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# PEH:Relative Permeability and Capillary Pressure

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Petroleum Engineering Handbook

Larry W. Lake, Editor-in-Chief

Volume I – General Engineering

John R. Fanchi, Editor

Chapter 15 – Relative Permeability and Capillary Pressure

Richard L. Christiansen, Colorado School of Mines

Pgs. 727-765

ISBN 978-1-55563-108-6
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Reservoir engineers use relative permeability and capillary pressure relationships for estimating the amount of oil and gas in a reservoir and for predicting the capacity for flow of oil, water, and gas throughout the life of the reservoir. Relative permeabilities and capillary pressure are complex functions of the structure and chemistry of the fluids and solids in a producing reservoir. As a result, they can vary from place to place in a reservoir. Most often, these relationships are obtained by measurements, but network models are emerging as viable routes for estimating capillary pressure and relative permeability functions.

Before defining relative permeability and capillary pressure, let us briefly review the definition of permeability. Permeability represents the capacity for flow through porous material. It is defined by Darcy’s law (without gravitational effects) as

....................(15.1)

Darcy’s law relates the flow rate q to the permeability k, cross-sectional area A, viscosity μ, pressure drop ΔP, and length L of the material. High permeability corresponds to increased capacity for flow. The dimensions of permeability are length squared, often expressed as darcies (1 darcy = 0.987×10–8 cm2), millidarcies, or micrometers squared. Some writers use "absolute permeability" or "intrinsic permeability" in place of permeability.

For multiple-phase flow, the following expressions define relative permeabilities, specifically written for oil and water flow (without gravitational effects) in the x direction:

....................(15.2)

and

....................(15.3)

where kro and krw are the relative permeabilities of oil and water, respectively. Relative permeabilities are dimensionless functions that usually range between 0 and 1. Eqs. 15.2 and 15.3 allow for differences in the pressure in the oil and water phases. The difference in pressure between the two phases is the capillary pressure:

....................(15.4)

Capillary pressure relationships are dimensional functions that range from large negative to large positive values. (Capillary pressure is often defined as the pressure of the less-dense phase minus the pressure of the more-dense phase.) Relative permeabilities and capillary pressures are usually viewed as functions of the saturation of phases in the porous sample—so, for oil/water flow in the absence of a gas phase, we have kro (Sw), krw (Sw), and Pc(Sw). Saturation is the fraction of pore space that is occupied by a phase. In the present example of oil/water flow, Sw +So =1.

In some discussions, the products of permeability and relative permeability (e.g., kkro and kkrw in Eqs. 15.2 and 15.3) are termed the effective permeabilities. Effective permeability of oil at irreducible water saturation, or ko(Swi), is sometimes used to normalize relative permeabilities in place of absolute permeability. With this normalization, kro(Swi ) equals 1. It is possible for water relative permeability to exceed 1 when ko(Swi) is the normalizing factor. One must be very careful when using data to note whether absolute permeability or an effective permeability is used for normalizing.

The remainder of this chapter includes discussion of typical capillary pressure relationships and capillary pressure models, typical relative permeability relationships and relative permeability models, measurement of rock/fluid relationships, and trends in rock/fluid properties.[1]

## Observations of Capillary Pressure Relationships

As defined in the Introduction, capillary pressure refers to the difference in pressure across the interface between two phases. With Laplace’s equation, the capillary pressure Pcow between adjacent oil and water phases can be related to the principal radii of curvature R1 and R2 of the shared interface and the interfacial tension σow for the oil/water interface:

....................(15.5)

The relationship between capillary pressure and fluid saturation could be computed in principle, but this is rarely attempted except for very idealized models of porous media. Methods for measuring the relationship are discussed later in this chapter.

Fig. 15.1 shows a sketch of a typical capillary pressure relationship for gas invading a porous medium that is initially saturated with water; the gas/water capillary pressure is defined as Pcgw=pg-pw. For this example, water is the wetting phase, and gas is the nonwetting phase. As shown in Figs. 15.2 and 15.3, a wetting phase spreads out on the solid, and a nonwetting phase does not. Wettability of a solid with respect to two phases is characterized by the contact angle. Popular terminology for saturation changes in porous media reflects wettability: "drainage" refers to the decreasing saturation of a wetting phase, and "imbibition" refers to the increasing wetting-phase saturation. Thus, the capillary pressure relationship in Fig. 15.1 is for drainage—specifically primary drainage, meaning that the wetting phase (water) is decreasing from an initial value of 100%.

Gas does not penetrate the medium in Fig. 15.1 until the capillary pressure exceeds the threshold pressure Pct, which depends on the size and shape of the pores and the wettability of the sample. As capillary pressure increases beyond this value, the saturation of the water continues to decrease. It is generally believed that the gas cannot flow until its saturation is greater than a critical level Sgc, which is often 5 to 15% of the total pore volume. If gas is not mobile below Sgc, then the capillary pressure relationship between Sw = 1–Sgc and Sw = 1 in Fig. 15.1 is fictitious, as suggested by Muskat[2]—a detail largely ignored in later literature.

Below Sw = 1– Sgc, the capillary pressure increases with decreasing water saturation, with water saturation approaching an irreducible level Swi at very high capillary pressures. Morrow and Melrose[3] argue that capillary pressure measurements have not reached equilibrium if the capillary pressure trend asymptotically approaches an irreducible water saturation. As the water saturation decreases during a measurement, the capacity for flow of water rapidly diminishes, so the time needed for equilibration often increases beyond practical limitations. Hence, a difference develops between the measured relationship and the hypothetical equilibrium relationship, as shown in Fig. 15.1.

After completing measurements of capillary pressure for primary drainage, the direction of saturation change can be reversed, and another capillary pressure relationship can be measured—it is usually called an imbibition relationship. Imbibition is often analogous to the waterflooding process. The primary drainage and imbibition relationships generally differ significantly, as shown in Fig. 15.4 for a gas/water system. This difference is called capillary pressure hysteresis—the magnitude of capillary pressure depends on the saturation and the direction of saturation change. For imbibition of a strongly wetting phase, the capillary pressure generally does not reach zero until the wetting-phase saturation is large, as shown in Fig. 15.4. For a less strongly wetting phase, the capillary pressure reaches zero at a lower saturation, as shown in Fig. 15.5. Capillary pressure behavior for secondary drainage is also shown in Figs. 15.4 and 15.5.

As shown in Figs. 15.4 and 15.5, the wettability of the porous material is an important factor in the shape of capillary pressure relationships. Wettabilities of reservoir systems are categorized by a variety of names. Some systems are strongly water-wet, while others are oil-wet or neutrally wet. Spotty (or "dalmation") wettability and mixed wettability describe systems with nonuniform wetting properties, in which portions of the solid surface are wet by one phase, and other portions are wet by the other phase. Mixed wettability, as proposed by Salathiel,[4] describes a nonuniform wetting condition that developed through a process of contact of oil with the solid surface. Salathiel hypothesized that the initial trapping of oil in a reservoir is a primary drainage process, as water (the wetting phase) is displaced by nonwetting oil. Then, those portions of the pore structure that experience intimate contact with the oil phase become coated with hydrocarbon compounds and change to oil-wet.

The drainage and imbibition terminology for saturation changes breaks down when applied to reservoirs with nonuniform wettability. Rather than using drainage and imbibition to refer to the decreasing and increasing saturation of the wetting phase, some engineers define these terms to mean decreasing and increasing water saturation, even if water is not the wetting phase for all surfaces.

Treiber et al.[5] reported a study of wettabilities of 55 oil reservoirs. Twenty-five of the reservoirs were carbonate, and the others were silicic (28 sandstone, 1 conglomerate, and 1 chert). To characterize wettability, they used the following ranges for the oil/water/solid contact angle as measured through the water phase:

0 to 75° = water-wet
75 to 105° = intermediate-wet
105 to 180° = oil-wet

Their wettability results are listed in Table 15.1. At the time of publication in 1972, it was surprising to readers that two-thirds of the reservoirs were oil-wet. Previously, reservoirs were believed to be mostly water-wet. Treiber et al.[5] also observed that calcium sulfate is strongly water-wet; thus, carbonate reservoirs with some calcium sulfate grains may have microscopic variations in wettability—dalmation wettability, as described previously.

