| Mathematical and physical model for the fluid flow in porous media is widely used in the exploration and production of oil and gas in petroleum reservoir [6] [9] [57]. Fluid flow in porous media follow the basic laws, which are based on the conservation laws for mass, momentum and energy of fluid.The purpose of petroleum reservoir research is to predict the future trend of oil and find out some way to enhance recovery ratio. The physical system that needs to be simulated is represented by an appropriate mathematical model,which is generally based on some necessary assumptions. From a practical point of view, in order to address the problem easily, these assumptions are definitely needed. The mathematical models are generally complicated and usually cannot be solved by analytic methods. So we have to approximate the solution by some numerical methods. Before petroleum reservoir simulation,it is necessary to establish a appropriate mathematical model.Mathematical and physical model for the fluid flow in porous media leads to a highly coupled time-dependent nonlinear partial differential equa-tions. Because of the complicated structure of the model for fluid flow in porous media, the fluid flow follow mass conservation law, which show the material balance, and perform for the volume injection and production and mass balance in real production; and the momentum conservation describe the mainly relation of velocity and pressure; and in real production, mass balance and pressure distribution are mainly concerned. Therefore, it is nec-essary to make further assumptions to simplify the model and reduce the coupling and nonlinear strength. Experimental formulas for example, the Darcy's law and other non-Darcy's law, the incompressibility, the slightly compressibility of the fluid. If assuming that the fluid is incompressible, the widely used mathematical model is the model which couple the divergence equation with Darcy's law. We can only solve the pressure equation of el-liptic type by eliminating the velocity. If assuming that the fluid is slightly compressible,we need to introduce a compressibility coefficient, eliminate the velocity, and solve a parabolic partial differential equation. By contrast,we can approximate the velocity and pressure simultaneously with remaining the velocity by mixed methods. There are a lot of work on the dual mixed formula in [16] [26] [17] [18] [51] [52]. And the primal mixed weak form can be referred to [28] [74].Darcy's law, i.e. the linear relationship between the velocity u of the creep flow and the gradient of the pressure p, describes the creeping flow of Newtonian fluids in porous media, which is valid when the Darcy velocity u,is extremely small [6]. Forchheimer in 1901 observed that the nonlinear rela-tionship between the pressure and Darcy velocity for the moderate Reynolds number ( Re > 1 approximately) [38]. The derivation of Forchheimer's model or the experimental evidence can be found in [66] [77] [21] [2] [35] [43]. While the mathematical theory of Forchheimer equation can be referred to [31] [69].Forchheimer equation is one of a class of monotone non-degenerate nonlin-ear equations. There are many other similar models, such like p-Laplacian model?quasi-Newtionian model [30] [29] [32] [33].In general, the structure of this kind mathematical model is very com-plicated, and the coupled model is difficult to solve. Meanwhile, due to the various types and multi-scale of porous media,the computational cost is very expensive and the rate of convergence is slow. Therefore, it is urgent to design fast solvers for large-scale computation.In recent years, many numerical methods of the Darcy-Forchheimer model have been developed. Girault and Wheeler in [38] proved the existence and uniqueness of the solution of the Darcy-Forchheimer model by proving the nonlinear operator A (v) =?/?K-1v+?/?|v|v is monotone, coercive and hemi-continuous, and establishing an appropriate inf-sup condition. Then they considered mixed finite element methods by approximating the velocity and the pressure by piecewise constant and nonconforming Crouzeix-Raviart(CR) elements, respectively. They proved a discrete inf-sup condition and the convergence of the mixed finite element scheme.They also proposed a Peaceman-Rachford (PR) type iterative method to solve the discretized non-linear system and proved convergence of this iterative solver. In the PR iteration, the nonlinear equation can be decoupled with the divergence con-straint and solved in a closed form. Lopez, Molina, and Salas in [49] carried out numerical tests of the methods proposed in [38], and made a compara-tive study between Newton's method and the PR iterative method. They pointed out that Newton's method is not competitive with the PR itera-tion. In each iteration, Newton's method needs to evaluate a Jacobian and solves a linear saddle point system, but the PR iteration computes an in-termediate solution for a decoupled nonlinear equation and then solves a simplified linear saddle point system. The cost of solving the decoupled nonlinear equation can be negligible in comparison with the Jacobian eval-uation. Furthermore the PR iteration required fewer iterations to converge than Newton's method with the same initial guess; see [49] for details. Park in [56] developed a mixed finite element method with a semi-discrete scheme for the time dependent Darcy-Forchheimer model. Pan and Rui in [54] gave a mixed element method for the Darcy-Forchheimer model based on the Raviart-Thomas (RT) element or the Brezzi-Douglas-Marini (BDM) element approximation of the velocity and piecewise constant (PO) approximation of the pressure.They got the nonlinear monotone non-degenerate elliptic partial differential equation for pressure by eliminating the velocity. The continu-ous and discrete Inf-Sup condition was proved based on the regularity of the monotone non-degenerate elliptic equation. The a priori error estimates in L2- and L3-norm for velocity and L2 norm for pressure were analyzed. Rui and Pan in [63] proposed a block-centered finite difference method for the Darcy-Forchheimer model, which was thought of as the lowest-order RT-PO mixed element with proper quadrature formula. Rui, Zhao and Pan in [64] p-resented a block-centered finite difference method for the Darcy-Forchheimer model with variable Forchheimer number ?