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Numerical Methods, Analysis, And Simulation For Porous Medium Flow

Posted on:2011-02-03Degree:DoctorType:Dissertation
Country:ChinaCandidate:K X WangFull Text:PDF
GTID:1100360305950906Subject:Computational Mathematics
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Porous medium flow and transport processes arise in many different areas and disciplines, including the exploration and production of oil and gas fields, the pre-diction and remediation of contaminants in groundwater, seawater intrusion, CO2 se-questration for the mitigation of global warming, the design of fuel cells for satelite and electric-power cars, the application in biology and medical sciences, manufacture of diapers, ink distribution in paper and photos, design of clothes and shoes, and so on. The mathematical models for describing the complex physical and chemical phe-nomena in these fluid flow and transport processes often lead to coupled systems of time-dependent nonlinear partial differential equations. The solutions of these systems often have complicated structures and moving steep fronts, especially near the steep front regions where important physical and chemical changes often take place and so needs to be resolved. Nevertheless, because of the complexity of these problems, their numerical simulations often encounter various difficulties.In this dissertation I carry out research on time-dependent convection-diffusion equations and the coupled systems arising from the mathematical models that de-scribe the porous medium flow and transport processes (and related problems). This includes the development of efficient and accurate numerical methods that retain the physical properties of the problem, the corresponding theoretical analysis, and im-plementation and scientific computing. In the development of the methods and the corresponding analysis we consider generic convection-diffusion equations and coupled systems (without preassuming that these problems have to arise from porous medium flow and transport). Nevertheless, I look at the porous medium flow and transport problems from time to time, since this will give us a better idea about the physical and mathematical properties of the problems that in turn helps us to develop, analyze, and implement the corresponding numerical methods.This dissertation consists of four parts:Part I contains the physical background and the presentation of a mathematical model, which will be used as the prototype mod-els to be solved numerically in this dissertation. Part II focuses on the development and analysis of several types of numerical methods for time-dependent linear convection-diffusion equations. In this part we prove several important error estimates, including the optimal-order error estimates that hold uniformly with respect to the vanishing parameter, optimal-order error estimates that hold uniformly respect to the degener-ate diffusion, and optimal-order error estimate for discontinuous Galerkin methods. In PartⅢ, we study an MMOC-MFEM time-stepping procedure for the coupled system of time-dependent nonlinear partial differential equations. We prove an optimal-order error estimate of the time-stepping procedure under reasonable conditions. In Part IV, we develop multiscale Eulerian-Lagrangian method for the transient linear convection-diffusion equations with oscillatory coefficients, an immersed Eulerian-Lagrangian lo-calized adjoint method for transient advection-diffusion equations with interfaces, and a fast finite difference method for fractional partial differential equations. Below we elaborate the contents in each chapter of the dissertation:In Chapter 1 we present mathematical models for describing porous medium flow and transport processes, which are used as the prototype models to be solved nu-merically in this dissertation. In§1.1 we derive a mathematical model for describing incompressible fluid flow processes in porous media that arise in petroleum reservoir simulation and subsurface contaminant transport and remediation. In§1.2 we address some recent developments in mathematical modeling, including multilscale problems and fractional partial differential equations.In Chapters 2-6, we focus on the development, analysis, and numerical perfor-mance of time-dependent convection-diffusion equations, which can be viewed as lin-earized analogue of the quasilinear convection-diffusion equation (1.1.11) from the cou-pled system (1.1.10)-(1.1.11). The decoupling and linearization of the system will be studied in Chapter 7, so all the numerical methods developed and analyzed in Chapters 2-6 can be used to solve the coupled system (1.1.10)-(1.1.11). This part contains the following several topics: In Chapter 2 we review several representative Eulerian-Lagrangian methods in-cluding the modified method of characteristics (MMOC), the modified method of char-acteristics with adjusted advection (MMOCAA), and the Eulerian-Lagrangian localized adjoint method (ELLAM).In Chapter 3 we study a classical problem in the area of numerical simulation of time-dependent convection-diffusion equations:ε-uniform error estimates for numeri-cal methods for convection-diffusion equations that contain a vanishing parameterε. Optimal-order error estimates were previously proved for various Eulerian-Lagrangian methods and standard Galerkin methods [2,30,31,106,109]. But the drawback of these estimates lies in the following:the generic constant and the smoothness norms of the true solutions on the right side depend inversely on the parameterε. Consequently, these estimates could blow up asεtends to zero.In this chapter we prove optimal-order error estimates in anε-weighted energy norm for several representative Eulerian-Lagrangian methods as well as Galerkin finite element methods for time-dependent convection-diffusion equations with a vanishing parameterε. The estimates depend only on certain Sobolev norms of the initial and right-hand side data but not onεor any norm of the true solution. We then use the interpolation of spaces and stability estimates to derive anε-uniform estimate for problems with minimal or intermediate regularity, where the convergence rates are proportional to certain Besov norms of the initial and right-hand side data.In Chapter 4 we study another classical problem in the area: uniform error esti-mates for numerical methods for degenerate convection-diffusion equations. Because subsurface geological