### Drainage and Imbibition for a Strongly Wet System

An example of capillary pressure relationships during drainage and imbibition for an unconsolidated dolomite powder is shown in Fig. 15.6.[6] The wetting phase is water, and the nonwetting phase is decane. The imbibition curve remains above zero capillary pressure, similar to the typical form of Fig. 15.4.

### Heterogeneity

Most naturally occurring porous media are heterogeneous, having laminations, fractures, vugs, and so forth. Such heterogeneities give rise to "bumps" in a capillary pressure relationship. An example of these bumps is shown in Fig. 15.7, as estimated with a simple model for a laminated material: the Brooks-Corey expression (Eq. 15.9 in the Capillary Pressure Models section of this chapter) for gas/oil capillary pressure was applied to rock consisting of alternate layers of two differing permeabilities. The permeabilities of the two layers differ by a factor of 4, and the threshold pressures differ by a factor of 2 (per the inverse-square-root proportionality to permeability that is suggested by Eq. 15.6 in the Capillary Pressure Models section). The threshold pressure for the higher-permeability layer is 1 psi. The residual oil saturation is 0.20, and the exponent λ is 2 for both layers. All layers have the same thickness. Starting at 100% oil saturation, the oil first drains from the high-permeability layers; when the capillary pressure reaches the threshold pressure for the low-permeability layers, oil drains from those layers. The consequence is a bump in the capillary pressure relationship at oil saturation equal to approximately 0.70. Heterogeneities other than laminations can cause bumps. Any porous material that is a composite of two types of pore structure should demonstrate bumps. Similar bumps are often seen for actual rock, as demonstrated with the mercury capillary pressure data in Fig. 15.8.

### Wettability

As reported by Bethel and Calhoun,[7] wettability affects the position of capillary pressure curves, as shown in Fig. 15.9 for displacement of oil (starting at So = 100%) by water from a glass-bead pack. The contact angles in the legend of Fig. 15.9 are as suggested by Bethel and Calhoun. The wettability moves from strongly water-wet at the top of the legend to strongly oil-wet at the bottom. With increasing oil wetness, the capillary pressure shifts upward, reflecting the increased pressure needed to push water into the pore spaces of the specimen. Fig. 15.9 also shows a variation in the residual oil saturation Sor with increasing wettability. When strongly water-wet, Sor is approximately 14%; when intermediate-wet, Sor rises to approximately 35%; and when strongly oil-wet, Sor returns to approximately 15%. Morrow[8] reports numerous examples of Sor between 6 and 10% for strongly oil-wet and intermediate oil-wet bead packs. For water-wet systems, the residual oil saturation is 14 to 16% for an unconsolidated sand with fairly uniform grain size, according to Chatzis et al.[9] These authors reported residual nonwetting saturations of 11% for clusters of smaller beads surrounded by larger beads. For larger beads surrounded by smaller beads, the residual nonwetting saturation rose to 36%.

Jerauld and Rathmell[10] report the imbibition and secondary-drainage data of Fig. 15.10 for a rock sample (permeability = 223 md, porosity = 0.257) from the Prudhoe Bay field, which they identify as a mixed-wet reservoir. As is typical of mixed-wet samples, the water saturation increases rapidly during imbibition for decreasing capillary pressure in the vicinity of zero. Similarly, water saturation decreases rapidly during the secondary-drainage cycle for increasing capillary pressure just above zero.

## Capillary Pressure Models

Leverett and coworkers,[11] based on the evaluation of gas/water capillary pressure data for drainage and imbibition in unconsolidated sands, proposed the following definition:

....................(15.6)

The function j(Sw), defined in Eq. 15.6, is known to many as the "Leverett j-function." The j-function is obtained from experimental data by plotting against Sw. The combination is often considered an estimate of the mean hydraulic radius of pore throats. However, the directional dependence of permeability complicates this interpretation: Which permeability should be used? While permeabilities for just one direction are one answer, this choice is often not available.

The j-function has been used for correlating capillary pressure data for rocks with similar pore types and wettability, but with different permeabilities.[12] Applications include allotment of oil reserves during unitization negotiations. However, one should be very careful when correlating data with Eq. 15.6 to use permeabilities that are measured in the same direction. Perhaps some of the scatter typical of j-function correlations results from inconsistent directions of permeability measurement.

The proportionalities of Eq. 15.6 summarize an intuitive expectation for the relationship between capillary pressure, interfacial tension (IFT), and permeability that is widely used. Occasionally, cos θ is included adjacent to σgw in the definition of the j-function, although Leverett did not write it that way. While the contact angle should affect capillary pressure, the cosine function may not be the correct function to include in the definition of the j-function. Indeed, the dependence of capillary pressure on the contact angle could be quite complex. Nevertheless, the traditional procedure for adjusting capillary pressures for wettability uses the cosine function:

....................(15.7)

This procedure of Eq. 15.7 has been applied to the conversion of mercury capillary pressures to oil/water, gas/oil, and gas/water capillary pressures. The results are mixed: sometimes it works well, and sometimes it does not.

### Thomeer Model

Thomeer[13] proposed a model of the following form for describing mercury-injection capillary pressure data:

....................(15.8)

in which SHg is the saturation of mercury. The Thomeer function has three parameters: the threshold pressure Pct, the pore geometric factor G, and the mercury saturation at infinite capillary pressure SHg∞. (Thomeer wrote Eq. 15.8 with bulk mercury saturation instead of mercury saturation. Bulk mercury saturation equals the product of porosity and mercury saturation. Bulk mercury saturation is appropriate for irregularly shaped samples as collected from drilling cuttings.) Thomeer related absolute permeability to the three parameters of Eq. 15.8.

### Brooks and Corey

Brooks and Corey,[14] extending the earlier work of Corey,[15] suggested the following relationship for capillary pressure during primary drainage of oil from an oil-saturated porous medium during gas invasion:

....................(15.9)

Sor is the residual oil saturation that remains trapped in the pore at high capillary pressure. Pct, the threshold pressure, corresponds approximately to the pressure at which the gas phase is sufficiently connected to allow flow. Brooks and Corey related the parameter λ to the distribution of pore sizes. For narrow distributions, λ is greater than 2; for wide distributions, λ is less than 2. Eq. 15.9 should be representative of any primary-drainage process as long as the porous medium is homogeneous and strongly wetted by the drainage phase; that is, the contact angle measured through the wetting phase must be small.

Brooks and Corey provide no suggestions for estimating the residual oil saturation and the threshold pressure in Eq. 15.9. Typical values of residual oil saturation vary from 8 to 40%. Thomas et al.[16] suggested the following expression for roughly estimating gas/water threshold pressures of low-permeability (less than 1 md), water-saturated sandstones and limestones:

....................(15.10)

(0.43 is near to the 0.5 in the Leverett function). For this expression, the appropriate units are millidarcies for permeability k and psi for threshold pressure Pcgwt. Eq. 15.10 should be applicable to fluid pairs other than gas and water if it is adjusted for IFT differences.

### Bentsen and Anli

Bentsen and Anli[17] proposed the following expression for capillary pressure for a primary-drainage process in which a porous sample initially saturated with water is invaded by an oil phase:

....................(15.11)

Pcs is a parameter with pressure units for controlling the shape of the capillary pressure function. Bentsen and Anli developed Eq. 15.11 following a qualitative argument. These authors reported a range of parameters for several rock/oil/water systems, but they did not suggest means for estimating those parameters.

### Alpak-Lake-Embid Model

An interesting model was proposed by Alpak et al.[18] for representing both capillary pressure and relative permeability relationships. For capillary pressure, they suggested an expression that can be obtained by applying basic thermodynamic arguments to capillary pressure concepts. Their model relates capillary pressure to the change of oil/water interfacial area and water/solid interfacial area with water saturation. Alpak et al.[18] applied their model to drainage and imbibition data with fair success. Research in the years to come may show whether this approach to interpreting capillary pressure is useful.

