, and ''M''<sub>''fl''</sub> are the mass of the solid, dry rock, saturated rock, buoyant rock, and fluid, respectively.


=== Relationships ===
=== Relationships ===


The density of a composite such as rocks (or drilling muds) can be calculated from the densities and volume fraction of each component. For a two-component system,
The density of a composite such as rocks (or drilling muds) can be calculated from the densities and volume fraction of each component. For a two-component system,<br/><br/>[[File:Vol1 page 0581 eq 001.png|RTENOTITLE]]....................(13.6)<br/><br/>where ''ρ''<sub>mix</sub> is the density of the mixture; ''ρ''<sub>''A''</sub> is the density of Component A; ''ρ''<sub>''B''</sub> is the density of B; A and B are the volume fractions of A and B respectively (and so B = 1− ''A'').<br/><br/>Expanding this into a general system with ''n'' components,<br/><br/>[[File:Vol1 page 0582 eq 001.png|RTENOTITLE]]....................(13.7)<br/><br/>For example, exploiting '''Eqs. 13.4''', '''13.5''', and '''13.6''' for a rock made up of two minerals, ''m''<sub>1</sub> and ''m''<sub>2</sub>, and two fluids, ''f''<sub>1</sub> and ''f''<sub>2</sub>, we find<br/><br/>[[File:Vol1 page 0582 eq 002.png|RTENOTITLE]]....................(13.8)<br/><br/>and<br/><br/>[[File:Vol1 page 0582 eq 003.png|RTENOTITLE]]....................(13.9)<br/><br/>'''Eq. 13.8''' is a fundamental relation used throughout the Earth sciences to calculate rock density. Given a porosity and specific fluid, density can be easily calculated if the mineral or grain density is known. Grain densities for common rock-forming minerals are shown in '''Table 13.3'''. The result of applying '''Eq. 13.9''' is shown in '''Fig. 13.9'''.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
[[File:Vol1 page 0581 eq 001.png]]....................(13.6)
<br>
<br>
where ''ρ''<sub>mix</sub> is the density of the mixture; ''ρ''<sub>''A''</sub> is the density of Component A; ''ρ''<sub>''B''</sub> is the density of B; A and B are the volume fractions of A and B respectively (and so B = 1− ''A'').  
<br>
<br>
Expanding this into a general system with ''n'' components,
<br>
<br>
[[File:Vol1 page 0582 eq 001.png]]....................(13.7)
<br>
<br>
For example, exploiting '''Eqs. 13.4''', '''13.5''', and '''13.6''' for a rock made up of two minerals, ''m''<sub>1</sub> and ''m''<sub>2</sub>, and two fluids, ''f''<sub>1</sub> and ''f''<sub> 2</sub>, we find
<br>
<br>
[[File:Vol1 page 0582 eq 002.png]]....................(13.8)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0582 eq 003.png]]....................(13.9)
<br>
<br>
'''Eq. 13.8''' is a fundamental relation used throughout the Earth sciences to calculate rock density. Given a porosity and specific fluid, density can be easily calculated if the mineral or grain density is known. Grain densities for common rock-forming minerals are shown in '''Table 13.3'''. The result of applying '''Eq. 13.9''' is shown in '''Fig. 13.9'''.
<br>
<br>
<gallery widths=300px heights=200px>
File:Vol1 Page 583 Image 0001.png|'''Table 13.3'''<ref name="r15" /><ref name="r16" /><ref name="r17" />
File:Vol1 Page 583 Image 0001.png|'''Table 13.3'''<ref name="r15" /><ref name="r16" /><ref name="r17" />
</gallery>
</gallery><br/>Note in '''Table 13.3''' that there are several densities reported for the same mineral group, such as feldspar or clay. The density will change systematically as composition varies. For example, in the plagioclase series, the density increases as sodium (albite, ''ρ'' = 2.61 g/cm<sup>3</sup>) is replaced by calcium (anorthite, ''ρ'' = 2.75 g/cm<sup>3</sup>). The most problematic minerals are clays, particularly expanding clays (montmorillonite or smectite) capable of containing large and variable amounts of water. In this case, densities can vary 40% or more. This is a particular problem, because clays are among the most common minerals in sedimentary rocks.<br/><br/>Reservoir rocks often contain significant amounts of semisolid organic material such as bitumen. These will typically have light densities similar in magnitude to those of coals.<br/><br/>Pore-fluid densities are covered in detail in the fluid property section (13.4).
<br>
Note in '''Table 13.3''' that there are several densities reported for the same mineral group, such as feldspar or clay. The density will change systematically as composition varies. For example, in the plagioclase series, the density increases as sodium (albite, ''ρ'' = 2.61 g/cm<sup>3</sup>) is replaced by calcium (anorthite, ''ρ'' = 2.75 g/cm<sup>3</sup>). The most problematic minerals are clays, particularly expanding clays (montmorillonite or smectite) capable of containing large and variable amounts of water. In this case, densities can vary 40% or more. This is a particular problem, because clays are among the most common minerals in sedimentary rocks.  
<br>
<br>
Reservoir rocks often contain significant amounts of semisolid organic material such as bitumen. These will typically have light densities similar in magnitude to those of coals.  
<br>
<br>
Pore-fluid densities are covered in detail in the fluid property section (13.4).


=== In-Situ Density and Porosity ===
=== In-Situ Density and Porosity ===


In general, density increases and porosity decreases monotonically with depth. This is expected, because differential pressures usually increase with depth. As pressure increases, grains will shift and rotate to reach a more dense packing. More force will be imposed on the grain contacts. Crushing and fracturing is a common result. In addition, diagenetic processes such as cementation work to fill the pore space. Material may be dissolved at point contacts or along styolites and then transported to fill pores. Some of the textures resulting from these processes were seen in the photomicrographs of the previous section. In '''Fig. 13.10''', generalized densities as a function of depth for shales are plotted. The shapes and overall behaviors for these curves are similar, even though they come from a wide variety of locations with different geologic histories. These kinds of curves are often fit with exponential functions in depth to define the local compaction trend.  
In general, density increases and porosity decreases monotonically with depth. This is expected, because differential pressures usually increase with depth. As pressure increases, grains will shift and rotate to reach a more dense packing. More force will be imposed on the grain contacts. Crushing and fracturing is a common result. In addition, diagenetic processes such as cementation work to fill the pore space. Material may be dissolved at point contacts or along styolites and then transported to fill pores. Some of the textures resulting from these processes were seen in the photomicrographs of the previous section. In '''Fig. 13.10''', generalized densities as a function of depth for shales are plotted. The shapes and overall behaviors for these curves are similar, even though they come from a wide variety of locations with different geologic histories. These kinds of curves are often fit with exponential functions in depth to define the local compaction trend.<br/><br/><gallery widths="300px" heights="200px">
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 584 Image 0001.png|'''Fig. 13.10 – Shale density as a function of depth from several sedimentary basins (after Castagna ''et al''.<ref name="r18" /> and Rieke and Chillingarian<ref name="r19" />). 1 = Gas saturated clastics: probable minimum density (McCulloh<ref name="r20" />). 2 = Po river valley mudstone (Storer<ref name="r21" />), 3 = average coastal Gulf of Mexico shales from geophysical measurements (Dickinson<ref name="r22" />), 4 = average coastal Gulf of Mexico shales from density logs (Eaton<ref name="r23" />), 5 = Marcaibo basin well (Dallmus<ref name="r24" />), 6 = Hungary calculated wet densities (Skeels<ref name="r25" />), 7 = Pennsylvanian and Permian dry shales (Dallmus<ref name="r24" />), 8 = Eastern Venezuela (Dallmus<ref name="r24" />).'''
File:vol1 Page 584 Image 0001.png|'''Fig. 13.10 – Shale density as a function of depth from several sedimentary basins (after Castagna ''et al''.<ref name="r18" /> and Rieke and Chillingarian<ref name="r19" />). 1 = Gas saturated clastics: probable minimum density (McCulloh<ref name="r20" />). 2 = Po river valley mudstone (Storer<ref name="r21" />), 3 = average coastal Gulf of Mexico shales from geophysical measurements (Dickinson<ref name="r22" />), 4 = average coastal Gulf of Mexico shales from density logs (Eaton<ref name="r23" />), 5 = Marcaibo basin well (Dallmus<ref name="r24" />), 6 = Hungary calculated wet densities (Skeels<ref name="r25" />), 7 = Pennsylvanian and Permian dry shales (Dallmus<ref name="r24" />), 8 = Eastern Venezuela (Dallmus<ref name="r24" />).'''
</gallery>
</gallery><br/>Differential or effective pressures do not always increase with increasing depth. Abnormally high pore fluid pressures ("overpressure") can occur because of rapid compaction, low permeability, mineral dewatering, or migration of high-pressure fluids. The high pore pressure results in an abnormally low differential of effective pressure. This can retard or even reverse the normal compaction trends. Such a situation is seen in '''Fig. 13.11'''. Porosities for both shales and sands show the expected porosity loss with increasing depth in the shallow portions. However, at about 3500 m, pore pressure rises and porosity actually increases with depth. This demonstrates why local calibration is needed. This behavior is also our first indication of the pressure dependence of rock properties, a topic covered in more detail in '''Section 13.5'''.<br/><br/><gallery widths="300px" heights="200px">
<br/>
Differential or effective pressures do not always increase with increasing depth. Abnormally high pore fluid pressures ("overpressure") can occur because of rapid compaction, low permeability, mineral dewatering, or migration of high-pressure fluids. The high pore pressure results in an abnormally low differential of effective pressure. This can retard or even reverse the normal compaction trends. Such a situation is seen in '''Fig. 13.11'''. Porosities for both shales and sands show the expected porosity loss with increasing depth in the shallow portions. However, at about 3500 m, pore pressure rises and porosity actually increases with depth. This demonstrates why local calibration is needed. This behavior is also our first indication of the pressure dependence of rock properties, a topic covered in more detail in '''Section 13.5'''.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 585 Image 0001.png|'''Fig. 13.11 – Shale and sandstone porosity with depth. Porosity decreases until high pore pressures (= geopressure) reduce the effective pressure and cause an increase in porosity (from Stuart<ref name="r26" />).'''
File:vol1 Page 585 Image 0001.png|'''Fig. 13.11 – Shale and sandstone porosity with depth. Porosity decreases until high pore pressures (= geopressure) reduce the effective pressure and cause an increase in porosity (from Stuart<ref name="r26" />).'''
</gallery>
</gallery>
<br/>


=== Measurement Techniques ===
=== Measurement Techniques ===


'''''Laboratory.''''' Numerous methods can be used in the laboratory to determine porosity and density. The most common are by saturation weight and Boyle’s law. For rocks without sensitive minerals such as smectites, the porosity and dry, grain, and saturated densities can be derived from the saturated mass, dry mass, and volume (or buoyant weight). These measurements allow calculation of saturated, dry, and grain density as well as porosity and mineral and pore volume by employing '''Eqs. 13.3''' through '''13.5'''.  
'''''Laboratory.''''' Numerous methods can be used in the laboratory to determine porosity and density. The most common are by saturation weight and Boyle’s law. For rocks without sensitive minerals such as smectites, the porosity and dry, grain, and saturated densities can be derived from the saturated mass, dry mass, and volume (or buoyant weight). These measurements allow calculation of saturated, dry, and grain density as well as porosity and mineral and pore volume by employing '''Eqs. 13.3''' through '''13.5'''.<br/><br/>The Boyle’s law technique measures the relative changes in gas pressures inside a chamber with and without a rock specimen. The internal (connected) pore volume is calculated from these variations in pressure, from which porosities and densities are extracted.<br/><br/>'''''Logging.''''' Several logging techniques are available to measure density or porosity.<ref name="r27">_</ref><ref name="r28">_</ref> These indirect techniques can have substantial errors depending on borehole conditions, but they do provide a measure of the in-situ properties. Gamma ray logs bombard the formation with radiation from an active source. Radiation is scattered back to the logging tool, depending on the electron density of the material. Formation density is extracted from the amplitude of these back-scattered gamma rays. The neutron log estimates porosity by particle interaction with hydrogen atoms. Neutrons lose energy when colliding with hydrogen atoms, thus giving a measure of the hydrogen content. Because most of the hydrogen in rocks resides in the pore space (water or oil), this is then related to the liquid-filled porosity. Note that the neutron log will include bound water within clays as porosity. In addition, when relatively hydrogen-poor gas is the pore fluid, the neutron log will underestimate porosity. In a similar fashion, the nuclear magnetic resonance (NMR) log will resolve the hydrogen content. This tool, however, has the ability to differentiate between free bulk water and bound water. Sonic logs are also used for porosity measurements, particularly when anomalous minerals (such as siderite) or borehole conditions render other tools less accurate. The technique involves inverting velocity to porosity using one of the relationships provided in the velocity section below. Gravimetry has also been used downhole to measure variations in density. Although this tool is insensitive to fine-scale changes, it permits density measurement far out into the formation.<br/><br/>'''''Seismic.''''' On a coarse scale, densities can sometimes be extracted from seismic data. This method requires separating the density component of impedance. This normally requires an analysis of the seismic data as a function of offset or reflection angle. This technique will probably see more use as seismic data improves and is further incorporated into reservoir description.
<br>
</div></div><div class="toccolours mw-collapsible mw-collapsed">
<br>
The Boyle’s law technique measures the relative changes in gas pressures inside a chamber with and without a rock specimen. The internal (connected) pore volume is calculated from these variations in pressure, from which porosities and densities are extracted.  
<br>
<br>
'''''Logging.''''' Several logging techniques are available to measure density or porosity.<ref name="r27" /><ref name="r28" /> These indirect techniques can have substantial errors depending on borehole conditions, but they do provide a measure of the in-situ properties. Gamma ray logs bombard the formation with radiation from an active source. Radiation is scattered back to the logging tool, depending on the electron density of the material. Formation density is extracted from the amplitude of these back-scattered gamma rays. The neutron log estimates porosity by particle interaction with hydrogen atoms. Neutrons lose energy when colliding with hydrogen atoms, thus giving a measure of the hydrogen content. Because most of the hydrogen in rocks resides in the pore space (water or oil), this is then related to the liquid-filled porosity. Note that the neutron log will include bound water within clays as porosity. In addition, when relatively hydrogen-poor gas is the pore fluid, the neutron log will underestimate porosity. In a similar fashion, the nuclear magnetic resonance (NMR) log will resolve the hydrogen content. This tool, however, has the ability to differentiate between free bulk water and bound water. Sonic logs are also used for porosity measurements, particularly when anomalous minerals (such as siderite) or borehole conditions render other tools less accurate. The technique involves inverting velocity to porosity using one of the relationships provided in the velocity section below. Gravimetry has also been used downhole to measure variations in density. Although this tool is insensitive to fine-scale changes, it permits density measurement far out into the formation.  
<br>
<br>
'''''Seismic.''''' On a coarse scale, densities can sometimes be extracted from seismic data. This method requires separating the density component of impedance. This normally requires an analysis of the seismic data as a function of offset or reflection angle. This technique will probably see more use as seismic data improves and is further incorporated into reservoir description.  
<br>
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" >
 
== Fluid Properties ==
== Fluid Properties ==
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
<br>
<br/>Hydrocarbons occur in a variety of conditions, in different phases, and with widely varying properties. In this section, we will cover the important geophysical properties of pore fluids. For more general information on the engineering properties of fluids, see the appropriate section in the Handbook. '''Fig. 13.12''' shows schematically the relation among the different mixtures. For a single, constant composition mixture, as we vary temperature and pressure over a wide range, we would encounter the boundary between the single and multiphase regions. In contrast, if we restrict the temperatures and pressures to those typical of reservoirs, we could again move in this phase "space" by changing compositions. Velocities and densities will be high (close to water) for heavy "black" oils to the left of the figure and decrease dramatically as we move right toward lighter compounds. In many cases, the hydrocarbons are greater than critical pressure and temperature conditions (greater than critical point). Properties then can vary continuously from liquid-like, for oils with gas in solution, to gas-like, for mixtures of light molecular weight. With changing pressure and temperature conditions, phase boundaries can be crossed, resulting in abrupt changes in fluid properties. Additional components are often injected during production, further complicating the distribution of compositions and properties.<br/><br/><gallery widths="300px" heights="200px">
Hydrocarbons occur in a variety of conditions, in different phases, and with widely varying properties. In this section, we will cover the important geophysical properties of pore fluids. For more general information on the engineering properties of fluids, see the appropriate section in the Handbook. '''Fig. 13.12''' shows schematically the relation among the different mixtures. For a single, constant composition mixture, as we vary temperature and pressure over a wide range, we would encounter the boundary between the single and multiphase regions. In contrast, if we restrict the temperatures and pressures to those typical of reservoirs, we could again move in this phase "space" by changing compositions. Velocities and densities will be high (close to water) for heavy "black" oils to the left of the figure and decrease dramatically as we move right toward lighter compounds. In many cases, the hydrocarbons are greater than critical pressure and temperature conditions (greater than critical point). Properties then can vary continuously from liquid-like, for oils with gas in solution, to gas-like, for mixtures of light molecular weight. With changing pressure and temperature conditions, phase boundaries can be crossed, resulting in abrupt changes in fluid properties. Additional components are often injected during production, further complicating the distribution of compositions and properties.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 586 Image 0001.png|'''Fig. 13.12 – General fluid phase behavior for hydrocarbon mixtures. Above the critical point, there is a continuum of compositions and properties from heavy oil to light gas.'''
File:vol1 Page 586 Image 0001.png|'''Fig. 13.12 – General fluid phase behavior for hydrocarbon mixtures. Above the critical point, there is a continuum of compositions and properties from heavy oil to light gas.'''
</gallery>
</gallery>
<br/>


=== Gas ===
=== Gas ===


The gas phase is the easiest to characterize. The compounds are usually relatively simple, and the thermodynamic properties have been thoroughly examined. Hydrocarbon gases usually consist of the light hydrocarbons of methane, butane, and propane. Additional gases, such as water vapor and heavier hydrocarbons, will occur in the gas depending on the pressure, temperature, and history of the deposit. The specific weight of these gases, as compared to air at standard temperature and pressure, will vary from about 0.6 for nearly pure methane to over 1.5 for gases with heavier components. Fortunately, when a rough idea of the gas weight is known, a fairly accurate estimate can be made of the gas properties at pressure and temperatures. Thomas ''et al''.<ref name="r29" /> did a complete analysis of the acoustic properties of natural gases, and we will follow a similar analysis here.  
The gas phase is the easiest to characterize. The compounds are usually relatively simple, and the thermodynamic properties have been thoroughly examined. Hydrocarbon gases usually consist of the light hydrocarbons of methane, butane, and propane. Additional gases, such as water vapor and heavier hydrocarbons, will occur in the gas depending on the pressure, temperature, and history of the deposit. The specific weight of these gases, as compared to air at standard temperature and pressure, will vary from about 0.6 for nearly pure methane to over 1.5 for gases with heavier components. Fortunately, when a rough idea of the gas weight is known, a fairly accurate estimate can be made of the gas properties at pressure and temperatures. Thomas ''et al''.<ref name="r29">_</ref> did a complete analysis of the acoustic properties of natural gases, and we will follow a similar analysis here.<br/><br/>The important seismic characteristics of a fluid (the bulk modulus, density, and sonic velocity) are all related to primary thermodynamic properties. Therefore, for gases, we are obliged to start with the ideal gas law.<br/><br/>[[File:Vol1 page 0587 eq 001.png|RTENOTITLE]]....................(13.10)<br/><br/>where ''P'' is pressure, ''V'' is volume, ''n'' is the number of moles of the gas, ''R'' is the gas constant, and ''T''<sub>''a''</sub> the absolute temperature. This leads to a density ''ρ'', of<br/><br/>[[File:Vol1 page 0587 eq 002.png|RTENOTITLE]]....................(13.11)<br/><br/>where ''M'' is the molecular weight. The isothermal compressibility ''β''<sub>''T''</sub> is<br/><br/>[[File:Vol1 page 0587 eq 003.png|RTENOTITLE]]....................(13.12)<br/><br/>for compressibility defined as a positive number.<br/><br/>If we calculate the "isothermal" velocity ''V''<sub>''T''</sub>, we find<br/><br/>[[File:Vol1 page 0587 eq 004.png|RTENOTITLE]]....................(13.13)<br/><br/>for an ideal gas. The acoustic velocity is controlled by the stiffness of the material and its density (see the derivation in '''Section 13.5.5'''). Therefore, velocity would increase with temperature and be independent of pressure.<br/><br/>Two mitigating factors bring the relationship closer to reality. First, because there are rapid temperature changes associated with the passage of an acoustic wave, we must use the adiabatic compressibility, ''β''<sub>''S''</sub>, rather than the isothermal compressibility ''γ'' ''β''<sub>''S''</sub> = ''β''<sub>''T''</sub>.<br/><br/>Here, ''γ'' is the ratio of heat capacity at constant pressure to the heat capacity at constant volume. In most solid materials, the difference between the isothermal and adiabatic compressibilities is negligible. However, in fluid phases, particularly gases, the isothermal compressibility can be twice the adiabatic value.<br/><br/>The second, more obvious factor stems from the inadequacies of the ideal gas law ('''Eq. 13.10'''). The gas law can be corrected by adding a compressibility factor (''Z''). The relationships are thus modified:<br/><br/>[[File:Vol1 page 0587 eq 005.png|RTENOTITLE]]....................(13.14)<br/><br/>[[File:Vol1 page 0587 eq 006.png|RTENOTITLE]]....................(13.15)<br/><br/>and<br/><br/>[[File:Vol1 page 0587 eq 007.png|RTENOTITLE]]....................(13.16)<br/><br/>The heat capacity ratio can itself be derived if the equations of state of the material are known. The seismic characteristics of the gas can, therefore, be described if we have an adequate description of ''Z'' with pressure, temperature, and composition.<br/><br/>Thomas ''et al''.<ref name="r29">_</ref> made use of the Benedict-Webb-Rubin (BWR) equation to define the gas behavior. The BWR equation of state is a rational equation, with numerous constants based on the behavior of natural gas mixtures. These gas mixtures range in gravity G (relative to air) from about 0.5 to 1.8. The results of the density calculations are shown in '''Fig. 13.13'''. As would be expected, the gas densities increase with pressure and decrease with temperature. However, the densities also strongly depend on the gas gravity, which is composition-dependent.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
The important seismic characteristics of a fluid (the bulk modulus, density, and sonic velocity) are all related to primary thermodynamic properties. Therefore, for gases, we are obliged to start with the ideal gas law.
<br>
<br>
[[File:Vol1 page 0587 eq 001.png]]....................(13.10)
<br>
<br>
where ''P'' is pressure, ''V'' is volume, ''n'' is the number of moles of the gas, ''R'' is the gas constant, and ''T''<sub>''a''</sub> the absolute temperature. This leads to a density ''ρ'', of
<br>
<br>
[[File:Vol1 page 0587 eq 002.png]]....................(13.11)
<br>
<br>
where ''M'' is the molecular weight. The isothermal compressibility ''β''<sub>''T''</sub> is
<br>
<br>
[[File:Vol1 page 0587 eq 003.png]]....................(13.12)
<br>
<br>
for compressibility defined as a positive number.  
<br>
<br>
If we calculate the "isothermal" velocity ''V''<sub>''T''</sub>, we find
<br>
<br>
[[File:Vol1 page 0587 eq 004.png]]....................(13.13)
<br>
<br>
for an ideal gas. The acoustic velocity is controlled by the stiffness of the material and its density (see the derivation in '''Section 13.5.5'''). Therefore, velocity would increase with temperature and be independent of pressure.  
<br>
<br>
Two mitigating factors bring the relationship closer to reality. First, because there are rapid temperature changes associated with the passage of an acoustic wave, we must use the adiabatic compressibility, ''β''<sub>''S''</sub>, rather than the isothermal compressibility ''γ'' ''β''<sub>''S''</sub> = ''β''<sub>''T''</sub>.  
<br>
<br>
Here, ''γ'' is the ratio of heat capacity at constant pressure to the heat capacity at constant volume. In most solid materials, the difference between the isothermal and adiabatic compressibilities is negligible. However, in fluid phases, particularly gases, the isothermal compressibility can be twice the adiabatic value.  
<br>
<br>
The second, more obvious factor stems from the inadequacies of the ideal gas law ('''Eq. 13.10'''). The gas law can be corrected by adding a compressibility factor (''Z''). The relationships are thus modified:
<br>
<br>
[[File:Vol1 page 0587 eq 005.png]]....................(13.14)
<br>
<br>
[[File:Vol1 page 0587 eq 006.png]]....................(13.15)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0587 eq 007.png]]....................(13.16)
<br>
<br>
The heat capacity ratio can itself be derived if the equations of state of the material are known. The seismic characteristics of the gas can, therefore, be described if we have an adequate description of ''Z'' with pressure, temperature, and composition.  
<br>
<br>
Thomas ''et al''.<ref name="r29" /> made use of the Benedict-Webb-Rubin (BWR) equation to define the gas behavior. The BWR equation of state is a rational equation, with numerous constants based on the behavior of natural gas mixtures. These gas mixtures range in gravity G (relative to air) from about 0.5 to 1.8. The results of the density calculations are shown in '''Fig. 13.13'''. As would be expected, the gas densities increase with pressure and decrease with temperature. However, the densities also strongly depend on the gas gravity, which is composition-dependent.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 588 Image 0001.png|'''Fig. 13.13 – Hydrocarbon gas densities as a function of pressure and temperature for gas gravities of ''G'' = 0.6 and ''G'' = 1.2 (from Batzle and Wang<ref name="r30" />).'''
File:vol1 Page 588 Image 0001.png|'''Fig. 13.13 – Hydrocarbon gas densities as a function of pressure and temperature for gas gravities of ''G'' = 0.6 and ''G'' = 1.2 (from Batzle and Wang<ref name="r30" />).'''
</gallery>
</gallery><br/>The adiabatic gas modulus ''K'' (the inverse of ''β'') also strongly depends on the composition as well as the pressure and temperature conditions. '''Fig. 13.14''' shows the calculated modulus from the Thomas relationships. Again, the modulus increases with pressure and decreases with temperature, but the relationship is not as linear. The impact of variable composition (gravity) is again obvious.<br/><br/><gallery widths="300px" heights="200px">
<br/>
The adiabatic gas modulus ''K'' (the inverse of ''β'') also strongly depends on the composition as well as the pressure and temperature conditions. '''Fig. 13.14''' shows the calculated modulus from the Thomas relationships. Again, the modulus increases with pressure and decreases with temperature, but the relationship is not as linear. The impact of variable composition (gravity) is again obvious.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 589 Image 0001.png|'''Fig. 13.14 – Hydrocarbon gas modulus as a function of pressure and temperature for gas gravities of 0.6 and 1.2 (from Batzle and Wang<ref name="r30" />).'''
File:vol1 Page 589 Image 0001.png|'''Fig. 13.14 – Hydrocarbon gas modulus as a function of pressure and temperature for gas gravities of 0.6 and 1.2 (from Batzle and Wang<ref name="r30" />).'''
</gallery>
</gallery>
<br/>


=== Oil ===
=== Oil ===


Crude oils can be mixtures of extremely complex organic compounds. Natural oils range from the lightest condensate liquids of low carbon number to very heavy tars. At the heavy extreme are bitumen and kerogen, which may be denser than water and act essentially like solids. At the light extreme are condensates that may become gas with decreasing pressure. Oils can absorb large quantities of hydrocarbon gases under pressure, thus significantly decreasing the moduli. Under room conditions, the densities can vary from 0.5 to greater than 1 gm/cc with most produced oils in the 0.7 to 0.8 gm/cc range. The American Petroleum Institute (API) number is defined as
Crude oils can be mixtures of extremely complex organic compounds. Natural oils range from the lightest condensate liquids of low carbon number to very heavy tars. At the heavy extreme are bitumen and kerogen, which may be denser than water and act essentially like solids. At the light extreme are condensates that may become gas with decreasing pressure. Oils can absorb large quantities of hydrocarbon gases under pressure, thus significantly decreasing the moduli. Under room conditions, the densities can vary from 0.5 to greater than 1 gm/cc with most produced oils in the 0.7 to 0.8 gm/cc range. The American Petroleum Institute (API) number is defined as<br/><br/>[[File:Vol1 page 0588 eq 001.png|RTENOTITLE]]....................(13.17)<br/><br/>This results in API numbers of about 5 for very heavy oils to near 100 for light condensates. The extreme variations in composition and ability to absorb gases produce greater variations in the seismic properties of oils.<br/><br/>If we had a general equation of state for oils, we could calculate the moduli and densities as we did for the gases. Such equations abound in the petroleum engineering literature. Unfortunately, the equations are almost always strongly dependent on the exact composition of a given oil. For the purposes of this ''Handbook'', we will develop only very general relations. Often, in petrophysical analysis we only have a rough idea of what the oils may be like. In some reservoirs, individual yet adjacent zones will have quite distinct oil types. We will, therefore, proceed along empirical lines based on the density of the oil.<br/><br/>The acoustic properties of numerous organic fluids have been studied as a function of pressure or temperature (see, for example, Rao and Rao<ref name="r31">_</ref>). Generally, the velocities, densities, and moduli are quite linear with pressure and temperature away from phase boundaries. In organic fluids typical of crude oils, the moduli decrease with increasing temperature and increase with increasing pressure. Wang and Nur<ref name="r32">_</ref> did an extensive study of several light alkanes, alkenes, and cycloparaffins and found simple relationships among the density, moduli, temperature, and carbon number or molecular weight. For velocity they found<br/><br/>[[File:Vol1 page 0589 eq 001.png|RTENOTITLE]]....................(13.18)<br/><br/>where ''V''<sub>''o''</sub> is the initial velocity, ''V''<sub>''T''</sub> is the velocity at temperature ''T'', Δ''T'' is the temperature change, and ''b'' is a constant for each compound of molecular weight ''M'':<br/><br/>[[File:Vol1 page 0589 eq 002.png|RTENOTITLE]]....................(13.19)<br/><br/>Similarly, the velocities are related in molecular weight by<br/><br/>[[File:Vol1 page 0590 eq 001.png|RTENOTITLE]]....................(13.20)<br/><br/>where ''V''<sub>''TM''</sub> is the velocity of oil of weight ''M'', and ''V''<sub>TOMO</sub> is the velocity of a reference oil of weight ''M''<sub>''o''</sub> at temperature ''T''<sub>''o''</sub>. The variable ''a''<sub>''m''</sub> is a positive function of temperature. We can see from the rightmost term in '''Eq. 13.20''' that the velocity of the fluid will increase with increasing molecular weight. When compounds are mixed, Wang and Nur<ref name="r32">_</ref> found that the resulting velocity is a simple fractional average of the end components. This is roughly equivalent to a fractional average of the bulk moduli of the end components. Pure simple hydrocarbons, therefore, behave in a simple predictable way. We must extend this analysis to include crude oils, which are generally much heavier and have more complex compositions. The influence of pressure must also be determined. In the petroleum engineering literature, broad empirical relationships are available. By empirically fitting equations to these data, we can get density as functions of initial density (or API number), temperature, and pressure<br/><br/>[[File:Vol1 page 0590 eq 002.png|RTENOTITLE]]....................(13.21)<br/><br/>These densities are shown in '''Fig. 13.15'''.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
[[File:Vol1 page 0588 eq 001.png]]....................(13.17)
<br>
<br>
This results in API numbers of about 5 for very heavy oils to near 100 for light condensates. The extreme variations in composition and ability to absorb gases produce greater variations in the seismic properties of oils.  
<br>
<br>
If we had a general equation of state for oils, we could calculate the moduli and densities as we did for the gases. Such equations abound in the petroleum engineering literature. Unfortunately, the equations are almost always strongly dependent on the exact composition of a given oil. For the purposes of this ''Handbook'', we will develop only very general relations. Often, in petrophysical analysis we only have a rough idea of what the oils may be like. In some reservoirs, individual yet adjacent zones will have quite distinct oil types. We will, therefore, proceed along empirical lines based on the density of the oil.  
<br>
<br>
The acoustic properties of numerous organic fluids have been studied as a function of pressure or temperature (see, for example, Rao and Rao<ref name="r31" />). Generally, the velocities, densities, and moduli are quite linear with pressure and temperature away from phase boundaries. In organic fluids typical of crude oils, the moduli decrease with increasing temperature and increase with increasing pressure. Wang and Nur<ref name="r32" /> did an extensive study of several light alkanes, alkenes, and cycloparaffins and found simple relationships among the density, moduli, temperature, and carbon number or molecular weight. For velocity they found
<br>
<br>
[[File:Vol1 page 0589 eq 001.png]]....................(13.18)
<br>
<br>
where ''V''<sub>''o''</sub> is the initial velocity, ''V''<sub>''T''</sub> is the velocity at temperature ''T'', Δ''T'' is the temperature change, and ''b'' is a constant for each compound of molecular weight ''M'':
<br>
<br>
[[File:Vol1 page 0589 eq 002.png]]....................(13.19)
<br>
<br>
Similarly, the velocities are related in molecular weight by
<br>
<br>
[[File:Vol1 page 0590 eq 001.png]]....................(13.20)
<br>
<br>
where ''V''<sub>''TM''</sub> is the velocity of oil of weight ''M'', and ''V''<sub>TOMO</sub> is the velocity of a reference oil of weight ''M''<sub>''o''</sub> at temperature ''T''<sub>''o''</sub>. The variable ''a''<sub>''m''</sub> is a positive function of temperature. We can see from the rightmost term in '''Eq. 13.20''' that the velocity of the fluid will increase with increasing molecular weight. When compounds are mixed, Wang and Nur<ref name="r32" /> found that the resulting velocity is a simple fractional average of the end components. This is roughly equivalent to a fractional average of the bulk moduli of the end components. Pure simple hydrocarbons, therefore, behave in a simple predictable way. We must extend this analysis to include crude oils, which are generally much heavier and have more complex compositions. The influence of pressure must also be determined. In the petroleum engineering literature, broad empirical relationships are available. By empirically fitting equations to these data, we can get density as functions of initial density (or API number), temperature, and pressure
<br>
<br>
[[File:Vol1 page 0590 eq 002.png]]....................(13.21)
<br>
<br>
These densities are shown in '''Fig. 13.15'''.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 590 Image 0001.png|'''Fig. 13.15 – Oil density as a function of pressure and temperature for three weights of oil: API = 10, 30, and 50 (from Batzle and Wang<ref name="r30" />).'''
File:vol1 Page 590 Image 0001.png|'''Fig. 13.15 – Oil density as a function of pressure and temperature for three weights of oil: API = 10, 30, and 50 (from Batzle and Wang<ref name="r30" />).'''
</gallery>
</gallery><br/>By differentiating '''Eq. 13.21''', we obtain the isothermal compressibility ''β''<sub>''T''</sub>,<br/><br/>[[File:Vol1 page 0590 eq 003.png|RTENOTITLE]]....................(13.22)<br/><br/>If we assume a reasonable and constant heat capacity ratio ''γ'' of 1.15, we obtain the adiabatic bulk moduli ''K''.<br/><br/>[[File:Vol1 page 0591 eq 001.png|RTENOTITLE]]....................(13.23)<br/><br/>The ultrasonic velocities of a variety of crude oils measured recently are reported in Wang ''et al''.<ref name="r33">_</ref> A general relationship of oil velocity was derived.<br/><br/>[[File:Vol1 page 0591 eq 002.png|RTENOTITLE]]....................(13.24)<br/><br/>where ''V'' is in m/s, ''T'' in °C, ''P'' in bars, and API is the API degree of the oil, or<br/><br/>[[File:Vol1 page 0591 eq 003.png|RTENOTITLE]]....................(13.25)<br/><br/>for ''V'' in ft/s, ''T'' in °F, and ''P'' in psi.<br/><br/>Using these velocities and the densities as shown in '''Fig. 13.15''', we find the moduli shown in '''Fig. 13.16'''.<br/><br/><gallery widths="300px" heights="200px">
<br/>
By differentiating '''Eq. 13.21''', we obtain the isothermal compressibility ''β''<sub>''T''</sub>,
<br>
<br>
[[File:Vol1 page 0590 eq 003.png]]....................(13.22)
<br>
<br>
If we assume a reasonable and constant heat capacity ratio ''γ'' of 1.15, we obtain the adiabatic bulk moduli ''K''.
<br>
<br>
[[File:Vol1 page 0591 eq 001.png]]....................(13.23)
<br>
<br>
The ultrasonic velocities of a variety of crude oils measured recently are reported in Wang ''et al''.<ref name="r33" /> A general relationship of oil velocity was derived.
<br>
<br>
[[File:Vol1 page 0591 eq 002.png]]....................(13.24)
<br>
<br>
where ''V'' is in m/s, ''T'' in °C, ''P'' in bars, and API is the API degree of the oil, or
<br>
<br>
[[File:Vol1 page 0591 eq 003.png]]....................(13.25)
<br>
<br>
for ''V'' in ft/s, ''T'' in °F, and ''P'' in psi.  
<br>
<br>
Using these velocities and the densities as shown in '''Fig. 13.15''', we find the moduli shown in '''Fig. 13.16'''.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 591 Image 0001.png|'''Fig. 13.16 – Oil modulus as a function of pressure and temperature for three weights of oil: API = 10, 30, and 50 (from Batzle and Wang<ref name="r30" />).'''
File:vol1 Page 591 Image 0001.png|'''Fig. 13.16 – Oil modulus as a function of pressure and temperature for three weights of oil: API = 10, 30, and 50 (from Batzle and Wang<ref name="r30" />).'''
</gallery>
</gallery><br/>Very large amounts of gas or light hydrocarbons can go into solution in crude oils. In fact, the lighter crudes are condensates from the gas phase. We would expect the "live" or gas-saturated oils to have significantly different properties than the "dead" or gas-free oils commonly available and measured. The amount of gas that can be dissolved is a function of pressure, temperature, and the composition of both the gas and the oil.<ref name="r34">_</ref><br/><br/>[[File:Vol1 page 0592 eq 001.png|RTENOTITLE]]....................(13.26)<br/><br/>where ''R'' is the gas-oil ratio in liters/liter (1 liter/liter = 5.615 cu ft/bbl) at atmospheric pressure and at 15.5°C and ''G'' is the gas gravity. '''Eq. 13.26''' indicates that much larger amounts of gas can go into the light (high API number) oils. In fact, heavy oils may precipitate heavy compounds if much gas goes into solution.<br/><br/>The effect of this gas in solution on the oil acoustic properties has not been well documented. Sergeev<ref name="r35">_</ref> noted that gas in solution will reduce both oil and brine velocities. He calculated that this mix would change some reservoir reflection coefficients by more than a factor of two. A rough estimate of this dissolved gas effect can be made by assuming that the relationship in '''Eq. 13.26''' remains valid and by adjusting the oil density to include the gas component. We are assuming here that the gas is a liquid component with its own volume and density and that the result is an ideal liquid mixture. The simple additive relations found in Wang and Nur<ref name="r32">_</ref> support this concept. The estimated density becomes<br/><br/>[[File:Vol1 page 0592 eq 002.png|RTENOTITLE]]....................(13.27)<br/><br/>where ''ρ''<sub>''O''</sub> is the dead oil density and ''ρ''<sub>''G''</sub> is the gas saturated live oil density. The factor ''F'' is derived from the gas/oil ratio<br/><br/>[[File:Vol1 page 0592 eq 003.png|RTENOTITLE]]....................(13.28)<br/><br/>'''Fig. 13.17''' shows the live and dead oil velocities measured in Wang ''et al''.<ref name="r33">_</ref> along with the estimates using '''Eqs. 13.25''', '''13.27''', and '''13.28'''.<br/><br/><gallery widths="300px" heights="200px">
<br/>
Very large amounts of gas or light hydrocarbons can go into solution in crude oils. In fact, the lighter crudes are condensates from the gas phase. We would expect the "live" or gas-saturated oils to have significantly different properties than the "dead" or gas-free oils commonly available and measured. The amount of gas that can be dissolved is a function of pressure, temperature, and the composition of both the gas and the oil.<ref name="r34" />
<br>
<br>
[[File:Vol1 page 0592 eq 001.png]]....................(13.26)
<br>
<br>
where ''R'' is the gas-oil ratio in liters/liter (1 liter/liter = 5.615 cu ft/bbl) at atmospheric pressure and at 15.5°C and ''G'' is the gas gravity. '''Eq. 13.26''' indicates that much larger amounts of gas can go into the light (high API number) oils. In fact, heavy oils may precipitate heavy compounds if much gas goes into solution.  
<br>
<br>
The effect of this gas in solution on the oil acoustic properties has not been well documented. Sergeev<ref name="r35" /> noted that gas in solution will reduce both oil and brine velocities. He calculated that this mix would change some reservoir reflection coefficients by more than a factor of two. A rough estimate of this dissolved gas effect can be made by assuming that the relationship in '''Eq. 13.26''' remains valid and by adjusting the oil density to include the gas component. We are assuming here that the gas is a liquid component with its own volume and density and that the result is an ideal liquid mixture. The simple additive relations found in Wang and Nur<ref name="r32" /> support this concept. The estimated density becomes
<br>
<br>
[[File:Vol1 page 0592 eq 002.png]]....................(13.27)
<br>
<br>
where ''ρ''<sub>''O''</sub> is the dead oil density and ''ρ''<sub>''G''</sub> is the gas saturated live oil density. The factor ''F'' is derived from the gas/oil ratio
<br>
<br>
[[File:Vol1 page 0592 eq 003.png]]....................(13.28)
<br>
<br>
'''Fig. 13.17''' shows the live and dead oil velocities measured in Wang ''et al''.<ref name="r33" /> along with the estimates using '''Eqs. 13.25''', '''13.27''', and '''13.28'''.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 593 Image 0001.png|'''Fig. 13.17 – Acoustic velocity of a 23 API oil both “dead” or gas-free and “live” with 85 L/L gas in solution. Dissolved lowers both the effective density and velocity of the live oil mixture.'''
File:vol1 Page 593 Image 0001.png|'''Fig. 13.17 – Acoustic velocity of a 23 API oil both “dead” or gas-free and “live” with 85 L/L gas in solution. Dissolved lowers both the effective density and velocity of the live oil mixture.'''
</gallery>
</gallery>
<br/>


=== Brines ===
=== Brines ===


The great bulk of the pore fluids consists of brines. Their composition can range from almost pure water to saturated saline solutions. Gulf of Mexico area brines often have rapid increases in concentration with increasing depth. In other areas, the concentrations are often lower but can vary drastically between adjacent fields.  
The great bulk of the pore fluids consists of brines. Their composition can range from almost pure water to saturated saline solutions. Gulf of Mexico area brines often have rapid increases in concentration with increasing depth. In other areas, the concentrations are often lower but can vary drastically between adjacent fields.<br/><br/>The thermodynamic properties of aqueous solutions have been studied in detail. Keenan ''et al''.<ref name="r36">_</ref> give a relation for pure water that can be iteratively solved to give densities at pressure and temperature. Helgeson and Kirkham<ref name="r37">_</ref> use this and other data to calculate a wide variety of water properties over an extensive temperature and pressure range. One obvious effect of salinity is to increase the density of the fluid. Rowe and Chou<ref name="r38">_</ref> presented a polynomial to calculate both specific volume and compressibility of various salt solutions at pressure over a limited temperature range. Extensive additional data on sodium chloride solutions is provided in Zarembo and Fedorov<ref name="r39">_</ref> and Potter and Brown.<ref name="r40">_</ref> Using all these data, a simple polynomial can be constructed that will adequately calculate the density of sodium chloride solutions:<br/><br/>[[File:Vol1 page 0592 eq 004.png|RTENOTITLE]]....................(13.29a)<br/><br/>and<br/><br/>[[File:Vol1 page 0593 eq 001.png|RTENOTITLE]]....................(13.29b)<br/><br/>Here, ''T'' and ''P'' are in °C and bars, respectively; ''x'' is the weight fraction of sodium chloride; and ''ρ''<sub>''B''</sub> is the density of the brine in gm/cm<sup>3</sup>. The calculated brine densities, along with selected data from Zarembo and Federov,<ref name="r39">_</ref> are plotted in '''Fig. 13.18'''. The accuracy of this relationship is limited largely to the extent that other mineral salts, particularly divalent ions, are in solution.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
The thermodynamic properties of aqueous solutions have been studied in detail. Keenan ''et al''.<ref name="r36" /> give a relation for pure water that can be iteratively solved to give densities at pressure and temperature. Helgeson and Kirkham<ref name="r37" /> use this and other data to calculate a wide variety of water properties over an extensive temperature and pressure range. One obvious effect of salinity is to increase the density of the fluid. Rowe and Chou<ref name="r38" /> presented a polynomial to calculate both specific volume and compressibility of various salt solutions at pressure over a limited temperature range. Extensive additional data on sodium chloride solutions is provided in Zarembo and Fedorov<ref name="r39" /> and Potter and Brown.<ref name="r40" /> Using all these data, a simple polynomial can be constructed that will adequately calculate the density of sodium chloride solutions:
<br>
<br>
[[File:Vol1 page 0592 eq 004.png]]....................(13.29a)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0593 eq 001.png]]....................(13.29b)
<br>
<br>
Here, ''T'' and ''P'' are in °C and bars, respectively; ''x'' is the weight fraction of sodium chloride; and ''ρ''<sub>''B''</sub> is the density of the brine in gm/cm<sup>3</sup>. The calculated brine densities, along with selected data from Zarembo and Federov,<ref name="r39" /> are plotted in '''Fig. 13.18'''. The accuracy of this relationship is limited largely to the extent that other mineral salts, particularly divalent ions, are in solution.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 594 Image 0001.png|'''Fig. 13.18 – Brine density as a function of temperature pressure and salinity (ppm = parts per million NaCl). Solid circles are from Zarembo and Fedorov.<ref name="r39" />'''
File:vol1 Page 594 Image 0001.png|'''Fig. 13.18 – Brine density as a function of temperature pressure and salinity (ppm = parts per million NaCl). Solid circles are from Zarembo and Fedorov.<ref name="r39" />'''
</gallery>
</gallery><br/>A vast amount of acoustic data is available for brines, but generally for pressure, temperature, and salinity expected under oceanic conditions. Wilson<ref name="r41">_</ref> provides a relationship for the velocity ''V<sub>w</sub>'' of pure water to 100°C and about 1000 bars<br/><br/>[[File:Vol1 page 0593 eq 002.png|RTENOTITLE]]....................(13.30)<br/><br/>Millero ''et al''.<ref name="r42">_</ref> and Chen ''et al''.<ref name="r43">_</ref> give additional factors to be added to the velocity of water to calculate the effects of salinity. Their corrections, unfortunately, are limited to 55°C and 1 molal ionic strength (55,000 ppm). We can extend their results by using the data of Wyllie ''et al''.<ref name="r44">_</ref> to 100°C and 150,000 ppm NaCl. Still, this leaves the high-temperature and -pressure region with no data. Here we can use the isothermal modulus calculated from '''Eq. 13.29''' to estimate the adiabatic moduli. We can also use the velocity function provided in Chen ''et al''.<ref name="r43">_</ref> but with the constants modified to fit the additional data. The heat capacity ratio for the brine can be estimated from the PVT relationship in '''Eq. 13.29''' and estimates of the isobaric heat capacity from Helgeson and Kirkham<ref name="r37">_</ref>:<br/><br/>[[File:Vol1 page 0594 eq 001.png|RTENOTITLE]]....................(13.31)<br/><br/>and<br/><br/>[[File:Vol1 page 0594 eq 002.png|RTENOTITLE]]....................(13.32)<br/><br/>In this equation, ''m'' is the molal salt concentration and ''c''<sub>''ij''</sub>, ''d''<sub>''ij''</sub>, and ''e''<sub>''i''</sub> are constants. Using the calculated density and velocity of brine produces the modulus, and this is shown in '''Fig. 13.19'''.<br/><br/><gallery widths="300px" heights="200px">
<br/>
A vast amount of acoustic data is available for brines, but generally for pressure, temperature, and salinity expected under oceanic conditions. Wilson<ref name="r41" /> provides a relationship for the velocity ''V<sub>w</sub>'' of pure water to 100°C and about 1000 bars
<br>
<br>
[[File:Vol1 page 0593 eq 002.png]]....................(13.30)
<br>
<br>
Millero ''et al''.<ref name="r42" /> and Chen ''et al''.<ref name="r43" /> give additional factors to be added to the velocity of water to calculate the effects of salinity. Their corrections, unfortunately, are limited to 55°C and 1 molal ionic strength (55,000 ppm). We can extend their results by using the data of Wyllie ''et al''.<ref name="r44" /> to 100°C and 150,000 ppm NaCl. Still, this leaves the high-temperature and -pressure region with no data. Here we can use the isothermal modulus calculated from '''Eq. 13.29''' to estimate the adiabatic moduli. We can also use the velocity function provided in Chen ''et al''.<ref name="r43" /> but with the constants modified to fit the additional data. The heat capacity ratio for the brine can be estimated from the PVT relationship in '''Eq. 13.29''' and estimates of the isobaric heat capacity from Helgeson and Kirkham<ref name="r37" />:
<br>
<br>
[[File:Vol1 page 0594 eq 001.png]]....................(13.31)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0594 eq 002.png]]....................(13.32)
<br>
<br>
In this equation, ''m'' is the molal salt concentration and ''c''<sub>''ij''</sub>, ''d''<sub>''ij''</sub>, and ''e''<sub>''i''</sub> are constants. Using the calculated density and velocity of brine produces the modulus, and this is shown in '''Fig. 13.19'''.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 595 Image 0001.png|'''Fig. 13.19 – Brine modulus as a function of temperature, pressure, and salinity (ppm = parts per million NaCl). Water and brines are peculiar fluids in that they have a modulus (and velocity) maximum around 70°C.'''
File:vol1 Page 595 Image 0001.png|'''Fig. 13.19 – Brine modulus as a function of temperature, pressure, and salinity (ppm = parts per million NaCl). Water and brines are peculiar fluids in that they have a modulus (and velocity) maximum around 70°C.'''
</gallery>
</gallery><br/>
<br/>
</div></div><div class="toccolours mw-collapsible mw-collapsed">
<br>
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" >
 
== Elasticity, Stress-Strain, and Elastic Waves ==
== Elasticity, Stress-Strain, and Elastic Waves ==
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
<br>
<br/>We will begin this section with an introduction to stress-strain relations. These form the foundation for several rock properties, such as elastic moduli (incompressibility), effective media theory, elastic wave velocity, and rock strength.
We will begin this section with an introduction to stress-strain relations. These form the foundation for several rock properties, such as elastic moduli (incompressibility), effective media theory, elastic wave velocity, and rock strength.  


=== Stress and Pressure – Definition ===
=== Stress and Pressure – Definition ===


Stress is the force per unit area.
Stress is the force per unit area.<br/><br/>[[File:Vol1 page 0594 eq 003.png|RTENOTITLE]]....................(13.33)<br/><br/>The metric units of stress or pressure are N/m<sup>2</sup> or Pascals (Pa). Other units that are commonly used are bars, megapascals (MPa), and lbm/in.<sup>2</sup> (psi) (see '''Table 13.4'''). These stresses can take various forms such as a homogeneous pressure ''P'', normal stress ''σ''<sub>''n''</sub>, or stress applied at a general angle ''σ''<sub>''g''</sub> ('''Fig. 13.20'''). This general stress can be decomposed into normal and tangential components. We usually refer to balanced stresses because, under quasistatic conditions, they produce no net acceleration. Stress is a second-order tensor denoted by ''σ''<sub>''ij''</sub>, where the first index denotes the surface and the second the direction of the applied force (see '''Fig. 13.21'''). In Earth sciences and engineering, compressive stresses are usually considered positive, whereas most material sciences consider tensional stress positive. More details on the influence of stresses and the stress tensor can be found in Jeager and Cook<ref name="r46">_</ref> and Nye.<ref name="r47">_</ref><br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
[[File:Vol1 page 0594 eq 003.png]]....................(13.33)
<br>
<br>
The metric units of stress or pressure are N/m<sup>2</sup> or Pascals (Pa). Other units that are commonly used are bars, megapascals (MPa), and lbm/in.<sup>2</sup> (psi) (see '''Table 13.4'''). These stresses can take various forms such as a homogeneous pressure ''P'', normal stress ''σ''<sub>''n''</sub>, or stress applied at a general angle ''σ''<sub>''g''</sub> ('''Fig. 13.20'''). This general stress can be decomposed into normal and tangential components. We usually refer to balanced stresses because, under quasistatic conditions, they produce no net acceleration. Stress is a second-order tensor denoted by ''σ''<sub>''ij''</sub>, where the first index denotes the surface and the second the direction of the applied force (see '''Fig. 13.21'''). In Earth sciences and engineering, compressive stresses are usually considered positive, whereas most material sciences consider tensional stress positive. More details on the influence of stresses and the stress tensor can be found in Jeager and Cook<ref name="r46" /> and Nye.<ref name="r47" />  
<br>
<br>
<gallery widths=300px heights=200px>
File:Vol1 Page 597 Image 0001.png|'''Table 13.4'''
File:Vol1 Page 597 Image 0001.png|'''Table 13.4'''


Line 530: Line 131:


File:vol1 Page 596 Image 0002.png|'''Fig. 13.21 – Stresses acting on the elemental cube. The stresses must be balanced so that there is no acceleration of the body.'''
File:vol1 Page 596 Image 0002.png|'''Fig. 13.21 – Stresses acting on the elemental cube. The stresses must be balanced so that there is no acceleration of the body.'''
</gallery>
</gallery><br/>Several standard stress conditions are either assumed for the Earth for analysis or modeling, or applied in the laboratory:<br/><br/>Hydrostatic stress: all confining stresses are equal
<br/>
 
Several standard stress conditions are either assumed for the Earth for analysis or modeling, or applied in the laboratory:  
Uniaxial stress: one stress applied along a single axis (other stresses are zero or held constant during an experiment)
<br>
<br>
Hydrostatic stress: all confining stresses are equal  


Uniaxial stress: one stress applied along a single axis (other stresses are zero or held constant during an experiment)  
Biaxial stress: two nonequal stresses applied (third direction is equal to one of the others)


Biaxial stress: two nonequal stresses applied (third direction is equal to one of the others)  
Triaxial stress: (1) Common usage—separate vertical and two equal horizontal stresses (e.g., biaxial); (2) better—three independent principal stresses.


Triaxial stress: (1) Common usage—separate vertical and two equal horizontal stresses (e.g., biaxial); (2) better—three independent principal stresses.  
Anisotropic stresses are usually responsible for rock deformation and failure (see '''Section 13.7'''). In much of this section, however, we will concern ourselves primarily with mean stress (''σ''<sub>''m''</sub>) or pressure (''P'').<br/><br/>[[File:Vol1 page 0596 eq 001.png|RTENOTITLE]]....................(13.34)<br/><br/>It is important to distinguish among the various kinds of pressure, because the combination often determines any specific rock property and influences the response to any production procedure.


