Convenient Formulations For Immiscible Displacement In Porous Media
Authors: Civan Faruk 
Published: December 2007

Abstract
Convenient formulations of the immiscible displacement in porous media are presented and applied for waterflooding The macroscopic equation of continuity for immiscible displacement is derived by porous media averaging. Richardson's approach and the fractional flow formulation are extended and generalized for anisotropic and heterogeneous porous media. The integral transformations according to Douglas et al and the coordinate transformations presented in this paper lead to differential equations which do not involve the variable fluid and porous media properties explicitly in the differential operators. Fractional flow and unit end-point mobility ratio formulations are also derived for specific applications to reduce the computational requirements and accomplish rapid simulation of waterflooding of petroleum reservoirs. It is demonstrated by typical examples that the resulting equations can be discretized and solved more conveniently and accurately than the conventional formulation which require cumbersome discretization formulae for mixed derivatives involving the fluid and porous media properties. Therefore, the convenient formulations offer potential advantages over the usual formulation used in the simulation of waterflooding such as improved accuracy and reduced computational effort.

Introduction
Formulations of the governing equations and boundary conditions for immiscible displacement in porous media are of continuing interest. Neglecting the capillary pressure effect, Buckley and Leverett obtained a simplified formulation. This equation can be solved analytically using Welge's method in an isotropic homogeneous media for one-dimensional linear, radial, and spherical flow. When the capillary pressure effect is included, the formulation results in a convection-dispersion type equation implying that two boundary conditions are necessary. Therefore, the majority of the reported studies have attempted formulating and implementing two boundary conditions. Whereas, as shown in the present study, two boundary conditions are only necessary for compressible fluids and one boundary condition is sufficient for incompressible fluids.
In this study, first, a rigorous derivation of the macroscopic equation of continuity for immiscible displacement in porous media is carried out. The resulting macroscopic equation contains a new term expressing dispersive fluid transport due to hydraulic gradients. This term disappears only for incompressible fluids. Second, the formulations are applied to water/oil systems and the generalized boundary conditions are derived. Third, convenient formulations are presented by transforming the governing equations by means of an integral transformation of the function and a transformation of the coordinates. In the convenient formulations, "the variable properties do not appear explicitly" and numerical solutions of "better accuracy with less computational effort" can be obtained, as stated by Civan and Sliepcevich. The fractional flow and unit end-point mobility ratio formulations are also presented for specific applications.

Macroscopic Equation Of Continuity
The microscopic equation of continuity for a phase in a pore volume surrounded by a porous matrix is given by:
(1)
For fluids in porous media, a macroscopic equation of continuity can be derived by averaging Eq. 1 over a representative elemental volume of porous media (Civan). Thus, consider the following averaging rules, according to Slattery and Marle:
(2)
(3)
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