The Numerical Simulations And Analysis For The Models Of Some Underground Fluid

The numerical simulations of the models for the underground fluid are the most important members of the numerical simulations of energy source. The research of the underground fluid has deeply influenced on the development of society and economics, and has been producing great benefits to the economy.A large number of the underground fluid of significant interest are modelled by convection-dominated diffusion systems. Examples include the simulation of multiphase and multicom-ponent flows in porous media coming from petroleum and environmental engineering, of certain semiconductor devices, and of various atmospheric and fluid flows. The numerical simulation of convection-dominated diffusion problems requires special treatment. Many scholars have carried out a great deal of research, and developed a variety of numerical techniques which have obtained better approximations.In this dissertation, two sorts of the models for the underground fluid are considered, first, incompressible two-phase miscible displacement in porous media which the diffusion matrix is positive semi-definite or positive definite; second, underground water contamination problems which contain the non-Lipschitz difficulty. We present the schemes which are suit for the models and maintain the mass conservation law, and give rigorous convergence analysis and experiments which verifying the theoretical results and indicating the efficiency and validity of these schemes. The work in this dissertation have important improvements, especially for the application of the finite volume element method, the upwind-mixed finite element scheme, and the methods on dynamically changing mesh.The work consists of three chapters.In chapter 1, the finite volume element and the upwind-mixed methods are presented for the multidimensional incompressible two-phase flow displacement in porous media, which is governed by a system of two equation, one of elliptic form for the pressure and the other of parabolic form for the concentration of one of the fluids. The problem is one of the most important basic fields of oil reservoir numerical simulation. J.Douglas, et al. developed this research field firstly [1,2,3]. They have presented many famous numerical methods, such asthe characteristic finite difference method, the characteristic finite element and the characteristic mixed finite element method etc., and have completed theoretical analysis and numerical experiments which have become the basic theories and experiments on oil reservoir numerical simulation [4-13].In physical oil engineering, the concentration is convection-dominated. It is well known that for the convection-dominated problems the standard methods, such as finite difference or finite element method, produce excessive numerical diffusion, or they introduce nonphys-ical oscillations into the numerical solution. The method of characteristics [6,8,9,10,13] has proved effective in treating convection-dominated diffusion problems. Error estimates and numerical experiments have shown that this method permits the use of large time steps, and avoids or sharply reduces the usual numerical difficulties: numerical diffusion and nonphysi-cal oscillations. Unfortunately, in practical application, the characteristics scheme are quite complicated, because temporal and spatial interpolation are needed especially for computing boundary. Furthermore, it can fail to preserve natural conservation, and because of using the characteristic direction, it requires that the model be fi-periodic.Besides the method of characteristics, a variety of numerical techniques have been introduced to obtain better approximations, such as the streamline diffusion method, higher-order Godunov scheme [14,15], upstream-weighted finite difference schemes, and the Eulerian-Lagrangian localized adjoint method (ELLAM) [16-18]. Each method has its advantages and disadvantages. Explicit characteristic and Godunov schemes require that a CFL time step constraint be imposed. Upstream weighting tends to introduce into the solution an excessive amount of numerical diffusion near the sharp fronts. Compared with upstream weighting, the streamline diffusion method reduces the amount of numerical diffusion. It adds a user-defined amount biased in the direction of the streamline. In ELLAM. it can be difficult to evaluate the resulting integrals.The finite volume element (the box method or generalized difference method abbr. as GDM) was created by Li Ronghua firstly [19-27]. FVE methods use two spaces: the trial space of piecewise polynomial functions over the primal partition and the test space of piece-wise constant functions over the dual partition, and similar as the finite element methods the unknowns are approximated by a Galerkin expansion. Realistic computations show that FVE not only keep the computational simplicity of difference methods, but also enjoy the accuracy of finite element methods. In addition to, they maintain the mass conservation law, which is very important for fluid and underground fluid computations. In [26], the upwind generalized difference schemes were analyzed for miscible displacement problem, and obtained the energy-norm when dispersion was not included. At first, in §1.2 a finite volume method on triangular subdivisions is presented and analyzed. The amount of computing workis smaller than that of finite element methods. Optimal order H1-error estimate is derived when dispersion is included (see Theorem 1.1) in §1.2.4. Under strong condition, we derive L2-error estimate (see Theorem 1.2) in §1.2.5. A numerical experiment is presented.In general, assume that the tensor matrix of the concentration equation is positive definite. But in many actual problems, the matrix is only positive semi-definite. It is very difficult to analyze the semi-definite problems. Some characteristic finite element methods [10,28] or characteristic finite difference methods [29] have been constructed by Dawson and Yuan Yirang to solve such case. But the finite element methods can't keep the local conservation properties of the original problems. In §1.3 and §1.4, the positive semi-definite problem (D(u) ≥ 0) is considered. We propose the upwind-mixed finite element method in §1.3, and the upwind-mixed multistep method in §1.4. The upwind-mixed method is better suited for the convection-dominated problems, which can conquer numerical diffusion and nonphysical oscillations. In particular, it keep the two important properties of original problems: the extremum principle and the principle of mass conservation. The conservation of mass for this scheme can be proved in §1.3.3, and give the important lemmas that error estimates require. We obtain a priori error estimates (see Theorem 1.3) in §1.3.4 and obtain the optimal error estimate for positive definite problem in §1.3.5. In §1.4 the upwind-mixed multistep method is presented for the semi-definite problem, and improve the precision in time. In §1.4.3, we analyze this method and state an error estimate (see Theorem 1.4). Finally, in §1.4.4 we present some numerical results