Numerical modelling of fluid flow in microscopic images of granular materials

A program for the simulation of two‐dimensional (2‐D) fluid flow at the microstructural level of a saturated anisotropic granular medium is presented. The program provides a numerical solution to the complete set of Navier–Stokes equations without a priori assumptions on the viscous or convection components. This is especially suited for the simulation of the flow of fluids with different density and viscosity values and for a wide range of granular material porosity. The analytical solution for fluid flow in a simple microstructure of porous medium is used to verify the computer program. Subsequently, the flow field is computed within microscopic images of granular material that differ in porosity, particle size and particle shape. The computed flow fields are shown to follow certain paths depending on air void size and connectivity. The permeability tensor coefficients are derived from the flow fields, and their values are shown to compare well with laboratory experimental data on glass beads, Ottawa sand and silica sands. The directional distribution of permeability is expressed in a functional form and its anisotropy is quantified. Permeability anisotropy is found to be more pronounced in the silica sand medium that consists of elongated particles. Copyright © 2001 John Wiley & Sons, Ltd.     

孔隙尺度多孔介质流体流动与溶质运移高性能模拟

深入探究孔隙尺度下的流体流动特性和溶质运移规律对石油开采、农田养分管理、地下水污染修复有着重要意义。以人工构建的多孔介质结构和同步辐射X射线显微CT扫描的土壤团聚体(分辨率3.7μm)为研究对象,在空间节点数多达64 000 000的情况下,基于格子Boltzmann模型和GPU并行技术计算得到多孔介质流体运动和溶质运移过程的关键参数,并据此探究多孔介质空间异质性对水力学特性的影响。通过对3组不同结构的多孔介质比较发现,结构复杂程度最高的土壤样品和不规则堆叠的圆球结构的渗透率在100 mD(即10^-13m^2)量级,远低于规则堆叠的圆球结构(>20 000 mD);土壤的迂曲度为1.40~1.60,明显高于规则堆叠的圆球结构。研究结果表明,渗透率大的样品具有较小的迂曲度,这与结构的空间异质性有较强的关系;土壤的渗透率和迂曲度呈现各向异性;在水力梯度一定的前提下,渗透率较大的样品,纵向弥散系数也较大;同时,结构的异质性也会影响溶质的穿透曲线。本研究提出的模拟方法可在土壤结构中进行高效的水流运动和溶质运移模拟,可用于土壤多孔介质在孔隙尺度下的水力学特性研究。     

About Darcy's law in non-Galilean frame

This paper is aimed towards investigating the filtration law of an incompressible viscous Newtonian fluid through a rigid non-inertial porous medium (e.g. a porous medium placed in a centrifuge basket). The filtration law is obtained by upscaling the flow equations at the pore scale. The upscaling technique is the homogenization method of multiple scale expansions which rigorously gives the macroscopic behaviour and the effective properties without any prerequisite on the form of the macroscopic equations. The derived filtration law is similar to Darcy's law, but the tensor of permeability presents the following remarkable properties: it depends upon the angular velocity of the porous matrix, it verifies Hall–Onsager's relationship and it is a non-symmetric tensor. We thus deduce that, under rotation, an isotropic porous medium leads to a non-isotropic effective permeability. In this paper, we present the results of numerical simulations of the flow through rotating porous media. This allows us to highlight the deviations of the flow due to Coriolis effects at both the microscopic scale (i.e. the pore scale), and the macroscopic scale (i.e. the sample scale). The above results confirm that for an isotropic medium, phenomenological laws already proposed in the literature fails at reproducing three-dimensional Coriolis effects in all types of pores geometry. We show that Coriolis effects may lead to significant variations of the permeability measured during centrifuge tests when the inverse Ekman number Ek⁻¹ is 𝒪(1). These variations are estimated to be less than 5% if Ek⁻¹<0.2, which is the case of classical geotechnical centrifuge tests. We finally conclude by showing that available experimental data from tests carried out in centrifuges are not sufficient to determining the effective tensor of permeability of rotating porous media. Copyright © 2004 John Wiley & Sons, Ltd.     