(x). Wang and Rui in [76] con-structed a stabilized CR element for the Darcy-Forchheimer model. Rui and Liu in [62] introduced a two-grid block-centered finite difference method for the Darcy-Forchheimer model. Salas, Lopez, and Molina in [67] presented a theoretical study of the mixed finite element method proposed in [49], and showed the well-posedness and convergence.Most of work mentioned above mainly focus on the discretization of the Darcy-Forchheimer model. Except the PR iteration presented in [38], no other work concentrates on fast solvers of the discretized nonlinear saddle point system which will be the topic of this paper.Multigrid method is one of the most efficient methods on solving the linear and nonlinear elliptic systems. It should be clarified that for nonlinear problems we no longer have a simple linear residual equation, which is the most significant difference between linear and nonlinear systems. The multi-grid scheme we used here is the most commonly used nonlinear version of multigrid. It is called the full approximation scheme (FAS) [20] because the problem in the coarse grid is solved for the full approximation rather than the correction.An efficient nonlinear multigrid method for a mixed finite element method of the Darcy-Forchheimer model is constructed in this paper. A Peaceman-Rachford type iteration is used as a smoother to decouple the nonlineari-ty from the divergence constraint. The nonlinear equation can be solved element-wise with a closed formulae. The linear saddle point system for the constraint is reduced into a symmetric positive definite system of Poisson type. Furthermore an empirical choice of the parameter used in the split-ting is proposed and the resulting multigrid method is robust to the so-called Forchheimer number which controls the strength of the nonlinearity. By com-paring the number of iterations and CPU time of different solvers in several numerical experiments, our multigrid method is shown to convergent with a rate independent of the mesh size and the Forchheimer number and with a nearly linear computational cost.The outline of the dissertation is as follows.In Chapter 1, we briefly introduce the Darcy-Forchheimer model and mass conservation law and its variants with some assumptions.In Chapter 2, we recall the basic numerical methods for solving dis-cretized equations, including direct solver and linear iterative methods for linear equation and nonlinear iterations. We also demonstrate the advantage and disadvantage of these solves. Many standard iterative methods possess the smoothing property. This property makes these methods very effective at eliminating the high-frequency or oscillatory components of the error, while leaving the low-frequency or smooth components relatively unchanged. We use different iterative methods to solve two-dimensional Poisson problem,which is used to demonstrate the smoothing property.In Chapter 3, we introduce the basic idea and algorithm of multigrid methods. We begin with linear multigrid methods. In linear case, error satisfied the residual equation. But in nonlinear case, it doesn't work. A new approach should be used for constructing nonlinear multigrid methods.We mainly introduce the two common used nonlinear multigrid methods.In Chapter 4,an efficient nonlinear multigrid method for a conforming mixed finite element method of the Darcy-Forchheimer model is constructed in this chapter. A Peaceman-Rachford type iteration is used as a smoother to decouple the nonlinearity from the divergence constraint. The linear saddle point system for the constraint is reduced into a symmetric positive definite system of Poisson type. Furthermore an empirical choice of the parame-ter used in the splitting is proposed and the resulting multigrid method is robust to the so-called Forchheimer number. By comparing the number of iterations and CPU time of different solvers in several numerical experiments,our multigrid method is shown to convergent with a rate independent of the mesh size and the Forchheimer number and with a nearly linear computa-tional cost. Results in this chapter are part of the author's paper [42],which has been published online by Journal of Scientific Computing (SCI).In Chapter 5,an efficient nonlinear multigrid method for a noncon-forming mixed finite element method of the Darcy-Forchheimer model is constructed in this chapter. The important difference between the noncon-forming case and the conforming case is that the finite element spaces are non-nested spaces. Hence we can no longer simply use the natural injection for the intergrid transfer of grid functions. The key point is how to define the transfer operator. As same as the conforming case, by comparing the number of iterations and CPU time of different solvers in several numerical experiments, our multigrid method is shown to convergent with a rate in-dependent of the mesh size and the Forchheimer number and with a nearly linear computational cost. |
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