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.     

热-水-应力耦合条件下三项因素对核素迁移影响的有限元分析

目前在放射性核素随地下水迁移的研究中还很少涉及温度场和应力场的影响。针对这一问题,引入渗透迁移方程,将所开发的饱和-非饱和孔隙介质中热-水-应力耦合弹塑性模型及其二维有限元程序进行了扩展和改进,使之可以同步地对温度场、渗流场、应力场和放射性核素浓度场的变化进行解析。对一个假想的核废料处置库算例,通过分别改变影响地下水中核素迁移的3个因素：渗透系数、分配系数和分子扩散系数,考察了在热-水-应力-迁移耦合作用条件下近场放射性核素浓度的分布及变化,从而认识到：当缓冲层的渗透系数小到一定程度后,缓冲层中核素迁移基本不再受渗流场的影响;缓冲层分配系数越小,分子扩散系数越大,核素迁移越快。     

近饱和条件下非饱和多孔介质渗流过程的数值分析

在非饱和多孔介质渗流分析中,近饱和条件下物理模型与数值模型之间的差异会导致数值不稳定问题。为解决这一问题,并保证模拟结果的可靠性,提出了3种方法,并在有限元分析程序U-DYSAC2中分别进行了程序代码的实施。通过数值试验与试验数据的比较,证实了在近饱和条件下土-水特征曲线和水力传导函数的高非线性可引起数值收敛性、稳定性和精度问题,而且在不同条件下含水率和基质吸力的预测结果差异明显。在3种方法中,修正的Van Genuchten模型（MVGM）方法对含水率的预测较为准确,而Line方法对基质吸力的预测较为合理。因此,解决在分析近饱和条件下非饱和多孔介质渗流问题时,为获得接近真实的模拟结果,采用合适的数值方法进行预测是非常关键的。     

Coupled theory of mixtures for clayey soils

In this work, elasto-plastic coupled equations are formulated in order to describe the time-dependent deformation of saturated cohesive soils (two-phase state). Formulation of these equations is based on the principle of virtual work and the theory of mixtures for inelastic porous media. The theory of mixtures for a linear elastic porous skeleton was first developed by Biot (Theory of elasticity and consolidation for a porous anisotropic solid, Journal of Applied Physics, 1955, 26, 188–185). An extension of Biot's theory into a nonlinear inelastic media was performed by Prevost (Mechanics of continuous porous media, International Journal of Engineering Science, 1980, 18, 787–800). The saturated soil is considered as a mixture of two deformable media, the solid grains and the water. Each medium is regarded as a continuum and follows its own motion. The flow of pore-water through the voids is assumed to follow Darcy's law. The coupled equations are developed for large deformations with finite strains in an updated Lagrangian reference frame. The coupled behavior of the two-phase materials (soil-water state) is implemented in a finite element program. A modified Cam-clay model is adopted and implemented in the finite element program in order to describe the plastic behavior of clayey soils. Penetration of a piezocone penetrometer in soil is numerically simulated and implemented into a finite element program. The piezocone penetrometer is assumed to be infinitely stiff. The continuous penetration of the cone is simulated by applying an incremental vertical movement of the cone tip boundary. Results of the finite element numerical simulation are compared with experimental measurements conducted at Louisiana State University using the calibration chamber. The numerical simulation is carried out for two cases. In the first case, the interface friction between the soil and the piezocone penetrometer is neglected. In the second case, interface friction is assumed between the soil and the piezocone. The results of the numerical simulations are compared with experimental laboratory measurements.     

边坡渗流耦合变形分析方法的研究及其应用

边坡在渗流作用下应力场会发生变化,是渗流场耦合应力场的作用结果。给出了一种考虑饱和-非饱和流渗影响的边坡弹塑性有限元分析方法,采用了不同含水量对抗剪强度影响的模型,求解带有渗透力作用项的土体弹塑性平衡方程得到边坡的应力应变分布,并应用于某滑坡的实际降雨分析。该实例分析证实,由于渗流作用影响,边坡变形加大,最后形成潜在滑动面。利用渗流耦合变形的弹塑性有限元法分析大气降雨造成土体滑坡的变形和发展是可行的,其在实际工程中的应用是成功的。     

多孔介质渗透率的分形描述

针对土壤、岩石等多孔介质结构的复杂性,从其结构形成的物理机制和达西定律出发,利用分形几何理论,将土壤等作为在统计意义上具有分形特征的多孔介质来研究其水力参数与结构之间的关系,建立了饱和多孔介质渗透率与其分维数之间的定量化的函数式。试验应用扫描电镜法研究了多孔介质断面微结构并算出分维数。试验结果表明:利用该模型预测的多孔介质渗透率与实测值基本吻合,能够比较精确地预测多孔介质水力参数。     

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.     

