多孔介质中两相不可压缩流体的时间局部网格加密有限差分格式

引入Charent压力变量,对于多孔介质中两相不可压缩流体的非混溶驱动问题,其模型表现为耦合的非线性偏微分方程组,一个是压力方程,另一个为饱和度方程．文中考虑一维问题且假定达西速度“已知,建立了在时间上进行局部加密的有限差分格式,给出了饱和度的最大模误差估计．最后给出了数值算例．     

A nonlinear test case for the finite element method in two-phase flow

The ability of the finite element method to compute the motion of sharp interfaces in two-phase flow is examined by applying it to a test problem for which an analytical solution can be found. The problem is one of imbibition, the nonlinear diffusion of a fixed amount of water into an oil filled porous medium and can be solved exactly by similarity and the separation of variables method used by Boyer. The finite element program used was of the Galerkin type and employed a self-adaptive time stepping algorithm with both linear and quadratic isoparametric triangular elements. Results are presented for both elements and show that there is little difficulty in this type of diffusion problem in following the oil-water interface to accuracies of 2 or 3 percent.     

The numerical simulation of multiphase flow through a porous medium and its application to reservoir engineering

The analysis of multiphase flow in porous media is of considerable significance in the field of petroleum reservoir simulation, where accurate predictions of fluid flow are important in assessing the performance of oil and gas fields. The specific case of two-phase immiscible flow is considered by first deriving the governing nonlinear partial differential equations. The space discretization is then carried out making use of the additional versatility of the finite element method compared with originally used finite differences. By using a pair of dependent variables P and R, the bandwidth of the discrete space-continuous time equations may be reduced to increase significantly the speed of the algorithm. A discussion of time-stepping methods is followed by an application of the technique to a five-spot,an extraction pattern used in the field. The boundary conditions used to simulate this flow pattern are also discussed.     

Error estimates for the finite volume discretization for the porous medium equation

We analyze the convergence of a numerical scheme for a class of degenerate parabolic problems modelling reactions in porous media, and involving a nonlinear, possibly vanishing diffusion. The scheme involves the Kirchhoff transformation of the regularized nonlinearity, as well as an Euler implicit time stepping and triangle based finite volumes. We prove the convergence of the approach by giving error estimates in terms of the discretization and regularization parameter.     

A hybrid numerical model for multiphase fluid flow in a deformable porous medium

In this paper, a fully coupled finite volume-finite element model for a deforming porous medium interacting with the flow of two immiscible pore fluids is presented. The basic equations describing the system are derived based on the averaging theory. Applying the standard Galerkin finite element method to solve this system of partial differential equations does not conserve mass locally. A non-conservative method may cause some accuracy and stability problems. The control volume based finite element technique that satisfies local mass conservation of the flow equations can be an appropriate alternative. Full coupling of control volume based finite element and the standard finite element techniques to solve the multiphase flow and geomechanical equilibrium equations is the main goal of this paper. The accuracy and efficiency of the method are verified by studying several examples for which analytical or numerical solutions are available. The effect of mesh orientation is investigated by simulating a benchmark water-flooding problem. A representative example is also presented to demonstrate the capability of the model to simulate the behavior in heterogeneous porous media.     

Degenerate two-phase incompressible flow III. Sharp error estimates

Summary. This is the third paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic-parabolic partial differential system which describes the flow of two incompressible, immiscible fluids in porous media. In this paper we consider a finite element approximation for this system. The elliptic equation for the pressure and velocity is approximated by a mixed finite element method, while the degenerate parabolic equation for the saturation is approximated by a Galerkin finite element method. A fully discrete approximation is analyzed. Sharp error estimates in energy norms are obtained for this approximation. The error analysis does not use any regularization of the saturation equation; the error estimates are derived directly from the degenerate equation. Also, the analysis does not impose any restriction on the nature of degeneracy. Finally, it respects the minimal regularity on the solution of the differential system. Received March 9, 1998 / Revised version received July 17, 2000 / Published online May 30, 2001     

Python framework for hp-adaptive discontinuous Galerkin methods for two-phase flow in porous media

In this paper we present a framework for solving two-phase flow problems in porous media. The discretization is based on a Discontinuous Galerkin method and includes local grid adaptivity and local choice of polynomial degree. The method is implemented using the new Python frontend Dune-FemPy to the open source framework Dune. The code used for the simulations is made available as Jupyter notebook and can be used through a Docker container. We present a number of time stepping approaches ranging from a classical IMPES method to a fully coupled implicit scheme. The implementation of the discretization is very flexible allowing to test different formulations of the two-phase flow model and adaptation strategies.     