孔隙尺度多孔介质流体流动与溶质运移高性能模拟

深入探究孔隙尺度下的流体流动特性和溶质运移规律对石油开采、农田养分管理、地下水污染修复有着重要意义。以人工构建的多孔介质结构和同步辐射X射线显微CT扫描的土壤团聚体（分辨率3.7 μm）为研究对象，在空间节点数多达64 000 000的情况下，基于格子Boltzmann模型和GPU并行技术计算得到多孔介质流体运动和溶质运移过程的关键参数，并据此探究多孔介质空间异质性对水力学特性的影响。通过对3组不同结构的多孔介质比较发现，结构复杂程度最高的土壤样品和不规则堆叠的圆球结构的渗透率在100 mD（即10^-13m²）量级，远低于规则堆叠的圆球结构（>20 000 mD）；土壤的迂曲度为1.40~1.60，明显高于规则堆叠的圆球结构。研究结果表明，渗透率大的样品具有较小的迂曲度，这与结构的空间异质性有较强的关系；土壤的渗透率和迂曲度呈现各向异性；在水力梯度一定的前提下，渗透率较大的样品，纵向弥散系数也较大；同时，结构的异质性也会影响溶质的穿透曲线。本研究提出的模拟方法可在土壤结构中进行高效的水流运动和溶质运移模拟，可用于土壤多孔介质在孔隙尺度下的水力学特性研究。     

Wave propagation in a general anisotropic poroelastic medium: Biot’s theories and homogenisation theory

Anisotropic wave propagation is studied in a fluid-saturated porous medium, using two different approaches. One is the dynamic approach of Biot’s theories. The other approach known as homogenisation theory, is based on the averaging process to derive macroscopic equations from the microscopic equations of motion. The medium considered is a general anisotropic poroelastic (APE) solid with a viscous fluid saturating its pores of anisotropic permeability. The wave propagation phenomenon in a saturated porous medium is explained through two relations. One defines modified Christoffel equations for the propagation of plane harmonic waves in the medium. The other defines a matrix to relate the relative displacement of fluid particles to the displacement of solid particles. The modified Christoffel equations are solved further to get a quartic equation whose roots represent complex velocities of the four attenuating quasi-waves in the medium. These complex velocities define the phase velocities of propagation and quality factors for attenuation of all the quasi-waves propagating along a given phase direction in three-dimensional space. The derivations in the mathematical models from different theories are compared in order to work out the equivalence between them. The variations of phase velocities and attenuation factors with the direction of phase propagation are computed, for a realistic numerical model. Differences between the velocities and attenuations of quasi-waves from the two approaches are exhibited numerically.     

高岭-蒙脱混合黏土渗透各向异性模型研究

完善的渗透各向异性模型有助于指导地下建筑施工期和运营期的排水注浆方案的选定,为了建立完善的渗透各向异性模型,利用三轴渗透仪对高岭-蒙脱混合黏土开展一系列渗透试验,并从宏观和微观两个角度尝试建立渗透各向异性模型。研究结果表明：随着膨润土掺入量增加,混合黏土中小孔隙含量不变,中孔隙含量增多,而大孔隙含量减少;随着固结应力的增大,混合黏土中孔隙均减少,且孔隙最大孔径减小;宏观参数孔隙比和液限孔隙比与渗透各向异性比之间的规律性较弱,说明孔隙比和液限孔隙比不是决定渗透各向异性比的关键参数;用微观参数表述的渗透各向异性模型有较强的规律性,说明从微观角度探究渗透各向异性模型不仅可行,而且更能揭示混合黏土渗透各向异性的本质,且微观参数模型中,用能反映较大孔隙变化的孔径指数 ～ 表述的渗透各向异性模型相关性最好。     

