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.     

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.     

Numerical simulation of liquefaction in porous media using nonlinear fluid flow law

It is well known that for a sufficiently high seepage velocity, the governing flow law of porous media is nonlinear (J. Computers & Fluids 2010; 39 : 2069–2077). However, this fact has not been considered in the studies of soil‐pore fluid interaction and in conventional soil mechanics. In the present paper, a fully explicit dynamic finite element method is developed for nonlinear Darcy law. The governing equations are expressed for saturated porous media based on the extension of the Biot (J. Appl. Phys. 1941; 12 : 155–164) formulation. The elastoplastic behavior of soil under earthquake loading is simulated using a generalized plasticity theory that is composed of a yield surface along with non‐associated flow rule. Numerical simulations of porous media subjected to horizontal and vertical components of ground motion excitations with different permeability coefficients are carried out; while computed maximum pore water pressure is specially taken into consideration to make the difference between Darcy and non‐Darcy flow regimes tangible. Finally, the effect of non‐Darcy flow on the evaluated liquefaction potential of sand in comparison to conventional Darcy law is examined. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Application of the representer method for parameter estimation in numerical reservoir models

A data assimilation method was applied to estimate poorly known parameters (permeabilities) in a numerical reservoir model. Most variational methods for data assimilation are based on the assumption that the model is perfect except for the poorly known parameters. The representer method allows also for model errors, i.e. for uncertainties in the state variables (pressures and saturations). The method is based on minimizing a cost functional, assuming all the errors and parameters to be multivariate Gaussian random variables with given mean and covariances. The uncertain parameters and variables are expanded into a finite sum of basis functions called representers, and the gradients of the cost functional are obtained with an adjoint method. This approach gives an optimal parametrization in the sense that the final result is equal to the solution of the full inverse problem. The method was tested on a simple one-dimensional model to simulate two-phase (oil-water) flow through a heterogeneous reservoir. The results show that the method is able to provide an acceptable estimate of the permeability field. We used pressure measurements from a small number of observation wells in between the injection and production wells, but the representer method could be used equally well to assimilate data from other sources. The method appears to be a promising data assimilation tool for applications in reservoir engineering.     

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.     

饱和多孔介质中水力-力学-传质耦合过程的混合有限元法

对饱和多孔介质提出了一个含溶混污染物输运（传质）过程的混合元方法,其中污染物输运过程数学模型包含了对流、机械逸散、分子弥散和吸附等机制。固相位移、应变和有效应力,孔隙水压力、压力空间梯度和Darcy速度,污染物浓度、浓度空间梯度和浓度流量在单元内均为独立变量分别插值。基于胡海昌-Washizu三变量广义变分原理,结合可以滤掉虚假振荡的特征线方法,推导出饱和土中水力-力学-传质耦合问题控制方程的单元弱形式,并导出了混合元计算公式。数值模拟证明了所提出的方法可以提供与传统4点积分方案同样精度,同时能够提高计算效率。     

复杂水力路径下非饱和流动的数值分析

多孔介质中的流动问题,与孔隙介质的特性,含水量状态以及含水量的变化历史密切相关。基于毛细循环滞回理论模型,考虑含水量变化历史对土水特征关系的影响,在开发的U-DYSAC2有限元程序中进行了相应的数值实施。在试验给定的初边值条件下进行了非饱和渗流模拟分析,并将模拟结果与实测数据比较,表明在压力边界条件反复变化下,考虑滞回效应能获得更接近实测的结果,证实该模型在模拟各种循环变化条件下非饱和土渗流初边值问题的适用性与必要性。对入渗重分布反复变化条件下非饱和土柱流动的数值模拟表明,考虑滞回与不考虑滞回条件下,含水量、孔隙水压力和湿峰的迁移的预测在入渗后的重分布过程差异较大。考虑滞回效应时,土柱上部的脱湿速率、下部的吸湿速率比不考虑滞回时要低。从而证实了非饱和多孔介质中的土水状态依赖于含水量变化,而且强烈依赖于土体的水力路径变化。因此,循环边界条件变化下,毛细滞回效应在非饱和渗流模拟中的影响显著,必须加以考虑。     

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。     

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.     

