PEH:Rock Properties

Publication Information

Petroleum Engineering Handbook

Larry W. Lake, Editor-in-Chief

Volume I – General Engineering

John R. Fanchi, Editor

Copyright 2007, Society of Petroleum Engineers

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
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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 provides an extensive list of mineral and rock properties; Birch presents tables of compressional velocities, and Gregory gives a detailed overview of the use of rock property information in seismic interpretation. Castagna et al. focused on rock properties for use in amplitude versus offset analyses. Useful handbooks on this topic include Carmichael and Lama and Vutukuri. Probably the best reference covering a wide range of rock property formulas and models is Mavko et al. 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.

• 

  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. 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.

• 

  Fig. 13.1 – Common sandstone textures include point contacts, cements, and microfractures. These microstructures determine the properties of the rock on a whole.

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.

• 

  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.

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.

• 

  Fig. 13.3 – SEM image of sandstone AT41 showing a progressing calcite cementation front.

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.

• 

  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.

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.

• 

  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).

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.

• 

  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.

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.

• 

  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.

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.

• 

  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.

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.

Density and Porosity

Basics and Definitions

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

....................(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).

• 

  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.

....................(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:


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

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

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

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

• 

  Fig. 13.9 – Calculated density vs. porosity for sandstone, limestone, and dolomite.

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

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

where M_s, M_d, M_sat, M_b, and M_fl 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,

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

where ρ_mix is the density of the mixture; ρ_A is the density of Component A; ρ_B 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,

....................(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

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

and

....................(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.

• 

  Table 13.3

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.

• 

  Fig. 13.10 – Shale density as a function of depth from several sedimentary basins (after Castagna et al. and Rieke and Chillingarian). 1 = Gas saturated clastics: probable minimum density (McCulloh). 2 = Po river valley mudstone (Storer), 3 = average coastal Gulf of Mexico shales from geophysical measurements (Dickinson), 4 = average coastal Gulf of Mexico shales from density logs (Eaton), 5 = Marcaibo basin well (Dallmus), 6 = Hungary calculated wet densities (Skeels), 7 = Pennsylvanian and Permian dry shales (Dallmus), 8 = Eastern Venezuela (Dallmus).

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.

• 

  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).

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. 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.

• 

  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.

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. 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.

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

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

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

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

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

for compressibility defined as a positive number.

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

....................(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, β_S, rather than the isothermal compressibility γ β_S = β_T.

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:

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

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

and

....................(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. 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.

• 

  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).

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.

• 

  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).

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

....................(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). 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 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

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

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

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

Similarly, the velocities are related in molecular weight by

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

where V_TM is the velocity of oil of weight M, and V_TOMO is the velocity of a reference oil of weight M_o at temperature T_o. The variable a_m 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 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

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

These densities are shown in Fig. 13.15.

• 

  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).

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

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

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

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

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

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

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

....................(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.

• 

  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).

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.

....................(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 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 support this concept. The estimated density becomes

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

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

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

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

• 

  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.

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. give a relation for pure water that can be iteratively solved to give densities at pressure and temperature. Helgeson and Kirkham 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 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 and Potter and Brown. Using all these data, a simple polynomial can be constructed that will adequately calculate the density of sodium chloride solutions:

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

and

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

Here, T and P are in °C and bars, respectively; x is the weight fraction of sodium chloride; and ρ_B is the density of the brine in gm/cm³. The calculated brine densities, along with selected data from Zarembo and Federov, 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.

• 

  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.

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

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

Millero et al. and Chen et al. 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. 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. 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:

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

and

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

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

• 

  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.

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.

....................(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 σ_n, or stress applied at a general angle σ_g (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_ij, 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 and Nye.

• 

  Table 13.4

• 

  Fig. 13.20 – Types of stresses: (a) pressure, (b) tensile, (c) general with normal and tangential components, and (d) shearing stress (modified from Hubbert).

• 

  Fig. 13.21 – Stresses acting on the elemental cube. The stresses must be balanced so that there is no acceleration of the body.

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 (σ_m) or pressure (P).

