Taking into account general boundary conditions in upscaling absolute permeability

Upscaling methods that need to solve local problems subject to boundary conditions are addressed in this article. We define a new upscaling method based on optimization problems, which can take into account general boundary conditions applied to local problems. The determination of upscaled permeability leads to minimizing the difference of dissipated energies (or averaged velocity) at fine and large scale. Using optimal control techniques, we obtain an effective computing algorithm that allows us to recover, with classical boundary conditions, the well-known results. The uniqueness issue is tackled for the optimization problems introduced in our approach. We show that the method is stable with respect to G-convergence, a property that establishes a link with homogenization theory, and finally, 2D numerical experiments are presented.     

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.     

A continuum‐discrete model using Darcy's law: formulation and verification

This paper presents a numerical scheme for fluid‐particle coupled discrete element method (DEM), which is based on poro‐elasticity. The motion of the particles is resolved by means of DEM. While within the proposition of Darcian regime, the fluid is assumed as a continuum phase on a Eulerian mesh, and the continuity equation on the fluid mesh for a compressible fluid is solved using the FEM. Analytical solutions of traditional soil mechanics examples, such as the isotropic compression and one‐dimensional upward seepage flow, were used to validate the proposed algorithm quantitatively. The numerical results showed very good agreement with the analytical solutions, which show the correctness of this algorithm. Sensitivity studies on the effect of some influential factors of the coupling scheme such as pore fluid bulk modulus, volumetric strain calculation, and fluid mesh size were performed to display the accuracy, efficiency, and robustness of the numerical algorithm. It is revealed that the pore fluid bulk modulus is a critical parameter that can affect the accuracy of the results. Because of the iterative coupling scheme of these algorithms, high value of fluid bulk modulus can result in instability and consequently reduction in the maximum possible time‐step. Furthermore, the increase of the fluid mesh size reduces the accuracy of the calculated pore pressure. This study enhances our current understanding of the capacity of fluid‐particle coupled DEM to simulate the mechanical behavior of saturated granular materials. Copyright © 2014 John Wiley & Sons, Ltd.     

Delineation of Microscale Regimes of Fully-Developed Drainage and Implications for Continuum Models

Despite significant progress made in recent years, a fundamental understanding of immiscible displacements at the macroscale is lacking. In this paper we use a version of percolation theory, based on invasion percolation in a gradient, to connect drainage processes at the pore-network scale with the displacement at the macroscale. When the mobility ratio M is sufficiently small, the displacement is stabilized and can be described by invasion percolation in a stabilizing gradient. In the opposite case, it has common features with invasion percolation in a destabilizing gradient. A diagram delineating the regimes of fully developed drainage is developed. The transition between stabilized displacement and fingering is controlled by the change of the sign of the gradient of the percolation probability, and the transition boundary is described by a scaling law involving the capillary number and the viscosity ratio. We review recent work for random networks and extend the method to correlated pore networks. As the regimes of stabilized displacement are also those for which conventional theories (such as the Buckley–Leverett equation) are expected to apply, the phase diagram helps to delineate their validity.     

Equivalent Permeability and Simulation of Two-Phase Flow in Heterogeneous Porous Media

The problem of calculating equivalent grid block permeability tensors for heterogeneous porous media is addressed. The homogenization method used involves solving Darcy's equation subject to linear boundary conditions with flux conservation in subregions of the reservoir and can be readily applied to unstructured grids. The resulting equivalent permeability tensor is stable as defined relative to G-convergence. It is proposed to use both conforming and mixed finite elements to solve the local problems and compute approximations from above and below of the equivalent permeability, respectively. Comparisons with results obtained using periodic, pressure and no-flux boundary conditions and the renormalization method are presented. A series of numerical examples demonstrates the effectiveness of the methodology for two-phase flow in heterogeneous reservoirs.     

