A micromechanics model for molecular diffusion in materials with complex pore structure

Molecular diffusion in fully saturated porous materials is strongly influenced by the pore space, which, in general, is characterized by a complex topological structure. Hence, information on macroscopic diffusion properties requires up‐scaling of transport processes within nano‐pores and micro‐pores over several spatial scales. A new model in the framework of continuum micromechanics is proposed for predicting the effective molecular diffusivity in porous materials. Considering a representative volume element, characterizing a porous material without any information about the pore space microstructure complexity, the uniform flux is perturbed by recursively embedding shape information hierarchically in the form of the ESHELBY matrix‐inclusion morphology to obtain the effective diffusivity as a function of the recurrence level and the porosity. The model predicts a threshold value for the porosity, below which no molecular diffusion can occur because of the presence of isolated pore clusters that are not connected and unavailable for transport. The maximum porosity, below which no molecular transport is possible, is predicted as one‐third for spherical inclusions. The model allows for extensions to more complex morphologies of the inclusions. We also identify, that the effects of the micro‐structure on molecular transport are characterized by porosity dependent long‐range and short‐range interactions. The developed framework is extended to incorporate realistic pore size distributions across several spatial scales by means of a distribution function within the hierarchical homogenization scheme. Available experimental results assert the model predictions. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Multiphase SPH modeling of free surface flow in porous media with variable porosity

This paper presents a numerical model for simulating free surface flow in porous media with spatially varying porosity. The governing equations are based on the mixture theory. The resistance forces between solid and fluid is assumed to be nonlinear. A multiphase SPH approach is presented to solve the governing equations. In the multiphase SPH, water is modeled as a weakly compressible fluid, and solid phase is discretized by fixed solid particles carrying information of porosity. The model is validated by several numerical examples including seepage through specimen, fast flow through rockfill dam and wave interaction with porous structure. Good agreements between numerical results and experimental data are obtained in terms of flow rate and evolution of free surface. Parameter study shows that (1) the nonlinear resistance law provides more accurate results; (2) particle size and porosity have significant influence on the porous flow.     

煤与瓦斯突出中单个煤壳解体突出的突变理论分析

在煤与瓦斯突出中,煤体自组织地形成最具抗力的球盖壳形式来抵抗被突出,煤壳一旦解体便丧失承载能力,高位势煤岩系统中与单个煤壳突出相应的各种能量得以释放。根据反映煤岩破坏特性的塑性变形局部化理论,导得突出前煤壳形变、位移时相应于偏应力的功、能增量关系,建立了煤壳失稳解体的突变模型,并对由于偏应力作用导致煤壳失稳解体机制进行了研究,给出失稳瞬间释放的偏应力应变能表达式、煤岩系统体积应变能释放量表达式和孔隙瓦斯粉碎解体后的碎煤及与煤层瓦斯渗流形成瓦斯-煤粉两相流描述。所得结果可深化对煤、地应力与瓦斯压力三因素对煤与突出演化规律综合作用的认识。     

多孔介质孔隙率与体积模量的关系

用两相等效体的概念,根据Walsh公式和球形孔隙的弹性公式导出了多孔介质的孔隙率和体积模量之间的近似公式和精确公式,近似关系式不适用于较大的孔隙率。精确公式可以较好的适用于较大孔隙率。试验结果表明,精确公式能够很好地表达多孔介质孔隙率与体积模量之间的关系。     

Double Porosity Finite Element Method for Borehole Modeling

Summary. This paper considers the mechanical and hydraulic response around an arbitrary oriented borehole drilled in a naturally fractured formation. The formation is treated as a double porosity medium consisting of the primary rock matrix as well as the fractured systems, which are each distinctly different in porosity and permeability. The poro-mechanical formulations that couple matrix and fracture deformations as well as fluid flow aspects are presented. A double porosity and double permeability finite element solution for any directional borehole drilled in the fractured porous medium is given. Compared with conventional single-porosity analyses, the proposed double-porosity solution has a larger pore pressure in the matrix and a smaller tensile stress in the near-wellbore region. The effects of time, fracture, mud weight, and borehole inclination in the double-porosity solution are parametrically studied to develop a better understanding of physical characteristics governing borehole problems.     

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.     

Implication of surface modified NZVI particle retention in the porous media: Assessment with the help of 1-D transport model

Retention of surface-modified nanoscale zero-valent iron (NZVI) particles in the porous media near the point of injection has been reported in the recent studies. Retention of excess particles in porous media can alter the media properties. The main objectives of this study are, therefore, to evaluate the effect of particle retention on the porous media properties and its implication on further NZVI particle transport under different flow conditions. To achieve the objectives, a one-dimensional transport model is developed by considering particle deposition, detachment, and straining mechanisms along with the effect of changes in porosity resulting from retention of NZVI particles. Two different flow conditions are considered for simulations. The first is a constant Darcy’s flow rate condition, which assumes a change in porosity, causes a change in pore water velocity and the second, is a constant head condition, which assumes the change in porosity, influence the permeability and hydraulic conductivity (thus Darcy’s flow rate). Overall a rapid decrease in porosity was observed as a result of high particle retention near the injection points resulting in a spatial distribution of deposition rate coefficient. In the case of constant head condition, the spatial distribution of Darcy’s velocities is predicted due to variation in porosity and hydraulic conductivity. The simulation results are compared with the data reported from the field studies; which suggests straining is likely to happen in the real field condition.     