....................(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 (P_c) alone will result in a decrease of rock volume, or compaction. In contrast, increasing the pore pressure (P_p) tends to increase rock volume. P_p counteracts the effects of P_c. Thus, rock properties are controlled largely by the difference between P_c and P_p, or the differential pressure P_d. 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_e) law

....................(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

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

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

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

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

• 

  Fig. 13.22 – Deformation of a material under vertical uniaxial stress (σ_zz) giving rise to vertical (ΔL) and horizontal (ΔW) deformation.

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

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

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

....................(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 ε_v:

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

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

....................(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:

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

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

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

Stress and strain are both tensors with 9 components. C_ijkl 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). For isotropic materials, this reduces to

....................(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

....................(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.

• 

  Table 13.5

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.

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

where M_V is the effective Voigt modulus, M_A and M_B 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.

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

where M_R 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

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

Note that in the case for minerals plus pores, M_pore = 0 and M_V decreases linearly with porosity. M_R equals zero for all porosities.

• 

  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.

An alternative approach, known as the Hasin-Shtrikman technique, 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

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

and

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

where K_i, μ_i, and f_i 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."

• 

  Fig. 13.24 – Schematic view of the composite material modeled by Hasin-Shtrikman method.

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.

• 

  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.

• 

  Table 13.6A

• 

  Table 13.6A (continued)

• 

  Table 13.6A (continued)

• 

  Table 13.6B

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_p, 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, Wang et al., and Prasad et al. 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:

....................(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

....................(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):

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

For constant elastic components, this simplifies to

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

The solution to this equation gives the compressional velocity

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

Similarly, for shear motion

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

and we get the shear velocity:

....................(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," Φ_c (Yin et al. and Nur et al.). At this high porosity limit, we reach the threshold of grain contacts and grain support (Han et al.).

• 

  Fig. 13.26 – (a) Compressional velocity vs. porosity in sandstones: laboratory data with elastic bounds (modified from Marion). (b) Sandstone velocities divided into classes.

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 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.

• 

  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). (b) Brine-saturated sandstone shear velocities as a function of porosity. Behavior is similar to the V_p trends seen in (a) above (after Vernik and Nur).

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_p/V_s ratios). V_p−V_s 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.).

• 

  Table 13.7A

• 

  Table 13.7B

• 

  Table 13.7C

Measured data for carbonates are less abundant. A systematic investigation of samples from several wells was reported by Rafavich et al. A plot of their results for carbonate V_p 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.

• 

  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.).

• 

  Table 13.8A

• 

  Table 13.8B

• 

  Table 13.8B (continued)

A similar set of measurements by Wang et al. 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. 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.

• 

  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.). (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.).

• 

  Table 13.9

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.

• 

  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.

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

....................(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).

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

Examining a similar set of data allowed Eberhart-Phillips et al. to develop a pair of relations for both V_p and V_s (see also Table 13.7)

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

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

where P_e 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_p from Rafavich et al. 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.

• 

  Fig. 13.31 – Compressional velocities normalized by the velocity at 40 MPa. For many sandstones, the average trend can be used.

• 

  Fig. 13.32 – Generalized compressional velocities dependence on pressure seen in carbonates by Rafavich et al. Pressure dependence is a function of porosity.

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

....................(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.

• 

  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.).

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, Nur and Simmons, Yin, and Scott et al.).

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.

• 

  Fig. 13.24 – Typical stress conditions that might be found in a compacting basin with no lateral deformation and no applied tectonic stresses.

A simple estimate of the horizontal stress, σ_h, can be made from the axial stress, σ_v, by

....................(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:

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

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

• 

  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 .

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_p and V_s surfaces for Woodbine sandstone. Figures such as 13.36 demonstrate that the V_p, V_s, and V_p/V_s 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.

• 

  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_p and V_s. Both absolute velocities and velocity ratios will be affected by anisotropic stresses.

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.

• 

  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.

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.

• 

  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).

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, Mavko and Mukerji, Mavko et al., Sengupta and Mavko, and Nolen-Hoeksema). 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.

• 

  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.

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:

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

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

and

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

where K₀, K_f, K_d, and K_s, are the bulk moduli of the mineral, fluid, dry rock, and saturated rock frame, respectively; Φ is porosity; and μ_s and μ_d are the saturated and dry rock shear moduli. ΔK_d 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, 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 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.). Recently published laboratory data (Batzle et al.) 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 K_n: the ratio of dry bulk modulus to that of the mineral.