一种适用于孔隙体积应变的有效应力方程

土力学奠基石Terzaghi有效应力原理被广泛应用于油藏孔隙和渗透率应力敏感研究中,然而其对于岩石孔隙体积应变的适用性存在争议。对颗粒不可压缩和颗粒可压缩的多孔介质分别进行了受力分析,推导了总体积、颗粒骨架、孔隙体积的有效应力表达式,与Biot、Skepmton有效应力方程对比,建立了适用于孔隙体积应变的新型有效应力方程,并进行了试验论证和应用举例。研究表明:在颗粒不可压缩多孔介质中,有效应力为超出平衡孔隙流压之外的颗粒间宏观等效应力;在颗粒可压缩变形多孔介质中,有效应力为其相同应变下的等效应力,有3种有效应力分别适用于总体积应变、颗粒体积应变、孔隙体积应变;新提出的孔隙体积有效应力方程与孔隙度、岩石总体积压缩系数、颗粒压缩系数、总应力和流压相关,4个理论计算式计算结果在3种多孔介质试验测试结果中的偏差均在5%以内;孔隙体积有效应力系数解决了如何定量增总应力来等效模拟储层降流压生产过程这一关键问题,3个压缩系数关系式理论计算准确方便。     

基于岩石颗粒排列方式的低渗透储层应力敏感性分析

以颗粒堆积模型为基础,考虑了低渗透岩心颗粒不同排列方式和不同变形方式,建立了毛管束模型,并通过颗粒Hertz接触变形原理对毛管变形量进行计算,研究毛管和多孔介质应力敏感性定量表征关系,通过有效毛管分数和毛管变形规律探讨了低渗透储层应力敏感性的作用机制。研究表明,低渗透储层的应力敏感性主要表现为渗透率的应力敏感性,相比于渗透率应力敏感性,孔隙度应力敏感性较弱;低渗透储层应力敏感性与岩石颗粒排列方式、颗粒变形方式、岩石微观孔隙结构、固液界面作用力和启动压力梯度效应等密切相关;考虑有效毛管分数和毛管变形量的多孔介质应力敏感性量化模型可从应力敏感性微观作用机制角度解释低渗透储层应力敏感性。     

Two methods for pore network of porous media

Two approaches of generating pore networks of porous media are presented to capture the pore fabric. The first methodology extracted pore structure from a computer simulated packing of spheres. The modified Delaunay tessellation was used to describe the porous media, and modified Nelder–Mead method in conjunction with three pore‐merging algorithms was used to generate the pore size and coordination number distributions of the randomly packed spheres. The Biconical Abscissa Asymmetric CONcentric bond was used to describe the connection between two adjacent voids. This algorithm was validated by predicting pore structure of a cubic array of spheres of equal radius with known pore sizes, throat sizes and coordination number distributions. The predicted distributions of pore structure agreed well with the measured. Then, the algorithm was used to predict pore structure and permeability of randomly packed spherical particles, and predicted permeability values were compared with published experimental data. The results showed that the predicted permeability values were in good agreement with those measured, confirming the proposed algorithm can capture the main flow paths of packed beds. The second methodology generated an equivalent pore network of porous media, of which the centers of voids were located in a regular lattice with constant pore center distance. However, this network allowed for matching both main geometrical and topological characteristics of the porous media. A comparison of the two approaches suggested that the second approach can also be used as a predictive tool to quantitatively study the microscopic properties of flow through 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.     

Flow Study in a Double Porosity Medium Containing Low Concentrations of Ellipsoidal Occluded Macro-voids

This paper presents an analytical study of fluid flow in a porous medium presenting pores of two different length scales: at the smallest or microscopic scale, the presence of connected voids confers a porous medium structure to the material investigated, while at the upper or mesoscopic scale, macro-pores are present. This microstructure is employed to represent the progressive opening of inter-aggregate pore spaces observed in natural compacted montmorillonites polluted by heavy metal ions. Three-dimensional analytical expressions are rigorously derived for pore fluid velocity and excess pore fluid pressure within the porous matrix, around an occluded ellipsoidal inter-aggregate void. The eccentricity ratio is employed to characterize the geometrical shape of the ellipsoidal void, while its orientation with respect to the inflow in the far field is determined by the dip angle θ. As an application, we investigate the flow focusing effect for varying eccentricity ratios and dip angles.     