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.     

Groutability of cement‐based grout with consideration of viscosity and filtration phenomenon

The groutability depends on the properties of the grout, its injection processes, and on the mechanical properties of the soil formation. During the process of pouring cement‐based grouting into a porous medium, a variation with time occurs in the viscosity of grout suspension. In addition, the particle filtration phenomenon will limit the expansion of the grouted zone because cement particles are progressively stagnant within the soil matrix. In this paper, a closed‐form solution was derived by implementing the mass balance equations and the generalized phenomenological filtration law, which can be used to evaluate the deposition of cement‐based grout in the soil matrix. The closed‐form solution relevant to a particular spherical flow was modified by a step‐wise numerical calculation, considering the variable viscosity caused by a chemical reaction, and the decrease in porosity resulting from grout particle deposition in the soil pores. A series of pilot‐scale chamber injection tests was performed to verify that the developed step‐wise numerical calculation is able to evaluate the injectable volume of grout and the deposition of grout particles. The results of the chamber injection tests concurred well with that of the step‐wise numerical calculation. Based on the filtration phenomenon, a viable approach for estimating the groutability of cement‐based grout in a porous medium was also suggested, which might facilitate a new insight in the design of the grouting process. Copyright © 2009 John Wiley & Sons, Ltd.     

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。     

Elastic consolidation around a point sink embedded in a half-space with anisotropic permeability

The complete solution is presented for the transient effects of pumping fluid from a point sink embedded in a saturated, porous elastic half-space. It is assumed that the medium is homogeneous and isotropic with respect to its elastic properties and homogeneous but anisotropic with respect to the flow of pore fluid. The soil skeleton is modelled as a linear elastic material obeying Hooke's law, while the pore fluid is assumed to be incompressible with its flow governed by Darcy's law. The solution has been evaluated for a particular value of Poisson's ratio of the solid skeleton, i.e. 0.25, and the results have been presented graphically in the form of isochrones of excess pore pressure and surface profile for the half-space. The solutions presented may have application in practical problems such as dewatering operations in compressible soil and rock masses.     

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.     

基于分形理论的赫巴流体在多孔介质中的渗流模型

泥浆是一种用途广泛的工程浆液，在工程施工过程中会渗流进入地层，给工程带来一定的影响。为深入研究泥浆的渗流机理，把地层模拟为多孔介质，泥浆模拟为能更全面地反映其流变性能的赫巴流体，基于分形理论，建立赫巴流体在多孔介质中的渗流模型。基于渗流模型计算结果，详细分析压力梯度、流性指数、稠度系数及孔隙率等参数对多孔介质中赫巴流体瞬时平均流速的影响，指出流速变化与压力梯度、流性指数及稠度系数的变化均呈幂指数关系，与孔隙率变化呈二项式关系，且流性指数是一个影响赫巴流体渗流速度的极敏感因素。渗流模型计算结果为相关工程中泥浆方案的设计与施工奠定了一定的理论基础。      

Physical and chemical steady-state compaction in deep-sea sediments: Role of mineral reactions

Marine sediments typically exhibit steep porosity gradients in their uppermost centimeters. Although the decrease in porosity with depth below the sediment-water interface is primarily due to compression arising from the accumulation of overlying sediment, early diagenetic mineral dissolution and precipitation reactions may potentially also affect the porosity gradient. Here, we present a steady state compaction model, based on the mass and momentum conservation of total fluid and solid phases, in order to quantify the relative contributions of mineral reactions and physical compaction on porosity changes. The compaction model is applied to estimate hydraulic conductivity and compressive response coefficients of deep-sea sediments from measured porosity depth profiles. The results suggest an inverse relation between the compressive response coefficient and the lithogenic content of marine sediments. For deep-sea sediments exhibiting high rates of dissolution of siliceous shell fragments, the compaction model ignoring mineral reactions overestimates the hydraulic conductivity and compressive response coefficients. In contrast to non-compacting porous media, mineral dissolution in surficial sediments can lead to lower porosity. However, as illustrated for a deep-sea sediment in the equatorial Atlantic characterized by extensive dissolution of calcareous shell fragments, the effect of mineral dissolution and precipitation reactions on porosity gradients is, in most cases, negligible.     