....................(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 K_n can be determined empirically or theoretically. For relatively clean sandstone at high differential pressure (>20 MPa), the complex dependence of K_n (x, y, z, …) can be simplified as a function of porosity.

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

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

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

Here [1-K_n (Φ)] is also the Biot parameter α_b (Biot). Furthermore, because usually K₀ >> K_f, it is reasonable to assume

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

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

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

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

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

Thus, fluid saturation effects on the bulk modulus are proportional to the gain function G(Φ) and the fluid modulus K_f. 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 K_s 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 K_f are completely independent. K_s, K_d, and porosity Φ are actually closely correlated. Bounds on K_d as a function of porosity, for example, constrain the bounds of K_s.

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

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

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

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

and

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

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

• 

  Fig. 13.40 – Elastic bounds on bulk modulus for sandstone. Stippled regions represent the extremes in changes in modulus upon water saturation.

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

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

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

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

....................(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.

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

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

....................(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, Φ_c, 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).

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

This is consistent with the results of Mavko and Mukerji.

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

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

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

....................(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,

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

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

where we assume the density of these sands is equal to

....................(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

....................(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%):

....................(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.

• 

  Table 13.10

• 

  Fig. 13.41 – Elastic bounds (Voigt and Reuss) and approximations to the normalized dry modulus (K_n = K_d/K₀) for sandstone using the D factor.

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

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

Solid Mineral Bulk Modulus

The mineral modulus (solid grain bulk modulus) K₀ is an independent parameter, and the rock texture controls K_d. However, as mentioned previously, the normalized modulus K_n controls the fluid saturation effect rather than K_d or K_s individually. The mineral modulus K₀ is equally as important as K_d. However, in most applications of the Gassmann’s equation, only K_d 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), and the moduli of clays are a particular problem (Wang et al. and Katahara; 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 K_d and K_f, 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 K_f.

• 

  Fig. 13.42 – Effect of varying mineral modulus (K₀) on the calculated saturated bulk modulus (K_s) for water-saturated sandstone.

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.

• 

  Table 13.11

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

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

where K_f1 and K_f2 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.

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

where ρ₁, ρ₂, V_p1, and V_p2 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_s. 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.

• 

  Fig. 13.43 – Effect of varying fluid modulus (K₀) on the calculated saturated bulk modulus (K_s) for sandstones.

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 examined the elastic deformation of such elliptical inclusions, and these results were then applied to the compressibility of rocks by Walsh. 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:

....................(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 specifically includes the effect of anisotropic crack distributions.

One particularly useful result was derived by Kuster and Toksoz. 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_m), mineral (K₀, μ₀), and inclusion or crack moduli (K′, μ′) (Cheng and Toksoz).

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

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

Here, T₁ and T₂ are scalar functions of K₀, μ₀, K′, and μ′, and correspond to T_iijj and T_ijij in Kuster and Toksoz. 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 (α_ms) can be included to force such a good fit. The actual population of cracks in rocks remains unknown.

• 

  Fig. 13.44 – Normalized compressional and shear velocities for cracked rocks (V_rock / V_mineral) both dry and saturated using the Kunster-Toksoz method (from Cheng and Toksoz). Velocities are controlled by the crack aspect ratio (a = axis_minor / axis_major) and crack porosity.

• 

  Fig. 13.45 – Measured and modeled velocities on the Bedford Limestone (Wang et al.).

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.

....................(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. 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.

• 

  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).

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. 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.

• 

  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).

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). For "weak" anisotropy (Thomsen), the dependence of velocities as a function of angle (θ) from the symmetry axis can be written as

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

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

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

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

The Thomsen anisotropic parameter ε can be defined as

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

where V_P0 is the compressional velocity parallel to the axis of symmetry, and V_P90 is the velocity perpendicular to this axis. The parameter γ can be defined as

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

where V_S0 is the shear velocity parallel to axis of symmetry, and V_S||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.

....................(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.

....................(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 + iα_l). Note that here α_l is a loss parameter, and not an aspect ratio. Therefore, we can rewrite Eq. 13.101 as

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

Another common measure is the loss decrement δ:

....................(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.

