Random porosity fields and their influence on the stability of granular media

It is well established that the mechanical behavior of granular media is strongly influenced by the media's microstructure. In this work, the influence of the microstructure is studied by integrating advances in the areas of geostatistics and computational plasticity, by spatially varying the porosity on samples of sand. In particular, geostatistical tools are used to characterize and simulate random porosity fields that are then fed into a nonlinear finite element model. The underlying effective mechanical response of the granular medium is governed by a newly developed elastoplastic model for sands, which readily incorporates spatial variability in the porosity field at the meso‐scale. The objective of this study is to assess the influence of heterogeneities in the porosity field on the stability of sand samples. One hundred and fifty isotropic and anisotropic samples of dense sand are failed under plane‐strain compression tests using Monte Carlo techniques. Results from parametric studies indicate that the axial strength of a specimen is affected by both the degree and orientation of anisotropy in heterogeneous porosity values with anisotropy orientation having a dominant effect, especially when the bands of high porosity are aligned with the natural orientation of shear banding in the specimen. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

An expression for the permeability of anisotropic granular media

There are many expressions proposed for the permeability of isotropic media based on flow channel and pore size distribution concepts, but there are no such expressions for anisotropic media. In this paper an expression for the permeability of an anisotropic medium is proposed, which has been verified in the laboratory. The mechanism behind fluid flow through soil was investigated using microscopic computer simulations to propose an expression for macroscopic permeability. The soil was assumed to be a spatially periodic porous medium, and the Navier-Stokes equation was solved using the FEM with appropriate boundary conditions for several different arrangements of the porous medium. The basic variables influencing flow through soil at the microscopic level were identified as specific surface area, void ratio, particle shape, material heterogeneity and the arrangement of particles in a porous medium. A sensitivity analysis was carried out to obtain an expression for the permeability in terms of the above variables. The corresponding macroscopic variables for the above microscopic variables are average specific surface area, average void ratio, anisotropy, tortuosity due to material heterogeneity, and the arrangement of particles respectively. An expression for the directional permeability is proposed in terms of these variables for the most common occurrence of particles in a porous medium. For the verification of the proposed equation, the permeability values of a fine-grained sand were measured at different void ratios and were compared with those predicted by the proposed equation. The results show that the predicted permeability values from the proposed equation are very close to the measured values.     

不同粒径级砂土渗透特性试验研究

渗透性是砂土的重要工程性质之一,影响砂土渗透性的因素有很多,比如土体密实程度、土颗粒自身特性、流体性质等,其中孔隙率与颗粒粒径是两个重要影响因素。而以往基于混合粒径的天然砂土的研究很难分别对这两个因素进行独立的研究。基于此,首先开展了单一粒径级砂土的常水头渗透试验,分别研究了同一粒径级砂土渗透系数随孔隙率的变化和同一孔隙率下不同粒径级砂土渗透系数随均值粒径的变化规律。试验结果显示,渗透系数随着孔隙率的增加而线性增加、随均值粒径二次方的增加而线性增加,其中均值粒径的影响较大,其变化能导致渗透系数量级上的差异。在此基础上,开展了多粒径混合砂土的渗透试验,讨论了曲率系数、不均匀系数等级配参数对渗透性的影响,从而将单一粒径级的研究成果推广到天然砂土,最终拟合出渗透系数与相关影响因素的经验公式,以便工程实践参考使用。     

3D finite element modeling of sand particle fracture based on in situ X‐Ray synchrotron imaging

Compressive loading of granular materials causes inter‐particle forces to develop and evolve into force chains that propagate through the granular body. At high‐applied compressive stresses, inter‐particle forces will be large enough to cause particle fracture, affecting the constitutive behavior of granular materials. The first step to modeling particle fracture within force chains in granular mass is to understand and model the fracture of a single particle using actual three‐dimensional (3D) particle shape. In this paper, the fracture mode of individual silica sand particles was captured using 3D x‐ray radiography and Synchrotron Micro‐computed Tomography (SMT) during in situ compression experiments. The SMT images were used to reconstruct particle surfaces through image processing techniques. Particle surface was then imported into Abaqus finite element (FE) software where the experimental loading setup was modeled using the extended finite element method (XFEM) where particle fracture was compared to experimental fracture mode viewed in radiograph images that were acquired during experimental loading. Load‐displacement relationships of the FE analysis were also compared with experimental measurements. 3D FE modeling of particle fracture offers an excellent tool to map stress distribution and monitors crack initiation and propagation within individual sand particles. Copyright © 2015 John Wiley & Sons, Ltd.     