Numerical simulations of water–gas flow in heterogeneous porous media with discontinuous capillary pressures by the concept of global pressure

We present an approach and numerical results for a new formulation modeling immiscible compressible two-phase flow in heterogeneous porous media with discontinuous capillary pressures. The main feature of this model is the introduction of a new global pressure, and it is fully equivalent to the original equations. The resulting equations are written in a fractional flow formulation and lead to a coupled degenerate system which consists of a nonlinear parabolic (the global pressure) equation and a nonlinear diffusion–convection one (the saturation equation) with nonlinear transmission conditions at the interfaces that separate different media. The resulting system is discretized using a vertex-centred finite volume method combined with pressure and flux interface conditions for the treatment of heterogeneities. An implicit Euler approach is used for time discretization. A Godunov-type method is used to treat the convection terms, and the diffusion terms are discretized by piecewise linear conforming finite elements. We present numerical simulations for three one-dimensional benchmark tests to demonstrate the ability of the method to approximate solutions of water–gas equations efficiently and accurately in nuclear underground waste disposal situations.     

Solution of double nonlinear problems in porous media by a combined finite volume–finite element algorithm

The combined finite volume–finite element scheme for a double nonlinear parabolic convection-dominated diffusion equation which models the variably saturated flow and contaminant transport problems in porous media is extended. Whereas the convection is approximated by a finite volume method (Multi-Point Flux Approximation), the diffusion is approximated by a finite element method. The scheme is fully implicit and involves a relaxation-regularized algorithm. Due to monotonicity and conservation properties of the approximated scheme and in view of the compactness theorem we show the convergence of the numerical scheme to the weak solution. Our scheme is applied for computing two dimensional examples with different degrees of complexity. The numerical results demonstrate that the proposed scheme gives good performance in convergence and accuracy.     

Extended finite element modeling of deformable porous media with arbitrary interfaces

In this paper, an enriched finite element method is presented for numerical simulation of saturated porous media. The arbitrary discontinuities, such as material interfaces, are encountered via the extended finite element method (X-FEM) by enhancing the standard FEM displacements. The X-FEM technique is applied to the governing equations of porous media for the spatial discretization, followed by a generalized Newmark scheme used for the time domain discretization. In X-FEM, the material interfaces are represented independently of element boundaries and the process is accomplished by partitioning the domain with some triangular sub-elements whose Gauss points are used for integration of the domain of elements. Finally, several numerical examples are analyzed, including the dynamic analysis of the failure of lower San Fernando dam, to demonstrate the efficiency of the X-FEM technique in saturated porous soils.     

Slightly compressible and immiscible two-phase flow in porous media

We describe the flow of two compressible phases in a porous medium. We consider the case of slightly compressible phases for which the density of each phase follows an exponential law with a small compressibility factor. A nonlinear parabolic system including quadratic velocity terms is derived to describe compressible and immiscible two-phase flow in porous media. In one-dimensional space, we establish the existence and uniqueness of a local strong solution for the regularized system. We show also that the saturation is physically admissible. We describe the asymptotic behavior of the solutions when the compressibility factor goes to zero.     

A PRIORI L_2 ERROR ESTIMATES FOR A NONLINEAR PARABOLIC SYSTEM BASED ON COMBINING THE METHOD OF CHARACTERISTICS WITH MIXED FINITE ELEMENT PROCEDURE

A nonlinear parabolic system is derived to describe incompressible nuclear waste-disposal contamination in porous media. A sequential implicit tirne-stepping is defined, in which the pressure and Darcy velocity of the mixture are approximated simultaneously by a mixed finite element method and the brine, radionuclid and heat are treated by a combination of a Galerkin finite element method and the method of characteristics. Optimal-order convergence in L2 is proved. Time-truncation errors of standard procedures are reduced by time stepping along the characteristics of the hyperbolic part of the brine, radionuclide and heal equalios, temporal and spatial error are lossened by direct compulation of the velocity in the mixed method, as opposed to differentiation of the pressure.     

An upwind-mixed method on changing meshes for two-phase miscible flow in porous media

Two-phase miscible flow in porous media is governed by a system of nonlinear partial differential equations. In this paper, the upwind-mixed method on dynamically changing meshes is presented for the problem in two dimensions. The pressure is approximated by a mixed finite element method and the concentration by a method which upwinds the convection and incorporates diffusion using an expanded mixed finite element method. The method developed is shown to obtain almost optimal rate error estimate. When the method is modified we can obtain the optimal rate error estimate that is well known for static meshes. The modification of the scheme is the construction of a linear approximation to the solution, which is used in projecting the solution from one mesh to another. Finally, numerical experiments are given.     