Anisotropic stresses are usually responsible for rock deformation and failure (see '''Section 13.7'''). In much of this section, however, we will concern ourselves primarily with mean stress (''σ''<sub>''m''</sub>) or pressure (''P'').
<br>
<br>
[[File:Vol1 page 0596 eq 001.png]]....................(13.34)
<br>
<br>
It is important to distinguish among the various kinds of pressure, because the combination often determines any specific rock property and influences the response to any production procedure.
<br>
{|
{|
|Confining pressure
|= ''P''<sub>''c''</sub>
|= Overburden pressure on rock frame
|-
|-
|Pore pressure
| Confining pressure
|= ''P''<sub>''p''</sub>
| = ''P''<sub>''c''</sub>
|= Fluid pressure inside pore space
| = Overburden pressure on rock frame
|-
| Pore pressure
| = ''P''<sub>''p''</sub>
| = Fluid pressure inside pore space
|-
|-
|Differential (or net) pressure
| Differential (or net) pressure
|= ''P''<sub>''d''</sub>
| = ''P''<sub>''d''</sub>
|= Difference between ''P''<sub>''c''</sub> and ''P''<sub>''p''</sub>
| = Difference between ''P''<sub>''c''</sub> and ''P''<sub>''p''</sub>
|-
|-
|Effective pressure
| Effective pressure
|= ''P''<sub>''e''</sub>
| = ''P''<sub>''e''</sub>
|= Combination of ''P''<sub>''c''</sub> and ''P''<sub>''p''</sub> controlling a property
| = Combination of ''P''<sub>''c''</sub> and ''P''<sub>''p''</sub> controlling a property
|}
|}
<br>


Increasing confining pressure (''P''<sub>''c''</sub>) alone will result in a decrease of rock volume, or compaction. In contrast, increasing the pore pressure (''P''<sub>''p''</sub>) tends to increase rock volume. ''P''<sub>''p''</sub> counteracts the effects of ''P''<sub>''c''</sub>. Thus, rock properties are controlled largely by the difference between ''P''<sub>''c''</sub> and ''P''<sub>''p''</sub>, or the differential pressure ''P''<sub>''d''</sub>. A more exact form will account for the interaction of the fluid pressure with the pore space and minerals and result in an effective stress (''P''<sub>''e''</sub>) law
 
<br>
 
<br>
Increasing confining pressure (''P''<sub>''c''</sub>) alone will result in a decrease of rock volume, or compaction. In contrast, increasing the pore pressure (''P''<sub>''p''</sub>) tends to increase rock volume. ''P''<sub>''p''</sub> counteracts the effects of ''P''<sub>''c''</sub>. Thus, rock properties are controlled largely by the difference between ''P''<sub>''c''</sub> and ''P''<sub>''p''</sub>, or the differential pressure ''P''<sub>''d''</sub>. A more exact form will account for the interaction of the fluid pressure with the pore space and minerals and result in an effective stress (''P''<sub>''e''</sub>) law<br/><br/>[[File:Vol1 page 0597 eq 001.png|RTENOTITLE]]....................(13.35)<br/><br/>where ''n'' is a term that can be derived theoretically or defined experimentally for each property.
[[File:Vol1 page 0597 eq 001.png]]....................(13.35)
<br>
<br>
where ''n'' is a term that can be derived theoretically or defined experimentally for each property.


=== Deformation, Strain, and Modulus ===
=== Deformation, Strain, and Modulus ===


Application of a single (vertical) stress is one typical experiment run to measure material mechanical properties ('''Fig. 13.22'''). If this stress continues to increase, eventually the material will fail when the uniaxial compressive strength is reached (see '''Section 13.7'''). For the rest of this chapter, however, we will deal only with small deformations and stresses such that the rock remains in the elastic region. Under this restriction, several important material properties can be defined. For an isotropic, homogeneous material, there is a vertical deformation (Δ''L'') associated with the vertical stress. Normalizing this deformation by the original length of the sample, ''L'', gives the vertical strain
Application of a single (vertical) stress is one typical experiment run to measure material mechanical properties ('''Fig. 13.22'''). If this stress continues to increase, eventually the material will fail when the uniaxial compressive strength is reached (see '''Section 13.7'''). For the rest of this chapter, however, we will deal only with small deformations and stresses such that the rock remains in the elastic region. Under this restriction, several important material properties can be defined. For an isotropic, homogeneous material, there is a vertical deformation (Δ''L'') associated with the vertical stress. Normalizing this deformation by the original length of the sample, ''L'', gives the vertical strain<br/><br/>[[File:Vol1 page 0597 eq 002.png|RTENOTITLE]]....................(13.36)<br/><br/>By definition, Young’s modulus, ''E'', is the ratio of the applied stress (''σ''<sub>''zz''</sub>) to this strain<br/><br/>[[File:Vol1 page 0597 eq 003.png|RTENOTITLE]]....................(13.37)<br/><br/>Because strain is dimensionless, ''E'' is in units of stress.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
[[File:Vol1 page 0597 eq 002.png]]....................(13.36)
<br>
<br>
By definition, Young’s modulus, ''E'', is the ratio of the applied stress (''σ''<sub>''zz''</sub>) to this strain
<br>
<br>
[[File:Vol1 page 0597 eq 003.png]]....................(13.37)
<br>
<br>
Because strain is dimensionless, ''E'' is in units of stress.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 598 Image 0001.png|'''Fig. 13.22 – Deformation of a material under vertical uniaxial stress (''σ<sub>zz</sub>'') giving rise to vertical (Δ''L'') and horizontal (Δ''W'') deformation.'''
File:vol1 Page 598 Image 0001.png|'''Fig. 13.22 – Deformation of a material under vertical uniaxial stress (''σ<sub>zz</sub>'') giving rise to vertical (Δ''L'') and horizontal (Δ''W'') deformation.'''
</gallery>
</gallery><br/>This same stress will generally result in a lateral or horizontal deformation, Δ''W''. The lateral strain can then be defined<br/><br/>[[File:Vol1 page 0598 eq 001.png|RTENOTITLE]]....................(13.38)<br/><br/>One important parameter relating the vertical and horizontal strains is Poisson’s ratio<br/><br/>[[File:Vol1 page 0598 eq 002.png|RTENOTITLE]]....................(13.39)<br/><br/>The minus sign is attached because the signs of the deformations are opposite for the horizontal vs. vertical strains in this simple case.<br/><br/>If instead we applied a pressure, we would get a volumetric strain ''ε''<sub>''v''</sub>:<br/><br/>[[File:Vol1 page 0599 eq 001.png|RTENOTITLE]]....................(13.40)<br/><br/>The bulk modulus of a material is then defined as the ratio of applied pressure to volumetric strain<br/><br/>[[File:Vol1 page 0599 eq 002.png|RTENOTITLE]]....................(13.41)<br/><br/>Bulk modulus is equivalent to the inverse of compressibility, ''β''.<br/><br/>In a similar way, shear modulus, ''μ'' (often "G" in many publications), can be defined as the ratio of shear stress to shear strain:<br/><br/>[[File:Vol1 page 0599 eq 003.png|RTENOTITLE]]....................(13.42)<br/><br/>These various equations are special cases of Hooke’s Law, which can be written for the general case<br/><br/>[[File:Vol1 page 0599 eq 004.png|RTENOTITLE]]....................(13.43)<br/><br/>Stress and strain are both tensors with 9 components. ''C''<sub>''ijkl''</sub> would then be a tensor with 81 components. However, because of symmetry considerations, only a maximum of 21 can be independent (a thorough treatment of the tensor relations is provided in Nye<ref name="r47">_</ref>). For isotropic materials, this reduces to<br/><br/>[[File:Vol1 page 0599 eq 005.png|RTENOTITLE]]....................(13.44)<br/><br/>where ''λ'' is Lame’s constant. In fact, for isotropic materials, there are only two independent elastic parameters. Any isotropic elastic constant can be written in terms of two others. For example, ''λ'' can be defined as<br/><br/>[[File:Vol1 page 0599 eq 006.png|RTENOTITLE]]....................(13.45)<br/><br/>The possible combinations among various isotropic elastic constants are shown in '''Table 13.5'''. This becomes important in applications, because restricting one term, say ''ν'', fixes the ratio of other moduli such as ''μ'' and ''K''.<br/><br/><gallery widths="300px" heights="200px">
<br/>
This same stress will generally result in a lateral or horizontal deformation, Δ''W''. The lateral strain can then be defined
<br>
<br>
[[File:Vol1 page 0598 eq 001.png]]....................(13.38)
<br>
<br>
One important parameter relating the vertical and horizontal strains is Poisson’s ratio
<br>
<br>
[[File:Vol1 page 0598 eq 002.png]]....................(13.39)
<br>
<br>
The minus sign is attached because the signs of the deformations are opposite for the horizontal vs. vertical strains in this simple case.  
<br>
<br>
If instead we applied a pressure, we would get a volumetric strain ''ε''<sub>''v''</sub>:
<br>
<br>
[[File:Vol1 page 0599 eq 001.png]]....................(13.40)
<br>
<br>
The bulk modulus of a material is then defined as the ratio of applied pressure to volumetric strain
<br>
<br>
[[File:Vol1 page 0599 eq 002.png]]....................(13.41)
<br>
<br>
Bulk modulus is equivalent to the inverse of compressibility, ''β''.  
<br>
<br>
In a similar way, shear modulus, ''μ'' (often "G" in many publications), can be defined as the ratio of shear stress to shear strain:
<br>
<br>
[[File:Vol1 page 0599 eq 003.png]]....................(13.42)
<br>
<br>
These various equations are special cases of Hooke’s Law, which can be written for the general case
<br>
<br>
[[File:Vol1 page 0599 eq 004.png]]....................(13.43)
<br>
<br>
Stress and strain are both tensors with 9 components. ''C''<sub>''ijkl''</sub> would then be a tensor with 81 components. However, because of symmetry considerations, only a maximum of 21 can be independent (a thorough treatment of the tensor relations is provided in Nye<ref name="r47" />). For isotropic materials, this reduces to
<br>
<br>
[[File:Vol1 page 0599 eq 005.png]]....................(13.44)
<br>
<br>
where ''λ'' is Lame’s constant. In fact, for isotropic materials, there are only two independent elastic parameters. Any isotropic elastic constant can be written in terms of two others. For example, ''λ'' can be defined as
<br>
<br>
[[File:Vol1 page 0599 eq 006.png]]....................(13.45)
<br>
<br>
The possible combinations among various isotropic elastic constants are shown in '''Table 13.5'''. This becomes important in applications, because restricting one term, say ''ν'', fixes the ratio of other moduli such as ''μ'' and ''K''.  
<br>
<br>
<gallery widths=300px heights=200px>
File:Vol1 Page 600 Image 0001.png|'''Table 13.5'''
File:Vol1 Page 600 Image 0001.png|'''Table 13.5'''
</gallery>
</gallery>
<br>


=== Effective Media, Bounds ===
=== Effective Media, Bounds ===


Rocks are usually not homogeneous, but are made up of multiple components such as mineral grains and pore space. On a larger scale, the bulk properties of rocks will be some weighted combination of the small-scale components. This averaging or upscaling step is needed if we wish to understand the behavior of our laboratory data or extract useful parameters from field data such as logs or seismic measurements.
Rocks are usually not homogeneous, but are made up of multiple components such as mineral grains and pore space. On a larger scale, the bulk properties of rocks will be some weighted combination of the small-scale components. This averaging or upscaling step is needed if we wish to understand the behavior of our laboratory data or extract useful parameters from field data such as logs or seismic measurements.<br/><br/>The simplest bounds are provided by the constant strain and constant stress limits. This method is equivalent to the series vs. parallel effective resistance of a resistor network. In the case that strains of the two materials making up our material are equal, as with the parallel plates in '''Fig. 13.23a''', we get the upper s(Voigt) limit. The response is controlled by the stiffer component.<br/><br/>[[File:Vol1 page 0601 eq 001.png|RTENOTITLE]]....................(13.46)<br/><br/>where ''M''<sub>''V''</sub> is the effective Voigt modulus, ''M''<sub>''A''</sub> and ''M''<sub>''B''</sub> are the component moduli, and ''A'' is the volume fraction of component ''A''. In contrast, with the constant-stress case ('''Fig. 13.23b'''), the soft component dominates the deformation and we get the lower (Reuss) limit.<br/><br/>[[File:Vol1 page 0601 eq 002.png|RTENOTITLE]]....................(13.47)<br/><br/>where ''M''<sub>''R''</sub> is the lower Reuss effective modulus. The average value between these two limits is often used in property estimation and is termed the Voigt-Reuss-Hill relation<br/><br/>[[File:Vol1 page 0601 eq 003.png|RTENOTITLE]]....................(13.48)<br/><br/>Note that in the case for minerals plus pores, ''M''<sub>pore</sub> = 0 and ''M''<sub>''V''</sub> decreases linearly with porosity. ''M''<sub>''R''</sub> equals zero for all porosities.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
The simplest bounds are provided by the constant strain and constant stress limits. This method is equivalent to the series vs. parallel effective resistance of a resistor network. In the case that strains of the two materials making up our material are equal, as with the parallel plates in '''Fig. 13.23a''', we get the upper s(Voigt) limit. The response is controlled by the stiffer component.
<br>
<br>
[[File:Vol1 page 0601 eq 001.png]]....................(13.46)
<br>
<br>
where ''M''<sub>''V''</sub> is the effective Voigt modulus, ''M''<sub>''A''</sub> and ''M''<sub>''B''</sub> are the component moduli, and ''A'' is the volume fraction of component ''A''. In contrast, with the constant-stress case ('''Fig. 13.23b'''), the soft component dominates the deformation and we get the lower (Reuss) limit.
<br>
<br>
[[File:Vol1 page 0601 eq 002.png]]....................(13.47)
<br>
<br>
where ''M''<sub>''R''</sub> is the lower Reuss effective modulus. The average value between these two limits is often used in property estimation and is termed the Voigt-Reuss-Hill relation
<br>
<br>
[[File:Vol1 page 0601 eq 003.png]]....................(13.48)
<br>
<br>
Note that in the case for minerals plus pores, ''M''<sub>pore</sub> = 0 and ''M''<sub>''V''</sub> decreases linearly with porosity. ''M''<sub>''R''</sub> equals zero for all porosities.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 601 Image 0001.png|'''Fig. 13.23 – Constraints leading to bounds on the elastic properties of a composite material. For rocks, one component is usually considered the mineral matrix, the other component the pore space: (a) constant strain condition, Voigt bound; (b) constant stress condition, Reuss bound.'''
File:vol1 Page 601 Image 0001.png|'''Fig. 13.23 – Constraints leading to bounds on the elastic properties of a composite material. For rocks, one component is usually considered the mineral matrix, the other component the pore space: (a) constant strain condition, Voigt bound; (b) constant stress condition, Reuss bound.'''
</gallery>
</gallery><br/>An alternative approach, known as the Hasin-Shtrikman technique,<ref name="r48">_</ref> is to fill space with concentric spheres. Material 1 is in the interior, and Material 2 forms a surrounding shell. Spheres such as these but of varying size are packed together to fill the entire medium ('''Fig. 13.24'''). The resulting upper and lower bounds ("+" vs. "–" respectively) for bulk and shear modulus are given by<br/><br/>[[File:Vol1 page 0602 eq 001.png|RTENOTITLE]]....................(13.49)<br/><br/>and<br/><br/>[[File:Vol1 page 0602 eq 002.png|RTENOTITLE]]....................(13.50)<br/><br/>where ''K''<sub>''i''</sub>, ''μ''<sub>''i''</sub>, and ''f''<sub>''i''</sub> refer to the bulk and shear moduli and volume fraction of component ''i'', respectively. The upper and lower bounds are derived by exchanging the stiff and soft components as "1" or "2."<br/><br/><gallery widths="300px" heights="200px">
<br/>
An alternative approach, known as the Hasin-Shtrikman technique,<ref name="r48" /> is to fill space with concentric spheres. Material 1 is in the interior, and Material 2 forms a surrounding shell. Spheres such as these but of varying size are packed together to fill the entire medium ('''Fig. 13.24'''). The resulting upper and lower bounds ("+" vs. "–" respectively) for bulk and shear modulus are given by  
<br>
<br>
[[File:Vol1 page 0602 eq 001.png]]....................(13.49)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0602 eq 002.png]]....................(13.50)
<br>
<br>
where ''K''<sub>''i''</sub>, ''μ''<sub>''i''</sub>, and ''f''<sub>''i''</sub> refer to the bulk and shear moduli and volume fraction of component ''i'', respectively. The upper and lower bounds are derived by exchanging the stiff and soft components as "1" or "2."  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 602 Image 0001.png|'''Fig. 13.24 – Schematic view of the composite material modeled by Hasin-Shtrikman method.'''
File:vol1 Page 602 Image 0001.png|'''Fig. 13.24 – Schematic view of the composite material modeled by Hasin-Shtrikman method.'''
</gallery>
</gallery><br/>The results of using '''Eqs. 13.46''' through '''13.50''' are shown in '''Fig. 13.25'''. Using quartz as the first component and porosity as the second, the composite bulk modulus is plotted in '''Fig. 13.25a''' as a function of porosity. In one case, the pores are empty (black), in the other, water fills the pores and is the second component (blue). Because we used quartz as the solid component ('''Table 13.6'''), these bounds should contain all possible values for sandstones (remember: for isotropic and homogeneous sandstones). If, on the other hand, our rock was made up of only quartz and calcite, we get bounds that appear in '''Fig. 13.25b'''. Note that the bounds have collapsed and produce only a narrow spread. This is a result of the two end components both being stiff and closer together. In cases such as these, a simple linear average can work well.<br/><br/><gallery widths="300px" heights="200px">
<br/>
The results of using '''Eqs. 13.46''' through '''13.50''' are shown in '''Fig. 13.25'''. Using quartz as the first component and porosity as the second, the composite bulk modulus is plotted in '''Fig. 13.25a''' as a function of porosity. In one case, the pores are empty (black), in the other, water fills the pores and is the second component (blue). Because we used quartz as the solid component ('''Table 13.6'''), these bounds should contain all possible values for sandstones (remember: for isotropic and homogeneous sandstones). If, on the other hand, our rock was made up of only quartz and calcite, we get bounds that appear in '''Fig. 13.25b'''. Note that the bounds have collapsed and produce only a narrow spread. This is a result of the two end components both being stiff and closer together. In cases such as these, a simple linear average can work well.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 603 Image 0001.png|'''Fig. 13.25 – (a) General bounds of a porous material made of quartz, both dry and saturated with water (HS = Hasin-Shtrikman). With extreme differences in material properties, the bounds can be very wide. (b) Example of bounds for a quartz-calcite mixture. Because the properties are comparable between the two minerals, the bounds act much more like simple linear averages.'''
File:vol1 Page 603 Image 0001.png|'''Fig. 13.25 – (a) General bounds of a porous material made of quartz, both dry and saturated with water (HS = Hasin-Shtrikman). With extreme differences in material properties, the bounds can be very wide. (b) Example of bounds for a quartz-calcite mixture. Because the properties are comparable between the two minerals, the bounds act much more like simple linear averages.'''


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File:Vol1 Page 608 Image 0001.png|'''Table 13.6B'''
File:Vol1 Page 608 Image 0001.png|'''Table 13.6B'''
</gallery>
</gallery>
<br>


=== Mineral Properties ===
=== Mineral Properties ===


There are numerous ways to measure mineral moduli. The most obvious is by deforming single crystals. Alternatively, elastic velocities can be measured and moduli extracted for zero porosity aggregates. '''Tables 13.6a''' and '''13.6b''' present lists of "isotropic" densities, mineral bulk and shear moduli, and elastic velocities. In reality, minerals are anisotropic, and the values listed in the table are averages derived from the effective media fomulas presented above to represent polygrained isotropic composites. The highest-velocity, highest-moduli are for such minerals as almandine and rutile. Velocities can reach 9 km/s for ''V''<sub>''p''</sub>, and moduli can be in the hundreds of GPa. Clays are a particular problem. As noted before, they are among the most abundant minerals on the surface of the Earth, and are common in most sedimentary rocks. Their small size, variable composition, and chemical activity make them difficult to characterize from a mechanical point of view. The results of Katahara,<ref name="r49" /> Wang ''et al''.,<ref name="r50" /> and Prasad ''et al''.<ref name="r51" /> are given in '''Table 13.6b'''.  
There are numerous ways to measure mineral moduli. The most obvious is by deforming single crystals. Alternatively, elastic velocities can be measured and moduli extracted for zero porosity aggregates. '''Tables 13.6a''' and '''13.6b''' present lists of "isotropic" densities, mineral bulk and shear moduli, and elastic velocities. In reality, minerals are anisotropic, and the values listed in the table are averages derived from the effective media fomulas presented above to represent polygrained isotropic composites. The highest-velocity, highest-moduli are for such minerals as almandine and rutile. Velocities can reach 9 km/s for ''V''<sub>''p''</sub>, and moduli can be in the hundreds of GPa. Clays are a particular problem. As noted before, they are among the most abundant minerals on the surface of the Earth, and are common in most sedimentary rocks. Their small size, variable composition, and chemical activity make them difficult to characterize from a mechanical point of view. The results of Katahara,<ref name="r49">_</ref> Wang ''et al''.,<ref name="r50">_</ref> and Prasad ''et al''.<ref name="r51">_</ref> are given in '''Table 13.6b'''.<br/><br/>Mineral properties can also be extracted from the numerous empirical trends developed for rocks, as we will see below.
<br>
<br>
Mineral properties can also be extracted from the numerous empirical trends developed for rocks, as we will see below.  


=== Elastic Wave Velocities ===
=== Elastic Wave Velocities ===


So far, we have considered only the static elastic deformation of materials. By adding the dynamic behavior, we arrive at how elastic waves propagate through materials. If a body is changing its speed as well as deforming, there will be an unbalanced force because of the acceleration described through Newton’s Second Law:
So far, we have considered only the static elastic deformation of materials. By adding the dynamic behavior, we arrive at how elastic waves propagate through materials. If a body is changing its speed as well as deforming, there will be an unbalanced force because of the acceleration described through Newton’s Second Law:<br/><br/>[[File:Vol1 page 0604 eq 001.png|RTENOTITLE]]....................(13.51)<br/><br/>where ''ρ'' is density, ''a'' is acceleration, ''u'' is displacement, and ''t'' is time. Combining this with Hook’s Law ('''Eq. 13.43''') gives the general wave equation. For a plane wave in the ''xx'' direction, this can be written as<br/><br/>[[File:Vol1 page 0604 eq 002.png|RTENOTITLE]]....................(13.52)<br/><br/>However, if the material is being deformed, we will have strains associated with the change of displacement with position. In turn, these strains can be related to the stresses through the appropriate modulus, M (for example, '''Eq. 13.37'''):<br/><br/>[[File:Vol1 page 0604 eq 003.png|RTENOTITLE]]....................(13.53)<br/><br/>For constant elastic components, this simplifies to<br/><br/>[[File:Vol1 page 0604 eq 004.png|RTENOTITLE]]....................(13.54)<br/><br/>The solution to this equation gives the compressional velocity<br/><br/>[[File:Vol1 page 0604 eq 005.png|RTENOTITLE]]....................(13.55)<br/><br/>Similarly, for shear motion<br/><br/>[[File:Vol1 page 0604 eq 006.png|RTENOTITLE]]....................(13.56)<br/><br/>and we get the shear velocity:<br/><br/>[[File:Vol1 page 0607 eq 001.png|RTENOTITLE]]....................(13.57)<br/>
<br>
<br>
[[File:Vol1 page 0604 eq 001.png]]....................(13.51)
<br>
<br>
where ''ρ'' is density, ''a'' is acceleration, ''u'' is displacement, and ''t'' is time. Combining this with Hook’s Law ('''Eq. 13.43''') gives the general wave equation. For a plane wave in the ''xx'' direction, this can be written as
<br>
<br>
[[File:Vol1 page 0604 eq 002.png]]....................(13.52)
<br>
<br>
However, if the material is being deformed, we will have strains associated with the change of displacement with position. In turn, these strains can be related to the stresses through the appropriate modulus, M (for example, '''Eq. 13.37'''):
<br>
<br>
[[File:Vol1 page 0604 eq 003.png]]....................(13.53)
<br>
<br>
For constant elastic components, this simplifies to
<br>
<br>
[[File:Vol1 page 0604 eq 004.png]]....................(13.54)
<br>
<br>
The solution to this equation gives the compressional velocity
<br>
<br>
[[File:Vol1 page 0604 eq 005.png]]....................(13.55)
<br>
<br>
Similarly, for shear motion
<br>
<br>
[[File:Vol1 page 0604 eq 006.png]]....................(13.56)
<br>
<br>
and we get the shear velocity:
<br>
<br>
[[File:Vol1 page 0607 eq 001.png]]....................(13.57)
<br>
<br>


=== Porosity Dependence ===
=== Porosity Dependence ===


The bounding relations we examined above can be applied directly to rock acoustic velocities. Some dolomites with vuggy pores may approach the Voigt bound. Highly fractured rocks may approach the Reuss bound. However, there is often a great difference between these idealized bounds and real rocks. For sandstones, we would expect to begin with quartz velocity at zero porosity and have decreasing velocity with increasing porosity. By combining '''Eqs. 13.46''' and '''13.47''' for moduli in '''Eq. 13.55''', we can plot expected velocity bounds, as in '''Fig. 13.26a'''. Observed distributions for sandstones are also plotted, and we see a systematic discrepancy with the upper (Voigt) bound. At high porosities, grains separate, and the mixture acts as a suspension. The majority of rocks have an upper limit to their porosity usually termed "critical porosity," ''Φ''<sub>''c''</sub> (Yin ''et al''.<ref name="r53" /> and Nur ''et al''.<ref name="r54" />). At this high porosity limit, we reach the threshold of grain contacts and grain support (Han ''et al''.<ref name="r55" />).  
The bounding relations we examined above can be applied directly to rock acoustic velocities. Some dolomites with vuggy pores may approach the Voigt bound. Highly fractured rocks may approach the Reuss bound. However, there is often a great difference between these idealized bounds and real rocks. For sandstones, we would expect to begin with quartz velocity at zero porosity and have decreasing velocity with increasing porosity. By combining '''Eqs. 13.46''' and '''13.47''' for moduli in '''Eq. 13.55''', we can plot expected velocity bounds, as in '''Fig. 13.26a'''. Observed distributions for sandstones are also plotted, and we see a systematic discrepancy with the upper (Voigt) bound. At high porosities, grains separate, and the mixture acts as a suspension. The majority of rocks have an upper limit to their porosity usually termed "critical porosity," ''Φ''<sub>''c''</sub> (Yin ''et al''.<ref name="r53">_</ref> and Nur ''et al''.<ref name="r54">_</ref>). At this high porosity limit, we reach the threshold of grain contacts and grain support (Han ''et al''.<ref name="r55">_</ref>).<br/><br/><gallery widths="300px" heights="200px">
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 609 Image 0001.png|'''Fig. 13.26 – (a) Compressional velocity vs. porosity in sandstones: laboratory data with elastic bounds (modified from Marion<ref name="r52" />). (b) Sandstone velocities divided into classes.'''
File:vol1 Page 609 Image 0001.png|'''Fig. 13.26 – (a) Compressional velocity vs. porosity in sandstones: laboratory data with elastic bounds (modified from Marion<ref name="r52" />). (b) Sandstone velocities divided into classes.'''
</gallery>
</gallery><br/>Brine-saturated sandstone velocities can be separated into classes based on their velocity-porosity relations ('''Fig. 13.26b'''). Very clean sandstones (Class I) decrease in a simple linear trend from the 6 km/s velocity of quartz as porosity increases. Most consolidated rocks (Class II) have somewhat lower velocities, still decreasing with increasing porosity. Poorly cemented sands (Class III) approach the lower Reuss bound for velocity. Pure suspensions are dominated by the modulus of water (Class IV) and are almost independent of the porosity. However, such suspensions are rare. Another important class is dominated by fractures (Class V). As we shall see later, fractures have a far greater effect on velocity than might be expected for their low porosity, and may approach the Reuss bound.
<br/>
Brine-saturated sandstone velocities can be separated into classes based on their velocity-porosity relations ('''Fig. 13.26b'''). Very clean sandstones (Class I) decrease in a simple linear trend from the 6 km/s velocity of quartz as porosity increases. Most consolidated rocks (Class II) have somewhat lower velocities, still decreasing with increasing porosity. Poorly cemented sands (Class III) approach the lower Reuss bound for velocity. Pure suspensions are dominated by the modulus of water (Class IV) and are almost independent of the porosity. However, such suspensions are rare. Another important class is dominated by fractures (Class V). As we shall see later, fractures have a far greater effect on velocity than might be expected for their low porosity, and may approach the Reuss bound.  


=== Measured Velocity-Porosity Relations ===
=== Measured Velocity-Porosity Relations ===


Numerous systematic investigations into the relationship of velocity, porosity, and lithology (usually clay content) have been conducted. The results of Vernik and Nur<ref name="r56" /> for brine-saturated sandstones are shown in '''Fig. 13.27''' for compressional and shear velocities, respectively. Very clean sands (clean arenites) show the linear decrease from quartz velocity. However, even small amounts of clays will substantially lower the trend. Increasing clay content will then continue to lower velocities.  
Numerous systematic investigations into the relationship of velocity, porosity, and lithology (usually clay content) have been conducted. The results of Vernik and Nur<ref name="r56">_</ref> for brine-saturated sandstones are shown in '''Fig. 13.27''' for compressional and shear velocities, respectively. Very clean sands (clean arenites) show the linear decrease from quartz velocity. However, even small amounts of clays will substantially lower the trend. Increasing clay content will then continue to lower velocities.<br/><br/><gallery widths="300px" heights="200px">
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 610 Image 0001.png|'''Fig. 13.27 – (a) Brine-saturated sandstone compressional velocities as a function of porosity. Sandstones are segregated into lithologic type, and show decreasing velocity as rocks depart from pure quartz content (after Vernik and Nur<ref name="r56" />). (b) Brine-saturated sandstone shear velocities as a function of porosity. Behavior is similar to the ''V<sub>p</sub>'' trends seen in (a) above (after Vernik and Nur<ref name="r56" />).'''
File:vol1 Page 610 Image 0001.png|'''Fig. 13.27 – (a) Brine-saturated sandstone compressional velocities as a function of porosity. Sandstones are segregated into lithologic type, and show decreasing velocity as rocks depart from pure quartz content (after Vernik and Nur<ref name="r56" />). (b) Brine-saturated sandstone shear velocities as a function of porosity. Behavior is similar to the ''V<sub>p</sub>'' trends seen in (a) above (after Vernik and Nur<ref name="r56" />).'''
</gallery>
</gallery><br/>Numerous examples of general porosity/velocity/clay content relations for sandstones are given in '''Table 13.7 a and b''' (symbol definitions for these relations are in '''Table 13.7c'''). These types of relations have proved very useful in giving velocities under general conditions, providing the overall effects of clay, and establishing the relation of compressional to shear velocity (''V''<sub>''p''</sub>/''V''<sub>''s''</sub> ratios). ''V''<sub>''p''</sub>−''V''<sub>''s''</sub> relations are extremely important, because shear logs are relatively rare, yet shear velocities are critical in determining seismic direct hydrocarbon indicators such as reflection Amplitude-Versus-Offset (AVO) trends (Castagna ''et al''.<ref name="r18">_</ref>).<br/><br/><gallery widths="300px" heights="200px">
<br/>
Numerous examples of general porosity/velocity/clay content relations for sandstones are given in '''Table 13.7 a and b''' (symbol definitions for these relations are in '''Table 13.7c'''). These types of relations have proved very useful in giving velocities under general conditions, providing the overall effects of clay, and establishing the relation of compressional to shear velocity (''V''<sub>''p''</sub>/''V''<sub>''s''</sub> ratios). ''V''<sub>''p''</sub>−''V''<sub>''s''</sub> relations are extremely important, because shear logs are relatively rare, yet shear velocities are critical in determining seismic direct hydrocarbon indicators such as reflection Amplitude-Versus-Offset (AVO) trends (Castagna ''et al''.<ref name="r18" />).
<br>
<br>
<gallery widths=300px heights=200px>
File:Vol1 Page 611 Image 0001.png|'''Table 13.7A'''
File:Vol1 Page 611 Image 0001.png|'''Table 13.7A'''


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File:Vol1 Page 612 Image 0002.png|'''Table 13.7C'''
File:Vol1 Page 612 Image 0002.png|'''Table 13.7C'''
</gallery>
</gallery>
<br>


Measured data for carbonates are less abundant. A systematic investigation of samples from several wells was reported by Rafavich ''et al''.<ref name="r57" /> A plot of their results for carbonate ''V''<sub>''p''</sub> as functions of porosity and composition is shown in '''Fig. 13.28'''. They collected detailed information on fabric and texture as well as porosity and mineralogy. Performing regressions on their extensive data set produced the relations given in '''Table 13.8a'''. The coefficients associated with these equations are given in '''Table 13.8b'''. Note that the relations are dependent on the effective pressure.  
Measured data for carbonates are less abundant. A systematic investigation of samples from several wells was reported by Rafavich ''et al''.<ref name="r57">_</ref> A plot of their results for carbonate ''V''<sub>''p''</sub> as functions of porosity and composition is shown in '''Fig. 13.28'''. They collected detailed information on fabric and texture as well as porosity and mineralogy. Performing regressions on their extensive data set produced the relations given in '''Table 13.8a'''. The coefficients associated with these equations are given in '''Table 13.8b'''. Note that the relations are dependent on the effective pressure.<br/><br/><gallery widths="300px" heights="200px">
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 613 Image 0001.png|'''Fig. 13.28 – Carbonate compressional velocity as functions of porosity and mineralogy. The compositions are plotted by number according to the Anhydrite-Calcite-Dolomite triangular diagram in the lower-left portion of the figure (from Rafavich ''et al''.<ref name="r57" />).'''
File:vol1 Page 613 Image 0001.png|'''Fig. 13.28 – Carbonate compressional velocity as functions of porosity and mineralogy. The compositions are plotted by number according to the Anhydrite-Calcite-Dolomite triangular diagram in the lower-left portion of the figure (from Rafavich ''et al''.<ref name="r57" />).'''


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File:Vol1 Page 617 Image 0001.png|'''Table 13.8B (continued)'''
File:Vol1 Page 617 Image 0001.png|'''Table 13.8B (continued)'''
</gallery>
</gallery><br/>A similar set of measurements by Wang ''et al''.<ref name="r58">_</ref> are shown in '''Fig. 13.29'''. For carbonates, the data can be quite scattered, but can still show the general velocity decrease with increasing porosity. These results were summarized in a set of relations ('''Table 13.9''') again showing pressure dependence. Their data, however, includes measurements made with samples not only brine-saturated, but hydrocarbon-saturated and after simulated reservoir floods. <ref name="r59">_</ref> They demonstrate that the overall velocity and impedance changes were strongly dependent on the imposed sequence of flooding. The ability to observe a particular reservoir process will be more complicated than simply completely substituting fluids into the rocks.<br/><br/><gallery widths="300px" heights="200px">
<br>
A similar set of measurements by Wang ''et al''.<ref name="r58" /> are shown in '''Fig. 13.29'''. For carbonates, the data can be quite scattered, but can still show the general velocity decrease with increasing porosity. These results were summarized in a set of relations ('''Table 13.9''') again showing pressure dependence. Their data, however, includes measurements made with samples not only brine-saturated, but hydrocarbon-saturated and after simulated reservoir floods. <ref name="r59" /> They demonstrate that the overall velocity and impedance changes were strongly dependent on the imposed sequence of flooding. The ability to observe a particular reservoir process will be more complicated than simply completely substituting fluids into the rocks.
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 614 Image 0001.png|'''Fig. 13.29 – (a) Dry carbonate velocities. Although there is considerable scatter, perhaps because of the heterogeneous nature of the porosity, a systematic decrease in both compressional and shear velocities with increasing porosity is obvious (from Wang ''et al''.<ref name="r58" />). (b) Carbonate velocities when saturated with a light refined oil. Compressional velocities are higher and shear velocities slightly lower. The porosity dependence is similar to the dry case (from Wang ''et al''.<ref name="r58" />).'''
File:vol1 Page 614 Image 0001.png|'''Fig. 13.29 – (a) Dry carbonate velocities. Although there is considerable scatter, perhaps because of the heterogeneous nature of the porosity, a systematic decrease in both compressional and shear velocities with increasing porosity is obvious (from Wang ''et al''.<ref name="r58" />). (b) Carbonate velocities when saturated with a light refined oil. Compressional velocities are higher and shear velocities slightly lower. The porosity dependence is similar to the dry case (from Wang ''et al''.<ref name="r58" />).'''


File:Vol1 Page 618 Image 0001.png|'''Table 13.9'''
File:Vol1 Page 618 Image 0001.png|'''Table 13.9'''
</gallery>
</gallery>
<br>


=== Pressure ===
=== Pressure ===


Rock moduli (compressibility) and elastic velocities are strongly influenced by pressure. With increasing effective pressure, compliant pores within a rock will deform, contract, or close. The rock becomes stiffer, and, as a result, velocities increase. Two examples are shown in '''Fig. 13.30'''. The typical behavior is rapid increase in velocity, with increasing pressure at low pressures, followed by a flattening of the curve at higher pressures. Presumably, compliant pores and cracks are closed at higher pressure, and velocities asymptotically approach a relatively constant velocity. This specific behavior at high pressures leads to the simple velocity-porosity transforms and probably is responsible for our ability to use sonic tools as in-situ porosity indicators with little regard to local pressures.  
Rock moduli (compressibility) and elastic velocities are strongly influenced by pressure. With increasing effective pressure, compliant pores within a rock will deform, contract, or close. The rock becomes stiffer, and, as a result, velocities increase. Two examples are shown in '''Fig. 13.30'''. The typical behavior is rapid increase in velocity, with increasing pressure at low pressures, followed by a flattening of the curve at higher pressures. Presumably, compliant pores and cracks are closed at higher pressure, and velocities asymptotically approach a relatively constant velocity. This specific behavior at high pressures leads to the simple velocity-porosity transforms and probably is responsible for our ability to use sonic tools as in-situ porosity indicators with little regard to local pressures.<br/><br/><gallery widths="300px" heights="200px">
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 618 Image 0002.png|'''Fig. 13.30 – Examples of dry and water-saturated sandstone velocities as a function of hydrostatic differential pressure. As pressure increases, compliant pores close, making the rock stiffer with higher velocity and lower pore fluid sensitivity.'''
File:vol1 Page 618 Image 0002.png|'''Fig. 13.30 – Examples of dry and water-saturated sandstone velocities as a function of hydrostatic differential pressure. As pressure increases, compliant pores close, making the rock stiffer with higher velocity and lower pore fluid sensitivity.'''
</gallery>
</gallery><br/>The stress dependence of granular material has been examined extensively. For example, Gassmann<ref name="r60">_</ref> and Duffy and Mindlin<ref name="r61">_</ref> modeled various packings of spheres. In general, they found that<br/><br/>[[File:Vol1 page 0613 eq 001.png|RTENOTITLE]]....................(13.58)<br/><br/>where ''f'' is approximately linear. This type of relation is particularly useful for poorly consolidated sands.<br/><br/>Although the absolute pressure dependences shown in '''Fig. 13.30a''' vs '''13.30b''' are in significant contrast, for most sandstones, relative changes are more consistent. By normalizing the velocities to those at high pressure (40 MPa), we get a much more consistent behavior ('''Fig. 13.31''').<br/><br/>[[File:Vol1 page 0613 eq 002.png|RTENOTITLE]]....................(13.59)<br/><br/>Examining a similar set of data allowed Eberhart-Phillips ''et al''.<ref name="r62">_</ref> to develop a pair of relations for both ''V''<sub>''p''</sub> and ''V''<sub>''s''</sub> (see also '''Table 13.7''')<br/><br/>[[File:Vol1 page 0614 eq 001.png|RTENOTITLE]]....................(13.60a)<br/><br/>[[File:Vol1 page 0615 eq 001.png|RTENOTITLE]]....................(13.60b)<br/><br/>where ''P''<sub>''e''</sub> is the effective pressure. For carbonates, the explicit pressure dependence given in '''Tables 13.8a''' and '''13.9''' allow the pressure dependence to be evaluated. The pressure dependence for carbonate ''V''<sub>''p''</sub> from Rafavich ''et al''.<ref name="r57">_</ref> is shown in '''Fig. 13.32'''. Note that pressure sensitivity increases with increasing porosity. These types of relations permit velocity changes associated with pressure changes in the reservoir to be modeled.<br/><br/><gallery widths="300px" heights="200px">
<br/>
The stress dependence of granular material has been examined extensively. For example, Gassmann<ref name="r60" /> and Duffy and Mindlin<ref name="r61" /> modeled various packings of spheres. In general, they found that
<br>
<br>
[[File:Vol1 page 0613 eq 001.png]]....................(13.58)
<br>
<br>
where ''f'' is approximately linear. This type of relation is particularly useful for poorly consolidated sands.  
<br>
<br>
Although the absolute pressure dependences shown in '''Fig. 13.30a''' vs '''13.30b''' are in significant contrast, for most sandstones, relative changes are more consistent. By normalizing the velocities to those at high pressure (40 MPa), we get a much more consistent behavior ('''Fig. 13.31''').
<br>
<br>
[[File:Vol1 page 0613 eq 002.png]]....................(13.59)
<br>
<br>
Examining a similar set of data allowed Eberhart-Phillips ''et al''.<ref name="r62" /> to develop a pair of relations for both ''V''<sub>''p''</sub> and ''V''<sub>''s''</sub> (see also '''Table 13.7''')
<br>
<br>
[[File:Vol1 page 0614 eq 001.png]]....................(13.60a)
<br>
<br>
[[File:Vol1 page 0615 eq 001.png]]....................(13.60b)
<br>
<br>
where ''P''<sub>''e''</sub> is the effective pressure. For carbonates, the explicit pressure dependence given in '''Tables 13.8a''' and '''13.9''' allow the pressure dependence to be evaluated. The pressure dependence for carbonate ''V''<sub>''p''</sub> from Rafavich ''et al''.<ref name="r57" /> is shown in '''Fig. 13.32'''. Note that pressure sensitivity increases with increasing porosity. These types of relations permit velocity changes associated with pressure changes in the reservoir to be modeled.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 619 Image 0001.png|'''Fig. 13.31 – Compressional velocities normalized by the velocity at 40 MPa. For many sandstones, the average trend can be used.'''
File:vol1 Page 619 Image 0001.png|'''Fig. 13.31 – Compressional velocities normalized by the velocity at 40 MPa. For many sandstones, the average trend can be used.'''


File:vol1 Page 620 Image 0001.png|'''Fig. 13.32 – Generalized compressional velocities dependence on pressure seen in carbonates by Rafavich ''et al''.<ref name="r57" /> Pressure dependence is a function of porosity.'''
File:vol1 Page 620 Image 0001.png|'''Fig. 13.32 – Generalized compressional velocities dependence on pressure seen in carbonates by Rafavich ''et al''.<ref name="r57" /> Pressure dependence is a function of porosity.'''
</gallery>
</gallery><br/>It is important to note that all these relations involve either differential pressure (''P''<sub>''d''</sub>) or effective pressure (''P''<sub>''e''</sub>). Pore pressure (''P''<sub>''p''</sub>) counters the influence of confining pressure (''P''<sub>''c''</sub>), so the difference between these two controls rock properties. This has been expressed simply in the Terzaghi<ref name="r63">_</ref> relation for the pressure dependence of a given porous material property ''S'',<br/><br/>[[File:Vol1 page 0615 eq 002.png|RTENOTITLE]]....................(13.61)<br/><br/>This kind of behavior has been seen in numerous cases, as in '''Fig. 13.33'''. This is one reason why properties such as density, resistivity, and velocity can decrease with increasing depth when "overpressure" or when increased pore pressure is encountered. Changes in reservoir pore pressure will have a similar influence. More precisely, it is the effective pressure ('''Eq. 13.35''') that controls properties rather than just the differential. However, the magnitude of effective pressure is often found to be close to the simpler differential pressure.<br/><br/><gallery widths="300px" heights="200px">
<br/>
It is important to note that all these relations involve either differential pressure (''P''<sub>''d''</sub>) or effective pressure (''P''<sub>''e''</sub>). Pore pressure (''P''<sub>''p''</sub>) counters the influence of confining pressure (''P''<sub>''c''</sub>), so the difference between these two controls rock properties. This has been expressed simply in the Terzaghi<ref name="r63" /> relation for the pressure dependence of a given porous material property ''S'',
<br>
<br>
[[File:Vol1 page 0615 eq 002.png]]....................(13.61)
<br>
<br>
This kind of behavior has been seen in numerous cases, as in '''Fig. 13.33'''. This is one reason why properties such as density, resistivity, and velocity can decrease with increasing depth when "overpressure" or when increased pore pressure is encountered. Changes in reservoir pore pressure will have a similar influence. More precisely, it is the effective pressure ('''Eq. 13.35''') that controls properties rather than just the differential. However, the magnitude of effective pressure is often found to be close to the simpler differential pressure.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 621 Image 0001.png|'''Fig. 13.33 – Compressional velocities through a water-saturated oil-wet sandstone sample at various confining and pore pressures. When confining and pore pressures are varied together to give constant differential pressure, the velocity stays almost constant (after Wylie ''et al''.<ref name="r64" />).'''
File:vol1 Page 621 Image 0001.png|'''Fig. 13.33 – Compressional velocities through a water-saturated oil-wet sandstone sample at various confining and pore pressures. When confining and pore pressures are varied together to give constant differential pressure, the velocity stays almost constant (after Wylie ''et al''.<ref name="r64" />).'''
</gallery>
</gallery>
<br/>


=== In-Situ Stresses ===
=== In-Situ Stresses ===


The in-situ "lithostatic" stresses are usually unequal. Such different stresses are required or faults, folds, and other structural features would never be developed. In contrast, most laboratory data are collected under equal stress or "hydrostatic" conditions. Differential or triaxial measurements are comparatively rare (e.g., Gregory,<ref name="r65" /> Nur and Simmons,<ref name="r66" /> Yin,<ref name="r67" /> and Scott ''et al''.<ref name="r68" />).  
The in-situ "lithostatic" stresses are usually unequal. Such different stresses are required or faults, folds, and other structural features would never be developed. In contrast, most laboratory data are collected under equal stress or "hydrostatic" conditions. Differential or triaxial measurements are comparatively rare (e.g., Gregory,<ref name="r65">_</ref> Nur and Simmons,<ref name="r66">_</ref> Yin,<ref name="r67">_</ref> and Scott ''et al''.<ref name="r68">_</ref>).<br/><br/>In a simple compacting basin with neither lateral deformation nor tectonic stresses, the vertical stress will be largest. Lateral stresses will be developed in a basin as sediments are buried and compacted but are constrained horizontally. Both uniform hydrostatic and unequal lithostatic stress conditions are shown in '''Fig. 13.34'''.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
In a simple compacting basin with neither lateral deformation nor tectonic stresses, the vertical stress will be largest. Lateral stresses will be developed in a basin as sediments are buried and compacted but are constrained horizontally. Both uniform hydrostatic and unequal lithostatic stress conditions are shown in '''Fig. 13.34'''.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 622 Image 0001.png|'''Fig. 13.24 – Typical stress conditions that might be found in a compacting basin with no lateral deformation and no applied tectonic stresses.'''
File:vol1 Page 622 Image 0001.png|'''Fig. 13.24 – Typical stress conditions that might be found in a compacting basin with no lateral deformation and no applied tectonic stresses.'''
</gallery>
</gallery><br/>A simple estimate of the horizontal stress, ''σ''<sub>''h''</sub>, can be made from the axial stress, ''σ''<sub>''v''</sub>, by<br/><br/>[[File:Vol1 page 0616 eq 001.png|RTENOTITLE]]....................(13.62)<br/><br/>where ''ν'' is Poisson’s ratio. Calculated stresses typical for sands (''ν'' = 0.1) and more clay-rich rocks (''ν'' = 0.25) are also shown in '''Fig. 13.34'''. This basic relation ('''Eq. 13.62''') is an oversimplification of actual conditions, but it does provide a useful conceptual model, and lateral stresses indeed are found to be lower in sandstones than in shaly sections in most places.<br/><br/>From a matrix of velocities measured over axial and lateral stress conditions, velocity surfaces could be calculated for a given rock sample. Data such as those shown in '''Fig. 13.35''' were fitted to a form based on that of '''Eq. 13.58''':<br/><br/>[[File:Vol1 page 0617 eq 001.png|RTENOTITLE]]....................(13.63)<br/><br/>where ''σ''<sub>''e''</sub> is the effective stress. Fits are usually very good even for consolidated rocks with regression factors of around 0.98.<br/><br/><gallery widths="300px" heights="200px">
<br/>
A simple estimate of the horizontal stress, ''σ''<sub>''h''</sub>, can be made from the axial stress, ''σ''<sub>''v''</sub>, by
<br>
<br>
[[File:Vol1 page 0616 eq 001.png]]....................(13.62)
<br>
<br>
where ''ν'' is Poisson’s ratio. Calculated stresses typical for sands (''ν'' = 0.1) and more clay-rich rocks (''ν'' = 0.25) are also shown in '''Fig. 13.34'''. This basic relation ('''Eq. 13.62''') is an oversimplification of actual conditions, but it does provide a useful conceptual model, and lateral stresses indeed are found to be lower in sandstones than in shaly sections in most places.  
<br>
<br>
From a matrix of velocities measured over axial and lateral stress conditions, velocity surfaces could be calculated for a given rock sample. Data such as those shown in '''Fig. 13.35''' were fitted to a form based on that of '''Eq. 13.58''':
<br>
<br>
[[File:Vol1 page 0617 eq 001.png]]....................(13.63)
<br>
<br>
where ''σ''<sub>''e''</sub> is the effective stress. Fits are usually very good even for consolidated rocks with regression factors of around 0.98.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 623 Image 0001.png|'''Fig. 13.35 – Chocolate Sandstone (porosity = 0.225) compressional and shear velocities, both under hydrostatic and more realistic lithostatic stress conditions. In both cases, the velocities are fit well by the form [[File:Vol1 page 0623 inline 001.png]].'''
File:vol1 Page 623 Image 0001.png|'''Fig. 13.35 – Chocolate Sandstone (porosity = 0.225) compressional and shear velocities, both under hydrostatic and more realistic lithostatic stress conditions. In both cases, the velocities are fit well by the form [[File:Vol1 page 0623 inline 001.png]].'''
</gallery>
</gallery><br/>Velocities can vary substantially over the stress field shown in '''Fig. 13.34''', not only among samples but also between compressional and shear waves. '''Fig. 13.36''' shows the ''V''<sub>''p''</sub> and ''V''<sub>''s''</sub> surfaces for Woodbine sandstone. Figures such as 13.36 demonstrate that the ''V''<sub>''p''</sub>, ''V''<sub>''s''</sub>, and ''V''<sub>''p''</sub>/''V''<sub>''s''</sub> ratio will all be strongly dependent on the exact stress tensor at depth. Laboratory measurements under hydrostatic conditions are at best a first-order approximation.<br/><br/><gallery widths="300px" heights="200px">
<br/>
Velocities can vary substantially over the stress field shown in '''Fig. 13.34''', not only among samples but also between compressional and shear waves. '''Fig. 13.36''' shows the ''V''<sub>''p''</sub> and ''V''<sub>''s''</sub> surfaces for Woodbine sandstone. Figures such as 13.36 demonstrate that the ''V''<sub>''p''</sub>, ''V''<sub>''s''</sub>, and ''V''<sub>''p''</sub>/''V''<sub>''s''</sub> ratio will all be strongly dependent on the exact stress tensor at depth. Laboratory measurements under hydrostatic conditions are at best a first-order approximation.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 624 Image 0001.png|'''Fig. 13.36 – Iso-velocity contours for the Woodbine sandstone related to the anisotropic stress condition described in Fig. 13.34. Propagation is vertical for both ''V<sub>p</sub>'' and ''V<sub>s</sub>''. Both absolute velocities and velocity ratios will be affected by anisotropic stresses.'''
File:vol1 Page 624 Image 0001.png|'''Fig. 13.36 – Iso-velocity contours for the Woodbine sandstone related to the anisotropic stress condition described in Fig. 13.34. Propagation is vertical for both ''V<sub>p</sub>'' and ''V<sub>s</sub>''. Both absolute velocities and velocity ratios will be affected by anisotropic stresses.'''
</gallery>
</gallery>
<br/>


=== Temperature ===
=== Temperature ===


For consolidated rocks (Classes I, II, and V, '''Fig. 13.26b'''), the elastic mineral frame properties are usually only weakly dependent on temperature. This is true for most reservoir operations with the exception of some thermal recovery procedures. In the case of poorly consolidated sands containing heavy oils, velocities show that a strong temperature dependence is observed ('''Fig. 13.37'''). Several factors can combine to produce such large effects. First, in heavy-oil sands, the material may actually be a suspension of minerals in tar ('''Fig. 13.26b''', Class IV). The framework is basically a fluid, not solid. In addition, during many measurements, pore pressure cannot reach equilibrium. The large coefficient of thermal expansion of oils combined with the high viscosity often results in high pore pressures within the rock samples. Thus, effective pressures can drop substantially ('''Eq. 13.61'''). Care needs to be taken during such measurements that equilibrium pressures are reached.  
For consolidated rocks (Classes I, II, and V, '''Fig. 13.26b'''), the elastic mineral frame properties are usually only weakly dependent on temperature. This is true for most reservoir operations with the exception of some thermal recovery procedures. In the case of poorly consolidated sands containing heavy oils, velocities show that a strong temperature dependence is observed ('''Fig. 13.37'''). Several factors can combine to produce such large effects. First, in heavy-oil sands, the material may actually be a suspension of minerals in tar ('''Fig. 13.26b''', Class IV). The framework is basically a fluid, not solid. In addition, during many measurements, pore pressure cannot reach equilibrium. The large coefficient of thermal expansion of oils combined with the high viscosity often results in high pore pressures within the rock samples. Thus, effective pressures can drop substantially ('''Eq. 13.61'''). Care needs to be taken during such measurements that equilibrium pressures are reached.<br/><br/><gallery widths="300px" heights="200px">
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 625 Image 0001.png|'''Fig. 13.37 – Compressional velocity as a function of temperature for a sandstone sample testing the effects of thermal flooding. The large drop in velocity is a combined effect of grain separation, pore pressure, and fluid modulus changes.'''
File:vol1 Page 625 Image 0001.png|'''Fig. 13.37 – Compressional velocity as a function of temperature for a sandstone sample testing the effects of thermal flooding. The large drop in velocity is a combined effect of grain separation, pore pressure, and fluid modulus changes.'''
</gallery>
</gallery><br/>The primary influence of temperature is through the pore fluid properties (refer to the Fluid Properties section). '''Fig. 13.38''' demonstrates this general temperature dependence. For dry (gas-saturated) rock, or rock saturated with brine, almost no change in velocity is observed, even for changes of almost 150°C. At elevated pore pressures, both gas and brine have only weak temperature dependence. Mineral properties are almost unchanged. However, when the rocks are even partially saturated with oil, dramatic temperature dependence is observed. Such changes can be understood by first calculating fluid properties with temperature, then using a Gassmann substitution to calculate the bulk rock properties. Note that for heavy viscous oils, velocity dispersion (velocity dependence on frequency) can be significant, and measured ultrasonic data may not agree with seismic results.<br/><br/><gallery widths="300px" heights="200px">
<br/>
The primary influence of temperature is through the pore fluid properties (refer to the Fluid Properties section). '''Fig. 13.38''' demonstrates this general temperature dependence. For dry (gas-saturated) rock, or rock saturated with brine, almost no change in velocity is observed, even for changes of almost 150°C. At elevated pore pressures, both gas and brine have only weak temperature dependence. Mineral properties are almost unchanged. However, when the rocks are even partially saturated with oil, dramatic temperature dependence is observed. Such changes can be understood by first calculating fluid properties with temperature, then using a Gassmann substitution to calculate the bulk rock properties. Note that for heavy viscous oils, velocity dispersion (velocity dependence on frequency) can be significant, and measured ultrasonic data may not agree with seismic results.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 626 Image 0001.png|'''Fig. 13.38 – Compressional velocity as a function of temperature for Venezuelan (a) and Kern River California (b) oil sand samples. When either dry (gas-saturated) or brine-saturated, there is little temperature dependence. With increasing oil saturation, temperature dependence becomes strong. The effects of fluid phase changes can be seen by the drop in velocity in (b) with increasing gas saturation (from Tosaya<ref name="r69" />).'''
File:vol1 Page 626 Image 0001.png|'''Fig. 13.38 – Compressional velocity as a function of temperature for Venezuelan (a) and Kern River California (b) oil sand samples. When either dry (gas-saturated) or brine-saturated, there is little temperature dependence. With increasing oil saturation, temperature dependence becomes strong. The effects of fluid phase changes can be seen by the drop in velocity in (b) with increasing gas saturation (from Tosaya<ref name="r69" />).'''
</gallery>
</gallery><br/>Fluid phase changes may also occur as temperature is raised. These phase changes can have a strong influence, particularly for high-porosity rocks at low pressures. The effect can be seen in '''Fig. 13.38b''', where exsolving a gas phase could reduce the velocity from nearly 3.2 km/s to around 2.1 km/s. In several thermal recovery monitoring projects, the strongest seismic expression was a result of gas coming out of solution to form a separate phase, rather than the thermal effects themselves.
<br/>
Fluid phase changes may also occur as temperature is raised. These phase changes can have a strong influence, particularly for high-porosity rocks at low pressures. The effect can be seen in '''Fig. 13.38b''', where exsolving a gas phase could reduce the velocity from nearly 3.2 km/s to around 2.1 km/s. In several thermal recovery monitoring projects, the strongest seismic expression was a result of gas coming out of solution to form a separate phase, rather than the thermal effects themselves.  