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

One of the most straightforward descriptions of the relation of velocity to attenuation was developed by Cole and Cole and applied to attenuation measurements by Spencer. 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

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

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

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

This would lead to a general attenuation of

....................(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.

• 

  Fig. 13.48 – Generalized rock velocity (V_p) 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.

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, O’Connell and Budiansky, Walsh, and Dvorcik and Nur. Unfortunately, the different mechanisms proposed often give contradictory results.

Wave attenuation and dispersion in vacuum dry rock is relatively negligible. 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, Winkler et al., Murphy, Tittmann et al., Jones, and Tutuncu et al.). 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, Yamamato et al.).

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 and the squirt-flow mechanism (see, for example, O’Connell and Budiansky, or Dvorcik and Nur). Biot gives a characteristic frequency, ω_c (roughly, the boundary between high and low frequency range) with the viscosity dependence, η, in the numerator:

....................(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:

....................(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 (V_p) and shear (V_s) 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_p and V_s 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_p increases with viscosity also. More importantly, the dispersion curve shows a systematic shift to lower frequencies with increasing velocities, consistent with squirt flow.

• 

  Fig. 13.49 – Compressional (V_p) and shear (V_s) 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_p and V_s. This dispersion is shifted to lower frequencies at lower temperatures because of the increased glycerine viscosity.

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). More recently, Klimentos 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.

• 

  Fig. 13.50 – Extensional (or Young’s modulus) attenuation (1/Q_e) as a function of water saturation (a) and frequency (b) for Berea sandstone. With increasing saturation, 1/Q_e increases and reaches a peak at approximately 95% saturation (a). Attenuation decreases with increasing pressure, as shown in (b).

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). 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, σ_m, and radius of this circle, r, are simple sums and differences of the principal stresses.

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

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

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

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

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

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

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

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

• 

  Fig. 13.51 – Schematic diagram defining stress orientations along and across a plane.

• 

  Fig. 13.52 – Generalized rock failure with a linear envelope.

From Eqs. 13.114 and 13.115, the maximum shear occurs along a plane oriented at (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

....................(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)

C_u = Cohesive strength or the intercept of the envelope with σ_n = 0.

C_t = Tensional strength.

The failure envelope is then defined by the line

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

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

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

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

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

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

which leads to

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

Using the general solution to a second-order polynomial gives

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

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

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

After some algebraic manipulation, we find

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

and

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

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

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

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

....................(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 C_u 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).

• 

  Fig. 13.53 – Measured failure strengths for the B-d #1 sandstone with straight-line segments and curved envelope fits to failure.

• 

  Fig. 13.54 – Curved rock failure envelope with schematic failure styles at different mean pressure ranges.

• 

  Fig. 13.55 – Failure mechanisms as a function of porosity for sandstones (after Scott).

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 and Hoek and Brown 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.

• 

  Table 13.12

Hoek and Brown 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

....................(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

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

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

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

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

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

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

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

• 

  Fig. 13.56 – Failure envelope for sandstones after Hoek and Brown. Stresses are normalized by rock uniaxial compressive strength.

• 

  Table 13.13

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 C_t is defined as a positive number).

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

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

....................(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 and Bienlawski:

....................(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,

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

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

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

Therefore, the general form for an envelope is

....................(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 defined by Eq. 13.123 for the stress condition, σ_m, r we can subtract the same from a slightly different stress condition, σ_m’, r’. Solving for α gives

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

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

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

....................(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

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

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

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

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

....................(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

....................(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 σ_n such as σ_n = 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,

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

This value falls within the range of 0.65 to 0.75 suggested by Yudhbir et al. 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.

• 

  Table 13.14

• 

  Fig. 13.57 – Examples of reported uniaxial compressive strength (C_o) relations for different rock types.

• 

  Fig. 13.58 – Uniaxial compressive strength (C_o) relations for sandstones.

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

....................(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 Φ_c. This limiting porosity was shown as a crossover porosity from rock to a slurry by Raymer et al. and was referred to as "critical porosity" elsewhere. Jizba used such a concept to derive a general strength relationship for sandstones:

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

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

The 0.36 within the parentheses is her presumed value for Φ_c. 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. Dobereiner and DeFreitas 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

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

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

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_c, 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.