formations typically exhibit strong spatial heterogeneity, in re-alistic modeling and simulation of subsurface porous medium flow and transport, the governing transport equations are time-dependent convection-diffusion transport equa-tions with degenerate diffusion. In§4.2 we prove optimal-order error estimates in a diffusion-weighted energy norm for a family of Eulerian-Lagrangian methods for time-dependent convection-diffusion equations with degenerate diffusion and these estimates hold uniformly with respect to the degenerate diffusion. For an unstructured mesh these estimates are suboptimal but sharp when the Courant number is less than unity and optimal otherwise. In§4.3 we use the interpolation of spaces and stability estimates to derive an estimate for problems with minimal or intermediate regularity, where the convergence rates are proportional to certain Besov norms of the initial and right side data.In Chaper 5 we prove an optimal-order error estimate in the L2 norm and a superconvergence estimate in the energy norm for a family of ELLAM schemes for time-dependent convection-diffusion equations with general flux boundary conditions on unstructured meshes on a general d-dimensional domain. The ELLAM schemes use finite element approximations of degree m≥1 on unstructured meshes over the d-dimensional domain.In Chapter 6 we prove optimal-order error estimate for the characteristic dis-continuous Galerkin methods for both convection-diffusion equations and convection-reaction equations. In§6.2 we construct several kinds of characteristic discontinuous Galerkin methods for time-dependent convection-diffusion equations and prove their optimal-order error estimates in the L2 norm.§6.3 presents a CFL-free explicit charac-teristic interior penalty scheme for linear convection-reaction equations and illustrates its performance by numerical experiments. Furthermore, we analyze the nonsymmet-ric discontinuous Galerkin methods for linear parabolic equations in multiple spatial dimensions and derive its L2-optimal estimates.Chapter 7 is devoted to the coupled system of nonlinear partial differential equa-tions (1.1.10)-(1.1.11). In 1983 Ewing ed. had proposed an MMOC-MFEM time-stepping procedure to solve the coupled system (1.1.10)-(1.1.11) and then conducted a delicate and rigorous mathematical analysis to prove an optimal-order error estimate. However, a primary shortcoming of these estimates is that they are value only if the Courant number of the numerical discretization tends to zero asymptotically. This con-straint is numerically very restrictive and does not fully reflect the numerical strength of the MMOC-MFEM time-stepping procedure. In§7.2 we prove an optimal-order error estimate for a family of MMOC-MFEM approximation to the coupled system, which holds even if the Courant number tends to infinity asymptotically. In§7.3, we also prove a superconvergence estimate for the Galerkin FEM-MFEM scheme.In Chapter 8 we develop a multiscale Eulerian-Lagrangian method for time-dependent convection-diffusion equations with oscillatory coefficients. Numerical experiments demonstrate the strong potential of the developed method.In Chapter 9 we develop an immersed Eulerian-Lagrangian localized adjoint method for transient advection-diffusion equations with interfaces and prove the optimal-order error estimate. In Chapter 10 we study numerical methods for fractional partial differential equa-tions. Because the fractional differential operator is non-local, the corresponding nu-merical scheme will have a full coefficient matrix, which requires significantly more computational cost and storage. In this chapter we develop a fast finite difference method for fractional partial differential equations, which reduces the storage require-ment from O(I2) to O(I) and computational cost from O(I3) to O(I log2 I) with I being the number of unknowns, but retains the same accuracy as the existing numeri-cal methods.The main contribution of this dissertation can be summarized as follows:We have proved optimal-order error estimates in a weighted energy norm for sev-eral representative Eulerian-Lagrangian methods for time-dependent convection-diffusion equations that contain a vanishing parameter. The derived estimates hold uniformly with respect to the vanishing parameter.We have proved optimal-order error estimates in a weighted energy norm for sev-eral representative Eulerian-Lagrangian methods for time-dependent convection-diffusion equations with degenerate diffusion. The derived estimates hold uniformly with respect to the degenerate diffusion.We have proved optimal-order error estimates in a weighted energy norm for Galerkin finite element methods for time-dependent convection-diffusion equations that contain a vanishing parameter. The derived estimates hold uniformly with respect to the vanishing parameter.We have proved optimal-order error estimates for a family of Eulerian-Lagrangian localized adjoint methods for time-dependent convection-diffusion equations with general boundary conditions on a general domain with an unstructured mesh.We have proved optimal-order error estimates for characteristic discontinuous Galer-kin methods for time-dependent convection-diffusion equations.We have proved an optimal-order error estimate for nonsymmetric interior penalty discontinuous Galerkin methods for multidimensional diffusion equations.We have proved an optimal-order error estimate for a family of mixed finite element methods for the coupled system of miscible porous medium flow. We have developed a multiscale Lagrangian methods for transient linear convection-diffusion equations with oscillatory coefficients and tested it by numerical experi-ments.We have developed an immersed Eulerian-Lagrangian localized adjoint method for transient advection-diffusion equations with interfaces and proved the optimal-order error estimate.We have developed a fast finite difference method for fractional convection-diffusion equations, which could significantly reduce the storage requirement and computa-tion cost.To keep the dissertation within a reasonable length, I choose to present some presentative works of my research publications during the course of my PhD study in this dissertation while citing the rest during the presentation.
Keywords/Search Tags:convection-diffusion equation, coupled system, Eulerian-Lagrangian methods, optimal-order error estimate, porous medium flow
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