基于四参数随机生长法重构土体的格子玻尔兹曼细观渗流研究

多孔介质模型的重构问题是土体细观渗流机理研究的基础和关键。由四参数随机生长法(QSGS)构建土体模型,采用格子玻尔兹曼方法(LBM),通过MATLAB自编程序研究重构土在不同条件下的细观渗流机理。结果表明:随模型尺寸增大,孔隙连通程度显著提高,300×300格点大小的模型连通孔隙率增长幅度(34.38%)最大,继续扩大模型尺寸发现增加不明显;流体粒子在孔隙连通性好、孔径大的区域,会形成主渗流通道,且存在指进效应,孔道中间流速最大,可达0.0324,越靠近孔壁流速越小;大孔隙率土的流速比小孔隙率土大,而低孔隙率土中的流速相比大孔隙土更稳定;LBM模拟渗透率与经典K-C公式计算结果对比发现,孔隙率越高计算渗透率越准确(n=0.78,误差为10.22%);土颗粒越小,渗流孔道越细窄、分布越密集,对应的速度场分布更为均匀,同时流速也更小。该研究成果能较好地揭示重构土的细观渗流机理,也可为现有细观土体孔隙流研究提供一定借鉴。     

二维饱和土体动态孔隙率及相关动力响应特性研究

饱和多孔介质的动力响应研究在众多工程领域具有重要意义。充分考虑孔隙率的变化规律与影响因素,有利于合理揭示饱和多孔介质的相关力学行为。为此,将动态孔隙率模型与用于表征饱和多孔介质动力特性的u-U-p型方程结合,构建相应的非线性力学模型,利用Comsol Multiphysis PDE求取相应的数值解,以此研究不同透水条件下,受谐波载荷激励的二维饱和土体的孔隙率、变形量及孔隙水压力的变化规律。结果表明：孔隙率的变化与土骨架的体应变及孔隙水压力直接相关,土体压缩过程中,孔隙率相应减小,土骨架与孔隙流体的相互作用增强,土体运动时所受阻力增大,其无量纲竖向位移小于孔隙率被视为常数时的情况,在此条件下,由于土体的变形量减小,其孔隙水压力也相对减小。故充分考虑动态孔隙率,有利于更加精确地研究等饱土体和多孔介质的相关力学行为。此外,土体上表面透水条件下,孔隙流体可以从土体表面自由排出,土骨架承受的载荷更大,与不透水条件相比,土体孔隙率、竖向位移、孔隙水压力等变化更为显著。     

高水力梯度条件下颗粒堆积型多孔介质渗流规律试验研究

煤矿开采面临的水文地质条件越来越复杂,尤其是遭遇承压含水层的水压力越来越大,突水灾害发生时必然会带来高水力梯度引起的破碎岩体突水通道内高速非线性渗流问题。据此,研制高水力梯度（最大600）条件下堆积型多孔介质中高速非线性渗流试验装置,采用堆积型钢球模拟破碎岩体,对粒径为1、2、3、4、5、6 mm共6种光滑钢球分别开展了一维均质圆柱渗流试验。试验结果表明：对于由1～6 mm钢球堆积而成的孔隙率为0.44～0.45的多孔介质,当水力梯度大于145时,通过分析水力梯度-平均流速（J-v）曲线和水力梯度-雷诺数（J-Re）关系曲线,将流动状态划分为3个模式：线性层流、非线性层流、紊流,并获得了从线性层流过渡到非线性层流的临界流速为0.23～0.78 cm/s、临界水力梯度为3～8;从层流到紊流转捩的临界流速为1.6～4.8 cm/s、临界水力梯度为90～145。从小粒径多孔介质到大粒径多孔介质的渗流过程中,临界流速越来越大,而临界水力梯度逐渐减小。 渗透率与粒径的平方、非达西流影响系数与粒径的倒数均呈线性正相关,非达西流影响系数随着渗透率的增加呈指数减小。该研究对多孔介质非线性渗流的理论研究以及实际工程中高承压含水层突涌水问题有重要借鉴意义。     

Scattering of elastic wave by a partially sealed cylindrical shell embedded in poroelastic medium

Scattering of an elastic wave by a cylindrical shell embedded in poroelastic medium is investigated theoretically with the assumption that the shell material is also a porous elastic medium. The porous medium is modellized via Biot's theory and the scattering by cylindrical shell is expressed by the definition of scattering matrix. The normal mode expansion technique is employed for analysing the scattering field, and the asymptotic solutions of displacements, stresses and pore pressure are derived. Two limiting cases‐scattering by a poroelastic cylinder in Biot medium and a elastic cylindrical shell in elastic medium are obtained from the general solutions. The dispersion curves of displacement amplitude at the interface of shell and medium is compared with the case of elastic shell. The scattering amplitude associated with the fast, slow and transverse waves are identified by numerical simulation. Furthermore, the influence of the poroelastic property of shell material on scattering amplitude is analysed. Copyright © 2005 John Wiley & Sons, Ltd.