A Model for Two-Phase Flow in Consolidated Materials

Abstract

A consolidated porous medium is mathematically modeled by networks of irregularly shaped interconnected pore channels. Mechanisms are described that form residual saturations during immiscible displacement both by entire pore channels being bypassed and by fluids being isolated by the movement of an interface within individual pore channels. This latter mechanism is shown to depend strongly on pore channel irregularity. Together, these mechanisms provide an explanation for the drainage-imbibition-hysteresis effect. The calculation of steady-state relative permeabilities, based on a pore-size distribution permeabilities, based on a pore-size distribution obtained from a Berea sandstone, is described. These relative permeability curves agree qualitatively with curves that are generally accepted to be typical for highly consolidated materials. In situations where interfacial effects predominate over viscous and gravitational effects, the following conclusions are reached.Relative permeability at a given saturation is everywhere independent of flow rate.Relative permeability is independent of viscosity ratio everywhere except at very low values of wetting phase relative permeability.Irreducible wetting-phase saturation following steady-state drainage decreases with increasing ratio of nonwetting- to wetting-phase viscosity.Irreducible wetting-phase saturation following unsteady-state drainage is lower than for steady-state drainage.Irreducible nonwetting-phase saturation following imbibition is independent of viscosity ratio, whether or not the imbibition is carried out under steady- or unsteady-state conditions. Experiments qualitatively verify the conclusions regarding unsteady-state residual wetting-phase saturation. Implications of these conclusions are discussed. Introduction Natural and artificial porous materials are generally composed of matrix substance brought together in a more or less random manner. This leads to the creation of a network of interconnected pore spaces of highly irregular shape. Since the pore spaces of highly irregular shape. Since the geometry of such a network is impossible to describe, we can never obtain a complete description of its flow behavior. We can, however, abstract those properties of the porous medium pertinent to the type of flow under consideration, and thus obtain an adequate description of that flow. Thus, the Kozeny-Carmen equation, by considering a porous medium as a bundle of noninterconnecting capillary tubes, provides an adequate description of single-phase provides an adequate description of single-phase flow. With the addition of a saturation-dependent tortuosity parameter in two-phase flow to account for flow path elongation, the Kozeny-Carmen equation has been used to predict relative permeabilities for the displacement of a wetting permeabilities for the displacement of a wetting liquid by a gas. It has long been recognized that relative permeability depends not only on saturation but permeability depends not only on saturation but also on saturation history as well. Naar and Henderson described a mathematical model in which differences between drainage and imbibition behavior are explained in terms of a bypassing mechanism by which oil is trapped during imbibition. Fatt proposed a model for a porous medium that consisted of regular networks of cylindrical tubes of randomly distributed radii. From this he calculated the drainage relative permeability curves. Moore and Slobod, Rose and Witherspoon, and Rose and Cleary each considered flow in a pore doublet (a parallel arrangement of a small and pore doublet (a parallel arrangement of a small and large diameter cylindrical capillary tube). They concluded that, because of the different rates of flow in each tube, trapping would occur in one of the tubes; the extent of which would depend upon viscosity ratio and capillary pressure. SPEJ p. 221

AN EXAMINATION OF RELATIVE PERMEABILITY RELATIONS FOR TWO‐PHASE FLOW IN POROUS MEDIA1

“…Odeh (1959), on the other hand, found a sizeable enhancement of km at residual saturations of the wetting phase (see Figure 4), and this effect was further amplified in porous media having low intrinsic permeability. Ehrlich and Crane's (1969) data showed a similar effect. Lefebvre du Prey's (1973) results also showed evidence of an enhancement of the kr of the more viscous phase (regardless of which was the wetting phase), but the enhancement was not as dramatic as that found by Odeh (1959).…”

“…That interfacial tension should have an effect on two-phase flow can be seen from the Laplace-Young equation: Leverett (1939) 0.057-90.0 3.2.6 .8 Leverett and Lewis (1941) 1.86-20.2 (for liquids) 5.4-16.2 Sandberg (1958) 0.48-2.02 0.413-0.757 Odeh (1959) 0.44-82.7 0.0021-0.405 Donaldson, et al (1966) 35-' 78 0.76-1.20 Ehrlich and Crane (1969) 0 Equation (6) states that the capillary pressure is directly proportional to the interfacial tension. Capillary pressure is responsible for the segregation of the wetting liquid in the small pores and the nonwetting in the large pores, so the interfacial tension would also affect the liquid distribution and thus the relative permeability.…”

Three‐dimensional, randomized, network model for two‐phase flow through porous media

“…Rather than permitting the individual pores to be randomly oriented in space, these models identify the directions of the pores with the directions of bonds in a network, although equality of pore pressures and conservation of mass at nodes are not recognized. Ehrlich and Crane (1969) demonstrated that hysteresis can be described qualitatively with a network of capillaries having interconnections. They constructed a simple four-pore series-parallel network to compute drainage and imbibition relative permeability curves from a given pore size distribution.…”

Alkaline Waterflooding for Wettability Alteration-Evaluating a Potential Field Application

“…21 This argument indicates that the oil recovery improvement obtainable from wettabiIity alteration depends on the amount of continuous oil present at the time the wetting change occurs. Altering contact angle before or during formation of these droplets alters their configuration and causes residual oil saturation to be different from that of an unaltered displacement.…”