Numerical Computation For Darcy-Forchheimer Fluid Flow In Porous Media

Mathematical and physical model for the fluid flow and transport pro-cesses in porous media is widely used in the exploration and production of oil and gas in petroleum reservoir [4][6][40]. The model is based on the conserva-tion laws for mass, momentum and energy of fluid or their approximate form. The engineers focus on physical quantities including pressure, strain, stress, fluid velocity, temperature and concentration of one component in miscible flow or saturation of one phase in multiphase flow. Following the conservation law and considering the physical quantities above, many geological and fluid parameters are introduced such as porosity, absolute permeability, relative permeability, capillary pressure and mass density, viscosity, compressibility. Heterogeneity and anisotropy enable the parameters vary steely, thus many physical quantities are in average sense and the model with their approximate form coincides the practical physical process in certain range.Mathematical and physical model for the fluid flow and transport pro-cesses in porous media leads to a highly coupled time-dependent nonlinear partial differential equations. Because of the complex structure of the par-tial differential equations, analytical solution formulas exist in some special cases. Therefore, large-scale, high-accuracy simulation for the model is a urgent necessity in science and engineering. The design, analysis and simu-lation verification of the numerical scheme is the problem in advance. In sight of the complexity of the model for fluid flow in porous media, mass conservation results in the mass balance of the injection and produc-tion and the momentum conservation is often approximated by some simpli-fied velocity and pressure relation, experimental formulas for example, the Darcy's law and other non-Darcy's law. Furthermore, some assumptions is brought about to simplify, decouple or linearize the equations, such as the incompressibility, the slightly compressibility, the miscibility or immiscibility of the two different fluids.If assuming that the fluid is incompressible, the divergence free equa-tion of the velocity combined with Darcy's law is the widely used classical model. Eliminating the velocity, we can derive a second order elliptic par-tial differential equation. If assuming that the fluid is slightly compressible and introducing a compressibility, the simplified mass conservation equation combines with the Darcy's law. Also eliminating the velocity, we can derive a second order parabolic partial differential equation. If retaining the veloc-ity, we can approximate the velocity and pressure simultaneously and this method is called mixed method. There are a lot of work on the dual mixed weak form with its element construction and they arc summarized in [8][13]. The earlier literatures can trace back to [9][10][36][37]. The primal mixed weak form with its element construction can be referred to [16][53].Darcy's law, i.e. the linear relation between the pressure gradient and Darcy velocity, describes the creeping flow of Newtonian fluids in porous me-dia. Forchheimer observed (1901) that the nonlinear relationship between the pressure and Darcy velocity for the moderate Reynolds number (Re>1approximately)[27]. The derivation of Forchheimer's model or the experi-mental evidence can be found in [46][54][12][2][23][31]. While the mathemat-ical theory of Forchheimer equation can be referred as [20][50][3][30]. The monotone non-degenerate nonlinear property owned by Forchheimer equa-tion must be emphasized. Some similar models, for example p-Laplacian model,quasi-Newtionian model, provide the same mathematical tools and the references are [21][22][28][48][18][17]. Numerical methods are proposed recently for Forchheimer equation.[27] and [35] propose and verify a primal nonconforming mixed scheme;[47] adopts a primal conforming mixed scheme;[39] utilizes the dual mixed el-ement to deal with the slightly compressible case. Girault and Wheeler (2008)[27] prove existence and uniqueness of the primal mixed weak form for Darcy-Forchhcimcr equation. They propose a primal nonconforming mixed scheme:piecewisd constant element for velocity and Crouzeix-Raviart ele-ment for pressure; they also mention a Peaceman-Rachford type alternating-direction iterative algorithm to solve the nonlinear equation; they give the a priori error estimate of the the primal nonconforming mixed scheme and the convergence of the iterative algorithm. Lopez ct al.(2009)[35] carry out numerical tests for the primal nonconforming mixed scheme [27], especially for the convergence of the iterative algorithm and the first order convergence rate for the numerical scheme; they also compare the Peaceman-Rachford type alternating-direction iterative algorithm with the Newton iterative al-gorithm; otherwise, they propose another approximate scheme for pressure, i.e. first order Lagrange interpolation. Salas et al.