Micromechanics of saturated and unsaturated porous media

The homogenization method is used to determine the formulation of the behaviour of both saturated and unsaturated porous media. This approach makes it possible to assess the validity of the effective stress concept as a function of the properties of the porous media at the microscopic scale. Furthermore, the influence of the morphologies of the solid and fluid phases on the macroscopic behaviour is studied. The strain induced by drying is examined as a function of the morphological properties. Copyright © 2002 John Wiley & Sons, Ltd.     

Transport phenomena in kaolinite clay: Molecular simulation,homogenization analysis and similitude law

Kaolinite is a common clay mineral. It is a nanomaterial with a platelet crystalline structure. In order to analyze the behavior of kaolinite, its microscopic structure and material properties must be specified correctly. A molecular dynamics (MD) simulation is used for determining the microscale properties of hydrated kaolinite, and these properties are introduced into a multiscale homogenization analysis (HA). We previously developed such an MD/HA technique to investigate seepage, diffusion, sorption and consolidation in bentonite clay (Proceedings of the Science Basis for Nuclear Waste Management, Davos, Switzerland, vol. XXI. Material Research Society: Warrendale, PA, 1997; 359–366; Eng. Geol. 1999; 54 :21–31; Eng. Geol. 2001; 60 :127–138; Coupled Thermo‐Hydro‐Mechanical‐Chemical Processes in Geo‐systems. Elsevier: Amsterdam, 2005; 457–464). We here apply the method to kaolinite clay to investigate the permeability, diffusion and related similitude law. The obtained results are supported by existing experimental data. Copyright © 2008 John Wiley & Sons, Ltd.     

A bridge between dual porosity and multiscale models of heterogeneous deformable porous media

In this paper, a multiscale homogenization approach is developed for fully coupled saturated porous media to represent the idealized sugar cube model, which is generally employed in fractured porous media on the basis of dual porosity models. In this manner, an extended version of the Hill-Mandel theory that incorporates the microdynamic effects into the multiscale analysis is presented, and the concept of the deformable dual porosity model is demonstrated. Numerical simulations are performed employing the multiscale analysis and dual porosity model, and the results are compared with the direct numerical simulation through 2 numerical examples. Finally, a combined multiscale-dual porosity technique is introduced by employing a bridge between these 2 techniques as an alternative approach that reduces the computational cost of numerical simulation in modeling of heterogeneous deformable porous media.     

Mortar Upscaling for Multiphase Flow in Porous Media

In mortar space upscaling methods, a reservoir is decomposed into a series of subdomains (blocks) in which independently constructed numerical grids and possibly different physical models and discretization techniques can be employed in each block. Physically meaningful matching conditions are imposed on block interfaces in a numerically stable and accurate way using mortar finite element spaces. Coarse mortar grids and fine subdomain grids provide two-scale approximations. In the resulting effective solution flow is computed in subdomains on the fine scale while fluxes are matched on the coarse scale. In addition the flexibility to vary adaptively the number of interface degrees of freedom leads to more accurate multiscale approximations. This methodology has been implemented in the Center for Subsurface Modeling's multiphysics multiblock simulator IPARS (Integrated Parallel Accurate reservoir Simulator). Computational experiments demonstrate that this approach is scalable in parallel and it can be applied to non-matching grids across the interface, multinumerics and multiphysics models, and mortar adaptivity. Moreover unlike most upscaling approaches the underlying systems can be treated fully implicitly.     

