Studies Of Mortar Finite Volume Element Methods And Fractional Flow Formulation For Two-Phase Flow In Porous Media

With the rapid development of computational mathematics, discretization methods using domain decomposition concept are becoming powerful tools to handle real-life problems with complicated domains or complex processes, and have the capabilities of high performance computation to large-scale problems. Mortar finite element method is one of the domain decomposition methods which allows the coupling of different physical models, numerical schemes and non-matching grids along interior interfaces of the computational domain. This offers the advantages to approximate the problems with abruptly changing diffusion coefficients, local anisotropies or singularities, and different governing processes (such as advection-dominated or diffusion-dominated processes) in different parts of the domain.This method was first introduced by Bernardi, Maday and Patera in [12] for the Possion problem, originally to couple finite element and spectral element approximations. And it was extended to three dimensional case in Belgacem and Maday [9]. By relaxing the continuity constraints at the vertices of subdomains without loss of stability, the original method in [12] has been developed into a standard mortar finite element method which is now extensively used and analyzed. The standard method consists of two kinds of formulations: nonconforming positive definite formulation, e.g. Marcinkowski [47] or [48], and saddle point formulation which was introduced by Belgacem [8] and is also called the mortar finite element method with Lagrange multipliers. In [66], Wohlmuth has proposed a modified Lagrange multiplier space, i.e. dual space for which it is much easier to implement the matching condition at the interface. For more details, see Wohlmuth [67].Initially, the mortar finite element method was constructed based on the decomposition of computational domain into nonoverlapping subdomains. There are two versions of the decomposition: geometrically conforming one which satisfies that the intersection of two different closed subdomains is either empty, a vertex or an edge in two-dimensional case, see Figure 1 (left); otherwise, geometrically nonconforming one, see Figure 1 (right). Furthermore, the case of overlapping subdomains is considered in e.g. Cai et al. [18]. Different algorithms for solving the algebraic systems arising from the mortar finite element method can be found in e.g. Achdou, Maday and Widlund [2], Casarin and Widlund [23], Gopalakrishnan and Pasciak [37], Wohlmuth [68].Moreover, mortar finite element has also been combined with adaptive method in e.g. Bergam et al. [10]. Bernardi and Hecht [11]; with finite volume element method in e.g. Ewing et al. [35], Bi and Li [13]; and with mixed finite element method in e.g. Arbogast et al. [4], Wheeler and Yotov [65]. In this dissertation, we focus on the mortar finite volume element method which is based on the saddle point formulation in Chapter 1 and on the nonconforming positive definite formulation in Chapter 2.Finite volume element method is a discretization technique which can inherit the physical conservation laws of the original problems locally. Thus it is popular in computational fluid mechanics since the partial differential equations studied in this field arise from mass, momentum and energy conservation laws. The finite volume element approach combines finite difference method with the technique of finite element method in a new development which can be used on general triangular and quadrilateral grids and allows more general construction of the control volumes (see e.g. [19, 20. 29, 36, 41, 45]) in comparison with the early finite volume approach. The combination preserves the advantages of both methods including simple discretization stencils, good accuracy and discrete local mass conservation, the very important property for many applications.The goals and structure of this dissertation arc as follows.Porous media flow problems widely occur in our daily life, ranging from the treatment of polluted ground water, paper (diaper, female pad, etc.) manufacturing, petroleum exploiting, carbon dioxide storage in deep geological formations to medical fields, such as the study of blood flow processes in the human body. These problems arouse many researchers' (from engineering, mathematics, physics, biology fields, etc.) interest since the improvements of solution techniques to these problems are closely linked to the improvements of our living qualities.On one hand, there has been growing interest in the mortar finite element method due to its flexibility and great potential for large-scale parallel computation. For example, the porous medium for ground water flow is soil which is normally heterogeneous. And heterogeneities have an important impact on the flow behaviors and processes. However, it is difficult to describe the heterogeneities using standard conforming numerical methods. Therefore, the mortar finite element method is expected to handle this difficulty by decomposing the heterogeneous problem on the whole domain with different permeabilities into homogeneous subproblems on each subdomain with the same soil property, for which suitable discretization techniques are available.On the other hand, the finite volume element methods are popular in computational fluid mechanics since they can keep the properties of original problems, i.e., satisfy discrete local mass conservation which is the most desirable feature of the numerical methods for many applications.In terms of the above two reasons, the first goal of the dissertation is to study the mortar finite volume element method for simple models including stationary elliptic problems in Chapter 1 and time-dependent parabolic problems in Chapter 2. Furthermore, we can consider the porous media flow applications using this method based on the theory analyses for simple models in the future work because the advantages combination of flexibility and local conservation in this method is attractive to these applications. The second goal of the dissertation is generated from the idea to implement the mortar finite volume element method into multi-phase flow problems in porous media, initially, into two-phase flow problems. There are two major mathematical formulations for two-phase flow. One is the fully coupled formulation, and the other is the fractional flow formulation which consists of two weakly coupled equations - the elliptic or parabolic pressure equation and the advection-diffusion saturation equation, in detail, see