Streamline method for resolving sharp fronts for complex two-phase flow in porous media

In this paper, we present a fast streamline-based numerical method for the two-phase flow equations in high-rate flooding scenarios for incompressible fluids in heterogeneous and anisotropic porous media. A fractional flow formulation is adopted and a discontinuous Galerkin method (DG) is employed to solve the pressure equation. Capillary effects can be neglected in high-rate flooding scenarios. This allows us to present an improved streamline approach in combination with the one-dimensional front tracking method to solve the transport equation. To handle the high computational costs of the DG approximation, domain decomposition is applied combined with an algebraic multigrid preconditioner to solve the linear system. Special care at the interior interfaces is required and the streamline tracer has to include a dynamic communication strategy. The method is validated in various two- and three-dimensional tests, where comparisons of the solutions in terms of approximation of flow front propagation with standard fully implicit finite-volume methods are provided.     

Sequential implicit vertex approximate gradient discretization of incompressible two-phase Darcy flows with discontinuous capillary pressure

This work deals with sequential implicit schemes for incompressible and immiscible two-phase Darcy flows which are commonly used and well understood in the case of spatially homogeneous capillary pressure functions. To our knowledge, the stability of this type of splitting schemes solving sequentially a pressure equation followed by the saturation equation has not been investigated so far in the case of discontinuous capillary pressure curves at different rock type interfaces. It will be shown here to raise severe stability issues for which stabilization strategies are investigated in this work. To fix ideas, the spatial discretization is based on the Vertex Approximate Gradient (VAG) scheme accounting for unstructured polyhedral meshes combined with an Hybrid Upwinding (HU) of the transport term and an upwind positive approximation of the capillary and gravity fluxes. The sequential implicit schemes are built from the total velocity formulation of the two-phase flow model and only differ in the way the conservative VAG total velocity fluxes are approximated. The stability, accuracy and computational cost of the sequential implicit schemes studied in this work are tested on oil migration test cases in 1D, 2D and 3D basins with a large range of capillary pressure parameters for the drain and barrier rock types. It will be shown that usual splitting strategies fail to capture the right solutions for highly contrasted rock types and that it can be fixed by maintaining locally the pressure saturation coupling at different rock type interfaces in the definition of the conservative total velocity fluxes. The numerical investigation of the sequential schemes is also extended to the widely used finite volume Two-Point Flux Approximation spatial discretization.

     

Immiscible two-phase Darcy flow model accounting for vanishing and discontinuous capillary pressures: application to the flow in fractured porous media

Fully implicit time-space discretizations applied to the two-phase Darcy flow problem leads to the systems of nonlinear equations, which are traditionally solved by some variant of Newton’s method. The efficiency of the resulting algorithms heavily depends on the choice of the primary unknowns since Newton’s method is not invariant with respect to a nonlinear change of variable. In this regard, the role of capillary pressure/saturation relation is paramount because the choice of primary unknowns is restricted by its shape. We propose an elegant mathematical framework for two-phase flow in heterogeneous porous media resulting in a family of formulations, which apply to general monotone capillary pressure/saturation relations and handle the saturation jumps at rocktype interfaces. The presented approach is applied to the hybrid dimensional model of two-phase water-gas Darcy flow in fractured porous media for which the fractures are modelled as interfaces of co-dimension one. The problem is discretized using an extension of vertex approximate gradient scheme. As for the phase pressure formulation, the discrete model requires only two unknowns by degree of freedom.     

Modelling Two-Phase Incompressible Flow in Porous Media Using Mixed Hybrid and Discontinuous Finite Elements

In this paper, we present a numerical model for simulating two-phase (oil–water and air–water) incompressible and immiscible flow in porous media. The mathematical model which is based on a fractional flow formulation is formed of two nonlinear partial differential equations: a mean pressure equation and a water saturation equation. These two equations can be solved in a sequential manner. Two numerical methods are used to discretize the equations of the two-phase flow model: mixed hybrid finite elements are used to treat the pressure equation, h-based Richards' equation and the diffusion term in the saturation equation, the advection term in the saturation equation is treated with the discontinuous finite elements. We propose a better way to calculate the nonlinear coefficients contained in our equations on each element of the discretized domain. In heterogeneous porous media, the saturation becomes discontinuous at the interface between two porous media. We show in this paper how to use the capillary pressure–saturation relationship in order to handle the saturation jump in the mixed hybrid finite element method. The two-phase flow simulator is verified against analytical solutions for some flow problems treated by other authors.     