• 

  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).

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

....................(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.

• 

  Fig. 13.60 – Generalized failure envelope and Roscoe (crushing) surface at different porosities. Hoek-Brown parameter m = 15 used.

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. 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.

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

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

• 

  Fig. 13.61 – Effect of smectite content on the coefficient of internal friction of Austin Chalk (from Corbett et al.).

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

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

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

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

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

• 

  Fig. 13.62 – Effect of increasing clay content on strength.

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

....................(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 and has been documented by numerous investigators. The most common form for the effective stress law is

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

where σ_e is the effective stress, σ_a the applied stress on the rock surface, P_p, 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.

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

where K_d is the dry rock bulk modulus and K_o 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 P_p 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.

• 

  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.

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 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). 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 and Kirby) have shown that even small amounts of water or brine can have a substantial influence on rock mechanical properties. Colback and Wiid 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 and Dunning and Huff saw a direct relationship between the loss in rock strength and the chemical activity of the pore fluid. Meredith and Atkinson, Freeman, and others have shown increased crack velocities and acoustic emissions at constant crack intensity factors when water is introduced. Ujtai et al. 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.

• 

  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).

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 and Morgenstern et al. 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 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 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_c, of

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

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

• 

  Fig. 13.65 – Critical or crushing pressure as a function of the grain size-grain radius product (after Zhang).

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. 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 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 to provide estimates of compressive strength from dynamic measurements. Simplified forms of these relations are:

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

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

....................(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.

....................(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 V_p-V_s 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:

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

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

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

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

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

....................(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.

• 

  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).

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 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.

• 

  Fig. 13.67 – Distribution of relative radioactivity level for various rock types (from Bigelow after Russell).

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.

• 

  Fig. 13.68 – Gamma-ray energy levels resulting from disintegration of unstable isotopes (adapted from Tittman et al.).

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.

• 

  Table 13.15

• 

  Table 13.16

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.

• 

  Fig. 13.69 – Gamma-ray energy windows used for spectral gamma-ray logging (from Bigelow).

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.

• 

  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.).

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

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

where R is the measured radiation level, R_cleansand is the baseline level through a reference sand, and R_shale 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.

....................(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.

• 

  Fig. 13.71 – Reported gamma-ray index to shale volume conversions (from Bigelow).

• 

  Table 13.17

A more likely presumption is that the radiation level is dependent on the mixture densities and not volumes (Wahl and Katahara). In this case, a fabric analysis can also be performed. Katahara 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.

• 

  Fig. 13.72 – Modeled gamma-ray response to different clay distributions within a shaley sand series (modified from Katahara).

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.

• 

  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.

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.

• 

  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).

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

General References

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, 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.

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.

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.

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.

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.

Brill, J.P. and Beggs, H.D. 1977. Two-Phase Flow in Pipes. Tulsa, Oklahoma: University of Tulsa Press.

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.

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.

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.

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.

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.

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.

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.

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. 1977. Elastic properties of unconsolidated porous sand reservoirs. Geophysics 42 (7): 1339-1368. 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.

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.

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.

Engelder, T. and Plumb, R. 1984. The relation between ultrasonic properties and strain relaxation. Int. J. Rock Mech. Min. Sci. & Geomech. Abstracts 21: 7.

Fertl, W.H. 1976. Abnormal Formation Pressures. New York: Elsevier.

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.

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.

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.

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.

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.

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. 1981. Geothermics-using temperature in hydrocarbon exploration, No. 17. Tulsa, Oklahoma: AAPG Continuing Education Course Note Series, American Association of Petroleum Geologists.

Griffith, A.A. 1924. The theory of rupture. Proc., First International Congress for Applied Mechanics, Delft, The Netherlands, 54–63.

Griggs, D.T. 1967. Hydrologic weakening of quartz and other silicates. Geophys. J. R. Astron. Soc. 14: 19–31.

Hadley, K. 1976. Comparison of calculated and observed crack densities and seismic velocities in westerly granite. J. Geophys. Res. 81 (20): 3484-3494. http://dx.doi.org/10.1029/JB081i020p03484.

