PEH:Single-Phase Permeability

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 14 – Single-Phase Permeability

Philip H. Nelson, U.S. Geological Survey and Michael L. Batzle, Colorado School of Mines

Pgs. 687-726

ISBN 978-1-55563-108-6
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The capacity to flow fluids is one of the most important properties of reservoir rocks. As a result, extensive research has been applied to describe and understand the permeability of rocks to fluid flow. In this chapter, only single-phase or absolute permeability will be considered. Multiphase relative permeabilities must be derived using relations described in the chapter on relative permeability and capillary pressure.

Permeability (k) is a rock property relating the flow per unit area to the hydraulic gradient by Darcy’s law,

....................(14.1)

where p is pressure, ρ is fluid density, g is gravitational acceleration, z is elevation, and μ is the dynamic viscosity. The ratio q/A has the units of velocity and is sometimes referred to as the "Darcy velocity" to distinguish it from the localized velocity of flow within pore channels. The natural unit of k is length squared; however, petroleum usage casts Eq. 14.1 in mixed units, so that the unit of k is the darcy, which is defined as the permeability of a porous medium filled with a single-phase fluid of 1-cp viscosity flowing at a rate of 1 cm³/s per cross-sectional area of 1 cm² under a gradient of 1 atm pressure per 1 cm. Reservoir rocks are usually characterized in millidarcies (md), a unit that is 1/1000 of a darcy. Conversion factors are 1 darcy = 0.9869×10^-12 m² or 1 md = 0.9869×10 -11 cm². Bass noted that Darcy’s law holds only for viscous flow and that the medium must be 100% saturated with the flowing fluid when the determination of permeability is made. Furthermore, the medium and the fluid must not react by chemical reaction, absorption, or adsorption; otherwise, the permeability changes as the fluid flows through the sample. Darcy’s law (Eq. 14.1) has many practical applications, including determination of permeability in the laboratory and wellbore.

In hydrological applications, the fluid is assumed to be water at near-surface conditions. The viscosity of water is factored into the transport term, which is called hydraulic conductivity (K) and has the units of velocity. Darcy’s law is then written as

....................(14.2)

This version of Darcy’s law is not useful to the petroleum engineer, but it is sometimes handy to be able to convert from hydraulic conductivity units to permeability. To obtain k in darcies, multiply K in m/s by 1.04×10⁵.

Permeability is a property of pore space geometry; specifically, it has been found to be proportional to (RΦ)², where R is a pore throat dimension and Φ is porosity. However, a measure of R is not available unless capillary pressure determinations have been made, in which case permeability has also been determined in the laboratory. Because permeability can be measured only on a restricted set of samples or from a limited number of well tests, it must often be derived from other properties or measurements. Porosity and permeability are routinely measured on core plugs; for this reason, the systematics of permeability and porosity are reviewed in this chapter.

The topics of fluid sensitivity and stress also deserve consideration. Many rocks contain clays or other minerals that are sensitive to the pore fluid. If an incompatible pore fluid is introduced during a production process, these minerals can change form, swell, or migrate. Permeability can then decrease by orders of magnitude. As effective pressure is increased, pore space decreases and permeability is lowered. The change in permeability with pressure is greater at low effective pressures. This pressure dependence is also strong in poorly consolidated rocks or rocks where flow is dominated by fractures. As rocks become consolidated or well cemented, the pressure dependence may become negligible.

Finally, the problem of quantitatively predicting permeability from porosity and other measurements that can be made with well logs is examined. To predict permeability, one needs a physical model and a method of zoning or clustering the data. Models used to predict permeability from porosity and other measurable rock parameters fall into classes based on grain size, mineralogy, surface area, or pore dimension parameters. Zonation techniques include database, statistical, clustering, and neural network approaches. Ultimately, the choices of model and zonation method depend on the problem at hand, the data available, and the resources devoted to the task.

Petrologic Controls

Permeability values of rocks range over many factors of 10; therefore, permeability is plotted on a logarithmic scale. Values commonly encountered in petroleum reservoirs range from a fraction of a millidarcy to several darcies. The log₁₀(k)-Φ plot of Fig. 14.1 shows four data sets from sands and sandstones, illustrating the reduction in permeability and porosity that occurs as pore dimensions are reduced with compaction and alteration of minerals (diagenesis). In these examples, k of newly deposited beach sands exceeds 30 darcies, k of partially consolidated sandstones ranges from 300 to 2,000 md, k of consolidated sandstones ranges from 0.01 to 100 md, and k of tight gas sandstones is <0.01 md. Porosity is reduced from a maximum of 52% in newly deposited sandstones to as low as 1% in consolidated sandstones. In this section, some of the causes of variability in log₁₀(k)-Φ space are examined.

• 

  Fig. 14.1 – Permeability/porosity plot showing newly deposited beach sands and three sample suites from wells in oil and gas fields. Permeability scale ranges from 1,000 darcies to 0.1 μd.

The permeability and porosity of a rock are the result of both depositional and diagenetic factors (Fig. 14.2) that combine to produce a unique set of pore space geometries as the rock is formed. Consequently, the heavy line in Fig. 14.2 represents only one of many possible evolutionary paths in log(k)-Φ space. First, consider the depositional factors. Better sorting increases both k and Φ. Gravel and coarse grain size produce anomalously high k even though decreasing Φ. Very fine grains of silt and detrital clay produce low permeability at high porosity. High quartz content can produce efficient systems having good permeability even at low porosity, whereas sandstones with feldspar or lithic grains may develop significant noneffective porosity content. Diagenetic effects, starting with compaction and followed by cementation and alteration of depositional minerals to clays, tend to decrease log(k) proportionately as Φ is decreased. Several examples are presented to illustrate these controls.

• 

  Fig. 14.2 – Sketch of the impact of primary depositional features (such as quartz content and sorting, in italics) and diagenetic processes (such as compaction and cementation) on permeability/porosity trends in sandstones.

In newly deposited sands and poorly consolidated sandstones, grain size correlates well with pore size and hence is a primary control on permeability. Grain size ranges for sandstones are defined by factors of two (Fig. 14.3). For example, a sand with grain diameters between 250 and 500 μm is classed as a medium-grained sand. The sedimentological phi scale provides a convenient label to the size classes, and D=2^-phi is the grain diameter in mm (for example, 2 -3 =0.125 mm=125 μm). Also shown in Fig. 14.3 are size ranges for various types of carbonates. Note that grain diameters can be as large as 2,000 μm in very coarse-grained sands and as small as 1 μm in chalks.

• 

  Fig. 14.3 – Grain size categories in sandstone, limestone, and dolostone. Sedimentological scale (phi scale) is logarithmic in powers of two. Grain diameter (d) (crystal size in case of dolostones) is given in micrometers. Limestone and dolostone classes are from Lucia; chalk data are from Mortensen et al. See Discussion for explanation of vertical bars showing grain and pore throat sizes in a consolidated sandstone.

Unconsolidated Sandpacks

One laboratory study deserves examination because it illustrates the relationships among grain size, sorting, k, and Φ. Using sand from two Texas rivers, Beard and Weyl sieved 48 sand samples into 8 size classes and 6 sorting classes. Each data point shown in Fig. 14.4 represents the permeability and porosity of a sample with a unique grain size and sorting. Median grain size ranges from 0.840 mm for the coarse sample to 0.074 mm for the very fine sample. The authors present photomicrographs of thin-section comparators for each of the 48 samples to document the wide range of size and sorting represented by the sample suite. The maximum permeability value for a well-compacted, unconsolidated sandpack is about 500 darcies. Porosity ranges from 23.4% to 43.5%. Note the general increase in permeability as grain size increases from very fine to coarse and the increase in both porosity and permeability as sorting progresses from very poor to well sorted.

• 

  Fig. 14.4 – Permeability/porosity data from unconsolidated artificial sandpacks by Beard and Weyl. Symbols linked with solid lines denote size ranges; dotted lines distinguish sorting classes.

Consider the extremely well-sorted samples represented by the open-circle data points along the right-hand edge of Fig. 14.4. For these extremely well-sorted samples, porosity is independent of grain size, as it should be for a packing of uniform spheres. However, for those samples that are not well sorted, an increase in coarse grain content results in somewhat decreased Φ even as k increases. This pattern is preserved in some consolidated samples. The extremely well-sorted samples of Fig. 14.4 also show that log(k) increases in equal increments as grain size increases. The samples were sized so that the mean grain diameter of each adjacent size interval increases by the square root of 2. Permeability increases by a factor of 2 for each increment of grain size. Thus, Beard and Weyl’s data show that permeability is proportional to the square of grain size. Because theoretical models show that flow is proportional to the square of the radius of a pore opening, it can be said that Beard and Weyl’s data demonstrate that pore size is proportional to grain size in sandpacks.