=== Gassmann Fluid Substitution ===
=== Gassmann Fluid Substitution ===


To extract fluid types or saturations from seismic, crosswell, or borehole sonic data, we need a procedure to model fluid effects on rock velocity and density. Numerous techniques have been developed. Gassmann’s equations are by far the most widely used relations to calculate seismic velocity changes because of different fluid saturations in reservoirs. Gassmann’s formulation is straightforward, and the simple input parameters typically can be directly measured from logs or assumed based on rock type. This is a prime reason for its importance in geophysical techniques such as time-lapse reservoir monitoring and direct hydrocarbon indicators (DHI) such as amplitude "bright spots," and amplitude vs. offset (AVO). Because of the dominance of this technique, we will describe it at length.  
To extract fluid types or saturations from seismic, crosswell, or borehole sonic data, we need a procedure to model fluid effects on rock velocity and density. Numerous techniques have been developed. Gassmann’s equations are by far the most widely used relations to calculate seismic velocity changes because of different fluid saturations in reservoirs. Gassmann’s formulation is straightforward, and the simple input parameters typically can be directly measured from logs or assumed based on rock type. This is a prime reason for its importance in geophysical techniques such as time-lapse reservoir monitoring and direct hydrocarbon indicators (DHI) such as amplitude "bright spots," and amplitude vs. offset (AVO). Because of the dominance of this technique, we will describe it at length.<br/><br/>Despite the popularity of Gassmann’s equations and their incorporation within most software packages for seismic reservoir interpretation, important aspects of these equations are usually not observed. Many of the basic assumptions are invalid for common reservoir rocks and fluids. Many efforts have been made to understand the operation and application of Gassmann’s equations (Han,<ref name="r70">_</ref> Mavko and Mukerji,<ref name="r71">_</ref> Mavko ''et al''.,<ref name="r8">_</ref> Sengupta and Mavko,<ref name="r72">_</ref> and Nolen-Hoeksema<ref name="r73">_</ref>). Most of these works have attempted to isolate individual parameter effects. We will extend this analysis to incorporate mechanical bounds for porous media (see previous) and the magnitude of the fluid effect.<br/><br/>Compressional (P-wave) and shear (S-wave) velocities along with densities directly control the seismic response of reservoirs at any single location. '''Fig. 13.39a''' shows measured dry and water saturated P- and S-wave velocities of sandstones as a function of differential pressure. P-wave velocity increases, while S-wave velocity decreases slightly with water saturation. However, both P- and S-wave velocities are generally not the best indicators for any fluid saturation effect. This is a function of coupling between P- and S-wave through the shear modulus and bulk density. In contrast, if we plot bulk and shear modulus as functions of pressure ('''Fig. 13.39b'''), the water-saturation effect shows the following:
<br>
 
<br>
#Bulk modulus increases about 50%.
Despite the popularity of Gassmann’s equations and their incorporation within most software packages for seismic reservoir interpretation, important aspects of these equations are usually not observed. Many of the basic assumptions are invalid for common reservoir rocks and fluids. Many efforts have been made to understand the operation and application of Gassmann’s equations (Han,<ref name="r70" /> Mavko and Mukerji,<ref name="r71" /> Mavko ''et al''.,<ref name="r8" /> Sengupta and Mavko,<ref name="r72" /> and Nolen-Hoeksema<ref name="r73" />). Most of these works have attempted to isolate individual parameter effects. We will extend this analysis to incorporate mechanical bounds for porous media (see previous) and the magnitude of the fluid effect.  
#Shear modulus remains almost constant.
<br>
 
<br>
Compressional (P-wave) and shear (S-wave) velocities along with densities directly control the seismic response of reservoirs at any single location. '''Fig. 13.39a''' shows measured dry and water saturated P- and S-wave velocities of sandstones as a function of differential pressure. P-wave velocity increases, while S-wave velocity decreases slightly with water saturation. However, both P- and S-wave velocities are generally not the best indicators for any fluid saturation effect. This is a function of coupling between P- and S-wave through the shear modulus and bulk density. In contrast, if we plot bulk and shear modulus as functions of pressure ('''Fig. 13.39b'''), the water-saturation effect shows the following:  
<br>
#Bulk modulus increases about 50%.  
#Shear modulus remains almost constant.  
<br>


Bulk modulus is more strongly sensitive to water saturation. The bulk volume deformation produced by a passing seismic wave results in a pore volume change, and causes a pressure increase of pore fluid (water). This has the effect of stiffening the rock and increasing the bulk modulus. Shear deformation usually does not produce pore volume change, and differing pore fluids often do not affect shear modulus.  
 
<br/>
Bulk modulus is more strongly sensitive to water saturation. The bulk volume deformation produced by a passing seismic wave results in a pore volume change, and causes a pressure increase of pore fluid (water). This has the effect of stiffening the rock and increasing the bulk modulus. Shear deformation usually does not produce pore volume change, and differing pore fluids often do not affect shear modulus.<br/><br/><gallery widths="300px" heights="200px">
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 627 Image 0001.png|'''Fig. 13.39 – (a) Compressional and shear velocity as a function of pressure for a dry and brine-saturated sandstone; (b) the same sandstone, but plotted in terms of bulk and shear moduli. The change in bulk modulus upon saturation is more dramatic than velocities.'''
File:vol1 Page 627 Image 0001.png|'''Fig. 13.39 – (a) Compressional and shear velocity as a function of pressure for a dry and brine-saturated sandstone; (b) the same sandstone, but plotted in terms of bulk and shear moduli. The change in bulk modulus upon saturation is more dramatic than velocities.'''
</gallery>
</gallery><br/>Gassmann’s equations provide a simple model to estimate fluid saturation effect on bulk modulus. '''Eqs. 13.64a''' through '''13.65''' are convenient forms for Gassmann’s relations that show the physical meaning:<br/><br/>[[File:Vol1 page 0621 eq 001.png|RTENOTITLE]]....................(13.64a)<br/><br/>[[File:Vol1 page 0622 eq 001.png|RTENOTITLE]]....................(13.64b)<br/><br/>and<br/><br/>[[File:Vol1 page 0622 eq 002.png|RTENOTITLE]]....................(13.65)<br/><br/>where ''K''<sub>0</sub>, ''K''<sub>''f''</sub>, ''K''<sub>''d''</sub>, and ''K''<sub>''s''</sub>, are the bulk moduli of the mineral, fluid, dry rock, and saturated rock frame, respectively; ''Φ'' is porosity; and ''μ''<sub>''s''</sub> and ''μ''<sub>''d''</sub> are the saturated and dry rock shear moduli. Δ''K''<sub>''d''</sub> is an increment of bulk modulus caused by fluid saturation. These equations indicate that fluid in pores will affect bulk modulus but not shear modulus, consistent with the earlier discussion. As pointed out by Berryman,<ref name="r74">_</ref> a shear modulus independent of fluid saturation is a direct result of the assumptions used to derive Gassmann’s equation.<br/><br/>Numerous assumptions are involved in the derivation of Gassmann’s equation:
<br/>
 
Gassmann’s equations provide a simple model to estimate fluid saturation effect on bulk modulus. '''Eqs. 13.64a''' through '''13.65''' are convenient forms for Gassmann’s relations that show the physical meaning:
#The porous material is isotropic, elastic, monomineralic, and homogeneous.
<br>
#The pore space is well connected and in pressure equilibrium (zero frequency limit).
<br>
#The medium is a closed system with no pore fluid movement across boundaries.
[[File:Vol1 page 0621 eq 001.png]]....................(13.64a)
#There is no chemical interaction between fluids and rock frame (shear modulus remains constant).
<br>
 
<br>
 
[[File:Vol1 page 0622 eq 001.png]]....................(13.64b)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0622 eq 002.png]]....................(13.65)
<br>
<br>
where ''K''<sub>0</sub>, ''K''<sub>''f''</sub>, ''K''<sub>''d''</sub>, and ''K''<sub>''s''</sub>, are the bulk moduli of the mineral, fluid, dry rock, and saturated rock frame, respectively; ''Φ'' is porosity; and ''μ''<sub>''s''</sub> and ''μ''<sub>''d''</sub> are the saturated and dry rock shear moduli. Δ''K''<sub>''d''</sub> is an increment of bulk modulus caused by fluid saturation. These equations indicate that fluid in pores will affect bulk modulus but not shear modulus, consistent with the earlier discussion. As pointed out by Berryman,<ref name="r74" /> a shear modulus independent of fluid saturation is a direct result of the assumptions used to derive Gassmann’s equation.  
<br>
<br>
Numerous assumptions are involved in the derivation of Gassmann’s equation:  
<br>
#The porous material is isotropic, elastic, monomineralic, and homogeneous.  
#The pore space is well connected and in pressure equilibrium (zero frequency limit).  
#The medium is a closed system with no pore fluid movement across boundaries.  
#There is no chemical interaction between fluids and rock frame (shear modulus remains constant).  
<br>


Many of these assumptions may not be valid for hydrocarbon reservoirs, and they depend on rock and fluid properties and in-situ conditions. For example, most rocks are anisotropic to some degree. The work of Brown and Korringa<ref name="r75" /> provides an explicit form for an anisotropic fluid substitution. In seismic applications, it is normally assumed that Gassmann’s equation works best for seismic data at frequencies less than 100 Hz (Mavko ''et al''.<ref name="r8" />). Recently published laboratory data (Batzle ''et al''.<ref name="r76" />) show that acoustic waves may be dispersive in rocks within the typical seismic band, invalidating assumption 2. In such cases, seismic frequencies may still be too high for application of Gassmann’s equation. Pore pressures may not have enough time to reach equilibrium. The rock remains unrelaxed or only partially relaxed.  
Many of these assumptions may not be valid for hydrocarbon reservoirs, and they depend on rock and fluid properties and in-situ conditions. For example, most rocks are anisotropic to some degree. The work of Brown and Korringa<ref name="r75">_</ref> provides an explicit form for an anisotropic fluid substitution. In seismic applications, it is normally assumed that Gassmann’s equation works best for seismic data at frequencies less than 100 Hz (Mavko ''et al''.<ref name="r8">_</ref>). Recently published laboratory data (Batzle ''et al''.<ref name="r76">_</ref>) show that acoustic waves may be dispersive in rocks within the typical seismic band, invalidating assumption 2. In such cases, seismic frequencies may still be too high for application of Gassmann’s equation. Pore pressures may not have enough time to reach equilibrium. The rock remains unrelaxed or only partially relaxed.<br/><br/>The primary measure of the sensitivity of rock to fluids is its normalized modulus ''K''<sub>''n''</sub>: the ratio of dry bulk modulus to that of the mineral.<br/><br/>[[File:Vol1 page 0623 eq 001.png|RTENOTITLE]]....................(13.66)<br/><br/>This function can be complicated and depends on rock texture (porosity, clay content, pore geometry, grain size, grain contact, cementation, mineral composition, and so on) and reservoir conditions (pressure and temperature). This ''K''<sub>''n''</sub> can be determined empirically or theoretically. For relatively clean sandstone at high differential pressure (>20 MPa), the complex dependence of ''K''<sub>''n''</sub> (''x'', ''y'', ''z'', …) can be simplified as a function of porosity.<br/><br/>[[File:Vol1 page 0624 eq 001.png|RTENOTITLE]]....................(13.67)<br/><br/>From '''Eq. 13.66''', bulk modulus increment is then equal to<br/><br/>[[File:Vol1 page 0624 eq 002.png|RTENOTITLE]]....................(13.68)<br/><br/>Here [1-''K''<sub>''n''</sub> (''Φ'')] is also the Biot parameter ''α''<sub>''b''</sub> (Biot<ref name="r77">_</ref>). Furthermore, because usually ''K''<sub>0</sub> >> ''K''<sub>''f''</sub>, it is reasonable to assume<br/><br/>[[File:Vol1 page 0624 eq 003.png|RTENOTITLE]]....................(13.69)<br/><br/>for sedimentary rocks with high porosity (>15%). Therefore,<br/><br/>[[File:Vol1 page 0624 eq 004.png|RTENOTITLE]]....................(13.70)<br/><br/>where ''G''(''Φ'') is the saturation gain function defined as<br/><br/>[[File:Vol1 page 0624 eq 005.png|RTENOTITLE]]....................(13.71)<br/><br/>Thus, fluid saturation effects on the bulk modulus are proportional to the gain function ''G''(''Φ'') and the fluid modulus ''K''<sub>''f''</sub>. The ''G''(''Φ'') in turn depends directly on dry rock properties: the normalized modulus and porosity. In general, ''G''(''Φ'') is independent of fluid properties (ignoring interactions between rock frame and pore fluid). We must know both gain function of dry rock frame and pore fluid modulus to evaluate the fluid saturation effect on seismic properties. Note that the normalized modulus must be a smooth function of porosity or ''G''(''Φ'') can be unstable, particularly at small porosities.<br/><br/>At high differential pressure (>20 MPa), the ''K''<sub>''s''</sub> of water-saturated sands calculated using simplified form is 3% overestimated for porous rock (porosity > 15%). Those errors will decrease significantly with low fluid modulus (gas and light oil saturation). For low-porosity sands with high clay content, the simplified Gassmann’s equation overestimates water saturation effects substantially.<br/><br/>In '''Eq. 13.64b''', there are five parameters, and usually the only applied constraint is that the parameters are physically meaningful (>0). Incompatible or mismatched data might generate wrong or even unphysical results such as a negative modulus. In reality, only ''K''<sub>0</sub> and ''K''<sub>''f''</sub> are completely independent. ''K''<sub>''s''</sub>, ''K''<sub>''d''</sub>, and porosity ''Φ'' are actually closely correlated. Bounds on ''K''<sub>''d''</sub> as a function of porosity, for example, constrain the bounds of ''K''<sub>''s''</sub>.<br/><br/>Assuming porous media is a Voigt material, which is a high bound for ''K''<sub>''d''</sub> ('''Fig. 13.40'''),<br/><br/>[[File:Vol1 page 0625 eq 001.png|RTENOTITLE]]....................(13.72)<br/><br/>Putting this equation (13.72) into Gassmann’s Equation (13.64) gives<br/><br/>[[File:Vol1 page 0625 eq 002.png|RTENOTITLE]]....................(13.73)<br/><br/>and<br/><br/>[[File:Vol1 page 0626 eq 001.png|RTENOTITLE]]....................(13.74)<br/><br/>Because this Voigt bound is the stiffest upper limit, the fluid saturation effect on bulk modulus here (Δ''K''<sub>''d''min</sub>) will be a minimum (see '''Fig. 13.40''').<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
The primary measure of the sensitivity of rock to fluids is its normalized modulus ''K''<sub>''n''</sub>: the ratio of dry bulk modulus to that of the mineral.
<br>
<br>
[[File:Vol1 page 0623 eq 001.png]]....................(13.66)
<br>
<br>
This function can be complicated and depends on rock texture (porosity, clay content, pore geometry, grain size, grain contact, cementation, mineral composition, and so on) and reservoir conditions (pressure and temperature). This ''K''<sub>''n''</sub> can be determined empirically or theoretically. For relatively clean sandstone at high differential pressure (>20 MPa), the complex dependence of ''K''<sub>''n''</sub> (''x'', ''y'', ''z'', …) can be simplified as a function of porosity.
<br>
<br>
[[File:Vol1 page 0624 eq 001.png]]....................(13.67)
<br>
<br>
From '''Eq. 13.66''', bulk modulus increment is then equal to
<br>
<br>
[[File:Vol1 page 0624 eq 002.png]]....................(13.68)
<br>
<br>
Here [1-''K''<sub>''n''</sub> (''Φ'')] is also the Biot parameter ''α''<sub>''b''</sub> (Biot<ref name="r77" />). Furthermore, because usually ''K''<sub>0</sub> >> ''K''<sub>''f''</sub>, it is reasonable to assume
<br>
<br>
[[File:Vol1 page 0624 eq 003.png]]....................(13.69)
<br>
<br>
for sedimentary rocks with high porosity (>15%). Therefore,
<br>
<br>
[[File:Vol1 page 0624 eq 004.png]]....................(13.70)
<br>
<br>
where ''G''(''Φ'') is the saturation gain function defined as
<br>
<br>
[[File:Vol1 page 0624 eq 005.png]]....................(13.71)
<br>
<br>
Thus, fluid saturation effects on the bulk modulus are proportional to the gain function ''G''(''Φ'') and the fluid modulus ''K''<sub>''f''</sub>. The ''G''(''Φ'') in turn depends directly on dry rock properties: the normalized modulus and porosity. In general, ''G''(''Φ'') is independent of fluid properties (ignoring interactions between rock frame and pore fluid). We must know both gain function of dry rock frame and pore fluid modulus to evaluate the fluid saturation effect on seismic properties. Note that the normalized modulus must be a smooth function of porosity or ''G''(''Φ'') can be unstable, particularly at small porosities.  
<br>
<br>
At high differential pressure (>20 MPa), the ''K''<sub>''s''</sub> of water-saturated sands calculated using simplified form is 3% overestimated for porous rock (porosity > 15%). Those errors will decrease significantly with low fluid modulus (gas and light oil saturation). For low-porosity sands with high clay content, the simplified Gassmann’s equation overestimates water saturation effects substantially.  
<br>
<br>
In '''Eq. 13.64b''', there are five parameters, and usually the only applied constraint is that the parameters are physically meaningful (>0). Incompatible or mismatched data might generate wrong or even unphysical results such as a negative modulus. In reality, only ''K''<sub>0</sub> and ''K''<sub>''f''</sub> are completely independent. ''K''<sub>''s''</sub>, ''K''<sub>''d''</sub>, and porosity ''Φ'' are actually closely correlated. Bounds on ''K''<sub>''d''</sub> as a function of porosity, for example, constrain the bounds of ''K''<sub>''s''</sub>.  
<br>
<br>
Assuming porous media is a Voigt material, which is a high bound for ''K''<sub>''d''</sub> ('''Fig. 13.40'''),  
<br>
<br>
[[File:Vol1 page 0625 eq 001.png]]....................(13.72)
<br>
<br>
Putting this equation (13.72) into Gassmann’s Equation (13.64) gives
<br>
<br>
[[File:Vol1 page 0625 eq 002.png]]....................(13.73)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0626 eq 001.png]]....................(13.74)
<br>
<br>
Because this Voigt bound is the stiffest upper limit, the fluid saturation effect on bulk modulus here (Δ''K''<sub>''d''min</sub>) will be a minimum (see '''Fig. 13.40''').  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 628 Image 0001.png|'''Fig. 13.40 – Elastic bounds on bulk modulus for sandstone. Stippled regions represent the extremes in changes in modulus upon water saturation.'''
File:vol1 Page 628 Image 0001.png|'''Fig. 13.40 – Elastic bounds on bulk modulus for sandstone. Stippled regions represent the extremes in changes in modulus upon water saturation.'''
</gallery>
</gallery><br/>As we have seen, the low modulus bound for porous media is the Reuss bound.<br/><br/>[[File:Vol1 page 0626 eq 002.png|RTENOTITLE]]....................(13.75)<br/><br/>[[File:Vol1 page 0626 eq 003.png|RTENOTITLE]]....................(13.76)<br/><br/>For completely empty (dry) rocks, the fluid modulus ''K''<sub>''f''</sub> is equal to zero, and both the Reuss bound and the normalized modulus (''K''<sub>''nR''</sub>) for a dry rock in this limit equals zero (for nonzero porosity).<br/><br/>[[File:Vol1 page 0626 eq 004.png|RTENOTITLE]]....................(13.77)<br/><br/>Substituting '''Eq. 13.77''' into Gassmann’s Equation (13.64), we find the fluid saturation effect on bulk modulus when the frame is at this lower bound.<br/><br/>[[File:Vol1 page 0627 eq 001.png|RTENOTITLE]]....................(13.78)<br/><br/>For this case, the modulus increment Δ''K'' from dry to fluid saturation is equal to the Reuss bound.<br/><br/>[[File:Vol1 page 0628 eq 001.png|RTENOTITLE]]....................(13.79)<br/><br/>Again, Gassmann’s equation is consistent with the dry and fluid-saturated Reuss bounds. Physically, for rocks with the weakest frame, fluids have a maximum effect.<br/><br/>Critical porosity, ''Φ''<sub>''c''</sub>, can be used to give tighter constraints for dry- and fluid-saturated bulk modulus for sands. A new triangle is formed which provides a linear formulation and a graphic procedure for Gassmann’s calculation: the fluid saturation effect on bulk modulus proportional to normalized porosity and the maximum fluid saturation effect on bulk modulus (Reuss bound) at the critical porosity ('''Fig. 13.40''').<br/><br/>[[File:Vol1 page 0628 eq 002.png|RTENOTITLE]]....................(13.80)<br/><br/>This is consistent with the results of Mavko and Mukerji.<ref name="r71">_</ref><br/><br/>For typical sandstones, the critical porosity ''Φ''<sub>''c''</sub> is around 40%. Thus, we also can generate a simplified numerical formula of the normalized modulus ''K''<sub>''n''</sub> for modified Voigt model:<br/><br/>[[File:Vol1 page 0628 eq 003.png|RTENOTITLE]]....................(13.81)<br/><br/>Using this in Gassmann’s Equation (13.64) yields fluid saturation effect<br/><br/>[[File:Vol1 page 0628 eq 004.png|RTENOTITLE]]....................(13.82)<br/><br/>Extending our empirical approach to first order, both P- and S-wave velocity can correlate linearly with porosity at high differential pressure. From '''Table 13.7''', for dry clean sands,<br/><br/>[[File:Vol1 page 0629 eq 001.png|RTENOTITLE]]....................(13.83)<br/><br/>[[File:Vol1 page 0629 eq 002.png|RTENOTITLE]]....................(13.84)<br/><br/>where we assume the density of these sands is equal to<br/><br/>[[File:Vol1 page 0629 eq 003.png|RTENOTITLE]]....................(13.85)<br/><br/>Since the modulus is the product of the density and square of velocity, we get an equation that is cubic in terms of porosity. The bulk modulus can be derived as<br/><br/>[[File:Vol1 page 0629 eq 004.png|RTENOTITLE]]....................(13.86)<br/><br/>where ''A'' = 3.206, ''B'' = 3.349, and ''C'' = 1.143. '''Eq. 13.86''' can be further simplified if porosity ''Φ'' is not too large (<30%):<br/><br/>[[File:Vol1 page 0629 eq 005.png|RTENOTITLE]]....................(13.87)<br/><br/>where ''D'' for clean sandstone is equal to 1.52. This includes an empirical expression of the normalized modulus as a direct dependence on porosity and "''D''" parameter. '''Table 13.10''' and '''Fig. 13.41''' show empirical relations generated from dry velocity data of relatively clean rocks. The parameter ''D'' is related to rock texture and should be calibrated for local reservoir conditions. In general, it has a narrow range from 1.45 to slightly more than 2.0, primarily depending on rock consolidation.<br/><br/><gallery widths="300px" heights="200px">
<br/>
As we have seen, the low modulus bound for porous media is the Reuss bound.
<br>
<br>
[[File:Vol1 page 0626 eq 002.png]]....................(13.75)
<br>
<br>
[[File:Vol1 page 0626 eq 003.png]]....................(13.76)
<br>
<br>
For completely empty (dry) rocks, the fluid modulus ''K''<sub>''f''</sub> is equal to zero, and both the Reuss bound and the normalized modulus (''K''<sub>''nR''</sub>) for a dry rock in this limit equals zero (for nonzero porosity).
<br>
<br>
[[File:Vol1 page 0626 eq 004.png]]....................(13.77)
<br>
<br>
Substituting '''Eq. 13.77''' into Gassmann’s Equation (13.64), we find the fluid saturation effect on bulk modulus when the frame is at this lower bound.
<br>
<br>
[[File:Vol1 page 0627 eq 001.png]]....................(13.78)
<br>
<br>
For this case, the modulus increment Δ''K'' from dry to fluid saturation is equal to the Reuss bound.
<br>
<br>
[[File:Vol1 page 0628 eq 001.png]]....................(13.79)
<br>
<br>
Again, Gassmann’s equation is consistent with the dry and fluid-saturated Reuss bounds. Physically, for rocks with the weakest frame, fluids have a maximum effect.  
<br>
<br>
Critical porosity, ''Φ''<sub>''c''</sub>, can be used to give tighter constraints for dry- and fluid-saturated bulk modulus for sands. A new triangle is formed which provides a linear formulation and a graphic procedure for Gassmann’s calculation: the fluid saturation effect on bulk modulus proportional to normalized porosity and the maximum fluid saturation effect on bulk modulus (Reuss bound) at the critical porosity ('''Fig. 13.40''').
<br>
<br>
[[File:Vol1 page 0628 eq 002.png]]....................(13.80)
<br>
<br>
This is consistent with the results of Mavko and Mukerji.<ref name="r71" />  
<br>
<br>
For typical sandstones, the critical porosity ''Φ''<sub>''c''</sub> is around 40%. Thus, we also can generate a simplified numerical formula of the normalized modulus ''K''<sub>''n''</sub> for modified Voigt model:
<br>
<br>
[[File:Vol1 page 0628 eq 003.png]]....................(13.81)
<br>
<br>
Using this in Gassmann’s Equation (13.64) yields fluid saturation effect
<br>
<br>
[[File:Vol1 page 0628 eq 004.png]]....................(13.82)
<br>
<br>
Extending our empirical approach to first order, both P- and S-wave velocity can correlate linearly with porosity at high differential pressure. From '''Table 13.7''', for dry clean sands,
<br>
<br>
[[File:Vol1 page 0629 eq 001.png]]....................(13.83)
<br>
<br>
[[File:Vol1 page 0629 eq 002.png]]....................(13.84)
<br>
<br>
where we assume the density of these sands is equal to
<br>
<br>
[[File:Vol1 page 0629 eq 003.png]]....................(13.85)
<br>
<br>
Since the modulus is the product of the density and square of velocity, we get an equation that is cubic in terms of porosity. The bulk modulus can be derived as
<br>
<br>
[[File:Vol1 page 0629 eq 004.png]]....................(13.86)
<br>
<br>
where ''A'' = 3.206, ''B'' = 3.349, and ''C'' = 1.143. '''Eq. 13.86''' can be further simplified if porosity ''Φ'' is not too large (<30%):
<br>
<br>
[[File:Vol1 page 0629 eq 005.png]]....................(13.87)
<br>
<br>
where ''D'' for clean sandstone is equal to 1.52. This includes an empirical expression of the normalized modulus as a direct dependence on porosity and "''D''" parameter. '''Table 13.10''' and '''Fig. 13.41''' show empirical relations generated from dry velocity data of relatively clean rocks. The parameter ''D'' is related to rock texture and should be calibrated for local reservoir conditions. In general, it has a narrow range from 1.45 to slightly more than 2.0, primarily depending on rock consolidation.  
<br>
<br>
<gallery widths=300px heights=200px>
File:Vol1 Page 631 Image 0002.png|'''Table 13.10'''
File:Vol1 Page 631 Image 0002.png|'''Table 13.10'''


File:vol1 Page 630 Image 0001.png|'''Fig. 13.41 – Elastic bounds (Voigt and Reuss) and approximations to the normalized dry modulus (''K<sub>n</sub>'' = ''K<sub>d</sub>''/''K''<sub>0</sub>) for sandstone using the ''D'' factor.'''
File:vol1 Page 630 Image 0001.png|'''Fig. 13.41 – Elastic bounds (Voigt and Reuss) and approximations to the normalized dry modulus (''K<sub>n</sub>'' = ''K<sub>d</sub>''/''K''<sub>0</sub>) for sandstone using the ''D'' factor.'''
</gallery>
</gallery><br/>By inserting this ''D'' function into '''Eq. 13.71''', we find<br/><br/>[[File:Vol1 page 0629 eq 006.png|RTENOTITLE]]....................(13.88)<br/>
<br/>
By inserting this ''D'' function into '''Eq. 13.71''', we find
<br>
<br>
[[File:Vol1 page 0629 eq 006.png]]....................(13.88)
<br>
<br>


=== Solid Mineral Bulk Modulus ===
=== Solid Mineral Bulk Modulus ===


The mineral modulus (solid grain bulk modulus) ''K''<sub>0</sub> is an independent parameter, and the rock texture controls ''K''<sub>''d''</sub>. However, as mentioned previously, the normalized modulus ''K''<sub>''n''</sub> controls the fluid saturation effect rather than ''K''<sub>''d''</sub> or ''K''<sub>''s''</sub> individually. The mineral modulus ''K''<sub>0</sub> is equally as important as ''K''<sub>''d''</sub>. However, in most applications of the Gassmann’s equation, only ''K''<sub>''d''</sub> is measured. Properties of the mineral modulus ''K''<sub>0</sub> are often poorly understood and oversimplified. ''K''<sub>0</sub> is the modulus of the solid material that includes grains, cements, and pore fillings ('''Figs. 13.1''' through '''13.8'''). If clays or other minerals are present with complicated distributions and structures, ''K''<sub>0</sub> can vary over a wide range. Unfortunately, few measurements of ''K''<sub>0</sub> have been made on sedimentary rocks (Coyner<ref name="r78" />), and the moduli of clays are a particular problem (Wang ''et al''.<ref name="r50" /> and Katahara<ref name="r49" />; see '''Table 13.6b'''). These data show that at a high pressure, ''K''<sub>0</sub> for sandstone samples range from 33 to 39 MPa. ''K''<sub>0</sub> is not a constant and can increase more than 10% with increasing effective pressure. '''Fig. 13.42''' shows the influence of ''K''<sub>0</sub> on Gassmann’s calculation. This case uses a dry bulk modulus calculated with the mineral modulus of 40 GPa, ''D'' = 2, and a water modulus of 2.8 GPa. The water saturation effect was calculated for three mineral moduli of 65, 40, and 32 GPa. Results show that for the same ''K''<sub>''d''</sub> and ''K''<sub>''f''</sub>, bulk modulus increment Δ''K'' because of fluid saturation increases with increasing mineral modulus ''K''<sub>0</sub>. Errors caused by uncertainty of ''K''<sub>0</sub> decrease with increasing porosity and fluid modulus ''K''<sub>''f''</sub>.  
The mineral modulus (solid grain bulk modulus) ''K''<sub>0</sub> is an independent parameter, and the rock texture controls ''K''<sub>''d''</sub>. However, as mentioned previously, the normalized modulus ''K''<sub>''n''</sub> controls the fluid saturation effect rather than ''K''<sub>''d''</sub> or ''K''<sub>''s''</sub> individually. The mineral modulus ''K''<sub>0</sub> is equally as important as ''K''<sub>''d''</sub>. However, in most applications of the Gassmann’s equation, only ''K''<sub>''d''</sub> is measured. Properties of the mineral modulus ''K''<sub>0</sub> are often poorly understood and oversimplified. ''K''<sub>0</sub> is the modulus of the solid material that includes grains, cements, and pore fillings ('''Figs. 13.1''' through '''13.8'''). If clays or other minerals are present with complicated distributions and structures, ''K''<sub>0</sub> can vary over a wide range. Unfortunately, few measurements of ''K''<sub>0</sub> have been made on sedimentary rocks (Coyner<ref name="r78">_</ref>), and the moduli of clays are a particular problem (Wang ''et al''.<ref name="r50">_</ref> and Katahara<ref name="r49">_</ref>; see '''Table 13.6b'''). These data show that at a high pressure, ''K''<sub>0</sub> for sandstone samples range from 33 to 39 MPa. ''K''<sub>0</sub> is not a constant and can increase more than 10% with increasing effective pressure. '''Fig. 13.42''' shows the influence of ''K''<sub>0</sub> on Gassmann’s calculation. This case uses a dry bulk modulus calculated with the mineral modulus of 40 GPa, ''D'' = 2, and a water modulus of 2.8 GPa. The water saturation effect was calculated for three mineral moduli of 65, 40, and 32 GPa. Results show that for the same ''K''<sub>''d''</sub> and ''K''<sub>''f''</sub>, bulk modulus increment Δ''K'' because of fluid saturation increases with increasing mineral modulus ''K''<sub>0</sub>. Errors caused by uncertainty of ''K''<sub>0</sub> decrease with increasing porosity and fluid modulus ''K''<sub>''f''</sub>.<br/><br/><gallery widths="300px" heights="200px">
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 631 Image 0001.png|'''Fig. 13.42 – Effect of varying mineral modulus (''K''<sub>0</sub>) on the calculated saturated bulk modulus (''K<sub>s</sub>'') for water-saturated sandstone.'''
File:vol1 Page 631 Image 0001.png|'''Fig. 13.42 – Effect of varying mineral modulus (''K''<sub>0</sub>) on the calculated saturated bulk modulus (''K<sub>s</sub>'') for water-saturated sandstone.'''
</gallery>
</gallery><br/>Because of lack of measurements on bulk mineral modulus, we often must use measured velocity/porosity/clay-content relationships for shaly sandstone to estimate the mineral modulus. Assuming zero porosity and grain bulk modulus of 2.65 gm/cc, we can derive mineral bulk and shear modulus from measured P- and S-wave velocity. The results are shown in '''Table 13.11'''.
<br/>
 
Because of lack of measurements on bulk mineral modulus, we often must use measured velocity/porosity/clay-content relationships for shaly sandstone to estimate the mineral modulus. Assuming zero porosity and grain bulk modulus of 2.65 gm/cc, we can derive mineral bulk and shear modulus from measured P- and S-wave velocity. The results are shown in '''Table 13.11'''.  
#For relatively clean sandstone (with few percent clay content), mineral bulk modulus is 39 GPa, which is stable for differential pressures higher than 20 MPa. Mineral shear modulus is around 33 GPa, which is significantly less than 44 GPa for a pure quartz aggregate. Shear modulus is more sensitive to differential pressure and clay content.
<br>
#For shaly sandstone, mineral bulk modulus decreases 1.7 GPa per 10% increment of clay content.
#For relatively clean sandstone (with few percent clay content), mineral bulk modulus is 39 GPa, which is stable for differential pressures higher than 20 MPa. Mineral shear modulus is around 33 GPa, which is significantly less than 44 GPa for a pure quartz aggregate. Shear modulus is more sensitive to differential pressure and clay content.  
 
#For shaly sandstone, mineral bulk modulus decreases 1.7 GPa per 10% increment of clay content.  
 
<br>


Such derived mineral bulk moduli can be used for Gassmann’s calculation if there are no directly measured data or reliable models for calculation.  
Such derived mineral bulk moduli can be used for Gassmann’s calculation if there are no directly measured data or reliable models for calculation.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
<gallery widths=300px heights=200px>
File:Vol1 Page 632 Image 0001.png|'''Table 13.11'''
File:Vol1 Page 632 Image 0001.png|'''Table 13.11'''
</gallery>
</gallery><br/>With a change of fluid saturation from Fluid 1 to Fluid 2, the bulk modulus increment (Δ''K'') is equal to<br/><br/>[[File:Vol1 page 0630 eq 001.png|RTENOTITLE]]....................(13.89)<br/><br/>where ''K''<sub>''f''1</sub> and ''K''<sub>''f''2</sub> are the moduli of Fluids 1 and 2, respectively, and Δ''K''<sub>21</sub> represents the change in the saturation increment that results from substituting Fluid 2 for Fluid 1. '''Eq. 13.89''' uses the fact that the gain function ''G''(''Φ'') of the dry rock frame remains constant as fluid modulus changed (this may not be true for real rocks). The fluid substitution effect on bulk modulus is simply proportional to the difference of fluid bulk modulus.<br/><br/>If we know the gain function for a rock formation, we can estimate the fluid substitution effect without knowing shear modulus.<br/><br/>[[File:Vol1 page 0631 eq 001.png|RTENOTITLE]]....................(13.90)<br/><br/>where ''ρ''<sub>1</sub>, ''ρ''<sub>2</sub>, ''V''<sub>''p''1</sub>, and ''V''<sub>''p''2</sub> are the density and velocity of rock with Fluid 1 and 2 saturation. Both '''Eqs. 13.89''' and '''13.90''' are direct results from simplified Gassmann’s equation ('''Eq. 13.64'''). In '''Fig. 13.43''', we show the typical fluid modulus effect on the saturated bulk modulus ''K''<sub>''s''</sub>. Even at a modest porosity of 15%, changes can be substantial. At in-situ conditions, pore fluids are often multiphase mixtures. Dynamic fluid modulus may also depend on fluid mobility, fluid distribution, rock compressibility, and seismic wavelength.<br/><br/><gallery widths="300px" heights="200px">
<br>
With a change of fluid saturation from Fluid 1 to Fluid 2, the bulk modulus increment (Δ''K'') is equal to
<br>
<br>
[[File:Vol1 page 0630 eq 001.png]]....................(13.89)
<br>
<br>
where ''K''<sub>''f''1</sub> and ''K''<sub>''f''2</sub> are the moduli of Fluids 1 and 2, respectively, and Δ''K''<sub>21</sub> represents the change in the saturation increment that results from substituting Fluid 2 for Fluid 1. '''Eq. 13.89''' uses the fact that the gain function ''G''(''Φ'') of the dry rock frame remains constant as fluid modulus changed (this may not be true for real rocks). The fluid substitution effect on bulk modulus is simply proportional to the difference of fluid bulk modulus.  
<br>
<br>
If we know the gain function for a rock formation, we can estimate the fluid substitution effect without knowing shear modulus.
<br>
<br>
[[File:Vol1 page 0631 eq 001.png]]....................(13.90)
<br>
<br>
where ''ρ''<sub>1</sub>, ''ρ''<sub>2</sub>, ''V''<sub>''p''1</sub>, and ''V''<sub>''p''2</sub> are the density and velocity of rock with Fluid 1 and 2 saturation. Both '''Eqs. 13.89''' and '''13.90''' are direct results from simplified Gassmann’s equation ('''Eq. 13.64'''). In '''Fig. 13.43''', we show the typical fluid modulus effect on the saturated bulk modulus ''K''<sub>''s''</sub>. Even at a modest porosity of 15%, changes can be substantial. At in-situ conditions, pore fluids are often multiphase mixtures. Dynamic fluid modulus may also depend on fluid mobility, fluid distribution, rock compressibility, and seismic wavelength.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:Vol1 Page 632 Image 0002.png|'''Fig. 13.43 – Effect of varying fluid modulus (''K''<sub>0</sub>) on the calculated saturated bulk modulus (''K<sub>s</sub>'') for sandstones.'''
File:Vol1 Page 632 Image 0002.png|'''Fig. 13.43 – Effect of varying fluid modulus (''K''<sub>0</sub>) on the calculated saturated bulk modulus (''K<sub>s</sub>'') for sandstones.'''
</gallery>
</gallery>
<br/>


=== Cracked Rock ===
=== Cracked Rock ===


For some cracked rocks, different methods of calculating velocities and the effects of pore fluids are preferable. Numerous theories have been developed to describe the effects of crack-like pores. Most view cracks as ellipsoids with their aspect ratio, ''α'', defined as the ratio of the semiminor to semimajor axes. Eshelby<ref name="r79" /> examined the elastic deformation of such elliptical inclusions, and these results were then applied to the compressibility of rocks by Walsh.<ref name="r80" /> In concept, long, narrow cracks are compliant and can be very effective at reducing the rock moduli at low crack porosities. The primary controlling factor for these elliptical fractures is the aspect ratio, ''α'', defined as the ratio of the ellipse semiminor (''a'') to semimajor (''b'') axes:
For some cracked rocks, different methods of calculating velocities and the effects of pore fluids are preferable. Numerous theories have been developed to describe the effects of crack-like pores. Most view cracks as ellipsoids with their aspect ratio, ''α'', defined as the ratio of the semiminor to semimajor axes. Eshelby<ref name="r79">_</ref> examined the elastic deformation of such elliptical inclusions, and these results were then applied to the compressibility of rocks by Walsh.<ref name="r80">_</ref> In concept, long, narrow cracks are compliant and can be very effective at reducing the rock moduli at low crack porosities. The primary controlling factor for these elliptical fractures is the aspect ratio, ''α'', defined as the ratio of the ellipse semiminor (''a'') to semimajor (''b'') axes:<br/><br/>[[File:Vol1 page 0633 eq 001.png|RTENOTITLE]]....................(13.91)<br/><br/>The smaller the value of α, the softer the crack and cracked rock, resulting in lower velocities and stronger pressure dependence.<br/><br/>Numerous assumptions are made in the derivation and application of cracked media models, such as the following:
<br>
<br>
[[File:Vol1 page 0633 eq 001.png]]....................(13.91)
<br>
<br>
The smaller the value of α, the softer the crack and cracked rock, resulting in lower velocities and stronger pressure dependence.  
<br>
<br>
Numerous assumptions are made in the derivation and application of cracked media models, such as the following:  
<br>
#The porous material is isotropic, elastic, monomineralic, and homogeneous.
#The fracture population is dilute, and few, or only first-order, mechanical interactions occur among fractures.
#Fractures can be described by simple shapes.
#The pore-fluid system is closed, and there is no chemical interaction between fluids and rock frame (however, shear modulus need not remain constant).
<br>


Some of these assumptions may be dropped, depending on the model involved. For example, Hudson<ref name="r81" /> specifically includes the effect of anisotropic crack distributions.  
#The porous material is isotropic, elastic, monomineralic, and homogeneous.
<br>
#The fracture population is dilute, and few, or only first-order, mechanical interactions occur among fractures.
<br>
#Fractures can be described by simple shapes.
One particularly useful result was derived by Kuster and Toksoz.<ref name="r82" /> Using scattering theory, they derived the general relation of bulk and shear moduli of the cracked rock (''K''*, ''μ''*) to the crack porosity (''c''), aspect ratio (''α''<sub>''m''</sub>), mineral (''K''<sub>0</sub>, ''μ''<sub>0</sub>), and inclusion or crack moduli (''K''′, ''μ''′) (Cheng and Toksoz<ref name="r83" />).
#The pore-fluid system is closed, and there is no chemical interaction between fluids and rock frame (however, shear modulus need not remain constant).
<br>
 
<br>
 
[[File:Vol1 page 0633 eq 002.png]]....................(13.92)
 
<br>
Some of these assumptions may be dropped, depending on the model involved. For example, Hudson<ref name="r81">_</ref> specifically includes the effect of anisotropic crack distributions.<br/><br/>One particularly useful result was derived by Kuster and Toksoz.<ref name="r82">_</ref> Using scattering theory, they derived the general relation of bulk and shear moduli of the cracked rock (''K''*, ''μ''*) to the crack porosity (''c''), aspect ratio (''α''<sub>''m''</sub>), mineral (''K''<sub>0</sub>, ''μ''<sub>0</sub>), and inclusion or crack moduli (''K''′, ''μ''′) (Cheng and Toksoz<ref name="r83">_</ref>).<br/><br/>[[File:Vol1 page 0633 eq 002.png|RTENOTITLE]]....................(13.92)<br/><br/>[[File:Vol1 page 0633 eq 003.png|RTENOTITLE]]....................(13.93)<br/><br/>Here, ''T''<sub>1</sub> and ''T''<sub>2</sub> are scalar functions of ''K''<sub>0</sub>, ''μ''<sub>0</sub>, ''K''′, and ''μ''′, and correspond to ''T''<sub>''iijj''</sub> and ''T''<sub>''ijij''</sub> in Kuster and Toksoz.<ref name="r82">_</ref> This formulation allows the effects of several populations (several values of ''m'') of cracks to be summed. The general limitation is that the porosity for any particular aspect ratio cannot exceed the value of the aspect ratio itself.<br/><br/>The results of the Kuster-Toksoz model are shown in '''Fig. 13.44'''. Numerous important features should be noted. Velocities drop rapidly for long, narrow cracks (small ''α''), with even small crack porosities. For such soft cracks, the increase in velocity is dramatic. At a shape close to spherical (''α'' above about 0.5), the pores are stiff, and the change in density dominates. Thus, with ''α''s close to unity, going from dry to water-saturated actually decreases the velocity. Notice also that for small aspect ratios, the shear velocity increases with water saturation. This requires a changing shear modulus with saturation, in direct violation of a primary assumption of Gassmann’s relations. This changing shear modulus is one reason why Gassmann’s relations may not work well in fractured rocks. An example of a rock modeled by both Gassmann’s and Kuster-Toksoz techniques is shown in '''Fig. 13.45'''. For this limestone, Gassmann’s relations substantially under estimate the effect of liquid saturation. The Kuster-Toksoz prediction for oil saturation is close to the experimental observed values. However, the success of this model is not quite as spectacular as it seems, because an arbitrary population of fractures and aspect ratios (''α''<sub>''m''</sub>''s'') can be included to force such a good fit. The actual population of cracks in rocks remains unknown.<br/><br/><gallery widths="300px" heights="200px">
<br>
[[File:Vol1 page 0633 eq 003.png]]....................(13.93)
<br>
<br>
Here, ''T''<sub>1</sub> and ''T''<sub>2</sub> are scalar functions of ''K''<sub>0</sub>, ''μ''<sub>0</sub>, ''K''′, and ''μ''′, and correspond to ''T''<sub>''iijj''</sub> and ''T''<sub>''ijij''</sub> in Kuster and Toksoz.<ref name="r82" /> This formulation allows the effects of several populations (several values of ''m'') of cracks to be summed. The general limitation is that the porosity for any particular aspect ratio cannot exceed the value of the aspect ratio itself.  
<br>
<br>
The results of the Kuster-Toksoz model are shown in '''Fig. 13.44'''. Numerous important features should be noted. Velocities drop rapidly for long, narrow cracks (small ''α''), with even small crack porosities. For such soft cracks, the increase in velocity is dramatic. At a shape close to spherical (''α'' above about 0.5), the pores are stiff, and the change in density dominates. Thus, with ''α''s close to unity, going from dry to water-saturated actually decreases the velocity. Notice also that for small aspect ratios, the shear velocity increases with water saturation. This requires a changing shear modulus with saturation, in direct violation of a primary assumption of Gassmann’s relations. This changing shear modulus is one reason why Gassmann’s relations may not work well in fractured rocks. An example of a rock modeled by both Gassmann’s and Kuster-Toksoz techniques is shown in '''Fig. 13.45'''. For this limestone, Gassmann’s relations substantially under estimate the effect of liquid saturation. The Kuster-Toksoz prediction for oil saturation is close to the experimental observed values. However, the success of this model is not quite as spectacular as it seems, because an arbitrary population of fractures and aspect ratios (''α''<sub>''m''</sub>''s'') can be included to force such a good fit. The actual population of cracks in rocks remains unknown.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 634 Image 0001.png|'''Fig. 13.44 – Normalized compressional and shear velocities for cracked rocks (''V''<sub>rock</sub> / ''V''<sub>mineral</sub>) both dry and saturated using the Kunster-Toksoz<ref name="r82" /> method (from Cheng and Toksoz<ref name="r83" />). Velocities are controlled by the crack aspect ratio (''a'' = axis<sub>minor</sub> / axis<sub>major</sub>) and crack porosity.'''
File:vol1 Page 634 Image 0001.png|'''Fig. 13.44 – Normalized compressional and shear velocities for cracked rocks (''V''<sub>rock</sub> / ''V''<sub>mineral</sub>) both dry and saturated using the Kunster-Toksoz<ref name="r82" /> method (from Cheng and Toksoz<ref name="r83" />). Velocities are controlled by the crack aspect ratio (''a'' = axis<sub>minor</sub> / axis<sub>major</sub>) and crack porosity.'''


File:vol1 Page 635 Image 0001.png|'''Fig. 13.45 – Measured and modeled velocities on the Bedford Limestone (Wang ''et al''.<ref name="r58" />).'''
File:vol1 Page 635 Image 0001.png|'''Fig. 13.45 – Measured and modeled velocities on the Bedford Limestone (Wang ''et al''.<ref name="r58" />).'''
</gallery>
</gallery><br/>The expressions in '''Eqs. 13.92''' and '''13.93''' are complicated and difficult to apply. The linear relation of normalized velocities to crack aspect ratio and porosity suggests that a simplified form can be derived to give a first-order approximation.<br/><br/>[[File:Vol1 page 0634 eq 001.png|RTENOTITLE]]....................(13.94)<br/>
<br/>
The expressions in '''Eqs. 13.92''' and '''13.93''' are complicated and difficult to apply. The linear relation of normalized velocities to crack aspect ratio and porosity suggests that a simplified form can be derived to give a first-order approximation.
<br>
<br>
[[File:Vol1 page 0634 eq 001.png]]....................(13.94)
<br>
<br>


=== Anisotropy ===
=== Anisotropy ===


To this point, we have usually considered rocks to be isotropic. In reality, most rocks are anisotropic to some degree. Some dominant lithologies, such as shales, are by definition anisotropic (otherwise, they are mudstones). In addition, many ubiquitous sedimentary features such as bedding will lead to anisotropy on a larger scale. In-situ stresses are anisotropic ('''Fig. 13.34'''), resulting in an anisotropy in rock properties. Anisotropy in transport properties such as permeability is a common concern in describing reservoir flow. Fractured reservoirs typically have a preferred fracture and flow direction, and these directions often can be ascertained from oriented borehole or surface seismic data.  
To this point, we have usually considered rocks to be isotropic. In reality, most rocks are anisotropic to some degree. Some dominant lithologies, such as shales, are by definition anisotropic (otherwise, they are mudstones). In addition, many ubiquitous sedimentary features such as bedding will lead to anisotropy on a larger scale. In-situ stresses are anisotropic ('''Fig. 13.34'''), resulting in an anisotropy in rock properties. Anisotropy in transport properties such as permeability is a common concern in describing reservoir flow. Fractured reservoirs typically have a preferred fracture and flow direction, and these directions often can be ascertained from oriented borehole or surface seismic data.<br/><br/>An interesting aspect of anisotropy is the phenomenon of shear-wave splitting. Elastic anisotropy means that the stiffness or effective moduli in one direction will be different from that in another. For shear waves, their particle motion will be approximately normal to the direction of propagation. The velocity will depend on the orientation of the particle motion. The shear wave will then "split" into two shear waves with orthogonal particle motion, each traveling with the velocity determined by the stiffness in that direction. An example of this is shown in '''Fig. 13.46''' from Sondergeld and Rai.<ref name="r84">_</ref> The recorded waveform can be seen as two distinct shear waves traveling at their own velocities. Note that when these distinct waves are examined in isolation, their velocity is independent of direction. A single input wave has been split into two waves. This is similar to the image splitting in optics when light travels through an anisotropic medium. On the other hand, because compressional waves have particle motion only along the direction of propagation, they have no splitting.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
An interesting aspect of anisotropy is the phenomenon of shear-wave splitting. Elastic anisotropy means that the stiffness or effective moduli in one direction will be different from that in another. For shear waves, their particle motion will be approximately normal to the direction of propagation. The velocity will depend on the orientation of the particle motion. The shear wave will then "split" into two shear waves with orthogonal particle motion, each traveling with the velocity determined by the stiffness in that direction. An example of this is shown in '''Fig. 13.46''' from Sondergeld and Rai.<ref name="r84" /> The recorded waveform can be seen as two distinct shear waves traveling at their own velocities. Note that when these distinct waves are examined in isolation, their velocity is independent of direction. A single input wave has been split into two waves. This is similar to the image splitting in optics when light travels through an anisotropic medium. On the other hand, because compressional waves have particle motion only along the direction of propagation, they have no splitting.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 636 Image 0001.png|'''Fig. 13.46 – Transmitter and receiver are rotated simultaneously through an azimuth aperture of 180°. When particle motion is either parallel or perpendicular to the shale fabric, only one arriving wave is seen. At other angles, both slow and fast waves are present (after Sondergeld and Rai<ref name="r84" />).'''
File:vol1 Page 636 Image 0001.png|'''Fig. 13.46 – Transmitter and receiver are rotated simultaneously through an azimuth aperture of 180°. When particle motion is either parallel or perpendicular to the shale fabric, only one arriving wave is seen. At other angles, both slow and fast waves are present (after Sondergeld and Rai<ref name="r84" />).'''
</gallery>
</gallery><br/>Although the split shear waves may travel each with a constant velocity, the amplitude within each will be strongly dependent on angle. The energy of the initial single shear wave is partitioned as vector components in each of the principal directions. This amplitude dependence on angle is shown in '''Fig. 13.47''', also from Sondergeld and Rai.<ref name="r84">_</ref> '''Figs. 13.46''' and '''13.47''' demonstrate that measurement of seismic shear waves at the surface will be useful in delineating in-situ anisotropy directions. This anisotropy can then be related to factors such as oriented fractures and in-situ stress directions.<br/><br/><gallery widths="300px" heights="200px">
<br/>
Although the split shear waves may travel each with a constant velocity, the amplitude within each will be strongly dependent on angle. The energy of the initial single shear wave is partitioned as vector components in each of the principal directions. This amplitude dependence on angle is shown in '''Fig. 13.47''', also from Sondergeld and Rai.<ref name="r84" /> '''Figs. 13.46''' and '''13.47''' demonstrate that measurement of seismic shear waves at the surface will be useful in delineating in-situ anisotropy directions. This anisotropy can then be related to factors such as oriented fractures and in-situ stress directions.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 637 Image 0001.png|'''Fig. 13.47 – Measured phase amplitudes for fast (circles) and slow (squares) shear-wave arrivals. Change in maximum amplitude for the two fast wave cycles is caused by variations in acoustic coupling (after Sondergeld and Rai<ref name="r84" />).'''
File:vol1 Page 637 Image 0001.png|'''Fig. 13.47 – Measured phase amplitudes for fast (circles) and slow (squares) shear-wave arrivals. Change in maximum amplitude for the two fast wave cycles is caused by variations in acoustic coupling (after Sondergeld and Rai<ref name="r84" />).'''
</gallery>
</gallery><br/>A typical homogeneous but bedded sedimentary unit would have a horizontal plane of symmetry as well as a vertical symmetry axis of rotation. This situation is commonly referred to as Vertical Transverse Isotropy (VTI), although the term "Polar Anisotropy" has also been suggested (Thomsen<ref name="r85">_</ref>). For "weak" anisotropy (Thomsen<ref name="r86">_</ref>), the dependence of velocities as a function of angle (''θ'') from the symmetry axis can be written as<br/><br/>[[File:Vol1 page 0636 eq 001.png|RTENOTITLE]]....................(13.95)<br/><br/>[[File:Vol1 page 0636 eq 002.png|RTENOTITLE]]....................(13.96)<br/><br/>[[File:Vol1 page 0636 eq 003.png|RTENOTITLE]]....................(13.97)<br/><br/>where ''V''<sub>''p''</sub>(''θ'') is the compressional velocity and ''V''<sub>''S-''</sub>(''θ'') and ''V''<sub>''S||''</sub>(''θ'') are the shear velocities with particle polarizations perpendicular and parallel to the symmetry plane (e.g., bedding), respectively.
<br/>
A typical homogeneous but bedded sedimentary unit would have a horizontal plane of symmetry as well as a vertical symmetry axis of rotation. This situation is commonly referred to as Vertical Transverse Isotropy (VTI), although the term "Polar Anisotropy" has also been suggested (Thomsen<ref name="r85" />). For "weak" anisotropy (Thomsen<ref name="r86" />), the dependence of velocities as a function of angle (''θ'') from the symmetry axis can be written as
<br>
<br>
[[File:Vol1 page 0636 eq 001.png]]....................(13.95)
<br>
<br>
[[File:Vol1 page 0636 eq 002.png]]....................(13.96)
<br>
<br>
[[File:Vol1 page 0636 eq 003.png]]....................(13.97)
<br>
<br>
where ''V''<sub>''p''</sub>(''θ'') is the compressional velocity and ''V''<sub>''S-''</sub>(''θ'') and ''V''<sub>''S||''</sub>(''θ'') are the shear velocities with particle polarizations perpendicular and parallel to the symmetry plane (e.g., bedding), respectively.  