Hamilton, E.L. 1956. Low Sound Velocities in High-Porosity Sediments. The Journal of the Acoustical Society of America 28 (1): 16-19. http://dx.doi.org/10.1121/1.1908208.

Hamilton, E.L. 1972. Compressional-wave attenuation in marine sediments. Geophysics 37 (4): 620-646. http://dx.doi.org/10.1190/1.1440287.

Hamilton, E.L. 1976. Variations of density and porosity with depth in deep-sea sediments. J. Sediment. Res. 46 (2): 280-300.

Han, D.-H, Nur, A., and Morgan, D. 1986. Effects of porosity and clay content on wave velocities in sandstones. Geophysics 51 (11): 2093-2107. http://dx.doi.org/10.1190/1.1442062.

Han, D.-H. and Batzle, M. 2000a. Velocity, Density And Modulus of Hydrocarbon Fluids --Data Measurement. Presented at the 2000 SEG Annual Meeting, Calgary, 6-11 August. Paper No. 2000-1862.

Han, D.-H. and Batzle, M. 2000b. Velocity, Density And Modulus of Hydrocarbon Fluids --Empirical Modeling. Presented at the 2000 SEG Annual Meeting, Calgary, 6-11 August. Paper No. 2000-1867.

Harrison, A.R., Randall, C.J., Aron, J.B. et al. 1990. Acquisition and Analysis of Sonic Waveforms From a Borehole Monopole and Dipole Source for the Determination of Compressional and Shear Speeds and Their Relation to Rock Mechanical Properties and Surface Seismic Data. Presented at the SPE Annual Technical Conference and Exhibition, New Orleans, 23-26 September. SPE-20557-MS. http://dx.doi.org/10.2118/20557-MS.

Hauge, P.S. 1981. Measurement of attenuation from vertical seismic profiles. Geophysics 46 (11): 1548-1558. http://dx.doi.org/10.1190/1.1441161.

Hedberg, H.D. 1936. Gravitational compaction of clays and shales. Am. J. Sci. Series 5 Vol. 31 (184): 241-287. http://dx.doi.org/10.2475/ajs.s5-31.184.241.

Hill, R. 1952. The Elastic Behavior of Crystalline Aggregate. Proc. Phys. Soc. London, Sect. A 65 (5): 349–354. http://dx.doi.org/10.1088/0370-1298/65/5/307.

Hinz, K., Dostmann, H., and Fritsch, J. 1982. The continental margin of Morocco: Seismic sequences, structural elements, and geological development. In Geology of the Northwest African Continental Margin, ed. U. von Rad et al. New York: Springer-Verlag.

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/10.1016/0148-9062(87)90177-X.

Jachens, R.C. and Griscom, A. 1985. An isostatic residual gravity map of Califomia—a residual map for interpretation of anomalies from intracrustal sources. In The Utility of Regional Gravity and Magnetic Anomaly Maps, ed. W.J. Hinze. Tulsa, Oklahoma: SEG.

Johnston, D.H. 1986. Shale properties at temperature and pressure. Oil Gas J. 84 (13): 60.

Jones, T. and Nur, A. 1983. Velocity and attenuation in sandstone at elevated temperatures and pressures. Geophys. Res. Lett. 10 (2): 140-143. http://dx.doi.org/10.1029/GL010i002p00140.

Kan, T.K., Batzle, M.L., and Gaiser, J.E. 1983. Attenuation Measured From VSP: Evidence of Frequency-Dependent Q. Presented at the 1983 SEG Annual Meeting, Las Vegas, Nevada, USA, 11-15 September. Paper No. 1983-0589.

Keller, G.R. et al. 1985. Regional gravity and magnetic study of west Texas. In The Utility of Regional Gravity and Magnetic Anomaly Maps, ed. W.J. Hinze. Tulsa, Oklahoma: SEG.

Kowallis, B.J., Jones, L.E.A., and Wang, H.F. 1984. Velocity-porosity-clay content systematics of poorly consolidated sandstones. Journal of Geophysical Research: Solid Earth 89 (B12): 10355-10364. http://dx.doi.org/10.1029/JB089iB12p10355.

Garat, J., Stellingwerf, J., Ventre, J. et al. 1990. A Petrophysical Interpretation Using The Velocities Of P And S Waves (full-waveform Sonic). The Log Analyst 31 (6): 354–369. Paper No. 1990-v31n6a2.