## Observations of Relative Permeability Relationships

As defined in the Introduction of this chapter, relative permeabilities are dimensionless functions of saturation with values generally ranging between 0 and 1. Figs. 15.11 and 15.12 show typical behavior for a gas/oil system. The semilog scale of Fig. 15.12 is convenient for reading the relative permeabilities less than 0.05. Although the curves are labeled "Gas" and "Oil" in these figures, the phase identity of a curve can be deduced without the labels. For example, the relative permeability that increases in the direction of increasing oil saturation must be the oil relative permeability. The endpoints of the relative permeabilities in Figs. 15.11 and 15.12 are defined by the critical gas saturation Sgc and the residual oil saturation Sor. Common names and symbols for some saturation endpoints are listed in Table 15.2.

### Hysteresis

As is the case for capillary pressure, the relative permeabilities depend on the direction of saturation change, as shown schematically in Fig. 15.13. For this gas/oil system, hysteresis is much greater for the gas relative permeability. Usually, the hysteresis of the wetting phase (oil, in this example) is very small. The trapped-gas saturation Sgt that remains at the end of the imbibition process is a key feature of hysteresis.

Actual observations of hysteresis for water/oil systems are shown in Figs. 15.14 [19]through 15.16. These three figures share some common characteristics. For example, one phase shows large hysteresis, while the other phase shows small hysteresis. Interestingly, the imbibition tracks in Figs. 15.15 and 15.16[20] are above the secondary-drainage trends. Jones and Roszelle[21] report large variations in krw and small variations in kro in what they consider to be a water-wet sample.

### Wettaility

Wettability affects the position of relative permeabilities, as shown in Fig. 15.17 (from Owens and Archer[22]). The authors measured oil/water relative permeabilities for varying wettabilities with a Torpedo sandstone sample. Wettability was controlled by the concentration of additives in the oil and water. Advancing contact angles were measured on a flat quartz surface.

Fig. 15.17 shows two important trends. With increasing wetting by the water, the intersection of the oil and water relative permeabilities shifts to the right, and the maximum krw decreases. Similar trends were documented by Morrow et al.[23] and by McCaffery and Bennion.[24] Reservoir engineers use these trends as indicators of wettability.

As mentioned previously, Treiber et al.[5] reported wettabilities for 55 oil-producing reservoirs. A rock was deemed water-wet if krw at Sor is less than 15% of kro at Swi; intermediate-wet if krw at Sor is between 15 and 50% of kro at Swi; and oil-wet if krw at Sor is greater than 50% of kro at Swi.

In addition to the shape of the relative permeability relationships, the authors used connate water saturations, gas/oil and gas/water relative permeabilities, and contact-angle measurements to supplement their judgment of wettability. The judgments of Treiber et al.[5] relied heavily on the results of Schneider and Owens[25] and Owens and Archer.[22] Treiber et al.[5] emphasized that interpretation of wettability from relative permeability behavior is subject to large error because the relative permeabilities depend on connate water saturations and pore-size distribution in addition to wettability. Furthermore, the authors recognized that laminations and other heterogeneities can dramatically alter the relative permeability behavior and, hence, the interpretation of wettability. To prevent such mistaken interpretations, the authors selected rock samples with a high degree of homogeneity.

### IFT

Relative permeabilities change with decreasing IFT, especially when IFT falls below 0.1 dyne/cm2. The sensitivity of relative permeability to decreasing IFT is of great interest for enhanced-oil-recovery processes, such as miscible-gas processes and surfactant processes, and for the recovery of fluids from retrograde gas reservoirs.

The change in gas/oil relative permeabilities with decreasing gas/oil IFT as reported by Bardon and Longeron[26] is shown in Fig. 15.18. At very low IFT, the relative permeabilities approach an "X" shape, with endpoints close to oil saturations of 0 and 1, while at higher IFT, the relative permeabilities display more curvature and have endpoints more distant from the edges of the water-saturation scale. Significant changes in relative permeabilities are not usually observed until the IFT falls below approximately 0.1 dyne/cm2. Another example of the effect of IFT on relative permeabilities as reported by Haniff and Ali[27] is shown in Fig. 15.19. Asar and Handy[28] also reported on the changes in relative permeabilities for gas/condensate systems as the gas/condensate IFT decreased from approximately 10 to 0.01 dyne/cm2. Amaefule and Handy[29] reported relative permeabilities for low-IFT oil/water displacements.

### Endpoint Saturation Relationships

Residual oil saturation, irreducible water saturation, trapped-oil and -gas saturations, and critical gas and condensate saturations are the most frequently encountered saturation endpoints. Residual oil, irreducible water, and trapped-gas and trapped-oil saturations all refer to the remaining saturation of those phases after extensive displacement by other phases. Critical saturation, whether gas or condensate, refers to the minimum saturation at which a phase becomes mobile.

The endpoint saturation of a phase for a specific displacement process depends on the structure of the porous material, the wettabilities with respect to the various phases, the previous saturation history of the phases, and the extent of the displacement process (the number of pore volumes injected). The endpoint saturation also can depend on IFTs when they are very low, and on the rate of displacement when it is very high.

Results reported by Chatzis et al.[9] give general insight on the combined effects of wettability and porous structure on residual saturations. In tests with an unconsolidated sand of nonuniform grain size, the wetting phase (oil) was displaced by a nonwetting phase (air) from an initial saturation of 100% to a residual value. The authors observed residual wetting-phase saturations Swr of 7 to 8%. They also found that heterogeneities in the porous medium can lead to Swr greater or less than 7 to 8%, depending on the nature of the heterogeneities. Chatzis et al.[9] also reported residual nonwetting-phase (air) saturations Snwr for displacements by a wetting phase (oil). They reported that Snwr is approximately 14% for an unconsolidated sand of fairly uniform size. In tests on sandpacks of distributed grain size, Snwr rose to an average of 16%. Chatzis et al.[9] also measured Snwr for glass-bead packs consisting of lightly consolidated clusters of glass beads of one grain size distributed in unconsolidated glass beads of another size. They reported that Snwr was 11% for clusters of smaller beads surrounded by larger beads. For larger beads surrounded by smaller beads, Snwr rose to 36%. These results suggest two general conclusions. First, the residual saturation of a wetting phase is less than the residual saturation of a nonwetting phase. Second, the residual saturation of a nonwetting phase is much more sensitive to heterogeneities in the porous structure.

General conclusions on the effects of wettability are useful, but the diverse array of wetting alternatives suggests caution, especially in oil/water reservoir systems. This wide range of wetting possibilities is an obstacle to interpreting or predicting the effect of wettability on endpoint saturations. Indeed, conflicting results for different porous media are likely. For example, Jadhunandan and Morrow[30] report that residual oil saturation displays a minimum value for mixed-wet media as wettability shifts from water-wet to oil-wet—counter to the results of Bethel and Calhoun,[7] who reported a maximum for media of uniform wettability.

In the subsections below, specific relationships for endpoints of the oil, gas, and water phases are discussed.

Critical Gas Saturation. The critical gas saturation is that saturation at which gas first becomes mobile during a gasflood in a porous material that is initially saturated with oil and/or water. If, for example, the critical gas saturation is 5%, then gas does not flow until its saturation exceeds 5%. Values of Sgc range from zero to 20%.

Critical Gas Saturation. Interest in the mobility of condensates in retrograde gas reservoirs developed in the 1990s, as it was observed that condensates could hamper gas production severely in some reservoirs, particularly those with low permeability. The trend of increasing critical condensate saturations with decreasing permeability, as summarized by Barnum et al.,[31] is reproduced in Fig. 15.20.

Trapped, or Residual, Gas Saturation. As shown in Fig. 15.21[32], the remaining gas saturation after a waterflood depends on the gas saturation before the waterflood. The relationship of Fig. 15.21 is often called a "trapping relationship." The amount of gas that is trapped in gas reservoirs is of considerable economic significance. For example, in a gas reservoir, encroachment of the aquifer will lead to trapping of some portion of the gas.

Several correlations and summaries for residual gas saturation are found in the literature. Katz and Lee[33] provide a summary of residual gas saturations in a graphical form that is useful for estimates. According to the model presented by Naar and Henderson[34] for multiphase flow through rock, the trapped or residual gas saturation is one-half of its initial saturation; this Naar-Henderson rule is the simplest correlation for residual gas. Agarwal[35] correlated a large collection of residual gas saturations for consolidated and unconsolidated sandstones, for unconsolidated sands, and for limestones. The ranges of parameters in the correlations are summarized in Table 15.3. The correlations may be erroneous outside of these ranges. Three of the Agarwal correlations are listed below:

....................(15.12)

....................(15.13)

....................(15.14)

In these expressions, residual gas saturation Sgr, initial gas saturation Sgi, and porosity Φ are fractional quantities, not percents. Permeability k is in millidarcies.