孔隙介质的渗透特性初探

孔隙的存在是岩土类介质材料结构的本质特征,它不但改变了岩土体的力学特性,而且严重影响着岩土体的渗透特性。大多数经典的渗流理论中,多孔介质模型都假定孔隙率和渗透系数是与时间无关的材料常数。实际上由于淘涮、侵蚀、冲刷等原因,它们是随时间和坐标变化的,同时又与孔隙中的压力、流速等因素有关。基于孔隙率和损伤变量之间的定量关系,从连续损伤力学的角度对多孔介质岩土材料的渗流力学特性进行了研究。首先,对传统的达西定律形式进行修正,提出了孔隙介质完备有效的达西定律（模型）;然后,对该模型中渗透参数的特性进行了讨论和分析,得出了一些有益的结论。     

Numerical Homogenization of the Rigidity Tensor in Hooke''s Law Using the Node-Based Finite Element Method

Combining a geological model with a geomechanical model, it generally turns out that the geomechanical model is built from units that are at least a 100 times larger in volume than the units of the geological model. To counter this mismatch in scales, the geological data model's heterogeneous fine-scale Young's moduli and Poisson's ratios have to be upscaled to one equivalent homogeneous coarse-scale rigidity. This coarse-scale rigidity relates the volume-averaged displacement, strain, stress, and energy to each other, in such a way that the equilibrium equation, Hooke's law, and the energy equation preserve their fine-scale form on the coarse scale. Under the simplifying assumption of spatial periodicity of the heterogeneous fine-scale rigidity, homogenization theory can be applied. However, even then the spatial variability is generally so complex that exact solutions cannot be found. Therefore, numerical approximation methods have to be applied. Here the node-based finite element method for the displacement as primary variable has been used. Three numerical examples showing the upper bound character of this finite element method are presented.     

冻土水热耦合模型数值求解及结果检验

首先对作者所建立的基于多孔介质理论的季节冻土水热迁移耦合模型进行数值求解;对模型方程进行修正, 并给出了模型方程中参数的确定方法。然后以长春松原公路段土体为研究对象, 对实际工程中冻结情况下水分迁移的情况进行预测;给定模型边界条件对模型求解, 将结果与野外实际监测结果进行对比。温度变化对比数据表明, 模型可以较好地预测终值情况, 而中间过程的误差较大, 但是趋势基本一致。水分迁移方向及量的对比数据表明, 模型计算结果要小于实测结果, 但是整体上计算结果与实测结果的变化趋势较一致, 且同样是和最终值吻合较好, 误差最小。结果表明, 模型计算结果可较好地模拟参数最终值, 但存在一定误差。     

Modelling of steady‐state fluid flow in 3D fractured isotropic porous media: application to effective permeability calculation

In this paper, 3D steady‐state fluid flow in a porous medium with a large number of intersecting fractures is derived numerically by using collocation method. Fluid flow in the matrix and fractures is described by Darcy's law and Poiseuille's law, respectively. The recent theoretical development presented a general potential solution to model the steady‐state flow in fractured porous media under a far‐field condition. This solution is a hypersingular integral equation with pressure field in the fracture surfaces as the main unknown. The numerical procedure can resolve the problem for any form of fractures and also takes into account the interactions and the intersection between fractures. Once the pressure field and then the flux field in fractures have been determined, the pressure field in the porous matrix is computed completely. The basic problem of a single fracture is investigated, and a semi‐analytical solution is presented. Using the solution obtained for a single fracture, Mori‐Tanaka and self‐consistent schemes are employed for upscaling the effective permeability of 3D fractured porous media. Copyright © 2012 John Wiley & Sons, Ltd.     

Modeling permeability in imperfectly layered porous media. I. Derivation of block-scale permeability tensor for thin grid-blocks

Practical expressions are given for the nine components of the block-scale permeability tensor of a thin block. These expressions are derived from the local-scale continuity equation and Darcy's law in an anisotropic layered porous medium. The flow problem is separated in a bottom-flux problem and a top-flux problem, both of which can be solved in essentially the same way. The bottom-flux problem has been worked out in detail, and has been separated in two parts: a vertical potential difference and a horizontal potential difference part. Each is solved with a different approach specially designed for it. Depth-averaged expressions are obtained first, after which block-scale expressions are obtained by assuming a constant depth-averaged flux. In the zeroth order, this results in the well-known Dupuit approximation in geohydrology, and the vertical equilibrium (VE) approximation in petroleum reservoir engineering. The novelty of the theory presented here stems from the application of a perturbation technique to obtain first-order corrections to these well-known results. The local-scale laws are applied in the coordinate system coinciding with the principal axes of the local-scale permeability tensor. Only in this coordinate system the local-scale permeability tensor has zero off-diagonal components. However, since the porous medium is imperfectly layered, the first-order corrections show that the off-diagonal components of the block-scale permeability tensor are not zero. Furthermore, the block-scale permeability tensor is generally nonsymmetric, which implies that a coordinate system in which the off-diagonal terms disappear does not exist.     

