Mixed Finite Element Method for Miscible Displacement Problems in Porous Media

Abstract

Abstract 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 porous medium. The new mixed finite-element procedure solves for both the pressure and velocity of the 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 = (-k/mu) delta p partial differential equations. By solving for u = (-k/mu) delta p as one term, we minimize the difficulties occurring in standard methods caused by differentiation or differencing of p and multiplication by rough coefficients k/mu. 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 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 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 Omega R2 over a time period J=[0, T]. If p is the pressure of the total fluid with viscosity mu in a medium with permeability k, we define the Darcy velocity of the total permeability k, we define the Darcy velocity of the total fluid by (1) Then, letting c denote the concentration of the invading fluid and phi denote the porosity of the medium, the coupled quasilinear system of partial differential equations describing the fluid flow is given by (2)(3) Hen q=q(x, t) represents the total flow into or out of the region omega (q greater than 0 at injection wells and q less than 0 at production wells in this setting), c=c(x, t) is equal to the value of c production wells in this setting), c=c(x, t) is equal to the value of c at a producing well and the specified inlet concentration at an injection well, and D=D(x, u) is the diffusion-dispersion tensor given by (4) where dm, a small molecular diffusion coefficient, and dc and dt, the magnitudes of longitudinal and transverse dispersion, are given constants. Here absolute value of mu is the standard Euclidean norm of the vector mu. To complete the description of the flow we augment the system (Eqs. 2 and 3) with a prescription of an initial concentration distribution of the invading fluid and no-flow conditions across the boundary, delta omega, given by (5)(6)(7) where vi are components of the outward normal vector to delta omega. Incompressibility requires that (8) SPEJ P. 391