Covolume Method For Solute Spread In Porous Media Flow

Consider the single-pha.se flow in the porous media. The: solution contains certain solute which moves along with the fluid flow. The movement includes both convection and diffusion. The mathematical model to discribe the process is as follows:where p represents the fluid pressure and c. the solute concentration. The two equationsabove are called pressure equation and concentration equation respectively, while the concentration equation is relevant to the Darcy velocity u. In the pratical computation, the convergence rate decreases if we choose to obtain the approximate solution of velocity via the pressure gradient. Since the concentration equation em-ploys the Darcy velocity directly, the accuracy of Darcy velocity affects the overall computing result significantly. The MFEM (mixed finite element method) solves the pressure and the Darcy velocity simultaneously, while the accuracy of the approxi-mate velocity is guaranteed. There are many numerical methods for the model such as Galerkin method, characteristie FEM(finite element method), MFEM, etc.With the enlightment of MFEM, this paper transforms the pressure equation above into a div-curl system formally, then employs the covolume method to get the numerical solution of Darcy velocity, and proves the existence and uniqueness of the numerical solution and gives the error estimate. This method solves the velocity directly and needn't the work of solving the pressure any more. Then, solve the concentration equation from the conservative form using the approximate velocity obtained by the covolume method.[1] introduced a covolume method to handle with the div-curl equations which enipolyed the dual mesh and the idea of control volume. Different with FVEM(rinite volume element method), covolume method introduce a discrete vector field as well as the relevant field theory. Covolume method implements the properties of the field theory as tool to research the existence, uniqueness and error estimate of the numerical solution of div-curl system.This paper generalizes the covolume method in [1] to some extent to deal with the .solute spread problem in porous media flow and apply the covolume method to the pressure equation to obtain the numerical solution of Darcy velocity. Since the numerical solution is a vector, an appropriate transition is needed between the discrete solution and the continuous form when we use it to solve concentration equation. This paper carries out this transition by an interpolation operator via an analogue with the construction of MFEM space.This paper consists four diapers:The first chapter gives the introduction of covolume method for div-curl systemand the problem of solute spread in porous media flow and its usual numerical methods.The second chapter gives the notations and instructions of the meshes in planararea and introduces some important matrixes which represent the geometrical properties of the triangulation. The discretization scheme of the covolume method is indicated by these matrixes which set the basis to prove the existence and uniquenessof the numerical solution by the algebra techniques.The third chapter generalized the covolume method dealing with constant coefficientin [1] to that with variable coefficient. We employ the covolume method to the pressure equation to obtain the numerical solution u_h to the variable u and achieve error estimates. In the first section, we transform the pressure equation into the div-curl form by variable transition and give the relevant boundary conditions. The second section forms the algebra equations of the discretization scheme depending on the basic idea of covolume method. The third section gives the proof of existence and uniqueness of the covolume solution based on the discrete vector field theory. The fourth section gives the error estimate in the measure of discrete norm.In the fouth chapter, the fully-discrete scheme of the concentration equation is established and the error estimates are given. In the first section, we introduced an interpolation operator I_h, then got the interpolation I_hu_h, of Darcy velocity in the MFEM space (vector part) and analysed the error estimate of ||I_hu-I_hu_h||. In the second section we implement I_hu_h. into the computing scheme of the concentration equation to construct the fully-discrete scheme. In the error analysis, the following error decomposition is employed:Finally, we obtain the optimal L₂ error estimate with respect to the concentration triangulation parameter h_c.