Internal erosion of chemically reinforced granular materials: a mathematical modeling approach

Internal erosion (IE) affects the stability of natural and reinforced materials by causing instability within their granular structure. The dislodgement and transport of eroded particles affect both the particulate concentration of eroding fluid and the pore network of eroded material. In this study, we examined these modifications using a transport model with a finite element code. First, IE tests on chemically reinforced sand columns were performed to obtain information about eroded material loss of mass, particulate concentration of effluent, porosity and permeability modifications, and existing IE stages. Second, based on experimental results, a mathematical one‐dimensional model has been formulated to monitor the evolution and spatial distribution of erodible solids, fluidized particles, porosity, permeability, and seepage stresses. The model consists of a set of coupled nonlinear differential equations solved in sequence. It provides valuable information about the extent and the dynamics of structural changes, which can be used to estimate an IE time for the hydraulic work to reach failure. Copyright © 2011 John Wiley & Sons, Ltd.     

基于数字图像相关法的砂土细观直剪试验及其颗粒流数值模拟

砂土等天然颗粒材料具有的剪胀性、压硬性、各向异性等特殊力学性质受控于其内部的微细结构及其演化,如何利用试验与数值模拟等手段对砂土细观组构进行量化,对深入研究砂土变形机制尤为重要。基于数字图像相关方法的光学测量技术与土工试验相结合的新手段,通过对直剪仪的可视化改造,开发了能初步实现土体宏细观力学性状联合测量的“砂土变形细观瞬时光学测量系统”,并利用该系统对福建标准砂进行了室内细观直剪试验。随后进行了基于颗粒流软件PFC2D直剪数值试验,提取室内试验还难以获得的细观信息,作为对室内试验的补充。通过对宏观力学性质和位移场、应变场、颗粒定向、速度场、颗粒配位数和接触力链等细观的室内与数值试验结果进行剖析,探讨了砂土力学性质的细观机制。     

Pore scale study of permeability and tortuosity for flow through particulate media using Lattice Boltzmann method

In this paper, Lattice Boltzmann method (LBM) has been used to study the effects of permeability and tortuosity on flow through saturated particulate media and identify the relationships between permeability and tortuosity with other parameters such as particles diameter, grain specific surface, and porosity. LBM is a simple kinematic model that can incorporate the essential physics of microscopic and mesoscopic processes involved in flow through granular soils. The obtained results indicate that the 2D LB model, due to its inherent theoretical advantages, is capable of demonstrating that the porosity and specific surface are the most influential parameters in determining the intrinsic permeability of granular media. The obtained results show that particles' size diameter has a two‐fold effect on the coefficient of permeability: one is through specific surface and the other is by tortuosity factor. Numerical study also reveals that tortuosity of granular soils decreases almost linearly with increasing the porosity. Copyright © 2010 John Wiley & Sons, Ltd.     

Cavity expansion analyses of crushable granular materials with state‐dependent dilatancy

Crushability is one of the important behaviors of granular materials particularly under high stress states, and affects both the deformability and strength of the materials that are in essence associated with state‐dependent dilatancy. In this presentation, first, a new critical state model is proposed to take into account the three different modes of compressive deformation of crushable granular materials, i.e. particle rearrangement, particle crushing and pseudo‐elastic deformation. Second, the governing equations for cavity expansion in crushable granulates are introduced, in which the state‐dependent dilatancy as well as the bounding surface plasticity model are used. Then, the procedure to obtain semi‐analytical solutions to cavity expansion in the material is described in detail, in which a commercial differential equation solver is employed. Finally, cavity expansion analyses are carried out on Toyoura sand, a well‐documented granular material, to demonstrate the effects of crushability and state‐dependent dilatancy. The study shows that particle crushing does occur at both high stress and critical states and affects the stress fields and the deformation behavior of the material surrounding the cavity in association with state‐dependent dilatancy. This leads to conclusion that particle crushing and state‐dependent dilatancy have to be taken into account when cavity expansion theory is used to interpret cone penetration tests and pressuremeter tests. Copyright © 2011 John Wiley & Sons, Ltd.     

