An iterative perturbation method for the pressure equation in the simulation of miscible displacement in porous media

NOTE: THE MATHEMATICAL SYMBOLS IN THIS ABSTRACT CANNOT BE DISPLAYED CORRECTLY ON THIS PAGE. PLEASE REFER TO THE ABSTRACT IN THE PUBLISHER’S WEBSITE FOR AN ACCURATE DISPLAY. The miscible displacement problem in porous media is modeled by a nonlinear coupled system of two partial differential equations: the pressure-velocity equation and the concentration equation. An iterative perturbation procedure is proposed and analyzed for the pressure-velocity equation, which is capable of producing as accurate a velocity approximation as the mixed finite element method, and which requires the solution of symmetric positive definite linear systems. Only the velocity variable is involved in the linear systems, and the pressure variable is obtained by substitution. Trivially applying perturbation methods can only give an error $O(\epsilon)$, while our iterative scheme can improve the error to $O(\epsilon^m)$ at the $m$th iteration level, where $\epsilon$ is a small positive number. Thus the convergence rate of our iterative procedure is $O(\epsilon)$, and consequently a small number of iterations is required. Theoretical convergence analysis and numerical experiments are presented to show the efficiency and accuracy of our method. ©1998 Society for Industrial and Applied Mathematics