Displacement Problem In Porous Media Research Articles

A two-grid characteristic block-centered finite difference method for Darcy-Forchheimer slightly compressible miscible displacement problem

In this paper, a two-grid characteristic block-centered finite difference method is developed for solving a nonlinear Darcy-Forchheimer slightly compressible miscible displacement problem in porous media. The method is proposed for solving the nonlinear system in two steps. In the first step, the nonlinear equations are approximated on a coarse grid using the characteristic block-centered finite difference method. In the second step, the nonlinear system is linearized using Newton's method with a small positive parameter to ensure the differentiability of the nonlinear term |u|u in the Darcy-Forchheimer equation. The proposed two-grid method is rigorously analyzed, in which a priori error estimates of the velocity, pressure, concentration and its flux are provided, and the rates of convergence are O(ε+Δt+H3+h2). Finally, numerical experiments are conducted to demonstrate the effectiveness of the proposed methods by comparing the efficiency, especially in terms of CPU time.

A two‐grid discontinuous Galerkin method for an incompressible miscible displacement problem

AbstractAn incompressible miscible displacement problem in porous media is studied. A two‐grid algorithm based on an interior penalty discontinuous Galerkin method is proposed. With this algorithm, solving a nonlinear system on the discontinuous finite element space is reduced to solving a nonlinear problem on a coarse grid and a linear problem on a fine grid. The error estimate for the concentration in H1‐norm and the error estimate for the velocity in L2‐norm are obtained. The numerical experiments are provided to confirm our theoretical analysis.

The weak Galerkin method for the miscible displacement of incompressible fluids in porous media on polygonal mesh

In this paper, we develop a robust weak Galerkin discretization for the incompressible miscible displacement problem in porous media, which is a nonlinear time-dependent system. The scheme we constructed is positive definite and flexible by arbitrarily shaped polygonal mesh without imposing extra conditions on the stabilization. The other feature of our method is that there is no limitation on the ratio of time-step and spatial mesh size. Error estimates are derived for concentration and pressure in the L2 norm and energy norm, respectively. Extensive numerical experiments, including realistic test cases, have been conducted to verify our theoretical results and the efficiency of the method.

A characteristic block-centered finite difference method for Darcy–Forchheimer compressible miscible displacement problem

In this paper, we construct, analyze, and numerically validate a characteristic block-centered finite difference method for Darcy–Forchheimer compressible miscible displacement problem in porous media. The block-centered finite difference method is used to discretize the miscible problem, where the pressure equation is described by the nonlinear Darcy–Forchheimer model, and the transport equation is addressed with the help of the characteristic method. A decoupled algorithm is designed for solving the resulting nonlinear system combined with Picard iteration. Our method is shown to be effective with the established theoretical analysis and several supporting numerical experiments.

A Two-Grid Combined Mixed Finite Element and Discontinuous Galerkin Method for an Incompressible Miscible Displacement Problem in Porous Media

A Two-Grid Combined Mixed Finite Element and Discontinuous Galerkin Method for an Incompressible Miscible Displacement Problem in Porous Media

A new discontinuous Galerkin mixed finite element method for compressible miscible displacement problem

A new combined discontinuous Galerkin method is proposed for compressible miscible displacement problem in porous media. Here, a splitting positive definite mixed finite element method is used for the pressure and Darcy velocity, while an interior penalty discontinuous Galerkin (IPDG) method is used for the transport equation. The stability and convergence of this algorithm are considered, and the optimal a priori error estimate in l∞(L2) for velocity, pressure and concentration are given. Finally we provide some numerical results to confirm our theoretical analysis, and simulate compressible fluid flows through homogeneous and isotropic porous media.

Hp-discontinuous Galerkin Finite Element Method for Incompressible Miscible Displacement in Porous Media

We used (hp-DGFEM) for solving the coupled system of incompressible miscible displacement problem in porous media. Flow and transport equations are solved by hp-symmetric interior penalty Discontinuous Galerkin (hp-SIPG) method. We prove the error analysis in the energy norm to the coupled system of incompressible one-phase flow problem.