(2011)[47] prove the existence, uniqueness and the convergence of the primal conforming mixed scheme in [35]; two new approximate scheme are demonstrated, one uses the Crouzeix-Raviart element for velocity and first order Lagrange element for pressure, the other approximates both velocity and pressure with first order Lagrange clement.Above all, solving the the flow equation using the mixed element is reviewed. Furthermore, misdble displacement of one fluid by another is considered by introducing the component and concentration.[45][19] give the a priori error estimate for the numerical scheme of the incompressible miscible displacement model based on the Darcy's law.[34][14][32][33] prove the a priori error estimate for the numerical scheme of the slightly compressible miscible displacement model based on the Darcy's law.This thesis investigates the Darcy-Forehheimer flow model, miscible dis-placement model in porous media, including the numerical scheme, its con- vergence analysis and simulation. The key point for the flow equation is the handling of the monotone nonlinear part. The a priori error estimate of the dual conforming mixed scheme is presented. Then, the numerical scheme for incompressible, slightly compressible miscible displacement problem based on the Darcy-Forchheimer' law is considered and the a priori error estimate is also demonstrated. Numerical simulation is to tested to compare the Darcy's law and Darcy-Forchheimer's law for both incompressible and slightly com-pressible flow. While the incompressible and slightly compressible miscible displacement model based on Darcy-Forchheimer's law is also simulated.The outline of the dissertation is as follows.In Chapter1, the mathematical models describing the fluid flow and transport process in porous media. The numerical schemes are designed to solve the equations which are the combination of some basic equations, including mass conservation law with its simplifications, mechanical mecha-nism and mathematical properties of Darcy's law and non-Darcy's law. Some signs and physical quantities are stated, including the function space with its norm definition, space triangulation and discrete space definition, time discretization. The interpolation and projection, initial and boundary con-dition, regularity assumption is demonstrated.In Chapter2, dual conforming mixed element scheme is designed for the Darcy-Forchhciemr equation combined with the incompressible mass con-servation equation, i.e. the velocity and pressure are approximated by the Raviart-Thomas or Brczzi-Douglas-Marini element. Eliminating the veloc-ity, we get the nonlinear monotone non-degenerate elliptic partial differen-tial equation for pressure. The continuous and discrete Inf-Sup condition is proved based on the regularity of the monotone non-degenerate elliptic equation. The existence and uniqueness arc easy to get based on the Inf-Sup condition. The a priori error estimates in L2-and L3-norm for velocity and L2norm for pressure are analyzed utilizing the monotonicity owned by Darcy-Forchheimer operator. The convergence rates of the approximate scheme and convergence of the linear iterative scheme are verified numerically, which is in consistent with the theoretical results. Results in this chapter are part of the author's paper [38], which has been published online by Journal of Scientific Computing (SCI).In Chapter3, dual conforming mixed element scheme is extended to the Darcy-Forchheiemr equation combined with the slightly compressible mass conservation equation. The a priori error estimates for both semi-discrete scheme and fully-discrete scheme are presented using the solution of incom-pressible Darcy-Forchheimer equation as projection. Numerical examples are constructed to show the convergence rates of the scheme, which is in consis-tent with the theoretical results.In Chapter4, the mixed element-Galerkin scheme is used to solve in-compressible miscible displacement model based on Darcy-Forchheeimr's law. Decouple the velocity-pressure equation and the concentration equation us-ing the extrapolation used in [45][19]. The a priori error estimate is analyzed utilizing the monotonicity owned by Darcy-Forchheimer operator. Numeri-cal examples are constructed to show the convergence rates of the scheme, which is in consistent with the theoretical results.In Chapter5, the semi-discrete and fully-discrete mixed element-Galerkin schemes are applied to solve slightly compressible miscible displacement model using Darcy-Forchheimer's law. Based on the slightly compressible model [34] and introducing the assumption that different components have the same compressibility, the extrapolation in [45][19] can be used to decouple the velocity-pressure equation and the concentration equation. The a priori error estimate is analyzed utilizing the monotonicity owned by Darcy-Forchheimer operator. Numerical examples are constructed to show the convergence rates of the scheme, which is in consistent with the theoretical results.