Fully coupled hydro-mechanical numerical manifold modeling of porous rock with dominant fractures

Coupled hydro-mechanical (HM) processes are significant in geological engineering such as oil and gas extraction, geothermal energy, nuclear waste disposal and for the safety assessment of dam foundations and rock slopes, where the geological media usually consist of fractured rock masses. In this study, we developed a model for the analysis of coupled hydro-mechanical processes in porous rock containing dominant fractures, by using the numerical manifold method (NMM). In the current model, the fractures are regarded as different material domains from surrounding rock, i.e., finite-thickness fracture zones as porous media. Compared with the rock matrix, these fractured porous media are characterized with nonlinear behavior of hydraulic and mechanical properties, involving not only direct (poroelastic) coupling but also indirect (property change) coupling. By combining the potential energy associated with mechanical responses, fluid flow and solid–fluid interactions, a new formulation for direct HM coupling in porous media is established. For indirect coupling associated with fracture opening/closure, we developed a new approach implicitly considering the nonlinear properties by directly assembling the corresponding strain energy. Compared with traditional methods with approximation of the nonlinear constitutive equations, this new formulation achieves a more accurate representation of the nonlinear behavior. We implemented the new model for coupled HM analysis in NMM, which has fixed mathematical grid and accurate integration, and developed a new computer code. We tested the code for direct coupling on two classical poroelastic problems with coarse mesh and compared the results with the analytical solutions, achieving excellent agreement, respectively. Finally, we tested for indirect coupling on models with a single dominant fracture and obtained reasonable results. The current poroelastic NNM model with a continuous finite-thickness fracture zone will be further developed considering thin fractures in a discontinuous approach for a comprehensive model for HM analysis in fractured porous rock masses.     

Fully coupled elastoplastic hydro-mechanical analysis of unsaturated porous media using a meshfree method

This paper presents the first application of an advanced meshfree method, ie, the edge-based smoothed point interpolation method (ESPIM), in simulation of the coupled hydro-mechanical behaviour of unsaturated porous media. In the proposed technique, the problem domain is spatially discretised using a triangular background mesh, and the polynomial point interpolation method combined with a simple node selection scheme is adopted for creating nodal shape functions. Smoothing domains are formed on top of the background mesh, and a constant smoothed strain, created by applying the smoothing operation over the smoothing domains, is assigned to each smoothing domain. The deformation and flow models are developed based on the equilibrium equation of the mixture, and linear momentum and mass balance equations of the fluid phases, respectively. The effective stress approach is followed to account for the coupling between the flow and deformation models. Further coupling among the phases is captured through a hysteretic soil water retention model that evolves with changes in void ratio. An advanced elastoplastic constitutive model within the context of the bounding surface plasticity theory is employed for predicting the nonlinear behaviour of soil skeleton. Time discretisation is performed by adopting a three-point discretisation method with growing time steps to avoid temporal instabilities. A modified Newton-Raphson framework is designed for dealing with nonlinearities of the discretised system of equations. The performance of the numerical model is examined through a number of numerical examples. The state-of-the-art computational scheme developed is useful for simulation of geotechnical engineering problems involving unsaturated soils.     

多孔介质渗透系数的空间尺度效应研究进展

多孔介质渗透系数的空间尺度问题是一个与地下流体运动和溶质运移的数值模拟密切相关的应用性课题,广泛的应用需求和新的计算方法使其成为近年的热门课题之一。它涉及到相互联系的两个方面：①非均质介质场渗透系数空间尺度行为的分析与模拟;②将局部测量尺度下的试验参数转化为数值模拟网格尺度下的参数输入值的升尺度（ｕｐｓｃａｌｉｎｇ）方法和计算模型。首先介绍了该课题在概念上的拓展及其物理含义,进而以方法为主线,对这一领域具有代表性的研究成果进行了分类和评述,讨论了该课题的研究对地下水流和溶质运移的模拟分析乃至整个多孔介质流体运动研究的意义。     

孔隙液体的可压缩性对非饱和孔隙介质弹塑性分析的影响

首先在多相孔隙介质的混合物理论框架内探讨了如何利用固-液两相孔隙介质模型描述介质的非饱和问题。其次,讨论了一个实用的描述孔隙液体可压缩性的方法,并建立了小变形条件下固-液两相孔隙介质的弹塑性本构方程。最后,采用俞茂宏统一强度理论,针对一个具体的结构例子,分析了孔隙液体的可压缩性对结构弹塑性分析的影响。     

Numerical analysis of frost effects in porous media. Benefits and limits of the finite element poroelasticity formulation

The aim of this paper is to analyse the performance of a finite element formulation usable for predicting the mechanical consequences of frost effects on porous media. It considers the characteristics of porous media and how the frost action can be assessed. The problem is then separated into two parts: thermal and poromechanical calculations. The constitutive equations developed in the framework of poromechanics are presented and the implementation in a usual finite element poroelasticity formulation based on Zuber's method is adopted. An analysis of the time‐step influence on the convergence rate is given and leads us to propose a simple method in order to obtain objectivity of the finite element response and avoid over‐long calculations. Frost effect simulations are carried out on real porous media (two fired clays) as a case study. Although the experimental behaviour of the porous media subjected to frost action is in accordance with some observations, the calculated strains appear to be overestimated compared with measurements. The problem could be largely attributable to the difficulty of assessing permeability evolution during frost development. Copyright © 2011 John Wiley & Sons, Ltd.     