A high order hybridizable discontinuous Galerkin method for incompressible miscible displacement in heterogeneous media

An hybridizable discontinuous Galerkin method of arbitrary high order is formulated to solve the miscible displacement problem in porous media. The spatial discretization is combined with a sequential algorithm that decouples the flow and the transport equations. Hybridization produces a linear system for the globally coupled degrees of freedom, that is smaller in size compared to the system resulting from the interior penalty discontinuous Galerkin methods. We study the impact of increasing the polynomial order on the accuracy of the solution. Numerical experiments show that the method converges optimally and that it is robust for highly heterogeneous porous media in two and three dimensions.

A Multipoint Flux Mixed Finite Element Method for Darcy–Forchheimer Incompressible Miscible Displacement Problem

We consider a numerical scheme for incompressible miscible displacement problem in porous media. A multipoint flux mixed finite element method is used to handle the velocity–pressure equation. The standard finite element method is used to approximate the concentration equation. Error estimates for pressure and velocity and concentration are presented. Numerical experiments show that the convergence rates of this scheme are in agreement with the theoretical analysis.

A fully conservative block‐centered finite difference method for Darcy–Forchheimer incompressible miscible displacement problem

AbstractIn this paper, a block‐centered finite difference method is derived for the Darcy–Forchheimer incompressible miscible displacement problem in porous media by introducing an auxiliary flux variable to guarantee full mass conservation. Error estimates for the pressure, velocity, concentration, and auxiliary flux in different discrete norms are established rigorously and carefully on nonuniform grids. Finally, some numerical experiments are presented to show that the convergence rates are in agreement with the theoretical analysis.

Discontinuous finite volume method for compressible miscible displacement problems in porous media

The compressible miscible displacement problem in porous media is modeled by a coupled system of two nonlinear partial differential equations: the pressure equation and the concentration equation are parabolic equation. In this article, we present discontinuous finite volume method for the concentration equation and the pressure equation. The optimal order error estimates for pressure and concentration are obtained in a mesh dependent norm.

Local Discontinuous Galerkin Method with Implicit–Explicit Time Marching for Incompressible Miscible Displacement Problem in Porous Media

In this paper, we shall present two fully-discrete local discontinuous Galerkin methods, coupled with multi-step implicit–explicit time discretization up to second order, for solving the two-dimensional incompressible miscible displacement problem. To avoid the solving of nonlinear algebraic systems, the extrapolation linearization is adopted to diffusion–dispersion tensor. Under weak temporal-spatial conditions, the optimal error estimates in $$L^{\infty }(L^{2})$$ norm for both concentration and velocity are derived. Numerical experiments are also given to demonstrate the theoretical results.

Convergence Analysis for Mixed Finite Element Method of Positive Semi-definite Problems

A mixed element-characteristic finite element method is put forward to approximate three-dimensional incompressible miscible positive semi-definite displacement problems in porous media. The mathematical model is formulated by a nonlinear partial differential system. The flow equation is approximated by a mixed element scheme, and the pressure and Darcy velocity are computed at the same time. The concentration equation is treated by the method of characteristic finite element, where the convection term is discretized along the characteristics and the diffusion term is computed by the scheme of finite element. The method of characteristics can confirm strong computation stability at the sharp fronts and avoid numerical dispersion and nonphysical oscillation. Furthermore, a large step is adopted while small time truncation error and high order accuracy are obtained. It is an important feature in numerical simulation of seepage mechanics that the mixed volume element can compute the pressure and Darcy velocity simultaneously and the accuracy of Darcy velocity is improved one order. Using the form of variation, energy method, $L^2$ projection and the technique of priori estimates of differential equations, we show convergence analysis for positive semi-definite problems. Then a powerfu tool is given to solve international famous problems.