Land[36] suggested the following form for estimating trapped-gas saturation Sgr as a function of initial gas saturation Sgi:

....................(15.15)

To calculate C, a limited data set is needed, consisting of the maximum trapped-gas saturation Sgr,max for Sgi=1-Swi . Then,

....................(15.16)

Land[37] reported C = 1.27 for four Berea sandstone samples.

Residual Oil Relationships. Residual oil saturations after waterflooding or gasflooding are clearly significant for oil recovery. Here, the dependence of residual oil saturation on initial oil saturation and capillary number for a waterflood will be considered.

The relationship between initial and residual oil saturation is termed the oil-trapping relationship. For strongly water-wet rocks, the oil-trapping relationship should be identical to the gas-trapping relationship. Indeed, because of this analogy and because it is easier to measure gas-trapping relationships, few oil-trapping relationships have been measured. A set of oil-trapping relationships reported by Pickell et al.[38] are shown in Fig. 15.22. Oil-trapping relationships are important for estimating reserves in transition zones. In conventional reservoir engineering, residual oil saturation refers to the remaining oil saturation after a displacement that starts near the maximum initial oil saturation, which generally equals one minus the initial water saturation.

In the remainder of this section, the dependence of residual oil saturation on capillary number is discussed for processes starting with initial oil saturation at a maximum value: So = 1– Swi. This topic has received much more attention in the literature than oil-trapping functions. The capillary number is the ratio of viscous forces to capillary forces. It is represented quantitatively with various expressions, as summarized by Lake.[39] These expressions are derived from the ratio of pressure drop in the water phase to the capillary pressure between the oil and water phases. A popular definition of the capillary number is as follows:

....................(15.17)

with v representing the velocity of the water. The capillary number is small (less than 0.00001) when capillary forces dominate the flow processes. The following example shows just how small capillary numbers can be.

Example 15.1

Use the following quantities to estimate a capillary number for a waterflood with Eq. 15.17, where

μw = 1 cp = 0.01 g/cm/s
v = 1 ft/D = 30.48 cm/(24 × 3,600 s) = 0.00035 cm/s
σow = 30 dynes/cm

Therefore, the capillary number is as follows:

Capillary forces do indeed dominate flow processes for waterfloods. Even in high-velocity regions, such as the vicinity of a well that is producing oil and water, the capillary number will remain very small.

Having defined the capillary number, the relationship between residual oil saturation and capillary number will be discussed next. As the capillary number for an oil-displacing process increases, residual oil saturation decreases in the manner sketched in Fig. 15.23[40]. Above the "critical capillary number," the rate of decrease of Sor is particularly rapid. The critical capillary number is 10–5 to 10–4 for porous media with fairly uniform pore sizes. With increasing distribution of pore sizes, the critical capillary number decreases, the Sor at low Nc increases, and the domain for decreasing S or becomes broader. Extensive discussion of these relationships is available elsewhere.[41] King et al.[42] suggested centrifuge methods for measuring these relationships. Pope et al.[43] correlated residual phase saturation with a modified form of the capillary number, which was termed the "trapping number." Adjusting a parameter in their correlation fits the effects of wetting on residual saturation.

Residual (Irreducible) Water Saturation. Residual, or irreducible, water saturation Swi is the lowest water saturation that can be achieved by a displacement process, and it varies with the nature of the process—gas displacement or oil displacement. Also, Swi varies with the extent of the displacement, as measured by pore volumes of oil or gas injected or by time allowed for drainage.

To be more specific, the results of Chatzis et al.[9] (discussed in the introduction to this section) can be extended to suggest irreducible water saturations of 7 to 9% for displacements in unconsolidated sand and glass beads that are water-wet. Furthermore, Swi should increase slightly with increasing breadth of grain-size distribution. Significant variations in Swi should occur when small clusters of consolidated media of one grain size are surrounded by media of another grain size. If the grains of the clusters are smaller than those of the surrounding media, Swi increases; if the grains of the clusters are larger than those of the surrounding media, Swi decreases.

The saturation of water in an oil or gas reservoir at discovery is called the connate water saturation, or Swc. The connate water saturation and the irreducible water saturation can differ. If the reservoir processes that produced the connate water saturation can be replicated, then the Swi for the replicated processes should be the same as Swc. Swc is significant for its connection to initial oil or gas saturation in a reservoir. For an oil reservoir, So = 1– Swc; for a gas reservoir, Sg = 1– Swc. The connate water saturation will also affect initial oil or gas relative permeability and, hence, the economic viability of a reservoir. Bulnes and Fitting[44] concluded that low-permeability limestone reservoirs are more viable than sandstone reservoirs of the same permeability because the connate water saturation is lower in the limestones than in the sandstones; as a result, the relative permeabilities to oil are higher in the limestones than in the sandstones.

Salathiel[4] observed that the connate water saturations in carefully retrieved rock samples from some oil reservoirs are substantially lower than can be achieved when the rock is waterflooded and then oilflooded. He attributed this effect to the mixed-wettability condition. When the reservoir was first invaded by oil, the rock was water-wet, and low water saturations were obtained. However, the wettability of the rock surfaces that were now in contact with oil changed from water-wet to oil-wet as portions of the hydrocarbons adsorbed onto the solid surfaces. So, when such a rock is waterflooded and then oilflooded, the connate water saturation is not obtained because the water in the oil-wet portions of the rock becomes trapped.

### Temperature

The effects of temperature on relative permeability have been studied primarily for applications to steamflooding and in-situ combustion. Mechanistically speaking, temperature can affect relative permeability by altering the IFT between flowing phases or by altering the wettability of the porous material. IFT between water and oil should decrease with increasing temperature, but to substantially influence relative permeability, the IFT would need to decrease to 0.1 dyne/cm2 or less, according to the discussions in previous sections. Such reductions would be possible only at very high temperatures with light oils. Therefore, temperature-related IFT reductions could influence relative permeabilities for in-situ combustion processes, but they would not be important for typical steamflooding.

The influence of temperature on wettability and, hence, on relative permeability is more likely to be important for most applications. With increasing temperature, the wettability could shift either to more water-wet or more oil-wet conditions, depending on the reservoir fluids and the chemical composition of the porous medium.

Akin et al.[45] reviewed a wide variety of published studies of relative permeabilities for heavy oil and water at different temperatures. Some of the studies concluded that these relative permeabilities were unaffected by temperature changes, while other studies concluded the opposite. In the light of the previous paragraph, these contradictory observations in the literature are not surprising. However, Akin et al.[45] concluded that viscous instability—not wettability change—is the cause of most reported changes in relative permeability with increasing temperature.* With increasing temperature, the viscosity of the heavy oil decreases, and the water/oil displacement process becomes more stable. The changing stability of the displacement (estimated with the expression of Peters and Flock[46]) causes the apparent relative permeabilities to change with temperature. Nevertheless, it is possible that relative permeabilities do change with temperature for some systems. As Akin et al.[45] conclude, further study of this subject is needed.

*

Viscous instability results from displacement of a viscous (low-mobility) phase by a less-viscous (high-mobility) phase. The high-mobility phase is prone to bypass or "finger" through the low-mobility phase. With "viscous fingering," the displacement must be 2D or 3D rather than 1D. One-dimensional displacements are preferred for measurement of relative permeabilities.

## Relative Permeability Models

### Brooks-Corey and Related Models

In 1954, Corey[15] combined predictions of a tube-bundle model with his empirical expression for capillary pressure to obtain expressions for gas and oil relative permeabilities. In 1964, Brooks and Corey[47] extended Corey’s results using Eq. 15.9 for capillary pressure to obtain the following expressions for oil and gas relative permeabilities:

....................(15.18)

and ....................(15.19)

Eqs. 15.18 and 15.19 apply to a porous material that is initially fully saturated with oil and then invaded by gas. These equations do not allow for nonzero critical gas saturation. For λ=2, Eqs. 15.18 and 15.19 reduce to the 1954 Corey expressions. Brooks and Corey related the parameter λ to the distribution of pore sizes. For narrow distributions, λ is greater than 2; for wide distributions, λ is less than 2. They reported that λ =7.30 for an unconsolidated pack of glass beads of uniform diameter. For sandpacks with broader distributions of particle sizes, λ ranged from 1.8 to 3.7. For a particularly homogeneous consolidated sandstone, they reported λ =4.17.