Similarities between GSH,hypoplasticity and KCR

Accounting for elasto-plastic motion in granular media, hypoplasticity is a state-of-the-art constitutive model derived from data accumulated over many decades. In contrast, GSH, a hydrodynamic theory, is derived from general principles of physics, with comparatively few inputs from experiments, yet sporting an applicability ranging from static stress distribution via elasto-plastic motion to fast dense flow, including non-uniform ones such as a shear band. Comparing both theories, we find great similarities for uniform, slow, elasto-plastic motion. We also find that proportional paths and the Goldscheider rule used to construct barodesy, another, more recent constitutive model, are natural results of GSH’s equations. This is useful as it gives these constitutive relations a solid foundation in physics and, in reverse, GSH a robust connection to reality. The same symbiotic relation exists between GSH and KCR, or Kamrin’s non-local constitutive relation, a model that was successfully employed to account for a wide shear band in split-bottom cells.     

考虑毛细滞回效应的非饱和多孔介质渗流与变形耦合本构模型

在Wheeler本构模型框架的基础上,提出了一个水力与力学耦合的本构模型。该模型中的土-水特征曲线采用毛细滞回内变量模型,能够更好地描述水力历史变化下毛细滞回现象对非饱和多孔介质变形的影响,同时也可描述非饱和多孔介质变形对渗流的影响。非饱和土的强度不仅与吸力有关,而且受到饱和度的影响。相同的吸力下,土样经过吸湿和脱湿路径的饱和度不同,因此,非饱和土的强度也不同。此模型以体积含水率的塑性变化和体变的塑性变化为硬化参数,不仅能描述基质吸力对非饱和土的强化作用,而且考虑了饱和度对强度及变形的影响。试验结果与模型预测基本吻合,证明该模型能够模拟非饱和土的主要特性。为了简化,此模型是在各向同性荷载下推得的,有待于推广到一般的应力状态     

A meshless method for axisymmetric problems in continuously nonhomogeneous saturated porous media

A meshless method based on the local Petrov–Galerkin approach is proposed to analyze 3-d axisymmetric problems in porous functionally graded materials. Constitutive equations for porous materials possess a coupling between mechanical displacements for solid and fluid phases. The work is based on the u–u formulation and the incognita fields of the coupled analysis in focus are the solid skeleton displacements and the fluid displacements. Independent spatial discretization is considered for each phase of the model, rendering a more flexible and efficient methodology. Both displacements are approximated by the moving least-squares (MLS) scheme. The paper presents in the first time a general meshless method for the numerical analysis of axisymmetric problems in continuously nonhomogeneous saturated porous media. Numerical results are given for boreholes in continuously nonhomogeneous porous medium with prescribed misfit and exponential variation of material parameters in the excavation zone.     

Upscaling of the permeability by multiscale wavelet transformations and simulation of multiphase flows in heterogeneous porous media

We describe a new approach for simulation of multiphase flows through heterogeneous porous media, such as oil reservoirs. The method, which is based on the wavelet transformation of the spatial distribution of the single-phase permeabilities, incorporates in the upscaled computational grid all the relevant data on the permeability, porosity, and other important properties of a porous medium at all the length scales. The upscaling method generates a nonuniform computational grid which preserves the resolved structure of the geological model in the near-well zones as well as in the high-permeability sectors and upscales the rest of the geological model. As such, the method is a multiscale one that preserves all the important information across all the relevant length scales. Using a robust front-detection method which eliminates the numerical dispersion by a high-order total variation diminishing method (suitable for the type of nonuniform upscaled grid that we generate), we obtain highly accurate results with a greatly reduced computational cost. The speed-up in the computations is up to over three orders of magnitude, depending on the degree of heterogeneity of the model. To demonstrate the accuracy and efficiency of our methods, five distinct models (including one with fractures) of heterogeneous porous media are considered, and two-phase flows in the models are studied, with and without the capillary pressure.     