Parallel algorithm combined with mixed element procedure for compressible miscible displacement problem

Based on overlapping domain decomposition, we construct a parallel mixed finite element algorithm for solving the compressible miscible displacement problem in porous media. The algorithm is fully parallel. We consider the relation between the convergence rate and discretization parameters, including the overlapping degree of the subspaces. We give the corresponding error estimate, which tells us that only two iterations are needed to reach to given accuracy at each time level. Numerical results are presented to confirm our theoretical analysis.

Local Discontinuous Galerkin Method for Incompressible Miscible Displacement Problem in Porous Media

In this paper, we develop local discontinuous Galerkin method for the two-dimensional coupled system of incompressible miscible displacement problem. Optimal error estimates in $$L^{\infty }(0, T; ...

A new MCC–MFE method for compressible miscible displacement in porous media

This paper presents a new mass-conservative finite element method for solving the compressible miscible displacement problem in porous media. This new method combines characteristic tracking with the mixed finite element method. Specifically, the mass-conservative characteristic finite element method is applied to the concentration equation whereas the mixed finite element method with continuous flux approximation is used for the flow equation. This procedure conserves mass globally. Convergence of the fully discrete scheme is analyzed and optimal L2-norm and H1-norm error estimates for the concentration are obtained. Numerical experiments are presented to illustrate the theoretical analysis.

Error analysis of the semi-discrete local discontinuous Galerkin method for compressible miscible displacement problem in porous media

In this paper, local discontinuous Galerkin method for flow and transport is introduced for the coupled system of compressible miscible displacement problem. Optimal error estimates in L∞(0,T;L2) norm not only for the solution itself but also for the auxiliary variables are derived. The main technical difficulties in the analysis include the nonlinearity, the coupling of the models and the treatment of the inter-element jump terms which arise from the discontinuous nature of the numerical method. The numerical results illustrate the accuracy and capability of the method.

A combined mixed finite element method and local discontinuous Galerkin method for miscible displacement problem in porous media

A combined method consisting of the mixed finite element method for flow and the local discontinuous Galerkin method for transport is introduced for the one-dimensional coupled system of incompressible miscible displacement problem. Optimal error estimates in L ∞(0, T;L 2) for concentration c, in L 2(0, T; L 2) for c x and L ∞(0, T;L 2) for velocity u are derived. The main technical difficulties in the analysis include the treatment of the inter-element jump terms which arise from the discontinuous nature of the numerical method, the nonlinearity, and the coupling of the models. Numerical experiments are performed to verify the theoretical results. Finally, we apply this method to the one-dimensional compressible miscible displacement problem and give the numerical experiments to confirm the efficiency of the scheme.

Mixed finite element method and the characteristics-mixed finite element method for a slightly compressible miscible displacement problem in porous media

Single-phase, two-component slightly compressible miscible flow in porous media, is governed by a system of nonlinear partial differential equations. The density of the mixture is related to the pressure and the concentration. An approximate scheme is introduced for this system, which is constructed by two methods. A standard mixed finite element is used for the pressure equation. A characteristic-mixed finite element method is presented for the concentration equation. Characteristic approximation is applied to handle the convection part of the concentration equation, and a lowest order mixed finite element spatial approximation is adopted to deal with the diffusion part. Thus, the scalar unknown concentration and the diffusive flux can be approximated simultaneously. In order to derive the optimal L2-norm error estimates, a post-processing step is included in the approximation to the scalar unknown concentration. A numerical experiment is presented finally to validate the theoretical analysis.

Modified Method of Characteristics Combined with Finite Volume Element Methods for Incompressible Miscible Displacement Problems in Porous Media

The incompressible miscible displacement problem in porous media is modeled by a coupled system of two nonlinear partial differential equations, the pressure-velocity equation and the concentration equation. In this paper, we present a mixed finite volume element method (FVEM) for the approximation of the pressure-velocity equation. Since modified method of characteristics (MMOC) minimizes the grid orientation effect, for the approximation of the concentration equation, we apply a standard FVEM combined with MMOC. A priori error estimates in L∞(L2) norm are derived for velocity, pressure and concentration. Numerical results are presented to substantiate the validity of the theoretical results.