The Thomsen<ref name="r86" /> anisotropic parameter ''ε'' can be defined as
The Thomsen<ref name="r86">_</ref> anisotropic parameter ''ε'' can be defined as<br/><br/>[[File:Vol1 page 0636 eq 004.png|RTENOTITLE]]....................(13.98)<br/><br/>where ''V''<sub>''P''0</sub> is the compressional velocity parallel to the axis of symmetry, and ''V''<sub>''P''90</sub> is the velocity perpendicular to this axis. The parameter ''γ'' can be defined as<br/><br/>[[File:Vol1 page 0636 eq 005.png|RTENOTITLE]]....................(13.99)<br/><br/>where ''V''<sub>''S''0</sub> is the shear velocity parallel to axis of symmetry, and ''V''<sub>''S||''90</sub> is the velocity perpendicular to this axis.<br/><br/>The anisotropic parameter ''δ'' is more difficult to characterize, and is the primary component modifying the compressional moveout velocity from the isotropic case. To describe it, we must refer back to stiffness defined in the generalized Hooke’s law given in '''Eq. 13.43'''.<br/><br/>[[File:Vol1 page 0637 eq 001.png|RTENOTITLE]]....................(13.100)<br/><br/>The advantage of these formulations is that they can be extracted from observed shear-wave splitting or extracted from normal moveout (NMO) corrections during seismic processing. Thus, they provide a valuable tool to describe the anisotropic character of reservoirs from remote measurements.
<br>
<br>
[[File:Vol1 page 0636 eq 004.png]]....................(13.98)
<br>
<br>
where ''V''<sub>''P''0</sub> is the compressional velocity parallel to the axis of symmetry, and ''V''<sub>''P''90</sub> is the velocity perpendicular to this axis. The parameter ''γ'' can be defined as
<br>
<br>
[[File:Vol1 page 0636 eq 005.png]]....................(13.99)
<br>
<br>
where ''V''<sub>''S''0</sub> is the shear velocity parallel to axis of symmetry, and ''V''<sub>''S||''90</sub> is the velocity perpendicular to this axis.  
<br>
<br>
The anisotropic parameter ''δ'' is more difficult to characterize, and is the primary component modifying the compressional moveout velocity from the isotropic case. To describe it, we must refer back to stiffness defined in the generalized Hooke’s law given in '''Eq. 13.43'''.
<br>
<br>
[[File:Vol1 page 0637 eq 001.png]]....................(13.100)
<br>
<br>
The advantage of these formulations is that they can be extracted from observed shear-wave splitting or extracted from normal moveout (NMO) corrections during seismic processing. Thus, they provide a valuable tool to describe the anisotropic character of reservoirs from remote measurements.


=== Attenuation and Velocity Dispersion ===
=== Attenuation and Velocity Dispersion ===


As seismic acoustic waves pass through rock, some of their energy will be lost to heat. For tight, hard rocks, this loss can be negligible. However, for most sedimentary rocks, this loss will be significant, particularly on seismic scales. In reality, all rocks are anelastic to some degree. We must rewrite our wave equation to include this energy or amplitude loss with depth, ''z''.
As seismic acoustic waves pass through rock, some of their energy will be lost to heat. For tight, hard rocks, this loss can be negligible. However, for most sedimentary rocks, this loss will be significant, particularly on seismic scales. In reality, all rocks are anelastic to some degree. We must rewrite our wave equation to include this energy or amplitude loss with depth, ''z''.<br/><br/>[[File:Vol1 page 0637 eq 002.png|RTENOTITLE]]....................(13.101)<br/><br/>where ''A''(''z,t'') is the amplitude at some point of depth and time, ''A''<sub>0</sub> is the initial amplitude, and ''k''* is the complex wave number (''k''* = ''k'' + ''iα''<sub>l</sub>). Note that here ''α''<sub>l</sub> is a loss parameter, and not an aspect ratio. Therefore, we can rewrite '''Eq. 13.101''' as<br/><br/>[[File:Vol1 page 0638 eq 001.png|RTENOTITLE]]....................(13.102)<br/><br/>Another common measure is the loss decrement ''δ'':<br/><br/>[[File:Vol1 page 0638 eq 002.png|RTENOTITLE]]....................(13.103)<br/><br/>where the wavelength ''λ'' depends on the velocity ''V'' and frequency ''f'': ''λ'' = ''V''/''f''. However, the most common measure of attenuation is 1/''Q''.<br/><br/>[[File:Vol1 page 0638 eq 003.png|RTENOTITLE]]....................(13.104)<br/><br/>One of the most straightforward descriptions of the relation of velocity to attenuation was developed by Cole and Cole<ref name="r87">_</ref> and applied to attenuation measurements by Spencer.<ref name="r88">_</ref> The Cole-Cole relationships involve both a peak frequency or characteristic relaxation time, ''τ'', for the attenuation mechanism, and a spread factor, ''b'', which determines the distribution of relaxation times. The real and imaginary components, ''B′'' and ''B"'', of a general modulus, ''B'' = ''B′'' + ''iB"'', are<br/><br/>[[File:Vol1 page 0638 eq 004.png|RTENOTITLE]]....................(13.105)<br/><br/>[[File:Vol1 page 0638 eq 005.png|RTENOTITLE]]....................(13.106)<br/><br/>where ''y'' = ln(''ωτ''), ''B''<sub>0</sub> and ''B''<sub>∞</sub> are the zero and infinite frequency moduli.<br/><br/>This would lead to a general attenuation of<br/><br/>[[File:Vol1 page 0638 eq 006.png|RTENOTITLE]]....................(13.107)<br/><br/>These relations connecting velocity and attenuation are plotted in '''Fig. 13.48'''. This figure represents losses and velocity dispersion (frequency dependence) caused by a single relaxation mechanism. At high frequencies, the material is unrelaxed and stiffer, and it has a higher velocity. At low frequencies, the material has time to relax, and velocities are lower.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
[[File:Vol1 page 0637 eq 002.png]]....................(13.101)
<br>
<br>
where ''A''(''z,t'') is the amplitude at some point of depth and time, ''A''<sub>0</sub> is the initial amplitude, and ''k''* is the complex wave number (''k''* = ''k'' + ''iα''<sub>l</sub>). Note that here ''α''<sub>l</sub> is a loss parameter, and not an aspect ratio. Therefore, we can rewrite '''Eq. 13.101''' as
<br>
<br>
[[File:Vol1 page 0638 eq 001.png]]....................(13.102)
<br>
<br>
Another common measure is the loss decrement ''δ'':
<br>
<br>
[[File:Vol1 page 0638 eq 002.png]]....................(13.103)
<br>
<br>
where the wavelength ''λ'' depends on the velocity ''V'' and frequency ''f'': ''λ'' = ''V''/''f''. However, the most common measure of attenuation is 1/''Q''.
<br>
<br>
[[File:Vol1 page 0638 eq 003.png]]....................(13.104)
<br>
<br>
One of the most straightforward descriptions of the relation of velocity to attenuation was developed by Cole and Cole<ref name="r87" /> and applied to attenuation measurements by Spencer.<ref name="r88" /> The Cole-Cole relationships involve both a peak frequency or characteristic relaxation time, ''τ'', for the attenuation mechanism, and a spread factor, ''b'', which determines the distribution of relaxation times. The real and imaginary components, ''B′'' and ''B"'', of a general modulus, ''B'' = ''B′'' + ''iB"'', are
<br>
<br>
[[File:Vol1 page 0638 eq 004.png]]....................(13.105)
<br>
<br>
[[File:Vol1 page 0638 eq 005.png]]....................(13.106)
<br>
<br>
where ''y'' = ln(''ωτ''), ''B''<sub>0</sub> and ''B''<sub>∞</sub> are the zero and infinite frequency moduli.  
<br>
<br>
This would lead to a general attenuation of
<br>
<br>
[[File:Vol1 page 0638 eq 006.png]]....................(13.107)
<br>
<br>
These relations connecting velocity and attenuation are plotted in '''Fig. 13.48'''. This figure represents losses and velocity dispersion (frequency dependence) caused by a single relaxation mechanism. At high frequencies, the material is unrelaxed and stiffer, and it has a higher velocity. At low frequencies, the material has time to relax, and velocities are lower.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 639 Image 0001.png|'''Fig. 13.48 – Generalized rock velocity (''V<sub>p</sub>'') and attenuation (1/''Q'') relations as a function of frequency. Changes in the relaxation mechanism (e.g., by changing fluid mobility) will shift the frequency response.'''
File:vol1 Page 639 Image 0001.png|'''Fig. 13.48 – Generalized rock velocity (''V<sub>p</sub>'') and attenuation (1/''Q'') relations as a function of frequency. Changes in the relaxation mechanism (e.g., by changing fluid mobility) will shift the frequency response.'''
</gallery>
</gallery><br/>Fluid mobility also influences rock inelastic properties. Most of the observed losses are caused by relative motion of fluid in the pore space. For a constant pore fluid type, permeability will control the motion and dissipation, thus making attenuation a permeability indicator. For variations in viscosities, mobility also will be dependent on frequency, and attenuation and dispersion may indicate fluid type.<br/><br/>Many models have been proposed, such as those of Biot,<ref name="r89">_</ref> O’Connell and Budiansky,<ref name="r90">_</ref> Walsh,<ref name="r91">_</ref> and Dvorcik and Nur.<ref name="r92">_</ref> Unfortunately, the different mechanisms proposed often give contradictory results.<br/><br/>Wave attenuation and dispersion in vacuum dry rock is relatively negligible.<ref name="r93">_</ref> Porous rocks containing fluids show a strong frequency-dependent attenuation. Variations in fluid properties such as modulus, viscosity, and polarity have a strong influence on 1/''Q'' (Clark,<ref name="r93">_</ref> Winkler ''et al''.,<ref name="r94">_</ref> Murphy,<ref name="r95">_</ref> Tittmann ''et al''.,<ref name="r96">_</ref> Jones,<ref name="r97">_</ref> and Tutuncu ''et al''.<ref name="r98">_</ref>). These results indicate that the dominant 1/''Q'' mechanism is the interaction and motion of fluid in the rock frame rather than intrinsic losses either in the frame or the fluids themselves. Squirt flow is believed to be the primary loss mechanism in consolidated rocks, although the inertial Biot mechanism may be important in highly permeable rocks (Vo-Thant,<ref name="r99">_</ref> Yamamato ''et al''.<ref name="r100">_</ref>).<br/><br/>Fluid motion and pressure control velocity changes and seismic sensitivity to pore fluid types. One obvious factor is viscosity. The two most commonly used theoretical concepts are the inertial coupling of Biot<ref name="r89">_</ref> and the squirt-flow mechanism (see, for example, O’Connell and Budiansky,<ref name="r90">_</ref> or Dvorcik and Nur<ref name="r92">_</ref>). Biot gives a characteristic frequency, ''ω''<sub>''c''</sub> (roughly, the boundary between high and low frequency range) with the viscosity dependence, ''η'', in the numerator:<br/><br/>[[File:Vol1 page 0639 eq 001.png|RTENOTITLE]]....................(13.108)<br/><br/>Here, ''Φ'' is porosity, ''k'' is permeability, and ''ρ'' is fluid density. However, squirt-flow mechanisms lead to viscosity dependence in the denominator:<br/><br/>[[File:Vol1 page 0639 eq 002.png|RTENOTITLE]]....................(13.109)<br/><br/>Here, ''K'' is frame modulus, and ''α'' is crack aspect ratio. These contrasting dependencies indicate that viscosity can be an obvious test to ascertain which theory is applicable.<br/><br/>Compressional (''V''<sub>''p''</sub>) and shear (''V''<sub>''s''</sub>) velocities for a sample of the Upper Fox Hills Sandstone (Heather well) are shown in '''Fig. 13.49'''. Several features should be noted. For the dry sample (open symbols), ''V''<sub>''p''</sub> and ''V''<sub>''s''</sub> show little frequency or temperature influence. This confirms that the primary dispersive and temperature effects are dependent on pore fluids. When saturated with glycerine, strong temperature and frequency dependence is obvious. Shear velocity is not independent of the fluid, but increases with increasing fluid viscosity, indicating a viscous contribution to the shear modulus. ''V''<sub>''p''</sub> increases with viscosity also. More importantly, the dispersion curve shows a systematic shift to lower frequencies with increasing velocities, consistent with squirt flow.<br/><br/><gallery widths="300px" heights="200px">
<br/>
Fluid mobility also influences rock inelastic properties. Most of the observed losses are caused by relative motion of fluid in the pore space. For a constant pore fluid type, permeability will control the motion and dissipation, thus making attenuation a permeability indicator. For variations in viscosities, mobility also will be dependent on frequency, and attenuation and dispersion may indicate fluid type.  
<br>
<br>
Many models have been proposed, such as those of Biot,<ref name="r89" /> O’Connell and Budiansky,<ref name="r90" /> Walsh,<ref name="r91" /> and Dvorcik and Nur.<ref name="r92" /> Unfortunately, the different mechanisms proposed often give contradictory results.  
<br>
<br>
Wave attenuation and dispersion in vacuum dry rock is relatively negligible.<ref name="r93" /> Porous rocks containing fluids show a strong frequency-dependent attenuation. Variations in fluid properties such as modulus, viscosity, and polarity have a strong influence on 1/''Q'' (Clark,<ref name="r93" /> Winkler ''et al''.,<ref name="r94" /> Murphy,<ref name="r95" /> Tittmann ''et al''.,<ref name="r96" /> Jones,<ref name="r97" /> and Tutuncu ''et al''.<ref name="r98" />). These results indicate that the dominant 1/''Q'' mechanism is the interaction and motion of fluid in the rock frame rather than intrinsic losses either in the frame or the fluids themselves. Squirt flow is believed to be the primary loss mechanism in consolidated rocks, although the inertial Biot mechanism may be important in highly permeable rocks (Vo-Thant,<ref name="r99" /> Yamamato ''et al''.<ref name="r100" />).  
<br>
<br>
Fluid motion and pressure control velocity changes and seismic sensitivity to pore fluid types. One obvious factor is viscosity. The two most commonly used theoretical concepts are the inertial coupling of Biot<ref name="r89" /> and the squirt-flow mechanism (see, for example, O’Connell and Budiansky,<ref name="r90" /> or Dvorcik and Nur<ref name="r92" />). Biot gives a characteristic frequency, ''ω''<sub>''c''</sub> (roughly, the boundary between high and low frequency range) with the viscosity dependence, ''η'', in the numerator:
<br>
<br>
[[File:Vol1 page 0639 eq 001.png]]....................(13.108)
<br>
<br>
Here, ''Φ'' is porosity, ''k'' is permeability, and ''ρ'' is fluid density. However, squirt-flow mechanisms lead to viscosity dependence in the denominator:
<br>
<br>
[[File:Vol1 page 0639 eq 002.png]]....................(13.109)
<br>
<br>
Here, ''K'' is frame modulus, and ''α'' is crack aspect ratio. These contrasting dependencies indicate that viscosity can be an obvious test to ascertain which theory is applicable.  
<br>
<br>
Compressional (''V''<sub>''p''</sub>) and shear (''V''<sub>''s''</sub>) velocities for a sample of the Upper Fox Hills Sandstone (Heather well) are shown in '''Fig. 13.49'''. Several features should be noted. For the dry sample (open symbols), ''V''<sub>''p''</sub> and ''V''<sub>''s''</sub> show little frequency or temperature influence. This confirms that the primary dispersive and temperature effects are dependent on pore fluids. When saturated with glycerine, strong temperature and frequency dependence is obvious. Shear velocity is not independent of the fluid, but increases with increasing fluid viscosity, indicating a viscous contribution to the shear modulus. ''V''<sub>''p''</sub> increases with viscosity also. More importantly, the dispersion curve shows a systematic shift to lower frequencies with increasing velocities, consistent with squirt flow.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 640 Image 0001.png|'''Fig. 13.49 – Compressional (''V<sub>p</sub>'') and shear (''V<sub>s</sub>'') velocities as a function of frequency for Fox Hills sandstone both dry and saturated with glycerine. Dry velocities show almost no temperature dependence. After saturation with glycerine, dispersion occurs both in ''V<sub>p</sub>'' and ''V<sub>s</sub>''. This dispersion is shifted to lower frequencies at lower temperatures because of the increased glycerine viscosity.'''
File:vol1 Page 640 Image 0001.png|'''Fig. 13.49 – Compressional (''V<sub>p</sub>'') and shear (''V<sub>s</sub>'') velocities as a function of frequency for Fox Hills sandstone both dry and saturated with glycerine. Dry velocities show almost no temperature dependence. After saturation with glycerine, dispersion occurs both in ''V<sub>p</sub>'' and ''V<sub>s</sub>''. This dispersion is shifted to lower frequencies at lower temperatures because of the increased glycerine viscosity.'''
</gallery>
</gallery><br/>Attenuation (1/''Q'') and velocity dispersion are strongly dependent on pore phase and compressibility, particularly as controlled by partial gas saturation. Attenuation could become a valuable direct hydrocarbon indicator (e.g., Tanner and Sheriff<ref name="r101">_</ref>). More recently, Klimentos<ref name="r102">_</ref> used the ratio of compressional to shear attenuations as a hydrocarbon indicator in well logs. Unfortunately, application of these properties is not frequent because of incomplete understanding of the phenomena and lack of appropriate tools to extract the information. Laboratory measurements at frequencies and amplitudes encompassing the seismic range have confirmed the strong dependence on partial gas saturation ('''Fig. 13.50a'''). However, attenuation is decreased by confining pressure, dropping rapidly as pressure increases ('''Fig. 13.50b'''). Attenuation peaks will also depend on specific rock characteristics. Absorption peaks seen in one frequency band may not be apparent in others.<br/><br/><gallery widths="300px" heights="200px">
<br/>
Attenuation (1/''Q'') and velocity dispersion are strongly dependent on pore phase and compressibility, particularly as controlled by partial gas saturation. Attenuation could become a valuable direct hydrocarbon indicator (e.g., Tanner and Sheriff<ref name="r101" />). More recently, Klimentos<ref name="r102" /> used the ratio of compressional to shear attenuations as a hydrocarbon indicator in well logs. Unfortunately, application of these properties is not frequent because of incomplete understanding of the phenomena and lack of appropriate tools to extract the information. Laboratory measurements at frequencies and amplitudes encompassing the seismic range have confirmed the strong dependence on partial gas saturation ('''Fig. 13.50a'''). However, attenuation is decreased by confining pressure, dropping rapidly as pressure increases ('''Fig. 13.50b'''). Attenuation peaks will also depend on specific rock characteristics. Absorption peaks seen in one frequency band may not be apparent in others.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 641 Image 0001.png|'''Fig. 13.50 – Extensional (or Young’s modulus) attenuation (1/''Q<sub>e</sub>'') as a function of water saturation (a) and frequency (b) for Berea sandstone. With increasing saturation, 1/''Q<sub>e</sub>'' increases and reaches a peak at approximately 95% saturation (a). Attenuation decreases with increasing pressure, as shown in (b).'''
File:vol1 Page 641 Image 0001.png|'''Fig. 13.50 – Extensional (or Young’s modulus) attenuation (1/''Q<sub>e</sub>'') as a function of water saturation (a) and frequency (b) for Berea sandstone. With increasing saturation, 1/''Q<sub>e</sub>'' increases and reaches a peak at approximately 95% saturation (a). Attenuation decreases with increasing pressure, as shown in (b).'''
</gallery>
</gallery><br/>With the improving quality of seismic data, maps of the estimated attenuation are becoming a common displayed attribute. The relative values of 1/''Q'' measured through time-lapse reservoir monitoring are becoming robust. As indicated in '''Fig. 13.50a''', 1/''Q'' will be sensitive to many of the common recovery processes.
<br/>
</div></div><div class="toccolours mw-collapsible mw-collapsed">
With the improving quality of seismic data, maps of the estimated attenuation are becoming a common displayed attribute. The relative values of 1/''Q'' measured through time-lapse reservoir monitoring are becoming robust. As indicated in '''Fig. 13.50a''', 1/''Q'' will be sensitive to many of the common recovery processes.  
<br>
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" >
 
== Rock Failure Relationships ==
== Rock Failure Relationships ==
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
=== Introduction ===
=== Introduction ===


In this section, we will go through the various relationships describing mechanical failure in rocks. This is important because under reservoir pressure and stress conditions, production can induce rock failure, sometime with catastrophic effects. By applying strength criteria, within reservoir simulators we can predict when problems might occur. In '''Section 13.5''', we examined the elastic behavior, which was largely reversible. Here we deal with permanent deformation. By rock failure, we mean the formation of faults and fracture planes, crushing, and relative motion of individual mineral grains and cements. Failure can involve formation of discrete fracture zones and the more "ductile" or homogeneous deformation. The later deformation is caused by a broad distribution of fracture zones or general grain crushing during compaction. We will not consider deformation caused by plastic strains of the mineral components, as is common in salt and in calcite at higher temperatures. In our analysis, several assumptions are made: The material is isotropic and homogeneous; stresses are applied uniformly; textural characteristics such as grain size and sorting have no influence; temperature and strain rate are ignored; and the intermediate stresses are presumed to play no role. Each of these assumptions can be violated, and some have been demonstrated to have major influences on rock strength.  
In this section, we will go through the various relationships describing mechanical failure in rocks. This is important because under reservoir pressure and stress conditions, production can induce rock failure, sometime with catastrophic effects. By applying strength criteria, within reservoir simulators we can predict when problems might occur. In '''Section 13.5''', we examined the elastic behavior, which was largely reversible. Here we deal with permanent deformation. By rock failure, we mean the formation of faults and fracture planes, crushing, and relative motion of individual mineral grains and cements. Failure can involve formation of discrete fracture zones and the more "ductile" or homogeneous deformation. The later deformation is caused by a broad distribution of fracture zones or general grain crushing during compaction. We will not consider deformation caused by plastic strains of the mineral components, as is common in salt and in calcite at higher temperatures. In our analysis, several assumptions are made: The material is isotropic and homogeneous; stresses are applied uniformly; textural characteristics such as grain size and sorting have no influence; temperature and strain rate are ignored; and the intermediate stresses are presumed to play no role. Each of these assumptions can be violated, and some have been demonstrated to have major influences on rock strength.


=== Coulumb-Navier Failure ===
=== Coulumb-Navier Failure ===


To begin with, a brief review of the standard Mohr failure criteria will be examined to introduce concepts and define terms, as well as to establish the basic mathematics behind the strength relationships. Units of stress and strength are the same as pressure and were covered previously. More detailed descriptions can be found in standard textbooks (i.e., Jaeger<ref name="r103" /><ref name="r104" />). Mohr circles and a linear failure envelope are the most common methods used to plot stresses and indicate strength limits. This technique predicts failure when stresses surpass both the intrinsic strength of a rock and internal friction. The primary terms and characteristics are shown in '''Fig. 13.51'''. Normal stresses across any plane are plotted on the horizontal axis, and shear stresses are plotted on the vertical axis. Compressive stresses are defined as positive (as opposed to the mechanical engineering convention of tensional stresses being positive). For the hydrostatic case, all stresses are equal; this stress state is represented by a point on the horizontal axis. When stresses differ, the maximum principal stress, ''σ''<sub>1</sub>, and minimum stress, ''σ''<sub>3</sub>, are plotted on the horizontal axis and the possible shear stresses along any plane fall on a hemisphere connecting ''σ''<sub>1</sub> and ''σ''<sub>3</sub> ('''Fig. 13.52'''). The mean stress, ''σ''<sub>''m''</sub>, and radius of this circle, ''r'', are simple sums and differences of the principal stresses.  
To begin with, a brief review of the standard Mohr failure criteria will be examined to introduce concepts and define terms, as well as to establish the basic mathematics behind the strength relationships. Units of stress and strength are the same as pressure and were covered previously. More detailed descriptions can be found in standard textbooks (i.e., Jaeger<ref name="r103">_</ref><ref name="r104">_</ref>). Mohr circles and a linear failure envelope are the most common methods used to plot stresses and indicate strength limits. This technique predicts failure when stresses surpass both the intrinsic strength of a rock and internal friction. The primary terms and characteristics are shown in '''Fig. 13.51'''. Normal stresses across any plane are plotted on the horizontal axis, and shear stresses are plotted on the vertical axis. Compressive stresses are defined as positive (as opposed to the mechanical engineering convention of tensional stresses being positive). For the hydrostatic case, all stresses are equal; this stress state is represented by a point on the horizontal axis. When stresses differ, the maximum principal stress, ''σ''<sub>1</sub>, and minimum stress, ''σ''<sub>3</sub>, are plotted on the horizontal axis and the possible shear stresses along any plane fall on a hemisphere connecting ''σ''<sub>1</sub> and ''σ''<sub>3</sub> ('''Fig. 13.52'''). The mean stress, ''σ''<sub>''m''</sub>, and radius of this circle, ''r'', are simple sums and differences of the principal stresses.<br/><br/>[[File:Vol1 page 0642 eq 001.png|RTENOTITLE]]....................(13.110)<br/><br/>[[File:Vol1 page 0642 eq 002.png|RTENOTITLE]]....................(13.111)<br/><br/>The normal stress across any plane, ''σ''<sub>''n''</sub>, and the shear stress along the plane, ''τ'', are functions of the principal stresses and the plane orientation.<br/><br/>[[File:Vol1 page 0643 eq 001.png|RTENOTITLE]]....................(13.112)<br/><br/>[[File:Vol1 page 0643 eq 002.png|RTENOTITLE]]....................(13.113)<br/><br/>[[File:Vol1 page 0643 eq 003.png|RTENOTITLE]]....................(13.114)<br/><br/>[[File:Vol1 page 0643 eq 004.png|RTENOTITLE]]....................(13.115)<br/><br/>where ''θ'' is the angle between the plane and the ''σ''<sub>3</sub> direction.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
[[File:Vol1 page 0642 eq 001.png]]....................(13.110)
<br>
<br>
[[File:Vol1 page 0642 eq 002.png]]....................(13.111)
<br>
<br>
The normal stress across any plane, ''σ''<sub>''n''</sub>, and the shear stress along the plane, ''τ'', are functions of the principal stresses and the plane orientation.
<br>
<br>
[[File:Vol1 page 0643 eq 001.png]]....................(13.112)
<br>
<br>
[[File:Vol1 page 0643 eq 002.png]]....................(13.113)
<br>
<br>
[[File:Vol1 page 0643 eq 003.png]]....................(13.114)
<br>
<br>
[[File:Vol1 page 0643 eq 004.png]]....................(13.115)
<br>
<br>
where ''θ'' is the angle between the plane and the ''σ''<sub>3</sub> direction.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 642 Image 0001.png|'''Fig. 13.51 – Schematic diagram defining stress orientations along and across a plane.'''
File:vol1 Page 642 Image 0001.png|'''Fig. 13.51 – Schematic diagram defining stress orientations along and across a plane.'''


File:vol1 Page 643 Image 0001.png|'''Fig. 13.52 – Generalized rock failure with a linear envelope.'''
File:vol1 Page 643 Image 0001.png|'''Fig. 13.52 – Generalized rock failure with a linear envelope.'''
</gallery>
</gallery><br/>From '''Eqs. 13.114''' and '''13.115''', the maximum shear occurs along a plane oriented at [[File:Vol1 page 0643 inline 001.png|RTENOTITLE]] (45°). However, because of friction, rocks do not fail along this plane. Instead, failure occurs along some rotated plane where friction is lower, yet shear stress is still high. This failure point (or plane) is shown in '''Fig. 13.51''' as the nearly diagonal line. '''Fig. 13.51''' also shows the associated normal and shear stresses. If numerous failure tests are made and plotted, an envelope is defined as in '''Fig. 13.52'''. In this case, friction is assumed to be a simple linear function of normal stress, and the resulting envelope is also linear. The slope of this envelope is ''α'', and we define ''μ'' as the angle of internal friction<br/><br/>[[File:Vol1 page 0643 eq 005.png|RTENOTITLE]]....................(13.116)<br/><br/>Within this framework, we can define several important properties of the rock as shown in '''Fig. 13.52''':
<br/>
From '''Eqs. 13.114''' and '''13.115''', the maximum shear occurs along a plane oriented at [[File:Vol1 page 0643 inline 001.png]] (45°). However, because of friction, rocks do not fail along this plane. Instead, failure occurs along some rotated plane where friction is lower, yet shear stress is still high. This failure point (or plane) is shown in '''Fig. 13.51''' as the nearly diagonal line. '''Fig. 13.51''' also shows the associated normal and shear stresses. If numerous failure tests are made and plotted, an envelope is defined as in '''Fig. 13.52'''. In this case, friction is assumed to be a simple linear function of normal stress, and the resulting envelope is also linear. The slope of this envelope is ''α'', and we define ''μ'' as the angle of internal friction
<br>
<br>
[[File:Vol1 page 0643 eq 005.png]]....................(13.116)
<br>
<br>
Within this framework, we can define several important properties of the rock as shown in '''Fig. 13.52''':


''C''<sub>0</sub> = Uniaxial or unconfined compressive strength (''σ''<sub>3</sub> = 0)
''C''<sub>0</sub> = Uniaxial or unconfined compressive strength (''σ''<sub>3</sub> = 0)
Line 1,463: Line 362:
''C''<sub>''t''</sub> = Tensional strength.
''C''<sub>''t''</sub> = Tensional strength.


The failure envelope is then defined by the line
The failure envelope is then defined by the line<br/><br/>[[File:Vol1 page 0644 eq 001.png|RTENOTITLE]]....................(13.117)<br/><br/>If the rock has already been broken, or a fracture already exists, then both ''C''<sub>''u''</sub> and ''C''<sub>''t''</sub> will be close to zero.<br/><br/>Several useful equations can be derived from the geometric relationships shown so far. From the equation for a circle,<br/><br/>[[File:Vol1 page 0644 eq 002.png|RTENOTITLE]]....................(13.118)<br/><br/>At the intersection of the envelope and the circle, we must have<br/><br/>[[File:Vol1 page 0644 eq 003.png|RTENOTITLE]]....................(13.119)<br/><br/>which leads to<br/><br/>[[File:Vol1 page 0644 eq 004.png|RTENOTITLE]]....................(13.120)<br/><br/>Using the general solution to a second-order polynomial gives<br/><br/>[[File:Vol1 page 0644 eq 005.png|RTENOTITLE]]....................(13.121)<br/><br/>Because we want only a point where the circle touches the envelope, the square root term must vanish.<br/><br/>[[File:Vol1 page 0645 eq 001.png|RTENOTITLE]]....................(13.122)<br/><br/>After some algebraic manipulation, we find<br/><br/>[[File:Vol1 page 0645 eq 002.png|RTENOTITLE]]....................(13.123)<br/><br/>and<br/><br/>[[File:Vol1 page 0645 eq 003.png|RTENOTITLE]]....................(13.124)<br/><br/>Substitution of ''C''<sub>''u''</sub> (defined in '''Eq. 13.123''') into '''Eq. 13.121''' gives an expression for normal stress.<br/><br/>[[File:Vol1 page 0645 eq 004.png|RTENOTITLE]]....................(13.125)<br/><br/>If the envelope could be continued into the tensional region, the tensional strength could easily be obtained:<br/><br/>[[File:Vol1 page 0645 eq 005.png|RTENOTITLE]]....................(13.126)<br/><br/>Under tension, the stresses are negative, although the tensional strength is a positive number. Thus, if rocks could fail according to a constant internal friction, we would have a simple way to relate the stresses involved and need only a couple of material constants, such as ''C''<sub>''u''</sub> and ''α''.
<br>
<br>
[[File:Vol1 page 0644 eq 001.png]]....................(13.117)
<br>
<br>
If the rock has already been broken, or a fracture already exists, then both ''C''<sub>''u''</sub> and ''C''<sub>''t''</sub> will be close to zero.  
<br>
<br>
Several useful equations can be derived from the geometric relationships shown so far. From the equation for a circle,
<br>
<br>
[[File:Vol1 page 0644 eq 002.png]]....................(13.118)
<br>
<br>
At the intersection of the envelope and the circle, we must have
<br>
<br>
[[File:Vol1 page 0644 eq 003.png]]....................(13.119)
<br>
<br>
which leads to
<br>
<br>
[[File:Vol1 page 0644 eq 004.png]]....................(13.120)
<br>
<br>
Using the general solution to a second-order polynomial gives
<br>
<br>
[[File:Vol1 page 0644 eq 005.png]]....................(13.121)
<br>
<br>
Because we want only a point where the circle touches the envelope, the square root term must vanish.
<br>
<br>
[[File:Vol1 page 0645 eq 001.png]]....................(13.122)
<br>
<br>
After some algebraic manipulation, we find
<br>
<br>
[[File:Vol1 page 0645 eq 002.png]]....................(13.123)
<br>
<br>
and
<br>
<br>
[[File:Vol1 page 0645 eq 003.png]]....................(13.124)
<br>
<br>
Substitution of ''C''<sub>''u''</sub> (defined in '''Eq. 13.123''') into '''Eq. 13.121''' gives an expression for normal stress.
<br>
<br>
[[File:Vol1 page 0645 eq 004.png]]....................(13.125)
<br>
<br>
If the envelope could be continued into the tensional region, the tensional strength could easily be obtained:
<br>
<br>
[[File:Vol1 page 0645 eq 005.png]]....................(13.126)
<br>
<br>
Under tension, the stresses are negative, although the tensional strength is a positive number. Thus, if rocks could fail according to a constant internal friction, we would have a simple way to relate the stresses involved and need only a couple of material constants, such as ''C''<sub>''u''</sub> and ''α''.


=== Mohr Failure, Curved Envelopes and Hoek-Brown Relationships ===
=== Mohr Failure, Curved Envelopes and Hoek-Brown Relationships ===


We are immediately faced with two problems when we try to apply Coulomb-Navier failure criteria: (1) Rocks do not generally have a linear failure envelope, and (2) material properties controlling failure must be obtained either through logs or assumed behavior. '''Fig. 13.53''' shows the type of envelope commonly seen. In fact, we know that the slope must change as stresses are increased because rocks begin yielding and act more plastically. '''Fig. 13.54''' shows the generalized behavior expected. At normal stresses above the brittle-ductile transition, failure can no longer be maintained on a single plane, but is distributed more homogeneously throughout the sample. We must develop different failure criteria, one that produced an appropriately curved envelope, and we expect it to have a strong porosity dependence ('''Fig. 13.55''').  
We are immediately faced with two problems when we try to apply Coulomb-Navier failure criteria: (1) Rocks do not generally have a linear failure envelope, and (2) material properties controlling failure must be obtained either through logs or assumed behavior. '''Fig. 13.53''' shows the type of envelope commonly seen. In fact, we know that the slope must change as stresses are increased because rocks begin yielding and act more plastically. '''Fig. 13.54''' shows the generalized behavior expected. At normal stresses above the brittle-ductile transition, failure can no longer be maintained on a single plane, but is distributed more homogeneously throughout the sample. We must develop different failure criteria, one that produced an appropriately curved envelope, and we expect it to have a strong porosity dependence ('''Fig. 13.55''').<br/><br/><gallery widths="300px" heights="200px">
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 644 Image 0001.png|'''Fig. 13.53 – Measured failure strengths for the B-d #1 sandstone with straight-line segments and curved envelope fits to failure.'''
File:vol1 Page 644 Image 0001.png|'''Fig. 13.53 – Measured failure strengths for the B-d #1 sandstone with straight-line segments and curved envelope fits to failure.'''


Line 1,539: Line 372:


File:vol1 Page 647 Image 0001.png|'''Fig. 13.55 – Failure mechanisms as a function of porosity for sandstones (after Scott<ref name="r105" />).'''
File:vol1 Page 647 Image 0001.png|'''Fig. 13.55 – Failure mechanisms as a function of porosity for sandstones (after Scott<ref name="r105" />).'''
</gallery>
</gallery><br/>Numerous failure criteria have been proposed that are primarily empirically based. '''Table 13.12''' shows some of the criteria proposed both for general purposes and for specific rock types or conditions. Observed failure envelopes are smooth forms so simple exponential or power-law functions can usually be found that fit the data well. The relations of Bienlawski<ref name="r111">_</ref> and Hoek and Brown<ref name="r112">_</ref> are most common. Much of the recent work in rock mechanics has been directed toward ascertaining the constants of these relationships in terms of easily measurable rock properties. Note that these relationships apply primarily to the brittle failure regime and cannot be used for grain crushing or pore collapse (as we shall see later) or when substantial ductile or plastic deformation is involved. We will examine these proposed forms to interrelate terms and reduce unknowns to variables that can be derived from logs.<br/><br/><gallery widths="300px" heights="200px">
<br/>
Numerous failure criteria have been proposed that are primarily empirically based. '''Table 13.12''' shows some of the criteria proposed both for general purposes and for specific rock types or conditions. Observed failure envelopes are smooth forms so simple exponential or power-law functions can usually be found that fit the data well. The relations of Bienlawski<ref name="r111" /> and Hoek and Brown<ref name="r112" /> are most common. Much of the recent work in rock mechanics has been directed toward ascertaining the constants of these relationships in terms of easily measurable rock properties. Note that these relationships apply primarily to the brittle failure regime and cannot be used for grain crushing or pore collapse (as we shall see later) or when substantial ductile or plastic deformation is involved. We will examine these proposed forms to interrelate terms and reduce unknowns to variables that can be derived from logs.  
<br>
<br>
<gallery widths=300px heights=200px>
File:Vol1 Page 648 Image 0001.png|'''Table 13.12'''<ref name="r104" /><ref name="r105" /><ref name="r106" /><ref name="r107" /><ref name="r108" /><ref name="r109" /><ref name="r110" /><ref name="r111" /><ref name="r112" />
File:Vol1 Page 648 Image 0001.png|'''Table 13.12'''<ref name="r104" /><ref name="r105" /><ref name="r106" /><ref name="r107" /><ref name="r108" /><ref name="r109" /><ref name="r110" /><ref name="r111" /><ref name="r112" />
</gallery>
</gallery><br/>Hoek and Brown<ref name="r112">_</ref> compiled extensive data on a variety of rock types and produced relationships that are simple and can be developed into forms amenable to well-log analysis. A primary feature of this failure criterion is a relation between the maximum and minimum stresses when both are normalized by the uniaxial compressive strength<br/><br/>[[File:Vol1 page 0646 eq 001.png|RTENOTITLE]]....................(13.127)<br/><br/>This formulation was motivated by the systematic behavior seen in many tests as shown in '''Fig. 13.56'''. In '''Eq. 13.127''', ''m'' and ''s'' are material constants dependent on the overall quality of the rock mass, and ''m'' is also dependent on the rock type ('''Table 13.13'''). Note that we could derive the value for ''m'' from a mineralogic analysis. In our analysis, we will presume that the local rock mass of interest is intact, and thus<br/><br/>[[File:Vol1 page 0646 eq 002.png|RTENOTITLE]]....................(13.128)<br/><br/>For applications that are in sandstones, numeric results can often use<br/><br/>[[File:Vol1 page 0646 eq 003.png|RTENOTITLE]]....................(13.129)<br/><br/>'''Eq. 13.127''' can be rewritten to give one principal stress in terms of the other:<br/><br/>[[File:Vol1 page 0646 eq 004.png|RTENOTITLE]]....................(13.130)<br/><br/>[[File:Vol1 page 0647 eq 001.png|RTENOTITLE]]....................(13.131)<br/><br/>Such normalized stress states were used to construct the curved envelope in '''Fig. 13.54'''.<br/><br/><gallery widths="300px" heights="200px">
<br>
Hoek and Brown<ref name="r112" /> compiled extensive data on a variety of rock types and produced relationships that are simple and can be developed into forms amenable to well-log analysis. A primary feature of this failure criterion is a relation between the maximum and minimum stresses when both are normalized by the uniaxial compressive strength
<br>
<br>
[[File:Vol1 page 0646 eq 001.png]]....................(13.127)
<br>
<br>
This formulation was motivated by the systematic behavior seen in many tests as shown in '''Fig. 13.56'''. In '''Eq. 13.127''', ''m'' and ''s'' are material constants dependent on the overall quality of the rock mass, and ''m'' is also dependent on the rock type ('''Table 13.13'''). Note that we could derive the value for ''m'' from a mineralogic analysis. In our analysis, we will presume that the local rock mass of interest is intact, and thus
<br>
<br>
[[File:Vol1 page 0646 eq 002.png]]....................(13.128)
<br>
<br>
For applications that are in sandstones, numeric results can often use
<br>
<br>
[[File:Vol1 page 0646 eq 003.png]]....................(13.129)
<br>
<br>
'''Eq. 13.127''' can be rewritten to give one principal stress in terms of the other:
<br>
<br>
[[File:Vol1 page 0646 eq 004.png]]....................(13.130)
<br>
<br>
[[File:Vol1 page 0647 eq 001.png]]....................(13.131)
<br>
<br>
Such normalized stress states were used to construct the curved envelope in '''Fig. 13.54'''.
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 649 Image 0001.png|'''Fig. 13.56 – Failure envelope for sandstones after Hoek and Brown.<ref name="r112" /> Stresses are normalized by rock uniaxial compressive strength.'''
File:vol1 Page 649 Image 0001.png|'''Fig. 13.56 – Failure envelope for sandstones after Hoek and Brown.<ref name="r112" /> Stresses are normalized by rock uniaxial compressive strength.'''


File:Vol1 Page 650 Image 0001.png|'''Table 13.13'''
File:Vol1 Page 650 Image 0001.png|'''Table 13.13'''
</gallery>
</gallery><br/>The tensional strength, the stress at which an envelope would cross the horizontal axis, is found by equating ''σ''<sub>1</sub> to ''σ''<sub>3</sub> in '''Eq. 13.130''' (note that ''C''<sub>''t''</sub> is defined as a positive number).<br/><br/>[[File:Vol1 page 0647 eq 002.png|RTENOTITLE]]....................(13.132)<br/><br/>For sandstones, this results in [[File:Vol1 page 0647 inline 001.png|RTENOTITLE]], or 0.067 ''C''<sub>0</sub>. The 15 uniaxial tensional strength ''σ''<sub>''τ''</sub>* is slightly different and is defined as the value at which the maximum stress, ''σ''<sub>1</sub>, equals zero. From '''Eq. 13.130''', we get<br/><br/>[[File:Vol1 page 0647 eq 003.png|RTENOTITLE]]....................(13.133)<br/><br/>Other basic properties are not so simply derived.<br/><br/>We must produce from the stress relationships ('''Eqs. 13.127''' or '''13.130''') an equation for a failure envelope that permits us to resolve the shear and normal stresses on a failure plane, its orientation, and an approximation of the internal friction, and simply predict regions of instability. The general envelope shapes seen in '''Figs. 13.54''' and '''13.56''' suggest a form like that proposed by Murrell<ref name="r107">_</ref> and Bienlawski<ref name="r111">_</ref>:<br/><br/>[[File:Vol1 page 0648 eq 001.png|RTENOTITLE]]....................(13.134)<br/><br/>where ''A'', ''b'', and ''n'' are material constants. Because the envelope intersects the horizontal axis when the normal stress equals the tensional strength,<br/><br/>[[File:Vol1 page 0648 eq 002.png|RTENOTITLE]]....................(13.135)<br/><br/>When the normal stress is zero, the envelope intersects the vertical axis at the cohesion value ''C''<sub>''u''</sub>. From '''Eq. 13.134''', this requires<br/><br/>[[File:Vol1 page 0648 eq 003.png|RTENOTITLE]]....................(13.136)<br/><br/>Therefore, the general form for an envelope is<br/><br/>[[File:Vol1 page 0648 eq 004.png|RTENOTITLE]]....................(13.137)<br/><br/>To derive the slope, ''α'', at any point, we note that the envelope is only slowly varying over a small stress range and could be locally approximated by a line. If we use a pseudocohesion [[File:Vol1 page 0649 inline 001.png|RTENOTITLE]] defined by '''Eq. 13.123''' for the stress condition, ''σ''<sub>''m''</sub>, ''r'' we can subtract the same [[File:Vol1 page 0649 inline 001.png|RTENOTITLE]] from a slightly different stress condition, ''σ''<sub>''m’''</sub>, ''r’''. Solving for ''α'' gives<br/><br/>[[File:Vol1 page 0649 eq 001.png|RTENOTITLE]]....................(13.138)<br/><br/>The Hoek-Brown stress criteria allow us to redefine the mean, ''σ''<sub>''m''</sub>, and differential, ''r'', stresses<br/><br/>[[File:Vol1 page 0649 eq 002.png|RTENOTITLE]]....................(13.139)<br/><br/>[[File:Vol1 page 0649 eq 003.png|RTENOTITLE]]....................(13.140)<br/><br/>By substituting these relations into '''Eq. 13.138''' for two stresses ''σ''<sub>3</sub> and ''σ''<sub>3</sub> + ''δ'' ''σ''<sub>3</sub>, expanding the result and allowing the stress difference, ''δσ''<sub>3</sub>, to approach zero (what a pain!), we find<br/><br/>[[File:Vol1 page 0650 eq 001.png|RTENOTITLE]]....................(13.141)<br/><br/>As we found previously ('''Eq. 13.125'''), the normal stress is then<br/><br/>[[File:Vol1 page 0650 eq 002.png|RTENOTITLE]]....................(13.142)<br/><br/>The cohesion is the shear stress value when ''σ''<sub>''n''</sub> equals zero. This will occur for ''σ''<sub>3</sub> somewhere between zero and −''C''<sub>''t''</sub>. In other words, ''σ''<sub>''n''</sub> = 0 for<br/><br/>[[File:Vol1 page 0650 eq 003.png|RTENOTITLE]]....................(13.143)<br/><br/>where ''β'' is a value around 0.5. We could substitute this term into '''Eqs. 13.141''' and '''13.142''' and solve for ''β'' . However, this results in a rather complicated root to a third-order polynomial. Fortunately, by iteration, we can show that ''β'' is relatively constant at about 0.62 with little dependence on ''m''. Using this value of ''β'' in '''Eq. 13.143''' and substituting into the previous equations gives us our cohesion. For a sandstone with ''m'' = 15, we get<br/><br/>[[File:Vol1 page 0650 eq 004.png|RTENOTITLE]]....................(13.144)<br/><br/>The definition of our curved envelope in '''Eq. 13.137''' is not strictly compatible with the Hoek-Brown stress relations. However, we can get an estimate of the exponent, ''n'', by using our tensile and cohesion strengths and some reasonable value of ''σ''<sub>''n''</sub> such as ''σ''<sub>''n''</sub> = ''C''<sub>0</sub>. From '''Fig. 13.54''', we can see that ''τ'' is approximately 1.1 ''C''<sub>0</sub> at this point. From '''Eq. 13.137''', with ''m'' equal to 15,<br/><br/>[[File:Vol1 page 0651 eq 001.png|RTENOTITLE]]....................(13.145)<br/><br/>This value falls within the range of 0.65 to 0.75 suggested by Yudhbir ''et al''.<ref name="r113">_</ref> Thus, from a presumed simple relation between ''σ''<sub>1</sub> and ''σ''<sub>3</sub>, almost all the necessary parameters can be derived.
<br>
The tensional strength, the stress at which an envelope would cross the horizontal axis, is found by equating ''σ''<sub>1</sub> to ''σ''<sub>3</sub> in '''Eq. 13.130''' (note that ''C''<sub>''t''</sub> is defined as a positive number).
<br>
<br>
[[File:Vol1 page 0647 eq 002.png]]....................(13.132)
<br>
<br>
For sandstones, this results in [[File:Vol1 page 0647 inline 001.png]], or 0.067 ''C''<sub>0</sub>. The 15 uniaxial tensional strength ''σ''<sub>''τ''</sub>* is slightly different and is defined as the value at which the maximum stress, ''σ''<sub>1</sub>, equals zero. From '''Eq. 13.130''', we get
<br>
<br>
[[File:Vol1 page 0647 eq 003.png]]....................(13.133)
<br>
<br>
Other basic properties are not so simply derived.  
<br>
<br>
We must produce from the stress relationships ('''Eqs. 13.127''' or '''13.130''') an equation for a failure envelope that permits us to resolve the shear and normal stresses on a failure plane, its orientation, and an approximation of the internal friction, and simply predict regions of instability. The general envelope shapes seen in '''Figs. 13.54''' and '''13.56''' suggest a form like that proposed by Murrell<ref name="r107" /> and Bienlawski<ref name="r111" />:
<br>
<br>
[[File:Vol1 page 0648 eq 001.png]]....................(13.134)
<br>
<br>
where ''A'', ''b'', and ''n'' are material constants. Because the envelope intersects the horizontal axis when the normal stress equals the tensional strength,
<br>
<br>
[[File:Vol1 page 0648 eq 002.png]]....................(13.135)
<br>
<br>
When the normal stress is zero, the envelope intersects the vertical axis at the cohesion value ''C''<sub>''u''</sub>. From '''Eq. 13.134''', this requires
<br>
<br>
[[File:Vol1 page 0648 eq 003.png]]....................(13.136)
<br>
<br>
Therefore, the general form for an envelope is
<br>
<br>
[[File:Vol1 page 0648 eq 004.png]]....................(13.137)
<br>
<br>
To derive the slope, ''α'', at any point, we note that the envelope is only slowly varying over a small stress range and could be locally approximated by a line. If we use a pseudocohesion [[File:Vol1 page 0649 inline 001.png]] defined by '''Eq. 13.123''' for the stress condition, ''σ''<sub>''m''</sub>, ''r'' we can subtract the same [[File:Vol1 page 0649 inline 001.png]] from a slightly different stress condition, ''σ''<sub>''m’''</sub>, ''r’''. Solving for ''α'' gives
<br>
<br>
[[File:Vol1 page 0649 eq 001.png]]....................(13.138)
<br>
<br>
The Hoek-Brown stress criteria allow us to redefine the mean, ''σ''<sub>''m''</sub>, and differential, ''r'', stresses
<br>
<br>
[[File:Vol1 page 0649 eq 002.png]]....................(13.139)
<br>
<br>
[[File:Vol1 page 0649 eq 003.png]]....................(13.140)
<br>
<br>
By substituting these relations into '''Eq. 13.138''' for two stresses ''σ''<sub>3</sub> and ''σ''<sub>3</sub> + ''δ'' ''σ''<sub>3</sub>, expanding the result and allowing the stress difference, ''δσ''<sub>3</sub>, to approach zero (what a pain!), we find
<br>
<br>
[[File:Vol1 page 0650 eq 001.png]]....................(13.141)
<br>
<br>
As we found previously ('''Eq. 13.125'''), the normal stress is then
<br>
<br>
[[File:Vol1 page 0650 eq 002.png]]....................(13.142)
<br>
<br>
The cohesion is the shear stress value when ''σ''<sub>''n''</sub> equals zero. This will occur for ''σ''<sub>3</sub> somewhere between zero and −''C''<sub>''t''</sub>. In other words, ''σ''<sub>''n''</sub> = 0 for
<br>
<br>
[[File:Vol1 page 0650 eq 003.png]]....................(13.143)
<br>
<br>
where ''β'' is a value around 0.5. We could substitute this term into '''Eqs. 13.141''' and '''13.142''' and solve for ''β'' . However, this results in a rather complicated root to a third-order polynomial. Fortunately, by iteration, we can show that ''β'' is relatively constant at about 0.62 with little dependence on ''m''. Using this value of ''β'' in '''Eq. 13.143''' and substituting into the previous equations gives us our cohesion. For a sandstone with ''m'' = 15, we get
<br>
<br>
[[File:Vol1 page 0650 eq 004.png]]....................(13.144)
<br>
<br>
The definition of our curved envelope in '''Eq. 13.137''' is not strictly compatible with the Hoek-Brown stress relations. However, we can get an estimate of the exponent, ''n'', by using our tensile and cohesion strengths and some reasonable value of ''σ''<sub>''n''</sub> such as ''σ''<sub>''n''</sub> = ''C''<sub>0</sub>. From '''Fig. 13.54''', we can see that ''τ'' is approximately 1.1 ''C''<sub>0</sub> at this point. From '''Eq. 13.137''', with ''m'' equal to 15,
<br>
<br>
[[File:Vol1 page 0651 eq 001.png]]....................(13.145)
<br>
<br>
This value falls within the range of 0.65 to 0.75 suggested by Yudhbir ''et al''.<ref name="r113" /> Thus, from a presumed simple relation between ''σ''<sub>1</sub> and ''σ''<sub>3</sub>, almost all the necessary parameters can be derived.


