Incompressible Miscible Displacement Problem Research Articles

Hybrid mixed discontinuous Galerkin finite element method for incompressible miscible displacement problem

A new hybrid mixed discontinuous Galerkin finite element (HMDGFE) method is constructed for incompressible miscible displacement problem. In this method, the hybrid mixed finite element (HMFE) procedure is considered to solve the pressure and velocity equations, and a new hybrid mixed discontinuous Galerkin finite element procedure is constructed to solve the concentration equation with upwind technique. Compared with other traditional discontinuous Galerkin methods, the new method can reach a global system with less unknowns and sparser stencils. The consistency and conservation of the method are analyzed, the stability and optimal error estimates are also derived by the new technique. Meanwhile, numerical examples are provided to confirm the corresponding theoretical analysis.

A multipoint flux mixed finite element method with mass-conservative characteristic finite element method for incompressible miscible displacement problem

A multipoint flux mixed finite element method with mass-conservative characteristic finite element method for incompressible miscible displacement problem

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 expanded mixed finite element numerical method for incompressible miscible displacement problem involving dispersion term

A characteristic expanded mixed finite element method is presented for miscible displacement problem involving dispersion term. Two-grid algorithm is used to linearize the nonlinear coupling equations. The error estimations of the two-grid solution is conduced in detail by L q $L^q$ error estimates of characteristic expanded mixed finite element methods. Numerical experiment shows that this algorithm is more efficient when the dispersion term D ( u ) $\bm {D}(\bm {u})$ is very small.

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

Numerical simulation for a incompressible miscible displacement problem using a reduced-order finite element method based on POD technique

Numerical simulation for a incompressible miscible displacement problem using a reduced-order finite element method based on POD technique

A reduced-order finite element method based on POD for the incompressible miscible displacement problem

In this paper, we establish a reduced-order finite element (ROFE) method with very few degrees of freedom for the incompressible miscible displacement problem. Firstly, we construct the finite element (FE) method with second-order accuracy in time, where backward difference formulation is used for the time discretization and classical FE formulation is used for the space discretization. Optimal a priori error estimates for the FE solutions are proved. Secondly, we apply the proper orthogonal decomposition (POD) technique to develop the ROFE method, which can effectively reduce degrees of freedom and CPU time. Optimal a priori error estimates for the ROFE solutions are derived. Besides, we give the algorithm process of the ROFE method. Finally, some numerical examples are presented to verify the behavior of the ROFE method for piecewise linear element. And these examples imply the proposed method is feasible and effective for solving the incompressible miscible displacement problem.

A combined hybrid mixed element method for incompressible miscible displacement problem with local discontinuous Galerkin procedure

AbstractIn this article, we propose a combined hybrid discontinuous mixed finite element method for miscible displacement problem with local discontinuous Galerkin method. Here, to obtain more accurate approximation and deal with the discontinuous case, we use the hybrid mixed element method to approximate the pressure and velocity, and use the local discontinuous Galerkin finite element method for the concentration. Compared with other combined methods, this method can improve the efficiency of computation, deal with the discontinuous problem well and keep local mass balance. We study the convergence of this method and give the corresponding optimal error estimates in L∞(L2) for velocity and concentration and the super convergence in L∞(H1) for pressure. Finally, we also present some numerical examples to confirm our theoretical analysis.

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 two‐grid method for characteristic expanded mixed finite element solution of miscible displacement problem

SummaryA combined method consisting of mixed finite element method (MFEM) for the pressure equation and expanded mixed finite element method with characteristics(CEMFEM) for the concentration equation is presented to solve the coupled system of incompressible miscible displacement problem. To solve the resulting nonlinear system of equations efficiently, the two‐grid algorithm relegates all of the Newton‐like iterations to grids much coarser than the original one, with no loss in order of accuracy. It is shown that coarse space can be extremely coarse and our algorithm achieve asymptotically optimal approximation when the mesh sizes satisfy . Numerical experiment is provided to confirm our theoretical results.

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.

A two-grid method for incompressible miscible displacement problems by mixed finite element and Eulerian–Lagrangian localized adjoint methods

In this paper, we present a scheme for solving two-dimensional miscible displacement problems using Eulerian–Lagrangian localized adjoint methods and mixed finite element methods. Since only the velocity and not the pressure appears explicitly in the concentration equation, an Eulerian–Lagrangian localized adjoint method is used to solve the concentration equation and a mixed finite element method is used for the pressure equation. To linearize and decouple the mixed-method equations, we use a two-grid algorithm based on the Newton iteration method for this fully discrete problems. First, we solve the original nonlinear equations on the coarse grid, then, we solve the linearized problem on the fine grid using Newton iteration once. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(h1/2) in this paper.

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.

A priori error estimates of a combined mixed finite element and local discontinuous Galerkin method for an incompressible miscible displacement problem

A numerical approximation for a kind of incompressible miscible displacement problems in high dimension in porous media is studied. Mixed finite element method is applied to the flow equation, and the transport one is solved by the local discontinuous Galerkin method (LDG). Based on interpolation projection properties and the induction hypothesis, a priori hp error estimates are obtained. Numerical results are presented, which verify 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.

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; ...

Superconvergence Analysis of a Full-Discrete Combined Mixed Finite Element and Discontinuous Galerkin Approximation for an Incompressible Miscible Displacement Problem

For an incompressible miscible displacement problem, an effective time-stepping procedure is proposed. The mixed finite element method is applied to the flow equation with uniform meshes, and the transport equation is solved by a full discretized interior penalty discontinuous Galerkin method with regular partitions. Convolution of the Darcy velocity approximation with the Bramble-Schatz kernel function and averaging are applied in the evaluation of the coefficients in the Galerkin procedure for the concentration. A superconvergence estimate is presented.

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.