Operator Splitting Discontinuous Galerkin Method Of Miscible Displacement In Porous Media

This paper introduces the miscible displacement problem first, and then gives two operator-splitting methods: Viscosity-Splitting method for concentration equation of miscibledisplacement with solute absorption and Operator-Splitting discontinuous Galerkin method.A lot of chemical reagents, such as buck, polymer and external active solvent, have been used to improve recovery ratio in petroleum exploitation, this process refers to solute absorption problem. On the condition of incompressible fluid, its mathematical scheme can be put in the form which is coupling nonlinear partial difference equations about pressure p(x, t), concentration c(x,t), and solute concentration c_r(x,t) absorbing on the media surface[2,4]:In chapter two, the concretely Viscosity-Splitting scheme has been given. Based on error analysis, the following theorem has been gotten:Theorem 2.1 Let h_p, h_lc,h_mc is parameters to pressure and two concentration spaces respectively,s,l,m is the index of pressure Raviart-Thomas space and two concentrationGlerkin spaces respectively.. Let (Cⁿ⁺¹,(?),C_rⁿ⁺¹,Uⁿ⁺¹,Pⁿ⁺¹) is the solution of viscosity-splitting scheme, assume c(t),u(t),p(t),c_r(t)are sooth enough, and when l≥1, m≥1, s≥1, h_lc=o(â–³t),â–³t=o(h_2c), h_2c=O(h_p), we get:In chapter three, the first part is to extend Operator-Splitting method in convective-diffusionequations, and we get the computational scheme and error estimate. Based on the forgoing analysis, we apply Operator-Splitting discontinuous Galerkin method to miscible displacement problems, aiming at the problems:The concentration equation can be divided into two equations. One of them is first-orderhyperbolic equation which is solved by discontinuous Galerkin method, the other is heat-conduction equation which is solved by normal finite element method. L² error estimate has be analyzed and we get the conclusion:Theorem 3.5 Let h_p, h_c is space parameters to pressure and concentration spaces respectively,â–³t_p,â–³t_c is time parameters to pressure and concentration respectively, k, r is the index of pressure Raviart-Thomas space and concentration Glerkin space respectively. Let parameters satisfy:â–³t_c=O(h_ch_p),(â–³t_p)²=O(h_ch_p). Assume k＞1,r＞2, c(x,t)is the solution of (1.1), while {Cⁿ}₀^N is the solution of Operator-Splitting scheme, then the following error estimate is satisfied:...