Clays and Shales

The permeability of shales and mudstones determines the effectiveness of seals for many hydrocarbon reservoirs, but measurements are few. Neuzil compiled data sets from 12 laboratory studies and 7 field studies that provided ranges of permeability and porosity data in bottom muds, clay, unconsolidated sediment, glacial till, clayey siltstone and sandstone, claystone, mudstone, and argillite. Permeability is as high as 1 md in unconsolidated sediment with 70% porosity and as low as 0.01 nanodarcy (nd) in argillite with 5% porosity. With few exceptions, permeability ranges over 3 factors of 10 at a given porosity and decreases progressively as porosity decreases. For example, at a porosity of 20%, permeability ranges from 0.1 μd to 0.1 nd, a range well below the lower limit of k plotted in Fig. 14.1. Although it was expected that permeability would be scale dependent in clays and shales (regional permeability would be greater than laboratory sample permeability because of fractures), it was found that permeability ranges from the field studies are roughly the same as the laboratory studies, thereby indicating a lack of scale dependence.

Sandstones

Thomson describes continental sandstones from the Lower Cretaceous Hosston formation in Mississippi (Fig. 14.5): "Secondary quartz cement and compaction through pressure solution of grains are the principal causes of porosity reduction. The early introduction of large amounts of dolomite has inhibited compaction of framework grains. Kaolinite ranges from 5% to 15% of total rock volume. All samples contain a little illite. The permeability/porosity plots indicate a progressive and uniform loss of permeability as porosity is reduced, suggesting that the sandstones underwent a simple diagenetic history, uncomplicated by such late processes as leaching, development of authigenic clay minerals, and so forth." Thomson also suggests that the introduction of hydrocarbons caused a cessation in diagenesis in the lower part of the reservoir. The effect of grain size remains apparent in the data of Fig. 14.5, although diagenetic effects have blurred the separations seen in Fig. 14.4, so some of the medium-grained samples have k as low as in the very fine-grained samples.

• 

  Fig. 14.5 – Permeability/porosity data from the Lower Cretaceous Hosston Sandstone from Thomson.

Permeability in Oligocene-Miocene sandstones ranges from <1.0 to >1,000 md (Fig. 14.6). Bloch reports that the sandstones were deposited in a variety of environments and that lithology ranges from lithic arkoses to feldspathic litharenites, meaning that 25% or more of the primary grain composition is either lithic fragments or feldspar grains. Up to 30% of porosity is secondary porosity, formed by dissolution of potassium feldspar. Because permeability is >1 darcy at porosity values <20%, the secondary porosity is probably well connected and contributing to flow. Although the samples with coarsest grain sizes tend to have the highest permeability values, the symbols depicting different grain sizes are intermixed, another indication that secondary porosity is contributing to flow.

• 

  Fig. 14.6 – Permeability/porosity data from Oligocene and Miocene sandstones from Bloch.

Bos describes results from an exploration well that encountered (1) clean sandstone, (2) sandstone with pores filled with kaolinite, (3) laminated sandstone, part clean and part filled with kaolinite (indicated as "laminated" in Fig. 14.7), and (4) shale. Scanning electron microscope photographs document the extent to which kaolinite fills the pores, thereby reducing k as shown in Fig. 14.7. Here again we see a linear relationship between log(k) and Φ, with pore-infilling clays reducing both k and Φ in a fairly systematic fashion.

• 

  Fig. 14.7 – Permeability/porosity data from a Lower Cretaceous sandstone from the North Sea from Bos.

Wilson contends that many clay coatings, particularly on eolian sandstones, formed on the framework grains before deposition. Their presence actually preserves porosity because quartz overgrowths cannot readily form. According to Wilson, many of the largest petroleum reservoirs (North Sea, North Slope of Alaska) have retained good porosity because of detrital clay coatings. Samples in which kaolinite and illite occur as clay coatings fall within the boundaries of the three upper fields in Fig. 14.8. However, fibrous illite can form within the pore space in the Rotliegend sandstones (lower two fields in Fig. 14.8), reducing the permeability one to two orders of magnitude compared with rocks in which clay occurs as grain coatings. Under the scanning electron microscope, the appearance of numerous fine strands of illite within pores makes it obvious why permeability is so impaired. Special core preparation techniques are required to preserve clay textures so that laboratory measurements reflect the in-situ permeability values.

• 

  Fig. 14.8 – Permeability/porosity data fields from wells penetrating the Permian (Rotliegend) sandstones of the southern North Sea, measured at ambient conditions, from Wilson.

Carbonates

Samples from an oil-productive dolomite facies in the Williston basin of North Dakota were characterized in terms of the size of dolomite grains. Originally deposited as a carbonate mud, after burial this facies was altered to a sucrosic dolomite or calcareous dolomite with good intercrystalline porosity. At any given porosity, samples with the larger dolomite crystal sizes have the highest permeability (Fig. 14.9). At a given crystal size, an important control on porosity is the amount of calcite, which is believed to be recrystallized lime mud. Vuggy porosity is 5% or less. Different productive zones in the same field may have different dolomite textures, suggesting that original sediment texture and chemistry were the main factors determining the distribution of crystal sizes.

• 

  Fig. 14.9 – Permeability/porosity data from a sucrosic dolomite in the Williston basin of North Dakota, from Petty.

In the North Sea, oil is produced from Cretaceous and Tertiary chalks, even though permeability is <10 md (Fig. 14.10). From measurements of specific surface area, the equivalent grain diameter is computed to range from 1.0 to 2.7 μm. As indicated in Fig. 14.10, the separation between the two chalks is attributed to specific surface area, which is higher in the lower-permeability Danian samples than in the Maastrichtian samples. The addition of pore space produces modest gains in permeability (low slopes for the two data sets in Fig. 14.10), from which one can infer that a significant fraction of the pore space is poorly connected or of very small size. Mortensen et al. conclude that the intrafossil porosity behaves the same, in terms of flow, as interparticle porosity.

• 

  Fig. 14.10 – Permeability/porosity data from two chalk reservoirs of Cretaceous (Maastrichtian) and Paleocene (Danian) age, from Mortensen et al.

Lucia found a size effect in limestones and dolostones, as evidenced by dolostone data shown in Fig. 14.11. To obtain petrophysically viable groupings, Lucia grouped all dolomitized grainstones with mud-dominated samples having large dolomite crystals and grouped dolomitized packstones with mud-dominated samples having medium-sized dolomite crystals (key in Fig. 14.11). He suggests that the plot can be used to estimate permeability of a nonvuggy carbonate rock if the porosity and particle size are known. He points out that the effect of vugs is to increase porosity but not alter permeability much. In Fig. 14.11 we can see the quasilinear log(k)-Φ relationship and the decline in slope (and k) with decreasing grain size. It appears that the fundamental controls observed in the sandstones are also present in these selected carbonates, if care is exercised in categorizing the carbonates in terms of grain or crystal size.

• 

  Fig. 14.11 – Permeability/porosity data and classifications for selected nonvuggy dolostones from Lucia.

Summary: Empirical Trends

Figs. 14.5 through 14.11 exhibit a linear or piecewise-linear relationship between log(k) and Φ as determined in many consolidated sandstones and in carbonates if care is taken to isolate rock types. Such linear trends are often seen in samples from an individual rock unit or formation. These trends have the general form of

....................(14.3)

Eq. 14.3 is strictly an empirical relationship between log(k) and Φ. It is useful when data from an area of interest are available because log(k) can be predicted simply from Φ. However it masks the dependence of log(k) on pore throat size and thereby obscures the physics of flow in porous media, as is shown in a subsequent section.

Corrections to Core Measurements

Before selecting a method of determining permeability in a specific reservoir, one must first be assured that the core measurements are appropriate for reservoir conditions. Sample collection, selection, and preparation are important steps in ensuring that the data set represents the geology at in-situ conditions; some precautions are discussed in the chapter on relative permeability and capillary pressure. Adjustments may be necessary for the type of test fluid and for pressure effects.

