The MMOCAA Method For Compressible Miscible Displacement In Porous Media

Petroleum reservoir, seawater intrusion and many other physical phenomena are essentially convection-dominated flows in underground porous media. Numerical simulationsof these problems involve solving nonlinear convention-dominated diffusion equations and usually face considerable practical difficulties. Convection-dominated diffusion equation possesses property of hyperbolic ones. Standard numerical methodssolving these equations often produce nonphysical oscillations, and upwind methodsusually cause excessive numerical diffusion which smear the sharp fronts. To seek numerical methods for such problems that reflect their almost hyperbolic nature, Douglas et al. developed the so-called modified method of characteristics(MMOC), which involves time stepping along the characteristics of the hyperbolic part of the parabolic equations. The MMOC procedure, however, fails to preserve integral conservationlaws. Its variant, called the modified method of characteristics with adjust advection(MMMOCAA) preserves the integral conservation law at a minor additional computation cost.In this paper, we consider the miscible displacement of one compressible fluid by another in the domainΩ, which is modeled by the system:The main work of this paper is to apply the MMOCAA mixed finite element method to the above problem. We analyze the convergence properties and achieve the optimal error estimates in the H~1 norm and the suboptimal error estimates in the L~2 norm.This paper consists of four chapters.In Chapter 1, the mathmatical model is introduced and the fundamental thoughts are explained. In Chapter 2, we propose the MMOCAA formulation of the concentrationequation of compressible miscible displacement in porous media. In Chapter 3, a full-time MMOCAA mixed finite element, discretization scheme is presented and the convergence properties arc analyzed. In Chapter 4, we propose a MMOCAA economicaldifference scheme for linear convection-diffusion equations, analyze the stability and convergence and achieve the optimal error estimates in the H~1 norm based on linear interpolation.