The mathematical model for two-phase displacement in porous media is a coupled initial boundary value problem of nonlinear partial differential equations which consist of a pressure equation and a saturation equation....The mathematical model for two-phase displacement in porous media is a coupled initial boundary value problem of nonlinear partial differential equations which consist of a pressure equation and a saturation equation. In this paper, the mixed least-square weak form of pressure equation is got, and the positive definite characteristics of the weak form is proved. Based on this weak form, a new kind of numerical methods for two-phase displacement problems is proposed: the mixed least-square finite element method is used to solve pressure and Darcy velocity, and the saturation is solved by using characteristic finite element method. The main merits of the mixed least-square finite element method compared with mixed finite element method are: first, the structure of the mixed least-square finite el- ement spaces is just standard finite element spaces, it is simple and easy to use; second, the weak form of the mixed least-square finite element method for pressure is symetric and definite positive, thus there are many efficient methods to solve numerically; and the last, the Darcy velocity solved by mixed least-square finite element method is continuous. In numerical analyses, a very important inequality is obtained which is used to control the errors of the pressure and Darcy velocity, and the optimal error estomates of the proposed method are proved.展开更多
文摘The mathematical model for two-phase displacement in porous media is a coupled initial boundary value problem of nonlinear partial differential equations which consist of a pressure equation and a saturation equation. In this paper, the mixed least-square weak form of pressure equation is got, and the positive definite characteristics of the weak form is proved. Based on this weak form, a new kind of numerical methods for two-phase displacement problems is proposed: the mixed least-square finite element method is used to solve pressure and Darcy velocity, and the saturation is solved by using characteristic finite element method. The main merits of the mixed least-square finite element method compared with mixed finite element method are: first, the structure of the mixed least-square finite el- ement spaces is just standard finite element spaces, it is simple and easy to use; second, the weak form of the mixed least-square finite element method for pressure is symetric and definite positive, thus there are many efficient methods to solve numerically; and the last, the Darcy velocity solved by mixed least-square finite element method is continuous. In numerical analyses, a very important inequality is obtained which is used to control the errors of the pressure and Darcy velocity, and the optimal error estomates of the proposed method are proved.