Strength anisotropy of granular material consisting of perfectly round particles

The strength anisotropy of granular materials deposited under gravity has mostly been attributed to elongated particles' tendency to align long axes along the bedding plane direction. However, recent experiments on near‐spherical glass beads, for which preferred particle alignment is inapplicable, have exhibited surprisingly strong strength anisotropy. This study tests the hypothesis that certain amount of fabric anisotropy caused by the anisotropic stress during deposition under gravity can be locked in a circular‐particle deposit. Such locked‐in fabric anisotropy can withstand isotropic consolidation and leads to significant strength anisotropy. 2D discrete element method simulations of direct shear tests on circular‐particle deposits are conducted in this study, allowing for the monitoring of both stress and fabric. Simulations on both monodispersed and polydispersed circular‐particle samples generated under downward gravitational acceleration exhibit clear anisotropy in shear strength, thereby proving the hypothesis. When using contact normal‐based and void‐based fabric tensors to quantify fabric anisotropy in the material, we find that the intensity of anisotropy is discernible but low prior to shearing and is dependent on the consolidation process and the dispersity of the sample. The fact that samples with very low anisotropy intensity measurements still exhibit fairly strong strength anisotropy suggests that current typical contact normal‐based and void‐based second‐order fabric tensor formulations may not be very effective in reflecting the anisotropic peak shear strength of granular materials. Copyright © 2017 John Wiley & Sons, Ltd.     

Multi-scale modelling of the permeability evolution of fine sands during cement suspension grouting with filtration

Fluid flow during permeation grouting of fine sands with a microcement-based grout is studied by assuming that the heterogeneous medium composed of the initial granular skeleton, filtered cement and the interstitial fluid phase can be replaced by a continuous equivalent medium at the macroscopic level. Consequently, the method of Homogenization of Periodic Structures (HPS) is used to identify the effective permeability tensor evolution under the effect of cement filtration. The expression of the macroscopic permeability tensor derived through the HPS procedure is shown to depend on the permeating fluid viscosity and the geometrical arrangement of the sand grains and cement deposit within the microstructure. Numerical computations are made using various two-dimensional and three-dimensional microstructures, and the model results are confronted with grouting experiments performed on small scale columns in the laboratory.     

Study on the mechanical behavior of sands using 3D discrete element method with realistic particle models

Presented is a study on the geometrical characteristics of sand particles and the mechanical behavior of sand material under external loading. Based on computed tomography technique, a reconstruction method of granular particles was developed and used to build a database of 3D geometrical models for sand particles. The studied sand particles showed good regularities in morphological characteristics and thus were suitable to be used for the random generation of numerical samples. DEM tests using realistically shaped particles were proven to better simulate the mechanical behavior of the sample during elastoplastic loading stage, which was an issue for the simplified spherical particles. The generation, extension, and breakage of the force chains controlled the strain softening behavior of sands. Anisotropy analysis using the spherical harmonic series showed that the evolution of anisotropy directions and parameters corresponded well with the macroscopic mechanical behavior of the material. Pore volume computation based on Voronoi diagram was performed to illustrate the formation and evolution of localized shear zone. The mesoscopic analysis showed that particle shape significantly influences the mechanical behavior of sands and thus should be properly modeled in numerical simulations.

     

A continuum‐discrete hydromechanical analysis of granular deposit liquefaction

A coupled continuum‐discrete hydromechanical model was employed to analyse the liquefaction of a saturated loose deposit of cohesionless particles when subjected to a dynamic base excitation. The pore fluid flow was idealized using averaged Navier–Stokes equations and the discrete element method was employed to model the solid phase particles. A well established semi‐empirical relationship was utilized to quantify the fluid–particle interactions. The conducted simulations revealed a number of salient micro‐mechanical mechanisms and response patterns associated with the deposit liquefaction. Space and time variation of porosity was a major factor which affected the coupled response of the solid and fluid phases. Pore fluid flow was within Darcy's regime. The predicted response exhibited macroscopic patterns consistent with experimental results and case histories of the liquefaction of granular soil deposits. Copyright © 2004 John Wiley & Sons, Ltd.     

