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.     

A time‐discontinuous Galerkin method for the dynamical analysis of porous media

We present a time‐discontinuous Galerkin method (DGT) for the dynamic analysis of fully saturated porous media. The numerical method consists of a finite element discretization in space and time. The discrete basis functions are continuous in space and discontinuous in time. The continuity across the time interval is weakly enforced by a flux function. Two applications and several numerical investigations confirm the quality of the proposed space–time finite element scheme. Copyright © 2006 John Wiley & Sons, Ltd.     

Calculating the internodal transmissibilities using finite analytic method and its application for multi‐phase flow in heterogeneous porous media

In the traditional numerical reservoir simulations, the internodal transmissibility is usually defined as the harmonic mean of the permeabilities of the adjacent grids. This definition underestimates the phase flux and the speed of the saturation front, especially for the strong heterogeneous case. In this article, the internodal transmissibility is recalculated according to the nodal analytic solution. The redefined internodal transmissibility can be used directly to calculate the multiphase flow in the numerical reservoir simulations. Numerical examples show that, compared to the traditional numerical methods, the proposed scheme makes the convergences much faster as the refinement parameter increases, and the accuracy is independent of the heterogeneity. Copyright © 2016 John Wiley & Sons, Ltd.     

Multiscale finite-volume method for density-driven flow in porous media

The multiscale finite-volume (MSFV) method has been developed to solve multiphase flow problems on large and highly heterogeneous domains efficiently. It employs an auxiliary coarse grid, together with its dual, to define and solve a coarse-scale pressure problem. A set of basis functions, which are local solutions on dual cells, is used to interpolate the coarse-grid pressure and obtain an approximate fine-scale pressure distribution. However, if flow takes place in presence of gravity (or capillarity), the basis functions are not good interpolators. To treat this case correctly, a correction function is added to the basis function interpolated pressure. This function, which is similar to a supplementary basis function independent of the coarse-scale pressure, allows for a very accurate fine-scale approximation. In the coarse-scale pressure equation, it appears as an additional source term and can be regarded as a local correction to the coarse-scale operator: It modifies the fluxes across the coarse-cell interfaces defined by the basis functions. Given the closure assumption that localizes the pressure problem in a dual cell, the derivation of the local problem that defines the correction function is exact, and no additional hypothesis is needed. Therefore, as in the original MSFV method, the only closure approximation is the localization assumption. The numerical experiments performed for density-driven flow problems (counter-current flow and lock exchange) demonstrate excellent agreement between the MSFV solutions and the corresponding fine-scale reference solutions.     

模拟裂隙多孔介质中变饱和渗流的广义等效连续体方法

描述了一种计算裂隙多孔介质中变饱和渗流的广义等效连续体方法。这种方法忽略裂隙的毛细作用,设定一个与某孔隙饱和度相对应的综合饱和度极限值,并假定：（1）如果裂隙多孔介质的综合饱和度小于该极限值,水只在孔隙中存在并流动,而裂隙中则没有水的流动;（2）如果综合饱和度等于或大于该极限值,水将进入裂隙,并在裂隙内运动。分析比较了等效连续体模型的不同计算方法,并给出了一个模拟裂隙岩体中变饱和渗流与传热耦合问题的应用算例。结果表明,所述方法具有一般性,可以有效地模拟裂隙多孔介质中变饱和渗流的基本特征。     

采用复合单元法模拟裂隙多孔介质变饱和流动

采用复合单元法建立了模拟裂隙多孔介质变饱和流动的数值模型。该模型具有以下特点:裂隙不需要离散成特定单元,而是根据几何位置插入到孔隙基质单元中形成复合单元;在复合单元中,分别建立裂隙流和孔隙基质流的计算方程,二者通过裂隙-基质界面产生联系并整合成复合单元方程;复合单元方程具有和常规有限单元方程相同的格式,因此,可以使用常规有限单元方程的求解技术。采用欠松弛迭代、集中质量矩阵以及自适应时步调节等技术,开发了裂隙多孔介质变饱和流动计算程序。通过模拟一维干土入渗和复杂裂隙含水层内的流动问题,验证了该模型的合理性和适用性。模拟结果为进一步认识非饱和裂隙含水层地下水流动特性提供了理论依据。     

JHomogenizer: a computational tool for upscaling permeability for flow in heterogeneous porous media

This paper presents an object-oriented programming approach for the design of numerical homogenization programs, called JHomogenizer. It currently includes five functional modules to compute effective permeability and simple codes for computing solutions for flow in porous media. Examples with graphical output are shown to illustrate some functionalities of the program. A series of numerical examples demonstrates the effectiveness of the methodology for two-phase flow in heterogeneous reservoirs. The software is freely available, and the open architecture of the program facilitates further development and can adapt to suit specific needs easily and quickly.     