=== Uniaxial Compressive Strength ===
=== Uniaxial Compressive Strength ===


We have seen how a general rock failure criterion can be reduced to a few parameters dependent on lithology (''m'') and the uniaxial compressive strength (''C''<sub>0</sub>). Lithology is commonly derived during log analysis, so ''m'' may be estimated ('''Table 13.13'''). What is needed still is an initial measure of rock strength provided by ''C''<sub>0</sub>. ''C''<sub>0</sub> can be estimated from porosity or sonic velocities, but many factors such as grain size, clay content, or saturation have significant influences.  
We have seen how a general rock failure criterion can be reduced to a few parameters dependent on lithology (''m'') and the uniaxial compressive strength (''C''<sub>0</sub>). Lithology is commonly derived during log analysis, so ''m'' may be estimated ('''Table 13.13'''). What is needed still is an initial measure of rock strength provided by ''C''<sub>0</sub>. ''C''<sub>0</sub> can be estimated from porosity or sonic velocities, but many factors such as grain size, clay content, or saturation have significant influences.<br/><br/>A large amount of ''C''<sub>0</sub> data is available and, although there is considerable scatter, ''C''<sub>0</sub> usually varies systematically with other rock characteristics. We will concentrate on porosity as the primary controlling factor because it is routinely available from logs and is a fundamental input into reservoir simulators.<br/><br/>Numerous relationships have been developed to estimate ''C''<sub>0</sub>, often in conjunction with general rock strength relationships. '''Table 13.14''' lists many of the proposed relations for ''C''<sub>0</sub>, some of which are plotted for various rock types in '''Fig. 13.57''' and for sandstones in '''Fig. 13.58'''. We expect ''C''<sub>0</sub> to decrease as porosity increases. At some transition porosity, rocks will lose all initial strength and become merely a loose aggregate. No matter which relationship is chosen, variables such as cementation, alteration, texture, and so on can cause significant scatter.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
A large amount of ''C''<sub>0</sub> data is available and, although there is considerable scatter, ''C''<sub>0</sub> usually varies systematically with other rock characteristics. We will concentrate on porosity as the primary controlling factor because it is routinely available from logs and is a fundamental input into reservoir simulators.  
<br>
<br>
Numerous relationships have been developed to estimate ''C''<sub>0</sub>, often in conjunction with general rock strength relationships. '''Table 13.14''' lists many of the proposed relations for ''C''<sub>0</sub>, some of which are plotted for various rock types in '''Fig. 13.57''' and for sandstones in '''Fig. 13.58'''. We expect ''C''<sub>0</sub> to decrease as porosity increases. At some transition porosity, rocks will lose all initial strength and become merely a loose aggregate. No matter which relationship is chosen, variables such as cementation, alteration, texture, and so on can cause significant scatter.
<br>
<br>
<gallery widths=300px heights=200px>
File:Vol1 Page 653 Image 0001.png|'''Table 13.14'''<ref name="r114" /><ref name="r115" /><ref name="r116" /><ref name="r117" /><ref name="r118" />
File:Vol1 Page 653 Image 0001.png|'''Table 13.14'''<ref name="r114" /><ref name="r115" /><ref name="r116" /><ref name="r117" /><ref name="r118" />


Line 1,687: Line 388:


File:vol1 Page 652 Image 0001.png|'''Fig. 13.58 – Uniaxial compressive strength (''C<sub>o</sub>'') relations for sandstones.'''
File:vol1 Page 652 Image 0001.png|'''Fig. 13.58 – Uniaxial compressive strength (''C<sub>o</sub>'') relations for sandstones.'''
</gallery>
</gallery><br/>If we accept the restrictive relationships for failure of '''Eq. 13.130''' or '''13.134''', we can derive ''C''<sub>0</sub> from any such strength data:<br/><br/>[[File:Vol1 page 0652 eq 001.png|RTENOTITLE]]....................(13.146)<br/><br/>However, this equation predicts a finite strength even as porosity approaches 1.0. More realistic forms must be used so that strength vanishes at some porosity ''Φ''<sub>''c''</sub>. This limiting porosity was shown as a crossover porosity from rock to a slurry by Raymer ''et al''.<ref name="r119">_</ref> and was referred to as "critical porosity" elsewhere. Jizba<ref name="r108">_</ref> used such a concept to derive a general strength relationship for sandstones:<br/><br/>[[File:Vol1 page 0652 eq 002.png|RTENOTITLE]]....................(13.147)<br/><br/>where ''τ'' and ''σ''<sub>''n''</sub> are the shear and normal stresses at failure.<br/><br/>The 0.36 within the parentheses is her presumed value for ''Φ''<sub>''c''</sub>. Notice, however, that this form indicates that sandstones have no tensile or cohesive strength. We can obtain a better result by using Jizba’s relationship at elevated confining pressure (say, 50 MPa), where it is more valid, and recasting the trend in terms of '''Eq. 13.130''', as we did for the Scott relation.<ref name="r105">_</ref> Dobereiner and DeFreitas<ref name="r121">_</ref> measured several weak sandstones, and their results suggest that critical porosity is approximately 0.42. Using this critical porosity, we derive a uniaxial compressive strength<br/><br/>[[File:Vol1 page 0653 eq 001.png|RTENOTITLE]]....................(13.148)<br/><br/>This ''C''<sub>0</sub> equation is plotted in '''Fig. 13.58''' along with the modified Scott<ref name="r110">_</ref> and Jizba<ref name="r108">_</ref> equations and data of Dobereiner and DeFreitas.<ref name="r121">_</ref>
<br/>
If we accept the restrictive relationships for failure of '''Eq. 13.130''' or '''13.134''', we can derive ''C''<sub>0</sub> from any such strength data:
<br>
<br>
[[File:Vol1 page 0652 eq 001.png]]....................(13.146)
<br>
<br>
However, this equation predicts a finite strength even as porosity approaches 1.0. More realistic forms must be used so that strength vanishes at some porosity ''Φ''<sub>''c''</sub>. This limiting porosity was shown as a crossover porosity from rock to a slurry by Raymer ''et al''.<ref name="r119" /> and was referred to as "critical porosity" elsewhere. Jizba<ref name="r108" /> used such a concept to derive a general strength relationship for sandstones:
<br>
<br>
[[File:Vol1 page 0652 eq 002.png]]....................(13.147)
<br>
<br>
where ''τ'' and ''σ''<sub>''n''</sub> are the shear and normal stresses at failure.  
<br>
<br>
The 0.36 within the parentheses is her presumed value for ''Φ''<sub>''c''</sub>. Notice, however, that this form indicates that sandstones have no tensile or cohesive strength. We can obtain a better result by using Jizba’s relationship at elevated confining pressure (say, 50 MPa), where it is more valid, and recasting the trend in terms of '''Eq. 13.130''', as we did for the Scott relation.<ref name="r105" /> Dobereiner and DeFreitas<ref name="r121" /> measured several weak sandstones, and their results suggest that critical porosity is approximately 0.42. Using this critical porosity, we derive a uniaxial compressive strength
<br>
<br>
[[File:Vol1 page 0653 eq 001.png]]....................(13.148)
<br>
<br>
This ''C''<sub>0</sub> equation is plotted in '''Fig. 13.58''' along with the modified Scott<ref name="r110" /> and Jizba<ref name="r108" /> equations and data of Dobereiner and DeFreitas.<ref name="r121" />


=== Compaction Strength ===
=== Compaction Strength ===


As was indicated in '''Fig. 13.55''', at some elevated stress or confining pressure, the rock will begin to show ductile deformation. The grain structure begins to collapse, and the rock will compact and lose porosity. This compaction strength, ''C''<sub>''c''</sub>, is itself a function of porosity as well as mineralogy, diagenesis, and texture. In '''Figs. 13.59a and 13.59b''', the behavior of two rocks under hydrostatic pressure is shown. The high-porosity (33%) sandstone ('''Fig. 13.59a''') has a low "crush" strength of about 55 MPa. With a lower porosity of 19%, Berea sandstone has a much higher strength of 440 MPa ('''Fig. 13.59b'''). Notice that in both '''Figs. 13.59a''' and '''59b''', permanent deformation remains even after the stress is released. This hysteresis demonstrates the damage to the matrix structure caused by exceeding the crush strength.  
As was indicated in '''Fig. 13.55''', at some elevated stress or confining pressure, the rock will begin to show ductile deformation. The grain structure begins to collapse, and the rock will compact and lose porosity. This compaction strength, ''C''<sub>''c''</sub>, is itself a function of porosity as well as mineralogy, diagenesis, and texture. In '''Figs. 13.59a and 13.59b''', the behavior of two rocks under hydrostatic pressure is shown. The high-porosity (33%) sandstone ('''Fig. 13.59a''') has a low "crush" strength of about 55 MPa. With a lower porosity of 19%, Berea sandstone has a much higher strength of 440 MPa ('''Fig. 13.59b'''). Notice that in both '''Figs. 13.59a''' and '''59b''', permanent deformation remains even after the stress is released. This hysteresis demonstrates the damage to the matrix structure caused by exceeding the crush strength.<br/><br/><gallery widths="300px" heights="200px">
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 654 Image 0001.png|'''Fig. 13.59 – (a) Hydrostatic compaction of high-porosity Gulf of Mexico sandstone (T.E. Scott, personal communication); (b) hydrostatic compaction of Berea sandstone (T.E. Scott, personal communication).'''
File:vol1 Page 654 Image 0001.png|'''Fig. 13.59 – (a) Hydrostatic compaction of high-porosity Gulf of Mexico sandstone (T.E. Scott, personal communication); (b) hydrostatic compaction of Berea sandstone (T.E. Scott, personal communication).'''
</gallery>
</gallery><br/>In the cases in which studies are restricted to sandstones, an exponential dependence on porosity is usually observed ('''Fig. 13.59a'''). Scott<ref name="r110">_</ref> fit his and the Dunn ''et al''.<ref name="r106">_</ref> data to the form<br/><br/>[[File:Vol1 page 0653 eq 002.png|RTENOTITLE]]....................(13.149)<br/><br/>With a general relationship available for uniaxial compressive strength and the compaction limit, rock failure envelopes can be determined for sandstones at any porosity. '''Fig. 13.60''' shows the complete envelopes for the porosity range 0.15 to 0.35.<br/><br/><gallery widths="300px" heights="200px">
<br/>
In the cases in which studies are restricted to sandstones, an exponential dependence on porosity is usually observed ('''Fig. 13.59a'''). Scott<ref name="r110" /> fit his and the Dunn ''et al''.<ref name="r106" /> data to the form
<br>
<br>
[[File:Vol1 page 0653 eq 002.png]]....................(13.149)
<br>
<br>
With a general relationship available for uniaxial compressive strength and the compaction limit, rock failure envelopes can be determined for sandstones at any porosity. '''Fig. 13.60''' shows the complete envelopes for the porosity range 0.15 to 0.35.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 655 Image 0001.png|'''Fig. 13.60 – Generalized failure envelope and Roscoe (crushing) surface at different porosities. Hoek-Brown parameter ''m'' = 15 used.'''
File:vol1 Page 655 Image 0001.png|'''Fig. 13.60 – Generalized failure envelope and Roscoe (crushing) surface at different porosities. Hoek-Brown parameter ''m'' = 15 used.'''
</gallery>
</gallery>
<br/>


=== Clay Content ===
=== Clay Content ===


Most sandstones are mixtures of mineral such as feldspars, calcite, dolomite, micas, clays, and of course quartz. Clays are a very common component and can make up anywhere from 0 to nearly 100% of a clastic rock. Usually, at some point greater than 40% clay, the rock is considered a shale or mudstone rather than a sandstone (refer to '''Section 13.7'''). The structure of clay minerals and their typical methods of bonding are significantly different from those of quartz, so we would expect clays to strongly influence mechanical properties. Such influences depend on the nature of the clay, the amount and location within the rock framework, and the state of hydration.  
Most sandstones are mixtures of mineral such as feldspars, calcite, dolomite, micas, clays, and of course quartz. Clays are a very common component and can make up anywhere from 0 to nearly 100% of a clastic rock. Usually, at some point greater than 40% clay, the rock is considered a shale or mudstone rather than a sandstone (refer to '''Section 13.7'''). The structure of clay minerals and their typical methods of bonding are significantly different from those of quartz, so we would expect clays to strongly influence mechanical properties. Such influences depend on the nature of the clay, the amount and location within the rock framework, and the state of hydration.<br/><br/>There have been few systematic studies of clay effects on the mechanical properties of rocks. Corbett ''et al''.<ref name="r120">_</ref> demonstrated how the coefficient of internal friction and thus the strength of Austin chalk strongly depends on even a small clay fraction ('''Fig. 13.61'''). In particular, smectite content was found to have more influence in this case than other clays. This allows us to derive a general relationship between failure and clay content.<br/><br/>[[File:Vol1 page 0654 eq 001.png|RTENOTITLE]]....................(13.150)<br/><br/>where ''C'' is the smectite fraction. Unfortunately, this equation was developed for dry samples.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
There have been few systematic studies of clay effects on the mechanical properties of rocks. Corbett ''et al''.<ref name="r120" /> demonstrated how the coefficient of internal friction and thus the strength of Austin chalk strongly depends on even a small clay fraction ('''Fig. 13.61'''). In particular, smectite content was found to have more influence in this case than other clays. This allows us to derive a general relationship between failure and clay content.
<br>
<br>
[[File:Vol1 page 0654 eq 001.png]]....................(13.150)
<br>
<br>
where ''C'' is the smectite fraction. Unfortunately, this equation was developed for dry samples.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 656 Image 0001.png|'''Fig. 13.61 – Effect of smectite content on the coefficient of internal friction of Austin Chalk (from Corbett ''et al''.<ref name="r120" />).'''
File:vol1 Page 656 Image 0001.png|'''Fig. 13.61 – Effect of smectite content on the coefficient of internal friction of Austin Chalk (from Corbett ''et al''.<ref name="r120" />).'''
</gallery>
</gallery><br/>Jizba<ref name="r114">_</ref> tested several dry clay-rich samples and proposed a general linear envelope form for shales and shaley sandstones.<br/><br/>[[File:Vol1 page 0654 eq 002.png|RTENOTITLE]]....................(13.151)<br/><br/>More relevant data, however, comes from Steiger and Leung<ref name="r122">_</ref> with both dry and saturated shale measurements ('''Fig. 13.62'''). From these data, we derive an approximation for the wet shale uniaxial compressional strength.<br/><br/>[[File:Vol1 page 0654 eq 003.png|RTENOTITLE]]....................(13.152)<br/><br/>This relation, as well as those for the Austin chalk, suggests a strong clay dependence. Jizba,<ref name="r108">_</ref> however, reported only a slight dependence of ''C''<sub>0</sub> on clay content in shaley sands.<br/><br/><gallery widths="300px" heights="200px">
<br/>
Jizba<ref name="r114" /> tested several dry clay-rich samples and proposed a general linear envelope form for shales and shaley sandstones.
<br>
<br>
[[File:Vol1 page 0654 eq 002.png]]....................(13.151)
<br>
<br>
More relevant data, however, comes from Steiger and Leung<ref name="r122" /> with both dry and saturated shale measurements ('''Fig. 13.62'''). From these data, we derive an approximation for the wet shale uniaxial compressional strength.
<br>
<br>
[[File:Vol1 page 0654 eq 003.png]]....................(13.152)
<br>
<br>
This relation, as well as those for the Austin chalk, suggests a strong clay dependence. Jizba,<ref name="r108" /> however, reported only a slight dependence of ''C''<sub>0</sub> on clay content in shaley sands.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 657 Image 0001.png|'''Fig. 13.62 – Effect of increasing clay content on strength.'''
File:vol1 Page 657 Image 0001.png|'''Fig. 13.62 – Effect of increasing clay content on strength.'''
</gallery>
</gallery><br/>It is likely that in many sands, clays reside as pore-filling materials and have only a secondary effect on mechanical properties. At this point, we expect clays to have a significant effect even in fairly pure sands (this will be seen also in sonic velocity measurements). Thus, a more general form for uniaxial compressive strength of sandstones would be<br/><br/>[[File:Vol1 page 0655 eq 001.png|RTENOTITLE]]....................(13.153)<br/><br/>where the coefficient ''a'' has a value of approximately 100. The influence of clays on the mechanical properties of rocks needs much further investigation.
<br/>
It is likely that in many sands, clays reside as pore-filling materials and have only a secondary effect on mechanical properties. At this point, we expect clays to have a significant effect even in fairly pure sands (this will be seen also in sonic velocity measurements). Thus, a more general form for uniaxial compressive strength of sandstones would be
<br>
<br>
[[File:Vol1 page 0655 eq 001.png]]....................(13.153)
<br>
<br>
where the coefficient ''a'' has a value of approximately 100. The influence of clays on the mechanical properties of rocks needs much further investigation.


=== Pore Fluid Effects ===
=== Pore Fluid Effects ===


Fluids can alter rock mechanical properties of rocks through fluid pressure, chemical reactions with mineral surfaces, and by lubricating moving surfaces. The primary fluids encountered are brines and hydrocarbon oils and gases. Drilling, completion, and fracturing fluids can also be present, and their effects are typically studied to prevent formation damage. We will concentrate on the role of water and, in particular, how water saturation can influence rock strengths measured in the laboratory or derived from well logs.  
Fluids can alter rock mechanical properties of rocks through fluid pressure, chemical reactions with mineral surfaces, and by lubricating moving surfaces. The primary fluids encountered are brines and hydrocarbon oils and gases. Drilling, completion, and fracturing fluids can also be present, and their effects are typically studied to prevent formation damage. We will concentrate on the role of water and, in particular, how water saturation can influence rock strengths measured in the laboratory or derived from well logs.<br/><br/>'''''Effective Stress.''''' Pore fluid pressures will reduce the effective stress supported by the rock mineral frame. This effect has been well known since the publication of Terzaghi and Peck<ref name="r63">_</ref> and has been documented by numerous investigators. The most common form for the effective stress law is<br/><br/>[[File:Vol1 page 0655 eq 002.png|RTENOTITLE]]....................(13.154)<br/><br/>where ''σ''<sub>''e''</sub> is the effective stress, ''σ''<sub>''a''</sub> the applied stress on the rock surface, ''P''<sub>''p''</sub>, and the pore pressure. Note that this is the same as '''Eq. 13.35'''. The effective stress coefficient ''n'' is also called Biot’s poroelastic term.<br/><br/>[[File:Vol1 page 0656 eq 001.png|RTENOTITLE]]....................(13.155)<br/><br/>where ''K''<sub>''d''</sub> is the dry rock bulk modulus and ''K''<sub>''o''</sub> the mineral bulk modulus. Because the rock modulus is usually much lower than the mineral modulus, ''n'' is often close to unity. In many applications and when no other information is available, ''n'' is usually taken as one.<br/><br/>In our analyses, all of the stresses used to describe rock failure were actually effective stresses. Rock failure can be dramatically affected by pore pressure, as indicated in '''Fig. 13.63'''. An envelope is plotted for a sandstone with porosity of 25%. For applied principal stresses of 225 MPa for ''σ''<sub>1</sub>, 175 MPa for ''σ''<sub>3</sub>, and a ''P''<sub>''p''</sub> of 75 MPa, the effective Mohr circle plots well within the field of stability. The pore pressure has been subtracted from both applied stresses to give effective principal stresses of 150 and 100 MPa. If pore pressure is increased, the effective stresses decrease, and the Mohr circle is shifted left until eventually the envelope may be contacted and the rock fails by brittle fracture. On the other hand, if pore pressure decreases, the Mohr circle shifts right, and the rock may contract the Roscoe surface and fail by compaction or grain crushing. In any case, if pore pressures are known, their effects can be accounted for in a straightforward way.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
'''''Effective Stress.''''' Pore fluid pressures will reduce the effective stress supported by the rock mineral frame. This effect has been well known since the publication of Terzaghi and Peck<ref name="r63" /> and has been documented by numerous investigators. The most common form for the effective stress law is
<br>
<br>
[[File:Vol1 page 0655 eq 002.png]]....................(13.154)
<br>
<br>
where ''σ''<sub>''e''</sub> is the effective stress, ''σ''<sub>''a''</sub> the applied stress on the rock surface, ''P''<sub>''p''</sub>, and the pore pressure. Note that this is the same as '''Eq. 13.35'''. The effective stress coefficient ''n'' is also called Biot’s poroelastic term.
<br>
<br>
[[File:Vol1 page 0656 eq 001.png]]....................(13.155)
<br>
<br>
where ''K''<sub>''d''</sub> is the dry rock bulk modulus and ''K''<sub>''o''</sub> the mineral bulk modulus. Because the rock modulus is usually much lower than the mineral modulus, ''n'' is often close to unity. In many applications and when no other information is available, ''n'' is usually taken as one.  
<br>
<br>
In our analyses, all of the stresses used to describe rock failure were actually effective stresses. Rock failure can be dramatically affected by pore pressure, as indicated in '''Fig. 13.63'''. An envelope is plotted for a sandstone with porosity of 25%. For applied principal stresses of 225 MPa for ''σ''<sub>1</sub>, 175 MPa for ''σ''<sub>3</sub>, and a ''P''<sub>''p''</sub> of 75 MPa, the effective Mohr circle plots well within the field of stability. The pore pressure has been subtracted from both applied stresses to give effective principal stresses of 150 and 100 MPa. If pore pressure is increased, the effective stresses decrease, and the Mohr circle is shifted left until eventually the envelope may be contacted and the rock fails by brittle fracture. On the other hand, if pore pressure decreases, the Mohr circle shifts right, and the rock may contract the Roscoe surface and fail by compaction or grain crushing. In any case, if pore pressures are known, their effects can be accounted for in a straightforward way.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 657 Image 0002.png|'''Fig. 13.63 – Rock failure caused by either increasing or decreasing pore pressure. Increased pore pressure decreases effective stress, leading to fracture failure. Decreasing pore pressure increases effective stress, which can produce crushing.'''
File:vol1 Page 657 Image 0002.png|'''Fig. 13.63 – Rock failure caused by either increasing or decreasing pore pressure. Increased pore pressure decreases effective stress, leading to fracture failure. Decreasing pore pressure increases effective stress, which can produce crushing.'''
</gallery>
</gallery><br/>Problems can arise experimentally because of the inability of pore pressure to reach equilibrium. If fluid can flow freely and constant pore pressure is maintained, then an experiment is termed "drained." If deformation is too rapid, permeability low, fluid viscosity high, or boundaries are sealed, then fluid is trapped in the rock, and fluid pressure changes as a function of rock deformation. Brace and Martin<ref name="r123">_</ref> showed that strain rates must be very low in crystalline rocks of low permeability to maintain a uniform pore pressure and follow a macroscopically defined effective stress law such as '''Eq. 13.154'''. For most sandstones, permeability is sufficient to provide drained conditions. Problems usually occur in low-permeability rocks such as siltstone or shales. Considerable effort and time are usually needed to allow constant pore pressure, or merely to maintain pore pressure equilibrium (Steiger and Leung<ref name="r122">_</ref>). Tests are made under undrained conditions, but the resulting changes in pore pressure must then be measured or otherwise calculated. These effects are mechanical problems that are often difficult to deal with, but the processes are basically well understood.<br/><br/>'''''Chemical Effects.''''' A more subtle problem involves chemical effects of pore fluids. Water is an active, polar compound, and numerous investigations (Griggs<ref name="r124">_</ref> and Kirby<ref name="r125">_</ref>) have shown that even small amounts of water or brine can have a substantial influence on rock mechanical properties. Colback and Wiid<ref name="r126">_</ref> demonstrated how even changes in the relative humidity or partial pressure of water in the pores can lower rock strength dramatically ('''Fig. 13.64'''). Colback and Wiid<ref name="r126">_</ref> and Dunning and Huff<ref name="r127">_</ref> saw a direct relationship between the loss in rock strength and the chemical activity of the pore fluid. Meredith and Atkinson,<ref name="r128">_</ref> Freeman,<ref name="r129">_</ref> and others have shown increased crack velocities and acoustic emissions at constant crack intensity factors when water is introduced. Ujtai ''et al''.<ref name="r130">_</ref> saw substantial effects of water on all time-dependent tests for creep strain, fatigue, and slow crack growth. In general, uniaxial compressive strength is reduced by 20 to 25% in wet rocks. This implies that many laboratory measurements result in rock strengths that are systematically too high.<br/><br/><gallery widths="300px" heights="200px">
<br/>
Problems can arise experimentally because of the inability of pore pressure to reach equilibrium. If fluid can flow freely and constant pore pressure is maintained, then an experiment is termed "drained." If deformation is too rapid, permeability low, fluid viscosity high, or boundaries are sealed, then fluid is trapped in the rock, and fluid pressure changes as a function of rock deformation. Brace and Martin<ref name="r123" /> showed that strain rates must be very low in crystalline rocks of low permeability to maintain a uniform pore pressure and follow a macroscopically defined effective stress law such as '''Eq. 13.154'''. For most sandstones, permeability is sufficient to provide drained conditions. Problems usually occur in low-permeability rocks such as siltstone or shales. Considerable effort and time are usually needed to allow constant pore pressure, or merely to maintain pore pressure equilibrium (Steiger and Leung<ref name="r122" />). Tests are made under undrained conditions, but the resulting changes in pore pressure must then be measured or otherwise calculated. These effects are mechanical problems that are often difficult to deal with, but the processes are basically well understood.  
<br>
<br>
'''''Chemical Effects.''''' A more subtle problem involves chemical effects of pore fluids. Water is an active, polar compound, and numerous investigations (Griggs<ref name="r124" /> and Kirby<ref name="r125" />) have shown that even small amounts of water or brine can have a substantial influence on rock mechanical properties. Colback and Wiid<ref name="r126" /> demonstrated how even changes in the relative humidity or partial pressure of water in the pores can lower rock strength dramatically ('''Fig. 13.64'''). Colback and Wiid<ref name="r126" /> and Dunning and Huff<ref name="r127" /> saw a direct relationship between the loss in rock strength and the chemical activity of the pore fluid. Meredith and Atkinson,<ref name="r128" /> Freeman,<ref name="r129" /> and others have shown increased crack velocities and acoustic emissions at constant crack intensity factors when water is introduced. Ujtai ''et al''.<ref name="r130" /> saw substantial effects of water on all time-dependent tests for creep strain, fatigue, and slow crack growth. In general, uniaxial compressive strength is reduced by 20 to 25% in wet rocks. This implies that many laboratory measurements result in rock strengths that are systematically too high.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 658 Image 0001.png|'''Fig. 13.64 – Rock strength as a function of water content or humidity. Strength drops rapidly with the first few monolayers of water (from Colback and Wiid<ref name="r125" />).'''
File:vol1 Page 658 Image 0001.png|'''Fig. 13.64 – Rock strength as a function of water content or humidity. Strength drops rapidly with the first few monolayers of water (from Colback and Wiid<ref name="r125" />).'''
</gallery>
</gallery><br/>A strong influence of the chemical activity on rock mechanical properties is supported by other types of measurements. Seismic properties depend upon mineral grain stiffness and the stiffness of grain-to-grain contacts. In completely dry rocks (oven-dried under vacuum), there is almost no seismic attenuation, and rocks are stiff. Even small amounts of water, a few monolayers, can appreciably lower rock stiffness and seismic velocities.<br/><br/>'''''Bulk Lubrication.''''' Common experience leads us to expect many geologic materials, such as soils, to be substantially weaker when wet. We have already seen this effect in chalk and shales. Surface bonding energies and water surface tension result in strong capillary forces that draw and hold water in pore spaces. Water penetrates and separates grains. Grain movement is facilitated by motion in mobile fluid layers. This is a highly scientific way of saying "slippery when wet." Clay minerals in particular are well known for their ability to absorb large quantities of water. Swelling properties of clays and shales are often studied for drilling engineering purposes. Not only do clays have lower friction surfaces when wet, but water absorption and the resulting clay expansion can disaggregate the rock matrix. Loss of strength because of such mechanisms is more important in poorly consolidated or unconsolidated sediments. Dobereiner and DeFreitas<ref name="r121">_</ref> and Morgenstern ''et al''.<ref name="r131">_</ref> report a 60% reduction in strength for muddy sediments upon saturation. At this point, we have not developed a systematic way of including a lubrication factor except as an implicit part of the clay corrections mentioned previously or as a measured reduction of the shear or Young’s modulus. We would expect the loss of intergrain friction to reduce the shear modulus significantly.
<br/>
A strong influence of the chemical activity on rock mechanical properties is supported by other types of measurements. Seismic properties depend upon mineral grain stiffness and the stiffness of grain-to-grain contacts. In completely dry rocks (oven-dried under vacuum), there is almost no seismic attenuation, and rocks are stiff. Even small amounts of water, a few monolayers, can appreciably lower rock stiffness and seismic velocities.  
<br>
<br>
'''''Bulk Lubrication.''''' Common experience leads us to expect many geologic materials, such as soils, to be substantially weaker when wet. We have already seen this effect in chalk and shales. Surface bonding energies and water surface tension result in strong capillary forces that draw and hold water in pore spaces. Water penetrates and separates grains. Grain movement is facilitated by motion in mobile fluid layers. This is a highly scientific way of saying "slippery when wet." Clay minerals in particular are well known for their ability to absorb large quantities of water. Swelling properties of clays and shales are often studied for drilling engineering purposes. Not only do clays have lower friction surfaces when wet, but water absorption and the resulting clay expansion can disaggregate the rock matrix. Loss of strength because of such mechanisms is more important in poorly consolidated or unconsolidated sediments. Dobereiner and DeFreitas<ref name="r121" /> and Morgenstern ''et al''.<ref name="r131" /> report a 60% reduction in strength for muddy sediments upon saturation. At this point, we have not developed a systematic way of including a lubrication factor except as an implicit part of the clay corrections mentioned previously or as a measured reduction of the shear or Young’s modulus. We would expect the loss of intergrain friction to reduce the shear modulus significantly.  


=== Grain Size and Texture ===
=== Grain Size and Texture ===


In granular rocks, grain size also influences strength. For constant porosity, mineralogy, and texture, a smaller grain size means greater strength. This tendency has been observed in several sandstones and can be understood in terms of grain contact models. Nelson<ref name="r132" /> presents data on Navajo sandstone strength indicating a strong dependence on grain size. If a rock can be considered an aggregate of uniform spheres, smaller spheres will have more grain contacts per unit volume. Loads are distributed over more contracts, and each grain experiences lower stresses. Zhang<ref name="r133" /> used Hertzian contact theory to calculate critical crushing strengths of quartz sands and found that porosity and grain radius combine to determine strength ('''Fig. 13.65'''). By fixing grain size, Zhang’s relationships could also provide crushing or compaction limits (Roscoe surfaces, '''Fig. 13.60''') for sands at various porosities. For a grain size of 0.2 mm, we get a crushing strength, ''C''<sub>''c''</sub>, of
In granular rocks, grain size also influences strength. For constant porosity, mineralogy, and texture, a smaller grain size means greater strength. This tendency has been observed in several sandstones and can be understood in terms of grain contact models. Nelson<ref name="r132">_</ref> presents data on Navajo sandstone strength indicating a strong dependence on grain size. If a rock can be considered an aggregate of uniform spheres, smaller spheres will have more grain contacts per unit volume. Loads are distributed over more contracts, and each grain experiences lower stresses. Zhang<ref name="r133">_</ref> used Hertzian contact theory to calculate critical crushing strengths of quartz sands and found that porosity and grain radius combine to determine strength ('''Fig. 13.65'''). By fixing grain size, Zhang’s relationships could also provide crushing or compaction limits (Roscoe surfaces, '''Fig. 13.60''') for sands at various porosities. For a grain size of 0.2 mm, we get a crushing strength, ''C''<sub>''c''</sub>, of<br/><br/>[[File:Vol1 page 0661 eq 001.png|RTENOTITLE]]....................(13.156)<br/><br/>However, factors such as cementation and grain angularity will strongly alter this simple relationship.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
[[File:Vol1 page 0661 eq 001.png]]....................(13.156)
<br>
<br>
However, factors such as cementation and grain angularity will strongly alter this simple relationship.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 659 Image 0001.png|'''Fig. 13.65 – Critical or crushing pressure as a function of the grain size-grain radius product (after Zhang<ref name="r131" />).'''
File:vol1 Page 659 Image 0001.png|'''Fig. 13.65 – Critical or crushing pressure as a function of the grain size-grain radius product (after Zhang<ref name="r131" />).'''
</gallery>
</gallery><br/>If grains become cemented, not only does porosity decrease, but the effective area of intergranular contracts increases. Even small amounts of cement will increase strength substantially. Angularity of grains and sorting will also influence strength. More angular grains result in sharper point contacts, stress concentrations, and lower strength.<br/><br/>In general, if grain size is known to be smaller or cementation greater (for a given porosity and composition), then increased strength can be estimated by reducing the Hoek-Brown coefficient ''m''. A value of ''m'' = 0 for siltstones and shales was suggested by Hoek and Brown.<ref name="r134">_</ref> Notice that this leads to minor contradiction because clays, with very fine grain size, weaken rocks. It is possible that many of Hoek and Brown’s "shales" were well indurated (slightly metamorphosed?), and grain size and increased cementation account for the increased strength (and reduced ''m''). In rocks with low levels of diagenesis, clays reduce strength and require an increased ''m''.
<br/>
If grains become cemented, not only does porosity decrease, but the effective area of intergranular contracts increases. Even small amounts of cement will increase strength substantially. Angularity of grains and sorting will also influence strength. More angular grains result in sharper point contacts, stress concentrations, and lower strength.
<br>
<br>
In general, if grain size is known to be smaller or cementation greater (for a given porosity and composition), then increased strength can be estimated by reducing the Hoek-Brown coefficient ''m''. A value of ''m'' = 0 for siltstones and shales was suggested by Hoek and Brown.<ref name="r134" /> Notice that this leads to minor contradiction because clays, with very fine grain size, weaken rocks. It is possible that many of Hoek and Brown’s "shales" were well indurated (slightly metamorphosed?), and grain size and increased cementation account for the increased strength (and reduced ''m''). In rocks with low levels of diagenesis, clays reduce strength and require an increased ''m''.


=== Rock Strength From Logs ===
=== Rock Strength From Logs ===


Several techniques have been proposed for deriving rock strength from well log parameters. Coates and Denoo<ref name="r135" /> calculated stresses induced around a borehole and estimated failure from assumed linear envelopes with strength parameters derived from shear and compressional velocities. They relied on the work of Deere and Miller<ref name="r136" /> to provide estimates of compressive strength from dynamic measurements. Simplified forms of these relations are:
Several techniques have been proposed for deriving rock strength from well log parameters. Coates and Denoo<ref name="r135">_</ref> calculated stresses induced around a borehole and estimated failure from assumed linear envelopes with strength parameters derived from shear and compressional velocities. They relied on the work of Deere and Miller<ref name="r136">_</ref> to provide estimates of compressive strength from dynamic measurements. Simplified forms of these relations are:<br/><br/>[[File:Vol1 page 0662 eq 001.png|RTENOTITLE]]....................(13.157a)<br/><br/>[[File:Vol1 page 0662 eq 002.png|RTENOTITLE]]....................(13.157b)<br/><br/>[[File:Vol1 page 0662 eq 003.png|RTENOTITLE]]....................(13.157c)<br/><br/>where ''C''<sub>0</sub> is uniaxial compressive strength and ''E'' is dynamic Young’s modulus (see '''Section 13.5'''). Alternatively, we can include an empirical dependence of the internal friction angle, ''α'', or the porosity, ''Φ''.<br/><br/>[[File:Vol1 page 0664 eq 001.png|RTENOTITLE]]....................(13.158)<br/><br/>'''Eqs. 13.159''' and '''13.160''' provide a way to derive strengths assuming a linear envelope, and provided that compressional and shear velocity, lithology (e.g., gamma ray or SP), and density logs are available. If there is no shear log, one can be derived from the compressional velocity log and ''V''<sub>''p''</sub>-''V''<sub>''s''</sub> relationships previously shown in '''Table 13.7'''.<br/><br/>The strength-porosity trend shown in '''Eq. 13.146''' and modulus-porosity trends in '''Section 13.5''' imply a correlation between strength and shear modulus for sandstone:<br/><br/>[[File:Vol1 page 0665 eq 001.png|RTENOTITLE]]....................(13.159)<br/><br/>This leads to a velocity transform if the bulk density is known:<br/><br/>[[File:Vol1 page 0665 eq 002.png|RTENOTITLE]]....................(13.160)<br/><br/>[[File:Vol1 page 0665 eq 003.png|RTENOTITLE]]....................(13.161)<br/><br/>If we presume a simple relationship between compressional velocity of brine-saturated sandstones and shear velocity as developed by Castagna ''et al''.,<ref name="r4">_</ref> we get<br/><br/>[[File:Vol1 page 0665 eq 004.png|RTENOTITLE]]....................(13.162)<br/><br/>The shear modulus (or velocity) should be the most sensitive measure of strength, and shear properties are little affected by fluid saturations. Whenever possible, shear wave data should be collected and used in this analysis. If only compressional data is available, care must be used in translating the information into effective gas- or brine-saturated values (see '''Section 13.5.1 1'''). This is particularly true for partial oil saturations.<br/><br/>In our analysis, ''C''<sub>0</sub> was first determined from porosity. The influence of clay content was examined separately. The velocity-strength relationships above were derived from the porosity dependence, but clays are handled only indirectly through their effects on velocities. Strength parameters can be calculated directly from porosity ('''Eq. 13.148'''), but clays must then be included, as in '''Eq. 13.153'''. Calculated strengths based directly on porosity and clay content are shown in '''Fig. 13.66'''. These types of logs can be very valuable in detecting weak zones and units susceptible to failure. If at all possible, these kinds of logs should be calibrated with strength measurements directly on core samples.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
[[File:Vol1 page 0662 eq 001.png]]....................(13.157a)
<br>
<br>
[[File:Vol1 page 0662 eq 002.png]]....................(13.157b)
<br>
<br>
[[File:Vol1 page 0662 eq 003.png]]....................(13.157c)
<br>
<br>
where ''C''<sub>0</sub> is uniaxial compressive strength and ''E'' is dynamic Young’s modulus (see '''Section 13.5'''). Alternatively, we can include an empirical dependence of the internal friction angle, ''α'', or the porosity, ''Φ''.
<br>
<br>
[[File:Vol1 page 0664 eq 001.png]]....................(13.158)
<br>
<br>
'''Eqs. 13.159''' and '''13.160''' provide a way to derive strengths assuming a linear envelope, and provided that compressional and shear velocity, lithology (e.g., gamma ray or SP), and density logs are available. If there is no shear log, one can be derived from the compressional velocity log and ''V''<sub>''p''</sub>-''V''<sub>''s''</sub> relationships previously shown in '''Table 13.7'''.  
<br>
<br>
The strength-porosity trend shown in '''Eq. 13.146''' and modulus-porosity trends in '''Section 13.5''' imply a correlation between strength and shear modulus for sandstone:
<br>
<br>
[[File:Vol1 page 0665 eq 001.png]]....................(13.159)
<br>
<br>
This leads to a velocity transform if the bulk density is known:
<br>
<br>
[[File:Vol1 page 0665 eq 002.png]]....................(13.160)
<br>
<br>
[[File:Vol1 page 0665 eq 003.png]]....................(13.161)
<br>
<br>
If we presume a simple relationship between compressional velocity of brine-saturated sandstones and shear velocity as developed by Castagna ''et al''.,<ref name="r4" /> we get
<br>
<br>
[[File:Vol1 page 0665 eq 004.png]]....................(13.162)
<br>
<br>
The shear modulus (or velocity) should be the most sensitive measure of strength, and shear properties are little affected by fluid saturations. Whenever possible, shear wave data should be collected and used in this analysis. If only compressional data is available, care must be used in translating the information into effective gas- or brine-saturated values (see '''Section 13.5.1 1'''). This is particularly true for partial oil saturations.  
<br>
<br>
In our analysis, ''C''<sub>0</sub> was first determined from porosity. The influence of clay content was examined separately. The velocity-strength relationships above were derived from the porosity dependence, but clays are handled only indirectly through their effects on velocities. Strength parameters can be calculated directly from porosity ('''Eq. 13.148'''), but clays must then be included, as in '''Eq. 13.153'''. Calculated strengths based directly on porosity and clay content are shown in '''Fig. 13.66'''. These types of logs can be very valuable in detecting weak zones and units susceptible to failure. If at all possible, these kinds of logs should be calibrated with strength measurements directly on core samples.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 660 Image 0001.png|'''Fig. 13.66 – Strength analysis applied to a Gulf of Mexico suite of logs. Sand/shale fraction is derived from gamma ray and SP logs (left track). Porosity is extracted from the density log (see previous sections). Uniaxial compressive strength is derived from using Eq. 13.153. Weaker sands can be identified and failure predicted based on in-situ stresses around a borehole and a particular production scenario (red zones).'''
File:vol1 Page 660 Image 0001.png|'''Fig. 13.66 – Strength analysis applied to a Gulf of Mexico suite of logs. Sand/shale fraction is derived from gamma ray and SP logs (left track). Porosity is extracted from the density log (see previous sections). Uniaxial compressive strength is derived from using Eq. 13.153. Weaker sands can be identified and failure predicted based on in-situ stresses around a borehole and a particular production scenario (red zones).'''
</gallery>
</gallery><br/>
<br/>
</div></div><div class="toccolours mw-collapsible mw-collapsed">
<br>
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" >
== Gamma Ray Characteristics ==
== Gamma Ray Characteristics ==
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
=== Introduction ===
=== Introduction ===


The radioactivity of rocks has been used for many years to help derive lithologies. Natural occurring radioactive materials (NORM) include the elements uranium, thorium, potassium, radium, and radon, along with the minerals that contain them. There is usually no fundamental connection between different rock types and measured gamma ray intensity, but there exists a strong general correlation between the radioactive isotope content and mineralogy. Observed distributions have been available for numerous decades. In '''Fig. 13.67''', the distributions of radiation levels observed by Russell<ref name="r137" /> are plotted for numerous rock types. Evaporites (NaCl salt, anhydrites) and coals typically have low levels. In other rocks, the general trend toward higher radioactivity with increased shale content is apparent. At the high radioactivity extreme are organic-rich shales and potash (KCl). These plotted values can include beta as well as gamma radioactivity (collected with a Geiger counter). Modern techniques concentrate on gamma ray detection.  
The radioactivity of rocks has been used for many years to help derive lithologies. Natural occurring radioactive materials (NORM) include the elements uranium, thorium, potassium, radium, and radon, along with the minerals that contain them. There is usually no fundamental connection between different rock types and measured gamma ray intensity, but there exists a strong general correlation between the radioactive isotope content and mineralogy. Observed distributions have been available for numerous decades. In '''Fig. 13.67''', the distributions of radiation levels observed by Russell<ref name="r137">_</ref> are plotted for numerous rock types. Evaporites (NaCl salt, anhydrites) and coals typically have low levels. In other rocks, the general trend toward higher radioactivity with increased shale content is apparent. At the high radioactivity extreme are organic-rich shales and potash (KCl). These plotted values can include beta as well as gamma radioactivity (collected with a Geiger counter). Modern techniques concentrate on gamma ray detection.<br/><br/><gallery widths="300px" heights="200px">
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 661 Image 0001.png|'''Fig. 13.67 – Distribution of relative radioactivity level for various rock types (from Bigelow<ref name="r138" /> after Russell<ref name="r137" />).'''
File:vol1 Page 661 Image 0001.png|'''Fig. 13.67 – Distribution of relative radioactivity level for various rock types (from Bigelow<ref name="r138" /> after Russell<ref name="r137" />).'''
</gallery>
</gallery><br/>The primary radioactive isotopes in rocks are potassium-40 and the isotope series associated with the disintegration of uranium and thorium. '''Fig. 13.68''' shows the equilibrium distribution of energy levels associated with each of these groups. Potassium-40 (K<sup>40</sup>) produces a single gamma ray of energy of 1.46 MeV as it transforms into stable calcium. On the other hand, both thorium (Th) and uranium (U) break down to form a sequence of radioactive daughter products. Subsequent breakdown of these unstable isotopes produces a variety of energy levels. Standard gamma ray tools measure a very broad band of energy including all the primary peaks as well as lower-energy daughter peaks. As might be expected from '''Fig. 13.68''', the total count can be dominated by the low-energy decay radiation.<br/><br/><gallery widths="300px" heights="200px">
<br/>
The primary radioactive isotopes in rocks are potassium-40 and the isotope series associated with the disintegration of uranium and thorium. '''Fig. 13.68''' shows the equilibrium distribution of energy levels associated with each of these groups. Potassium-40 (K<sup>40</sup>) produces a single gamma ray of energy of 1.46 MeV as it transforms into stable calcium. On the other hand, both thorium (Th) and uranium (U) break down to form a sequence of radioactive daughter products. Subsequent breakdown of these unstable isotopes produces a variety of energy levels. Standard gamma ray tools measure a very broad band of energy including all the primary peaks as well as lower-energy daughter peaks. As might be expected from '''Fig. 13.68''', the total count can be dominated by the low-energy decay radiation.
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 662 Image 0001.png|'''Fig. 13.68 – Gamma-ray energy levels resulting from disintegration of unstable isotopes (adapted from Tittman ''et al''.<ref name="r96" />).'''
File:vol1 Page 662 Image 0001.png|'''Fig. 13.68 – Gamma-ray energy levels resulting from disintegration of unstable isotopes (adapted from Tittman ''et al''.<ref name="r96" />).'''
</gallery>
</gallery><br/>The radionuclides, including radium, may become more mobile in formation waters found in oil fields. Typically, the greater the ionic strength (salinity), the higher the radium content. Produced waters can have slightly higher radioactivity than background. In addition, the radionuclides are often concentrated in the solid deposits (scale) formed in oilfield equipment. When enclosed in flow equipment (pipes, tanks, etc.) this elevated concentration is not important. However, health risks may occur when equipment is cleaned for reuse or old equipment is put to different application.<br/><br/>'''Table 13.15''' lists some of the common rock types and their typical content of potassium, uranium, and thorium. Potassium is an abundant element, so the radioactive K<sup>40</sup> is widely distributed ('''Table 13.16'''). Potassium feldspars and micas are common components in igneous and metamorphic rocks. Immature sandstones can retain an abundance of these components. In addition, potassium is common in clays. Under extreme evaporitic conditions, KCl (sylvite) will be deposited and result in very high radioactivity levels. Uranium and thorium, on the other hand, are much less common. Both U and Th are found in clays (by absorption), volcanic ashes, and heavy minerals.<br/><br/><gallery widths="300px" heights="200px">
<br/>
The radionuclides, including radium, may become more mobile in formation waters found in oil fields. Typically, the greater the ionic strength (salinity), the higher the radium content. Produced waters can have slightly higher radioactivity than background. In addition, the radionuclides are often concentrated in the solid deposits (scale) formed in oilfield equipment. When enclosed in flow equipment (pipes, tanks, etc.) this elevated concentration is not important. However, health risks may occur when equipment is cleaned for reuse or old equipment is put to different application.  
<br>
<br>
'''Table 13.15''' lists some of the common rock types and their typical content of potassium, uranium, and thorium. Potassium is an abundant element, so the radioactive K<sup>40</sup> is widely distributed ('''Table 13.16'''). Potassium feldspars and micas are common components in igneous and metamorphic rocks. Immature sandstones can retain an abundance of these components. In addition, potassium is common in clays. Under extreme evaporitic conditions, KCl (sylvite) will be deposited and result in very high radioactivity levels. Uranium and thorium, on the other hand, are much less common. Both U and Th are found in clays (by absorption), volcanic ashes, and heavy minerals.  
<br>
<br>
<gallery widths=300px heights=200px>
File:Vol1 Page 663 Image 0001.png|'''Table 13.15'''
File:Vol1 Page 663 Image 0001.png|'''Table 13.15'''


File:Vol1 Page 664 Image 0001.png|'''Table 13.16'''
File:Vol1 Page 664 Image 0001.png|'''Table 13.16'''
</gallery>
</gallery>
<br>


=== Measurement ===
=== Measurement ===


Gamma ray logs are among the most common and useful tools in the oil and gas industry. Originally, measurements were reported in count rates, but all modern tools are calibrated to API units. Typical sedimentary response ranges from 0 to 200 in API units. Gamma ray log character is one of the primary methods used to correlate the stratigraphic section. For most engineering and geophysical applications, the gamma ray log is primarily used to extract lithologic, mineralogic, or fabric estimates.  
Gamma ray logs are among the most common and useful tools in the oil and gas industry. Originally, measurements were reported in count rates, but all modern tools are calibrated to API units. Typical sedimentary response ranges from 0 to 200 in API units. Gamma ray log character is one of the primary methods used to correlate the stratigraphic section. For most engineering and geophysical applications, the gamma ray log is primarily used to extract lithologic, mineralogic, or fabric estimates.<br/><br/>The log response depends on the radiation, tool characteristics, and logging parameters. A 30-cm sodium iodide scintillation crystal with a photomultiplier tube is a common detector configuration. Thin, highly radioactive beds may be detected, but cannot be resolved below about 0.25 m. Radiation is damped primarily by formation material electron density and Compton scattering. This limits the depth of investigation to around 30 cm, although it will depend on the energy levels. Because the radioactive decay is a statistical process, slower logging rates produce better results. The low number of counts resulting from logging too fast cannot be increased by logging rate correction factors. Most tools are usually out of calibration if they are not centered in the borehole. Heavy barite mud can also lower the overall count rate, particularly for low-energy gamma rays.<br/><br/>Rather than merely measuring total gamma radiation, the energy levels can be detected separately. This allows the concentrations of K, U, and Th to be derived as independent parameters. '''Fig. 13.69''' shows the energy windows used in a Baker-Atlas tool. This would allow, for example, the feldspars in immature sands to be separated from clays with adsorbed U or Th.<br/><br/><gallery widths="300px" heights="200px">
<br>
<br>
The log response depends on the radiation, tool characteristics, and logging parameters. A 30-cm sodium iodide scintillation crystal with a photomultiplier tube is a common detector configuration. Thin, highly radioactive beds may be detected, but cannot be resolved below about 0.25 m. Radiation is damped primarily by formation material electron density and Compton scattering. This limits the depth of investigation to around 30 cm, although it will depend on the energy levels. Because the radioactive decay is a statistical process, slower logging rates produce better results. The low number of counts resulting from logging too fast cannot be increased by logging rate correction factors. Most tools are usually out of calibration if they are not centered in the borehole. Heavy barite mud can also lower the overall count rate, particularly for low-energy gamma rays.  
<br>
<br>
Rather than merely measuring total gamma radiation, the energy levels can be detected separately. This allows the concentrations of K, U, and Th to be derived as independent parameters. '''Fig. 13.69''' shows the energy windows used in a Baker-Atlas tool. This would allow, for example, the feldspars in immature sands to be separated from clays with adsorbed U or Th.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 665 Image 0001.png|'''Fig. 13.69 – Gamma-ray energy windows used for spectral gamma-ray logging (from Bigelow<ref name="r138" />).'''
File:vol1 Page 665 Image 0001.png|'''Fig. 13.69 – Gamma-ray energy windows used for spectral gamma-ray logging (from Bigelow<ref name="r138" />).'''
</gallery>
</gallery><br/>The most common use of gamma ray logs is to estimate the shale "volume" in rocks. It is important to remember that the tool measures radioactivity, and the correlation to shale content is empirical. Shales are presumed to be composed of clay minerals. Thus, the gamma ray level is assumed to be correlated with grain size. In reality, shales may be composed of 30% or more of quartz and other minerals. The clays within the shales may not be radioactive, and the adjacent sands may contain radioactive isotopes. However, radioactivity levels typically are related to grain size, as seen in '''Fig. 13.70'''. Here, core plugs were analyzed for median grain size and radioactivity level measured directly; crosses are fine-grained sands, while dots are silts and clay-rich rocks.<br/><br/><gallery widths="300px" heights="200px">
<br/>
The most common use of gamma ray logs is to estimate the shale "volume" in rocks. It is important to remember that the tool measures radioactivity, and the correlation to shale content is empirical. Shales are presumed to be composed of clay minerals. Thus, the gamma ray level is assumed to be correlated with grain size. In reality, shales may be composed of 30% or more of quartz and other minerals. The clays within the shales may not be radioactive, and the adjacent sands may contain radioactive isotopes. However, radioactivity levels typically are related to grain size, as seen in '''Fig. 13.70'''. Here, core plugs were analyzed for median grain size and radioactivity level measured directly; crosses are fine-grained sands, while dots are silts and clay-rich rocks.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 666 Image 0001.png|'''Fig. 13.70 – Measured mean grain size vs. gamma-ray levels (calibrated to API value) for clastic samples. The rough correspondence of gamma ray value can be seen, but relationship is not simple (data from Georgi ''et al''.<ref name="r141" />).'''
File:vol1 Page 666 Image 0001.png|'''Fig. 13.70 – Measured mean grain size vs. gamma-ray levels (calibrated to API value) for clastic samples. The rough correspondence of gamma ray value can be seen, but relationship is not simple (data from Georgi ''et al''.<ref name="r141" />).'''
</gallery>
</gallery><br/>To extract the shale content in rocks, a linear or near-linear relation is used to convert a gamma ray index, ''I''<sub>''gr''</sub>, to shale volume ''V''<sub>''sh''</sub>. Because local sands can contain radioactive components, and the shales may vary with depth, local baseline levels are chosen near the zone of interest.<br/><br/>[[File:Vol1 page 0669 eq 001.png|RTENOTITLE]]....................(13.163)<br/><br/>where ''R'' is the measured radiation level, ''R''<sub>cleansand</sub> is the baseline level through a reference sand, and ''R''<sub>shale</sub> is the baseline through a representative shale. Several relations have been developed to derive shale volume ('''Fig. 13.71'''). A linear relation simply sets the shale content equal to the gamma ray index.<br/><br/>[[File:Vol1 page 0670 eq 001.png|RTENOTITLE]]....................(13.164)<br/><br/>Other proposed relations shown in '''Fig. 13.71''' are defined in '''Table 13.17'''. Several assumptions are made in these evaluations:
<br/>
 
To extract the shale content in rocks, a linear or near-linear relation is used to convert a gamma ray index, ''I''<sub>''gr''</sub>, to shale volume ''V''<sub>''sh''</sub>. Because local sands can contain radioactive components, and the shales may vary with depth, local baseline levels are chosen near the zone of interest.
*Compositions of sand and shale components are constant.
<br>
*Baselines are chosen on representative "shales" and "clean" sands (although these terms are very subjective).
<br>
*Simple mixture laws apply.
[[File:Vol1 page 0669 eq 001.png]]....................(13.163)
*Fabric is not important.
<br>
 
<br>
 
where ''R'' is the measured radiation level, ''R''<sub>cleansand</sub> is the baseline level through a reference sand, and ''R''<sub>shale</sub> is the baseline through a representative shale. Several relations have been developed to derive shale volume ('''Fig. 13.71'''). A linear relation simply sets the shale content equal to the gamma ray index.
<br>
<br>
[[File:Vol1 page 0670 eq 001.png]]....................(13.164)
<br>
<br>
Other proposed relations shown in '''Fig. 13.71''' are defined in '''Table 13.17'''. Several assumptions are made in these evaluations:
<br>
* Compositions of sand and shale components are constant.
* Baselines are chosen on representative "shales" and "clean" sands (although these terms are very subjective).
* Simple mixture laws apply.
* Fabric is not important.
<br>


Many of these assumptions may be poor approximations.  
Many of these assumptions may be poor approximations.<br/><br/><gallery widths="300px" heights="200px">
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 667 Image 0001.png|'''Fig. 13.71 – Reported gamma-ray index to shale volume conversions (from Bigelow<ref name="r138" />).'''
File:vol1 Page 667 Image 0001.png|'''Fig. 13.71 – Reported gamma-ray index to shale volume conversions (from Bigelow<ref name="r138" />).'''


File:Vol1 Page 668 Image 0001.png|'''Table 13.17'''
File:Vol1 Page 668 Image 0001.png|'''Table 13.17'''
</gallery>
</gallery><br/>A more likely presumption is that the radiation level is dependent on the mixture densities and not volumes (Wahl<ref name="r139">_</ref> and Katahara<ref name="r140">_</ref>). In this case, a fabric analysis can also be performed. Katahara<ref name="r139">_</ref> modeled the shale component of shaly sands as existing in three forms:
<br>
 
A more likely presumption is that the radiation level is dependent on the mixture densities and not volumes (Wahl<ref name="r139" /> and Katahara<ref name="r140" />). In this case, a fabric analysis can also be performed. Katahara<ref name="r139" /> modeled the shale component of shaly sands as existing in three forms:
*Structural—an original depositional granular form.
<br>
*Dispersed—clay distributed through the rock and pore space.
* Structural—an original depositional granular form.
*Laminated—thin layers of shale cutting the sand beds.
* Dispersed—clay distributed through the rock and pore space.
 
* Laminated—thin layers of shale cutting the sand beds.
 