Klinkenberg Effect

The permeability of a sample to a gas varies with the molecular weight of the gas and the applied pressure, as a consequence of gas slippage at the pore wall. Klinkenberg determined that liquid permeability (k_L) is related to gas permeability (k_g) by k_L = k_g/(1+b/p), where p is the mean flowing pressure and b is a constant for a particular gas in a given rock type. The correction parameter b is determined by conducting the test at several flowing pressures and extrapolating to infinite pressure. Alternatively, one can use an empirical correlation established by Jones to estimate b. The correlation, with R² of 0.90, is based on measurements on 384 samples (mostly sandstones) with permeabilities ranging from 0.01 to 2500 md. For helium, b_helium=44.6(k/Φ)^-0.447 and for air, b_air = 0.35 b_helium. The units of b are psi for permeability in units of md and porosity expressed as a fraction. Another empirical correlation was established by Jones and Owens for tight gas sandstones with permeabilities ranging from 0.0001 to 10 md: b=0.86k^-0.33, where b is in atm and k is in md. The Klinkenberg correction is quite important for low-permeability rocks and less important or unimportant for high-permeability rocks. The value of k_L obtained after applying the correction represents the permeability to a gas at infinite pressure or to a liquid that does not react with the component minerals of the rock.

Pore Fluid Sensitivity

The clays or other materials coating grain surfaces can be sensitive to pore fluid. This complicates the problem of describing permeability because flow properties depend not only on the lithology but also on pore fluid chemistry. This kind of reaction was seen in Fig. 13.6. Measured permeabilities on this sample as a function of salinity are shown in Fig. 14.12. Samples were obtained after drying and storage, and as a result, clays had collapsed (Fig. 13.6a). This collapsed state did not significantly change when the rock was saturated with very-high-salinity brine. As the pore fluid decreased in salinity, at a point near 30,000 ppm salt content, the clays expanded and effectively plugged the pore space. This result demonstrates the need to take special precautions in preserving and drying the core, as mentioned previously.

• 

  Fig. 14.12 – Permeability of a clay-rich sandstone sample containing fluid-sensitive smectitic clays decreases dramatically as pore fluid salinity decreases.

In low-permeability sandstones, permeability to water (k_w) is systematically less than the Klinkenberg-corrected permeability (k_L). A correction equation based on >100 samples with a permeability range of 0.0001< k_L <1 md is k_w = k_L^1.32. The correction is only approximate, as scatter on a graph of k_w vs. k_L is high at k_L <0.01 md. Surprisingly, the sensitivity to brine concentration in low-permeability sandstones is reported to be less than that in high-permeability sandstones.

Pressure Dependence

Permeabilities discussed so far were measured at constant effective pressure. However, as increasing pressure closes fractures and compresses the pore space, permeability will decrease. The magnitude of the change depends on the rock fabric. Weak, unconsolidated rocks will collapse easily, and the drop in permeability can be dramatic. As the rock becomes better consolidated, this pressure dependence decreases. On the other hand, even for tight rocks, as fractures are introduced and begin to dominate the fluid flow, this general trend is reversed and pressure dependence increases.

The pressure dependence can often be fit well with an empirical power law and negative exponential relation,

....................(14.4)

where k(p_e) is the permeability measured at effective pressure p_e, k_o is the permeability at zero pressure, and b is a parameter adjusted to fit each rock. This general relation for normalized permeability is shown in Fig. 14.13. A larger b factor yields a stronger pressure dependence.

• 

  Fig. 14.13 – General pressure dependence of permeability of the form k(P_e) = k_o exp in four sandstone samples that have undergone various levels of compaction and cementation.

In Fig. 14.14, sample D1452-281 seen previously in Fig. 13.1 is a clean sandstone with a high porosity of 35%. The pressure dependence is strong, as can be seen in Fig. 14.14. A b factor of 0.13 fits the general trend of the data (when pressure is expressed in MPa). In more compacted or cemented samples, such as in Fig. 14.15, with a lower porosity of 18%, the permeability decrease can be fit well with b=0.07. In general, as cementation increases, the pressure sensitivity declines, and the value of b approaches zero. On the other hand, in low-porosity, brittle rocks, flow often becomes fracture dominated. Because fractures are compliant and close easily with pressure, the pressure dependence of permeability again becomes high. In Fig. 14.16, permeability of a very-low-porosity (0.18%) crystalline rock is plotted on a logarithmic scale. Even though the absolute value of permeability is low to begin with, k drops to 1% of its unconfined value at 100 MPa, and the decline curve can be fit with a b factor of 0.6.

• 

  Fig. 14.14 – Dependence of permeability on effective pressure in a high-porosity (35%) sample.

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  Fig. 14.15 – Dependence of permeability on effective pressure in a medium-porosity (18%) sample.

• 

  Fig. 14.16 – Permeability of a crystalline sample (porosity equals 0.18%) showing dependence on pressure. Note logarithmic scale for permeability.

A more elaborate equation relating Klinkenberg-corrected permeability k to effective confining pressure p is



where k_o, the permeability at zero confining pressure, slope a, pressure coefficient P*, and coefficient C are determined from experimental data. However, if constant values of C=3×10^-6 psi^-1 and P*=3,000 psi are used, errors are about 5%, and the two remaining coefficients, k_o and a, can be determined if k is determined at only two values of p. Jones recommends the use of 1,500 and 5,000 psi, although very poorly consolidated samples may require that the higher value of p be reduced so as not to exceed the yield strength of the sample.

In practice, one may be required to correct values of k measured with air at ambient conditions to k for brine at reservoir conditions. The correction should be based on samples and conditions for the problem at hand. As a guide, Swanson established a correction of k_brine= 0.292k_air^1.186 for a collection of 24 sandstone and 32 limestone samples (0.002< k_brine <400 md), where k_brine was measured at 1,000-psi effective stress. Swanson’s empirical equation appears to incorporate all three factors discussed above: the gas slippage effect, presence of brine as opposed to an inert fluid, and effect of stress.

Petrophysical Models

The problem of predicting permeability is one of selecting a model expressing k in terms of other, measurable rock properties. Historically, the first approaches were based on a tube-like model of rock pore space known as the Kozeny-Carman relationship. The derivation of this "equivalent channel model" has been reworked by Paterson and Walsh and Brace. The model assumes that flow through a porous medium can be represented by flow through a bundle of tubes of different radii. Within each tube, the flow rate is low enough that flow is laminar rather than turbulent. A tube is assigned a shape factor f, a dimensionless number between 1.7 and 3, and length L_a that is greater than the sample length L. The assumption is that each flow path forms a twisted, tortuous, yet independent route from one end of the rock to the other. The tortuosity is defined as τ=(L_a/L)². From considerations of flow through tubes, the resulting equation is

....................(14.5a)

where the hydraulic radius, r_h, is defined as the reciprocal of Σ_p, the ratio of pore surface area to pore volume. The pore surface area normalized by a volume is often called the specific surface area. The form of Eq. 14.5a depends on which volume is used to normalize the pore surface area. If specific surface area is instead expressed as Σ_r, the ratio of pore surface area to rock volume, then Eq. 14.5a becomes

....................(14.5b)

If specific surface area is defined as the ratio of pore surface area to grain volume, Σ_g, the expression is

....................(14.5c)

Thus, the functional dependence of k on Φ, which differs among Eqs. 14.5a, 14.5b, and 14.5c, depends on the definition of specific surface area.

Paterson and Walsh and Brace establish a relationship between electrical properties and tortuosity, determining that formation factor F=(L_a/L)²/Φ=τ/Φ. They note that this expression differs from earlier incorrect formulations. With it, tortuosity can be eliminated from Eq. 14.5a to obtain

....................(14.5d)

Different approaches to porous media theory apply the concept of tortuosity in different ways. For purposes of this chapter, tortuosity is represented by electrical formation factor, as in Eq. 14.5d, or by porosity raised to an exponent.

As shown in the following sections, many models that relate k to a pore dimension r are derived, either in spirit or in rigor, from the Kozeny-Carman relationship, which recognizes explicitly the dependence of k on r².