which can claim the character and superiority of the approximation.In chapter 2, the Godunov-mixed method for the contaminant transport equations are considered. The problems of contaminant transport are divided into equilibrium and nonequi-Iibrium adsorption kinetics, which depend on the rate of reaction with respect to the rate of flow. The model itself is advection-dominated, and the equation is characterized by ad-vection, diffusion, and adsorption. We note that in many important cases, the problems are non-Lipschitz. The simulation of these problems are very important for the environment protection, sewage disposal and the resource of underground water.A number of publications on the numerical approximation of equilibrium and nonequi-librium adsorption have appeared; see. for example, [30-33]. In [30] and [31], Barrett and Knabner describe and analyze a piece wise-linear Galerkin finite element approximation to a model similar to these problems but with no advection term, assuming both nonequilib-rium and equilibrium adsorption kinetics. They solve a perturbed system, replacing φ{c) by a smooth function φε which converges to φ as ε → 0. Then the estimates in the norm L∞(0,T; L2(Ω)) were obtained for this procedure. The approach described here differs from their approach in that it is better suited to advection-dominated problems and use a mixedfinite element approximation rather than a Galerkin approximation. In [32], a Characteristic Galerkin approximation to contaminant transport with nonequilibrium adsorption kinetics is described and analyzed. The methodology described here also differs substantially from this approach, and in fact, the analysis given in [32] does not directly extend to the problem considered here. The Godunov-mixed method can preserve natural conservation, and can conquer the numerical diffusion produced by advection-dominated. In [33] the semi-discrete Godunov-mixed method has been used to contaminant transport with equilibrium adsorption kinetics in one dimension, and derived better result. In this chapter, we substantially extend the Godunov-mixed method to contaminant transport with equilibrium and nonequilibrium adsorption kinetics in two dimensions, and also consider the difficulty of non-Lipschitz non-linearities and obtain the convergence.In §2.2 we consider contaminant transport equations with equilibrium adsorption processes and the adsorption term is modeled by a Freundlich isotherm. The semi-discrete Godunov-mixed method for the problem is introduced in §2.2.2. In§2.2.3, we give the important lemmas that error estimates require; Error estimates for a semi-discrete formulation are derived in §2.2.4, and we obtain the convergence theorem (see Theorem 2.1). In §2.3. we consider the equation with nonlinear, nonequilibrium adsorption reactions, derive the fully discrete Godunov-mixed method in §2.3.2. and in §2.3.3 we analysis the scheme and obtain the error estimate (see Theorem 2.2). In §2.3.4, we give numerical experiment to verify the convergence analysis.In chapter 3, we consider two different finite element methods on dynamically changing mesh. As is well known, the smoothness of solutions to partial differential equations (PDEs) in general vary considerably in space and time with, for instance, initial transients where highly oscillatory components of the solution are rapidly decaying. Efficient computational methods for PDEs therefore require space and time steps which are variable, ideally in both space and time, with small steps in transients and larger steps as the exact solution becomes smoother. The numerical simulation of the problems which involve time-changing localized phenomena requires capabilities for dynamic adaptive local mesh refinements and basis function improvements. In the last few years adaptive finite element methods, whereby the mesh changes dynamically during a simulation, have been considered by a number of authors. In particular, we refer the reader to the discontinuous Galerkin methods proposed and analyzed in series of papers by Eriksson and Johnson [34-36]. the moving space-time finite elements studied by Bank and Santos [37]. the earlier work on mesh modification by Dupont [38]. and the moving mesh methods of Miller [39,40] and Liang Guoping [41.42].In this chapter, we consider the solution of PDEs by a standard time-stepping procedure such as a backward Euler method, combined with the alternating-direction finiteelement method [43,44], and with the " lowest-order " Raviart Thomas mixed finite element method [45] in space. The alternating-direction method has been firstly proposed by Douglas and Dupont [43,44] for parabolic and hyperbolic problems on rectangular domains, and has proved to be very valuable in the approximate solution of PDEs involving several space variables. The object of this method is to reduce a multi-dimensional problem to a collection of one-dimensional ones. So it has the advantage of decreasing the amount of calculating, natural parallelism and the advantage of higher accuracy of finite element method. It has been of much current interest [46-48]. In considering the advantages of the method of the alternating-direction method and the dynamically changing mesh, it is natural to combine these two methods to treat PDEs in multi-dimensional. This combination is better suited to many problems with character of large region, huge scale and long computing time. In §3.2 and §3.3, we consider the solution of time-dependent parabolic and hyperbolic PDEs by the alternating-direction method on dynamically moving mesh. The analysis of convergent error estimate is presented in §3.2.2. and §3.3.3 (see Theorem 3.1,3.2). Finally, we give the numerical examples.In §3.4 we consider two-phase miscible displacement in porous media, which has been discussed in §1. Many numerical methods, based on fixed mesh and fixed basis functions, have been developed [1-5,7-10,28,29] for solving the problem. Although many of them are excellent when the concentration solution is relatively smooth, they are inefficient and inappropriate for resolving sharp concentration fronts or other critical local features, which change with time. To overcome this difficulty, the dynamic finite element method should be applied for dynamically implementing local mesh refinement and basis function improvements in the neighborhood of sharp concentration fronts.In this section, for the two cases of positive semi-definite and positive definite, the up-winding procedure combined with the " lowest-order " mixed finite element method on dynamically changing meshes are presented. In §3.4.2 when D(u) ≥ 0 we construct the upwind-mixed method on a dynamically changing mesh, and obtain the almost optimal rate error estimate under very general changes in the mesh (see Theorem 3.3). In §3.4.3 the method in the former section is analyzed in the case of positive definite, i.e. D(u) ≥ 0. In §3.4.4 we introduce a modification to this method, which preserves the optimal rate error estimate under very general assumptions on the mesh modifications (see Theorem 3.4). Finally, we give some numerical results for the test problem in §3.4.5.