The following "power-law" relationships are often used to describe oil, water, and gas relative permeabilities, respectively:

....................(15.20)

....................(15.21)

....................(15.22)

The exponents no, nw, and ng range from 1 to 6. The maximum relative permeabilities, kro max, krw,max, and krg,max, are between 0 and 1. These expressions are often referred to as modified Brooks-Corey relations, reflecting their similarity to the Brooks-Corey expression for oil relative permeability.

### A Model for Heterogeneous Rock

In 1956, Corey and Rathjens[48] extended Corey’s earlier work to explain the effect of laminations on gas/oil relative permeability relations. They considered flow parallel and flow perpendicular to N laminations. For flow perpendicular to laminations,

....................(15.23)

and ....................(15.24)

For flow parallel to laminations,

....................(15.25)

and ....................(15.26)

In these expressions, ki is the permeability of each layer; kroi and krgi are oil and gas relative permeabilities for each layer, respectively; fi is the fraction of total sample volume for each layer; kT is the total permeability of the laminated materials; and kroT and krgT are the resulting total relative permeabilities of the laminated materials. The predictions of Eqs. 15.23 through 15.26 for N = 2 are shown in Fig. 15.24. The permeability of one region is one-fourth that of the other region. The relative permeabilities are given by Eqs. 15.18 and 15.19 with λ = 2 and with Sor = Sgc = 0.20. Corey and Rathjens[48] noted two consequences of laminations: bumps in the relative permeability relationships and a shifting of the critical gas saturation. As shown in Fig. 15.24, the critical gas saturation shifts from 0.20 for the uniform sample to 0.10 for flow parallel to strata, and to 0.33 for flow perpendicular to laminations. Also, the relative permeability for flow perpendicular to laminations is much less than that parallel to laminations. As shown in Fig. 15.25, measured relative permeabilities for a sample of Berea outcrop sandstone with visually apparent laminations show the predicted trends. The Corey-Rathjens observations were used by Ehrlich[49] to explain the relative permeability behavior of vuggy and fractured samples.

### Chierici Model

Chierici[50] proposed exponential expressions for fitting gas/oil relative permeabilities:

....................(15.27)

and ....................(15.28)

with gas saturation normalized as follows: . His expressions for oil/water relative permeabilities are

....................(15.29)

and ....................(15.30)

with water saturation normalized as follows: . The Chierici expressions were used in a recent discussion of three-phase relative permeabilities.[51]

### Correlations of Honarpour et al. and Ibrahim

Other researchers have offered correlations for laboratory measurements of relative permeabilities. Honarpour et al.[52] suggest correlations for two sets of rock samples—sandstones and conglomerates and limestones and dolomites—with varying wettabilities. Ibrahim[53] reported a more extensive set of correlations. Such correlations are useful for understanding general trends and for preliminary estimates; however, they can be far from correct when applied to a specific formation.

### Hysteresis Models

The effect of relative permeability hysteresis on reservoir performance can be significant for processes with variable directions of saturation change. For example, during coning of water toward an oil-producing well, the water saturation is increasing; however, if the production rate is decreased or set to zero, the water saturation can decrease. Hysteresis also can affect the performance of a waterflood if the relative production rate of a well in a pattern of producers is changed. To facilitate simulation of these processes, a number of models have been proposed for representing hysteresis effects on relative permeability and capillary pressure. The hysteresis models of Killough[54] and Carlson[55] are used in some commercial simulation software. Fayers et al.[51] proposed another model for incorporating hysteresis effects in reservoir simulation. These models extrapolate and interpolate from measured drainage and imbibition curves to generate "reasonable" estimates of relative permeabilities. Although these models may be satisfactory for preliminary estimates, many more experimental data on hysteresis of relative permeability and capillary pressure are needed.

### Carman-Kozeny Models

Alpak et al.[18] used concepts from a model for capillary pressure to build a relative permeability model with the form of the Carman-Kozeny expression for permeability. (See Alpak et al.[18] for references to other related models.) The Alpak et al.[18] model relates relative permeabilities to the total surface area of the solid AT, the oil/water interfacial area Aow, the water/solid interfacial area Aws, and the oil/solid interfacial area Aos:

....................(15.31)

and ....................(15.32)

This relative permeability model includes tortuosity relationships, τw/τ and τo/τ , that do not arise in the capillary pressure model of Alpak et al.[18] This interesting difference suggests that relative permeabilities cannot be estimated from capillary pressure information alone. Alpak et al.[18] suggested relationships for the tortuosity and the area functions. They used the model to fit relative permeability data for unconsolidated and consolidated media. Future research will test the merit of this approach to modeling relative permeability.

### Network Models

The advent of computers led to the development of models that represent porous structure as 2D and 3D networks of flow channels. Analysis of these network models leads to capillary pressure and relative permeability relationships. The capacity of these models to represent real behavior has increased with improved descriptions of the displacement mechanisms.[56]

### Models for Three-Phase Relative Permeabilities

Many reservoir processes, such as waterflooding below the bubblepoint pressure of the oil in place, involve simultaneous flow of three phases. To model these processes, three-phase relative permeabilities are mandatory. Measurements of three-phase relative permeabilities are much rarer than those for two-phase relative permeabilities, and there is more uncertainty in the reported three-phase data, as noted by Baker[57] in his 1988 review of three-phase correlations.

Current efforts in three-phase relative permeability studies are weighted toward identification of models for extrapolating two-phase relative permeability data to three-phase applications. Stone[58] started this trend in 1970 with a model that is now known as the Stone I model. In this model for water-wet porous media, the three-phase water relative permeability krw,wog depends only on water saturation and is identical to krw,wo measured in water/oil displacements:

....................(15.33)

Similarly, the three-phase gas relative permeability krg,wog depends only on gas saturation and is identical to krg,go measured in gas/oil displacements at irreducible water saturation.

....................(15.34)

The equality of water and gas relative permeabilities in two- and three-phase flow, as described by Eqs. 15.33 and 15.34, is supported by much of the three-phase data in the literature for water-wet media. On the other hand, the three-phase oil relative permeability kro,wog depends nonlinearly on water and gas saturations:*

....................(15.35)

with the oil, water, and gas saturations scaled, respectively, as follows:

....................(15.36)

....................(15.37)

and ....................(15.38)

According to Stone, the minimum oil saturation Som should be in the range of ¼ Swc to ½ Swc, and it should be less than or equal to the smaller of Sorw or Sorg, the residual oil saturations for waterflooding and gasflooding, respectively. Fayers and Matthews[59] proposed the following expression for Som:

....................(15.39)

with α = 1 - Sg/(1 - Swc - Sorg) . Aleman, as reported by Baker,[57] proposed an alternative expression:

....................(15.40)

To account for hysteresis effects in three-phase flow, Stone recommended use of the appropriate two-phase relative permeabilities. For example, in a water-wet system, if oil saturation is decreasing and gas and water saturations are increasing in the three-phase setting, then the following two-phase relative permeabilities should be used: kro,wo for decreasing oil saturation; krw,wo for increasing water saturation; and krg,go for increasing gas saturation.

In 1973, Stone[60] proposed another model, which has become known as the Stone II model. In this model, the three-phase water and gas relative permeabilities are again equal to those measured in two-phase flow, as expressed by Eqs. 15.33 and 15.34, while the oil relative permeability is as follows:

....................(15.41)

The water/oil and the gas/oil relative permeabilities in Eq. 15.41 are functions of water saturation and gas saturation, respectively.

