Efficient Time-Stepping Methods for Miscible Displacement Problems in Porous Media

Abstract

Efficient procedures for time-stepping Galerkin methods for approximating the solution of a coupled system for $c = c(x,t)$ and $p = p(x,t)$, with nonlinear Neumann boundary conditions, of the form \[ \begin{gathered} - \nabla \cdot [a(x,c)\{ \nabla p - y(x,c)\nabla g\} ] \equiv \nabla \cdot u = f_1 (x,t),\quad x \in \Omega ,\quad t \in (0,T], \hfill \\ \nabla \cdot [b(x,c,\nabla p)\nabla c] - u \cdot \nabla c = \phi (x)\frac{{\partial c}} {{\partial t}} - f_2 (x,t,c),\quad x \in \Omega ,\quad t \in (0,T], \hfill \\ u \cdot v = q_1 (x,t),\quad x \in \partial \Omega ,\quad t \in (0,T], \hfill \\ b\frac{{\partial c}} {{\partial v}} = q(x,t,c),\quad x \in \partial \Omega ,\quad t \in (0,T], \hfill \\ c(x,0) = c_0 (x),\quad x \in \Omega , \hfill \\ \end{gathered} \] where $\Omega \subset \mathbb{R}^d $, $2 \leqq d \leqq 3$, are presented and analyzed. This system is a possible model system for describing the miscible displacement of one incompressible fluid by another in a porous medium when flow conditions are prescribed on the boundary. The procedures involve the use of a preconditioned iterative method for approximately solving the algebraic problem at each time step. The iteration need be performed only long enough to stabilize the scheme. Motivated by the fact that the pressure is smoother in time than the concentration, larger time steps are used for the pressure than for the concentration. Under certain smoothness assumptions on the solution, optimal order convergence rates and almost optimal order work estimates are obtained.