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.     

多孔介质中溶质有效扩散系数预测的分形模型

依据分形理论和方法,探索溶质在多孔介质中的有效扩散系数的替代预测方法。在多孔介质溶质扩散的弯曲毛细管束模型的基础上,以分形维数作为介质的基本几何特性参数,建立了多孔介质中溶质扩散的分形毛细管束模型,推导出了溶质有效扩散系数与介质孔隙度之间的幂定律关系式,幂指数是介质孔隙分维和表面分维的函数,反映了介质孔隙体积的层次分布与孔隙通道曲折程度对扩散的影响。对粘性土的分形维数测定数据和有效扩散系数试验测定数据的分析表明,利用该关系式预测多孔介质中溶质的有效扩散系数是较为准确可靠的。     

碎(块)石路堤孔隙空气对流运动的Darcy定律适用性

针对青藏高原多年冻土区路基工程中采用的碎石路堤地温调控技术,分析了碎石层孔隙空气对流运动符合Darcy定律的适用范围及其可能偏离Darcy定律的非线性效应,并给出了考虑非线性效应的Darcy定律修正方程.同时,讨论了适合于路堤碎石层的空气渗透系数及其表征空气流动非线性效应的Ergun常数的估算公式.     

Adaptive hierarchical upscaling of flow in heterogeneous reservoirs based on an a posteriori error estimate

This paper treats the upscaling of the absolute permeability in a heterogeneous reservoir. By replacing the fine scale permeability tensor with an upscaled, or effective permeability tensor, a modelling error is introduced. An a posteriori error estimate on this modelling error is formulated and tested. An implementation of the theory, based on domain decomposition coupled with a hierarchical representation of the absolute permeability field, is given. As hierarchical basis functions we have chosen the Haar system, which leads to a wavelet representation of the permeability. The wavelet representation offers a natural upscaling technique which resembles the highcut filters commonly used in signal analysis. This procedure represents an adaptive upscaling method. The numerical results show that this method conserves both the dissipation and the mean velocity in the problem fairly well. The a posteriori error estimate on the modelling error coupled with domain decomposition methods constitutes a powerful modelling tool.     

不同类型海岸带海水入侵数值模拟研究进展

海水入侵作为一个全球化的问题,正随着沿海地区对地下淡水需求的增加而不断加剧,其不断发展引起地下水水质 恶化、土壤盐渍化等一系列生态环境问题。海水入侵的相关研究既具有重要的理论意义,对沿海地区的可持续发展也有重 要的实际价值,因此逐渐成为了国际研究的热点。随着数值计算方法及计算平台的不断发展,数值模拟方法已经成为研究 海水入侵问题最有效的工具之一。文章总结归纳了多种海岸带类型的划分方法与标准,从海岸带水文地质学的角度将海岸 带概括为松散岩类和基岩类两大类,并将其含水介质分别概化为等效多孔介质和裂隙岩溶介质。在此基础上,分析阐述这 两类含水介质海水入侵的数值模拟方法及其适用性,对当前海水入侵数值模拟方法进行了较为全面的概括。此外,对海水 入侵数值模拟方法的发展方向及趋势进行了展望,指出考虑密度变化的过渡带模型和离散-连续介质耦合模型将成为今后 研究中的重点发展方向。