Models Based On Grain Size

Krumbein and Monk’s Equation. Krumbein and Monk’s Equation. Using experimental procedures that were later adopted by Beard and Weyl, Krumbein and Monk measured permeability in sandpacks of constant 40% porosity for specified size and sorting ranges. Analysis of their data, coupled with dimensional analysis of the definition of permeability, led to

....................(14.6)

where k is given in darcies, d_g is the geometric mean grain diameter (in mm), and σ_D is the standard deviation of grain diameter in phi units, where phi=-log₂(d) and d is expressed in millimeters. Although the Krumbein and Monk equation is based on sandpacks of 40% porosity and does not include porosity as a parameter, Beard and Weyl showed that Eq. 14.6 fits their own data fairly well even though porosity of the Beard and Weyl samples ranges from 23% to 43%. In fact, because of difficulties in obtaining homogeneous sandpacks, Beard and Weyl chose to use computed k values from Eq. 14.6 rather than their measured data in tabulating values for fine and very fine samples with poor or very poor sorting. If Eq. 14.6 can predict k for a varying Φ in unconsolidated sandpacks, then the exponential dependence on sorting must be adequate to describe all the effects associated with porosity reduction. In other words, both k and Φ reduction pictured in Fig. 14.2 are due primarily to a decline in degree of sorting.

The laboratory studies of Krumbein and Monk and Beard and Weyl dealt with sieved sands from a common source, so such grain properties as angularity, sphericity, and surface texture did not vary much. Moreover, sorting was purposely controlled to be log normal. In situations where these ideal conditions are not met, other techniques must be invoked to predict permeability in unconsolidated sands. A disproportionate amount of fines can drastically reduce k in unconsolidated sands. Morrow et al., using statistical techniques on data from Gulf Coast sands, found that permeability correlated best with the logarithm of grain size times sorting if the fines fraction, taken to be <44 μm, was accounted for.

Berg’s Model. An interesting model linking petrological variables—grain size, shape, and sorting—to permeability is that of Berg. Berg considers "rectilinear pores," defined as those pores that penetrate the solid without change in shape or direction, in various packings of spheres. Simple expressions for k are derived from each packing, which form a linear trend when log(k) is plotted against log(Φ). From these geometrical considerations comes an expression relating k to Φ raised to a power and to the square of the grain diameter,

....................(14.7a)

where k is given in darcies, d (in mm) is the median grain diameter, Φ is porosity in percent, and p, a sorting term, is explained below. If permeability is expressed in millidarcies, d in micrometers, and Φ as fractional porosity, this expression becomes

....................(14.7b)

To account for a range of grain sizes, Berg considered two mixtures of spheres and assumed that k will be controlled primarily by the smaller grains. This introduces a sorting term p=P₉₀-P₁₀, called the percentile deviation, to account for the spread in grain size. The p term is expressed in phi units, where phi=-log₂d (in mm). For a sample with a median grain diameter of 0.177 mm, a value of 1.0 for p implies that 10% of the grains are >0.25 mm and 10% are <0.125 mm.

Berg’s expression (Eq. 14.7b) is illustrated in Fig. 14.17 for p=1 and varying d. Permeability increases rapidly with increasing porosity, depending on Φ to the fifth power, and the curves migrate downward and to the right with decreasing grain size. Nelson finds that Fig. 14.17 is remarkably concordant with several published data sets. Berg’s model appears to be a usable means of estimating permeability in unconsolidated sands and in relatively clean consolidated quartzose rocks. This is true even though Berg did not expect his model to be applicable for porosity values <30%.

• 

  Fig. 14.17 – Theoretical model by Berg relating permeability to porosity with varying median grain size (d).

Van Baaren’s Model. Proceeding along more empirical lines, Van Baaren obtains a result nearly identical to that of Berg. Van Baaren begins with the Kozeny-Carman expression of Eq. 14.5b and makes a series of substitutions (see summary by Nelson) that result in

....................(14.8a)

where d_d (in μm) is the dominant grain size from petrological observation, m is the cementation exponent, and C is a sorting index that ranges from 0.7 for very well sorted to 1.0 for poorly sorted sandstones. Consequently, Eq. 14.8a can be used to estimate k from petrological observations on dominant grain diameter d_d and degree of sorting, along with a porosity estimate obtained from either core or logs.

Assuming that the dominant grain size d_d is equivalent to Berg’s median grain diameter d, then Eq. 14.8a is very similar in form to Eq. 14.7a. For example, a sorting parameter p=1 in Berg’s Eq. 14.7b results in

....................(14.7c)

where k is given in millidarcies, whereas for a well-sorted sandstone, C=0.84 and Eq. 14.8a becomes

....................(14.8b)

Van Baaren’s Eq. 14.8b is so close to Berg’s Eq. 14.7c that a separate log(k)-Φ plot is not warranted here. Van Baaren’s expression is probably easier to use because the parameters are directly related to practical petrological variables. Both models display a porosity exponent >5, and both are compatible with the data of Beard and Weyl on unconsolidated sands in that k increases with the square of grain size.

Models With Mineralogical Factors

Several models have been devised to accommodate the influence of mineralogical textures on k. The first two described below are based on the Kozeny-Carman equation; the third uses a network topology that is independent of the Kozeny-Carman equation.

Herron uses Eq. 14.5c as a starting point for a model using mineralogical abundances in place of specific surface area. He obtains

....................(14.9)

where M_i is the weight fraction of each mineral component in the solid rock and B_i is a constant for each mineral, so that quartz produces high k and clay minerals produce low k. Mineral abundances are obtained by performing an element-to-mineral transform on data from a logging tool that measures chemical elemental concentrations by means of neutron-induced gamma ray spectroscopy. The coefficient A_f is a textural maturity indicator; it can be used to reflect the amount of feldspar alteration to clay minerals. Nelson gives further details on the model.

Panda and Lake extended the Kozeny-Carman expression to include the effect of grain-size sorting. They assume a sandpack of spherical grains having a log-normal distribution of grain diameters d, characterized by mean diameter, standard deviation, and skewness. Their expression is based on Eq. 14.5c, substituting Σ_g=6/D and incorporating an additional term including standard deviation and skewness. With the additional term accounting for sorting, their extended model agrees well with Beard and Weyl’s data for sandpacks but overpredicts k in consolidated sandstones. To make the model applicable to consolidated sandstones, three types of cement filling the pore space were considered: pore-bridging, pore-lining, and pore-filling cement. Their resulting equation for k includes the sum of surface areas contributed by each of the cement geometries and a tortuosity factor that is a function of cement type, as well as statistical terms describing the log-normal distribution of grain diameters d.

Bryant et al. and Cade et al. performed numerical modeling on a pack of (initially) equally sized spheres in a geometry based on a laboratory random pack. Permeability is computed directly by considering flow across the faces of individual linked tetrahedra; thus, the method is independent of the Kozeny-Carman equation. Their method simulates a quartz system in which all surfaces participate in compaction and cementation processes, causing k and Φ to decrease along a characteristic curve in log(k)-Φ space. At some point in this process, clay minerals are introduced in grain-rimming or pore-filling geometries, and k decreases more sharply with a continuing decrease in Φ. The resulting computed curve in log(k)-Φ space tracks the effects of progressive diagenesis of a single pack as burial progresses.

Models Based on Surface Area and Water Saturation

Two ideas inherent in Eq. 14.5 are important for later developments: the dependence of k on a power of porosity and on the inverse square of surface area. Eq. 14.5 has been used as a starting point for predicting permeability from well log data by assuming that residual water saturation is proportional to specific surface area, Σ.

Granberry and Keelan’s Chart. Granberry and Keelan published a set of curves relating permeability, porosity, and "critical water" saturation (S_ciw) for Gulf Coast Tertiary sands that frequently are poorly consolidated. Their chart, originally presented with S_ciw as a function of permeability with porosity as a parameter, is transposed into log(k)-Φ coordinates in Fig. 14.18. The S_ciw parameter is taken from the "knee" of a capillary pressure curve and is greater than irreducible water saturation, S_wi. It is said that if the water saturation in the formation is less than this critical value, the well will produce water free. Because S_ciw is taken from the capillary pressure curve, it is a function of the size of interconnected pores. Fig. 14.18 cannot be used to estimate permeability from porosity and water saturation as determined from well logs because it reflects only the critical water saturation. It was determined from reservoirs in which oil viscosity was approximately twice that of water and requires adjustment for low- or high-gravity oils.

• 

  Fig. 14.18 – Empirical chart relating permeability to porosity with critical interstitial water S_ciw as a parameter, after Granberry and Keelan.