Flow Study in a Double Porosity Medium Containing Ellipsoidal Occluded Macro-voids

This paper presents an analytical study of fluid flow in a porous medium presenting pores of two different length scales: at the smallest or microscopic scale, the presence of connected voids confers a porous medium structure to the material investigated, while at the upper or mesoscopic scale, macro-pores are present. This microstructure is employed to represent the progressive opening of inter-aggregate pore spaces observed in natural compacted montmorillonites polluted by heavy metal ions. Three-dimensional analytical expressions are rigorously derived for pore fluid velocity and excess pore fluid pressure within the porous matrix, around an occluded ellipsoidal inter-aggregate void. The eccentricity ratio is employed to characterize the geometrical shape of the ellipsoidal void, while its orientation with respect to the inflow in the far field is determined by the dip angle θ. As an application, we investigate the flow focusing effect for varying eccentricity ratios and dip angles.     

Numerical modeling of multiphase fluid flow in deforming porous media: A comparison between two- and three-phase models for seismic analysis of earth and rockfill dams

In this paper, a fully coupled numerical model is presented for the finite element analysis of the deforming porous medium interacting with the flow of two immiscible compressible wetting and non-wetting pore fluids. The governing equations involving coupled fluid flow and deformation processes in unsaturated soils are derived within the framework of the generalized Biot theory. The displacements of the solid phase, the pressure of the wetting phase and the capillary pressure are taken as the primary unknowns of the present formulation. The other variables are incorporated into the model using the experimentally determined functions that define the relationship between the hydraulic properties of the porous medium, i.e. saturation, relative permeability and capillary pressure. It is worth mentioning that the imposition of various boundary conditions is feasible notwithstanding the choice of the primary variables. The modified Pastor–Zienkiewicz generalized constitutive model is introduced into the mathematical formulation to simulate the mechanical behavior of the unsaturated soil. The accuracy of the proposed mathematical model for analyzing coupled fluid flows in porous media is verified by the resolution of several numerical examples for which previous solutions are known. Finally, the performance of the computational algorithm in modeling of large-scale porous media problems including the large elasto-plastic deformations is demonstrated through the fully coupled analysis of the failure of two earth and rockfill dams. Furthermore, the three-phase model is compared to its simplified one which simulates the unsaturated porous medium as a two-phase one with static air phase. The paper illustrates the shortcomings of the commonly used simplified approach in the context of seismic analysis of two earth and rockfill dams. It is shown that accounting the pore air as an independent phase significantly influences the unsaturated soil behavior.     

基于格子博尔兹曼方法表征体元尺度土体细观渗流场的数值模拟

为了研究土体的细观渗流特性,假设土体是完全饱和且在渗流过程中水分的流动始终处于层流状态。考虑宏观统计参数（孔隙率、渗透率及有效黏滞系数等）的影响,基于表征体元（REV）尺度的格子博尔兹曼（Boltzmann）方法,建立了压力作用下土体细观渗流的数值模型。采用D2Q9模型考虑水分流动的离散速度分布,宏观边界条件为左右侧面为不透水边界 ,上下边界设置不同的密度来控制压力边界,在微观边界条件上采用非平衡态外推格式。编制相应的计算程序,将计算区域内的多孔介质材料设置成流体（孔隙率 1.0,渗透率 ）,验证了经典的Poiseuille流。此外,结合算例分别讨论了土体在压力作用下孔隙率、渗透率及渗透压力等影响因素与渗流速度的相互关系,研究表明该数值方法与Darcy定律得到的计算结果较为吻合。因此,基于REV尺度的格子Boltzmann方法可以有效地模拟土体的渗流机制,为进一步研究土体渗流特性提供了一种新的研究手段。     

