Mixed Finite Element Method for Miscible Displacement Problems in Porous Media

Effective numerical simulation of many EOR problems requires very accurate approximation of the Darcy velocities of the respective fluids. In this paper we describe a new method for the accurate determination of the Darcy velocity of the total fluid in the miscible displacement of one incompressible fluid by another in a porous medium. The new mixed finite-element procedure solves for both the pressure and velocity of the total fluid simultaneously as a system of first-order partial differential equations. By solving for u=( -klfJ-) \7p as one term, we minimize the difficulties occurring in standard methods caused by differentiation or differencing of p and multiplication by rough coefficients kl fJ-. By using mixed finite elements for the pressure equation coupled in a sequential method with a finite element procedure for the concentration of the invading fluid, we are able to treat a variety of problems with variable permeabilities, different mobility ratios, and a fairly general location of injection and production wells. Mixed finite-element methods also· produce minimal grid-orientation effect. Computational results on a variety of two-dimensional (2D) problems are presented. Introduction This paper considers the miscible displacement of one incompressible fluid by another in a horizontal reservoir n c R2 over a time period 1 = [0, T]. If p is the pressure of the total fluid with viscosity fJin a medium with permeability k, we define the Darcy velocity of the total fluid by k u=--\7p . ............................. (1) It Then, letting c denote the concentration of the invading fluid and rjJ denote the porosity of the medium, the coupled quasilinear system of partial differential equations describing the fluid flow is given by 1-3 \7. ( ~ \7p) == \7'u=q, XEn, tEl, ......... (2) 0197·7520/84/0081-0501$00.25 Copyright 1984 Society of Petroleum Engineers of AIME