Chapter 3. In recent years, there is a lot of literature contributed to the mortar finite element method for elliptic (e.g. Marcinkowski [47], Belgacem [8]), parabolic (e.g. Chen and Xu [27], Bergam et al. [10]) and advection-diffusion problems (e.g. Bourgault and Boukili [14], Achdou [1]). However, we are currently not aware of the existence of the mortar finite element method for the strongly coupled, parabolic type equations in the fully coupled formulation. Hence, the fractional flow formulation for two-phase flow is concerned and studied in Chapter 3 of the dissertation under some assumptions in order to keep the model simple in the beginning. Actually, a good understanding of the fractional flow formulation will be helpful for us to design appropriate mortar finite volume element methods for porous media flow problems in our future work. Thus a specific motivation is given for a comparative study of the fractional flow formulation. Based on the local motivation of Chapter 3, more adaptive possibilities are proposed for the fractional flow system according to the differences between the two different mathematical formulations. Furthermore, the fractional flow formulation is combined with time-of-flight calculation using a discontinuous Galerkin method which gives another application of the weakly coupled system.The dissertation is divided into three chapters.Chapter 1 focuses on the accuracy of the mortar finite volume element method with Lagrange multipliers based on locally conforming P₁ finite elements for second order elliptic boundary value problems. Here, we analyze one of the mortar finite volume element schemes proposed in Ewing et al. [35] which uses finite volume element approximations only on the subdomains and finite elements on the interfaces for Lagrange multipliers. Two main theoretical results arc stated and strictly proved. One is the general L²-norm error estimate for the scheme to show the dependency of convergence rate of the discrete solution on the regularities of the exact solution and the source term f. Ewing et al. [35] only give some numerical experiments for the L²-norm error without proof. Our analysis indicates that the error estimate is optimal only if f∈H^βwithβ≥1 The additional smoothness requirement on f is necessary due to the formulation of finite volume element method The other one is the uniform convergence of the discrete solution under a minimal regularity assumption. We are currently not aware of the existence of a study for the uniform convergence of the mortar finite volume element method. The results in this chapter have been published in Cao and Rui [21].In Chapter 2, we construct and analyze two full-discrete symmetric mortar finite volume element schemes for parabolic problems, which use mortar finite volume element discretization in space based on nonconforming Crouzeix-Raviart (CR) elements and backward Euler or Crank-Nicolson discretization in time Bi and Li have proposed the mortar finite volume method for elliptic problems based on CR elements in [13], however, we are currently not aware of the existence of such results devoted to parabolic problems. Moreover, both schemes we propose are the modifications of the standard mortar finite volume element method which satisfy the symmetry of discrete algebraic systems. This property is important since many efficient algorithms for solving large-scale linear algebraic equations, e.g., the conjugate gradient method, rely on the symmetry of the discrete algebraic systems, and in many cases it is the fundamental physical principle of reciprocity. To obtain error estimates for the discretization schemes, a Ritz projection operator and a symmetric mortar finite volume element operator are given. Some auxiliary lemmas are set up to get properties of the two operators which are necessary for convergence analysis. In the last section of this chapter, two main theorems, Theorem 2.1 and Theorem 2.2, are stated and proved to present optimal discrete L²(H¹)-norm. error estimates for backward Euler and Crank Nicolson symmetric mortar finite volume element schemes, respectively.Chapter 3 is devoted to a systematic and comparative study of the fractional flow formulation for two-phase flow in porous media, and these studies will be published in Cao et al. [22] as a technical report.In Section§3.1, we present the motivation and goal of this chapterIn Section§3.2, the fundamentals of the physical background and two main kinds of mathematical formulations (fully coupled formulation and fractional flow formulation) for two-phase flow are presented. The implicit discretization schemes and IMPES (IMplicit-Pressure-Explicit-Saturation) concept based on a vertex-centered finite volume element method are given for both formulations, respectively.Based on the differences between the fully coupled (FC) and fractional flow (FF) formulation, more adaptive possibilities for FF system are proposed in Section§3.3. First, an overview of different adaptive techniques which have been developed is given. Some new ideas of the combination of these techniques are then presented for the fractional flow formulation including three main strategies: using different discretization grids for the pressure and saturation equations; combining error indicators with error estimators to balance the global error on the whole domain and save the computing time at the same time; using Peclet number to control the upwind method in discretization schemes and adapt discretization grids.In Section§3.4, one- and two-dimensional test cases are presented for two-phase flow in homogeneous porous media. Numerical investigations are carried out for both the accuracy and the computing time of simulations. We first implement numerical simulations using the same uniform grids for the two equations in each mathematical formulation. Numerical results for both formulations are compared which proves that the FF formulation is more efficient for the solution of advection-dominated two-phase flow problems. And the numerical simulation for fractional flow formulation using different uniform grids is also shown to investigate its feasibility.Section§3.5 shows two real-life applications based on the fractional flow formulation combined with time-of-flight (TOF) calculation, and these two novel applications have great practical value to ground water problems. Here, a discontinuous Galerkin method is used and described for TOF equation. Direct information for practical use is obtained from numerical simulations, and these applications are only possible for the fractional flow formulation of two-phase flow.