Homogenization of a Darcy–Stokes system modeling vuggy porous media

We derive a macroscopic model for single-phase, incompressible, viscous fluid flow in a porous medium with small cavities called vugs. We model the vuggy medium on the microscopic scale using Stokes equations within the vugular inclusions, Darcy's law within the porous rock, and a Beavers–Joseph–Saffman boundary condition on the interface between the two regions. We assume periodicity of the medium and obtain uniform energy estimates independent of the period. Through a two-scale homogenization limit as the period tends to zero, we obtain a macroscopic Darcy's law governing the medium on larger scales. We also develop some needed generalizations of the two-scale convergence theory needed for our bimodal medium, including a two-scale convergence result on the Darcy–Stokes interface. The macroscopic Darcy permeability is computable from the solution of a cell problem. An analytic solution to this problem in a simple geometry suggests that: (1) flow along vug channels is primarily Poiseuille with a small perturbation related to the Beavers–Joseph slip, and (2) flow that alternates from vug to matrix behaves as if the vugs have infinite permeability.     

土工织物-非稳定土体反滤系统中滤饼的研究

在土-土工织物组成的反滤系统中,如果被保护土是非稳定土,则在渗流作用下被保护土中会有大量细粒土发生移动,很容易在织物与土层的交界面上形成一层薄滤饼,滤饼的厚度及渗透系数成为影响系统反滤效果的重要因素。根据多孔介质中的渗流与沉积理论,研究了在非稳定土体渗滤过程中,随着细粒土的迁移滤饼的厚度与织物厚度及渗透系数、外部压力等因素之间的关系,并给出了解析表达式。     

初始孔隙比对高吸力下非饱和土土水特性的影响

在我国西北干旱半干旱地区,处于高吸力范围内的非饱和土广泛存在于各种岩土工程中。非饱和土土水特性的研究是非饱和土渗流、强度与变形研究的基础。为研究高吸力下不同初始孔隙比非饱和土的土水特性,以一种粉土试样作为研究对象,采用饱和盐溶液蒸汽平衡法控制土试样的吸力,利用阿基米德原理量测不规则土试样的体积,通过修正的Van-Gen...     

非饱和含砂细粒土的气体渗透特性研究

为了探讨非饱和含砂细粒土的气体渗透规律,制备不同含水率和干密度试样进行气体渗透试验,并基于多孔介质渗流理论分析进气压力、孔隙比和饱和度等对气体渗透率的影响规律,总结了非饱和土中渗透率的经验公式。试验结果表明,随着进气压力的增加,土样气体渗透率测试值逐渐趋于稳定值;含水率较高时,进气压力对土样气体渗透率的影响程度较大;含水率较低时,土样气体渗透率随进气压力增加而变化幅值较小。气体渗透率随土样含水率的增加（或饱和度的增加）呈减小趋势;当含水率低于最优含水率时,气体渗透率变化较小,但当含水率大于最优含水率后,气体渗透率急剧降低;在最优含水率两侧,土样微观结构的差异（絮凝结构和分散结构）是导致气体渗透率发生突变的主要原因;气隙比可有效表征土体孔隙比与饱和度对气体渗透率的影响规律,气体渗透率与气隙比呈良好的幂函数关系,并采用已有文献中试验数据验证了经验公式的合理性。     

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.     

浅析隧洞围岩宏微观结构与岩爆

文章分别分析了隧洞围岩的宏观结构因素（结构面发育情况，结构面与硐室或隧道轴线的关系及与最大主应力的关系等）及微观结构因素（岩石矿物颗粒的排列、颗粒间的联接以及粒间的空隙特征等）对岩爆的影响。分析表明，围岩的宏、微观结构与岩爆发生及发生的强烈程度有紧密的关系。在实际工程中，这一认识对岩爆问题的正确评价无疑是十分有益的。