Timur’s Model. Timur used a database of 155 sandstone samples from three oil fields (Fig. 14.19) . The three sandstones exhibit varying degrees of sorting, consolidation, and ranges of porosity. Timur measured irreducible water saturation (S_wi) using a centrifuge and then held k proportional to S_wi⁻² in the general power-law relationship,

....................(14.10)

Coefficients a and b were determined statistically. Timur’s statistical results show that the exponent b can range between 3 and 5 and still give reasonable results. Results for b=4.4 produced a fit somewhat better than other values; it was obtained by taking the logarithm of both sides of Eq. 14.6 and testing the correlation coefficient with respect to Φ^b/S_wi². For b=4.4, the value of a is 0.136 if Φ and S_wi are in percent and 8,581 if Φ and S_wi are fractional values. There is no theoretical basis for the substitution of S_wi for specific surface area Σ, so although the form of Eq. 14.10 is similar to that of Eq. 14.5, it is strictly an empirical relationship. The effectiveness of Eq. 14.10 as a predictor of permeability is shown in Fig. 14.20, and its form on a log₁₀(k)-Φ plot is shown in Fig. 14.21.

• 

  Fig. 14.19 – Permeability/porosity data from three U.S. oil fields by Timur.

• 

  Fig. 14.20 – Permeability data from three U.S. oil fields as a function of ϕ^4.4/S_wi², after Timur. Two bounding lines represent the standard error band that includes 68% of the sample points.

• 

  Fig. 14.21 – Permeability/porosity relationship with irreducible water saturation as a parameter, after Timur.

It is not easy to apply Eq. 14.10, which is based totally on core data, to an oil reservoir. The S_wi core data used to establish Eq. 14.10 were obtained for a fixed value of capillary pressure (P_c). In a reservoir, P_c varies with height, and because S_wi varies with P_c, it is necessary to assume a functional dependence of S_wi on P_c. There are also some practical difficulties in establishing the coefficients a and b in a reservoir in which the oil/water contact cuts across lithologies because of regional dip or structure. In particular, within the transition zone, only part of the water is irreducible (S_wi); the remainder is movable. Thus, a log-based estimate of saturation immediately above the oil/water contact will overestimate S_wi.

Dual-Water Model. An algorithm discussed by Ahmed et al. is attributed to Coates. An extension of Eq. 14.10 and Fig. 14.21, it assumes that permeability declines to zero as S_wi increases to fill the entire pore space:

....................(14.11)

A further refinement incorporates the presence of clay minerals and is based on the dual-water model. It requires log-based estimates of the total porosity (Φ_t) and either effective porosity (Φ_e) or bound water saturation (S_bw). Effective porosity is defined as Φ_e=Φ_t(1-S_bw). The fractional volume of bound water, V_bw=S_bwΦ_t, is computed, and an estimate of a parameter V_bi=S_wiΦ_t called the (fractional) bulk volume irreducible water in clean wet rock must also be provided. Then, computed as a function of depth is the total immovable water,

....................(14.12)

and the permeability,

....................(14.13)

The algorithm of Eqs. 14.12 and 14.13 uses a pair of parameters, V_bi and V_bw, which in effect sweep out a broad region of the log(k)-Φ crossplot (Fig. 14.22). For the solid curves, V_bw has been set to 0.0 as if the rock were entirely clay free. As irreducible water V_bi increases, the curves shift downward and to the right, into the regime populated by fine-grained rock. The dashed curve is drawn for V_bw and V_bi each equal to 0.05, thereby representing one of a second family of curves for a fine-grained dirty sandstone. Note how S_bw increases with decreasing Φ.

• 

  Fig. 14.22 – Permeability/porosity relationship based on the dual-water model incorporating irreducible (V_bi ) and bound water (V_bw) as parameters. Four solid curves show effect of increasing V_bi with V_bw = 0.0. Two dashed curves show effect of increasing V_bw with V_bi = 0.05.

This algorithm produces reasonable results in sandstones if V_bi is chosen judiciously. One difficulty is choosing a value for V_bi in coarse-grained and gravel-bearing sandstones.

"Tight" Sandstones. Predicting permeability becomes much more difficult in formations with small grain size and an abundance of clay minerals. Such rocks are called "tight gas sands" or "submillidarcy reservoirs" (see example in Fig. 14.1). Kukal and Simons show that the Timur equation produces k values much too high in such formations and establish some predictive equations that decrease the porosity by multiplying Φ by 1-V_cl, where V_cl is the clay fraction. They show that the water saturation term S_wi is not so important in these high-clay rocks. Although their predictive equation is a welcome improvement, the scatter shows the difficulty in dealing with such low-porosity systems.

Nuclear Magnetic Resonance. Eq. 14.5a indicates that other measures of specific surface area could be correlated with permeability. A study by Sen et al. provides laboratory data on 100 sandstone samples on exchange cation molarity (Q_v), nuclear magnetic resonance (NMR) longitudinal decay time (t₁), and pore-surface-area-to-pore-volume ratio (Σ_p) from the gas adsorption method. Borgia et al. provide data on Σp and t₁ on 32 samples. Both studies include measurements of k, Φ, and formation factor (F) on their samples. Both sample suites are made up of samples from different formations, so the log(k)-Φ plots exhibit scatter, as shown by Fig. 14.23.

• 

  Fig. 14.23 – Permeability/porosity data from Sen et al. Inserts show log (k) plotted agains log (ϕ^m/Σ_p) and against log(ϕ^mt₁).

Both groups of experimenters found that k correlated best with measures of specific surface when it formed a product with Φ^m or Φ². For example, Sen et al. found k to be strongly correlated (R around 0.9) with (Φ^m/Σ_p)^2.08, with (Φ^mt₁)^2.15, and with (Φ^m/Q_v)^2.11. Two of these correlations are shown as insets in Fig. 14.23. Borgia et al. did not incorporate m into their regression equations but found k to be best correlated with (Φ⁴/Σ_p²)^0.76 and with (Φ⁴t₁₂)^0.72. As an example of these statistical fits, the expression from Sen et al.,

....................(14.14)

where k is in millidarcies, t₁ is in milliseconds, and Φ is fractional porosity, is plotted in Fig. 14.24. Because the porosity exponent is very close to that established by Timur (Eq. 14.9), the curves in Fig. 14.24 are quite similar to those in Fig. 14.21.

• 

  Fig. 14.24 – Empirical relationship among permeability, porosity, and t₁ from NMR measurement by Sen et al.

Later work showed that the transverse decay time t₂, which is a more practical parameter to detect with a logging tool than t₁, could also be used to estimate permeability (consult the literature for further details on NMR):

....................(14.15)

where k is in millidarcies, t_2gm is the geometric mean of t₂ in milliseconds, and c=4.5 in sandstones and 0.1 in carbonates. The value of k obtained from Eq. 14.15 is referred to as k_SDR. Better results are obtained if a cutoff can be selected for t_2L so that only the pores contributing to permeability are included. Kenyon notes that the NMR measurement is inherently responsive to pore size, whereas permeability depends on pore throat size. He suggests that the experimentally determined Φ⁴ dependence somehow accounts for the way in which the ratio (pore throat size to pore size) varies with porosity.

The Coates equation for estimating permeability is

....................(14.16)

where k is in millidarcies, V_bvi is the bulk volume irreducible fluid fraction, V_ffi is the free fluid fraction and is equal to Φ-V_bvi, and porosity Φ is taken from the NMR tool. Eq. 14.16 closely resembles Eq. 14.11, which is written in terms of irreducible water saturation; V_bvi is computed from the portion of the t₂ spectrum with the smallest times. Except for the porosity term Φ⁴, there is little obvious resemblance between Eqs. 14.15 and 14.16. However, Sigal argues that a t₂ cutoff time is implicit in Eq. 14.16 and that its value is incorporated in the constant c. Even so, the two equations are not equivalent because the two choices of t₂ result in different weightings of the pore size distribution spectrum. Sigal relates the two choices of t₂, one explicit in Eq. 14.15 and the other implicit in Eq. 14.16, for different distributions of t₂ and for several experimental data sets.

As Sigal points out, the problem of selecting a value of t₂ from NMR data is analogous to the problem of selecting a value of R from capillary pressure data (which is reviewed in the next section): One must capture the length scale appropriate to the estimation of permeability.

Summary. Timur’s equation and its corresponding chart offer a viable method of permeability estimation in which porosity and irreducible water saturation can be estimated. Difficulties arise if there is uncertainty in S_wi, as there is within an extensive transition zone. The dual-water predictor is an interesting embellishment that can include a clay content parameter. Laboratory data show that, when combined with Φ^m, specific surface area, cation exchange molarity, and NMR decay time all correlate well with k. Because it responds to the pore size spectrum, NMR is a particularly effective tool in obtaining log-based estimates of permeability.

