%0 Journal Article
%T Parallel Nonoverlapping DDM Combined with the Characteristic Method for Incompressible Miscible Displacements in Porous Media
%A Keying Ma
%A Tongjun Sun
%J Advances in Numerical Analysis
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/303952
%X Two types of approximation schemes are established for incompressible miscible displacements in porous media. First, standard mixed finite element method is used to approximate the velocity and pressure. And then parallel non-overlapping domain decomposition methods combined with the characteristics method are presented for the concentration. These methods use the characteristic method to handle the material derivative term of the concentration equation in the subdomains and explicit flux calculations on the interdomain boundaries by integral mean method or extrapolation method to predict the inner-boundary conditions. Thus, the velocity and pressure can be approximated simultaneously, and the parallelism can be achieved for the concentration equation. The explicit nature of the flux prediction induces a time step limitation that is necessary to preserve stability. These schemes hold the advantages of nonoverlapping domain decomposition methods and the characteristic method. Optimal error estimates in -norm are derived for these two schemes, respectively. 1. Introduction The two-phase fluid displacements in porous media is one of the most important basic problems in the oil reservoir numerical simulation. It is governed by a nonlinear coupled system of partial differential equations with initial and boundary values. In this paper, we will consider the following incompressible miscible case: the pressure is governed by an elliptic equation and the concentration is governed by a convection-diffusion equation [1¨C5]. where is a bounded domain in , , and is nonzero at injection wells only. The variables in (1a)¨C(1d) are the pressure in the fluid mixture, the Darcy velocity , and the relative concentration of the injected fluid. The is the unit outward normal vector on boundary . The coefficients and data in (1a)¨C(1d) are : the permeability of the porous media; , the viscosity of the fluid mixture; : representing flow rates at wells; and , the gravity coefficient and vertical coordinate; : the porosity of the rock; , the injected concentration at injection wells ( ) and the resident concentration at production wells ( ). Here, is a tensor matrix and generally has the form where matrix satisfies , , is the molecular diffusivity, and , are longitudinal and transverse dispersivities, respectively. Furthermore, a compatibility condition must be imposed to determine the pressure. The pressure equation is elliptic and easily handled by standard mixed finite element method, which has been proven to be an effective numerical method for solving fluid problems. It has
%U http://www.hindawi.com/journals/ana/2013/303952/