<br>


In '''Fig. 13.72''', his results show a surprisingly simple form. The conclusion is that in most cases, the simple linear relation is appropriate.  
In '''Fig. 13.72''', his results show a surprisingly simple form. The conclusion is that in most cases, the simple linear relation is appropriate.<br/><br/><gallery widths="300px" heights="200px">
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 669 Image 0001.png|'''Fig. 13.72 – Modeled gamma-ray response to different clay distributions within a shaley sand series (modified from Katahara<ref name="r140" />).'''
File:vol1 Page 669 Image 0001.png|'''Fig. 13.72 – Modeled gamma-ray response to different clay distributions within a shaley sand series (modified from Katahara<ref name="r140" />).'''
</gallery>
</gallery><br/>As an example of this process, the shale content of a zone in a Gulf of Mexico well is estimated. In '''Fig. 13.73''', a sand-shale sequence gives a gamma ray range of approximately 20 to 90 API units. A baseline of approximately 25 is chosen through the sand, and a baseline of approximately 98 is chosen for the shale. Using the relations in '''Eqs. 13.163''' and '''13.164''' result in the shale volume estimates scaled at the bottom of the logged zone.<br/><br/><gallery widths="300px" heights="200px">
<br/>
As an example of this process, the shale content of a zone in a Gulf of Mexico well is estimated. In '''Fig. 13.73''', a sand-shale sequence gives a gamma ray range of approximately 20 to 90 API units. A baseline of approximately 25 is chosen through the sand, and a baseline of approximately 98 is chosen for the shale. Using the relations in '''Eqs. 13.163''' and '''13.164''' result in the shale volume estimates scaled at the bottom of the logged zone.  
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 670 Image 0001.png|'''Fig. 13.73 – Typical shale volume extraction from a Gulf of Mexico log. 100% “sand” and 100% “shale” lines are locally established; then, shale content is a linear interpolation between the two.'''
File:vol1 Page 670 Image 0001.png|'''Fig. 13.73 – Typical shale volume extraction from a Gulf of Mexico log. 100% “sand” and 100% “shale” lines are locally established; then, shale content is a linear interpolation between the two.'''
</gallery>
</gallery><br/>Gamma radiation levels can also be measured on core. This technique provides a profile of levels along the length of the core. The primary use is to correlate core depths to logged depths. An example is shown in '''Fig. 13.74'''. This procedure can be used to identify log features or positioning of the cored interval. Especially when core recovery is poor, this method is very useful in tying the core fragments to true depths. Core plugs can also be measured, although special equipment must be used to record the low levels of radiation associated with the small samples. In general, property correlations to the measured gamma ray levels are much better for cores than for the log because of the depth averaging in the log.<ref name="r141">_</ref><br/><br/><gallery widths="300px" heights="200px">
<br/>
Gamma radiation levels can also be measured on core. This technique provides a profile of levels along the length of the core. The primary use is to correlate core depths to logged depths. An example is shown in '''Fig. 13.74'''. This procedure can be used to identify log features or positioning of the cored interval. Especially when core recovery is poor, this method is very useful in tying the core fragments to true depths. Core plugs can also be measured, although special equipment must be used to record the low levels of radiation associated with the small samples. In general, property correlations to the measured gamma ray levels are much better for cores than for the log because of the depth averaging in the log.<ref name="r141" />
<br/>
<br/>
<gallery widths="300px" heights="200px">
File:vol1 Page 671 Image 0001.png|'''Fig. 13.74 – A measured core gamma-ray profile vs. logged data in a carbonate section. Comparison of peaks shows the offset in measured depth (adapted from Core Labs data).'''
File:vol1 Page 671 Image 0001.png|'''Fig. 13.74 – A measured core gamma-ray profile vs. logged data in a carbonate section. Comparison of peaks shows the offset in measured depth (adapted from Core Labs data).'''
</gallery>
</gallery><br/>
<br/>
</div></div><div class="toccolours mw-collapsible mw-collapsed">
<br>
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" >
== Nomenclature ==
== Nomenclature ==
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
<br>
 
 
{|
{|
|''a''<sub>''ij''</sub>
|=
|water density coefficients
|-
|-
|''A''  
| ''a''<sub>''ij''</sub>
|=  
| =
|bulk modulus/porosity factor, '''Eq. 13.86'''  
| water density coefficients
|-
| ''A''
| =
| bulk modulus/porosity factor, '''Eq. 13.86'''
|-
|-
|''A''  
| ''A''
|=  
| =
|strength material constant, '''Eq. 13.134'''  
| strength material constant, '''Eq. 13.134'''
|-
|-
|''A''<sub>''o''</sub>  
| ''A''<sub>''o''</sub>
|=  
| =
|initial wave amplitude  
| initial wave amplitude
|-
|-
|''A''(''z'',''t'')  
| ''A''(''z'',''t'')
|=  
| =
|wave amplitude with distance and time  
| wave amplitude with distance and time
|-
|-
|''A''<sub>''f'' 1</sub>, ''A''<sub>''f'' 2</sub>  
| ''A''<sub>''f'' 1</sub>, ''A''<sub>''f'' 2</sub>
|=  
| =
|fraction fluid component 1, 2, etc.  
| fraction fluid component 1, 2, etc.
|-
|-
|''A''<sub>''m''1</sub>, ''A''<sub>''m''2</sub>  
| ''A''<sub>''m''1</sub>, ''A''<sub>''m''2</sub>
|=  
| =
|fraction mineral component 1, 2, etc.  
| fraction mineral component 1, 2, etc.
|-
|-
|''A''<sub>1</sub>, ''A''<sub>2</sub>  
| ''A''<sub>1</sub>, ''A''<sub>2</sub>
|=  
| =
|fraction component 1, 2, etc.  
| fraction component 1, 2, etc.
|-
|-
|''b''  
| ''b''
|=  
| =
|velocity/temperature constant, m/sC, '''Eq. 13.18'''  
| velocity/temperature constant, m/sC, '''Eq. 13.18'''
|-
|-
|''b''  
| ''b''
|=  
| =
|strength envelope intercept, GPa or MPa, '''Eq. 13.134'''  
| strength envelope intercept, GPa or MPa, '''Eq. 13.134'''
|-
|-
|''b''<sub>''ij''</sub>  
| ''b''<sub>''ij''</sub>
|=  
| =
|brine density coefficients  
| brine density coefficients
|-
|-
|''B''  
| ''B''
|=  
| =
|brine compressional velocity factor, m/s, '''Eq. 13.32b'''  
| brine compressional velocity factor, m/s, '''Eq. 13.32b'''
|-
|-
|''B''  
| ''B''
|=  
| =
|bulk modulus/porosity factor, '''Eq. 13.86'''  
| bulk modulus/porosity factor, '''Eq. 13.86'''
|-
|-
|''B′''  
| ''B′''
|=  
| =
|rock modulus, real component, GPa or MPa, '''Eq. 13.105'''  
| rock modulus, real component, GPa or MPa, '''Eq. 13.105'''
|-
|-
|''B"''  
| ''B"''
|=  
| =
|rock modulus, imaginary component, GPa or MPa  
| rock modulus, imaginary component, GPa or MPa
|-
|-
|''B''<sub>''o''</sub>  
| ''B''<sub>''o''</sub>
|=  
| =
|rock modulus, zero frequency, GPa or MPa  
| rock modulus, zero frequency, GPa or MPa
|-
|-
|''B''<sub>''oo''</sub>  
| ''B''<sub>''oo''</sub>
|=  
| =
|rock modulus, infinite frequency, GPa or MPa  
| rock modulus, infinite frequency, GPa or MPa
|-
|-
|''C''  
| ''C''
|=  
| =
|bulk modulus/porosity factor, '''Eq. 13.86'''  
| bulk modulus/porosity factor, '''Eq. 13.86'''
|-
|-
|''C''  
| ''C''
|=  
| =
|clay content, '''Eq. 13.150'''
| clay content, '''Eq. 13.150'''
|-
|-
|''C''<sub>''ijkl''</sub>  
| ''C''<sub>''ijkl''</sub>
|=  
| =
|stiffness tensor components, GPa or MPa
| stiffness tensor components, GPa or MPa
|-
|-
|''C''<sub>0</sub>  
| ''C''<sub>0</sub>
|=  
| =
|uniaxial or unconfined compressive strength, GPa or MPa  
| uniaxial or unconfined compressive strength, GPa or MPa
|-
|-
|''C''<sub>''t''</sub>  
| ''C''<sub>''t''</sub>
|=  
| =
|tensional strength, GPa or MPa  
| tensional strength, GPa or MPa
|-
|-
|''C''<sub>''u''</sub>  
| ''C''<sub>''u''</sub>
|=  
| =
|cohesive strength, GPa or MPa  
| cohesive strength, GPa or MPa
|-
|-
|''D''  
| ''D''
|=  
| =
|bulk modulus/porosity factor  
| bulk modulus/porosity factor
|-
|-
|''E''  
| ''E''
|=  
| =
|Young’s modulus, GPa or MPa  
| Young’s modulus, GPa or MPa
|-
|-
|''f''  
| ''f''
|=  
| =
|frequency, s<sup>–1</sup>, Hz (cycles/s)  
| frequency, s<sup>–1</sup>, Hz (cycles/s)
|-
|-
|''F''  
| ''F''
|=  
| =
|volume factor  
| volume factor
|-
|-
|''G''  
| ''G''
|=  
| =
|shear modulus, GPa or MPa  
| shear modulus, GPa or MPa
|-
|-
|''G''(''Φ'')  
| ''G''(''Φ'')
|=  
| =
|gain factor  
| gain factor
|-
|-
|''I''<sub>''gr''</sub>  
| ''I''<sub>''gr''</sub>
|=  
| =
|gamma ray index  
| gamma ray index
|-
|-
|''k''  
| ''k''
|=  
| =
|permeability, m<sup>2</sup>, '''Eq. 13.108'''  
| permeability, m<sup>2</sup>, '''Eq. 13.108'''
|-
|-
|''k''  
| ''k''
|=  
| =
|wave number, m<sup>–1</sup>, '''Eq. 13.102'''  
| wave number, m<sup>–1</sup>, '''Eq. 13.102'''
|-
|-
|''k''*  
| ''k''*
|=  
| =
|complex wave number, m<sup>–1</sup>  
| complex wave number, m<sup>–1</sup>
|-
|-
|''K''  
| ''K''
|=  
| =
|bulk modulus, GPa or MPa  
| bulk modulus, GPa or MPa
|-
|-
|''K''<sub>''d''</sub>  
| ''K''<sub>''d''</sub>
|=  
| =
|dry bulk modulus, GPa or MPa  
| dry bulk modulus, GPa or MPa
|-
|-
|''K''<sub>''d''</sub> <sub>min</sub>  
| ''K''<sub>''d''</sub> <sub>min</sub>
|=  
| =
|minimum bulk modulus, GPa or MPa  
| minimum bulk modulus, GPa or MPa
|-
|-
|''K''<sub>''f''</sub>  
| ''K''<sub>''f''</sub>
|=  
| =
|fluid bulk modulus, GPa or MPa  
| fluid bulk modulus, GPa or MPa
|-
|-
|''K''<sub>''f'' 1</sub>, ''K''<sub>''f'' 2</sub>  
| ''K''<sub>''f'' 1</sub>, ''K''<sub>''f'' 2</sub>
|=  
| =
|bulk modulus of fluid 1, 2, etc., GPa or MPa  
| bulk modulus of fluid 1, 2, etc., GPa or MPa
|-
|-
|''K''<sub>''HS''</sub>  
| ''K''<sub>''HS''</sub>
|=  
| =
|Hasin-Shtrikman bound bulk modulus, GPa or MPa  
| Hasin-Shtrikman bound bulk modulus, GPa or MPa
|-
|-
|''K''<sub>''n''</sub>  
| ''K''<sub>''n''</sub>
|=  
| =
|normalized bulk modulus, numeric  
| normalized bulk modulus, numeric
|-
|-
|''K''<sub>''n R''</sub>  
| ''K''<sub>''n R''</sub>
|=  
| =
|normalized Reuss bound bulk modulus, numeric  
| normalized Reuss bound bulk modulus, numeric
|-
|-
|''K''<sub>''o''</sub>  
| ''K''<sub>''o''</sub>
|=  
| =
|mineral bulk modulus, GPa or MPa  
| mineral bulk modulus, GPa or MPa
|-
|-
|''K''<sub>''R''</sub>  
| ''K''<sub>''R''</sub>
|=  
| =
|Reuss bound bulk modulus, GPa or MPa  
| Reuss bound bulk modulus, GPa or MPa
|-
|-
|''K''<sub>''s''</sub>  
| ''K''<sub>''s''</sub>
|=  
| =
|saturated bulk modulus, GPa or MPa  
| saturated bulk modulus, GPa or MPa
|-
|-
|''K''<sub>1</sub>, ''K''<sub>2</sub>  
| ''K''<sub>1</sub>, ''K''<sub>2</sub>
|=  
| =
|bulk modulus of component 1, 2, etc., GPa or MPa  
| bulk modulus of component 1, 2, etc., GPa or MPa
|-
|-
|''K''*  
| ''K''*
|=  
| =
|effective bulk modulus, GPa or MPa  
| effective bulk modulus, GPa or MPa
|-
|-
|''K′''  
| ''K′''
|=  
| =
|effective crack bulk modulus, GPa or MPa  
| effective crack bulk modulus, GPa or MPa
|-
|-
|Δ''K''<sub>''d''</sub>  
| Δ''K''<sub>''d''</sub>
|=  
| =
|change in bulk modulus, GPa or MPa  
| change in bulk modulus, GPa or MPa
|-
|-
|Δ''K''<sub>''d''max</sub>  
| Δ''K''<sub>''d''max</sub>
|=  
| =
|maximum change in bulk modulus, GPa or MPa  
| maximum change in bulk modulus, GPa or MPa
|-
|-
|Δ''K''<sub>12</sub>  
| Δ''K''<sub>12</sub>
|=  
| =
|change in bulk modulus, fluid 1 to fluid 2, GPa or MPa  
| change in bulk modulus, fluid 1 to fluid 2, GPa or MPa
|-
|-
|''L''  
| ''L''
|=  
| =
|length, m  
| length, m
|-
|-
|Δ''L''  
| Δ''L''
|=  
| =
|change in length, m  
| change in length, m
|-
|-
|''m''  
| ''m''
|=  
| =
|Hoek-Brown strength coefficient  
| Hoek-Brown strength coefficient
|-
|-
|''M''  
| ''M''
|=  
| =
|molecular weight, g/mole  
| molecular weight, g/mole
|-
|-
|''M''<sub>''A''</sub>, ''M''<sub>''B''</sub>  
| ''M''<sub>''A''</sub>, ''M''<sub>''B''</sub>
|=  
| =
|modulus of component ''a'', ''b'', etc., GPa or MPa  
| modulus of component ''a'', ''b'', etc., GPa or MPa
|-
|-
|''M''<sub>''O''</sub>  
| ''M''<sub>''O''</sub>
|=  
| =
|reference oil molecular weight, g/mole  
| reference oil molecular weight, g/mole
|-
|-
|''M''<sub>''R''</sub>  
| ''M''<sub>''R''</sub>
|=  
| =
|Reuss bound modulus, GPa or MPa  
| Reuss bound modulus, GPa or MPa
|-
|-
|''M''<sub>''V''</sub>  
| ''M''<sub>''V''</sub>
|=  
| =
|Voigt bound modulus, GPa or MPa  
| Voigt bound modulus, GPa or MPa
|-
|-
|''M''<sub>''VRH''</sub>  
| ''M''<sub>''VRH''</sub>
|=  
| =
|Voigt-Reuss-Hill bound modulus, GPa or MPa  
| Voigt-Reuss-Hill bound modulus, GPa or MPa
|-
|-
|''n''  
| ''n''
|=  
| =
|number of moles, '''Eq. 13.10'''
| number of moles, '''Eq. 13.10'''
|-
|-
|''n''  
| ''n''
|=  
| =
|effective stress coefficient, '''Eq. 13.35'''  
| effective stress coefficient, '''Eq. 13.35'''
|-
|-
|''n''  
| ''n''
|=  
| =
|strength envelope exponent, '''Eq. 13.134'''  
| strength envelope exponent, '''Eq. 13.134'''
|-
|-
|''P''  
| ''P''
|=  
| =
|pressure, MPa  
| pressure, MPa
|-
|-
|''P''<sub>''c''</sub>  
| ''P''<sub>''c''</sub>
|=  
| =
|confining pressure, MPa  
| confining pressure, MPa
|-
|-
|''P''<sub>''d''</sub>  
| ''P''<sub>''d''</sub>
|=  
| =
|differential pressure, MPa  
| differential pressure, MPa
|-
|-
|''P''<sub>''e''</sub>  
| ''P''<sub>''e''</sub>
|=  
| =
|effective pressure, MPa  
| effective pressure, MPa
|-
|-
|''P''<sub>''p''</sub>  
| ''P''<sub>''p''</sub>
|=  
| =
|pore pressure, MPa  
| pore pressure, MPa
|-
|-
|''Q''  
| ''Q''
|=  
| =
|seismic quality factor, numeric  
| seismic quality factor, numeric
|-
|-
|''r''  
| ''r''
|=  
| =
|radius of stress "circle," GPa or MPa  
| radius of stress "circle," GPa or MPa
|-
|-
|''R''  
| ''R''
|=  
| =
|gas constant, (L MPa)/(K mole), '''Eq. 13.10'''  
| gas constant, (L MPa)/(K mole), '''Eq. 13.10'''
|-
|-
|''R''  
| ''R''
|=  
| =
|gas/oil ratio, '''Eq. 13.26'''  
| gas/oil ratio, '''Eq. 13.26'''
|-
|-
|''R''  
| ''R''
|=  
| =
|measured gamma radiation, API units  
| measured gamma radiation, API units
|-
|-
|''R''<sub>cleansand</sub>  
| ''R''<sub>cleansand</sub>
|=  
| =
|gamma radiation in a "clean" sand zone, API units  
| gamma radiation in a "clean" sand zone, API units
|-
|-
|''R''<sub>shale</sub>  
| ''R''<sub>shale</sub>
|=  
| =
|gamma radiation in a shale zone, API units  
| gamma radiation in a shale zone, API units
|-
|-
|''s''  
| ''s''
|=  
| =
|Hoek-Brown strength coefficient  
| Hoek-Brown strength coefficient
|-
|-
|''S'', ''S′''  
| ''S'', ''S′''
|=  
| =
|general rock property  
| general rock property
|-
|-
|''t''  
| ''t''
|=  
| =
|time, s  
| time, s
|-
|-
|''T''  
| ''T''
|=  
| =
|temperature, °C  
| temperature, °C
|-
|-
|''T''<sub>''a''</sub>  
| ''T''<sub>''a''</sub>
|=  
| =
|absolute temperature, K  
| absolute temperature, K
|-
|-
|''T''<sub>1</sub>, ''T''<sub>2</sub>  
| ''T''<sub>1</sub>, ''T''<sub>2</sub>
|=  
| =
|Kunster-Toksoz coefficients  
| Kunster-Toksoz coefficients
|-
|-
|Δ''T''  
| Δ''T''
|=  
| =
|change in temperature, K  
| change in temperature, K
|-
|-
|''V''<sub>''B''</sub>  
| ''V''<sub>''B''</sub>
|=  
| =
|brine compressional velocity, m/s  
| brine compressional velocity, m/s
|-
|-
|''V''<sub>''fx''</sub> or ''V''<sub>''cx''</sub>  
| ''V''<sub>''fx''</sub> or ''V''<sub>''cx''</sub>
|=  
| =
|fracture or crack volume, m<sup>3</sup> or cm<sup>3</sup>  
| fracture or crack volume, m<sup>3</sup> or cm<sup>3</sup>
|-
|-
|''V''<sub>''f''1</sub>, ''V''<sub>''f''2</sub>  
| ''V''<sub>''f''1</sub>, ''V''<sub>''f''2</sub>
|=  
| =
|fluid 1, 2, etc. volume, m<sup>3</sup> or cm<sup>3</sup>  
| fluid 1, 2, etc. volume, m<sup>3</sup> or cm<sup>3</sup>
|-
|-
|''V''<sub>''g''</sub> or ''V''<sub>''m''</sub>  
| ''V''<sub>''g''</sub> or ''V''<sub>''m''</sub>
|=  
| =
|grain or mineral volume, m<sup>3</sup> or cm<sup>3</sup>  
| grain or mineral volume, m<sup>3</sup> or cm<sup>3</sup>
|-
|-
|''V''<sub>mineral</sub>  
| ''V''<sub>mineral</sub>
|=  
| =
|mineral velocity, m/s  
| mineral velocity, m/s
|-
|-
|''V''<sub>''o''</sub>  
| ''V''<sub>''o''</sub>
|=  
| =
|reference compressional velocity, m/s  
| reference compressional velocity, m/s
|-
|-
|''V''<sub>''p''</sub>  
| ''V''<sub>''p''</sub>
|=  
| =
|compressional velocity, m/s  
| compressional velocity, m/s
|-
|-
|''V''<sub>''po''</sub>  
| ''V''<sub>''po''</sub>
|=  
| =
|vertical compressional velocity, m/s  
| vertical compressional velocity, m/s
|-
|-
|''V''<sub>''por''</sub>  
| ''V''<sub>''por''</sub>
|=  
| =
|total pore volume, m<sup>3</sup> or cm<sup>3</sup>  
| total pore volume, m<sup>3</sup> or cm<sup>3</sup>
|-
|-
|''V''<sub>''p-con''</sub>  
| ''V''<sub>''p-con''</sub>
|=  
| =
|connected pore volume, m<sup>3</sup> or cm<sup>3</sup>  
| connected pore volume, m<sup>3</sup> or cm<sup>3</sup>
|-
|-
|''V''<sub>''p-iso''</sub>  
| ''V''<sub>''p-iso''</sub>
|=  
| =
|isolated pore volume, m<sup>3</sup> or cm<sup>3</sup>  
| isolated pore volume, m<sup>3</sup> or cm<sup>3</sup>
|-
|-
|''V''<sub>rock</sub>  
| ''V''<sub>rock</sub>
|=  
| =
|rock velocity, m/s  
| rock velocity, m/s
|-
|-
|''V''<sub>''s''</sub>  
| ''V''<sub>''s''</sub>
|=  
| =
|shear velocity, m/s  
| shear velocity, m/s
|-
|-
|''V''<sub>''sh''</sub>  
| ''V''<sub>''sh''</sub>
|=  
| =
|shale volume, fractional  
| shale volume, fractional
|-
|-
|''V''<sub>''so''</sub>  
| ''V''<sub>''so''</sub>
|=  
| =
|vertical shear velocity, m/s  
| vertical shear velocity, m/s
|-
|-
|''V''<sub>''T''</sub>  
| ''V''<sub>''T''</sub>
|=  
| =
|isothermal fluid compressional velocity, m/s  
| isothermal fluid compressional velocity, m/s
|-
|-
|''V''<sub>''T''</sub> or ''V''<sub>''rx''</sub>  
| ''V''<sub>''T''</sub> or ''V''<sub>''rx''</sub>
|=  
| =
|total rock volume, m<sup>3</sup> or cm<sup>3</sup>  
| total rock volume, m<sup>3</sup> or cm<sup>3</sup>
|-
|-
|''V''<sub>''TM''</sub>  
| ''V''<sub>''TM''</sub>
|=  
| =
|oil weight m compressional velocity, m/s  
| oil weight m compressional velocity, m/s
|-
|-
|''V''<sub>''TOMO''</sub>  
| ''V''<sub>''TOMO''</sub>
|=  
| =
|oil weight m compressional velocity at ''t''<sub>''o''</sub>, m/s  
| oil weight m compressional velocity at ''t''<sub>''o''</sub>, m/s
|-
|-
|''V''<sub>''W''</sub>  
| ''V''<sub>''W''</sub>
|=  
| =
|water compressional velocity, m/s  
| water compressional velocity, m/s
|-
|-
|''w''<sub>''ij''</sub>  
| ''w''<sub>''ij''</sub>
|=  
| =
|water compressional velocity coefficients  
| water compressional velocity coefficients
|-
|-
|''x''  
| ''x''
|=  
| =
|weight fraction of NaCl, ppm, '''Eq. 13.29b'''  
| weight fraction of NaCl, ppm, '''Eq. 13.29b'''
|-
|-
|''x''  
| ''x''
|=  
| =
|directional component, m  
| directional component, m
|-
|-
|''y''  
| ''y''
|=  
| =
|directional component, m  
| directional component, m
|-
|-
|''z''  
| ''z''
|=  
| =
|directional component, m  
| directional component, m
|-
|-
|''Z''  
| ''Z''
|=  
| =
|compressibility factor  
| compressibility factor
|-
|-
|''α''  
| ''α''
|=  
| =
|aspect ratio, '''Eq. 13.91'''  
| aspect ratio, '''Eq. 13.91'''
|-
|-
|''α''  
| ''α''
|=  
| =
|failure envelope slope, '''Eq. 13.116'''  
| failure envelope slope, '''Eq. 13.116'''
|-
|-
|''α''<sub>''m''</sub>  
| ''α''<sub>''m''</sub>
|=  
| =
|aspect ratio of fracture population ''m'', fractional
| aspect ratio of fracture population ''m'', fractional
|-
|-
|''α''<sub>l</sub>  
| ''α''<sub>l</sub>
|=  
| =
|logarithmic decrement (loss), nepers/m
| logarithmic decrement (loss), nepers/m
|-
|-
|''β''  
| ''β''
|=  
| =
|strength factor, numeric  
| strength factor, numeric
|-
|-
|''β''<sub>''S''</sub>  
| ''β''<sub>''S''</sub>
|=  
| =
|adiabatic compressibility, MPa<sup>–1</sup>  
| adiabatic compressibility, MPa<sup>–1</sup>
|-
|-
|''β''<sub>''T''</sub>  
| ''β''<sub>''T''</sub>
|=  
| =
|isothermal compressibility, MPa<sup>–1</sup>  
| isothermal compressibility, MPa<sup>–1</sup>
|-
|-
|''γ''  
| ''γ''
|=  
| =
|heat capacity ratio, '''Eq. 13.16'''  
| heat capacity ratio, '''Eq. 13.16'''
|-
|-
|''γ''  
| ''γ''
|=  
| =
|Thomsen ''V''<sub>''s''</sub> anisotropy factor, '''Eq. 13.95'''  
| Thomsen ''V''<sub>''s''</sub> anisotropy factor, '''Eq. 13.95'''
|-
|-
|''δ''  
| ''δ''
|=  
| =
|Thomsen anisotropy factor, '''Eq. 13.95'''  
| Thomsen anisotropy factor, '''Eq. 13.95'''
|-
|-
|''δ''  
| ''δ''
|=  
| =
|loss tangent, '''Eq. 13.103'''  
| loss tangent, '''Eq. 13.103'''
|-
|-
|''ε''  
| ''ε''
|=  
| =
|Thomsen ''V''<sub>''p''</sub> anisotropy factor, numeric  
| Thomsen ''V''<sub>''p''</sub> anisotropy factor, numeric
|-
|-
|''ε''<sub>''ij''</sub>  
| ''ε''<sub>''ij''</sub>
|=  
| =
|strain components, fractional  
| strain components, fractional
|-
|-
|''ε''<sub>''kl''</sub>  
| ''ε''<sub>''kl''</sub>
|=  
| =
|strain components, fractional  
| strain components, fractional
|-
|-
|''ε''<sub>shear</sub>  
| ''ε''<sub>shear</sub>
|=  
| =
|shear strain, fractional  
| shear strain, fractional
|-
|-
|''ε''<sub>''V''</sub>  
| ''ε''<sub>''V''</sub>
|=  
| =
|volumeteric strain, fractional  
| volumeteric strain, fractional
|-
|-
|''ε''<sub>''yy''</sub>  
| ''ε''<sub>''yy''</sub>
|=  
| =
|horizontal strain, fractional  
| horizontal strain, fractional
|-
|-
|''ε''<sub>''zz''</sub>  
| ''ε''<sub>''zz''</sub>
|=  
| =
|vertical strain, fractional  
| vertical strain, fractional
|-
|-
|''η''  
| ''η''
|=  
| =
|viscosity, Pa•s  
| viscosity, Pa•s
|-
|-
|''θ''  
| ''θ''
|=  
| =
|wave propagation angle to symmetry axis
| wave propagation angle to symmetry axis
|-
|-
|''λ''  
| ''λ''
|=  
| =
|Lame’s parameter, GPa or MPa, '''Eq. 13.45'''  
| Lame’s parameter, GPa or MPa, '''Eq. 13.45'''
|-
|-
|''λ''  
| ''λ''
|=  
| =
|wavelength, MPa<sup>−1</sup>, '''Eq. 13.103'''  
| wavelength, MPa<sup>−1</sup>, '''Eq. 13.103'''
|-
|-
|''μ''  
| ''μ''
|=  
| =
|shear modulus, GPa or MPa, '''Eq. 13.42'''  
| shear modulus, GPa or MPa, '''Eq. 13.42'''
|-
|-
|''μ''  
| ''μ''
|=  
| =
|coefficient of internal friction, '''Eq. 13.116'''  
| coefficient of internal friction, '''Eq. 13.116'''
|-
|-
|''μ''<sub>''o''</sub>  
| ''μ''<sub>''o''</sub>
|=  
| =
|mineral shear modulus, GPa or MPa  
| mineral shear modulus, GPa or MPa
|-
|-
|''μ''<sub>''s''</sub>  
| ''μ''<sub>''s''</sub>
|=  
| =
|saturated shear modulus, GPa or MPa  
| saturated shear modulus, GPa or MPa
|-
|-
|''μ''<sub>''sd''</sub>  
| ''μ''<sub>''sd''</sub>
|=  
| =
|dry shear modulus, GPa or MPa  
| dry shear modulus, GPa or MPa
|-
|-
|''μ''*
| ''μ''*
|=  
| =
|effective shear modulus, GPa or MPa  
| effective shear modulus, GPa or MPa
|-
|-
|''μ′''  
| ''μ′''
|=  
| =
|effective crack shear modulus, GPa or MPa  
| effective crack shear modulus, GPa or MPa
|-
|-
|''ν''  
| ''ν''
|=  
| =
|Poisson’s ratio, fractional  
| Poisson’s ratio, fractional
|-
|-
|''ρ''  
| ''ρ''
|=  
| =
|density, kg/m<sup>3</sup> or g/cm<sup>3</sup>  
| density, kg/m<sup>3</sup> or g/cm<sup>3</sup>
|-
|-
|''ρ''<sub>''b''</sub>  
| ''ρ''<sub>''b''</sub>
|=  
| =
|bulk density, kg/m<sup>3</sup> or g/cm<sup>3</sup>  
| bulk density, kg/m<sup>3</sup> or g/cm<sup>3</sup>
|-
|-
|''ρ''<sub>''B''</sub>  
| ''ρ''<sub>''B''</sub>
|=  
| =
|brine density, kg/m<sup>3</sup> or g/cm<sup>3</sup>  
| brine density, kg/m<sup>3</sup> or g/cm<sup>3</sup>
|-
|-
|''ρ''<sub>''d''</sub>  
| ''ρ''<sub>''d''</sub>
|=  
| =
|dry density, kg/m<sup>3</sup> or g/cm<sup>3</sup>  
| dry density, kg/m<sup>3</sup> or g/cm<sup>3</sup>
|-
|-
|''ρ''<sub>''fl''</sub>  
| ''ρ''<sub>''fl''</sub>
|=  
| =
|fluid density, kg/m<sup>3</sup> or g/cm<sup>3</sup>  
| fluid density, kg/m<sup>3</sup> or g/cm<sup>3</sup>
|-
|-
|''ρ''<sub>''g''</sub>  
| ''ρ''<sub>''g''</sub>
|=  
| =
|grain or mineral density, kg/m<sup>3</sup> or g/cm<sup>3</sup>  
| grain or mineral density, kg/m<sup>3</sup> or g/cm<sup>3</sup>
|-
|-
|''ρ''<sub>''G''</sub>  
| ''ρ''<sub>''G''</sub>
|=  
| =
|gas density, kg/m<sup>3</sup> or g/cm<sup>3</sup>  
| gas density, kg/m<sup>3</sup> or g/cm<sup>3</sup>
|-
|-
|''ρ''<sub>''O''</sub>  
| ''ρ''<sub>''O''</sub>
|=  
| =
|oil density, kg/m<sup>3</sup> or g/cm<sup>3</sup>  
| oil density, kg/m<sup>3</sup> or g/cm<sup>3</sup>
|-
|-
|''ρ''<sub>sat</sub>  
| ''ρ''<sub>sat</sub>
|=  
| =
|saturated density, kg/m<sup>3</sup> or g/cm<sup>3</sup>  
| saturated density, kg/m<sup>3</sup> or g/cm<sup>3</sup>
|-
|-
|''ρ''<sub>''W''</sub>  
| ''ρ''<sub>''W''</sub>
|=  
| =
|water density, kg/m<sup>3</sup> or g/cm<sup>3</sup>  
| water density, kg/m<sup>3</sup> or g/cm<sup>3</sup>
|-
|-
|''σ''<sub>''h''</sub>  
| ''σ''<sub>''h''</sub>
|=  
| =
|horizontal stress, GPa or MPa  
| horizontal stress, GPa or MPa
|-
|-
|''σ''<sub>''ij''</sub>  
| ''σ''<sub>''ij''</sub>
|=  
| =
|stress components, GPa or MPa  
| stress components, GPa or MPa
|-
|-
|''σ''<sub>''m''</sub>  
| ''σ''<sub>''m''</sub>
|=  
| =
|mean stress, GPa or MPa  
| mean stress, GPa or MPa
|-
|-
|''σ''<sub>''n''</sub>  
| ''σ''<sub>''n''</sub>
|=  
| =
|normal stress, GPa or MPa  
| normal stress, GPa or MPa
|-
|-
|''σ''<sub>shear</sub>  
| ''σ''<sub>shear</sub>
|=  
| =
|shear stress components, GPa or MPa  
| shear stress components, GPa or MPa
|-
|-
|''σ''<sub>''v''</sub>  
| ''σ''<sub>''v''</sub>
|=  
| =
|axial (vertical) stress, GPa or MPa  
| axial (vertical) stress, GPa or MPa
|-
|-
|''σ''<sub>''zz''</sub>  
| ''σ''<sub>''zz''</sub>
|=  
| =
|vertical stress component, GPa or MPa  
| vertical stress component, GPa or MPa
|-
|-
|''σ''<sub>1</sub>  
| ''σ''<sub>1</sub>
|=  
| =
|stress in direction 1, GPa or MPa  
| stress in direction 1, GPa or MPa
|-
|-
|''σ''<sub>3</sub>  
| ''σ''<sub>3</sub>
|=  
| =
|stress in direction 3, GPa or MPa  
| stress in direction 3, GPa or MPa
|-
|-
|''τ''  
| ''τ''
|=  
| =
|shear stress, GPa or MPa  
| shear stress, GPa or MPa
|-
|-
|''τ''  
| ''τ''
|=  
| =
|relaxation time, s<sup>–1</sup> (radians/s), '''Eq. 13.106'''  
| relaxation time, s<sup>–1</sup> (radians/s), '''Eq. 13.106'''
|-
|-
|''Φ''  
| ''Φ''
|=  
| =
|porosity  
| porosity
|-
|-
|''Φ''<sub>''fx''</sub>  
| ''Φ''<sub>''fx''</sub>
|=  
| =
|fracture porosity  
| fracture porosity
|-
|-
|''Φ''<sub>''p-e''</sub>  
| ''Φ''<sub>''p-e''</sub>
|=  
| =
|effective porosity  
| effective porosity
|-
|-
|''Φ''<sub>''p-iso''</sub>  
| ''Φ''<sub>''p-iso''</sub>
|=  
| =
|isolated, ineffective porosity  
| isolated, ineffective porosity
|-
|-
|''ω''  
| ''ω''
|=  
| =
|frequency (radian), s<sup>–1</sup> (radians/s)  
| frequency (radian), s<sup>–1</sup> (radians/s)
|-
|-
|''ω''<sub>''c''</sub>  
| ''ω''<sub>''c''</sub>
|=  
| =
|crossover frequency (radian), s<sup>–1</sup> (radians/s)  
| crossover frequency (radian), s<sup>–1</sup> (radians/s)
|}
|}
<br>
 
</div></div>
 
<div class="toccolours mw-collapsible mw-collapsed" >
</div></div><div class="toccolours mw-collapsible mw-collapsed">
== References ==
== References ==
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
<br>
<br/><references />
<references>
</div></div><div class="toccolours mw-collapsible mw-collapsed">
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<ref name="r109">Allison, R.J. 1987. Non-destructive determination of Young’s modulus and its relationship with compressive strength, porosity, and density. ''Deformation of Sediments and Sedimentary Rocks'', vol. 29, 63-69, M.E. Jones and R.M.F. Preston eds. London: Geological Society of London Special Publication 29. http://dx.doi.org/10.1144/GSL.SP.1987.029.01.06. </ref>
 
<ref name="r110">Kazi, A., Sen, Z., and  Sadagah, B.-E.H. 1983. Relationship Between Sonic Pulse Velocity And Uniaxial Compressive Strength Of Rocks. Presented at the 24th U.S. Symposium on Rock Mechanics (USRMS), College Station, Texas, USA, 20-23 June. ARMA-83-0409. </ref>
 
<ref name="r111">Bienawski, Z.T. 1974. Estimating the strength of rock materials. ''J. S. Afr. Inst. Min. Metall.'' '''74''' (8): 312-320. http://www.saimm.co.za/Journal/v074n08p312.pdf.</ref>
 
<ref name="r112">Hoek, E. and Brown, E.T. 1982. ''Underground Excavations in Rock''. Amsterdam, The Netherlands: Elsevier Applied Science. </ref>
 
<ref name="r113">Yudhbir, Lemanza, W., and  Prinzl, F. 1983. An empirical failure criterion for rock masses. Presented at the 5th ISRM Congress, Melbourne, Australia, 10-15 April. ISRM-5CONGRESS-1983-042. </ref>
 
<ref name="r114">Dowla, N., Hayatdavoudi, A., Ghalambor, A. et al. 1990. Laboratory investigation of saturation effect on mechanical properties of rocks. Presented at the 31st SPWLA Annual Logging Symposium, Lafayette, Louisiana, USA, 24–27 June. Paper EE. </ref>
 
<ref name="r115">Kowalski, W.C. 1966. The interdependence between the strength and voids ratio of limestones and marls in connection with their water saturation and anisotropy. ''Proc.'', 1966 Congress of the Intl. Society of Rock Mechanics, Lisbon, 143. </ref>
 
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<ref name="r118">Smordinov et al., reported in Howarth, D.F. 1987. The effect of pre-existing microcavities on mechanical rock performance in sedimentary and crystalline rocks. ''Int. J. Rock Mech. Min. Sci. & Geomech. Abstracts'' '''24''' (4): 223-233. http://dx.doi.org/http://dx.doi.org/10.1016/0148-9062(87)90177-X. </ref>
 
<ref name="r119">Raymer, L.L., Hunt, E.R., and  Gardner, J.S. 1980. An Improved Sonic Transit Time-To-Porosity Transform. Presented at the SPWLA 21st Annual Logging Symposium, Lafayette, Louisiana, USA, 8–11 July. Paper P. </ref>
 
<ref name="r120">Corbett, K., Friedman, M., and  Spang, J. 1987. Fracture Development and Mechanical Stratigraphy of Austin Chalk, Texas. ''AAPG Bull.'' '''71''' (1): 17-28. http://dx.doi.org/10.1306%2F94886D35-1704-11D7-8645000102C1865D. </ref>
 
<ref name="r121">Dobereiner, L. and De Freitas, M.H. 1986. Geotechnical properties of weak sandstones. ''Geotechnique'' '''36''' (1): 79-94. http://dx.doi.org/10.1680/geot.1986.36.1.79. </ref>
 
<ref name="r122">Steiger, R.P. and Leung, P.K. 1989. Predictions of wellbore stability in shale formations at great depth. In ''Rock at Great Depth: Rock Mechanics and Rock Physics at Great Depth—Proceedings of an International Symposium, Pau, 28–31 August 1989'', ed. V. Maury and D. Fourmaintraux, Vol. 3, 1209. London: Taylor & Francis. </ref>
 
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<ref name="r128">Meredith, P.G. and Atkinson, B.K. 1983. Stress corrosion and acoustic emission during tensile crack propagation in Whin Sill dolerite and other basic rocks. ''Geophys. J. R. Astron. Soc.'' '''75''' (1): 1-21. http://dx.doi.org/10.1111/j.1365-246X.1983.tb01911.x. </ref>
 
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</references>
<br>
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" >
 
== General References ==
== General References ==
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
<br>
<br/>Aldrich, M.J. 1969. Pore Pressure Effects on Berea Sandstone Subjected to Experimental Deformation. ''Geol. Soc. Am. Bull.'' '''80''' (8): 1577-1586. [http://dx.doi.org/10.1130/0016-7606(1969)80 http://dx.doi.org/10.1130/0016-7606(1969)80][1577:ppeobs]2.0.co;2.
Aldrich, M.J. 1969. Pore Pressure Effects on Berea Sandstone Subjected to Experimental Deformation. ''Geol. Soc. Am. Bull.'' '''80''' (8): 1577-1586. http://dx.doi.org/10.1130/0016-7606(1969)80[1577:ppeobs]2.0.co;2.


Anderson, O.L. and Lieberman, R.C. 1966. Sound velocities in rocks and minerals. VESIAC State-of-the-Art Report No. 7885-4-X, University of Michigan, Ann Arbor, Michigan.
Anderson, O.L. and Lieberman, R.C. 1966. Sound velocities in rocks and minerals. VESIAC State-of-the-Art Report No. 7885-4-X, University of Michigan, Ann Arbor, Michigan.
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Anderson, R.G. and Castagna, J.P. 1984. Analysis of sonic log compressional wave amplitudes using borehole compensation techniques. Presented at the SPWLA 25th Annual Logging Symposium, New Orleans, 10–13 June. SPWLA-1984-K.
Anderson, R.G. and Castagna, J.P. 1984. Analysis of sonic log compressional wave amplitudes using borehole compensation techniques. Presented at the SPWLA 25th Annual Logging Symposium, New Orleans, 10–13 June. SPWLA-1984-K.


Atkinson, B.K. 1984. Subcritical crack growth in geological materials. ''Journal of Geophysical Research: Solid Earth'' '''89''' (B6): 4077-4114. http://dx.doi.org/10.1029/JB089iB06p04077.
Atkinson, B.K. 1984. Subcritical crack growth in geological materials. ''Journal of Geophysical Research: Solid Earth'' '''89''' (B6): 4077-4114. [http://dx.doi.org/10.1029/JB089iB06p04077 http://dx.doi.org/10.1029/JB089iB06p04077].


Ateksandrov, K.S. and Ryzhova, T.V. 1961. The elastic properties of crystals. ''Soviet Physics-Crystallography'' '''6''': 228.
Ateksandrov, K.S. and Ryzhova, T.V. 1961. The elastic properties of crystals. ''Soviet Physics-Crystallography'' '''6''': 228.


Backus, G.E. 1962. Long-wave elastic anisotropy produced by horizontal layering. ''J. Geophys. Res.'' '''67''' (11): 4427-4440. http://dx.doi.org/10.1029/JZ067i011p04427.
Backus, G.E. 1962. Long-wave elastic anisotropy produced by horizontal layering. ''J. Geophys. Res.'' '''67''' (11): 4427-4440. [http://dx.doi.org/10.1029/JZ067i011p04427 http://dx.doi.org/10.1029/JZ067i011p04427].


Billings, M.P. 1972. ''Structural Geology''. New York: Prentice-Hall.
Billings, M.P. 1972. ''Structural Geology''. New York: Prentice-Hall.


Biot, M.A. 1962. Mechanics of Deformation and Acoustic Propagation in Porous Media. ''J. Appl. Phys.'' '''33''' (4): 1482-1498. http://dx.doi.org/10.1063/1.1728759.
Biot, M.A. 1962. Mechanics of Deformation and Acoustic Propagation in Porous Media. ''J. Appl. Phys.'' '''33''' (4): 1482-1498. [http://dx.doi.org/10.1063/1.1728759 http://dx.doi.org/10.1063/1.1728759].


Birch, F. 1966. Compressibility; elastic constants. In ''Handbook of Physical Constants'', revised edition, S.P. Clark, No. 87, 97–174. Boulder, Colorado: GSA Memoir, Geological Society of America.
Birch, F. 1966. Compressibility; elastic constants. In ''Handbook of Physical Constants'', revised edition, S.P. Clark, No. 87, 97–174. Boulder, Colorado: GSA Memoir, Geological Society of America.


Blatt, H., Middleton, G., and Murray, R. 1972. ''Origin of Sedimentary Rocks''. Englewood Cliffs, New Jersey: Prentice Hall.
Blatt, H., Middleton, G., and Murray, R. 1972. ''Origin of Sedimentary Rocks''. Englewood Cliffs, New Jersey: Prentice Hall.


Boozer, G.D., Hiller, K.H., and Serdengecti, A. 1962. Effects of pore fluids on the deformation behavior of rocks subjected to triaxial compression. Presented at the 1962 ISRM Annual Symposium, Fairhurst, Minnesota, USA, May 1962. Paper C.
Boozer, G.D., Hiller, K.H., and Serdengecti, A. 1962. Effects of pore fluids on the deformation behavior of rocks subjected to triaxial compression. Presented at the 1962 ISRM Annual Symposium, Fairhurst, Minnesota, USA, May 1962. Paper C.


Boretti-Onyszkiewicz, W. 1966. Joints in the flysch sandstones on the ground of strength examinations. Proc., First Congress of International Society of Rock Mechanics, Lisbon, Portugal, 25 September, Vol. I.
Boretti-Onyszkiewicz, W. 1966. Joints in the flysch sandstones on the ground of strength examinations. Proc., First Congress of International Society of Rock Mechanics, Lisbon, Portugal, 25 September, Vol. I.
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Brook, N. 1979. Estimating the triaxial strength of rocks. ''Int. J. Rock Mech. Min. Sci. & Geomech. Abstracts'' '''16''': 261.
Brook, N. 1979. Estimating the triaxial strength of rocks. ''Int. J. Rock Mech. Min. Sci. & Geomech. Abstracts'' '''16''': 261.


Brunauer, S., Kantro, D.L., and Weise, C.H. 1956. The surface energies of amorphous silica and hydrous amorphous silica. ''Can. J. Chem.'' '''34''' (10): 1483-1496. http://dx.doi.org/10.1139/v56-190.
Brunauer, S., Kantro, D.L., and Weise, C.H. 1956. The surface energies of amorphous silica and hydrous amorphous silica. ''Can. J. Chem.'' '''34''' (10): 1483-1496. [http://dx.doi.org/10.1139/v56-190 http://dx.doi.org/10.1139/v56-190].


Bulau, J.R., Tittmann, B.R., Abdel-Gawad, M. et al. 1984. The role of aqueous fluids in the internal friction of rock. ''Journal of Geophysical Research: Solid Earth'' '''89''' (B6): 4207-4212. http://dx.doi.org/10.1029/JB089iB06p04207.
Bulau, J.R., Tittmann, B.R., Abdel-Gawad, M. et al. 1984. The role of aqueous fluids in the internal friction of rock. ''Journal of Geophysical Research: Solid Earth'' '''89''' (B6): 4207-4212. [http://dx.doi.org/10.1029/JB089iB06p04207 http://dx.doi.org/10.1029/JB089iB06p04207].


Campbell, F.A. and Oliver, T.A. 1968. Mineralogic and chemical composition of Freton and Duvernay Formations, Central Alberta. ''Bull. Can. Petrol. Geol.'' '''16''': 40.
Campbell, F.A. and Oliver, T.A. 1968. Mineralogic and chemical composition of Freton and Duvernay Formations, Central Alberta. ''Bull. Can. Petrol. Geol.'' '''16''': 40.
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Castagna, J.P. 1985. Shear-wave Time-average Equation For Sandstones. Presented at the 55th Annual SEG Convention, Washington, DC, October 1985.
Castagna, J.P. 1985. Shear-wave Time-average Equation For Sandstones. Presented at the 55th Annual SEG Convention, Washington, DC, October 1985.


Culberson, O.L. and McKetta, J.J. 1951. Phase Equilibria In Hydrocarbon-Water Systems III—The Solubility of Methane in Water at Pressures to 10,000 PSIA. In ''Transactions of the American Institute of Mining, Metallurgical, & Petroleum Engineers'', Vol. 192, 223–226. Dallas, Texas: Society of Petroleum Engineers.  
Culberson, O.L. and McKetta, J.J. 1951. Phase Equilibria In Hydrocarbon-Water Systems III—The Solubility of Methane in Water at Pressures to 10,000 PSIA. In ''Transactions of the American Institute of Mining, Metallurgical, & Petroleum Engineers'', Vol. 192, 223–226. Dallas, Texas: Society of Petroleum Engineers.


Dandekar, D.P. 1968. Pressure Dependence of the Elastic Constants of Calcite. ''Physical Review'' '''172''' (3): 873-877. http://dx.doi.org/10.1103/PhysRev.172.873.PR.
Dandekar, D.P. 1968. Pressure Dependence of the Elastic Constants of Calcite. ''Physical Review'' '''172''' (3): 873-877. [http://dx.doi.org/10.1103/PhysRev.172.873.PR http://dx.doi.org/10.1103/PhysRev.172.873.PR].


D’Andrea, D.V., Fischer, R.L., and Fogelson, D.E. 1965. Prediction of compressive strength from other rock properties. Report of Investigations 6702, US Department of the Interior, Bureau of Mines, Washington, DC.
D’Andrea, D.V., Fischer, R.L., and Fogelson, D.E. 1965. Prediction of compressive strength from other rock properties. Report of Investigations 6702, US Department of the Interior, Bureau of Mines, Washington, DC.


Davey, F.J. and Cooper, A.K. 1987. Gravity studies of the Victoria Land Basin and Iselin Bank. In The Antarctic Continental Margin and Geophysics of the Western Ross Sea, A.K. Cooper and F.J. Davey. Houston, Texas: CPCEMR Earth Science Series, Circum-Pacific Council for Energy and Mineral Resources.
Davey, F.J. and Cooper, A.K. 1987. Gravity studies of the Victoria Land Basin and Iselin Bank. In The Antarctic Continental Margin and Geophysics of the Western Ross Sea, A.K. Cooper and F.J. Davey. Houston, Texas: CPCEMR Earth Science Series, Circum-Pacific Council for Energy and Mineral Resources.


DeVilbiss, J., Ito, H., and Nur, A. 1979. Measurement of compressional and shear wave velocities of water filled rocks during water-steam transition. 1979. ''Geophysics'' '''44''' (3): 407.  
DeVilbiss, J., Ito, H., and Nur, A. 1979. Measurement of compressional and shear wave velocities of water filled rocks during water-steam transition. 1979. ''Geophysics'' '''44''' (3): 407.


Dickey, P.A. 1966. Patterns of chemical composition in deep subsurface waters. ''AAPG Bull.'' '''50''' (11): 2472-2478. http://dx.doi.org/10.1306%2F5D25B775-16C1-11D7-8645000102C1865D.
Dickey, P.A. 1966. Patterns of chemical composition in deep subsurface waters. ''AAPG Bull.'' '''50''' (11): 2472-2478. [http://dx.doi.org/10.1306/5D25B775-16C1-11D7-8645000102C1865D http://dx.doi.org/10.1306%2F5D25B775-16C1-11D7-8645000102C1865D].


Dodson, C.R. and Standing, M.B. 1944. Pressure-volume-temperature and solubility relations for natural-gas-water mixtures. ''API Drilling and Production Practice'' (1944): 173–179.
Dodson, C.R. and Standing, M.B. 1944. Pressure-volume-temperature and solubility relations for natural-gas-water mixtures. ''API Drilling and Production Practice'' (1944): 173–179.


Domenico, S.N. 1976. Effect of brine-gas mixture on velocity in an unconsolidated sand reservoir. ''Geophysics'' '''41''' (5): 882-894. http://dx.doi.org/10.1190/1.1440670.
Domenico, S.N. 1976. Effect of brine-gas mixture on velocity in an unconsolidated sand reservoir. ''Geophysics'' '''41''' (5): 882-894. [http://dx.doi.org/10.1190/1.1440670 http://dx.doi.org/10.1190/1.1440670].


Domenico, S.N. 1977. Elastic properties of unconsolidated porous sand reservoirs. ''Geophysics'' '''42''' (7): 1339-1368. http://dx.doi.org/10.1190/1.1440797.
Domenico, S.N. 1977. Elastic properties of unconsolidated porous sand reservoirs. ''Geophysics'' '''42''' (7): 1339-1368. [http://dx.doi.org/10.1190/1.1440797 http://dx.doi.org/10.1190/1.1440797].


Domenico, S.N. 1984. Rock lithology and porosity determination from shear and compressional wave velocity. Geophysics 49 (8): 1188-1195. http://dx.doi.org/10.1190/1.1441748.
Domenico, S.N. 1984. Rock lithology and porosity determination from shear and compressional wave velocity. Geophysics 49 (8): 1188-1195. [http://dx.doi.org/10.1190/1.1441748 http://dx.doi.org/10.1190/1.1441748].


Dvorkin, J. and Nur, A. 1993. Dynamic poroelasticity: A unified model with the squirt and the Biot mechanisms. Geophysics 58 (4): 524-533. http://dx.doi.org/10.1190/1.1443435.
Dvorkin, J. and Nur, A. 1993. Dynamic poroelasticity: A unified model with the squirt and the Biot mechanisms. Geophysics 58 (4): 524-533. [http://dx.doi.org/10.1190/1.1443435 http://dx.doi.org/10.1190/1.1443435].


Eastwood, R.L. and Castagna, J.P. 1986. Interpretation of V/V ratios from sonic logs. In ''Shear Wave Exploration'', S.H. Danbom and S.N. Domenico, No. 1. Tulsa, Oklahoma: Geophysical Developments, SEG.
Eastwood, R.L. and Castagna, J.P. 1986. Interpretation of V/V ratios from sonic logs. In ''Shear Wave Exploration'', S.H. Danbom and S.N. Domenico, No. 1. Tulsa, Oklahoma: Geophysical Developments, SEG.
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Freed, R.L. 1980. Shale mineralogy of the No. 1 Pleasant Bayou geothermal test well: A progress report. ''Proc.'', 4th U.S. Gulf Coast Geopressured-Geothermal Energy Conference: Research and Development, Austin, Texas, 29-31 October, 153-167.
Freed, R.L. 1980. Shale mineralogy of the No. 1 Pleasant Bayou geothermal test well: A progress report. ''Proc.'', 4th U.S. Gulf Coast Geopressured-Geothermal Energy Conference: Research and Development, Austin, Texas, 29-31 October, 153-167.


Freund, D. 1992. Ultrasonic compressional and shear velocities in dry clastic rocks as a function of porosity, clay content, and confining pressure. ''Geophys. J. Int.'' '''108''' (1): 125-135. http://dx.doi.org/10.1111/j.1365-246X.1992.tb00843.x.
Freund, D. 1992. Ultrasonic compressional and shear velocities in dry clastic rocks as a function of porosity, clay content, and confining pressure. ''Geophys. J. Int.'' '''108''' (1): 125-135. [http://dx.doi.org/10.1111/j.1365-246X.1992.tb00843.x http://dx.doi.org/10.1111/j.1365-246X.1992.tb00843.x].


Ganley, D.C. and Kanasewich, E.R. 1980. Measurement of absorption and dispersion from check shot surveys. ''Journal of Geophysical Research: Solid Earth'' '''85''' (B10): 5219-5226. http://dx.doi.org/10.1029/JB085iB10p05219.
Ganley, D.C. and Kanasewich, E.R. 1980. Measurement of absorption and dispersion from check shot surveys. ''Journal of Geophysical Research: Solid Earth'' '''85''' (B10): 5219-5226. [http://dx.doi.org/10.1029/JB085iB10p05219 http://dx.doi.org/10.1029/JB085iB10p05219].


Gardner, G., Gardner, L., and Gregory, A. 1974. Formation velocity and density—The diagnostic basis for stratigraphic traps. ''Geophysics'' '''39''' (6): 770-780. http://dx.doi.org/10.1190/1.1440465.
Gardner, G., Gardner, L., and Gregory, A. 1974. Formation velocity and density—The diagnostic basis for stratigraphic traps. ''Geophysics'' '''39''' (6): 770-780. [http://dx.doi.org/10.1190/1.1440465 http://dx.doi.org/10.1190/1.1440465].


Gassmann, F. 1951. Über die elastizität poröser medien. ''Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich'' '''96''': 1–23.
Gassmann, F. 1951. Über die elastizität poröser medien. ''Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich'' '''96''': 1–23.


Greenberg, M.L. and Castagna, J.P. 1992. Shear wave velocity estimation in porous rocks: theoretical formulation, prelimining verification and applications. ''Geophys. Prospect.'' '''40''' (2): 195-209. http://dx.doi.org/10.1111/j.1365-2478.1992.tb00371.x.
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Greenhaigh, S.A. and Emerson, D.W. 1982. Physical properties of Permian bituminous coals from the Sydney Basin, New South Wales. Presented at the 1981 SEG Annual International Meeting, Los Angeles, California, USA, 11 October.
Greenhaigh, S.A. and Emerson, D.W. 1982. Physical properties of Permian bituminous coals from the Sydney Basin, New South Wales. Presented at the 1981 SEG Annual International Meeting, Los Angeles, California, USA, 11 October.


Gregory, A. 1976. Fluid saturation effects on dynamic elastic properties of sedimentary rocks. ''Geophysics'' '''41''' (5): 895-921. http://dx.doi.org/10.1190/1.1440671.
Gregory, A. 1976. Fluid saturation effects on dynamic elastic properties of sedimentary rocks. ''Geophysics'' '''41''' (5): 895-921. [http://dx.doi.org/10.1190/1.1440671 http://dx.doi.org/10.1190/1.1440671].


Gretener, P.E. 1979. ''Pore Pressure: Fundamentals, General Ramifications and Implications for Structural Geology (revised)'', No. 4. Tulsa, Oklahoma: AAPG Continuing Education Course Note Series, American Association of Petroleum Geologists.
Gretener, P.E. 1979. ''Pore Pressure: Fundamentals, General Ramifications and Implications for Structural Geology (revised)'', No. 4. Tulsa, Oklahoma: AAPG Continuing Education Course Note Series, American Association of Petroleum Geologists.
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Griggs, D.T. 1967. Hydrologic weakening of quartz and other silicates. ''Geophys. J. R. Astron. Soc.'' '''14''': 19–31.
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Yin, C.S. 1992. ''Wave attenuation in partially saturated porous solids''. PhD dissertation, Stanford University, Palo Alto, California.
Yin, C.S. 1992. ''Wave attenuation in partially saturated porous solids''. PhD dissertation, Stanford University, Palo Alto, California.
<br>
</div></div><div class="toccolours mw-collapsible mw-collapsed">
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" >
 
== SI Metric Conversion Factors ==
== SI Metric Conversion Factors ==
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
<br>
 
{|  
 
|°API  
{|
|
|-
|141.5/(131.5 + °API)  
| °API
|
|  
|=  
| 141.5/(131.5 + °API)
|g/cm<sup>3</sup>  
|  
| =
| g/cm<sup>3</sup>
|-
|-
|bbl  
| bbl
|×  
| ×
|1.589 873  
| 1.589 873
|E–01  
| E–01
|=  
| =
|m<sup>3</sup>  
| m<sup>3</sup>
|-
|-
|ft  
| ft
|×  
| ×
|3.048*  
| 3.048*
|E–01  
| E–01
|=  
| =
|m  
| m
|-
|-
|ft<sup>3</sup>  
| ft<sup>3</sup>
|×  
| ×
|2.831 685  
| 2.831 685
|E–02  
| E–02
|=  
| =
|m<sup>3</sup>  
| m<sup>3</sup>
|-
|-
|°F  
| °F
|
|  
|(°F−32)/1.8  
| (°F−32)/1.8
|
|  
|=  
| =
|°C  
| °C
|-
|-
|psi  
| psi
|×  
| ×
|6.894 757  
| 6.894 757
|E + 00  
| E + 00
|=  
| =
|kPa  
| kPa
|}
|}
<nowiki>*</nowiki>Conversion factor is exact.
</div></div>


[[Category:PEH]]


[[Category: 1.2.3 Rock properties]]
<nowiki>*</nowiki>
Conversion factor is exact.</div></div>[[Category:PEH]] [[Category:Volume I – General Engineering]]  [[Category:1.2.3 Rock properties]]

Latest revision as of 16:32, 26 April 2017

Publication Information

Vol1GECover.png

Petroleum Engineering Handbook

Larry W. Lake, Editor-in-Chief

Volume I – General Engineering

John R. Fanchi, Editor

Chapter 13 – Rock Properties

M. Batzle, Colorado School of Mines, D.-H. Han, U. of Houston, and R. Hofmann, Colorado School of Mines

Pgs. 571-685

ISBN 978-1-55563-108-6
Get permission for reuse



Introduction


Rock and fluid properties provide the common denominator around which we build the models, interpretations, and predictions of petroleum engineering, as well as geology and geophysics. We consider here the properties of sedimentary rocks, particularly those that make up hydrocarbon reservoirs. Usually, these consist of sandstones, limestones, and dolomites. We must be more inclusive, and consider rocks such as shales, evaporates, and diatomites because these provide the seals, bounding materials, or source rocks to our reservoirs. It is important to note that shales and claystones make up the most abundant rock type in the typical sedimentary column. Features such as seismic signature depend as much on the enclosing shale as on the reservoir sands.