Models Based on Pore Dimension

Capillary Pressure and Pore Size. Of course, the dimension of interconnected pores plays a major role in determining permeability (Eq. 14.5a). Thus, all the methods of estimating permeability discussed so far are indirect methods. A viable direct method requires both adequate theoretical underpinnings relating pore throat dimension to permeability and experimental determination of the critical pore dimension parameters. Many workers have made use of the capillary pressure curve, obtained experimentally by injecting mercury into a dried sample. As mercury pressure is increased, more mercury is forced into progressively smaller pores in the rock, and the resident pore fluid (air) is expelled. A length r, usually referred to as the pore throat radius, is related to the injection pressure by the Washburn equation,

....................(14.17)

where σ is the interfacial tension and θ is the wetting angle. The injection process can be visualized by examining the idealized capillary pressure curve of Fig. 14.25. A finite pressure is required to inject mercury into a 100% water-saturated sample (right side of Fig. 14.25). At the first inflection point (entry pressure), mercury occupies only a small fraction of the pore volume containing the largest pores. Next, much of the pore space becomes filled with mercury with a comparatively slight increase in pressure (progressing from the circle labeled Katz and Thompson to the circle labeled Swanson in Fig. 14.25). Finally, large pressure increases are required to force more mercury into the smallest pores (steep curve to left of Swanson circle).

• 

  Fig. 14.25 – Capillary pressure curve (idealized) showing measures used by different authors for determination of characteristic pore dimension.

Many authors have linked capillary pressure curves to permeability. Purcell derived an expression relating k to an integral of P_c^-2 over the entire saturation span, achieving a good match with core data. The relationships established by Timur and Granberry and Keelan, discussed previously, are represented at low water saturation in Fig. 14.25. Contributions by Swanson, Winland, and Katz and Thompson, symbolized by the circles in Fig. 14.25, are reviewed next.

Swanson’s Equation. Swanson provides a method of determining air and brine permeabilities from a single point on the capillary pressure curve. His regression relationships are based on permeability and capillary pressure data on 203 sandstone samples from 41 formations and 116 carbonates from 33 formations. His method picks the maximum ratio of mercury saturation to pressure, (S_b/P_c)_max, from the capillary pressure curve, arguing that at this point all the connected space is filled with mercury and "this capillary pressure corresponds to pore sizes effectively interconnecting the total major pore system and, thus, those that dominate fluid flow." From linear regression, Swanson obtains simple equations of the form,

....................(14.18)

where the constants a and c depend on rock type (carbonate vs. sandstone) and fluid type (air or brine). For carbonates and sandstones combined, c=2.005. Because S_b is defined as the mercury saturation as percent of bulk volume, it must be proportional to Φ(1-S_w); through Eq. 14.17, P_c can be linked with a pore throat radius r_apex. Thus, Swanson’s result shows that k is proportional to [Φ(1-S_wi)r_apex]², again demonstrating the dependence of k on the square of a pore throat size.

Winland’s Equation and Pittman’s Results. An empirical equation relating permeability, porosity, and a capillary pressure parameter is referred to as Winland’s equation. Based on laboratory measurements on 312 samples, Winland’s regression equation is

....................(14.19)

where r₃₅ is the pore throat radius at 35% mercury saturation, k is air permeability, and Φ is porosity in percent. A log(k)-Φ plot based on Eq. 14.19 and showing five characteristic lines for pore throat radius is shown in Fig. 14.26. Note that at a given porosity, permeability increases roughly as the square of the pore throat radius. And for a given throat size, the dependence of permeability on porosity is slightly less than Φ². Kolodzie states that a pore throat size of 0.5 μm was used as a cutoff for reserves determinations, in preference to the use of k or Φ. Hartmann and Coalson also present Winland’s equation in the same format as Fig. 14.26. They state that r₃₅ is a function of both entry size and pore throat sorting and is a good measure of the largest connected pore throats in a rock with intergranular porosity.

• 

  Fig. 14.26 – Empirical model based on regression attributed to Winland, from Kolodzie. Labels for four ranges of r₃₅ are taken from Martin et al.

Martin et al. used the r₃₅ parameter, along with other petrophysical, geological, and engineering data, to identify flow units in five carbonate reservoirs. With Eq. 14.19, r₃₅ can be computed from permeability and porosity measurements on core samples. Flow units are grouped by the size of pore throats using the designations of megaport, macroport, mesoport, and microport shown in Fig. 14.26. A completion analysis for the different r₃₅ size ranges in a reservoir of medium thickness and medium gravity oil yielded the following: megaport, tens of thousands of barrels of oil per day; macroport, thousands; mesoport, hundreds; and microport, nonreservoir. After flow units are identified, well logs and sequence stratigraphy are used to identify zones with similar properties where no core data exist. The method works well in carbonates where flow is controlled by intergranular, intercrystalline, or interparticle pore space but not so well if fractures or vugs are present.

Pittman sheds additional light on Winland’s equation, linking it to Swanson’s results. Pittman used a set of 202 sandstone samples from 14 formations on which k, Φ, and mercury injection data had been obtained. Using Eq. 14.17, he associated a pore size r_apex with the capillary pressure, P_c, determined by Swanson’s method and found that the mean value of r_apex has a mercury saturation of 36%. That is, on a statistical basis, the points denoted by circles labeled Swanson and Winland in Fig. 14.25 are practically identical, and the two methods are sampling the same fraction of the pore space.

Pittman also established regression equations for pore aperture sizes ranging from 10% to 75% mercury saturation. His expressions have been rearranged and displayed in Table 14.1 to show the exponents of r and Φ required to predict k. (Because r was used as the dependent variable in Pittman’s regressions, the coefficients in Table 14.1 differ somewhat from what would be obtained if k were the dependent variable; however the changes would not invalidate the point of this discussion.) Note that the r exponent decreases with increasing mercury saturation, while the Φ exponent increases. That is, the porosity term contributes relatively less to k than does r for mercury saturation values <35%. In fact, Pittman noted that the porosity term was statistically insignificant for r₁₀ through r₃₅.

• 

  Table 14.1

Katz and Thompson’s Equation. Another investigation on the influence of pore structure on flow properties comes from Katz and Thompson and Thompson et al. They use percolation theory to derive a deceptively simple relationship,

....................(14.20a)

where k is absolute permeability (same units as ), σ is electrical conductivity of the rock, and σ_o is the conductivity of the saturant. The value of the constant, given as 1/226, is dependent on the geometry assumed for the pore space. They substantiate Eq. 14.20a with experimental data on 60 sandstone and carbonate samples with permeabilities ranging from <1 md to 5 darcies.

The parameter l_c in Eq. 14.20a represents a dimension of a very particular subset of pores: "The arguments suggest that permeability can be estimated by assuming that the effective pore size is the smallest pore on the connected path of pores containing the largest pores. We call that effective pore size l_c." To obtain l_c, the pressure at the inflection point on a capillary pressure curve is converted to a diameter. The authors argue that the inflection point marks the pressure at which a sample is first filled continuously end to end with mercury and that the large pores first filled are those that control permeability.

The Katz and Thompson equation and its characteristic curves are given in Fig. 14.27. To plot curves on log(k)-Φ plots, we assumed the simplest relation between formation factor and porosity (cementation exponent of 2.0), σ/σ_o=Φ². Some data points from Katz and Thompson’s experiments are posted in Fig. 14.27 to indicate how well their measured l_c match the curves (This is not really a test of their model because they used formation factor in their correlations, not Φ²). Their result is similar to that of Swanson’s and Winland’s equations: Permeability is closely proportional to the square of rΦ.

• 

  Fig. 14.27 – Permeability equation with critical pore-size radius (R_c) as a parameter, from Katz and Thompson. Values of r_c posted next to data points are from mercury injection tests.

To obtain compatibility with other worker’s expressions, we define a critical radius r_c=l_c/2, keeping both permeability and r_c² in units of μm²:

....................(14.20b)

Eq. 14.20b is identical in form to Eq. 14.5d, but the percolation concepts used to derive Eq. 14.20b are quite different from the geometrical arguments used to derive the Kozeny-Carman expression. The coefficient in Eq. 14.5d, which is ≈0.4, is considerably greater than that (0.0177) in Eq. 14.20b. Consequently, the characteristic radius r_c is ≈4.7 times greater than the hydraulic radius, r_h. Although r_h is defined as the ratio of pore volume to pore surface area, it can be determined in a variety of ways, including the use of mercury injection. Conceptually, then, the Kozeny-Carman equation could also be represented by an extended horizontal line across Fig. 14.25; i.e., as a method that samples a broad spectrum of pore sizes.

