| The mathematical model for fluid flow in porous media is widely used in many scientific fields including groundwater,environment science, reservoir simulation and so on[11,62]. This model includes the fluid motion and mass transfer, which are based on conservation of mass, momentum and energy. Gravity, viscous, capillary and density play important role in this process, but we take more attention to fluid velocity, pressure, temperature and con-centration in reservoir simulation and engineering. The flow motion of fluid is described by Darcy’s law and the mass transfer is described by concen-tration equation. We can simplify these equations under some reasonable assumption, but it is still shown to be a nonlinear complex coupled system. It is valuable and significant to propose effective numerical scheme to keep physical properties of this system.The analytical model describing fluid flow in porous media is Darcy’s law[26,45], which shows the linear relation between average fluid velocity and pressure gradient. Mixed finite element scheme can be used to solve this model which coupled the Darcy’s law and mass conservation equation, as the velocity and pressure can be approximated accurately, simultaneously local mass conservation is kept. The classical Raviart-Thomas, Brezzi-Douglas-Marini mixed finite element methods keep the continuity of the approxi-mate velocity in the normal direction, and the existence, uniqueness and the optimal error estimate of discrete solution are obtained[17,18,19,65]. Some stabilization techniques can be used to solve this model, Masud and Hughes [60] added stabile term to the equation so that all continuous con-forming velocity-pressure spaces are viable. Least-square formulation and bubble function can also be used for stabilization. In some application, it is desirable that the velocity in the tangential direction is continuous. Arbo-gast and Wheeler [6] give a continuous velocity flux, though it loss the optimal convergence of the divergence norm, the L2error is optimal.We also consider coupling model which means that Darcy equation cou-pled anther Stokes equation in fluid region. They coupled with an Beavers-Joseph-Saffman condition[72] across the interface. The two parts have dif-ferent regularity properties, and the tangential velocity is discontinuous on the interface. Latyon[56] introduces the mathematical model, give the proof of existence of weak solutions by using Lagrange multipliers.[66] Yotov solve this coupled model with different finite methods on the two regions, they solve Stokes equation with DG method and Darcy equation with mixed fi-nite element. But it is not conducive difficulty to deal with the interface and programming. Burman[21] give the nonconforming stabilized Crouzeix-Raviart element, then Zhang[69] improve this scheme, and give a simpler stabilization penalty term to maintain element-wise mass conservation. Ar-bogast and Brunson[3] solve this coupled system refer to[6], give optimal L2convergence of velocity and pressure. Brinkman equation can be viewed as singular perturbation problem, Mardal[59] give the proof of existence and uniqueness of its weak solutions, introduce a robust finite element spaces, and gives the error estimate in detail. Following this analysis, more useful discrete schemes are discussed [61,83].Mathematically, the process of miscible displacement is described by a convection diffusion equation for the concentration of each chemical compo-nent. In the case of convection-dominated, classical finite element and finite difference methods do not work well because of the process of diffusion and dispersion. And the method of characteristics is efficient to treat this diffi-culty, as it transform primal equation to equivalent diffusion equation and have no strong constraint to time step. So we can use this skill to solve practical problem which need big time step. It is constructed initially as for-ward tracking method of characteristic [43], but it destroys the original space mesh with strong limitation. Douglas and Russell[54] introduce back tracking method of characteristic (modified method of characteristics MMOC) which overcome the restriction above. They combine finite element and finite differ-ence methods, moreover the optimal error estimates in L2, H1are derived for finite element scheme then this skill is widely used rapidly. Russell[71] apply it to incompressible miscible displacement, which Galerkin method is used to approximate the pressure, and optimal L2,H1convergence rates are given. In [35,36] MMOC is used to solve concentration equation and mixed finite ele-ment to pressure equation, in this way all the unknowns can be approximate optimally. In addition, for convection diffusion equation, an important prop-erty is mass conservation, which means that the total mass of component is balance along time direction regardless of sink and source, but general char-acteristics can’t meet it. Rui[68] construct a new characteristics scheme for convection diffusion equation which maintains mass conservation, which need continuity velocity. Celia[22] propose euler-lagrange localized adjoint method (ELLAM) scheme which is mass-balance, but bring trouble to computation. Wang[77,78] give the optimal error estimate of ELLAM scheme for convec-tion diffusion equation, while show computational experiments to simulate practical problem. For more complex compressible miscible displacement, Douglas and Roberts [53]give the nonlinear parabolic system, approximate the pressure with both Galerkin and mixed finite element method and give corresponding error for semi-discrete scheme. Chen and Ewing [23] analyze its full discrete scheme with finite element method, Cheng and Yuan also give the error estimate of finite element scheme base on MMOC [24,85].Han and Wu[47] give a mixed finite element with continue velocity flux on staggered mesh for Stokes equation, where the two components of the velocity and the pressure are defined on three different sets of grid-nodes. We modified it to obtain proper approximation for Darcy equation. Refer to [68], we obtain a new mass-balance scheme and error estimate for incompressible miscible displacement in porous media, and analyze the numerical scheme and error scheme for nonlinear coupled system that describes compressible miscible displacement. In this two cases, we all give numerical tests to verify our proof.The outline of the thesis is as follows.In Chapter1, we introduce the model which describes fluid flow in porous media, give Darcy’s law and mass conservation equation base on physical properties. Combine with state equation we give concentration equation describing mass transfer process of compressible and incompressible miscible displacement. Then we introduce some definitions of function space and corresponding norm, and show serval useful inequalities for this paper.In Chapter2, we introduce a mixed finite element method with contin-uous flux for an elliptic equation modeling Darcy flow in porous media. we present a better property of interpolation with the help of RT mixed finite element spaces and give convergence rate of L2norm. At last we give nu-merical examples to compare the new element with the mixed finite element space in [6]. We can see the convergence rates of this two scheme are almost identical, however we need less freedom of degree which need less time for computation.In Chapter3, we solve coupled Darcy-Stokes model with continue veloc-ity flux, give the error estimates and verify the convergence rate by numeri-cal experiments, then we consider Brinkman equation which is perturbation problem, we get convergence rate of L2norm. The numerical example val-idate the convergence rate of the unknowns which depend on perturbation parameter.In Chapter4, we analyze a numerical scheme for the coupled system of incompressible miscible displacement in porous media. Mass-conservative characteristic finite element is used for concentration equation refer to [68]. We prove the property of mass balance and decouple the velocity equation and concentration equation base on extrapolation. Under some reasonable assumption we obtain error estimate and convergence rate of L2norm. We also verify the mass balance and convergence rate by numerical examples. At last we give experiment of practical problem.In Chapter5, we discuss a numerical scheme for the coupled system of compressible miscible displacement in porous media, which is a strong non-linear and coupled system. Firstly we get the original value of the unknowns by projection. Secondly under some reasonable assumption we obtain error estimates for velocity and concentration equation respectively, then get the optimal convergence rate. Finally numerical examples verify the theoretical analysis. |
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