A discontinuous Galerkin method for two-phase flow in a porous medium enforcing H(div) velocityand continuous capillary pressure

We consider the slightly compressible two-phase flow problem in a porous medium with capillary pressure. The problem is solved using the implicit pressure, explicit saturation (IMPES) method, and the convergence is accelerated with iterative coupling of the equations. We use discontinuous Galerkin to discretize both the pressure and saturation equations. We apply two improvements, which are projecting the flux to the mass conservative H(div)-space and penalizing the jump in capillary pressure in the saturation equation. We also discuss the need and use of slope limiters and the choice of primary variables in discretization. The methods are verified with two- and three-dimensional numerical examples. The results show that the modifications stabilize the method and improve the solution.     

Efficient block preconditioners for the coupled equations of pressure and deformation in highly discontinuous media

Large‐scale simulations of flow in deformable porous media require efficient iterative methods for solving the involved systems of linear algebraic equations. Construction of efficient iterative methods is particularly challenging in problems with large jumps in material properties, which is often the case in geological applications, such as basin evolution at regional scales. The success of iterative methods for this type of problems depends strongly on finding effective preconditioners. This paper investigates how the block‐structured matrix system arising from single‐phase flow in elastic porous media should be preconditioned, in particular for highly discontinuous permeability and significant jumps in elastic properties. The most promising preconditioner combines algebraic multigrid with a Schur complement‐based exact block decomposition. The paper compares numerous block preconditioners with the aim of providing guidelines on how to formulate efficient preconditioners. Copyright © 2010 John Wiley & Sons, Ltd.     

基于OpenMP的非连续变形分析并行计算方法

非连续变形分析（DDA）方法严格满足平衡要求和能量守恒,具有完全的运动学及数值可靠性,但对大规模岩土工程问题的数值模拟耗时太长,尤其是线性方程组求解,并行计算可以很好地解决该问题。首先基于DDA方法的基本理论,阐述了适用于DDA方法中的基于块的行压缩法和基于“试验-误差”迭代格式的非零位置记录;其次,引入块雅可比迭代法并行求解DDA方法的线性方程组,并改进了相应的非零存储方法;最后,基于OpenMP实现了DDA线性方程组求解并行计算,并将其应用于地下洞室群的破坏过程分析,以加速比为并行效率的指标评价,结果表明,该并行计算策略可以极大提高DDA的计算效率,而且适合各种规模的问题。     

Numerical methods for flow in a porous medium with internal boundaries

Flow of fluids and transport of solutes in porous media are subjects of wide interest in several fields of applications: reservoir engineering, subsurface hydrology, chemical engineering, etc. In this paper we will study two-phase flow in a model consisting of two different types of sediments. Here, the absolute permeability, the relative permeabilities and the capillary pressure are discontinuous functions in space. This leads to interior boundary value problems at the interface between the sediments. The saturation S_w will be discontinuous or experience large gradients at the interface. A new solution procedure for such problems will be presented. The method combines the modified method of characteristics with a weak formulation where the basis functions are discontinuous at the interior boundary. The modified method of characteristics will provide a good first approximation for the jump in the discontinuous basis functions, which leads to a fast converging iterative solution scheme for the complete problem. The method has been implemented in a two-dimensional simulator, and results from numerical experiments will be presented. This revised version was published online in August 2006 with corrections to the Cover Date.     

Finite volume approximation for an immiscible two-phase flow in porous media with discontinuous capillary pressure

We consider an immiscible incompressible two-phase flow in a porous medium composed of two different rocks so that the capillary pressure field is discontinuous at the interface between the rocks. This leads us to apply a concept of multivalued phase pressures and a notion of weak solution for the flow which have been introduced in Cancès and Pierre (SIAM J Math Anal 44(2):966–992, 2012). We discretize the problem by means of a numerical algorithm which reduces to a standard finite volume scheme in each rock and prove the convergence of the approximate solution to a weak solution of the two-phase flow problem. The numerical experiments show in particular that this scheme permits to reproduce the oil-trapping phenomenon.     

A continuous/discontinuous Galerkin framework for modeling coupled subsurface and surface water flow

We consider conjunctive surface-subsurface flow modeling, where surface water flow is described by the shallow water equations and ground water flow by Richards’ equation for the vadose zone. Coupling between the models is based on the continuity of flux and water pressure. Numerical approximation of the coupled model using the framework of discontinuous Galerkin (DG) methods is formulated. In the subsurface, the local discontinuous Galerkin (LDG) method is used to approximate ground water velocity and hydraulic head; a DG method is also used to approximate surface water velocity and elevation. This approach allows for a weak coupling of the models and the use of different approximating spaces and/or meshes within each regime. A simplified LDG method based on continuous approximations to water head is also described. Numerical results that investigate physical and numerical aspects of surface–subsurface flow modeling are presented. This work was supported by National Science Foundation grant DMS-0411413.     