A new mixed finite element method for poro‐elasticity

Development of robust numerical solutions for poro‐elasticity is an important and timely issue in modern computational geomechanics. Recently, research in this area has seen a surge in activity, not only because of increased interest in coupled problems relevant to the petroleum industry, but also due to emerging applications of poro‐elasticity for modelling problems in biomedical engineering and materials science. In this paper, an original mixed least‐squares method for solving Biot consolidation problems is developed. The solution is obtained via minimization of a least‐squares functional, based upon the equations of equilibrium, the equations of continuity and weak forms of the constitutive relationships for elasticity and Darcy flow. The formulation involves four separate categories of unknowns: displacements, stresses, fluid pressures and velocities. Each of these unknowns is approximated by linear continuous functions. The mathematical formulation is implemented in an original computer program, written from scratch and using object‐oriented logic. The performance of the method is tested on one‐ and two‐dimensional classical problems in poro‐elasticity. The numerical experiments suggest the same rates of convergence for all four types of variables, when the same interpolation spaces are used. The continuous linear triangles show the same rates of convergence for both compressible and entirely incompressible elastic solids. This mixed formulation results in non‐oscillating fluid pressures over entire domain for different moments of time. The method appears to be naturally stable, without any need of additional stabilization terms with mesh‐dependent parameters. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical solutions for flow in porous media

A numerical approach is proposed to model the flow in porous media using homogenization theory. The proposed concept involves the analyses of micro‐true flow at pore‐level and macro‐seepage flow at macro‐level. Macro‐seepage and microscopic characteristic flow equations are first derived from the Navier–Stokes equation at low Reynolds number through a two‐scale homogenization method. This homogenization method adopts an asymptotic expansion of velocity and pressure through the micro‐structures of porous media. A slightly compressible condition is introduced to express the characteristic flow through only characteristic velocity. This characteristic flow is then numerically solved using a penalty FEM scheme. Reduced integration technique is introduced for the volumetric term to avoid mesh locking. Finally, the numerical model is examined using two sets of permeability test data on clay and one set of permeability test data on sand. The numerical predictions agree well with the experimental data if constraint water film is considered for clay and two‐dimensional cross‐connection effect is included for sand. Copyright © 2003 John Wiley & Sons, Ltd.     

Linear stability of one‐dimensional non‐Darcy flow in broken rocks

Considering the effect of non‐Darcy flow, the perturbation theory and normal mode method are introduced to analyze the linear stability of one‐dimensional non‐Darcy flow of gases in broken rocks. A stability criterion for linear systems is obtained theoretically when the steady states of pressure and velocity fields are perturbed, and the effects of the physical parameters on the linear governing system are studied theoretically and numerically. It is pointed out that the deviation coefficient from Darcy's law plays an important role in the governing system; the increasing absolute value of deviation coefficient from Darcy's law stabilizes the system, and the numerical results are shown graphically. Copyright © 2015 John Wiley & Sons, Ltd.     

A locally conservative variational multiscale method for the simulation of porous media flow with multiscale source terms

We present a variational multiscale mixed finite element method for the solution of Darcy flow in porous media, in which both the permeability field and the source term display a multiscale character. The formulation is based on a multiscale split of the solution into coarse and subgrid scales. This decomposition is invoked in a variational setting that leads to a rigorous definition of a (global) coarse problem and a set of (local) subgrid problems. One of the key issues for the success of the method is the proper definition of the boundary conditions for the localization of the subgrid problems. We identify a weak compatibility condition that allows for subgrid communication across element interfaces, a feature that turns out to be essential for obtaining high-quality solutions. We also remove the singularities due to concentrated sources from the coarse-scale problem by introducing additional multiscale basis functions, based on a decomposition of fine-scale source terms into coarse and deviatoric components. The method is locally conservative and employs a low-order approximation of pressure and velocity at both scales. We illustrate the performance of the method on several synthetic cases and conclude that the method is able to capture the global and local flow patterns accurately.     