It is interesting to compare the Katz and Thompson model (Fig. 14.27) with Winland’s empirical equation (Fig. 14.26). The shapes of the curves are comparable; i.e., the models agree on the approximate Φ² dependence. The pore radii given by the Winland equation are smaller than comparable radii in the Katz and Thompson model. This is expected because the Winland equation requires a saturation of 35%, a criterion of greater injection pressure than that of Katz and Thompson. What is noteworthy is the general agreement between the two models regarding the form of the log(k)-Φ relationship. They demonstrate that in the models invoking higher powers of Φ, which we have shown in previous graphs are not well grounded physically, the higher powers of Φ are required to compensate for lack of knowledge regarding the critical pore dimension. It does seem, however, that the empirical data that often show a "straight-line" log(k)-Φ relationship contain some fundamental information regarding how the critical pore dimension relates to porosity.

Flow Zone Indicator. Amaefule and Altunbay rearranged a version of the Kozeny-Carman equation (Eq. 14.5c) to obtain a parameter group named the flow zone indicator (I),

....................(14.21)

where the factor 0.0314 allows k to be expressed in millidarcies. As can be seen from Eq. 14.21, I has the units of pore size, in micrometers, and can be computed from core measurements of k and Φ, even though it is defined in terms of f, τ, and Σ_g, which are not easily measured. The choice of Eq. 14.5c over other forms of the Kozeny-Carman equation that use alternative definition of specific surface area (Eqs. 14.5a and 14.5b) seems a bit arbitrary and results in the particular combination of porosity terms used in Eq. 14.21.

Amaefule and Altunbay use I to define zones called "hydraulic flow units" on a doubly logarithmic plot incorporating the terms in Eq. 14.21. For compatibility with other plots in this chapter, a plot in log(k)-Φ coordinates is shown in Fig. 14.28. Each data point on a log(k)-Φ plot has an I value that associates it with a nearby curve of constant I value. The difficult step is deciding where the boundaries between adjacent I bands should be positioned and how to compute a value of I from well logs in uncored wells. Options for doing so are described next.

• 

  Fig. 14.28 – Zonation of permeability and porosity data based on a parameter called the flow zone indicator (I) in μm. Data from southeast Asia and algorithm taken from Amaefule and Altunbay.

Statistical Approaches and Reservoir Zonation

Having considered petrological controls on permeability/porosity patterns in Section 14.2 and various petrophysical (grain size, surface area, and pore size) models in Section 14.4, we now consider techniques for applying well logs and other data to the problem of predicting k or log(k) in uncored wells. If the rock formation of interest has a fairly uniform grain composition and a common diagenetic history, then log(k)-Φ patterns are simple, straightforward statistical prediction techniques can be used, and reservoir zonation is not required. However, if a field encompasses several lithologies, perhaps with varying diagenetic imprints resulting from varying mineral composition and fluid flow histories, then the log(k)-Φ patterns are scattered, and reservoir zonation is required before predictive techniques can be applied.

A widely used statistical approach is multiple linear regression. Linear regression techniques are popular for establishing predictors of geological variables because the methods are effective at predicting mean values, are fast computationally, are available in statistical software packages, and provide a means of assessing errors.

Predictors With One or Two Input Variables

When a straight-line relationship between log(k) and Φ exists, as it does in Figs. 14.5 and 14.6, the computation of a predictor for log(k) by Eq. 14.3 is straightforward and merits little discussion. Curvature in the log(k)-Φ relationship is treated by adopting a polynomial in Φ. Increased accuracy is also afforded by dividing the field by area or vertically and computing regression coefficients for each area. In one area, curvature in the statistical predictor may be rather pronounced; in another, curvature may be absent.

Predictors With Several Input Variables

The quality of the predictor can often be enhanced by adding a variable such as gamma ray response or depth normalized to top of formation. As variables are added to Eq. 14.3, families of curves are required to present graphically the effect of combinations of variables. When one or two parameters are varied, the curves sweep out a large area on the log(k)-Φ plot. Predictive power can be increased by adding other parameters. Predictive accuracy does not increase indefinitely as parameters are added but instead usually reaches a limit after several (anywhere from two to six) parameters are included in the regression (see Fig. 17 of Wendt et al. for an example).

Predictors Using Computed Parameters

Computed logs such as shale volume and differences between porosities from different logs can be included as independent variables. In this way, petrological information can also be incorporated into the predictive relationships. A petrological parameter (cement or gravel) is first "predicted" from well logs using core observations as "ground truth." The predicted petrological parameters can then be included in a relationship to estimate permeability.

As the complexity of the log(k)-Φ plot increases (i.e., as the data deviate from a linear trend), more variables must be incorporated into the predictive model to maintain predictive accuracy, although instability can result from having too many variables. The better the understanding is of petrological controls on permeability, the more effective the predictor and its application will be. Other complications with regression methods, including underestimation of high-permeability zones and overestimation of low-permeability zones, are mentioned by various authors. At some point, it becomes necessary to adopt a method of zoning the reservoir.

A database approach equivalent to an n-dimensional lookup table can also be used for predicting permeability within a field or common geology. In this approach, the user must first select the logs or log-derived variables that offer sufficient discriminating power for permeability. One must also choose a suitable bin size for each variable on the basis of its resolution. Then, a database is constructed from the core permeability values and associated log values. Each n-dimensional bin or volume is bounded by incremental log values and contains mean and standard deviation values of permeability plus the number of samples. In application, permeability estimates are extracted from a bin addressed by the log values. An interpolation scheme is used to extract an estimate from an empty bin. Like the regression method, the database approach can be used only when adequate core data are available to build the model, and results generally cannot be transferred to other areas.

Fuzzy clustering techniques provide a means of determining the number of clusters (bins in the preceding paragraph) and their domains. The term "fuzzy" indicates that a given input/output pair can belong (partially) to more than one cluster. Finol and Jing applied the technique to a shaly sandstone reservoir in which permeability ranged from 0.05 to 2,500 md. Six clusters were defined. In each cluster, permeability is determined by

....................(14.22)

where Φ is porosity and Q_v is the cation exchange capacity per unit pore volume. The final determination of log(k) is a weighted sum of the six log(k_i), with weights determined by the degree of membership of Φ and log(Q_v) in their respective clusters. An average correlation coefficient of 0.95 was obtained on test sets. Implementation of Eq. 14.22 in uncored wells requires that Q_v be determined from a porosity log and requires an estimate of grain density and shale fraction (V_sh).

Artificial neural networks are a third method of establishing a predictor specific to an area of interest. A back-propagation neural network is optimized on a training set in which the desired output (permeability at a given depth) is furnished to the network, along with a set of inputs chosen by the user. Rogers et al. established a predictor for a Jurassic carbonate field using only porosity and geographic coordinates as inputs. For each value of permeability to be predicted, porosity values spanning the depth of the desired permeability value were provided as inputs, rather than a single porosity value at a single depth. Permeability values predicted by the neural network in test wells were generally closer to the core measurements than were the values predicted by linear regression.

Discussion

The best petrophysical models show that pore throat size r is a prime control on k. Yet, r is missing from example data sets (Figs. 14.5 through 14.11) and from the diagram summarizing diagenetic processes (Fig. 14.2). With the models, it is possible to consider what the range of pore sizes might have been when sediments were deposited and what the range is in the present state of consolidation. As an example, consider the samples in Fig. 14.5 from the Hosston formation. The uppermost vertical bar in Fig. 14.3 shows the range of grain size reported for the Hosston formation. The range of initial pore throat sizes (vertical hachured bar in Fig. 14.3) was computed from the present-day grain size using data from Beard and Weyl (Fig. 14.4) to transform grain size to initial permeability, followed by application of the Katz-Thompson equation (Eq. 14.20) to obtain pore throat size from k and Φ. Present-day pore throat sizes (lowermost solid bar in Fig. 14.3) were computed from present-day k and Φ with the Katz-Thompson equation (Eq. 14.20). From this exercise, one can see that the largest pore throat size has diminished from 250 (initial) to 45 (present day) μm, that the smallest present-day sizes are <2 μm, and that the range of pore throat size has broadened considerably on the logarithmic phi scale.

Expressions relating permeability k to porosity Φ are summarized in Table 14.2. There, permeability is in millidarcies, and grain sizes (d) and pore sizes (r) are expressed in micrometers, so the coefficients may differ from the originating equation in the text. From these representative equations, it can be seen that (1) the predictive equations are simple in form, (2) k is related to a power of Φ (except for the Krumbein and Monk equation), and (3) k is related to the square of either a characteristic length or a measure of surface area.