Algebraic multigrid techniques for discontinuous Galerkin methods with varying polynomial order

We present a parallel algebraic multigrid (AMG) algorithm for the implicit solution of the Darcy problem discretized by the discontinuous Galerkin (DG) method that scales optimally for regular and irregular meshes. The main idea centers on recasting the preconditioning problem so that existing AMG solvers for nodal lower order finite elements can be leveraged. This is accomplished by a transformation operator which maps the solution from a Lagrange basis representation to a Legendre basis representation. While this mapping function must be user supplied, we demonstrate how easily it can be constructed for somepopular finite element representations includingquadrilateral/hexahedral and triangular/tetrahedral DG formulations. Furthermore, we show that the mapping does not depend on the Jacobian transformation between reference and physical space and so it can be constructed with very limited mesh information. Parallel performance studies demonstrate the versatility of this approach.     

Ordering-based nonlinear solver for fully-implicit simulation of three-phase flow

The Fully Implicit method (FIM) is often the method of choice for the temporal discretization of the partial differential equations governing multiphase flow in porous media. The FIM involves solving large coupled systems of nonlinear algebraic equations. Newton-based methods, which are employed to solve the nonlinear systems, can suffer from convergence problems—this is especially true for large time steps in the presence of highly nonlinear flow physics. To overcome such convergence problems, the time step is usually reduced, and the Newton steps are restarted from the solution of the previous (converged) time step. Recently, potential ordering and the reduced-Newton method were used to solve immiscible three-phase flow in the presence of buoyancy and capillary effects (e.g., Kwok and Tchelepi, J. Comput. Phys. 227(1), 706–727 2007). Here, we improve the robustness of the potential-based ordering method in the presence of gravity. Furthermore, we also extend this nonlinear approach to interphase mass transfer. Our algorithm deals effectively with mass transfer between the liquid and gas phases, including phase disappearance (e.g., gas going back in solution) and reappearance (e.g., gas coming out of solution and forming a separate phase), as a function of pressure and composition. Detailed comparisons of the robustness and efficiency of the potential-based solver with state-of-the-art nonlinear/linear solvers are presented for immiscible two-phase (Dead-Oil), Black-Oil, and compositional problems using heterogeneous models. The results show that for large time steps, our nonlinear ordering-based solver reduces the number of nonlinear iterations significantly, which leads to gains in the overall computational cost.     

Ordering-based nonlinear solver for fully-implicit simulation of three-phase flow

The Fully Implicit Method (FIM) is often the method of choice for the temporal discretization of the partial differential equations governing multiphase flow in porous media. The FIM involves solving large coupled systems of nonlinear algebraic equations. Newton-based methods, which are employed to solve the nonlinear systems, can suffer from convergence problems—this is especially true for large time steps in the presence of highly nonlinear flow physics. To overcome such convergence problems, the time step is usually reduced, and the Newton steps are restarted from the solution of the previous (converged) time step. Recently, potential ordering and the reduced-Newton method were used to solve immiscible three-phase flow in the presence of buoyancy and capillary effects (e.g., Kwok and Tchelepi, J. Comput. Phys. 227(1), 706–727 9). Here, we improve the robustness of the potential-based ordering method in the presence of gravity. Furthermore, we also extend this nonlinear approach to interphase mass transfer. Our algorithm deals effectively with mass transfer between the liquid and gas phases, including phase disappearance (e.g., gas going back in solution) and reappearance (e.g., gas coming out of solution and forming a separate phase), as a function of pressure and composition. Detailed comparisons of the robustness and efficiency of the potential-based solver with state-of-the-art nonlinear/linear solvers are presented for immiscible two-phase (Dead-Oil), Black-Oil, and compositional problems using heterogeneous models. The results show that for large time steps, our nonlinear ordering-based solver reduces the number of nonlinear iterations significantly, which leads to gains in the overall computational cost.     

非连续变形分析（DDA）线性方程组的高效求解算法

非连续变形分析(DDA)方法对大规模工程问题的数值模拟耗时太长,其中线性方程组求解耗时可占总计算时间的70%以上,因此,高效的线性方程组解法是重要研究课题。首先,阐述了适用于DDA方法的基于块的行压缩法和基于试验-误差迭代格式的非0位置记录;然后,针对DDA的子矩阵技术,将块雅可比迭代法 (BJ)、预处理的块共轭梯度法 (PCG,包括Jacobi-PCG、SSOR-PCG) 引入DDA方法,重点研究了线性方程组求解过程中的关键运算;最后,通过两个洞室开挖算例,分析了各线性方程组求解算法在DDA中的计算效率。研究表明：与迭代法相比,直解法无法满足大规模工程计算需要;BJ迭代法与块超松弛迭代法(BSOR)的效率差别不大,但明显不如PCG迭代法。因此,建议采用PCG迭代法求解DDA线性方程组,特别是SSOR-PCG值得推广;如果开展并行计算研究,Jacobi-PCG是较好的选择,当刚度矩阵惯性优势明显时,BJ迭代法同样有效。     

A space-time discontinuous Galerkin method applied to single-phase flow in porous media

A space-time discontinuous Galerkin finite element method is proposed and applied to a convection-dominant single-phase flow problem in porous media. The numerical scheme is based on a coupled space-time finite element discretization allowing for discontinuous approximations in space and in time. The continuities on the element interfaces are weakly enforced by the flux treatments, so that no extra penalty factor has to be determined. The resulting space-time formulation possesses the advantage of capturing the steep concentration front with sharp gradients efficiently. The stability and reliability of the proposed approach is demonstrated by numerical experiments. The author is grateful to the DFG (German Science Foundation—Deutsche Forschungsgemeinschaft) for the financial support under the grant number Di 430/4-2.     