In this chapter, we will tabulate important mineral and rock properties, and provide many of the mathematical models used to describe and predict properties. Much of this summary is drawn upon the extensive work and compellations already available. As examples, Clark[1] provides an extensive list of mineral and rock properties; Birch[2] presents tables of compressional velocities, and Gregory[3] gives a detailed overview of the use of rock property information in seismic interpretation. Castagna et al.[4] focused on rock properties for use in amplitude versus offset analyses. Useful handbooks on this topic include Carmichael[5] and Lama and Vutukuri.[6][7] Probably the best reference covering a wide range of rock property formulas and models is Mavko et al.[8] These references can be consulted for details not presented here.

Knowledge of Rock Properties Is Largely Empirical

Many theoretical models have been developed to predict or correlate specific physical properties of porous rock. Most theoretical models are built on simplified physical concepts: what are the properties of an ideal porous media. However, in comparison with real rocks, these models are always oversimplified (they must be, to be solvable). Most of these models are capable of "forward modeling" or predicting rock properties with one or more arbitrary parameters. However, as is typical in Earth science, our models cannot be inverted from measurements to predict uniquely real rock and pore-fluid properties. Many efforts have been made and will continue to be made to build porous rock models, but progress is very limited. Some of the most fundamental questions are still unanswered.

To establish the basic relationships between physical properties and rock parameters, laboratory investigations are made. Laboratory measurements of rock samples can provide controlled conditions and high data quality ("hard data"). These relationships can be extended to a larger scale, or can even be made scaleless. Typically, models and relations based on laboratory data are then applied to in-situ measurements to derive the parameters we actually need (say, permeability) from information we can actually collect (say, density and gamma ray radiation). The relative merits and problems associated with several rock and fluid measurement techniques are presented in Table 13.1.


Although many empirical relationships already have been established, when facing a frontier basin, new development areas, or untested portions of known formations, valid prediction of rock properties usually requires core data (including "sidewall" plugs). For many applications, standard trend data may not be adequate. A broad investigation is needed.

Philosophy for Rock Properties

Many of the factors affecting rock properties are incompletely ascertained. For example, acoustic velocities can be affected by numerous parameters, many of which cannot be measured. In addressing a rock physics problem, the following aspects should be remembered:

  • There may be no exact solution.
  • Rock properties are controlled by rock parameters, and these physical correlations can be examined and recognized (although perhaps not understood).
  • Often nature gives us a break. At certain conditions, relationships between the rock properties and rock parameters can be simplified (such as Archie’s Law).
  • We usually must settle on imperfect solutions with some uncertainty. Statistical trends or high and low bounds might be used to handle the uncertainty.
  • Every measurement is, to some degree, wrong. The question is: Can we tolerate the errors and understand how they propagate through our analyses?

We will begin this chapter with a suite of definitions and examples, then move on to data and models of individual properties. By necessity, we will be restricted in the material we can cover in a single chapter. As a result, we will not go into many details of rock fabrics and petrography. Also, with a few exceptions, the information provided here assumes that rocks are homogeneous and isotropic.


Rocks: Minerals Plus Pores


Rocks are defined for our purposes as aggregates or mixtures of minerals plus pores. The three general rock types are classified as igneous, metamorphic, and sedimentary. Although hydrocarbon reservoirs have been found in all three rock types, we will consider here primarily sedimentary rocks, by far the most common rocks associated with hydrocarbons.

Minerals are defined as naturally occurring solids: They have a definite structure, composition, and suite of properties that are either fixed or vary systematically within a definite range. Although there are dozens of elements and hundreds of described minerals available in the Earth’s crust, the actual number that we must concern ourselves with for reservoir engineering purposes is remarkably small. Classification can be broken into silicates, carbonates, sulfates, sulfides, and oxides. In addition, "solid" organic mixtures such as coal or bitumen can be abundant. Common sedimentary silicates include quartz, feldspars, micas, zeolites, and clays. Carbonates usually consist of calcite and dolomite, although siderite may be present. Gypsum and anhydrite are the most common sulfates, with pyrite the typical sulfide. Oxides are usually materials such as magnetite and hematite. For most of our purposes, we can further restrict our attention to the subset of quartz, feldspars, clays, calcite, dolomite, and anhydrite. A working knowledge of six or so minerals fulfills most engineering needs.

Clays represent an entire family of minerals with widely differing properties. This situation is compounded by the fact that clays are among the most abundant minerals in the sedimentary section. Clays are also problematic because their properties can vary with the in-situ pressure, temperature, and chemical environment. These issues have led to an unfortunate bias against clays when measuring or describing rocks. A "clean" sand, for example, is one that has little or no clay. "Dirty" sandstones or limestones have significant amounts of clay. Clays and their influence on rock properties remain poorly understood and continue to be an area requiring intensive research.

The properties of primary engineering interest are often controlled more by the rock fabric than by the bulk composition. The "holes" are usually more important than the mineral frame. With the following few examples, we will see many of the most common sedimentary rock forms and textures. Numerous attempts have been made to extract rock properties from images of the rock and pore space.[9][10][11] These techniques often work well, but depend on the observation scale, representative nature of the image, and internal heterogeneity.

A thin section of clean sandstone is shown in Fig. 13.1. Under plane-polarized light, quartz grains appear white and pores are stained blue. This is a high-porosity, friable sample that has not undergone substantial consolidation. Silica cement can be seen coating the individual grains and bonding the largely unchanged, rounded quartz grains. Grain-to-grain stress is indicated by the fractures radiating from points of grain contact. Although these fractures have a relatively small volume, they have a disproportionately large influence on the mechanical properties, particularly the pressure dependence. With continued diagenesis, quartz grains typically would become intergrown, and large amounts of cement would develop, reducing the pore volume.


A Scanning Electron Microscope (SEM) image of another sandstone is seen in Fig. 13.2. A higher degree of compaction is indicated here by the intergrown, sutured contacts of the quartz grains (gray areas). A grain undergoing alteration (a) as well as some of the matrix quartz (b) contain isolated, ineffective porosity. Fractures are again present, particularly near point of grain contact. Many of these fractures, however, may be caused by stress relief as the sample was cored, or by the cutting and polishing. The most obvious features are the contorted and rotated mica grains (d). These micas were crushed due to compaction, and now host numerous sets of parallel fractures. Some diagenetic clays are also beginning to grow in the pore spaces and act as a cement.


A cementation "front" is visible in Fig. 13.3. Cements come in a wide variety of forms. Open pores are black in the SEM image. In this case, the lighter gray calcite has filled the pores in the lower portion of the image. Unlike the dispersed silica and clay cements seen in the previous figures, the calcite is deposited with an abrupt front. This kind of texture is common for carbonate cements in sands and is probably caused by the availability of crystal nucleation sites available to a slightly supersaturated pore fluid. We would obviously expect vastly different properties of the uncemented vs. cemented portions separated by only a few grain diameters. This rock is an example of the extreme heterogeneity that can frequently occur even within the same small geologic unit of the same formation.


Carbonates can have extremely complex textures resulting form the mixture of fossils and matrix building the rock. In Fig. 13.4, an optical image demonstrates the multitude of forms that can be present. Shell fragments appear as crescent shapes in cross section. Much of the material between fragments can be filled with carbonate mud, reducing the porosity substantially. In this sample, bulk porosity is dominated by the larger disconnected vugs. Such vugs can occur as parts of fossils or as a result of chemical dissolution after deposition. Here, a coating of crystals has grown on the vug surfaces. Because of the wide range of sizes, shapes, and compositions that can occur in carbonate rocks, they are often difficult to characterize with core or even log sampling.


Dolomites are usually formed by recrystallization of original aragonite or calcite crystals in sediments. Magnesium in the pore fluids replace some of the calcium, forming a Mg-Ca carbonate structure. Because of the greater density of dolomite, this transformation can include a porosity increase. Sometimes, the replacement can be subtle, and original sedimentary structures and fossil forms can be preserved. Often, however, the recrystalliztion largely destroys the original rock fabric and rhombohedral dolomite crystals appear, as at (a) in Fig. 13.5. The other intergrown dolomite crystals form porosity that is polygonal. In this sample, many of the pores are coated (b) with pyrobitumin, a complex organic material similar to coal. This pyrobitumen is sometimes incased within dolomite crystals. In this case, it will lower the apparent grain density and strength of the rock.


As mentioned, clays are among the most abundant minerals. These minerals can influence or control physical properties to a major degree. In addition, many clays are sensitive to the environment and will change properties and forms under different conditions. An example of such "sensitive" clay fabrics is shown in Fig. 13.6. Note that the scale is much finer here than in previous figures. In Fig. 13.6a, chlorite originally coats the quartz grains. On top of the chlorite, a smectite coating was developed. This core sample was allowed to dry, and the smectite collapsed, forming long slender columns in the pore space. Resaturating the rock with distilled water allowed the smectite coating to expand and fill the pore space (Fig. 13.6b). The closed pores will obviously have different fluid-flow characteristics. In this case, we cannot assume the mineral in is a passive, inert solid. This rock will change properties according to pore fluid chemistry.


The most common sedimentary rock types are shales and silts. In Fig. 13.7, white quartz grains float in the surrounding clay matrix. Black organic material in thin layers indicates the horizontal bedding. As a result, this rock has properties that vary strongly with direction and are thus anisotropic. This material could serve as both a source rock and reservoir seal. This sample demonstrates how a mudstone or shale could have a complex composition. Although clays typically make up a large portion of fine-grained rocks, terms such as "clay" and "shale" are not synonymous.


Most sedimentary rocks have porosities under 0.50 (fractional). This is easy to understand, particularly with coarser clastic sediments, in which open grain packings that can support a matrix framework have maximum porosities around 0.45. Exceptions to this and other generalizations can occur, and an example is shown in Fig. 13.8. This globigerina "ooze" is composed largely of the small shells or tests of organisms. The matrix mud fills the region between tests, but interiors remain empty. In addition, the tests themselves are porous. As a result, porosities can be as high as 0.8. Despite these huge porosities, because of the isolated nature of the pores, permeability can be in the microdarcy range. A similar situation often occurs in shallow clay-rich sediments where the open clay plate structure results in initial very high porosities. In the remainder of this chapter, however, these types of sediments will be considered exceptional and will not be included in our analyses.


The rock images shown in these several figures are meant to convey a feel for the types of textures common in sedimentary rocks, and that influence physical properties. We will refer back to these images later in the chapter. These few images can in no way be considered a complete description of rock textures. For a more thorough treatment, the reader should consult one of the standard petrography texts or pertinent papers.[12][13][14]

Density and Porosity

Basics and Definitions

Density is defined as the mass per volume of a substance.

RTENOTITLE....................(13.1)

typically with units of g/cm3 or kg/m3. Other units that might be encountered are lbm/gallon or lbm/ft3 (see Table 13.2).


For simple, completely homogeneous (single-phase) material, this definition of density is straightforward. However, Earth materials involved in petroleum engineering are mixtures of several phases, both solids (minerals) and fluids. Rocks, in particular, are porous, and porosity is intimately related to density. For rocks, porosity (Φ) is defined as the nonsolid or pore-volume fraction.

RTENOTITLE....................(13.2)

Porosity is a volume ratio and thus dimensionless, and is usually reported as a fraction or percent. To avoid confusion, particularly when variable or changing porosities are involved, it is often reported in porosity units (1 PU = 1%).

Several volume definitions are required to describe porosity:

RTENOTITLE
RTENOTITLE....................(13.3)

From these we can define the various kinds of porosity encountered:

RTENOTITLE....................(13.4)

Fig. 13.9 shows the appearance of these types of porosity in a sandstone.


Similarly, the definitions of the standard densities associated with rocks then follows:

RTENOTITLE....................(13.5)

where Ms, Md, Msat, Mb, and Mfl are the mass of the solid, dry rock, saturated rock, buoyant rock, and fluid, respectively.

Relationships

The density of a composite such as rocks (or drilling muds) can be calculated from the densities and volume fraction of each component. For a two-component system,

RTENOTITLE....................(13.6)

where ρmix is the density of the mixture; ρA is the density of Component A; ρB is the density of B; A and B are the volume fractions of A and B respectively (and so B = 1− A).

Expanding this into a general system with n components,

RTENOTITLE....................(13.7)

For example, exploiting Eqs. 13.4, 13.5, and 13.6 for a rock made up of two minerals, m1 and m2, and two fluids, f1 and f2, we find

RTENOTITLE....................(13.8)

and

RTENOTITLE....................(13.9)

Eq. 13.8 is a fundamental relation used throughout the Earth sciences to calculate rock density. Given a porosity and specific fluid, density can be easily calculated if the mineral or grain density is known. Grain densities for common rock-forming minerals are shown in Table 13.3. The result of applying Eq. 13.9 is shown in Fig. 13.9.


Note in Table 13.3 that there are several densities reported for the same mineral group, such as feldspar or clay. The density will change systematically as composition varies. For example, in the plagioclase series, the density increases as sodium (albite, ρ = 2.61 g/cm3) is replaced by calcium (anorthite, ρ = 2.75 g/cm3). The most problematic minerals are clays, particularly expanding clays (montmorillonite or smectite) capable of containing large and variable amounts of water. In this case, densities can vary 40% or more. This is a particular problem, because clays are among the most common minerals in sedimentary rocks.

Reservoir rocks often contain significant amounts of semisolid organic material such as bitumen. These will typically have light densities similar in magnitude to those of coals.

Pore-fluid densities are covered in detail in the fluid property section (13.4).

In-Situ Density and Porosity

In general, density increases and porosity decreases monotonically with depth. This is expected, because differential pressures usually increase with depth. As pressure increases, grains will shift and rotate to reach a more dense packing. More force will be imposed on the grain contacts. Crushing and fracturing is a common result. In addition, diagenetic processes such as cementation work to fill the pore space. Material may be dissolved at point contacts or along styolites and then transported to fill pores. Some of the textures resulting from these processes were seen in the photomicrographs of the previous section. In Fig. 13.10, generalized densities as a function of depth for shales are plotted. The shapes and overall behaviors for these curves are similar, even though they come from a wide variety of locations with different geologic histories. These kinds of curves are often fit with exponential functions in depth to define the local compaction trend.


Differential or effective pressures do not always increase with increasing depth. Abnormally high pore fluid pressures ("overpressure") can occur because of rapid compaction, low permeability, mineral dewatering, or migration of high-pressure fluids. The high pore pressure results in an abnormally low differential of effective pressure. This can retard or even reverse the normal compaction trends. Such a situation is seen in Fig. 13.11. Porosities for both shales and sands show the expected porosity loss with increasing depth in the shallow portions. However, at about 3500 m, pore pressure rises and porosity actually increases with depth. This demonstrates why local calibration is needed. This behavior is also our first indication of the pressure dependence of rock properties, a topic covered in more detail in Section 13.5.

Measurement Techniques

Laboratory. Numerous methods can be used in the laboratory to determine porosity and density. The most common are by saturation weight and Boyle’s law. For rocks without sensitive minerals such as smectites, the porosity and dry, grain, and saturated densities can be derived from the saturated mass, dry mass, and volume (or buoyant weight). These measurements allow calculation of saturated, dry, and grain density as well as porosity and mineral and pore volume by employing Eqs. 13.3 through 13.5.

The Boyle’s law technique measures the relative changes in gas pressures inside a chamber with and without a rock specimen. The internal (connected) pore volume is calculated from these variations in pressure, from which porosities and densities are extracted.

Logging. Several logging techniques are available to measure density or porosity.[27][28] These indirect techniques can have substantial errors depending on borehole conditions, but they do provide a measure of the in-situ properties. Gamma ray logs bombard the formation with radiation from an active source. Radiation is scattered back to the logging tool, depending on the electron density of the material. Formation density is extracted from the amplitude of these back-scattered gamma rays. The neutron log estimates porosity by particle interaction with hydrogen atoms. Neutrons lose energy when colliding with hydrogen atoms, thus giving a measure of the hydrogen content. Because most of the hydrogen in rocks resides in the pore space (water or oil), this is then related to the liquid-filled porosity. Note that the neutron log will include bound water within clays as porosity. In addition, when relatively hydrogen-poor gas is the pore fluid, the neutron log will underestimate porosity. In a similar fashion, the nuclear magnetic resonance (NMR) log will resolve the hydrogen content. This tool, however, has the ability to differentiate between free bulk water and bound water. Sonic logs are also used for porosity measurements, particularly when anomalous minerals (such as siderite) or borehole conditions render other tools less accurate. The technique involves inverting velocity to porosity using one of the relationships provided in the velocity section below. Gravimetry has also been used downhole to measure variations in density. Although this tool is insensitive to fine-scale changes, it permits density measurement far out into the formation.

Seismic. On a coarse scale, densities can sometimes be extracted from seismic data. This method requires separating the density component of impedance. This normally requires an analysis of the seismic data as a function of offset or reflection angle. This technique will probably see more use as seismic data improves and is further incorporated into reservoir description.

Fluid Properties


Hydrocarbons occur in a variety of conditions, in different phases, and with widely varying properties. In this section, we will cover the important geophysical properties of pore fluids. For more general information on the engineering properties of fluids, see the appropriate section in the Handbook. Fig. 13.12 shows schematically the relation among the different mixtures. For a single, constant composition mixture, as we vary temperature and pressure over a wide range, we would encounter the boundary between the single and multiphase regions. In contrast, if we restrict the temperatures and pressures to those typical of reservoirs, we could again move in this phase "space" by changing compositions. Velocities and densities will be high (close to water) for heavy "black" oils to the left of the figure and decrease dramatically as we move right toward lighter compounds. In many cases, the hydrocarbons are greater than critical pressure and temperature conditions (greater than critical point). Properties then can vary continuously from liquid-like, for oils with gas in solution, to gas-like, for mixtures of light molecular weight. With changing pressure and temperature conditions, phase boundaries can be crossed, resulting in abrupt changes in fluid properties. Additional components are often injected during production, further complicating the distribution of compositions and properties.

Gas

The gas phase is the easiest to characterize. The compounds are usually relatively simple, and the thermodynamic properties have been thoroughly examined. Hydrocarbon gases usually consist of the light hydrocarbons of methane, butane, and propane. Additional gases, such as water vapor and heavier hydrocarbons, will occur in the gas depending on the pressure, temperature, and history of the deposit. The specific weight of these gases, as compared to air at standard temperature and pressure, will vary from about 0.6 for nearly pure methane to over 1.5 for gases with heavier components. Fortunately, when a rough idea of the gas weight is known, a fairly accurate estimate can be made of the gas properties at pressure and temperatures. Thomas et al.[29] did a complete analysis of the acoustic properties of natural gases, and we will follow a similar analysis here.

The important seismic characteristics of a fluid (the bulk modulus, density, and sonic velocity) are all related to primary thermodynamic properties. Therefore, for gases, we are obliged to start with the ideal gas law.

RTENOTITLE....................(13.10)

where P is pressure, V is volume, n is the number of moles of the gas, R is the gas constant, and Ta the absolute temperature. This leads to a density ρ, of

RTENOTITLE....................(13.11)

where M is the molecular weight. The isothermal compressibility βT is

RTENOTITLE....................(13.12)

for compressibility defined as a positive number.

If we calculate the "isothermal" velocity VT, we find

RTENOTITLE....................(13.13)

for an ideal gas. The acoustic velocity is controlled by the stiffness of the material and its density (see the derivation in Section 13.5.5). Therefore, velocity would increase with temperature and be independent of pressure.

Two mitigating factors bring the relationship closer to reality. First, because there are rapid temperature changes associated with the passage of an acoustic wave, we must use the adiabatic compressibility, βS, rather than the isothermal compressibility γ βS = βT.

Here, γ is the ratio of heat capacity at constant pressure to the heat capacity at constant volume. In most solid materials, the difference between the isothermal and adiabatic compressibilities is negligible. However, in fluid phases, particularly gases, the isothermal compressibility can be twice the adiabatic value.

The second, more obvious factor stems from the inadequacies of the ideal gas law (Eq. 13.10). The gas law can be corrected by adding a compressibility factor (Z). The relationships are thus modified:

RTENOTITLE....................(13.14)

RTENOTITLE....................(13.15)

and

RTENOTITLE....................(13.16)

The heat capacity ratio can itself be derived if the equations of state of the material are known. The seismic characteristics of the gas can, therefore, be described if we have an adequate description of Z with pressure, temperature, and composition.

Thomas et al.[29] made use of the Benedict-Webb-Rubin (BWR) equation to define the gas behavior. The BWR equation of state is a rational equation, with numerous constants based on the behavior of natural gas mixtures. These gas mixtures range in gravity G (relative to air) from about 0.5 to 1.8. The results of the density calculations are shown in Fig. 13.13. As would be expected, the gas densities increase with pressure and decrease with temperature. However, the densities also strongly depend on the gas gravity, which is composition-dependent.


The adiabatic gas modulus K (the inverse of β) also strongly depends on the composition as well as the pressure and temperature conditions. Fig. 13.14 shows the calculated modulus from the Thomas relationships. Again, the modulus increases with pressure and decreases with temperature, but the relationship is not as linear. The impact of variable composition (gravity) is again obvious.

Oil

Crude oils can be mixtures of extremely complex organic compounds. Natural oils range from the lightest condensate liquids of low carbon number to very heavy tars. At the heavy extreme are bitumen and kerogen, which may be denser than water and act essentially like solids. At the light extreme are condensates that may become gas with decreasing pressure. Oils can absorb large quantities of hydrocarbon gases under pressure, thus significantly decreasing the moduli. Under room conditions, the densities can vary from 0.5 to greater than 1 gm/cc with most produced oils in the 0.7 to 0.8 gm/cc range. The American Petroleum Institute (API) number is defined as

RTENOTITLE....................(13.17)

This results in API numbers of about 5 for very heavy oils to near 100 for light condensates. The extreme variations in composition and ability to absorb gases produce greater variations in the seismic properties of oils.

If we had a general equation of state for oils, we could calculate the moduli and densities as we did for the gases. Such equations abound in the petroleum engineering literature. Unfortunately, the equations are almost always strongly dependent on the exact composition of a given oil. For the purposes of this Handbook, we will develop only very general relations. Often, in petrophysical analysis we only have a rough idea of what the oils may be like. In some reservoirs, individual yet adjacent zones will have quite distinct oil types. We will, therefore, proceed along empirical lines based on the density of the oil.

The acoustic properties of numerous organic fluids have been studied as a function of pressure or temperature (see, for example, Rao and Rao[31]). Generally, the velocities, densities, and moduli are quite linear with pressure and temperature away from phase boundaries. In organic fluids typical of crude oils, the moduli decrease with increasing temperature and increase with increasing pressure. Wang and Nur[32] did an extensive study of several light alkanes, alkenes, and cycloparaffins and found simple relationships among the density, moduli, temperature, and carbon number or molecular weight. For velocity they found

RTENOTITLE....................(13.18)

where Vo is the initial velocity, VT is the velocity at temperature T, ΔT is the temperature change, and b is a constant for each compound of molecular weight M:

RTENOTITLE....................(13.19)

Similarly, the velocities are related in molecular weight by

RTENOTITLE....................(13.20)

where VTM is the velocity of oil of weight M, and VTOMO is the velocity of a reference oil of weight Mo at temperature To. The variable am is a positive function of temperature. We can see from the rightmost term in Eq. 13.20 that the velocity of the fluid will increase with increasing molecular weight. When compounds are mixed, Wang and Nur[32] found that the resulting velocity is a simple fractional average of the end components. This is roughly equivalent to a fractional average of the bulk moduli of the end components. Pure simple hydrocarbons, therefore, behave in a simple predictable way. We must extend this analysis to include crude oils, which are generally much heavier and have more complex compositions. The influence of pressure must also be determined. In the petroleum engineering literature, broad empirical relationships are available. By empirically fitting equations to these data, we can get density as functions of initial density (or API number), temperature, and pressure

RTENOTITLE....................(13.21)

These densities are shown in Fig. 13.15.


By differentiating Eq. 13.21, we obtain the isothermal compressibility βT,

RTENOTITLE....................(13.22)

If we assume a reasonable and constant heat capacity ratio γ of 1.15, we obtain the adiabatic bulk moduli K.

RTENOTITLE....................(13.23)

The ultrasonic velocities of a variety of crude oils measured recently are reported in Wang et al.[33] A general relationship of oil velocity was derived.

RTENOTITLE....................(13.24)

where V is in m/s, T in °C, P in bars, and API is the API degree of the oil, or

RTENOTITLE....................(13.25)

for V in ft/s, T in °F, and P in psi.

Using these velocities and the densities as shown in Fig. 13.15, we find the moduli shown in Fig. 13.16.


Very large amounts of gas or light hydrocarbons can go into solution in crude oils. In fact, the lighter crudes are condensates from the gas phase. We would expect the "live" or gas-saturated oils to have significantly different properties than the "dead" or gas-free oils commonly available and measured. The amount of gas that can be dissolved is a function of pressure, temperature, and the composition of both the gas and the oil.[34]

RTENOTITLE....................(13.26)

where R is the gas-oil ratio in liters/liter (1 liter/liter = 5.615 cu ft/bbl) at atmospheric pressure and at 15.5°C and G is the gas gravity. Eq. 13.26 indicates that much larger amounts of gas can go into the light (high API number) oils. In fact, heavy oils may precipitate heavy compounds if much gas goes into solution.

The effect of this gas in solution on the oil acoustic properties has not been well documented. Sergeev[35] noted that gas in solution will reduce both oil and brine velocities. He calculated that this mix would change some reservoir reflection coefficients by more than a factor of two. A rough estimate of this dissolved gas effect can be made by assuming that the relationship in Eq. 13.26 remains valid and by adjusting the oil density to include the gas component. We are assuming here that the gas is a liquid component with its own volume and density and that the result is an ideal liquid mixture. The simple additive relations found in Wang and Nur[32] support this concept. The estimated density becomes

RTENOTITLE....................(13.27)

where ρO is the dead oil density and ρG is the gas saturated live oil density. The factor F is derived from the gas/oil ratio

RTENOTITLE....................(13.28)

Fig. 13.17 shows the live and dead oil velocities measured in Wang et al.[33] along with the estimates using Eqs. 13.25, 13.27, and 13.28.

Brines

The great bulk of the pore fluids consists of brines. Their composition can range from almost pure water to saturated saline solutions. Gulf of Mexico area brines often have rapid increases in concentration with increasing depth. In other areas, the concentrations are often lower but can vary drastically between adjacent fields.

The thermodynamic properties of aqueous solutions have been studied in detail. Keenan et al.[36] give a relation for pure water that can be iteratively solved to give densities at pressure and temperature. Helgeson and Kirkham[37] use this and other data to calculate a wide variety of water properties over an extensive temperature and pressure range. One obvious effect of salinity is to increase the density of the fluid. Rowe and Chou[38] presented a polynomial to calculate both specific volume and compressibility of various salt solutions at pressure over a limited temperature range. Extensive additional data on sodium chloride solutions is provided in Zarembo and Fedorov[39] and Potter and Brown.[40] Using all these data, a simple polynomial can be constructed that will adequately calculate the density of sodium chloride solutions:

RTENOTITLE....................(13.29a)

and

RTENOTITLE....................(13.29b)

Here, T and P are in °C and bars, respectively; x is the weight fraction of sodium chloride; and ρB is the density of the brine in gm/cm3. The calculated brine densities, along with selected data from Zarembo and Federov,[39] are plotted in Fig. 13.18. The accuracy of this relationship is limited largely to the extent that other mineral salts, particularly divalent ions, are in solution.


A vast amount of acoustic data is available for brines, but generally for pressure, temperature, and salinity expected under oceanic conditions. Wilson[41] provides a relationship for the velocity Vw of pure water to 100°C and about 1000 bars

RTENOTITLE....................(13.30)

Millero et al.[42] and Chen et al.[43] give additional factors to be added to the velocity of water to calculate the effects of salinity. Their corrections, unfortunately, are limited to 55°C and 1 molal ionic strength (55,000 ppm). We can extend their results by using the data of Wyllie et al.[44] to 100°C and 150,000 ppm NaCl. Still, this leaves the high-temperature and -pressure region with no data. Here we can use the isothermal modulus calculated from Eq. 13.29 to estimate the adiabatic moduli. We can also use the velocity function provided in Chen et al.[43] but with the constants modified to fit the additional data. The heat capacity ratio for the brine can be estimated from the PVT relationship in Eq. 13.29 and estimates of the isobaric heat capacity from Helgeson and Kirkham[37]:

RTENOTITLE....................(13.31)

and

RTENOTITLE....................(13.32)

In this equation, m is the molal salt concentration and cij, dij, and ei are constants. Using the calculated density and velocity of brine produces the modulus, and this is shown in Fig. 13.19.


Elasticity, Stress-Strain, and Elastic Waves


We will begin this section with an introduction to stress-strain relations. These form the foundation for several rock properties, such as elastic moduli (incompressibility), effective media theory, elastic wave velocity, and rock strength.

Stress and Pressure – Definition

Stress is the force per unit area.

RTENOTITLE....................(13.33)

The metric units of stress or pressure are N/m2 or Pascals (Pa). Other units that are commonly used are bars, megapascals (MPa), and lbm/in.2 (psi) (see Table 13.4). These stresses can take various forms such as a homogeneous pressure P, normal stress σn, or stress applied at a general angle σg (Fig. 13.20). This general stress can be decomposed into normal and tangential components. We usually refer to balanced stresses because, under quasistatic conditions, they produce no net acceleration. Stress is a second-order tensor denoted by σij, where the first index denotes the surface and the second the direction of the applied force (see Fig. 13.21). In Earth sciences and engineering, compressive stresses are usually considered positive, whereas most material sciences consider tensional stress positive. More details on the influence of stresses and the stress tensor can be found in Jeager and Cook[45] and Nye.[46]


Several standard stress conditions are either assumed for the Earth for analysis or modeling, or applied in the laboratory:

Hydrostatic stress: all confining stresses are equal

Uniaxial stress: one stress applied along a single axis (other stresses are zero or held constant during an experiment)

Biaxial stress: two nonequal stresses applied (third direction is equal to one of the others)

Triaxial stress: (1) Common usage—separate vertical and two equal horizontal stresses (e.g., biaxial); (2) better—three independent principal stresses.

Anisotropic stresses are usually responsible for rock deformation and failure (see Section 13.7). In much of this section, however, we will concern ourselves primarily with mean stress (σm) or pressure (P).

RTENOTITLE....................(13.34)

It is important to distinguish among the various kinds of pressure, because the combination often determines any specific rock property and influences the response to any production procedure.

Confining pressure = Pc = Overburden pressure on rock frame
Pore pressure = Pp = Fluid pressure inside pore space
Differential (or net) pressure = Pd = Difference between Pc and Pp
Effective pressure = Pe = Combination of Pc and Pp controlling a property


Increasing confining pressure (Pc) alone will result in a decrease of rock volume, or compaction. In contrast, increasing the pore pressure (Pp) tends to increase rock volume. Pp counteracts the effects of Pc. Thus, rock properties are controlled largely by the difference between Pc and Pp, or the differential pressure Pd. A more exact form will account for the interaction of the fluid pressure with the pore space and minerals and result in an effective stress (Pe) law

RTENOTITLE....................(13.35)

where n is a term that can be derived theoretically or defined experimentally for each property.

Deformation, Strain, and Modulus

Application of a single (vertical) stress is one typical experiment run to measure material mechanical properties (Fig. 13.22). If this stress continues to increase, eventually the material will fail when the uniaxial compressive strength is reached (see Section 13.7). For the rest of this chapter, however, we will deal only with small deformations and stresses such that the rock remains in the elastic region. Under this restriction, several important material properties can be defined. For an isotropic, homogeneous material, there is a vertical deformation (ΔL) associated with the vertical stress. Normalizing this deformation by the original length of the sample, L, gives the vertical strain

RTENOTITLE....................(13.36)

By definition, Young’s modulus, E, is the ratio of the applied stress (σzz) to this strain

RTENOTITLE....................(13.37)

Because strain is dimensionless, E is in units of stress.


This same stress will generally result in a lateral or horizontal deformation, ΔW. The lateral strain can then be defined

RTENOTITLE....................(13.38)

One important parameter relating the vertical and horizontal strains is Poisson’s ratio

RTENOTITLE....................(13.39)

The minus sign is attached because the signs of the deformations are opposite for the horizontal vs. vertical strains in this simple case.

If instead we applied a pressure, we would get a volumetric strain εv:

RTENOTITLE....................(13.40)

The bulk modulus of a material is then defined as the ratio of applied pressure to volumetric strain

RTENOTITLE....................(13.41)

Bulk modulus is equivalent to the inverse of compressibility, β.

In a similar way, shear modulus, μ (often "G" in many publications), can be defined as the ratio of shear stress to shear strain:

RTENOTITLE....................(13.42)

These various equations are special cases of Hooke’s Law, which can be written for the general case

RTENOTITLE....................(13.43)

Stress and strain are both tensors with 9 components. Cijkl would then be a tensor with 81 components. However, because of symmetry considerations, only a maximum of 21 can be independent (a thorough treatment of the tensor relations is provided in Nye[46]). For isotropic materials, this reduces to

RTENOTITLE....................(13.44)

where λ is Lame’s constant. In fact, for isotropic materials, there are only two independent elastic parameters. Any isotropic elastic constant can be written in terms of two others. For example, λ can be defined as

RTENOTITLE....................(13.45)

The possible combinations among various isotropic elastic constants are shown in Table 13.5. This becomes important in applications, because restricting one term, say ν, fixes the ratio of other moduli such as μ and K.

Effective Media, Bounds

Rocks are usually not homogeneous, but are made up of multiple components such as mineral grains and pore space. On a larger scale, the bulk properties of rocks will be some weighted combination of the small-scale components. This averaging or upscaling step is needed if we wish to understand the behavior of our laboratory data or extract useful parameters from field data such as logs or seismic measurements.

The simplest bounds are provided by the constant strain and constant stress limits. This method is equivalent to the series vs. parallel effective resistance of a resistor network. In the case that strains of the two materials making up our material are equal, as with the parallel plates in Fig. 13.23a, we get the upper s(Voigt) limit. The response is controlled by the stiffer component.

RTENOTITLE....................(13.46)

where MV is the effective Voigt modulus, MA and MB are the component moduli, and A is the volume fraction of component A. In contrast, with the constant-stress case (Fig. 13.23b), the soft component dominates the deformation and we get the lower (Reuss) limit.

RTENOTITLE....................(13.47)

where MR is the lower Reuss effective modulus. The average value between these two limits is often used in property estimation and is termed the Voigt-Reuss-Hill relation

RTENOTITLE....................(13.48)

Note that in the case for minerals plus pores, Mpore = 0 and MV decreases linearly with porosity. MR equals zero for all porosities.


An alternative approach, known as the Hasin-Shtrikman technique,[48] is to fill space with concentric spheres. Material 1 is in the interior, and Material 2 forms a surrounding shell. Spheres such as these but of varying size are packed together to fill the entire medium (Fig. 13.24). The resulting upper and lower bounds ("+" vs. "–" respectively) for bulk and shear modulus are given by

RTENOTITLE....................(13.49)

and

RTENOTITLE....................(13.50)

where Ki, μi, and fi refer to the bulk and shear moduli and volume fraction of component i, respectively. The upper and lower bounds are derived by exchanging the stiff and soft components as "1" or "2."


The results of using Eqs. 13.46 through 13.50 are shown in Fig. 13.25. Using quartz as the first component and porosity as the second, the composite bulk modulus is plotted in Fig. 13.25a as a function of porosity. In one case, the pores are empty (black), in the other, water fills the pores and is the second component (blue). Because we used quartz as the solid component (Table 13.6), these bounds should contain all possible values for sandstones (remember: for isotropic and homogeneous sandstones). If, on the other hand, our rock was made up of only quartz and calcite, we get bounds that appear in Fig. 13.25b. Note that the bounds have collapsed and produce only a narrow spread. This is a result of the two end components both being stiff and closer together. In cases such as these, a simple linear average can work well.

Mineral Properties

There are numerous ways to measure mineral moduli. The most obvious is by deforming single crystals. Alternatively, elastic velocities can be measured and moduli extracted for zero porosity aggregates. Tables 13.6a and 13.6b present lists of "isotropic" densities, mineral bulk and shear moduli, and elastic velocities. In reality, minerals are anisotropic, and the values listed in the table are averages derived from the effective media fomulas presented above to represent polygrained isotropic composites. The highest-velocity, highest-moduli are for such minerals as almandine and rutile. Velocities can reach 9 km/s for Vp, and moduli can be in the hundreds of GPa. Clays are a particular problem. As noted before, they are among the most abundant minerals on the surface of the Earth, and are common in most sedimentary rocks. Their small size, variable composition, and chemical activity make them difficult to characterize from a mechanical point of view. The results of Katahara,[49] Wang et al.,[50] and Prasad et al.[51] are given in Table 13.6b.

Mineral properties can also be extracted from the numerous empirical trends developed for rocks, as we will see below.

Elastic Wave Velocities

So far, we have considered only the static elastic deformation of materials. By adding the dynamic behavior, we arrive at how elastic waves propagate through materials. If a body is changing its speed as well as deforming, there will be an unbalanced force because of the acceleration described through Newton’s Second Law:

RTENOTITLE....................(13.51)

where ρ is density, a is acceleration, u is displacement, and t is time. Combining this with Hook’s Law (Eq. 13.43) gives the general wave equation. For a plane wave in the xx direction, this can be written as

RTENOTITLE....................(13.52)

However, if the material is being deformed, we will have strains associated with the change of displacement with position. In turn, these strains can be related to the stresses through the appropriate modulus, M (for example, Eq. 13.37):

RTENOTITLE....................(13.53)

For constant elastic components, this simplifies to

RTENOTITLE....................(13.54)

The solution to this equation gives the compressional velocity

RTENOTITLE....................(13.55)

Similarly, for shear motion

RTENOTITLE....................(13.56)

and we get the shear velocity:

RTENOTITLE....................(13.57)

Porosity Dependence

The bounding relations we examined above can be applied directly to rock acoustic velocities. Some dolomites with vuggy pores may approach the Voigt bound. Highly fractured rocks may approach the Reuss bound. However, there is often a great difference between these idealized bounds and real rocks. For sandstones, we would expect to begin with quartz velocity at zero porosity and have decreasing velocity with increasing porosity. By combining Eqs. 13.46 and 13.47 for moduli in Eq. 13.55, we can plot expected velocity bounds, as in Fig. 13.26a. Observed distributions for sandstones are also plotted, and we see a systematic discrepancy with the upper (Voigt) bound. At high porosities, grains separate, and the mixture acts as a suspension. The majority of rocks have an upper limit to their porosity usually termed "critical porosity," Φc (Yin et al.[52] and Nur et al.[53]). At this high porosity limit, we reach the threshold of grain contacts and grain support (Han et al.[54]).


Brine-saturated sandstone velocities can be separated into classes based on their velocity-porosity relations (Fig. 13.26b). Very clean sandstones (Class I) decrease in a simple linear trend from the 6 km/s velocity of quartz as porosity increases. Most consolidated rocks (Class II) have somewhat lower velocities, still decreasing with increasing porosity. Poorly cemented sands (Class III) approach the lower Reuss bound for velocity. Pure suspensions are dominated by the modulus of water (Class IV) and are almost independent of the porosity. However, such suspensions are rare. Another important class is dominated by fractures (Class V). As we shall see later, fractures have a far greater effect on velocity than might be expected for their low porosity, and may approach the Reuss bound.

Measured Velocity-Porosity Relations

Numerous systematic investigations into the relationship of velocity, porosity, and lithology (usually clay content) have been conducted. The results of Vernik and Nur[56] for brine-saturated sandstones are shown in Fig. 13.27 for compressional and shear velocities, respectively. Very clean sands (clean arenites) show the linear decrease from quartz velocity. However, even small amounts of clays will substantially lower the trend. Increasing clay content will then continue to lower velocities.


Numerous examples of general porosity/velocity/clay content relations for sandstones are given in Table 13.7 a and b (symbol definitions for these relations are in Table 13.7c). These types of relations have proved very useful in giving velocities under general conditions, providing the overall effects of clay, and establishing the relation of compressional to shear velocity (Vp/Vs ratios). VpVs relations are extremely important, because shear logs are relatively rare, yet shear velocities are critical in determining seismic direct hydrocarbon indicators such as reflection Amplitude-Versus-Offset (AVO) trends (Castagna et al.[18]).

Measured data for carbonates are less abundant. A systematic investigation of samples from several wells was reported by Rafavich et al.[57] A plot of their results for carbonate Vp as functions of porosity and composition is shown in Fig. 13.28. They collected detailed information on fabric and texture as well as porosity and mineralogy. Performing regressions on their extensive data set produced the relations given in Table 13.8a. The coefficients associated with these equations are given in Table 13.8b. Note that the relations are dependent on the effective pressure.


A similar set of measurements by Wang et al.[58] are shown in Fig. 13.29. For carbonates, the data can be quite scattered, but can still show the general velocity decrease with increasing porosity. These results were summarized in a set of relations (Table 13.9) again showing pressure dependence. Their data, however, includes measurements made with samples not only brine-saturated, but hydrocarbon-saturated and after simulated reservoir floods. [59] They demonstrate that the overall velocity and impedance changes were strongly dependent on the imposed sequence of flooding. The ability to observe a particular reservoir process will be more complicated than simply completely substituting fluids into the rocks.

Pressure

Rock moduli (compressibility) and elastic velocities are strongly influenced by pressure. With increasing effective pressure, compliant pores within a rock will deform, contract, or close. The rock becomes stiffer, and, as a result, velocities increase. Two examples are shown in Fig. 13.30. The typical behavior is rapid increase in velocity, with increasing pressure at low pressures, followed by a flattening of the curve at higher pressures. Presumably, compliant pores and cracks are closed at higher pressure, and velocities asymptotically approach a relatively constant velocity. This specific behavior at high pressures leads to the simple velocity-porosity transforms and probably is responsible for our ability to use sonic tools as in-situ porosity indicators with little regard to local pressures.


The stress dependence of granular material has been examined extensively. For example, Gassmann[60] and Duffy and Mindlin[61] modeled various packings of spheres. In general, they found that

RTENOTITLE....................(13.58)

where f is approximately linear. This type of relation is particularly useful for poorly consolidated sands.

Although the absolute pressure dependences shown in Fig. 13.30a vs 13.30b are in significant contrast, for most sandstones, relative changes are more consistent. By normalizing the velocities to those at high pressure (40 MPa), we get a much more consistent behavior (Fig. 13.31).

RTENOTITLE....................(13.59)

Examining a similar set of data allowed Eberhart-Phillips et al.[62] to develop a pair of relations for both Vp and Vs (see also Table 13.7)

RTENOTITLE....................(13.60a)

RTENOTITLE....................(13.60b)

where Pe is the effective pressure. For carbonates, the explicit pressure dependence given in Tables 13.8a and 13.9 allow the pressure dependence to be evaluated. The pressure dependence for carbonate Vp from Rafavich et al.[57] is shown in Fig. 13.32. Note that pressure sensitivity increases with increasing porosity. These types of relations permit velocity changes associated with pressure changes in the reservoir to be modeled.


It is important to note that all these relations involve either differential pressure (Pd) or effective pressure (Pe). Pore pressure (Pp) counters the influence of confining pressure (Pc), so the difference between these two controls rock properties. This has been expressed simply in the Terzaghi[63] relation for the pressure dependence of a given porous material property S,

RTENOTITLE....................(13.61)

This kind of behavior has been seen in numerous cases, as in Fig. 13.33. This is one reason why properties such as density, resistivity, and velocity can decrease with increasing depth when "overpressure" or when increased pore pressure is encountered. Changes in reservoir pore pressure will have a similar influence. More precisely, it is the effective pressure (Eq. 13.35) that controls properties rather than just the differential. However, the magnitude of effective pressure is often found to be close to the simpler differential pressure.

In-Situ Stresses

The in-situ "lithostatic" stresses are usually unequal. Such different stresses are required or faults, folds, and other structural features would never be developed. In contrast, most laboratory data are collected under equal stress or "hydrostatic" conditions. Differential or triaxial measurements are comparatively rare (e.g., Gregory,[65] Nur and Simmons,[66] Yin,[67] and Scott et al.[68]).

In a simple compacting basin with neither lateral deformation nor tectonic stresses, the vertical stress will be largest. Lateral stresses will be developed in a basin as sediments are buried and compacted but are constrained horizontally. Both uniform hydrostatic and unequal lithostatic stress conditions are shown in Fig. 13.34.


A simple estimate of the horizontal stress, σh, can be made from the axial stress, σv, by

RTENOTITLE....................(13.62)

where ν is Poisson’s ratio. Calculated stresses typical for sands (ν = 0.1) and more clay-rich rocks (ν = 0.25) are also shown in Fig. 13.34. This basic relation (Eq. 13.62) is an oversimplification of actual conditions, but it does provide a useful conceptual model, and lateral stresses indeed are found to be lower in sandstones than in shaly sections in most places.

From a matrix of velocities measured over axial and lateral stress conditions, velocity surfaces could be calculated for a given rock sample. Data such as those shown in Fig. 13.35 were fitted to a form based on that of Eq. 13.58:

RTENOTITLE....................(13.63)

where σe is the effective stress. Fits are usually very good even for consolidated rocks with regression factors of around 0.98.


Velocities can vary substantially over the stress field shown in Fig. 13.34, not only among samples but also between compressional and shear waves. Fig. 13.36 shows the Vp and Vs surfaces for Woodbine sandstone. Figures such as 13.36 demonstrate that the Vp, Vs, and Vp/Vs ratio will all be strongly dependent on the exact stress tensor at depth. Laboratory measurements under hydrostatic conditions are at best a first-order approximation.

Temperature

For consolidated rocks (Classes I, II, and V, Fig. 13.26b), the elastic mineral frame properties are usually only weakly dependent on temperature. This is true for most reservoir operations with the exception of some thermal recovery procedures. In the case of poorly consolidated sands containing heavy oils, velocities show that a strong temperature dependence is observed (Fig. 13.37). Several factors can combine to produce such large effects. First, in heavy-oil sands, the material may actually be a suspension of minerals in tar (Fig. 13.26b, Class IV). The framework is basically a fluid, not solid. In addition, during many measurements, pore pressure cannot reach equilibrium. The large coefficient of thermal expansion of oils combined with the high viscosity often results in high pore pressures within the rock samples. Thus, effective pressures can drop substantially (Eq. 13.61). Care needs to be taken during such measurements that equilibrium pressures are reached.


The primary influence of temperature is through the pore fluid properties (refer to the Fluid Properties section). Fig. 13.38 demonstrates this general temperature dependence. For dry (gas-saturated) rock, or rock saturated with brine, almost no change in velocity is observed, even for changes of almost 150°C. At elevated pore pressures, both gas and brine have only weak temperature dependence. Mineral properties are almost unchanged. However, when the rocks are even partially saturated with oil, dramatic temperature dependence is observed. Such changes can be understood by first calculating fluid properties with temperature, then using a Gassmann substitution to calculate the bulk rock properties. Note that for heavy viscous oils, velocity dispersion (velocity dependence on frequency) can be significant, and measured ultrasonic data may not agree with seismic results.


Fluid phase changes may also occur as temperature is raised. These phase changes can have a strong influence, particularly for high-porosity rocks at low pressures. The effect can be seen in Fig. 13.38b, where exsolving a gas phase could reduce the velocity from nearly 3.2 km/s to around 2.1 km/s. In several thermal recovery monitoring projects, the strongest seismic expression was a result of gas coming out of solution to form a separate phase, rather than the thermal effects themselves.

Gassmann Fluid Substitution

To extract fluid types or saturations from seismic, crosswell, or borehole sonic data, we need a procedure to model fluid effects on rock velocity and density. Numerous techniques have been developed. Gassmann’s equations are by far the most widely used relations to calculate seismic velocity changes because of different fluid saturations in reservoirs. Gassmann’s formulation is straightforward, and the simple input parameters typically can be directly measured from logs or assumed based on rock type. This is a prime reason for its importance in geophysical techniques such as time-lapse reservoir monitoring and direct hydrocarbon indicators (DHI) such as amplitude "bright spots," and amplitude vs. offset (AVO). Because of the dominance of this technique, we will describe it at length.

Despite the popularity of Gassmann’s equations and their incorporation within most software packages for seismic reservoir interpretation, important aspects of these equations are usually not observed. Many of the basic assumptions are invalid for common reservoir rocks and fluids. Many efforts have been made to understand the operation and application of Gassmann’s equations (Han,[70] Mavko and Mukerji,[71] Mavko et al.,[8] Sengupta and Mavko,[72] and Nolen-Hoeksema[73]). Most of these works have attempted to isolate individual parameter effects. We will extend this analysis to incorporate mechanical bounds for porous media (see previous) and the magnitude of the fluid effect.

Compressional (P-wave) and shear (S-wave) velocities along with densities directly control the seismic response of reservoirs at any single location. Fig. 13.39a shows measured dry and water saturated P- and S-wave velocities of sandstones as a function of differential pressure. P-wave velocity increases, while S-wave velocity decreases slightly with water saturation. However, both P- and S-wave velocities are generally not the best indicators for any fluid saturation effect. This is a function of coupling between P- and S-wave through the shear modulus and bulk density. In contrast, if we plot bulk and shear modulus as functions of pressure (Fig. 13.39b), the water-saturation effect shows the following:

  1. Bulk modulus increases about 50%.
  2. Shear modulus remains almost constant.


Bulk modulus is more strongly sensitive to water saturation. The bulk volume deformation produced by a passing seismic wave results in a pore volume change, and causes a pressure increase of pore fluid (water). This has the effect of stiffening the rock and increasing the bulk modulus. Shear deformation usually does not produce pore volume change, and differing pore fluids often do not affect shear modulus.


Gassmann’s equations provide a simple model to estimate fluid saturation effect on bulk modulus. Eqs. 13.64a through 13.65 are convenient forms for Gassmann’s relations that show the physical meaning:

RTENOTITLE....................(13.64a)

RTENOTITLE....................(13.64b)

and

RTENOTITLE....................(13.65)

where K0, Kf, Kd, and Ks, are the bulk moduli of the mineral, fluid, dry rock, and saturated rock frame, respectively; Φ is porosity; and μs and μd are the saturated and dry rock shear moduli. ΔKd is an increment of bulk modulus caused by fluid saturation. These equations indicate that fluid in pores will affect bulk modulus but not shear modulus, consistent with the earlier discussion. As pointed out by Berryman,[74] a shear modulus independent of fluid saturation is a direct result of the assumptions used to derive Gassmann’s equation.

Numerous assumptions are involved in the derivation of Gassmann’s equation:
  1. The porous material is isotropic, elastic, monomineralic, and homogeneous.
  2. The pore space is well connected and in pressure equilibrium (zero frequency limit).
  3. The medium is a closed system with no pore fluid movement across boundaries.
  4. There is no chemical interaction between fluids and rock frame (shear modulus remains constant).


Many of these assumptions may not be valid for hydrocarbon reservoirs, and they depend on rock and fluid properties and in-situ conditions. For example, most rocks are anisotropic to some degree. The work of Brown and Korringa[75] provides an explicit form for an anisotropic fluid substitution. In seismic applications, it is normally assumed that Gassmann’s equation works best for seismic data at frequencies less than 100 Hz (Mavko et al.[8]). Recently published laboratory data (Batzle et al.[76]) show that acoustic waves may be dispersive in rocks within the typical seismic band, invalidating assumption 2. In such cases, seismic frequencies may still be too high for application of Gassmann’s equation. Pore pressures may not have enough time to reach equilibrium. The rock remains unrelaxed or only partially relaxed.

The primary measure of the sensitivity of rock to fluids is its normalized modulus Kn: the ratio of dry bulk modulus to that of the mineral.

RTENOTITLE....................(13.66)

This function can be complicated and depends on rock texture (porosity, clay content, pore geometry, grain size, grain contact, cementation, mineral composition, and so on) and reservoir conditions (pressure and temperature). This Kn can be determined empirically or theoretically. For relatively clean sandstone at high differential pressure (>20 MPa), the complex dependence of Kn (x, y, z, …) can be simplified as a function of porosity.

RTENOTITLE....................(13.67)

From Eq. 13.66, bulk modulus increment is then equal to

RTENOTITLE....................(13.68)

Here [1-Kn (Φ)] is also the Biot parameter αb (Biot[77]). Furthermore, because usually K0 >> Kf, it is reasonable to assume

RTENOTITLE....................(13.69)

for sedimentary rocks with high porosity (>15%). Therefore,

RTENOTITLE....................(13.70)

where G(Φ) is the saturation gain function defined as

RTENOTITLE....................(13.71)

Thus, fluid saturation effects on the bulk modulus are proportional to the gain function G(Φ) and the fluid modulus Kf. The G(Φ) in turn depends directly on dry rock properties: the normalized modulus and porosity. In general, G(Φ) is independent of fluid properties (ignoring interactions between rock frame and pore fluid). We must know both gain function of dry rock frame and pore fluid modulus to evaluate the fluid saturation effect on seismic properties. Note that the normalized modulus must be a smooth function of porosity or G(Φ) can be unstable, particularly at small porosities.

At high differential pressure (>20 MPa), the Ks of water-saturated sands calculated using simplified form is 3% overestimated for porous rock (porosity > 15%). Those errors will decrease significantly with low fluid modulus (gas and light oil saturation). For low-porosity sands with high clay content, the simplified Gassmann’s equation overestimates water saturation effects substantially.

In Eq. 13.64b, there are five parameters, and usually the only applied constraint is that the parameters are physically meaningful (>0). Incompatible or mismatched data might generate wrong or even unphysical results such as a negative modulus. In reality, only K0 and Kf are completely independent. Ks, Kd, and porosity Φ are actually closely correlated. Bounds on Kd as a function of porosity, for example, constrain the bounds of Ks.

Assuming porous media is a Voigt material, which is a high bound for Kd (Fig. 13.40),

RTENOTITLE....................(13.72)

Putting this equation (13.72) into Gassmann’s Equation (13.64) gives

RTENOTITLE....................(13.73)

and

RTENOTITLE....................(13.74)

Because this Voigt bound is the stiffest upper limit, the fluid saturation effect on bulk modulus here (ΔKdmin) will be a minimum (see Fig. 13.40).


As we have seen, the low modulus bound for porous media is the Reuss bound.

RTENOTITLE....................(13.75)

RTENOTITLE....................(13.76)

For completely empty (dry) rocks, the fluid modulus Kf is equal to zero, and both the Reuss bound and the normalized modulus (KnR) for a dry rock in this limit equals zero (for nonzero porosity).

RTENOTITLE....................(13.77)

Substituting Eq. 13.77 into Gassmann’s Equation (13.64), we find the fluid saturation effect on bulk modulus when the frame is at this lower bound.

RTENOTITLE....................(13.78)

For this case, the modulus increment ΔK from dry to fluid saturation is equal to the Reuss bound.

RTENOTITLE....................(13.79)

Again, Gassmann’s equation is consistent with the dry and fluid-saturated Reuss bounds. Physically, for rocks with the weakest frame, fluids have a maximum effect.

Critical porosity, Φc, can be used to give tighter constraints for dry- and fluid-saturated bulk modulus for sands. A new triangle is formed which provides a linear formulation and a graphic procedure for Gassmann’s calculation: the fluid saturation effect on bulk modulus proportional to normalized porosity and the maximum fluid saturation effect on bulk modulus (Reuss bound) at the critical porosity (Fig. 13.40).

