Fundamentals of Fluid Flow in Porous Media

Chapter 5

Miscible Displacement

The Equation of Continuity in Porous Media: Solutions to the One-Dimensional Convection-Dispersion Model

Factors Influencing Capacitance Model Parameters

Mass transfer into the stagnant volume is often assumed to take place by molecular diffusion, and, in some models, has been described by the unsteady diffusion equation (Turner, 1958; Gottschlich, 1963; Shearn and Wakeman, 1978). Jasti et al. (1987) have argued that mass transfer should be proportional to molecular diffusivity in the absence of eddies, and Stalkup (1970) has shown experimentally that mass transfer is strongly dependent on molecular diffusivity. However, a large amount of experimental work from different investigators has shown that the mass transfer coefficient, obtained by fitting the Coats and Smith model to experimental data, increases with increasing velocity (Baker, 1977; van Genuchten and Wierenga, 1977; Ramirez et al., 1980; Spence and Watkins, 1980; Batycky et al., 1982; Bretz and Orr, 1987; Patel and Greaves, 1987). The velocity dependence of the mass transfer coefficient indicates that mixing takes place by mechanisms other than pure molecular diffusion, and possibly indicates the inadequacy of a model that divides the pore space into a fraction that contributes to flow and one that is completely stagnant. Some flow which is likely to be present in the “stagnant” pore space or the presence of eddies may contribute to mixing. In the presence of a second, immiscible phase, the morphology of each phase depends on its saturation, and the mass transfer coefficient thus also depends on phase saturation (Ramirez et al., 1980; Batycky et al., 1982; Salter and Mohanty, 1982) and on interphase mass transfer (Shearn and Wakeman, 1978).

The flowing fraction depends on pore morphology and, in two-phase systems, on the morphology of the phase in question, this in turn being determined by wettability and phase saturation. The flowing fraction in sandstones containing a single fluid phase is frequently between 0.9 and 1.0 (Coats and Smith, 1964; Baker, 1977; Ramirez et al., 1980; Spence and Watkins, 1980; Batycky et al., 1982; Sorbie et al., 1985; Bretz et al., 1986; Patel and Greaves, 1987). The flowing fraction in carbonates can be much lower (as low as 0.38), reflecting the dual pore structure often encountered in carbonate rocks (Baker, 1977; Spence and Watkins, 1980; Batycky et al., 1982; Bretz et al., 1986; Bretz and Orr, 1987). Conflicting results regarding the dependence on velocity have been reported: some investigators have found f to be independent of velocity (Coats and Smith, 1964; Baker, 1977; Ramirez et al., 1980; Jasti et al., 1987; Patel and Greaves, 1987), while others have reported a decrease in f with increasing velocity (Spence and Watkins, 1980; Batycky et al., 1982; Bretz and Orr, 1987). Van Genuchten and Wierenga (1977) and Batycky et al. (1982) report an increase in f with increasing velocity for certain systems. The velocity dependence of f likely depends on porous medium structure. In a heterogeneous, poorly connected medium, such as a vuggy limestone, velocity contrasts in different regions of the pore space might be expected to increase with increasing velocity, resulting in a decrease in the flowing fraction. On the other extreme, in a relatively homogeneous porous medium an increase in velocity may lead to better mixing and a lower stagnant fraction.

In two-phase systems, the flowing fraction of a phase is expressed as a fraction of the phase saturation. Several investigators have found the flowing portion of one phase to decrease when the saturation of that phase decreases (Stalkup, 1970; Shuler, 1978; Batycky et al., 1982), while others found no correlation between flowing fraction and saturation (Ramirez et al., 1980). The most comprehensive study on this aspect of the capacitance model has been conducted by Salter and Mohanty (1982), who found that the flowing fraction of a non-wetting phase increases as its saturation increases, this phase becoming increasingly more connected. By contrast, the flowing fraction in the wetting phase goes through a minimum when the wetting phase saturation increases. The reason for this may be a transition of wetting phase flow through fine pores and films surrounding the grains, to simultaneous flow of wetting and non-wetting phases, and finally to flow of the wetting phase through the larger flow channels. These transitions between flow regimes parallel changes from irreducible wetting phase saturation at one extreme to residual non-wetting phase saturation at the other.