Krynine, P.D. 1948. The Megascopic Study and Field Classification of Sedimentary Rocks. The Journal of Geology 56 (2): 130-165. http://dx.doi.org/10.2307/30070803.

Lajtai, E.Z., Schmidtke, R.H., and Bielus, L.P. 1987. The effect of water on the time-dependent deformation and fracture of a granite. Int. J. Rock Mech. Min. Sci. & Geomech. Abstracts 24 (4): 247-255. http://dx.doi.org/10.1016/0148-9062(87)90179-3.

Martin, R.J. 1972. Time-dependent crack growth in quartz and its application to the creep of rocks. J. Geophys. Res. 77 (8): 1406-1419. http://dx.doi.org/10.1029/JB077i008p01406.

Matthews, J.C. and Matthews, W.R. 1985. Shale properties at temperature and pressure. Oil Gas J. 83.

Mavko, G.M. and Nur, A. 1978. The effect of nonelliptical cracks on the compressibility of rocks. Journal of Geophysical Research: Solid Earth 83 (B9): 4459-4468. http://dx.doi.org/10.1029/JB083iB09p04459.

McDonal, F., Angona, F., Mills, R. et al. 1958. Attenuation of shear and compressional waves in Pierre Shale. Geophysics 23 (3): 421-439. http://dx.doi.org/10.1190/1.1438489.

McSkimin, H.J., Andreatch Jr., P., and Thurston, R.N. 1965. Elastic Moduli of Quartz versus Hydrostatic Pressure at 25° and - 195.8°C. J. Appl. Phys. 36 (5): 1624-1632. http://dx.doi.org/10.1063/1.1703099.

Milholland, P., Manghnani, M.H., Schlanger, S.O. et al. 1980. Geoacoustic modeling of deep-sea carbonate sediments. The Journal of the Acoustical Society of America 68 (5): 1351-1360. http://dx.doi.org/10.1121/1.385102.

Murrell, S.A.F. 1965. The Effect of Triaxial Stress Systems on the Strength of Rocks at Atmospheric Temperatures. Geophys. J. R. Astron. Soc. 10 (3): 231-281. http://dx.doi.org/10.1111/j.1365-246X.1965.tb03155.x.

Murphy, W.F., Schwartz, L.M., and Homby, B. 1991. Interpretation of V p and V s in sedimentary rocks. Presented at the 32nd SPWLA Annual Logging Symposium, Midland, Texas, USA, 16–19 June. Paper FF.

Murphy, W., Reischer, A., and Hsu, K. 1993. Modulus decomposition of compressional and shear velocities in sand bodies. Geophysics 58 (2): 227-239. http://dx.doi.org/10.1190/1.1443408.

Murphy, W.F. 1982. Effects of Microstructure and Pore Fluids on the Acoustic Properties of Granular Sedimentary Materials. PhD dissertation, Stanford University, Stanford, California.

Nelson, R.A. 1983. Geological Analysis of Naturally Fractured Reservoirs. Houston, Texas: Gulf Publishing Company.

O'Connell, R.J. and Budiansky, B. 1974. Seismic Velocities in Dry and Saturated Cracked Solids. J. Geophys. Res. 79 (35): 5412-5426. http://dx.doi.org/10.1029/JB079i035p05412.

Parks, G.A. 1984. Surface and interfacial free energies of quartz. Journal of Geophysical Research: Solid Earth 89 (B6): 3997-4008. http://dx.doi.org/10.1029/JB089iB06p03997.

Peck, L. 1983. Stress corrosion and crack propagation in Sioux quartzite. Journal of Geophysical Research: Solid Earth 88 (B6): 5037-5046. http://dx.doi.org/10.1029/JB088iB06p05037.

Pettijohn, F.S. 1975. Sedimentary Rocks. New York: Harper and Row.

Pickett, G.R. 1963. Acoustic Character Logs and Their Applications in Formation Evaluation. J Pet Technol 15 (6): 659-667. SPE-452-PA. http://dx.doi.org/10.2118/452-PA.