Baker[57] compared four versions of the Stone models and various other models to available data for three-phase relative permeabilities. He concluded that models based on linear interpolation from the two-phase relative permeabilities perform as well as the other models. As an example of linear interpolation models, Baker suggested the following saturation weighting of two-phase relative permeabilities:

....................(15.42)

....................(15.43)

and ....................(15.44)

Blunt[61] showed how to extend this model to describe relative permeability of oil at very low saturations, often termed the "layer drainage" regime.

Another model for three-phase relative permeabilities and capillary pressures was proposed recently by Fayers et al.[51] The lasting value of this or any of the above-mentioned models, will be known only when more data become available for comparisons.

*

The factors that contain kro,wo (Swc) in Eqs. 15.35 and 15.41 reflect normalization of relative permeabilities with respect to absolute permeability, as is the convention in this chapter. For a comparison of these normalized equations to data, see Fayers and Matthews.[59] Originally, Stone normalized with respect to oil permeability at irreducible water saturation. The Stone versions of Eqs. 15.35 and 15.41 can be recovered exactly by omitting the extra factors from Eqs. 15.35 and 15.41.

## Measurement of Rock/Fluid Relationships

### Reliability of Measurements

The reliability of measurements of relative permeabilities and capillary pressures is an important issue for reservoir engineering. Although there are many factors that influence reliability, the following three topics are emphasized here:

• Sampling of rocks for measurements.
• Measurement methods.
• Treatment of data from the measurements.

Proper sampling of rocks for measurements is most important for ensuring the reliability of relative permeability and capillary pressure data. If samples are obtained improperly, costly and reliable methods for measuring rock/fluid properties may no longer be necessary or suitable. The goal of sampling should be to avoid or minimize mechanical and chemical damage to the rock. Mechanical and chemical damage can occur during any of four steps in the sampling process: coring and core retrieval, shipping and storing, cutting samples from the core, and cleaning and preparing the sample. With all of these opportunities, some damage is inevitable.

It should be obvious that reliable data require good measurement methods and correct treatment of the data obtained. But defining "good" and "correct" requires some not-so-obvious understanding—a goal of this discussion is to touch on the important issues. Measurement of relative permeability and capillary pressure relationships is complicated, particularly because of the intertwining nature of these rock/fluid properties. Indeed, it is the deciphering of these complications that largely defines what is meant by "good" and "correct."[62]

### Sample Handling and Preparation

From cutting a core from a formation to final preparations of a specimen for testing, improper practices will alter the porous structure and wettability of rock samples, which in turn will alter the quality of measurements. The most appropriate procedures depend on the ultimate objective of sample analysis. If the sample is to be used only for porosity and permeability measurement, wettability alterations during handling are not important. But for capillary pressure and relative permeability measurements with native-state wetting conditions, one must work to avoid contamination of the sample. Next, a distillation of industry experience with handling and preparation of samples is presented.

Cutting a Core From a Formation. Drilling fluids often contain oxygen, surfactants, polymers, clays, and other particulates in oil- or water-based slurries. To minimize contamination with drilling fluids, cores are sometimes cut with fresh formation oil, synthetic formation brines, or drilling fluids formulated to minimize invasion. Sometimes, the extent of invasion can be assessed later in the laboratory. Coring also can produce fractures in the rock, either by the mechanical stress of the drilling process, by relief of the in-situ mechanical and thermal stresses on the core as it travels to the surface, or by retrieval (often with a big hammer) from the core barrel at the surface.

Shipping. A variety of methods are used to ship core. Core has been shipped in containers filled with oil or water from the producing formation, it has been shipped in sealed polymeric bags with metallic barriers to oxygen permeation, and it has been shipped after wrapping with a metallic foil and a wax-coated fabric. Some core has been frozen immediately with dry ice or liquid nitrogen upon arrival at the surface, then shipped in a frozen condition to prevent oxidative and bacterial actions and to minimize evaporation of the water or oil.

Sample Collection. Selection of the locations in the core for taking samples is a critical step. For porosity and permeability testing, samples are usually collected at regular intervals (e.g., every foot). Sometimes, the most homogeneous samples are selected. Such a bias likely imposes an undesirable bias on the results. Selecting locations for sampling should follow a logical or a statistical thought process. For example, criteria for sample selection can include mineralogy, pore structure, homogeneity or heterogeneity, porosity, and permeability. Computerized tomography (CT) scanning can help the selection of samples without internal pebbles or fractures that might drastically affect the outcome of measurements. In general, the characteristics of a collected sample should represent a significant portion of the reservoir. After locations are chosen, collection can begin. Collecting a rock sample from the core generally entails a cutting action with diamond-coated saws and boring tools. The coolant for these cutting processes is a source of contamination. To reduce contamination, liquid nitrogen is sometimes used as a coolant, but most often kerosene or water is used—these coolants should be free of contaminants. For preserving water saturations in the sample, kerosene is a common choice for cooling. Excessive exposure of the core to air during sample collection can foster a shift of wettability from its native condition.

Sample Cleaning and Preparation. After collection from the core, rock samples are thoroughly cleaned in preparation for porosity, permeability, and other measurements. Cleaning by flushing the sample alternately with toluene, chloroform, acetone, and methanol (and mixtures of these) is common and often satisfactory. To preserve delicate clay structures in some samples, a cleaning process with miscible fluids and supercritical drying is sometimes used. Cleaning a sample with boiling solvent (such as toluene) in a reflux-extraction unit should be avoided. During such cleaning, water vaporizes, and any surface coated with water can become coated with high-molecular-weight hydrocarbons that are less soluble at elevated temperatures—this can impart a nearly permanent change in wettability.

Final Preparations for Testing.After suitable cleaning, the samples can be used in the clean condition for porosity, absolute permeability, and mercury capillary pressure measurements; alternatively, the condition of the samples can be returned to near reservoir condition by processes of flooding with brine and oil from the formation and aging in formation oil. Such restored-state samples can be suitable for capillary pressure and relative permeability measurements with reservoir fluids. Common practice includes centrifugation of brine-saturated samples in air to reduce the brine saturation to near reservoir levels. Some rock samples are not cleaned after collection from the core but are maintained instead in a preserved state by storing them in uncontaminated oil from the reservoir. Such samples are flooded with formation oil in preparation for measurements.

### Measuring Capillary Pressure Relationships

Most methods for measuring capillary pressure may be grouped under three headings: mercury methods, porous-plate methods, and centrifuge methods. Each of these is discussed below. Vapor-pressure and gravity-equilibrium methods also have been used, but they are not discussed here. The literature offers few examples of capillary pressure measurements with overburden stress matching that in the reservoir. The magnitude of error caused by this omission will vary, of course, from reservoir to reservoir, but the meaning and significance of capillary pressure data without the appropriate overburden stress are questionable. Some commercial laboratories offer capillary pressure measurements with overburden stress.

Mercury Methods. In the mercury method, a sample of rock is evacuated, and the volume of mercury that enters the sample at increasing pressures is measured, as shown in Fig. 15.26. Mercury methods are especially suited for samples of irregular shape, such as those found in drill cuttings. Mercury methods are useful for investigating the porous structure of the sample. Complete mercury capillary pressure curves can be determined within an hour or so, depending on the permeability of the sample. To apply the proper overburden stress, a cylindrical sample could be mounted in a confining sleeve in a variation of the apparatus of Fig. 15.26. Some commercial laboratories offer mercury measurements with overburden stress.

Use of the mercury method was first documented in the petroleum literature by Purcell in 1949.[63] Purcell showed that mercury capillary pressure and air/water capillary pressure relationships could be correlated. He outlined a method for estimating permeability from mercury measurements. Rose and Bruce[64] proposed a relationship between a rock’s threshold capillary properties and its permeability. Thomeer[13] proposed a correlation for estimating permeability from three parameters of the mercury capillary pressure curve, as mentioned previously. Swanson[65] proposed a correlation between permeability and the intersection of a 45° line with the mercury capillary pressure relationship plotted on a log-log scale.

Porous-Plate Methods. The porous-plate method can yield very accurate capillary pressure relationships. The following example describes the method. Consider a cylindrical rock sample that is first saturated with water. A flat face of the sample is then pressed against a flat porous plate (or membrane) in a chamber filled with gas, as shown in Fig. 15.27. The porous plate is also saturated with water. Often, a moist tissue is placed between the sample and the plate to produce good capillary contact.