多孔介质油水两相k~s~p关系数学模型的实验研究

数学模型是研究相对渗透率与饱和度关系曲线的重要方法。采用自行开发设计的人工平面多孔介质模型,测定了相对渗透率与饱和度的关系曲线。多孔介质选择粒径为0.5~1mm、1~2mm的标准砂,纯净的水为湿润相,用3号苏丹红染色的93#汽油为非湿润相,组成多孔介质油水两相流动系统。采用Van Genuchten and Mualeum(VGM)和Brooks-Corey-Burdine(BCB)两种数学模型计算相对渗透率与饱和度的关系曲线,通过比较两种数学模型计算结果之间和模型计算结果与实测结果的差异以及模型的应用、多相渗流系统自身特征,得出VGM、BCB两种数学模型计算结果符合实际情况,VGM模型应用过程更为简便,但VGM模型具有一定适用条件;在砂性多孔介质中,BCB模型计算相对渗透率与饱和度关系曲线更准确。     

A finite element method for modeling coupled flow and deformation in porous fractured media

Modeling the flow in highly fractured porous media by finite element method (FEM) has met two difficulties: mesh generation for fractured domains and a rigorous formulation of the flow problem accounting for fracture/matrix, fracture/fracture, and fracture/boundary fluid mass exchanges. Based on the recent theoretical progress for mass balance conditions in multifractured porous bodies, the governing equations for coupled flow and deformation in these bodies are first established in this paper. A weak formulation for this problem is then established allowing to build a FEM. Taking benefit from recent development of mesh‐generating tools for fractured media, this weak formulation has been implemented in a numerical code and applied to some typical problems of hydromechanical coupling in fractured porous media. It is shown that in this way, the FEM that has proved its efficiency to model hydromechanical phenomena in porous media is extended with all its performances (calculation time, couplings, and nonlinearities) to fractured porous media. Copyright © 2015 John Wiley & Sons, Ltd.     

The representer method for two-phase flow in porous media

The representer method is applied to a one-dimensional two-phase flow model in porous media; capillary pressure and gravity are neglected. The Euler–Lagrange equations must be linearized, and one such linearization is presented here. The representer method is applied to the linear system iteratively until convergence, though a rigorous proof of convergence is out of reach. The linearization chosen is easy to calculate but does not converge for certain weights; however, a simple damping restores convergence at the cost of extra iterations. Numerical experiments are performed that illustrate the method, and quick comparison to the ensemble Kalman smoother is made. This research was supported by NSF grant EIA-0121523.     

Modelling of cement suspension flow in granular porous media

A theoretical model of cement suspensions flow in granular porous media considering particle filtration is presented in this paper. Two phenomenological laws have been retained for the filtration rate and the intrinsic permeability evolution. A linear evolution with respect to the volume fraction of cement in the grout has been retained for the filtration rate. The intrinsic permeability of the porous medium is looked for in the form of a hyperbolic function of the porosity change. The model depends on two phenomenological parameters only. The equations of this model are solved analytically in the one‐dimensional case. Besides, a numerical resolution based on the finite element method is also presented. It could be implemented easily in situations where no analytical solution is available. Finally, the predictions of the model are compared to the results of a grout injection test on a long column of sand. Copyright © 2005 John Wiley & Sons, Ltd.     

Towards a thermodynamics of immiscible two-phase steady-state flow in porous media

We propose that steady-state two-phase flow in porous media may be described through a formalism closely resembling equilibrium thermodynamics. This leads to a Monte Carlo method that will be highly efficient in studying two-phase flow under steady-state conditions numerically. This work was partially supported by the Norwegian Research Council through grants nos. 154535/432 and 180296/S30.     

井内嵌入同轴多孔与固体圆柱地下水渗流流型分析

对二维无限大多孔介质内单向均匀水平流垂直绕过“固体小圆柱-多孔介质环-水环-多孔介质”复杂四层结构下的流场进行了解析求解。内、外多孔介质区域均采用Brinkman模型,纯流体水环采用Stokes模型,通过耦合界面间的质量、动量守恒关系得到了各区域流函数的通用表达式。在此基础上分析了不同几何参数,不同内、外多孔介质渗透系数情况下,圆柱外绕流的流型变化;着重研究了水环间隙以及内、外多孔介质渗透系数的变化对流型及横向、纵向速度分布的影响。结果表明：外部多孔区流型主要受控于外部渗透系数;水环间隙宽度对水环内速度峰值影响较大;内部渗透系数增加到某一临界值情况下,横截面速度分布从阶梯形变为抛物形,即“穿透”现象。研究结果对有类似结构的地埋管换热器、地下水污染物吸收装置、地下水测速装置等的设计研发有理论指导意义。     

A pore-scale numerical model for flow through porous media

A pore-scale numerical model based on Smoothed Particle Hydrodynamics (SPH) is described for modelling fluid flow phenomena in porous media. Originally developed for astrophysics applications, SPH is extended to model incompressible flows of low Reynolds number as encountered in groundwater flow systems. In this paper, an overview of SPH is provided and the required modifications for modelling flow through porous media are described, including treatment of viscosity, equation of state, and no-slip boundary conditions. The performance of the model is demonstrated for two-dimensional flow through idealized porous media composed of spatially periodic square and hexagonal arrays of cylinders. The results are in close agreement with solutions obtained using the finite element method and published solutions in the literature. Copyright © 1999 John Wiley & Sons, Ltd.     