Generalized EVI-space-time Galerkin method for dynamical modeling in porous media

We investigate a generalized space–time discontinuous Galerkin formulation for modeling dynamical phenomena in porous media. The finite element approximation is based on a coupled space–time discretization. An advanced Embedded Velocity Integration (EVI) technique is applied to circumvent the direction solution of the displacement fields but to solve the rate term of the unknowns. Moverover, a stabilization factor α is introduced to modify the integration scheme of the velocities, which further contributes to the numerical stabilization of the overall solution. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite element method for two‐phase immiscible flow

An explicit finite element method for numerically solving the two‐phase, immiscible, incompressible flow in a porous medium in two space dimensions is analyzed. The method is based on the use of a mixed finite element method for the approximation of the velocity and pressure a discontinuous upwinding finite element method for the approximation of the saturation. The mixed method gives an approximate velocity field in the precise form needed by the discontinuous method, which is trivially conservative and fully parallelizable in computation. It is proven that it converges to the exact solution. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 407–416, 1999     

Three-dimensional finite/infinite elements analysis of fluid flow in porous media

Fluid flow in naturally fractured porous media can always be regarded as an unbounded domain problem and be better solved by finite/infinite elements. In this paper, a three-dimensional two-direction mapped infinite element is generated and combined with conventional finite elements and one direction infinite element to simulate poroelasticity. Therefore, the entire semi-infinite domain can be included in the numerical analysis. Both single- and dual-porosity porous media are considered. For the purpose of validation, we compare the results of finite/infinite elements with those of finite elements under two extreme boundary conditions. The comparison indicated that mapped infinite element is an appropriate approach to model fluid flow in porous media and provides an intermediate solution.     

Time of complete displacement of an immiscible compressible fluid by water in porous media: Application to gas migration in a deep nuclear waste repository

A system of evolutionary partial differential equations (PDEs) describing the two-phase flow of immiscible fluids, such as water–gas, through porous media is studied. In this formulation, the wetting and nonwetting phases are treated to be incompressible and compressible, respectively. This treatment is indeed necessary when a compressible nonwetting phase is subjected to compression during confinement. The system of PDEs consists of an evolution equation for the wetting-phase saturation and an evolution equation for the pressure in the nonwetting phase. This system is applied to the problem of unsaturated flows to assess gas migration and two-phase flow through engineered and geological barriers for a deep repository for radioactive waste. This paper is primarily concerned with the large time behavior of solutions of this system. Under some realistic assumptions on the data, we derive estimates of the speed of propagation of the gas by water in porous media. Namely, we establish estimates of time stabilization for the water saturation to a constant limit profile. The analysis is based on the energy methods whose main idea involves deriving and studying suitable ordinary differential inequalities. We show that the time of complete displacement of a gas by water may be at most infinite or finite depending essentially on the power parameters defining the capillary pressure and the relative permeabilities. This result is then illustrated with two examples in the context of gas migration in a deep nuclear waste repository. We consider Van Genuchten’s and Brooks–Corey’s models for a two-phase water–gas system.     

可压可溶两相驱动问题的迎风混合元方法

研究迎风混合元方法求解多孔介质中含弥散的可压缩可混溶两相渗流驱动问题, 利用变分形式和先验估计的理论技巧,得到饱和度近似的L²模和压力近似的H¹模最优阶误差估计,数值实验证实该方法在克服数值扩散和非物理振荡方面是很有效的.     

Front tracking for two-phase flow in reservoir simulation by adaptive mesh

In this article, an algorithm for the numerical approximation of two-phase flow in porous media by adaptive mesh is presented. A convergent and conservative finite volume scheme for an elliptic equation is proposed, together with the finite difference schemes, upwind and MUSCL, for a hyperbolic equation on grids with local refinement. Hence, an IMPES method is applied in an adaptive composite grid to track the front of a moving solution. An object-oriented programmation technique is used. The computational results for different examples illustrate the efficiency of the proposed algorithm. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 673–697, 1997     

关于求解二相渗流的平面问题的一类新方法

本文提出一类求解二相平面渗流问题的新方法:用有限元法求解关于压力分布的椭圆型方程,然后利用所得的对压力梯度的半解析解,根据已有的饱和度沿流线传播的精确公式求得饱和度场.其主要特点和优点是能克服通常的数值模拟方法所具有的数值弥散,给出准确清晰的水驱油前沿饱和度间断面的位置,并且完全避免了通常必须与压力方程联立求解或交替求解的饱和度方程,从而使计算工作量大大减少.