RTENOTITLE....................(13.80)

This is consistent with the results of Mavko and Mukerji.[71]

For typical sandstones, the critical porosity Φc is around 40%. Thus, we also can generate a simplified numerical formula of the normalized modulus Kn for modified Voigt model:

RTENOTITLE....................(13.81)

Using this in Gassmann’s Equation (13.64) yields fluid saturation effect

RTENOTITLE....................(13.82)

Extending our empirical approach to first order, both P- and S-wave velocity can correlate linearly with porosity at high differential pressure. From Table 13.7, for dry clean sands,

RTENOTITLE....................(13.83)

RTENOTITLE....................(13.84)

where we assume the density of these sands is equal to

RTENOTITLE....................(13.85)

Since the modulus is the product of the density and square of velocity, we get an equation that is cubic in terms of porosity. The bulk modulus can be derived as

RTENOTITLE....................(13.86)

where A = 3.206, B = 3.349, and C = 1.143. Eq. 13.86 can be further simplified if porosity Φ is not too large (<30%):

RTENOTITLE....................(13.87)

where D for clean sandstone is equal to 1.52. This includes an empirical expression of the normalized modulus as a direct dependence on porosity and "D" parameter. Table 13.10 and Fig. 13.41 show empirical relations generated from dry velocity data of relatively clean rocks. The parameter D is related to rock texture and should be calibrated for local reservoir conditions. In general, it has a narrow range from 1.45 to slightly more than 2.0, primarily depending on rock consolidation.


By inserting this D function into Eq. 13.71, we find

RTENOTITLE....................(13.88)

Solid Mineral Bulk Modulus

The mineral modulus (solid grain bulk modulus) K0 is an independent parameter, and the rock texture controls Kd. However, as mentioned previously, the normalized modulus Kn controls the fluid saturation effect rather than Kd or Ks individually. The mineral modulus K0 is equally as important as Kd. However, in most applications of the Gassmann’s equation, only Kd is measured. Properties of the mineral modulus K0 are often poorly understood and oversimplified. K0 is the modulus of the solid material that includes grains, cements, and pore fillings (Figs. 13.1 through 13.8). If clays or other minerals are present with complicated distributions and structures, K0 can vary over a wide range. Unfortunately, few measurements of K0 have been made on sedimentary rocks (Coyner[78]), and the moduli of clays are a particular problem (Wang et al.[50] and Katahara[49]; see Table 13.6b). These data show that at a high pressure, K0 for sandstone samples range from 33 to 39 MPa. K0 is not a constant and can increase more than 10% with increasing effective pressure. Fig. 13.42 shows the influence of K0 on Gassmann’s calculation. This case uses a dry bulk modulus calculated with the mineral modulus of 40 GPa, D = 2, and a water modulus of 2.8 GPa. The water saturation effect was calculated for three mineral moduli of 65, 40, and 32 GPa. Results show that for the same Kd and Kf, bulk modulus increment ΔK because of fluid saturation increases with increasing mineral modulus K0. Errors caused by uncertainty of K0 decrease with increasing porosity and fluid modulus Kf.


Because of lack of measurements on bulk mineral modulus, we often must use measured velocity/porosity/clay-content relationships for shaly sandstone to estimate the mineral modulus. Assuming zero porosity and grain bulk modulus of 2.65 gm/cc, we can derive mineral bulk and shear modulus from measured P- and S-wave velocity. The results are shown in Table 13.11.
  1. For relatively clean sandstone (with few percent clay content), mineral bulk modulus is 39 GPa, which is stable for differential pressures higher than 20 MPa. Mineral shear modulus is around 33 GPa, which is significantly less than 44 GPa for a pure quartz aggregate. Shear modulus is more sensitive to differential pressure and clay content.
  2. For shaly sandstone, mineral bulk modulus decreases 1.7 GPa per 10% increment of clay content.


Such derived mineral bulk moduli can be used for Gassmann’s calculation if there are no directly measured data or reliable models for calculation.


With a change of fluid saturation from Fluid 1 to Fluid 2, the bulk modulus increment (ΔK) is equal to

RTENOTITLE....................(13.89)

where Kf1 and Kf2 are the moduli of Fluids 1 and 2, respectively, and ΔK21 represents the change in the saturation increment that results from substituting Fluid 2 for Fluid 1. Eq. 13.89 uses the fact that the gain function G(Φ) of the dry rock frame remains constant as fluid modulus changed (this may not be true for real rocks). The fluid substitution effect on bulk modulus is simply proportional to the difference of fluid bulk modulus.

If we know the gain function for a rock formation, we can estimate the fluid substitution effect without knowing shear modulus.

RTENOTITLE....................(13.90)

where ρ1, ρ2, Vp1, and Vp2 are the density and velocity of rock with Fluid 1 and 2 saturation. Both Eqs. 13.89 and 13.90 are direct results from simplified Gassmann’s equation (Eq. 13.64). In Fig. 13.43, we show the typical fluid modulus effect on the saturated bulk modulus Ks. Even at a modest porosity of 15%, changes can be substantial. At in-situ conditions, pore fluids are often multiphase mixtures. Dynamic fluid modulus may also depend on fluid mobility, fluid distribution, rock compressibility, and seismic wavelength.

Cracked Rock

For some cracked rocks, different methods of calculating velocities and the effects of pore fluids are preferable. Numerous theories have been developed to describe the effects of crack-like pores. Most view cracks as ellipsoids with their aspect ratio, α, defined as the ratio of the semiminor to semimajor axes. Eshelby[79] examined the elastic deformation of such elliptical inclusions, and these results were then applied to the compressibility of rocks by Walsh.[80] In concept, long, narrow cracks are compliant and can be very effective at reducing the rock moduli at low crack porosities. The primary controlling factor for these elliptical fractures is the aspect ratio, α, defined as the ratio of the ellipse semiminor (a) to semimajor (b) axes:

RTENOTITLE....................(13.91)

The smaller the value of α, the softer the crack and cracked rock, resulting in lower velocities and stronger pressure dependence.

Numerous assumptions are made in the derivation and application of cracked media models, such as the following:

  1. The porous material is isotropic, elastic, monomineralic, and homogeneous.
  2. The fracture population is dilute, and few, or only first-order, mechanical interactions occur among fractures.
  3. Fractures can be described by simple shapes.
  4. The pore-fluid system is closed, and there is no chemical interaction between fluids and rock frame (however, shear modulus need not remain constant).


Some of these assumptions may be dropped, depending on the model involved. For example, Hudson[81] specifically includes the effect of anisotropic crack distributions.

One particularly useful result was derived by Kuster and Toksoz.[82] Using scattering theory, they derived the general relation of bulk and shear moduli of the cracked rock (K*, μ*) to the crack porosity (c), aspect ratio (αm), mineral (K0, μ0), and inclusion or crack moduli (K′, μ′) (Cheng and Toksoz[83]).

RTENOTITLE....................(13.92)

RTENOTITLE....................(13.93)

Here, T1 and T2 are scalar functions of K0, μ0, K′, and μ′, and correspond to Tiijj and Tijij in Kuster and Toksoz.[82] This formulation allows the effects of several populations (several values of m) of cracks to be summed. The general limitation is that the porosity for any particular aspect ratio cannot exceed the value of the aspect ratio itself.

The results of the Kuster-Toksoz model are shown in Fig. 13.44. Numerous important features should be noted. Velocities drop rapidly for long, narrow cracks (small α), with even small crack porosities. For such soft cracks, the increase in velocity is dramatic. At a shape close to spherical (α above about 0.5), the pores are stiff, and the change in density dominates. Thus, with αs close to unity, going from dry to water-saturated actually decreases the velocity. Notice also that for small aspect ratios, the shear velocity increases with water saturation. This requires a changing shear modulus with saturation, in direct violation of a primary assumption of Gassmann’s relations. This changing shear modulus is one reason why Gassmann’s relations may not work well in fractured rocks. An example of a rock modeled by both Gassmann’s and Kuster-Toksoz techniques is shown in Fig. 13.45. For this limestone, Gassmann’s relations substantially under estimate the effect of liquid saturation. The Kuster-Toksoz prediction for oil saturation is close to the experimental observed values. However, the success of this model is not quite as spectacular as it seems, because an arbitrary population of fractures and aspect ratios (αms) can be included to force such a good fit. The actual population of cracks in rocks remains unknown.


The expressions in Eqs. 13.92 and 13.93 are complicated and difficult to apply. The linear relation of normalized velocities to crack aspect ratio and porosity suggests that a simplified form can be derived to give a first-order approximation.

RTENOTITLE....................(13.94)

Anisotropy

To this point, we have usually considered rocks to be isotropic. In reality, most rocks are anisotropic to some degree. Some dominant lithologies, such as shales, are by definition anisotropic (otherwise, they are mudstones). In addition, many ubiquitous sedimentary features such as bedding will lead to anisotropy on a larger scale. In-situ stresses are anisotropic (Fig. 13.34), resulting in an anisotropy in rock properties. Anisotropy in transport properties such as permeability is a common concern in describing reservoir flow. Fractured reservoirs typically have a preferred fracture and flow direction, and these directions often can be ascertained from oriented borehole or surface seismic data.

An interesting aspect of anisotropy is the phenomenon of shear-wave splitting. Elastic anisotropy means that the stiffness or effective moduli in one direction will be different from that in another. For shear waves, their particle motion will be approximately normal to the direction of propagation. The velocity will depend on the orientation of the particle motion. The shear wave will then "split" into two shear waves with orthogonal particle motion, each traveling with the velocity determined by the stiffness in that direction. An example of this is shown in Fig. 13.46 from Sondergeld and Rai.[84] The recorded waveform can be seen as two distinct shear waves traveling at their own velocities. Note that when these distinct waves are examined in isolation, their velocity is independent of direction. A single input wave has been split into two waves. This is similar to the image splitting in optics when light travels through an anisotropic medium. On the other hand, because compressional waves have particle motion only along the direction of propagation, they have no splitting.


Although the split shear waves may travel each with a constant velocity, the amplitude within each will be strongly dependent on angle. The energy of the initial single shear wave is partitioned as vector components in each of the principal directions. This amplitude dependence on angle is shown in Fig. 13.47, also from Sondergeld and Rai.[84] Figs. 13.46 and 13.47 demonstrate that measurement of seismic shear waves at the surface will be useful in delineating in-situ anisotropy directions. This anisotropy can then be related to factors such as oriented fractures and in-situ stress directions.


A typical homogeneous but bedded sedimentary unit would have a horizontal plane of symmetry as well as a vertical symmetry axis of rotation. This situation is commonly referred to as Vertical Transverse Isotropy (VTI), although the term "Polar Anisotropy" has also been suggested (Thomsen[85]). For "weak" anisotropy (Thomsen[86]), the dependence of velocities as a function of angle (θ) from the symmetry axis can be written as

RTENOTITLE....................(13.95)

RTENOTITLE....................(13.96)

RTENOTITLE....................(13.97)

where Vp(θ) is the compressional velocity and VS-(θ) and VS||(θ) are the shear velocities with particle polarizations perpendicular and parallel to the symmetry plane (e.g., bedding), respectively.

The Thomsen[86] anisotropic parameter ε can be defined as

RTENOTITLE....................(13.98)

where VP0 is the compressional velocity parallel to the axis of symmetry, and VP90 is the velocity perpendicular to this axis. The parameter γ can be defined as

RTENOTITLE....................(13.99)

where VS0 is the shear velocity parallel to axis of symmetry, and VS||90 is the velocity perpendicular to this axis.

The anisotropic parameter δ is more difficult to characterize, and is the primary component modifying the compressional moveout velocity from the isotropic case. To describe it, we must refer back to stiffness defined in the generalized Hooke’s law given in Eq. 13.43.

RTENOTITLE....................(13.100)

The advantage of these formulations is that they can be extracted from observed shear-wave splitting or extracted from normal moveout (NMO) corrections during seismic processing. Thus, they provide a valuable tool to describe the anisotropic character of reservoirs from remote measurements.

Attenuation and Velocity Dispersion

As seismic acoustic waves pass through rock, some of their energy will be lost to heat. For tight, hard rocks, this loss can be negligible. However, for most sedimentary rocks, this loss will be significant, particularly on seismic scales. In reality, all rocks are anelastic to some degree. We must rewrite our wave equation to include this energy or amplitude loss with depth, z.

RTENOTITLE....................(13.101)

where A(z,t) is the amplitude at some point of depth and time, A0 is the initial amplitude, and k* is the complex wave number (k* = k + l). Note that here αl is a loss parameter, and not an aspect ratio. Therefore, we can rewrite Eq. 13.101 as

RTENOTITLE....................(13.102)

Another common measure is the loss decrement δ:

RTENOTITLE....................(13.103)

where the wavelength λ depends on the velocity V and frequency f: λ = V/f. However, the most common measure of attenuation is 1/Q.

RTENOTITLE....................(13.104)

One of the most straightforward descriptions of the relation of velocity to attenuation was developed by Cole and Cole[87] and applied to attenuation measurements by Spencer.[88] The Cole-Cole relationships involve both a peak frequency or characteristic relaxation time, τ, for the attenuation mechanism, and a spread factor, b, which determines the distribution of relaxation times. The real and imaginary components, B′ and B", of a general modulus, B = B′ + iB", are

RTENOTITLE....................(13.105)

RTENOTITLE....................(13.106)

where y = ln(ωτ), B0 and B are the zero and infinite frequency moduli.

This would lead to a general attenuation of

RTENOTITLE....................(13.107)

These relations connecting velocity and attenuation are plotted in Fig. 13.48. This figure represents losses and velocity dispersion (frequency dependence) caused by a single relaxation mechanism. At high frequencies, the material is unrelaxed and stiffer, and it has a higher velocity. At low frequencies, the material has time to relax, and velocities are lower.


Fluid mobility also influences rock inelastic properties. Most of the observed losses are caused by relative motion of fluid in the pore space. For a constant pore fluid type, permeability will control the motion and dissipation, thus making attenuation a permeability indicator. For variations in viscosities, mobility also will be dependent on frequency, and attenuation and dispersion may indicate fluid type.

Many models have been proposed, such as those of Biot,[89] O’Connell and Budiansky,[90] Walsh,[91] and Dvorcik and Nur.[92] Unfortunately, the different mechanisms proposed often give contradictory results.

Wave attenuation and dispersion in vacuum dry rock is relatively negligible.[93] Porous rocks containing fluids show a strong frequency-dependent attenuation. Variations in fluid properties such as modulus, viscosity, and polarity have a strong influence on 1/Q (Clark,[93] Winkler et al.,[94] Murphy,[95] Tittmann et al.,[96] Jones,[97] and Tutuncu et al.[98]). These results indicate that the dominant 1/Q mechanism is the interaction and motion of fluid in the rock frame rather than intrinsic losses either in the frame or the fluids themselves. Squirt flow is believed to be the primary loss mechanism in consolidated rocks, although the inertial Biot mechanism may be important in highly permeable rocks (Vo-Thant,[99] Yamamato et al.[100]).

Fluid motion and pressure control velocity changes and seismic sensitivity to pore fluid types. One obvious factor is viscosity. The two most commonly used theoretical concepts are the inertial coupling of Biot[89] and the squirt-flow mechanism (see, for example, O’Connell and Budiansky,[90] or Dvorcik and Nur[92]). Biot gives a characteristic frequency, ωc (roughly, the boundary between high and low frequency range) with the viscosity dependence, η, in the numerator:

RTENOTITLE....................(13.108)

Here, Φ is porosity, k is permeability, and ρ is fluid density. However, squirt-flow mechanisms lead to viscosity dependence in the denominator:

RTENOTITLE....................(13.109)

Here, K is frame modulus, and α is crack aspect ratio. These contrasting dependencies indicate that viscosity can be an obvious test to ascertain which theory is applicable.

Compressional (Vp) and shear (Vs) velocities for a sample of the Upper Fox Hills Sandstone (Heather well) are shown in Fig. 13.49. Several features should be noted. For the dry sample (open symbols), Vp and Vs show little frequency or temperature influence. This confirms that the primary dispersive and temperature effects are dependent on pore fluids. When saturated with glycerine, strong temperature and frequency dependence is obvious. Shear velocity is not independent of the fluid, but increases with increasing fluid viscosity, indicating a viscous contribution to the shear modulus. Vp increases with viscosity also. More importantly, the dispersion curve shows a systematic shift to lower frequencies with increasing velocities, consistent with squirt flow.


Attenuation (1/Q) and velocity dispersion are strongly dependent on pore phase and compressibility, particularly as controlled by partial gas saturation. Attenuation could become a valuable direct hydrocarbon indicator (e.g., Tanner and Sheriff[101]). More recently, Klimentos[102] used the ratio of compressional to shear attenuations as a hydrocarbon indicator in well logs. Unfortunately, application of these properties is not frequent because of incomplete understanding of the phenomena and lack of appropriate tools to extract the information. Laboratory measurements at frequencies and amplitudes encompassing the seismic range have confirmed the strong dependence on partial gas saturation (Fig. 13.50a). However, attenuation is decreased by confining pressure, dropping rapidly as pressure increases (Fig. 13.50b). Attenuation peaks will also depend on specific rock characteristics. Absorption peaks seen in one frequency band may not be apparent in others.


With the improving quality of seismic data, maps of the estimated attenuation are becoming a common displayed attribute. The relative values of 1/Q measured through time-lapse reservoir monitoring are becoming robust. As indicated in Fig. 13.50a, 1/Q will be sensitive to many of the common recovery processes.

Rock Failure Relationships

Introduction

In this section, we will go through the various relationships describing mechanical failure in rocks. This is important because under reservoir pressure and stress conditions, production can induce rock failure, sometime with catastrophic effects. By applying strength criteria, within reservoir simulators we can predict when problems might occur. In Section 13.5, we examined the elastic behavior, which was largely reversible. Here we deal with permanent deformation. By rock failure, we mean the formation of faults and fracture planes, crushing, and relative motion of individual mineral grains and cements. Failure can involve formation of discrete fracture zones and the more "ductile" or homogeneous deformation. The later deformation is caused by a broad distribution of fracture zones or general grain crushing during compaction. We will not consider deformation caused by plastic strains of the mineral components, as is common in salt and in calcite at higher temperatures. In our analysis, several assumptions are made: The material is isotropic and homogeneous; stresses are applied uniformly; textural characteristics such as grain size and sorting have no influence; temperature and strain rate are ignored; and the intermediate stresses are presumed to play no role. Each of these assumptions can be violated, and some have been demonstrated to have major influences on rock strength.

Coulumb-Navier Failure

To begin with, a brief review of the standard Mohr failure criteria will be examined to introduce concepts and define terms, as well as to establish the basic mathematics behind the strength relationships. Units of stress and strength are the same as pressure and were covered previously. More detailed descriptions can be found in standard textbooks (i.e., Jaeger[103][104]). Mohr circles and a linear failure envelope are the most common methods used to plot stresses and indicate strength limits. This technique predicts failure when stresses surpass both the intrinsic strength of a rock and internal friction. The primary terms and characteristics are shown in Fig. 13.51. Normal stresses across any plane are plotted on the horizontal axis, and shear stresses are plotted on the vertical axis. Compressive stresses are defined as positive (as opposed to the mechanical engineering convention of tensional stresses being positive). For the hydrostatic case, all stresses are equal; this stress state is represented by a point on the horizontal axis. When stresses differ, the maximum principal stress, σ1, and minimum stress, σ3, are plotted on the horizontal axis and the possible shear stresses along any plane fall on a hemisphere connecting σ1 and σ3 (Fig. 13.52). The mean stress, σm, and radius of this circle, r, are simple sums and differences of the principal stresses.

RTENOTITLE....................(13.110)

RTENOTITLE....................(13.111)

The normal stress across any plane, σn, and the shear stress along the plane, τ, are functions of the principal stresses and the plane orientation.

RTENOTITLE....................(13.112)

RTENOTITLE....................(13.113)

RTENOTITLE....................(13.114)

RTENOTITLE....................(13.115)

where θ is the angle between the plane and the σ3 direction.


From Eqs. 13.114 and 13.115, the maximum shear occurs along a plane oriented at RTENOTITLE (45°). However, because of friction, rocks do not fail along this plane. Instead, failure occurs along some rotated plane where friction is lower, yet shear stress is still high. This failure point (or plane) is shown in Fig. 13.51 as the nearly diagonal line. Fig. 13.51 also shows the associated normal and shear stresses. If numerous failure tests are made and plotted, an envelope is defined as in Fig. 13.52. In this case, friction is assumed to be a simple linear function of normal stress, and the resulting envelope is also linear. The slope of this envelope is α, and we define μ as the angle of internal friction

RTENOTITLE....................(13.116)

Within this framework, we can define several important properties of the rock as shown in Fig. 13.52:

C0 = Uniaxial or unconfined compressive strength (σ3 = 0)

Cu = Cohesive strength or the intercept of the envelope with σn = 0.

Ct = Tensional strength.

The failure envelope is then defined by the line

RTENOTITLE....................(13.117)

If the rock has already been broken, or a fracture already exists, then both Cu and Ct will be close to zero.

Several useful equations can be derived from the geometric relationships shown so far. From the equation for a circle,

RTENOTITLE....................(13.118)

At the intersection of the envelope and the circle, we must have

RTENOTITLE....................(13.119)

which leads to

RTENOTITLE....................(13.120)

Using the general solution to a second-order polynomial gives

RTENOTITLE....................(13.121)

Because we want only a point where the circle touches the envelope, the square root term must vanish.

RTENOTITLE....................(13.122)

After some algebraic manipulation, we find

RTENOTITLE....................(13.123)

and

RTENOTITLE....................(13.124)

Substitution of Cu (defined in Eq. 13.123) into Eq. 13.121 gives an expression for normal stress.

RTENOTITLE....................(13.125)

If the envelope could be continued into the tensional region, the tensional strength could easily be obtained:

RTENOTITLE....................(13.126)

Under tension, the stresses are negative, although the tensional strength is a positive number. Thus, if rocks could fail according to a constant internal friction, we would have a simple way to relate the stresses involved and need only a couple of material constants, such as Cu and α.

Mohr Failure, Curved Envelopes and Hoek-Brown Relationships

We are immediately faced with two problems when we try to apply Coulomb-Navier failure criteria: (1) Rocks do not generally have a linear failure envelope, and (2) material properties controlling failure must be obtained either through logs or assumed behavior. Fig. 13.53 shows the type of envelope commonly seen. In fact, we know that the slope must change as stresses are increased because rocks begin yielding and act more plastically. Fig. 13.54 shows the generalized behavior expected. At normal stresses above the brittle-ductile transition, failure can no longer be maintained on a single plane, but is distributed more homogeneously throughout the sample. We must develop different failure criteria, one that produced an appropriately curved envelope, and we expect it to have a strong porosity dependence (Fig. 13.55).


Numerous failure criteria have been proposed that are primarily empirically based. Table 13.12 shows some of the criteria proposed both for general purposes and for specific rock types or conditions. Observed failure envelopes are smooth forms so simple exponential or power-law functions can usually be found that fit the data well. The relations of Bienlawski[106] and Hoek and Brown[107] are most common. Much of the recent work in rock mechanics has been directed toward ascertaining the constants of these relationships in terms of easily measurable rock properties. Note that these relationships apply primarily to the brittle failure regime and cannot be used for grain crushing or pore collapse (as we shall see later) or when substantial ductile or plastic deformation is involved. We will examine these proposed forms to interrelate terms and reduce unknowns to variables that can be derived from logs.


Hoek and Brown[107] compiled extensive data on a variety of rock types and produced relationships that are simple and can be developed into forms amenable to well-log analysis. A primary feature of this failure criterion is a relation between the maximum and minimum stresses when both are normalized by the uniaxial compressive strength

RTENOTITLE....................(13.127)

This formulation was motivated by the systematic behavior seen in many tests as shown in Fig. 13.56. In Eq. 13.127, m and s are material constants dependent on the overall quality of the rock mass, and m is also dependent on the rock type (Table 13.13). Note that we could derive the value for m from a mineralogic analysis. In our analysis, we will presume that the local rock mass of interest is intact, and thus

RTENOTITLE....................(13.128)

For applications that are in sandstones, numeric results can often use

RTENOTITLE....................(13.129)

Eq. 13.127 can be rewritten to give one principal stress in terms of the other:

RTENOTITLE....................(13.130)

RTENOTITLE....................(13.131)

Such normalized stress states were used to construct the curved envelope in Fig. 13.54.


The tensional strength, the stress at which an envelope would cross the horizontal axis, is found by equating σ1 to σ3 in Eq. 13.130 (note that Ct is defined as a positive number).

RTENOTITLE....................(13.132)

For sandstones, this results in RTENOTITLE, or 0.067 C0. The 15 uniaxial tensional strength στ* is slightly different and is defined as the value at which the maximum stress, σ1, equals zero. From Eq. 13.130, we get

RTENOTITLE....................(13.133)

Other basic properties are not so simply derived.

We must produce from the stress relationships (Eqs. 13.127 or 13.130) an equation for a failure envelope that permits us to resolve the shear and normal stresses on a failure plane, its orientation, and an approximation of the internal friction, and simply predict regions of instability. The general envelope shapes seen in Figs. 13.54 and 13.56 suggest a form like that proposed by Murrell[109] and Bienlawski[106]:

RTENOTITLE....................(13.134)

where A, b, and n are material constants. Because the envelope intersects the horizontal axis when the normal stress equals the tensional strength,

RTENOTITLE....................(13.135)

When the normal stress is zero, the envelope intersects the vertical axis at the cohesion value Cu. From Eq. 13.134, this requires

RTENOTITLE....................(13.136)

Therefore, the general form for an envelope is

RTENOTITLE....................(13.137)

To derive the slope, α, at any point, we note that the envelope is only slowly varying over a small stress range and could be locally approximated by a line. If we use a pseudocohesion RTENOTITLE defined by Eq. 13.123 for the stress condition, σm, r we can subtract the same RTENOTITLE from a slightly different stress condition, σm’, r’. Solving for α gives

RTENOTITLE....................(13.138)

The Hoek-Brown stress criteria allow us to redefine the mean, σm, and differential, r, stresses

RTENOTITLE....................(13.139)

RTENOTITLE....................(13.140)

By substituting these relations into Eq. 13.138 for two stresses σ3 and σ3 + δ σ3, expanding the result and allowing the stress difference, δσ3, to approach zero (what a pain!), we find

RTENOTITLE....................(13.141)

As we found previously (Eq. 13.125), the normal stress is then

RTENOTITLE....................(13.142)

The cohesion is the shear stress value when σn equals zero. This will occur for σ3 somewhere between zero and −Ct. In other words, σn = 0 for

RTENOTITLE....................(13.143)

where β is a value around 0.5. We could substitute this term into Eqs. 13.141 and 13.142 and solve for β . However, this results in a rather complicated root to a third-order polynomial. Fortunately, by iteration, we can show that β is relatively constant at about 0.62 with little dependence on m. Using this value of β in Eq. 13.143 and substituting into the previous equations gives us our cohesion. For a sandstone with m = 15, we get

RTENOTITLE....................(13.144)

The definition of our curved envelope in Eq. 13.137 is not strictly compatible with the Hoek-Brown stress relations. However, we can get an estimate of the exponent, n, by using our tensile and cohesion strengths and some reasonable value of σn such as σn = C0. From Fig. 13.54, we can see that τ is approximately 1.1 C0 at this point. From Eq. 13.137, with m equal to 15,

RTENOTITLE....................(13.145)

This value falls within the range of 0.65 to 0.75 suggested by Yudhbir et al.[113] Thus, from a presumed simple relation between σ1 and σ3, almost all the necessary parameters can be derived.

Uniaxial Compressive Strength

We have seen how a general rock failure criterion can be reduced to a few parameters dependent on lithology (m) and the uniaxial compressive strength (C0). Lithology is commonly derived during log analysis, so m may be estimated (Table 13.13). What is needed still is an initial measure of rock strength provided by C0. C0 can be estimated from porosity or sonic velocities, but many factors such as grain size, clay content, or saturation have significant influences.

A large amount of C0 data is available and, although there is considerable scatter, C0 usually varies systematically with other rock characteristics. We will concentrate on porosity as the primary controlling factor because it is routinely available from logs and is a fundamental input into reservoir simulators.

Numerous relationships have been developed to estimate C0, often in conjunction with general rock strength relationships. Table 13.14 lists many of the proposed relations for C0, some of which are plotted for various rock types in Fig. 13.57 and for sandstones in Fig. 13.58. We expect C0 to decrease as porosity increases. At some transition porosity, rocks will lose all initial strength and become merely a loose aggregate. No matter which relationship is chosen, variables such as cementation, alteration, texture, and so on can cause significant scatter.


If we accept the restrictive relationships for failure of Eq. 13.130 or 13.134, we can derive C0 from any such strength data:

RTENOTITLE....................(13.146)

However, this equation predicts a finite strength even as porosity approaches 1.0. More realistic forms must be used so that strength vanishes at some porosity Φc. This limiting porosity was shown as a crossover porosity from rock to a slurry by Raymer et al.[119] and was referred to as "critical porosity" elsewhere. Jizba[110] used such a concept to derive a general strength relationship for sandstones:

RTENOTITLE....................(13.147)

where τ and σn are the shear and normal stresses at failure.

The 0.36 within the parentheses is her presumed value for Φc. Notice, however, that this form indicates that sandstones have no tensile or cohesive strength. We can obtain a better result by using Jizba’s relationship at elevated confining pressure (say, 50 MPa), where it is more valid, and recasting the trend in terms of Eq. 13.130, as we did for the Scott relation.[105] Dobereiner and DeFreitas[120] measured several weak sandstones, and their results suggest that critical porosity is approximately 0.42. Using this critical porosity, we derive a uniaxial compressive strength

RTENOTITLE....................(13.148)

This C0 equation is plotted in Fig. 13.58 along with the modified Scott[112] and Jizba[110] equations and data of Dobereiner and DeFreitas.[120]

Compaction Strength

As was indicated in Fig. 13.55, at some elevated stress or confining pressure, the rock will begin to show ductile deformation. The grain structure begins to collapse, and the rock will compact and lose porosity. This compaction strength, Cc, is itself a function of porosity as well as mineralogy, diagenesis, and texture. In Figs. 13.59a and 13.59b, the behavior of two rocks under hydrostatic pressure is shown. The high-porosity (33%) sandstone (Fig. 13.59a) has a low "crush" strength of about 55 MPa. With a lower porosity of 19%, Berea sandstone has a much higher strength of 440 MPa (Fig. 13.59b). Notice that in both Figs. 13.59a and 59b, permanent deformation remains even after the stress is released. This hysteresis demonstrates the damage to the matrix structure caused by exceeding the crush strength.


In the cases in which studies are restricted to sandstones, an exponential dependence on porosity is usually observed (Fig. 13.59a). Scott[112] fit his and the Dunn et al.[108] data to the form

RTENOTITLE....................(13.149)

With a general relationship available for uniaxial compressive strength and the compaction limit, rock failure envelopes can be determined for sandstones at any porosity. Fig. 13.60 shows the complete envelopes for the porosity range 0.15 to 0.35.

Clay Content

Most sandstones are mixtures of mineral such as feldspars, calcite, dolomite, micas, clays, and of course quartz. Clays are a very common component and can make up anywhere from 0 to nearly 100% of a clastic rock. Usually, at some point greater than 40% clay, the rock is considered a shale or mudstone rather than a sandstone (refer to Section 13.7). The structure of clay minerals and their typical methods of bonding are significantly different from those of quartz, so we would expect clays to strongly influence mechanical properties. Such influences depend on the nature of the clay, the amount and location within the rock framework, and the state of hydration.

There have been few systematic studies of clay effects on the mechanical properties of rocks. Corbett et al.[121] demonstrated how the coefficient of internal friction and thus the strength of Austin chalk strongly depends on even a small clay fraction (Fig. 13.61). In particular, smectite content was found to have more influence in this case than other clays. This allows us to derive a general relationship between failure and clay content.

RTENOTITLE....................(13.150)

where C is the smectite fraction. Unfortunately, this equation was developed for dry samples.


Jizba[114] tested several dry clay-rich samples and proposed a general linear envelope form for shales and shaley sandstones.

RTENOTITLE....................(13.151)

More relevant data, however, comes from Steiger and Leung[122] with both dry and saturated shale measurements (Fig. 13.62). From these data, we derive an approximation for the wet shale uniaxial compressional strength.

RTENOTITLE....................(13.152)

This relation, as well as those for the Austin chalk, suggests a strong clay dependence. Jizba,[110] however, reported only a slight dependence of C0 on clay content in shaley sands.


It is likely that in many sands, clays reside as pore-filling materials and have only a secondary effect on mechanical properties. At this point, we expect clays to have a significant effect even in fairly pure sands (this will be seen also in sonic velocity measurements). Thus, a more general form for uniaxial compressive strength of sandstones would be

RTENOTITLE....................(13.153)

where the coefficient a has a value of approximately 100. The influence of clays on the mechanical properties of rocks needs much further investigation.

Pore Fluid Effects

Fluids can alter rock mechanical properties of rocks through fluid pressure, chemical reactions with mineral surfaces, and by lubricating moving surfaces. The primary fluids encountered are brines and hydrocarbon oils and gases. Drilling, completion, and fracturing fluids can also be present, and their effects are typically studied to prevent formation damage. We will concentrate on the role of water and, in particular, how water saturation can influence rock strengths measured in the laboratory or derived from well logs.

Effective Stress. Pore fluid pressures will reduce the effective stress supported by the rock mineral frame. This effect has been well known since the publication of Terzaghi and Peck[63] and has been documented by numerous investigators. The most common form for the effective stress law is

RTENOTITLE....................(13.154)

where σe is the effective stress, σa the applied stress on the rock surface, Pp, and the pore pressure. Note that this is the same as Eq. 13.35. The effective stress coefficient n is also called Biot’s poroelastic term.

RTENOTITLE....................(13.155)

where Kd is the dry rock bulk modulus and Ko the mineral bulk modulus. Because the rock modulus is usually much lower than the mineral modulus, n is often close to unity. In many applications and when no other information is available, n is usually taken as one.

In our analyses, all of the stresses used to describe rock failure were actually effective stresses. Rock failure can be dramatically affected by pore pressure, as indicated in Fig. 13.63. An envelope is plotted for a sandstone with porosity of 25%. For applied principal stresses of 225 MPa for σ1, 175 MPa for σ3, and a Pp of 75 MPa, the effective Mohr circle plots well within the field of stability. The pore pressure has been subtracted from both applied stresses to give effective principal stresses of 150 and 100 MPa. If pore pressure is increased, the effective stresses decrease, and the Mohr circle is shifted left until eventually the envelope may be contacted and the rock fails by brittle fracture. On the other hand, if pore pressure decreases, the Mohr circle shifts right, and the rock may contract the Roscoe surface and fail by compaction or grain crushing. In any case, if pore pressures are known, their effects can be accounted for in a straightforward way.


Problems can arise experimentally because of the inability of pore pressure to reach equilibrium. If fluid can flow freely and constant pore pressure is maintained, then an experiment is termed "drained." If deformation is too rapid, permeability low, fluid viscosity high, or boundaries are sealed, then fluid is trapped in the rock, and fluid pressure changes as a function of rock deformation. Brace and Martin[123] showed that strain rates must be very low in crystalline rocks of low permeability to maintain a uniform pore pressure and follow a macroscopically defined effective stress law such as Eq. 13.154. For most sandstones, permeability is sufficient to provide drained conditions. Problems usually occur in low-permeability rocks such as siltstone or shales. Considerable effort and time are usually needed to allow constant pore pressure, or merely to maintain pore pressure equilibrium (Steiger and Leung[122]). Tests are made under undrained conditions, but the resulting changes in pore pressure must then be measured or otherwise calculated. These effects are mechanical problems that are often difficult to deal with, but the processes are basically well understood.

Chemical Effects. A more subtle problem involves chemical effects of pore fluids. Water is an active, polar compound, and numerous investigations (Griggs[124] and Kirby[125]) have shown that even small amounts of water or brine can have a substantial influence on rock mechanical properties. Colback and Wiid[126] demonstrated how even changes in the relative humidity or partial pressure of water in the pores can lower rock strength dramatically (Fig. 13.64). Colback and Wiid[126] and Dunning and Huff[127] saw a direct relationship between the loss in rock strength and the chemical activity of the pore fluid. Meredith and Atkinson,[128] Freeman,[129] and others have shown increased crack velocities and acoustic emissions at constant crack intensity factors when water is introduced. Ujtai et al.[130] saw substantial effects of water on all time-dependent tests for creep strain, fatigue, and slow crack growth. In general, uniaxial compressive strength is reduced by 20 to 25% in wet rocks. This implies that many laboratory measurements result in rock strengths that are systematically too high.


A strong influence of the chemical activity on rock mechanical properties is supported by other types of measurements. Seismic properties depend upon mineral grain stiffness and the stiffness of grain-to-grain contacts. In completely dry rocks (oven-dried under vacuum), there is almost no seismic attenuation, and rocks are stiff. Even small amounts of water, a few monolayers, can appreciably lower rock stiffness and seismic velocities.

Bulk Lubrication. Common experience leads us to expect many geologic materials, such as soils, to be substantially weaker when wet. We have already seen this effect in chalk and shales. Surface bonding energies and water surface tension result in strong capillary forces that draw and hold water in pore spaces. Water penetrates and separates grains. Grain movement is facilitated by motion in mobile fluid layers. This is a highly scientific way of saying "slippery when wet." Clay minerals in particular are well known for their ability to absorb large quantities of water. Swelling properties of clays and shales are often studied for drilling engineering purposes. Not only do clays have lower friction surfaces when wet, but water absorption and the resulting clay expansion can disaggregate the rock matrix. Loss of strength because of such mechanisms is more important in poorly consolidated or unconsolidated sediments. Dobereiner and DeFreitas[120] and Morgenstern et al.[131] report a 60% reduction in strength for muddy sediments upon saturation. At this point, we have not developed a systematic way of including a lubrication factor except as an implicit part of the clay corrections mentioned previously or as a measured reduction of the shear or Young’s modulus. We would expect the loss of intergrain friction to reduce the shear modulus significantly.

Grain Size and Texture

In granular rocks, grain size also influences strength. For constant porosity, mineralogy, and texture, a smaller grain size means greater strength. This tendency has been observed in several sandstones and can be understood in terms of grain contact models. Nelson[132] presents data on Navajo sandstone strength indicating a strong dependence on grain size. If a rock can be considered an aggregate of uniform spheres, smaller spheres will have more grain contacts per unit volume. Loads are distributed over more contracts, and each grain experiences lower stresses. Zhang[133] used Hertzian contact theory to calculate critical crushing strengths of quartz sands and found that porosity and grain radius combine to determine strength (Fig. 13.65). By fixing grain size, Zhang’s relationships could also provide crushing or compaction limits (Roscoe surfaces, Fig. 13.60) for sands at various porosities. For a grain size of 0.2 mm, we get a crushing strength, Cc, of

RTENOTITLE....................(13.156)

However, factors such as cementation and grain angularity will strongly alter this simple relationship.


If grains become cemented, not only does porosity decrease, but the effective area of intergranular contracts increases. Even small amounts of cement will increase strength substantially. Angularity of grains and sorting will also influence strength. More angular grains result in sharper point contacts, stress concentrations, and lower strength.

In general, if grain size is known to be smaller or cementation greater (for a given porosity and composition), then increased strength can be estimated by reducing the Hoek-Brown coefficient m. A value of m = 0 for siltstones and shales was suggested by Hoek and Brown.[134] Notice that this leads to minor contradiction because clays, with very fine grain size, weaken rocks. It is possible that many of Hoek and Brown’s "shales" were well indurated (slightly metamorphosed?), and grain size and increased cementation account for the increased strength (and reduced m). In rocks with low levels of diagenesis, clays reduce strength and require an increased m.

Rock Strength From Logs

Several techniques have been proposed for deriving rock strength from well log parameters. Coates and Denoo[135] calculated stresses induced around a borehole and estimated failure from assumed linear envelopes with strength parameters derived from shear and compressional velocities. They relied on the work of Deere and Miller[136] to provide estimates of compressive strength from dynamic measurements. Simplified forms of these relations are:

RTENOTITLE....................(13.157a)

RTENOTITLE....................(13.157b)

RTENOTITLE....................(13.157c)

where C0 is uniaxial compressive strength and E is dynamic Young’s modulus (see Section 13.5). Alternatively, we can include an empirical dependence of the internal friction angle, α, or the porosity, Φ.

RTENOTITLE....................(13.158)

Eqs. 13.159 and 13.160 provide a way to derive strengths assuming a linear envelope, and provided that compressional and shear velocity, lithology (e.g., gamma ray or SP), and density logs are available. If there is no shear log, one can be derived from the compressional velocity log and Vp-Vs relationships previously shown in Table 13.7.

The strength-porosity trend shown in Eq. 13.146 and modulus-porosity trends in Section 13.5 imply a correlation between strength and shear modulus for sandstone:

RTENOTITLE....................(13.159)

This leads to a velocity transform if the bulk density is known:

RTENOTITLE....................(13.160)

RTENOTITLE....................(13.161)

If we presume a simple relationship between compressional velocity of brine-saturated sandstones and shear velocity as developed by Castagna et al.,[4] we get

RTENOTITLE....................(13.162)

The shear modulus (or velocity) should be the most sensitive measure of strength, and shear properties are little affected by fluid saturations. Whenever possible, shear wave data should be collected and used in this analysis. If only compressional data is available, care must be used in translating the information into effective gas- or brine-saturated values (see Section 13.5.1 1). This is particularly true for partial oil saturations.

In our analysis, C0 was first determined from porosity. The influence of clay content was examined separately. The velocity-strength relationships above were derived from the porosity dependence, but clays are handled only indirectly through their effects on velocities. Strength parameters can be calculated directly from porosity (Eq. 13.148), but clays must then be included, as in Eq. 13.153. Calculated strengths based directly on porosity and clay content are shown in Fig. 13.66. These types of logs can be very valuable in detecting weak zones and units susceptible to failure. If at all possible, these kinds of logs should be calibrated with strength measurements directly on core samples.


Gamma Ray Characteristics

Introduction

The radioactivity of rocks has been used for many years to help derive lithologies. Natural occurring radioactive materials (NORM) include the elements uranium, thorium, potassium, radium, and radon, along with the minerals that contain them. There is usually no fundamental connection between different rock types and measured gamma ray intensity, but there exists a strong general correlation between the radioactive isotope content and mineralogy. Observed distributions have been available for numerous decades. In Fig. 13.67, the distributions of radiation levels observed by Russell[137] are plotted for numerous rock types. Evaporites (NaCl salt, anhydrites) and coals typically have low levels. In other rocks, the general trend toward higher radioactivity with increased shale content is apparent. At the high radioactivity extreme are organic-rich shales and potash (KCl). These plotted values can include beta as well as gamma radioactivity (collected with a Geiger counter). Modern techniques concentrate on gamma ray detection.


The primary radioactive isotopes in rocks are potassium-40 and the isotope series associated with the disintegration of uranium and thorium. Fig. 13.68 shows the equilibrium distribution of energy levels associated with each of these groups. Potassium-40 (K40) produces a single gamma ray of energy of 1.46 MeV as it transforms into stable calcium. On the other hand, both thorium (Th) and uranium (U) break down to form a sequence of radioactive daughter products. Subsequent breakdown of these unstable isotopes produces a variety of energy levels. Standard gamma ray tools measure a very broad band of energy including all the primary peaks as well as lower-energy daughter peaks. As might be expected from Fig. 13.68, the total count can be dominated by the low-energy decay radiation.


The radionuclides, including radium, may become more mobile in formation waters found in oil fields. Typically, the greater the ionic strength (salinity), the higher the radium content. Produced waters can have slightly higher radioactivity than background. In addition, the radionuclides are often concentrated in the solid deposits (scale) formed in oilfield equipment. When enclosed in flow equipment (pipes, tanks, etc.) this elevated concentration is not important. However, health risks may occur when equipment is cleaned for reuse or old equipment is put to different application.

Table 13.15 lists some of the common rock types and their typical content of potassium, uranium, and thorium. Potassium is an abundant element, so the radioactive K40 is widely distributed (Table 13.16). Potassium feldspars and micas are common components in igneous and metamorphic rocks. Immature sandstones can retain an abundance of these components. In addition, potassium is common in clays. Under extreme evaporitic conditions, KCl (sylvite) will be deposited and result in very high radioactivity levels. Uranium and thorium, on the other hand, are much less common. Both U and Th are found in clays (by absorption), volcanic ashes, and heavy minerals.

Measurement

Gamma ray logs are among the most common and useful tools in the oil and gas industry. Originally, measurements were reported in count rates, but all modern tools are calibrated to API units. Typical sedimentary response ranges from 0 to 200 in API units. Gamma ray log character is one of the primary methods used to correlate the stratigraphic section. For most engineering and geophysical applications, the gamma ray log is primarily used to extract lithologic, mineralogic, or fabric estimates.

The log response depends on the radiation, tool characteristics, and logging parameters. A 30-cm sodium iodide scintillation crystal with a photomultiplier tube is a common detector configuration. Thin, highly radioactive beds may be detected, but cannot be resolved below about 0.25 m. Radiation is damped primarily by formation material electron density and Compton scattering. This limits the depth of investigation to around 30 cm, although it will depend on the energy levels. Because the radioactive decay is a statistical process, slower logging rates produce better results. The low number of counts resulting from logging too fast cannot be increased by logging rate correction factors. Most tools are usually out of calibration if they are not centered in the borehole. Heavy barite mud can also lower the overall count rate, particularly for low-energy gamma rays.

Rather than merely measuring total gamma radiation, the energy levels can be detected separately. This allows the concentrations of K, U, and Th to be derived as independent parameters. Fig. 13.69 shows the energy windows used in a Baker-Atlas tool. This would allow, for example, the feldspars in immature sands to be separated from clays with adsorbed U or Th.


The most common use of gamma ray logs is to estimate the shale "volume" in rocks. It is important to remember that the tool measures radioactivity, and the correlation to shale content is empirical. Shales are presumed to be composed of clay minerals. Thus, the gamma ray level is assumed to be correlated with grain size. In reality, shales may be composed of 30% or more of quartz and other minerals. The clays within the shales may not be radioactive, and the adjacent sands may contain radioactive isotopes. However, radioactivity levels typically are related to grain size, as seen in Fig. 13.70. Here, core plugs were analyzed for median grain size and radioactivity level measured directly; crosses are fine-grained sands, while dots are silts and clay-rich rocks.


To extract the shale content in rocks, a linear or near-linear relation is used to convert a gamma ray index, Igr, to shale volume Vsh. Because local sands can contain radioactive components, and the shales may vary with depth, local baseline levels are chosen near the zone of interest.

RTENOTITLE....................(13.163)

where R is the measured radiation level, Rcleansand is the baseline level through a reference sand, and Rshale is the baseline through a representative shale. Several relations have been developed to derive shale volume (Fig. 13.71). A linear relation simply sets the shale content equal to the gamma ray index.

RTENOTITLE....................(13.164)

Other proposed relations shown in Fig. 13.71 are defined in Table 13.17. Several assumptions are made in these evaluations:
  • Compositions of sand and shale components are constant.
  • Baselines are chosen on representative "shales" and "clean" sands (although these terms are very subjective).
  • Simple mixture laws apply.
  • Fabric is not important.


Many of these assumptions may be poor approximations.


A more likely presumption is that the radiation level is dependent on the mixture densities and not volumes (Wahl[140] and Katahara[141]). In this case, a fabric analysis can also be performed. Katahara[140] modeled the shale component of shaly sands as existing in three forms:
  • Structural—an original depositional granular form.
  • Dispersed—clay distributed through the rock and pore space.
  • Laminated—thin layers of shale cutting the sand beds.


In Fig. 13.72, his results show a surprisingly simple form. The conclusion is that in most cases, the simple linear relation is appropriate.


As an example of this process, the shale content of a zone in a Gulf of Mexico well is estimated. In Fig. 13.73, a sand-shale sequence gives a gamma ray range of approximately 20 to 90 API units. A baseline of approximately 25 is chosen through the sand, and a baseline of approximately 98 is chosen for the shale. Using the relations in Eqs. 13.163 and 13.164 result in the shale volume estimates scaled at the bottom of the logged zone.


Gamma radiation levels can also be measured on core. This technique provides a profile of levels along the length of the core. The primary use is to correlate core depths to logged depths. An example is shown in Fig. 13.74. This procedure can be used to identify log features or positioning of the cored interval. Especially when core recovery is poor, this method is very useful in tying the core fragments to true depths. Core plugs can also be measured, although special equipment must be used to record the low levels of radiation associated with the small samples. In general, property correlations to the measured gamma ray levels are much better for cores than for the log because of the depth averaging in the log.[139]


Nomenclature


aij = water density coefficients
A = bulk modulus/porosity factor, Eq. 13.86
A = strength material constant, Eq. 13.134
Ao = initial wave amplitude
A(z,t) = wave amplitude with distance and time
Af 1, Af 2 = fraction fluid component 1, 2, etc.
Am1, Am2 = fraction mineral component 1, 2, etc.
A1, A2 = fraction component 1, 2, etc.
b = velocity/temperature constant, m/sC, Eq. 13.18
b = strength envelope intercept, GPa or MPa, Eq. 13.134
bij = brine density coefficients
B = brine compressional velocity factor, m/s, Eq. 13.32b
B = bulk modulus/porosity factor, Eq. 13.86
B′ = rock modulus, real component, GPa or MPa, Eq. 13.105
B" = rock modulus, imaginary component, GPa or MPa
Bo = rock modulus, zero frequency, GPa or MPa
Boo = rock modulus, infinite frequency, GPa or MPa
C = bulk modulus/porosity factor, Eq. 13.86
C = clay content, Eq. 13.150
Cijkl = stiffness tensor components, GPa or MPa
C0 = uniaxial or unconfined compressive strength, GPa or MPa
Ct = tensional strength, GPa or MPa
Cu = cohesive strength, GPa or MPa
D = bulk modulus/porosity factor
E = Young’s modulus, GPa or MPa
f = frequency, s–1, Hz (cycles/s)
F = volume factor
G = shear modulus, GPa or MPa
G(Φ) = gain factor
Igr = gamma ray index
k = permeability, m2, Eq. 13.108
k = wave number, m–1, Eq. 13.102
k* = complex wave number, m–1
K = bulk modulus, GPa or MPa
Kd = dry bulk modulus, GPa or MPa
Kd min = minimum bulk modulus, GPa or MPa
Kf = fluid bulk modulus, GPa or MPa
Kf 1, Kf 2 = bulk modulus of fluid 1, 2, etc., GPa or MPa
KHS = Hasin-Shtrikman bound bulk modulus, GPa or MPa
Kn = normalized bulk modulus, numeric
Kn R = normalized Reuss bound bulk modulus, numeric
Ko = mineral bulk modulus, GPa or MPa
KR = Reuss bound bulk modulus, GPa or MPa
Ks = saturated bulk modulus, GPa or MPa
K1, K2 = bulk modulus of component 1, 2, etc., GPa or MPa
K* = effective bulk modulus, GPa or MPa
K′ = effective crack bulk modulus, GPa or MPa
ΔKd = change in bulk modulus, GPa or MPa
ΔKdmax = maximum change in bulk modulus, GPa or MPa
ΔK12 = change in bulk modulus, fluid 1 to fluid 2, GPa or MPa
L = length, m
ΔL = change in length, m
m = Hoek-Brown strength coefficient
M = molecular weight, g/mole
MA, MB = modulus of component a, b, etc., GPa or MPa
MO = reference oil molecular weight, g/mole
MR = Reuss bound modulus, GPa or MPa
MV = Voigt bound modulus, GPa or MPa
MVRH = Voigt-Reuss-Hill bound modulus, GPa or MPa
n = number of moles, Eq. 13.10
n = effective stress coefficient, Eq. 13.35
n = strength envelope exponent, Eq. 13.134
P = pressure, MPa
Pc = confining pressure, MPa
Pd = differential pressure, MPa
Pe = effective pressure, MPa
Pp = pore pressure, MPa
Q = seismic quality factor, numeric
r = radius of stress "circle," GPa or MPa
R = gas constant, (L MPa)/(K mole), Eq. 13.10
R = gas/oil ratio, Eq. 13.26
R = measured gamma radiation, API units
Rcleansand = gamma radiation in a "clean" sand zone, API units
Rshale = gamma radiation in a shale zone, API units
s = Hoek-Brown strength coefficient
S, S′ = general rock property
t = time, s
T = temperature, °C
Ta = absolute temperature, K
T1, T2 = Kunster-Toksoz coefficients
ΔT = change in temperature, K
VB = brine compressional velocity, m/s
Vfx or Vcx = fracture or crack volume, m3 or cm3
Vf1, Vf2 = fluid 1, 2, etc. volume, m3 or cm3
Vg or Vm = grain or mineral volume, m3 or cm3
Vmineral = mineral velocity, m/s
Vo = reference compressional velocity, m/s
Vp = compressional velocity, m/s
Vpo = vertical compressional velocity, m/s
Vpor = total pore volume, m3 or cm3
Vp-con = connected pore volume, m3 or cm3
Vp-iso = isolated pore volume, m3 or cm3
Vrock = rock velocity, m/s
Vs = shear velocity, m/s
Vsh = shale volume, fractional
Vso = vertical shear velocity, m/s
VT = isothermal fluid compressional velocity, m/s
VT or Vrx = total rock volume, m3 or cm3
VTM = oil weight m compressional velocity, m/s
VTOMO = oil weight m compressional velocity at to, m/s
VW = water compressional velocity, m/s
wij = water compressional velocity coefficients
x = weight fraction of NaCl, ppm, Eq. 13.29b
x = directional component, m
y = directional component, m
z = directional component, m
Z = compressibility factor
α = aspect ratio, Eq. 13.91
α = failure envelope slope, Eq. 13.116
αm = aspect ratio of fracture population m, fractional
αl = logarithmic decrement (loss), nepers/m
β = strength factor, numeric
βS = adiabatic compressibility, MPa–1
βT = isothermal compressibility, MPa–1
γ = heat capacity ratio, Eq. 13.16
γ = Thomsen Vs anisotropy factor, Eq. 13.95
δ = Thomsen anisotropy factor, Eq. 13.95
δ = loss tangent, Eq. 13.103
ε = Thomsen Vp anisotropy factor, numeric
εij = strain components, fractional
εkl = strain components, fractional
εshear = shear strain, fractional
εV = volumeteric strain, fractional
εyy = horizontal strain, fractional
εzz = vertical strain, fractional
η = viscosity, Pa•s
θ = wave propagation angle to symmetry axis
λ = Lame’s parameter, GPa or MPa, Eq. 13.45
λ = wavelength, MPa−1, Eq. 13.103
μ = shear modulus, GPa or MPa, Eq. 13.42
μ = coefficient of internal friction, Eq. 13.116
μo = mineral shear modulus, GPa or MPa
μs = saturated shear modulus, GPa or MPa
μsd = dry shear modulus, GPa or MPa
μ* = effective shear modulus, GPa or MPa
μ′ = effective crack shear modulus, GPa or MPa
ν = Poisson’s ratio, fractional
ρ = density, kg/m3 or g/cm3
ρb = bulk density, kg/m3 or g/cm3
ρB = brine density, kg/m3 or g/cm3
ρd = dry density, kg/m3 or g/cm3
ρfl = fluid density, kg/m3 or g/cm3
ρg = grain or mineral density, kg/m3 or g/cm3
ρG = gas density, kg/m3 or g/cm3
ρO = oil density, kg/m3 or g/cm3
ρsat = saturated density, kg/m3 or g/cm3
ρW = water density, kg/m3 or g/cm3
σh = horizontal stress, GPa or MPa
σij = stress components, GPa or MPa
σm = mean stress, GPa or MPa
σn = normal stress, GPa or MPa
σshear = shear stress components, GPa or MPa
σv = axial (vertical) stress, GPa or MPa
σzz = vertical stress component, GPa or MPa
σ1 = stress in direction 1, GPa or MPa
σ3 = stress in direction 3, GPa or MPa
τ = shear stress, GPa or MPa
τ = relaxation time, s–1 (radians/s), Eq. 13.106
Φ = porosity
Φfx = fracture porosity
Φp-e = effective porosity
Φp-iso = isolated, ineffective porosity
ω = frequency (radian), s–1 (radians/s)
ωc = crossover frequency (radian), s–1 (radians/s)


References


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SI Metric Conversion Factors


°API 141.5/(131.5 + °API) = g/cm3
bbl × 1.589 873 E–01 = m3
ft × 3.048* E–01 = m
ft3 × 2.831 685 E–02 = m3
°F (°F−32)/1.8 = °C
psi × 6.894 757 E + 00 = kPa


*

Conversion factor is exact.