Adjoint analysis of Buckley-Leverett and two-phase flow equations

This paper analyzes the adjoint equations and boundary conditions for porous media flow models, specifically the Buckley-Leverett equation, and the compressible two-phase flow equations in mass conservation form. An adjoint analysis of a general scalar hyperbolic conservation law whose primal solutions include a shock jump is initially presented, and the results are later specialized to the Buckley-Leverett equation. The non-convexity of the Buckley-Leverett flux function results in adjoint characteristics that are parallel to the shock front upstream of the shock and emerge from the shock front downstream of the shock. Thus, in contrast to the behavior of Burgers’ equation where the adjoint is continuous at a shock, the Buckley-Leverett adjoint, in general, contains a discontinuous jump across the shock. Discrete adjoint solutions from space-time discontinuous Galerkin finite element approximations of the Buckley-Leverett equation are shown to be consistent with the derived closed-form analytical solutions. Furthermore, a general result relating the adjoint equations for different (though equivalent) primal equations is used to relate the two-phase flow adjoints to the Buckley-Leverett adjoint. Adjoint solutions from space-time discontinuous Galerkin finite element approximations of the two-phase flow equations are observed to obey this relationship.     

Local problem-based <Emphasis Type="Italic">a posteriori</Emphasis> error estimators for discontinuous Galerkin approximations of reactive transport

We consider adaptive discontinuous Galerkin (DG) methods for solving reactive transport problems in porous media. To guide anisotropic and dynamic mesh adaptation, a posteriori error estimators based on solving local problems are established. These error estimators are efficient to compute and effective to capture local phenomena, and they apply to all the four primal DG schemes, namely, symmetric interior penalty Galerkin, nonsymmetric interior penalty Galerkin, incomplete interior penalty Galerkin, and the Oden–Babuška-Baumann version of DG. Numerical results are provided to illustrate the effectiveness of the proposed error estimators.     

High order discontinuous Galerkin method for simulating miscible flooding in porous media

We present a high-order method for miscible displacement simulation in porous media. The method is based on discontinuous Galerkin discretization with weighted average stabilization technique and flux reconstruction post processing. The mathematical model is decoupled and solved sequentially. We apply domain decomposition and algebraic multigrid preconditioner for the linear system resulting from the high-order discretization. The accuracy and robustness of the method are demonstrated in the convergence study with analytical solutions and heterogeneous porous media, respectively. We also investigate the effect of grid orientation and anisotropic permeability using high-order discontinuous Galerkin method in contrast with cell-centered finite volume method. The study of the parallel implementation shows the scalability and efficiency of the method on parallel architecture. We also verify the simulation result on highly heterogeneous permeability field from the SPE10 model.     

Numerical modeling of two-phase hysteresis combined with an interface condition for heterogeneous porous media

This paper presents a numerical implementation of two-phase capillary hysteresis and its combination with a capillary interface condition for the treatment of heterogeneities. The hysteresis concepts chosen in this work are first implemented in a node-centered FV discretization scheme and subsequently combined with the interface condition that predicts sharp saturation discontinuities at material interfaces, based on a pressure equilibrium concept. This approach allows for the approximation of history-dependent, and at the same time discontinuous, saturations at material interfaces. The resulting model provides a well-defined evolution of the hysteretic capillary pressure–saturation relationships at material interfaces that is independent of the grid spacing. As demonstrated with a simple 1-D example, this concept therefore offers the advantage that the solution of a two-phase flow problem involving hysteresis does not relate to the grid resolution at the material interfaces.     

Finite element methods for geothermal reservoir simulation

Two finite element algorithms suitable for long term simulation of geothermal reservoirs are presented. Both methods use a diagonal mass matrix and a Newton iteration scheme. The first scheme solves the 2N unsymmetric algebraic equations resulting from the finite element discretization of the equations governing the flow of heat and mass in porous media by using a banded equation solver. The second method, suitable for problems in which the transmissibility terms are small compared to the accumulation terms, reduces the set of N equations for the Newton corrections to a symmetric system. Comparison with finite difference schemes indicates that the proposed algorithms are competitive with existing methods.