Three‐dimensional modeling of problems in poro‐elasticity via a mixed least‐squares method using linear tetrahedral elements

In a previous publication we developed a new mixed least‐squares method for poro‐elasticity. The approximate solution was obtained via a minimization of a least‐squares functional, based upon the equations of equilibrium, the equations of continuity and weak forms of the constitutive relationships for elasticity and Darcy flow. The formulation involved four independent types of variables: displacements, stresses, pore pressures and velocities. All of them were approximated by linear continuous triangles. Encouraged by the computational results, obtained from the two‐dimensional implementation of the method, we extended our formulation to three dimensions. In this paper we present numerical examples for the performance of continuous linear tetrahedra within the context of the mixed least‐squares method. The initial results suggest that the method works well in the nearly and entirely incompressible limits for elasticity. For poro‐elasticity, the obtained pore pressures are stable without exhibiting the oscillations, which are observed when the standard Galerkin formulation is used. Copyright © 2010 John Wiley & Sons, Ltd.     

A comparison of multiscale methods for elliptic problems in porous media flow

We review and perform comparison studies for three recent multiscale methods for solving elliptic problems in porous media flow; the multiscale mixed finite-element method, the numerical subgrid upscaling method, and the multiscale finite-volume method. These methods are based on a hierarchical strategy, where the global flow equations are solved on a coarsened mesh only. However, for each method, the discrete formulation of the partial differential equations on the coarse mesh is designed in a particular fashion to account for the impact of heterogeneous subgrid structures of the porous medium. The three multiscale methods produce solutions that are mass conservative on the underlying fine mesh. The methods may therefore be viewed as efficient, approximate fine-scale solvers, i.e., as an inexpensive alternative to solving the elliptic problem on the fine mesh. In addition, the methods may be utilized as an alternative to upscaling, as they generate mass-conservative solutions on the coarse mesh. We therefore choose to also compare the multiscale methods with a state-of-the-art upscaling method – the adaptive local–global upscaling method, which may be viewed as a multiscale method when coupled with a mass-conservative downscaling procedure. We investigate the properties of all four methods through a series of numerical experiments designed to reveal differences with regard to accuracy and robustness. The numerical experiments reveal particular problems with some of the methods, and these will be discussed in detail along with possible solutions. Next, we comment on implementational aspects and perform a simple analysis and comparison of the computational costs associated with each of the methods. Finally, we apply the three multiscale methods to a dynamic two-phase flow case and demonstrate that high efficiency and accurate results can be obtained when the subgrid computations are made part of a preprocessing step and not updated, or updated infrequently, throughout the simulation. The research is funded by the Research Council of Norway under grant nos. 152732 and 158908.     

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.     

Numerical modeling of the flow and transport of radionuclides in heterogeneous porous media

This paper is concerned with numerical methods for the modeling of flow and transport of contaminant in porous media. The numerical methods feature the mixed finite element method over triangles as a solver to the Darcy flow equation and a conservative finite volume scheme for the concentration equation. The convective term is approximated with a Godunov scheme over the dual finite volume mesh, whereas the diffusion–dispersion term is discretized by piecewise linear conforming triangular finite elements. It is shown that the scheme satisfies a discrete maximum principle. Numerical examples demonstrate the effectiveness of the methodology for a coupled system that includes an elliptic equation and a diffusion–convection–reaction equation arising when modeling flow and transport in heterogeneous porous media. The proposed scheme is robust, conservative, efficient, and stable, as confirmed by numerical simulations.      

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.     

A model for moisture and heat flow in fractured unsaturated porous media

The present study investigates propagation of a cohesive crack in non‐isothermal unsaturated porous medium under mode I conditions. Basic points of skeleton deformation, moisture, and heat transfer for unsaturated porous medium are presented. Boundary conditions on the crack surface that consist of mechanical interaction of the crack and the porous medium, water, and heat flows through the crack are taken into consideration. For spatial discretization, the extended finite element method is used. This method uses enriched shape functions in addition to ordinary shape functions for approximation of displacement, pressure, and temperature fields. The Heaviside step function and the distance function are exploited as enrichment functions for representing the crack surfaces displacement and the discontinuous vertical gradients of the pressure and temperature fields along the crack, respectively. For temporal discretization, backward finite difference scheme is applied. Problems solved from the literature show the validity of the model as well as the dependency of structural response on the material properties and loading. Copyright © 2016 John Wiley & Sons, Ltd.