Unified convergence analysis of numerical schemes for a miscible displacement problem

This article performs a unified convergence analysis of a variety of numerical methods for a model of the miscible displacement of one incompressible fluid by another through a porous medium. The unified analysis is enabled through the framework of the gradient discretisation method for diffusion operators on generic grids. We use it to establish a novel convergence result in $L^\infty(0,T; L^2(\Omega))$ of the approximate concentration using minimal regularity assumptions on the solution to the continuous problem. The convection term in the concentration equation is discretised using a centred scheme. We present a variety of numerical tests from the literature, as well as a novel analytical test case. The performance of two schemes are compared on these tests; both are poor in the case of variable viscosity, small diffusion and medium to small time steps. We show that upstreaming is not a good option to recover stable and accurate solutions, and we propose a correction to recover stable and accurate schemes for all time steps and all ranges of diffusion

An HMM--ELLAM scheme on generic polygonal meshes for miscible incompressible flows in porous media

We design a numerical approximation of a system of partial differential equations modelling the miscible displacement of a fluid by another in a porous medium. The advective part of the system is discretised using a characteristic method, and the diffusive parts by a finite volume method. The scheme is applicable on generic (possibly non-conforming) meshes as encountered in applications. The main features of our work are the reconstruction of a Darcy velocity, from the discrete pressure fluxes, that enjoys a local consistency property, an analysis of implementation issues faced when tracking, via the characteristic method, distorted cells, and a new treatment of cells near the injection well that accounts better for the conservativity of the injected fluid

An Approximation to Miscible Fluid Flows in Porous Media With Point Sources and Sinks by an Eulerian-Lagrangian Localized Adjoint Method and Mixed Finite Element Methods

We develop an Eulerian–Lagrangian localized adjoint method (ELLAM)-mixed finite element method (MFEM) solution technique for accurate numerical simulation of coupled systems of partial differential equations (PDEs), which describe complex fluid flow processes in porous media. An ELLAM, which was shown previously to outperform many widely used methods in the context of linear convection-diffusion PDEs, is presented to solve the transport equation for concentration. Since accurate fluid velocities are crucial in numerical simulations, an MFEM is used to solve the pressure equation for the pressure and Darcy velocity. This minimizes the numerical difficulties occurring in standard methods for approximating velocities caused by differentiation of the pressure and then multiplication by rough coefficients. The ELLAM-MFEM solution technique significantly reduces temporal errors, symmetrizes the governing transport equation, eliminates nonphysical oscillation and/or excessive numerical dispersion in many simulators, conserves mass, and treats boundary conditions accurately. Numerical experiments show that the ELLAM-MFEM solution technique simulates miscible displacements of incompressible fluid flows in porous media accurately with fairly coarse spatial grids and very large time steps, which are one or two orders of magnitude larger than the time steps used in many methods. Moreover, the ELLAM-MFEM solution technique can treat large mobility ratios, discontinuous permeabilities and porosities, anisotropic dispersion in tensor form, and point sources and sinks

DISCONTINUOUS GALERKIN METHODS FOR COMPRESSIBLE MISCIBLE DISPLACEMENTS AND APPLICATIONS IN RESERVOIR SIMULATION

This dissertation contains research on discontinuous Galerkin (DG) methods applied to the system of compressible miscible displacements, which is widely adopted to model surfactant flooding in enhanced oil recovery (EOR) techniques. In most scenarios, DG methods can effectively simulate problems in miscible displacements.However, if the problem setting is complex, the oscillations in the numerical results can be detrimental, with severe overshoots leading to nonphysical numerical approximations. The first way to address this issue is to apply the bound-preservingtechnique. Therefore, we adopt a bound-preserving Discontinuous Galerkin methodwith a Second-order Implicit Pressure Explicit Concentration (SIPEC) time marchingmethod to compute the system of two-component compressible miscible displacement in our first work. The Implicit Pressure Explicit Concentration (IMPEC) method is one of the most prevalent time marching approaches used in reservoir simulation for solving coupled flow systems in porous media. The main idea of IMPEC is to treat the pressure equation implicitly and the concentration equations explicitly. However, this treatment results in a first-order accurate scheme. To improve the order of accuracy of the scheme, we propose a correction stage to compensate for the second-order accuracy in each time step, thus naming it the SIPEC method. The SIPEC method is a crucial innovation based on the traditional second-order strong-stability-preserving Runge-Kutta (SSP-RK2) method. However, the SIPEC method is limited to second-order accuracy and cannot efficiently simulate viscous fingering phenomena. High-order numerical methods are preferred to reduce numerical artifacts and mesh dependence. In our second work, we adopt the IMPEC method based on the implicit-explicit Runge-Kutta (IMEX-RK) Butcher tableau to achieve higher order temporal accuracy while also ensuring stability. The high-order discontinuous Galerkin method is employed to simulate the viscous fingering fluid instabilities in a coupled nonlinear system of compressible miscible displacements. Although the bound-preserving techniques can effectively yield physically relevant numerical approximations, their success depends heavily on theoretical analysis, which is not straightforward for high-order methods. Therefore, we introduce an oscillation-free damping term to effectively suppress the spurious oscillations near discontinuities in high-order DG methods. As indicated by the numerical experiments, the incorporation of the bound-preserving DG method with SIPEC time marching and high-order OFDG with IMPEC time marching provides satisfactory results for simulating fluid flow in reservoirs

Discontinuous Galerkin methods for convection-diffusion equations and applications in petroleum engineering

This dissertation contains research in discontinuous Galerkin (DG) methods applying to convection-diffusion equations. It contains both theoretical analysis and applications. Initially, we develop a conservative local discontinuous Galerkin (LDG) method for the coupled system of compressible miscible displacement problem in two space dimensions. The main difficulty is how to deal with the discontinuity of approximations of velocity, u, in the convection term across the cell interfaces. To overcome the problems, we apply the idea of LDG with IMEX time marching using the diffusion term to control the convection term. Optimal error estimates in Linfinity(0, T; L2) norm for the solution and the auxiliary variables will be derived. Then, high-order bound-preserving (BP) discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacements on triangular meshes will be developed. There are three main difficulties to make the concentration of each component between 0 and 1. Firstly, the concentration of each component did not satisfy a maximum-principle. Secondly, the first-order numerical flux was difficult to construct. Thirdly, the classical slope limiter could not be applied to the concentration of each component. To conquer these three obstacles, we first construct special techniques to preserve two bounds without using the maximum-principle-preserving technique. The time derivative of the pressure was treated as a source of the concentration equation. Next, we apply the flux limiter to obtain high-order accuracy using the second-order flux as the lower order one instead of using the first-order flux. Finally, L2-projection of the porosity and constructed special limiters that are suitable for multi-component fluid mixtures were used. Lastly, a new LDG method for convection-diffusion equations on overlapping mesh introduced in [J. Du, Y. Yang and E. Chung, Stability analysis and error estimates of local discontinuous Galerkin method for convection-diffusion equations on overlapping meshes, BIT Numerical Mathematics (2019)] showed that the convergence rates cannot be improved if the dual mesh is constructed by using the midpoint of the primitive mesh. They provided several ways to gain optimal convergence rates but the reason for accuracy degeneration is still unclear. We will use Fourier analysis to analyze the scheme for linear parabolic equations with periodic boundary conditions in one space dimension. To investigate the reason for the accuracy degeneration, we explicitly write out the error between the numerical and exact solutions. Moreover, some superconvergence points that may depend on the perturbation constant in the construction of the dual mesh were also found out