Robinson, B.E. 1991. Lithofacies and fracture potential of the upper Cretaceous Austin Chalk, Eagleford Shale, and Buda limestone along the Austin Chalk-Buda limestone production trend of south Texas. ARCO Technical Service Memo 91-0035, Atlantic Richfield Company (ARCO), Frisco, Texas.

Ruiz, M.D. 1966. Some technological characteristics of twenty-six Brazilian rock types. Proc., International Society of Rock Mechanics 1st International Congress, Lisbon, 25 September-01 October, 115.

Russell, W.L. 1951. Principles of Petroleum Geology. New York: McGraw-Hill.

Shaw, D.B. and Weaver, C.E. 1965. The mineralogical composition of shales. J. Sediment. Res. 35 (1): 213-222. http://dx.doi.org/10.1306/74d71221-2b21-11d7-8648000102c1865d.

Simmons, G. 1965. Ultrasonics in geology. Proc. IEEE 53 (10): 1337-1345. http://dx.doi.org/10.1109/proc.1965.4249.

Simmons, G. and Brace, W.F. 1965. Comparison of static and dynamic measurements of compressibility of rocks. J. Geophys. Res. 70 (22): 5649-5656. http://dx.doi.org/10.1029/JZ070i022p05649.

Stoll, R.D. and Houtz, R.E. 1983. Attenuation measurement with sonobuoys. The Journal of the Acoustical Society of America 73 (1): 163-172. http://dx.doi.org/10.1121/1.388834.

Sultanov, R.G., Skripka, V.G., and Namiot, A.Y. 1972. Solubility of methane in water at high temperatures and pressures. Gazovaia Promyshienmost (May 1972): 6.

Thomas, E.C. and Stieber, S.J. 1975. The distribution of shale in sandstones and its effect upon density. Presented at the SPWLA 16th Annual Logging Symposium, New Orleans, 4–7 June. Paper T.

Timur, A. 1977. Temperature dependence of compressional and shear wave velocities in rocks. Geophysics 42 (5): 950-956. http://dx.doi.org/10.1190/1.1440774.

Toksoz, M.N. and Johnston, D.H. 1981. Seismic Wave Attenuation. Tulsa, Oklahoma: SEG.

Tullos, F. and Reid, A. 1969. Seismic attenuation of Gulf Coast sediments. Geophysics 34 (4): 516-528. http://dx.doi.org/10.1190/1.1440028.

Van Voorhis, J.J., Craig, R.G., and Bartell, F.E. 1957. Free Energy of Immersion of Compressed Powders with Different Liquids. II. Silica Powder. The Journal of Physical Chemistry 61 (11): 1513-1519. http://dx.doi.org/10.1021/j150557a014.

Volarovich, M.P., Bayuk, E.I., and D’Yaur, J.J. 1985. Change in the velocity of longitudinal waves during deformation of samples. Izvestiya, Earth Physics 21: 302.

Watt, J.P., Davies, G.F., and O'Connell, R.J. 1976. The elastic properties of composite materials. Rev. Geophys. 14 (4): 541-563. http://dx.doi.org/10.1029/RG014i004p00541.

White, J.E. 1983. Underground Sound: Application of Seismic Waves. Amsterdam, The Netherlands: Methods in Geochemistry and Geophysics, Elsevier Science.

Wilkens, R., Simmons, G., and Caruso, L. 1984. The ratio as a discriminant of composition for siliceous limestones. Geophysics 49 (11): 1850-1860. http://dx.doi.org/doi:10.1190/1.1441598.

Winkler, K., Nur, A., and Gladwin, M. 1979. Friction and seismic attenuation in rocks. Nature 277 (5697): 528-531. http://dx.doi.org/10.1038/277528a0.10.1038/277528a0.

Winkler, K. 1986. Estimates of velocity dispersion between seismic and ultrasonic frequencies. Geophysics 51 (1): 183-189. http://dx.doi.org/10.1190/1.1442031.

Wood, A.B. 1941. A Textbook of Sound: Being an Account of the Physics of Vibrations With Special Reference to Recent Theoretical and Technical Developments, second edition. New York: Macmillan.

Yin, C.S. 1992. Wave attenuation in partially saturated porous solids. PhD dissertation, Stanford University, Palo Alto, California.

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.