Then, pressure in the gas phase above the porous plate is increased by a small step, forcing gas to displace the water from the sample through the plate. When the displacement ceases, the difference in pressure between the gas surrounding the sample and the water on the lower side of the plate is the capillary pressure corresponding to the saturation of water remaining in the sample. After a measurement is completed, the pressure in the gas is increased again, forcing more gas into the sample. This process is repeated, increasing the capillary pressure in a series of steps, yielding the capillary pressure relationship for decreasing water saturation.

If the pressure in the gas is raised too far, the gas will penetrate through the porous plate, terminating the test. The highest capillary pressure that can be achieved with the porous-plate method equals the threshold pressure of the plate—that pressure at which gas can penetrate through the plate. The plate threshold pressure depends on the size of pores in the plates and the interfacial properties of the two fluids separated by the plate. The process of gas injection is often reversed before the capillary pressure reaches the threshold pressure of the plate. By decreasing the pressure in the gas in small steps, the capillary pressure relationship for increasing water saturation can be determined.

Porous-plate methods have been applied to gas/water systems, gas/oil systems, and oil/water systems using porous plates with suitable wettability to prevent penetration through the plate of the phase surrounding the rock sample. Considerable time may be needed to reach equilibrium with porous-plate methods—often a week or more at each displacement step for oil/water systems. Christoffersen and Whitson[66] provide an example of the porous-plate technology with automated control of gas/oil displacements. To reduce equilibration time, their system features a thin membrane in place of a porous plate. They could complete the measurements on a 5-md chalk sample in 14 days.

As suggested earlier, capillary pressure relationships should be measured with an applied overburden stress that approximates the stress in the reservoir. The apparatus of Fig. 15.27 does not provide specifically for overburden stress, but by confining a cylindrical sample in a rubber (or similarly flexible) sleeve, one can apply approximately the needed stress. The downside of applying stress with the sleeve is that much of the surface area of the sample is blocked, which slows the pace of the test.

Centrifuge Methods. Centrifuge methods are increasingly favored for measuring capillary pressures. Although not as quick as mercury measurements, centrifuge measurements are much faster than porous-plate methods. To measure a gas/oil capillary pressure relationship with the centrifuge method, a cylindrical sample is first saturated with oil; next, it is mounted in a centrifuge, as shown in Fig. 15.28, and is spun in steps of increasing spin rate. The centrifugal forces throw oil from the sample while pulling surrounding gas into the sample. The duration of each spin step must be sufficient for production of oil to cease.

The average saturation of oil in the sample at each spin rate may be calculated from the volume of oil that is produced to the collector relative to the porous volume of the sample. Because 4 to 24 hours are needed to reach equilibrium at each spin rate, most centrifuge data sets consist of eight or fewer spin rates. Ruth and Chen[67] recommend at least 15 spin rates for accurate evaluation of capillary pressure.

The capillary pressure distribution Pc(r) at each spin-rate step depends on the gas/oil density difference Δρ, the spin rate ω, and the dimensions of the sample relative to the axis of rotation:

....................(15.45)

Here, ro is the radius from the axis of rotation to the outside face of the sample (see Fig. 15.28), and r is the radial distance to any point in the sample. At each spin rate, the capillary pressure at the inside face of the sample (at ri) is

....................(15.46)

The process for converting a set of average saturations and Pci from centrifuge measurements to capillary pressure relationships can involve differentiation of the centrifuge data, as first described by Hassler and Brunner.[68] This differentiation process compounds any error in the original data. So, although the centrifuge method is faster than the porous-plate method, it is not as accurate.[69]

Many variations of the centrifuge method are found in the literature and in industry practice. For example, to measure water/oil capillary pressure for increasing water saturation, the bucket in Fig. 15.28 is reversed so that the rock is on the outside, surrounded by water; the oil that is displaced by the water segregates toward the axis of rotation. Means for applying overburden stress also have been included in centrifuge designs. Baldwin and Spinler[70] and Spinler et al.[71] used magnetic resonance imaging to obtain capillary pressure relationships by direct measurement of fluid saturations in a rock sample taken from a centrifuge test.

### Measurement of Relative Permeabilities

There are many variations in methods for measuring relative permeabilities; only their general features will be discussed here. Usually, a cylindrical porous sample is mounted in a holder similar to that shown in Fig. 15.29. The cylindrical surfaces of the sample are sealed to prevent flow. The seal is accomplished in Fig. 15.29 with a rubber sleeve that also allows for application of a radial confining stress. Fluids are injected and produced from the sample through ports at each end. Often, additional ports are added for pressure measurement. In addition to the sample holder, other apparatus are needed to inject and collect fluids, measure pressures, apply confining pressure, measure saturations, and so forth. Some of these external features are shown in Fig. 15.30. Phase saturations can be estimated from the change in mass of the rock sample, the change of electrical conductivity, or the change of absorption of X-rays[72] or other radiation. Acoustic methods[73] and CT scanning[74][75] are also used. To measure the change in mass, the rock sample is quickly removed from the assembly, weighed, then returned to the assembly. Because this procedure could cause the saturation to change, in-situ techniques such as electrical conductivity or X-ray absorption have an advantage. Steady-state and unsteady-state methods for measuring relative permeabilities are discussed further in the subsections below.

Steady-State Methods.In steady-state methods, both phases (oil and water, gas and oil, or gas and water) are injected simultaneously at constant rates. Injection continues until a steady state is reached, as indicated by constant pressure drop and constant saturations. Four subcategories of the steady-state methods are introduced below.

Multiple-Core Method. In the multiple-core method, frequently called the Penn State method, a rock sample is sandwiched between two other rock samples to build a lengthened sample. The upstream and downstream rock samples distribute flow of the multiple phases over the cross section of the rock and reduce the influence of capillary end effects[76] on the central rock sample. Pressure drop is measured across the central sample while two fluid phases are pumped through the sample at constant flow rates. In some early applications of this method, the saturations in the central rock sample were measured by quickly removing the sample and measuring its mass.

High-Rate Method. In the high-rate method, two fluid phases are injected into a rock sample at high and constant flow rate. The actual magnitude of rate that is required for this method will depend on the length of the rock sample as well as its capillary pressure properties. The injection rate must be sufficient so that capillary end effects are negligible. Of the steady-state methods, the high-rate subcategory is used most frequently.

Stationary-Liquid Method. In the stationary-liquid method, relative permeability of one highly mobile phase is measured in the presence of an essentially immobile second phase. Typically, the immobile phase is a liquid phase, while the mobile phase is usually gas. Because of the high mobility of gas, the liquid phase can be essentially immobile as long as the pressure gradient is small.

Uniform-Capillary-Pressure Method. In the uniform-capillary-pressure method, often called the Hassler[77] method, the capillary pressure between two flowing phases is kept uniform throughout a rock sample by keeping the pressure gradients in both phases equal. This is accomplished by incorporating porous plates or membranes at the entrance and exit faces of the porous sample (not shown in Fig. 15.29). The membranes allow passage of just one of the injected fluids, so the pressure drop in each flowing phase can be measured separately. Although the Hassler method is rarely used, measurement methods with selective membranes are frequently encountered in the literature.

Unsteady-State Methods. In unsteady-state methods, just one phase is injected at either a constant flow rate or a constant pressure drop. Throughout the injection, the pressure drop and production of phases are measured. Three subcategories are described next.

High-Rate Methods. For measurements with high-rate unsteady-state methods, the injection rate must be sufficient so that capillary spreading effects and capillary end effects can be eliminated. The injection and production data and the differential pressure data must be differentiated to obtain the relative permeabilities. These high-rate methods are used most frequently in the oil and gas industry. They provide results for the least cost and with the least delay in time. The quality of the results has been questioned, but there is evidence in the literature that the methods can give results equivalent to those obtained with other methods.

Low-Rate Methods. The enormous increase in computing power and its availability in the last 20 years has facilitated measurement of relative permeabilities in low-rate unsteady-state tests. These tests are preferred to high-rate tests for samples that have fines that become mobile at high rates. The test equipment is identical to that used for high-rate methods, but numerical models are used for interpreting the production and pressure-drop data. The low-rate methods are not widely used.

