Expanded Mixed Method For Slightly Compressible Miscible Displacement In Heterogeneous Porous Media

The pressure equation with full tensor permeasibility is often derived by multiscale or upscaling skill when we solve miscible displacement in hetergeneous porous media.Traditional mixed finite element methods need the inverse matrix of permeasibility in the error analysis,and are available for miscilble displacement in homogeneous porous media,whose permeasiblity is dialognal.When porous media is hetergeneous,we will use expanded mixed method for pressure equaion.In this article,we will use expanded mixed method to solve the pressure equation with full tensor coefficients,and traditional finite element method to solve the concentrate equation.Expanded mixed method defines another variable-the divergence of pressure to solve the pressure equation,and get three equations:the equation for pressure and the divergence of pressure,the equation for the divergence of pressure and velocity,and velocity equation.In chapter 2,we get the semi-discrete schemes of expanded mixed method to solve the pressure equation and traditional finite element method to solve the concentrate equation.In chapter 3,we give the error analysis of the semi-discrete schemes given in chapter 2.We first define a project operator and then get the error for the project operator in and Lemma 3.1.Finally,in theorem 3.1,we get the error estimate in L~2 norm for pressure,the divergence pressure,the velocity,and the concentrate of expanded mixed method to solve the slightly compressible miscible displacement in heterogeneous porous media. Cell-centered 5-points finite difference scheme which is widely used in petroleum industry and is super-convergence in a discrete norm,can be derived from mixed finite element method if we use numerical integral to the inner product in traditional mixed finite element method.We can alse get cell-centered 9-points finite difference scheme if we use numerical integral to the inner product in expanded mixed method,which was also super-convergence in the discrete norm for elliptic equation[2].In chapter 4,we use the numerical integral to the inner product of the expanded mixed method for pressure equation in chapter 2,and get the integral scheme of cell-centered 9-points finite difference scheme for pressure equation with full tensor permeability.And then,we chooose the basic function as test function in the discrete inner product of integral scheme and get three difference scheme:the equation for pressure and the divergence of pressure,the equation for the divergence of pressure and velocity,and velocity equation.In the equation for pressure and the divergence of pressure,the coefficient matrix for the divergence of pressure is unitmatrix. In the equation for the divergence of pressure and velocity,the coefficient becomes dialoganal from 5-dialoganal beacause we institute the discrete inner product with the inner product.So the three finite difference equations can be merged easily into the cell-centered 9-points finite difference scheme with just pressure variable. we use traditional finite difference scheme to the concentrate equation.In chapter 5,we give the error analysis for intergral scheme of cell-centered 9-points finite difference method for pressure equation and traditional finite difference method for concentrate equation.We define the project operator in discrte inner product for pressure equation,and then use a differnent method to get the error of the project operator in Lemma 5.1-Lemma 5.4 and.Finally,in theorem 5.1,we get the error estimate of cell-centered 9-points finite difference scheme for pressure equation and traditional finite difference scheme for concentrate equation,which is super-convergence for pressure.In the end of this article,in chapter 6,we give an easy numerical experiment to varity the super-convergence for cell-centered 9-points finite difference scheme.