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# Capillary pressure

Relative permeability and capillary pressure defined capillary pressure as the difference in pressure across the interface between two phases, Such as capillary pressure in a soap bubble system. For immiscible fluids distribution in a porous medium, capillary pressure has been defined as the pressure differential between two immiscible fluid phases occupying the same pores caused by interfacial tension between the two phases that must be overcome to initiate flow. This page discusses capillary pressure forces.

## Capillary pressure equation

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:

....................(1)

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 in Measurement of capillary pressure and relative permeability.

## Capillary pressure behavior

Fig. 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. 2 and 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
• "Imbibition" refers to the increasing wetting-phase saturation

Thus, the capillary pressure relationship in Fig. 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. 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. 1 is fictitious, as suggested by Muskat[1]—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[2] 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. 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. 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. 4. For a less strongly wetting phase, the capillary pressure reaches zero at a lower saturation, as shown in Fig. 5. Capillary pressure behavior for secondary drainage is also shown in Figs. 4 and 5.

### Wettability of porous material

As shown in Figs. 4 and 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,[3] 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.[4] 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 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.[4] 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 above.

### 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. 6.[4] [5]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. 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. 7, as estimated with a simple model for a laminated material: the Brooks-Corey expression (Eq. 4 in Capillary pressure models) 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. 1 in Capillary pressure models). 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. 8.

### Wettability

As reported by Bethel and Calhoun,[6] wettability affects the position of capillary pressure curves, as shown in Fig. 9 for displacement of oil (starting at So = 100%) by water from a glass-bead pack. The contact angles in the legend of Fig. 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. 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%.
• When strongly oil-wet, Sor returns to approximately 15%.

Morrow[7] 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.[8] 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[9] report the imbibition and secondary-drainage data of Fig. 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.

## Nomenclature

 po = pressure in the oil phase, m/Lt 2, psi pw = pressure in the water phase, m/Lt 2, psi Pcow = capillary pressure between oil and water phases, m/Lt2, psi R1, R2 = principal radii of curvature, L σow = oil/water interfacial tension, m/t 2, dyne/cm

## References

1. Muskat, M. 1949. Calculation of Initial Fluid Distributions in Oil Reservoirs. Trans. of AIME 179 (1): 119-127. http://dx.doi.org/10.2118/949119-G
2. Morrow, N.R. and Melrose, J.C. 1991. Application of Capillary Pressure Measurements to the Determination of Connate Water Saturation. In Interfacial Phenomena in Petroleum Recovery, 257-287, ed. N.R. Morrow. New York City: Marcel Dekker Inc
3. Salathiel, R.A. 1973. Oil Recovery by Surface Film Drainage in Mixed-Wettability Rocks. J Pet Technol 25 (10): 1216–1224. SPE-4104-PA. http://dx.doi.org/10.2118/4104-PA
4. Treiber, L.E. and Owens, W.W. 1972. A Laboratory Evaluation of the Wettability of Fifty Oil-Producing Reservoirs. SPE J. 12 (6): 531–540. SPE-3526-PA. http://dx.doi.org/10.2118/3526-PA
5. Morrow, N.R., Cram, P.J., and McCaffery, F.G. 1973. Displacement Studies in Dolomite with Wettability Control by Octanoic Acid. SPE J. 13 (4): 221–232. SPE-3993-PA. http://dx.doi.org/10.2118/3993-PA
6. Bethel, F.T. and Calhoun, J.C. 1953. Capillary Desaturation in Unconsolidated Beads. J Pet Technol 5 (8): 197-202. SPE-953197-G. http://dx.doi.org/10.2118/953197-G
7. Morrow, N.R. 1970. Irreducible wetting-phase saturations in porous media. Chem. Eng. Sci. 25 (11): 1799–1818. http://dx.doi.org/10.1016/0009-2509(70)80070-7
8. Chatzis, I., Morrow, N.R., and Lim, H.T. 1983. Magnitude and Detailed Structure of Residual Oil Saturation. SPE J. 23 (2): 311–326. SPE-10681-PA. http://dx.doi.org/10.2118/10681-PA
9. Jerauld, G.R. and Rathmell, J.J. 1997. Wettability and Relative Permeability of Prudhoe Bay: A Case Study In Mixed-Wet Reservoirs. SPE Res Eng 12 (1): 58–65. SPE-28576-PA. http://dx.doi.org/10.2118/28576-PA