Application of infrared thermography for temperature distributions in fluid-saturated porous media

Infrared thermography has increasingly gained importance because of environmental and technological advancements of this method and is applied in a variety of disciplines related to non-isothermal flow. However, it has not been used so far for quantitative thermal analysis in saturated porous media. This article suggests infrared thermographic approach to obtain the entire surface temperature distribution(s) in water-saturated porous media. For this purpose, infrared thermal analysis is applied with in situ calibration for a better understanding of the heat transfer processes in porous media. Calibration is achieved with a combination of invasive sensors which are inserted into the medium and non-invasive thermal sensors in which sensors are not inserted to measure temperatures but it works through the detection of infrared radiation emitted from the surface. Thermocouples of relatively thin diameter are used to minimize the disturbance for flow. Thermocouples give the temperature values at specified positions inside the porous medium, and these values are compared with the values suggested by the infrared thermographic device at the same positions, in the calibration exercise. The calibration process was repeated for different temperatures and flow rates to get the temperature distributions of the whole material inside the system. This technique enables us to measure accurate two-dimensional temperature distributions, which is not possible by using thermocouples only. Continuous point heat sources at different flow rates and temperatures are studied experimentally. Additionally, it offers numerical simulations of the experiments utilizing a finite element-based model. A two-dimensional density and viscosity-dependent flow and transport model accounting for thermal dispersion is utilized to simulate the experimental results. Possible small heat losses from the surface are incorporated in the model according to the properties and thickness of the Plexiglass material used for the construction of the experiment tank. The numerical results agree well with the experimental observations.     

可压缩流体饱和孔隙介质中孔隙压力波传播数值分析

首次将高精度时空守恒元/解元方法推广到可压缩流体饱和孔隙介质中孔隙压力波传播的数值计算中。将孔隙度梯度从源（汇）项中分离,直接引入流通量,改进了理论模型。通过对孔隙介质激波问题的数值模拟,验证了方法的精度和有效性。在此基础上,提出了孔隙介质中二维黎曼问题,并揭示了孔隙压力波存在接触间断、激波、膨胀波、压缩波等复杂的结构特征。该成果对二氧化碳地质封存、二氧化碳提高石油采收率、页岩气压裂开采以及地震破裂过程的研究具有重要的理论与应用意义。     

堆石坝非达西渗流场分析

坝体的渗流计算常采用近似的线性渗流计算方法,然而对于堆石结构而言,在高水力梯度的作用下,由于流体惯性力与紊流的影响,水力梯度与渗流流速之间偏离了线性的达西定律,堆石体中不同孔隙介质的交界面错综复杂更增加了渗流场形态的复杂性。此时,使用非达西渗流的计算方法,可以提高渗流计算的精度,更加客观地描述渗流场中的渗流行为。基于伽辽金加权余量法推演了Forchheimer方程的非线性有限单元法公式,应用该方法对堆石坝渗流场问题进行了数值模拟计算,并给出了工程实例。研究表明,用该方法可以很好地研究渗流场中的非达西渗流现象。对比达西渗流计算与非达西渗流计算结果之间的差异,得出了非达西渗流计算的意义。对堆石坝结构而言,非达西渗流计算的结构显示渗流场的局部区域可能出现高水力坡降区域扩散的现象。此外,水力坡降峰值对比达西渗流计算结果有所增大,这说明真实渗流场的水力坡降峰值有可能超出材料的临界水力坡降允许值,从而引起坝体的渗透破坏。     

A fully coupled element‐free Galerkin model for hydro‐mechanical analysis of advancement of fluid‐driven fractures in porous media

Hydraulic fracturing (HF) of underground formations has widely been used in different fields of engineering. Despite the technological advances in techniques of in situ HF, the industry uses semi‐analytical tools to design HF treatment. This is due to the complex interaction among various mechanisms involved in this process, so that for thorough simulations of HF operations a fully coupled numerical model is required. In this study, using element‐free Galerkin (EFG) mesh‐less method, a new formulation for numerical modeling of hydraulic fracture propagation in porous media is developed. This numerical approach, which is based on the simultaneous solution of equilibrium and continuity equations, considers the hydro‐mechanical coupling between the crack and its surrounding porous medium. Therefore, the developed EFG model is capable of simulating fluid leak‐off and fluid lag phenomena. To create the discrete equation system, the Galerkin technique is applied, and the essential boundary conditions are imposed via penalty method. Then, the resultant constrained integral equations are discretized in space using EFG shape functions. For temporal discretization, a fully implicit scheme is employed. The final set of algebraic equations that forms a non‐linear equation system is solved using the direct iterative procedure. Modeling of cracks is performed on the basis of linear elastic fracture mechanics, and for this purpose, the so‐called diffraction method is employed. For verification of the model, a number of problems are solved. According to the obtained results, the developed EFG computer program can successfully be applied for simulating the complex process of hydraulic fracture propagation in porous media. Copyright © 2016 John Wiley & Sons, Ltd.