Fine particle deposition to initially starved,stationary, planar beds

Reported here are results from new flume experiments examining deposition and entrainment of inert, silt‐sized particles (with spherical diameters in the range from 20 to 60 μm) to and from planar, impermeable and initially starved beds underlying channel flows. Bed surfaces comprised smooth or fixed sand‐size granular roughness and provided hydraulically smooth to transitionally rough boundaries. Results of these experiments were analysed with a simple model that describes the evolution of vertically averaged concentration of suspended sediment and accommodates the simultaneous delivery to and entrainment of grains from the bed. The rate of particle arrival to a bed diminishes linearly, and the rate of particle entrainment increases by the 5/2 power, as the value of the dimensionless Saffman parameter S = u_*³/g’ν approaches a threshold value of order unity, where u is the conventional friction velocity of the turbulent channel flow, g’ is the acceleration due to gravity adjusted for the submerged buoyancy of individual particles and ν is the kinematic viscosity of the transporting fluid. This transport behaviour is consistent with the notion that non‐cohesive, silt‐sized particles can neither reach nor remain on an impermeable bed under flow conditions where mean lift imposed on stationary particles in the viscous sublayer equals or exceeds the submerged weight of individual particles. Within the size range of particles used in these experiments, particle size and the characteristic size of granular roughness, up to that of medium sand, did not affect rates of dimensionless arrival or entrainment to a significant degree. Instead, a new but consistent picture of fine‐particle transport is emerging. Silt‐sized material, at least, is subject to potentially significant interaction with the bed during intermittent suspension transport at intermediate flow speeds greater than the value required for initiation of transport (ca 20 cm sec^?1) but less than the value (ca 50 cm sec^?1) required by the Saffman criterion ensuring transport in fully passive suspension or, equivalently, ‘wash‐load’.     

Generation of a realistic 3D sand assembly using X‐ray micro‐computed tomography and spherical harmonic‐based principal component analysis

Three‐dimensional particle morphology is a significant problem in the discrete element modeling of granular sand. The major technical challenge is generating a realistic 3D sand assembly that is composed of a large number of random‐shaped particles containing essential morphological features of natural sands. Based on X‐ray micro‐computed tomography data collected from a series of image processing techniques, we used the spherical harmonics (SH) analysis to represent and reconstruct the multi‐scale features of real 3D particle morphologies. The SH analysis was extended to some highly complex particles with sharp corners and surface cavities. We then proposed a statistical approach for the generation of realistic particle assembly of a given type of sand based on the principle component analysis (PCA). The PCA aims to identify the major pattern of the coefficient matrix, which is made up of the SH coefficients of all the particles involved in the analysis. This approach takes into account the particle size effect on the variation of particle morphology, which is observed from the available results of micro‐computed tomography and QICPIC analyses of sand particle morphology. Using the aforementioned approach, two virtual sand samples were generated, whose statistics of morphological parameters were compared with those measured from real sand particles. The comparison shows that the proposed approach is capable of generating a realistic sand assembly that retains the major morphological features of the mother sand. Copyright © 2016 John Wiley & Sons, Ltd.     

New correlations of permeability and porosity versus confining pressure,cementation, and grain size and new quantitatively correlation relates permeability to porosity