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.     

基于SPH方法的非饱和多孔介质相变耦合研究

在考虑相变的热能平衡方程和非饱和水分迁移质量控制方程的基础上,建立温度场-水分场的耦合模型,并采用一种无网格粒子算法（SPH）进行数值求解。其中,耦合方程中考虑了水流传热以及温度势对水流的直接驱动,在不考虑相变的情况下,该耦合模型可退化为常温下的水-热耦合模型,故可用于模拟冻融循环的相关问题。从求解热能平衡方程中的含冰量出发,实现解耦并对半无限单向冻结条件下介质内非稳态温度场和体积含水率分布场进行模拟,将耦合作用下的温度场与不耦合的解析解进行对比,反映出水分迁移对温度场存在较大影响。最后,求解了路基边坡在季节性周期温度边界下,温度场、水分场分布的演变规律,并评估了边坡阴阳面受热不均对水热两场分布的影响。计算结果基本能反映土冻结相变的实际物理过程,光滑粒子算法可以用于尝试解决冻土领域的其他相关问题。     

A mixed solution for two-dimensional unsteady flow in fractured porous media

A mixed finite element–boundary element solution for the analysis of two-dimensional flow in porous media composed of rock blocks and discrete fractures is described. The rock blocks are modelled implicitly by using boundary elements whereas finite elements are adopted to model the discrete fractures. The computational procedure has been implemented in a hybrid code which has been validated first by comparing the numerical results with the closed-form solution for flow in a porous aquifer intercepted by a vertical fracture only. Then, a more complex problem has been solved where a pervious, homogeneous and isotropic matrix containing a net of fractures is considered. The results obtained are shown to describe satisfactorily the main features of the flow problem under study. © 1997 by John Wiley & Sons, Ltd.     

Numerical study of two‐phase fluid distributions in fractured porous media

Two‐phase fluid distributions in fractured porous media were studied using a single‐component multiphase (SCMP) lattice Boltzmann method (LBM), which was selected among three commonly used numerical approaches through a comparison against the available results of micro X‐ray computed tomography. The influence of the initial configuration and the periodic boundary conditions in the SCMP LBM for the fluid distribution analysis were investigated as well. It was revealed that regular porous media are sensitive to the initial distribution, whereas irregular porous media are insensitive. Moreover, to eliminate the influence of boundaries, the model's buffer size of an SCMP LBM simulation was suggested to be taken as approximately 12.5 times the average particle size. Then, the two‐phase fluid distribution of a porous medium was numerically studied using the SCMP LBM. Both detailed distribution patterns and macroscopic morphology parameters were reasonably well captured. Finally, the two‐phase fluid distributions in a fractured porous media were investigated. The influence of the degree of saturation, fracture length, and fracture width on the fluid distributions and migration was explored. Copyright © 2015 John Wiley & Sons, Ltd.     

Transverse and longitudinal fluid flow modelling in fractured porous media with non‐matching meshes

A new discrete fracture model is introduced to simulate the steady‐state fluid flow in discontinuous porous media. The formulation uses a multi‐layered approach to capture the effect of both longitudinal and transverse permeability of the discontinuities in the pressure distribution. The formulation allows the independent discretisation of mesh and discontinuities, which do not need to conform. Given that the formulation is developed at the element level, no additional degrees of freedom or special integration procedures are required for coupling the non‐conforming meshes. The proposed model is shown to be reliable regardless of the permeability of the discontinuity being higher or lower than the surrounding domain. Four numerical examples of increasing complexity are solved to demonstrate the efficiency and accuracy of the new technique when compared with results available in the literature. Results show that the proposed method can simulate the fluid pressure distribution in fractured porous media. Furthermore, a sensitivity analysis demonstrated the stability regarding the condition number for wide range values of the coupling parameter.     

Theoretical and numerical study of the steady‐state flow through finite fractured porous media

This paper investigates the two‐dimensional flow problem through an anisotropic porous medium containing several intersecting curved fractures. First, the governing equations of steady‐state fluid flow in a fractured porous body are summarized. The flow follows Darcy's law in matrix and Poiseuille's law in fractures. An infinite transversal permeability is considered for the fractures. A multi‐region boundary element method is used to derive a general pressure solution as a function of discharge through the fractures and the pressure and the normal flux on the domain boundary. The obtained solution fully accounts for the interaction and the intersection between fractures. A numerical procedure based on collocation method is presented to compute the unknowns on the boundaries and on the fractures. The numerical solution is validated by comparing with finite element solution or the results obtained for an infinite matrix. Pressure fields in the matrix are illustrated for domains containing several interconnected fractures, and mass balance at the intersection points is also checked. Copyright © 2013 John Wiley & Sons, Ltd.