Working Guide to Reservoir Engineering (eBook)

1.3.5 Capillary Pressure

Curvature at an interface between wetting and nonwetting phases causes a pressure difference that is called capillary pressure. This pressure can be viewed as a force per unit area that results from the interaction of surface forces and the geometry of the system.

Based on early work in the nineteenth century of Laplace, Young, and Plateau (e.g., Reference 94), a general expression for capillary pressure, Pc, as a function of interfacial tension, σ, and curvature of the interface is [19]:

c=σ1r1+1r2

where r1 and r2 are the principal radii of curvature at the interface. These radii are not usually measured, and a mean radius of curvature is given by the capillary pressure and interfacial tension.

For a cylindrical vertical capillary, such as a small tube, the capillary pressure for a spherical interface is [19]:

c=2σcosθcr=ghρ1−ρ2

where r is the radius of the tube, θ1 is the contact angle measured through the more dense phase that exists between the fluid and the wall of the tube, g is the gravitational constant, ρ is density, h is column height, and the subscripts refer to the fluids of interest. For a fluid that wets the wall of a capillary tube, the attraction between the fluid and the wall causes the fluid to rise in the tube. The extent of rise in the capillary is proportional to the interfacial tension between the fluids and the cosine of the contact angle and is inversely proportional to the tube radius.

An analogous situation can occur during two-phase flow in a porous medium. For example if capillary forces dominate in a water-wet rock, the existing pressure differential causes flow of the wetting fluid to occur through the smaller capillaries. However, if viscous forces dominate, flow will occur through the larger capillaries (from Pouiselle’s law, as a function of the 4th power of the radius).

Figure 1.46 depicts a typical capillary pressure curve for a core sample in which water is the wetting phase. Variation of capillary pressure is plotted as a function of water saturation. Initially, the core is saturated with the wetting phase (water). The nonwetting phase, oil in this case, is used to displace the water. As shown in the figure, a threshold pressure must be overcome before any oil enters the core. The initial (or primary) drainage curve represents the displacement of the wetting phase from 100% saturation to a condition where further increase in capillary pressure causes little or no change in water saturation. This condition is commonly termed the irreducible saturation, Siw. The imbibition curve reflects the displacement of the nonwetting phase (oil) from the irreducible water saturation to the residual oil saturation. Secondary drainage is the displacement of the wetting phase from the residual oil saturation to the irreducible water saturation. A hysteresis is always noted between the drainage and imbibition curves. Curves can be obtained within the hysteresis loop by reversing the direction of pressure change at some intermediate point along either the imbibition or secondary drainage curve. The nonuniform cross-section of the pores is the basic cause of the hysteresis in capillary pressure observed in porous media. Therefore, capillary pressure depends on pore geometry, interfacial tension between the fluids, wettability of the system (which will be discussed later in this chapter), and the saturation history in the medium.

Leverett [100] introduced a reduced capillary pressure function (subsequently termed the Leverett J function by Rose and Bruce [127]) that was suggested for correlating capillary pressure data:

Sw=Pcσcosθckϕ1/2

Sw=thecorrelatinggroupconsistingofthetermsofEquation1.75k=thepermeabilityϕ=porosityofthesample

The J function was originally proposed as means of converting all capillary pressure data for clean sand to a universal curve. A series of capillary pressure curves are shown as a function of permeability in Figure 1.47 [20]. An example of the J function curve generated from these data is shown in Figure 1.48 [20]. While the J function sometimes correlates capillary pressure data from a specific lithology within the same formation, significant variations can be noted for different formations.

Common laboratory methods of measuring capillary pressure include [19]: mercury injection, porous diaphragm or plate (restored state), centrifuge method, and steady-state flow in a dynamic method. While the restored state test is generally considered the most accurate, mercury injection is routinely used. However, it is necessary to correct the mercury injection data for wetting conditions before comparison to results from the restored state test.

A very valuable use of capillary pressure data is to indicate pore size distribution. Since the interfacial tension and contact angle remain constant during a test such as already described, pore sizes can be obtained from capillary pressures. For rocks with more uniform pore sizes, capillary pressure curves will be close to horizontal. The slope of the capillary pressure curve will generally increase with broader pore-size distribution.

If laboratory capillary pressure data are corrected to reservoir conditions, the results can be used for determining fluid saturations. Figure 1.49 shows a close agreement in water saturations obtained from capillary pressure and electric logs [48].

Capillary pressure data are helpful in providing a qualitative assessment of the transition zones in the reservoir. A transition zone is defined as the vertical thickness where saturation changes from 100% water to irreducible water for water-oil contact, or from 100% liquid to an irreducible water saturation for gas-oil contact.

1.3.6 Effective Permeability

In the previous section, “Absolute Permeability,” it was stated that permeability at 100% saturation of a fluid (other than gases at low pressure) is a characteristic of the rock and not a function of the flowing fluid. Of course, this implies that there is no interaction between the fluid and the rock (such as interaction between water and mobile or swelling clays). When permeabilities to gases are measured, corrections must be made for gas slippage which occurs when the capillary openings approach the mean free path of the gas. Klinkenberg [128] observed that gas permeability depends on the gas composition and is approximately a linear function of the reciprocal mean pressure. Figure 1.50 shows the variation in permeability as a function of mean pressure for hydrogen, nitrogen, and carbon dioxide. Klinkenberg found that by extrapolating all data to infinite mean pressure, the points converged at an equivalent liquid permeability (kℓ), which was the same as the permeability of the porous medium to a nonreactive single-phase liquid. From plots of this type, Klinkenberg showed that the equivalent liquid permeability could be obtained from the slope of the data, m, the measured gas permeability, kg, at a mean flowing pressure ¯, at which kg was observed:

ℓ=kg1+b/p¯=kg−mp¯

where b is a constant for a given gas in a given medium and is equal to m divided by kℓ. The amount of correction, known as the Klinkenberg effect, varies with permeability and is more significant in low permeability formations.

In studies [129, 130] with very low permeability sandstones, liquid per-meabilities were found to be less than gas permeabilities at infinite mean pressure, which is in contrast with the prior results of Klinkenberg. Furthermore, it has been shown [130] that liquid permeabilities decreased with increasing polarity of the liquid. For gas flow or brine flow in low-permeability sandstones, permeabilities were independent of temperature at all levels of confining pressure [130]. The data [130] showed that for a given permeability core sample at a given confining pressure, the Klinkenberg slip factors and slopes of the Klinkenberg plots were proportional to the product of viscosity and the square root of absolute temperature.

As shown in Figure 1.51 permeability of reservoir rocks can decrease when subjected to overburden pressure [131]. When cores are retrieved from a reservoir, the confining forces are removed and the rock can expand in all directions which can increase the dimensions of the available flow paths. In reservoirs where this is significant, it is imperative that permeability measured in the laboratory be conducted at the confining pressure that represents the overburden pressure of the formation tested.

As a general trend,...