Centrifuge Methods. Relative permeabilities can be measured in centrifuge tests using the same apparatus as that described for measurements of capillary pressure (Fig. 15.28). Standard practice provides for measurement of the relative permeability of the lowest-mobility phase. To obtain this relative permeability, the production of one phase as a function of drainage time must be measured. Then, differentiation of the data per the algorithm devised by Hagoort[78] gives the relative permeability.

### Measurement of Endpoint Saturations

Endpoint saturations are often valued more highly than capillary pressures and relative permeabilities for several reasons. First, the residual oil saturation for a waterflood defines the maximum amount of oil that can be recovered, so it is very useful for economics calculations. Irreducible water saturation is very useful for assessing the volume of oil in place in a reservoir. Furthermore, the endpoints can be measured more accurately than capillary pressure and relative permeability relationships. As such, some discussion of methods for measuring endpoint saturation is included here.

To measure residual oil saturations after a waterflood or gasflood, the apparatus of Figs. 15.29 and 15.30 can be used, but pressure-drop data are not needed—just the oil saturation at the end of the flood is required. As a result, residual oil saturations are much less costly to measure than relative permeabilities and capillary pressures. Still, care is needed to ensure proper wetting conditions for the measurements.

In principle, irreducible water saturation should be measured by oilflooding a rock sample that is initially saturated with water. (Of course, the wettability of the sample will influence the results, so care in handling and preparing the sample is important.) However, irreducible water saturations from such oilflooding are often greater than water saturations measured in retrieved cores from reservoirs. Therefore, alternative approaches have been explored. Satisfying estimates of Swi have been obtained from mercury-injection tests, with the mercury representing the oil phase and the vapor phase in the test representing the displaced water phase.[79]

## Trends and Cautions in Rock/Fluid Properties

At the conclusion of this discussion of capillary pressure and relative permeabilities, five important trends and cautions are worth emphasizing. Some of them were discussed in this chapter; others are offered as extrapolations from discussions of this chapter.

First, the need for accurate measurement of capillary pressure and relative permeability functions increases with the resolution of reservoir models. With low-resolution models, there is a need for algorithms to "upscale" permeabilities, relative permeabilities, and capillary pressures from the scale of measurement on a small sample of rock to the relatively huge size of blocks in reservoir models. The results of the averaging processes of upscaling are insensitive to the quality of measurements on small samples. The need for upscaling should diminish as increases in computer power permit higher-resolution models.

Second, to obtain accurate measurements of capillary pressure and relative permeabilities, tests with representative samples at representative conditions are critical. Much of the available data in our industry do not pass this standard.

Third, capillary end effects and viscous fingering have corrupted a significant portion of relative permeability data. If capillary pressure and relative permeabilities are available, the extent of this corruption for a sample can be assessed and sometimes corrected.

Fourth, we often interpret water/oil wettabilities from the shape of relative permeabilities. Such interpretations are particularly susceptible to error caused by the heterogeneity of the sample used for measurements. This susceptibility was conceded in the original literature on wettability interpretation, but it is not widely acknowledged.

Fifth, the quality of estimates of capillary pressure and relative permeability with network models is increasing. These models offer the hope of providing estimates for a large set of rock samples for any particular reservoir while avoiding the costs of measuring capillary pressure and relative permeability.

## Nomenclature

 A = area perpendicular to flow, L2 A, B = parameters in Chierici functions AT = total surface area of a porous sample, L2 Aow = area of oil/water interface in a porous sample, L2 Aos = area of oil/solid interface in a porous sample, L2 Aws = area of water/solid interface in a porous sample, L2 C = parameter in the Land function e = basis of natural logarithm (2.718...) fi = fraction of total sample volume occupied by layer i in a laminated sample G = pore geometric factor in Thomeer function j ( Sw) = Leverett j-function k = permeability, L2, md ki = permeability of layer i in a laminated sample, L, md kT = total permeability in a laminated sample, L2, md krg = relative permeability for gas krgi = relative permeability for gas in layer i of a laminated sample krg, max = maximum relative permeability for gas in modified Brooks-Corey functions krg,go = relative permeability for gas in gas/oil flow for three-phase models krg,gw = relative permeability for gas in gas/water flow for three-phase models krg,wog = relative permeability for gas in oil/water/gas flow for three-phase models krgo = relative permeability for oil in layer i of a laminated sample krgT = total relative permeability for gas in a laminated sample kro = relative permeability for oil kroi = relative permeability for oil in layer i of a laminated sample kro, max = maximum relative permeability for oil in modified Brooks-Corey functions kro,og = relative permeability for oil in oil/gas flow for three-phase models kro,wo = relative permeability for oil in oil/water flow for three-phase models kro,wog = relative permeability for oil in oil/water/gas flow for three-phase models kroT = total relative permeability for oil in a laminated sample krw = relative permeability for water krw,max = maximum relative permeability for water in modified Brooks-Corey functions krw,gw = relative permeability for water in gas/water flow for three-phase models krw,wo = relative permeability for water in oil/water flow for three-phase models krw,wog = relative permeability for water in oil/water/gas flow for three-phase models L = length L, M = parameters in Chierici functions ng = gas exponent for modified Brooks-Corey functions no = oil exponent for modified Brooks-Corey functions nw = water exponent for modified Brooks-Corey functions N = number of laminations Nc = capillary number Pc = capillary pressure, m/Lt2, psi Pcgo = capillary pressure between gas and oil phases, m/Lt2, psi Pcow = capillary pressure between oil and water phases, m/Lt2, psi Pcgw = capillary pressure between gas and water phases, m/Lt2, psi Pcgwt = threshold capillary pressure between gas and water phases, m/Lt2, psi Pci = capillary pressure at inside face of sample in centrifuge test, m/Lt 2, psi Pct = threshold capillary pressure, m/Lt 2, psi Pcs = shape parameter in Bentsen-Anli function , m/Lt 2, psi pg = pressure in the gas phase, m/Lt 2, psi po = pressure in the oil phase, m/Lt 2, psi pw = pressure in the water phase, m/Lt 2, psi ΔP = pressure drop, m/Lt 2, psi q = flow rate, L3/t qo = flow rate of oil, L3/t qw = flow rate of water, L3/t r = radius to a point in sample in centrifuge test, L ri = radius to inside face of sample in centrifuge test, L ro = radius to outside face of sample in centrifuge test, L R1, R2 = principal radii of curvature, L Scc = critical saturation of condensate Sg = saturation of gas Sgc = critical saturation of gas Sgi = initial saturation of gas SgN = normalized gas saturation Sgr = residual saturation of gas Sgr,max = maximum residual or trapped gas saturation SgS = scaled saturation of gas for Stone I model Sgt = trapped saturation of gas SHg = saturation of mercury Snwr = residual saturation of nonwetting phase So = saturation of oil Som = minimum saturation of oil for three-phase models Sor = residual saturation of oil Sorg = residual saturation of oil for a gas/oil displacement Sorw = residual saturation of oil for a water/oil displacement SoS = scaled saturation of oil for Stone I model Sw = saturation of water Swc = critical saturation of water Swi = irreducible or residual saturation of water Swn = normalized water saturation Swr = residual saturation of water or wetting phase SwS = scaled saturation of water for Stone I model v = velocity, L/t, cm/s x = position in the x direction, L α = parameter in Eqs. 15.39 and 15.40 β = parameter in Eq. 15.40 θ = contact angle, degrees λ = pore-size-distribution parameter in Corey functions μ = viscosity, m/Lt, cp μo = viscosity of oil, m/Lt, cp μw = viscosity of water, m/Lt, cp Δρ = density difference for fluids in centrifuge tests, m/L3 σgw = gas/water interfacial tension, m/t 2, dyne/cm σow = oil/water interfacial tension, m/t 2, dyne/cm τ = tortuosity of porous sample τo = tortuosity of oil phase in porous sample τw = tortuosity of water phase in porous sample Φ = porosity ω = spin rate for centrifuge tests, 1/t (ω=2π×RPM/60)

## References

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## SI Metric Conversion Factors

 dyne × 1.0* E – 02 = mN ft × 3.048* E – 01 = m in. × 2.54* E + 00 = cm in.2 × 6.451 6* E + 00 = cm2 psi × 6.894 757 E + 00 = kPa

*

Conversion factor is exact.