• 

  Table 14.2

We have seen that models relying on estimates of surface area, whether that estimate comes from irreducible water saturation, NMR, gas adsorption, or cation exchange data, require porosity raised to a power of ≈4. How can surface-area models requiring a porosity power of 4 (Figs. 14.21 and 14.24) be reconciled with pore dimension models requiring a power of 2 (Figs. 14.26 and 14.27)? The answer lies in the pore-size distribution. Because most of the surface area is contributed by the smallest grains (pores), measures of surface area emphasize the small end of the pore-size spectrum. Yet, the small pores contribute least to permeability. The high (≈4) power of porosity serves to unweight the contribution of the small pores. In the surface-area models, porosity serves a dual role, first as a measure of tortuosity and second as a measure of the pore-size distribution function.

A similar question arises with the grain size models. Both models by Berg (Fig. 14.17) and Van Baaren require a porosity power of ≈5, multiplied by the square of a dominant grain size. Why is the porosity power so high? The probable reason is that the dominant grain size becomes a progressively poorer measure of dominant pore size as the spread in grain size increases and small grains (pores) become more abundant. Again, porosity serves as both a measure of tortuosity and a weighting factor to compensate for the presence of small pores at lower porosities. Moreover, the retention of a sorting term in Eqs. 14.7a and 14.8a is inadequate compensation for small pores, even though a sorting term is all that is needed in sized samples (Eq. 14.6).

Models incorporating an estimator of pore size (Eqs. 14.5d, 14.19, and 14.20 in Table 14.2) include porosity raised to a power of m (≈2). Estimates of dominant or characteristic pore size are more effective at predicting k than estimates of grain size or surface area, so the higher exponent of porosity to compensate for the low end of the porosity spectrum is not required. Given a measure of r and Φ, the more information that r contains regarding the large through-going pores, the lower the dependence on Φ is. Indeed, the findings of Beard and Weyl, Swanson, Pittman, and Katz and Thompson all show that Φ is not so important as a predictor of k as long as the dominant r is well specified. Conversely, using Pittman’s findings of Table 14.1, as r decreases below r_apex, r becomes a progressively poorer estimator of the dominant r, and a higher exponent of Φ is required to compensate for the inclusion of pore throats that do not contribute to flow.

The preceding considerations hold for predicting k on individual samples from a wide range of rock formations, whereas the first part of this chapter shows that Φ can be a good predictor of k for samples from a given rock formation. Why is this? The pore-size models produce curves of constant pore size that transgress the steeper log(k)-Φ data trends. The cutting of the log(k)-Φ trends by the curves of constant pore size shows that porosity reduction is always accompanied by a reduction in characteristic pore size. As rocks from a common source are compacted and undergo diagenesis, pore space is reduced, and permeable pathways are progressively blocked in a systematic way that maintains a consistent relationship between Φ and r. Samples from different formations that have undergone different diagenetic processes follow different evolutionary paths in log(k)-Φ-r space and thus produce different trends on a log(k)-Φ plot.

Practical Applications

The problem of predicting permeability has been reviewed by compiling data and predictive algorithms from the literature. Which approach should be used to estimate permeability from core and well log data? As a practical matter, it depends on what data are available from a given well or field:

1. In cases in which no core data are available, one can proceed by analogy using data developed in formations with geological properties similar to the one under study. Figs. 14.5 through 14.8 are examples of the types of analog data that one might use.
2. When porosity and grain size estimates are available, refer to Fig. 14.17. This chart appears to give good estimates for many consolidated rocks. Exceptions will exist, such as rocks containing illite in pore space and low-permeability formations such as those shown in Fig. 14.8.
3. In situations in which porosity and water saturation can be estimated, permeability can be estimated from Timur’s relationship (Eq. 14.10 and Fig. 14.21). In clay-bearing rocks, the dual-water relationship for permeability (Eqs. 14.12 and 14.13) is an interesting enhancement, but the user is required to provide estimates of both interstitial and bound water.
4. When NMR logs are available, one can make use of the permeability transforms developed for such logs. Laboratory determination of a t₂ cutoff is advised.
5. Permeability is controlled by a pore dimension of a selected subset of the pore population and can be determined from capillary pressure by mercury injection (Figs. 14.26 and 14.27 and Eqs. 14.18 through 14.20). Mercury injection can be applied to determine the permeability of small or fragmented samples.
6. In field developments in which core data are abundant and a relatively simple (linear) log(k)-Φ relation is the result of a fairly uniform lithology and uncomplicated diagnetic history, then one can turn to regression methods to predict k from well log estimates of Φ.
7. In heterogeneous reservoirs, a high degree of scatter on a log(k)-Φ plot requires that the reservoir be zoned before k can be estimated. One must choose a petrophysical parameter with which to zone the reservoir rocks. Various practitioners have used r₃₅, the flow zone indicator (I), the square root of k/Φ, and even k itself (note that each of these four parameters has the dimension of length or length squared). One must also choose a method of zoning or clustering; among the candidates are linear regression, neural networks, data binning, and fuzzy clustering. A good set of core data is required to establish the zones or clusters. After the method is tested, then well logs are used to compute a value of the zonation parameter. The more geological information that can be incorporated into the zonation procedure, the better. In fact, the breadth of petrophysical, well test, and geological data is probably more important than the particular zonation parameter or clustering methodology chosen. Complex reservoirs require complex methods.
8. Fractured reservoirs are a special and difficult case. Fracture permeability cannot be measured with core samples, so it is difficult to establish ground truth. Methods of estimating fracture permeability from fracture aperture and fracture density are tenuous because aperture varies throughout the fracture plane, some fractures are sealed with mineral deposits, and some are open. Combinations of techniques seem to work well. Examination of core can provide orientation and number of fractures, facies descriptions, and mineralization on fracture surfaces. Borehole images provide fracture number, orientation, and aperture. Flow (spinner) logs reveal zones of fluid flow into the wellbore. Sonic waveform logs show fracture location, and if conditions permit, permeability estimates can be extracted from Stoneley waves. Other well logs provide porosity estimates. Well tests provide estimates of permeability over isolated intervals. Analysis of the state of stress can provide insight on fracture location and the probability of being open or closed. Analysis of complementary data sets can provide insights that cannot be obtained from isolated data sets.

In all cases, one must bear in mind that a permeability predictor will be unique to the field or formation for which it is developed. This unfortunate fact is a result of the multiple pathways that can be followed during burial and diagenesis, as seen in Section 14.2.

Nomenclature

A	=	area
d	=	grain diameter
f	=	shape factor
F	=	formation factor
g	=	gravitational acceleration
I	=	flow zone indicator
k	=	permeability
K	=	hydraulic conductivity
lc	=	pore-space dimension
m	=	Archie cementation exponent
Mi	=	weight fraction of mineral component
p	=	pressure
Pc	=	capillary pressure
q	=	volumetric flow rate
Qv	=	cation molarity
rh	=	hydraulic radius
r35	=	pore throat radius at 35% mercury saturation
R	=	pore throat dimension
Sb	=	mercury saturation
Sbw	=	bound water saturation
Swi	=	irreducible water saturation
Sciw	=	critical water saturation
t1	=	NMR longitudinal decay time
t2	=	NMR transverse decay time
Vffi	=	free fluid fraction
Vbw	=	volume of bound water, fraction
Vbi	=	bulk volume irreducible water, fraction
Vcl	=	clay fraction
Vsh	=	shale fraction
z	=	elevation
θ	=	wetting angle
μ	=	dynamic viscosity
ρ	=	density
σD	=	standard deviation
σ	=	electrical conductivity of rock
σo	=	electrical conductivity of saturant
σ	=	interfacial tension
Σp	=	ratio of pore surface area to pore volume
Σr	=	ratio of pore surface area to rock volume
Σg	=	ratio of pore surface area to grain volume
Σ	=	specific surface area
τ	=	tortuosity
Φ	=	porosity
Φt	=	total porosity
Φe	=	effective porosity

Subscripts

e	=	effective
l	=	liquid
o	=	oil
t	=	total
w	=	water

References

SI Metric Conversion Factors

atm	×	1.013 250*	E+05	=	Pa
cp	×	1.0*	E−03	=	Pa•s
in.	×	2.54*	E+00	=	cm
ft	×	3.048*	E−01	=	m
psi	×	6.894 757	E+00	=	kPa

*

Conversion factor is exact.