In sandstone, there is a trend between porosity (?) and permeability (k). It is a linear relationship having the form log (k)?=?a?+?(b ?). The slope, intercept, and degree of scatter of the log(k)???? trends vary from formation to another. These variations are attributed to differences in initial grain size and sorting, diagenetic history, cementation, clay content, pore geometry, and compaction history. In the literature, permeability and porosity modeling by using lab experiments was carried out by using unconsolidated sandstone, sand packs, or synthetic particles. Such models cannot be applied to predict flow properties of consolidated natural sandstone. Furthermore in these models, sand grain size, shape, and sorting factors were considered as the main factors that affect porosity and permeability. Hardly, any attention was paid to the confining pressure and the fraction of cementing material that bind the grain to form a coherent rock. If these two crucial aspects are not taken into consideration during the model development, the model cannot be applied to natural consolidated sandstone. The main objective of the present paper is to develop a new model for porosity versus permeability taking into account important factors such as sand grain size and sorting, compaction pressure, and concentration of cementing material that bind the sand grains. The effect for clay swelling or migration was however discarded, as the sand grains were washed prior to consolidation. The sand used in producing the sandstone cores was medium- to fine-sized well-sorted sand grains. The grain’s sphericity was measured to be in the range of (0.8–0.9) with little angularity. The fabricated cores have an average compressive strength of 5,700 psi, which is comparable with Bera sandstone strength. Also, the produced cores were stable in the fluid media as they were subjected to 300 °C to allow cementing material to be crystallized. The aspect of the present work was to analyze the dependence of both the permeability as well as the porosity on the variables of the present study that consist of grain size, cementation fraction, and the confining pressure. Using the experimental data, a linear relationship, in terms of each variable, was developed here that can eventually help researchers to fabricate cores with desired properties. The second step was to generate more general models to be used as references for scholars for further work in this research field. Nonlinear regression analysis was carried out on all the three variables of the present study to obtain two nonlinear correlations: one describes the behavior of permeability and the other describes porosity. In the third step, an advanced correlation that describes permeability versus porosity in a quantitative manner was developed by using nonlinear regression analysis. Permeability was studied accordingly as a function of all the three variables of the present study as well as porosity. This step represents the main objective of this paper.     

Permeability determination of porous media using large‐scale finite elements and iterative solver

The problem of finite element simulation of incompressible fluid flow in porous medium is considered. The porous medium is characterized by the X‐ray microtomography technique in three dimensions. The finite calculus‐based stabilization technique is reviewed to implement the equal order finite element interpolation functions for both velocity and pressure. A noble preconditioner, the nodal block diagonal preconditioner, is considered whose performance is thoroughly investigated. Combining this preconditioner with a standard iterative solver during the computational homogenization procedure, it is possible to carry out the large‐scale fluid flow simulation for estimating permeability of the porous medium with reasonable accuracy and reliability. Copyright © 2013 John Wiley & Sons, Ltd.     

Modelling strain localization in granular materials using micropolar theory: numerical implementation and verification

Implementation and applications for a constitutive numerical model on F‐75 silica sand, course silica sand and two sizes of glass beads compressed under plane strain conditions are presented in this work. The numerical model is used to predict the stress versus axial strain and volumetric strain versus axial strain relationships of those materials; moreover, comparisons between measured and predicted shear band thickness and inclination angles are discussed and the numerical results compare well with the experimental measurements. The numerical model is found to respond to the changes in confining pressure and the initial relative density of a given granular material. The mean particle size is used as an internal length scale. Increasing the confining pressure and the initial density is found to decrease the shear band thickness and increase the inclination angle. The micropolar or Cosserat theory is found to be effective in capturing strain localization in granular materials. The finite element formulations and the solution method for the boundary value problem in the updated Lagrangian frame (UP) are discussed in the companion paper. Copyright © 2006 John Wiley & Sons, Ltd.     

A fully implicit and consistent finite element framework for modeling reservoir compaction with large deformation and nonlinear flow model. Part I: theory and formulation

Reservoir depletion results in rock failure, wellbore instability, hydrocarbon production loss, oil sand production, and ground surface subsidence. Specifically, the compaction of carbonate reservoirs with soft rocks often induces large plastic deformation due to rock pore collapse. On the other hand, following the compaction of reservoirs and failure of rock formations, the porosity and permeability of formations will, in general, decrease. These bring a challenge for reservoir simulations because of high nonlinearity of coupled geomechanics and fluid flow fields. In this work, we present a fully implicit, fully coupled, and fully consistent finite element formulation for coupled geomechanics and fluid flow problems with finite deformation and nonlinear flow models. The Pelessone smooth cap plasticity model, an important material model to capture rock compaction behavior and a challenging material model for implicit numerical formulations, is incorporated in the proposed formulation. Furthermore, a stress-dependent permeability model is taken into account in the formulation. A co-rotational framework is adopted for finite deformation, and an implicit material integrator for cap plasticity models is consistently derived. Furthermore, the coupled field equations are consistently linearized including nonlinear flow models. The physical theories, nonlinear material and flow models, and numerical formulations are the focus of part I of this work. In part II, we verify the proposed numerical framework and demonstrate the performance of our numerical formulation using several numerical examples including a field reservoir with soft rocks undergoing serious compaction.