A fracture mapping and extended finite element scheme for coupled deformation and fluid flow in fractured porous media

This paper presents a fracture mapping (FM) approach combined with the extended finite element method (XFEM) to simulate coupled deformation and fluid flow in fractured porous media. Specifically, the method accurately represents the impact of discrete fractures on flow and deformation, although the individual fractures are not part of the finite element mesh. A key feature of FM‐XFEM is its ability to model discontinuities in the domain independently of the computational mesh. The proposed FM approach is a continuum‐based approach that is used to model the flow interaction between the porous matrix and existing fractures via a transfer function. Fracture geometry is defined using the level set method. Therefore, in contrast to the discrete fracture flow model, the fracture representation is not meshed along with the computational domain. Consequently, the method is able to determine the influence of fractures on fluid flow within a fractured domain without the complexity of meshing the fractures within the domain. The XFEM component of the scheme addresses the discontinuous displacement field within elements that are intersected by existing fractures. In XFEM, enrichment functions are added to the standard finite element approximation to adequately resolve discontinuous fields within the simulation domain. Numerical tests illustrate the ability of the method to adequately describe the displacement and fluid pressure fields within a fractured domain at significantly less computational expense than explicitly resolving the fracture within the finite element mesh. Copyright © 2013 John Wiley & Sons, Ltd.     

A finite element method for modeling coupled flow and deformation in porous fractured media

Modeling the flow in highly fractured porous media by finite element method (FEM) has met two difficulties: mesh generation for fractured domains and a rigorous formulation of the flow problem accounting for fracture/matrix, fracture/fracture, and fracture/boundary fluid mass exchanges. Based on the recent theoretical progress for mass balance conditions in multifractured porous bodies, the governing equations for coupled flow and deformation in these bodies are first established in this paper. A weak formulation for this problem is then established allowing to build a FEM. Taking benefit from recent development of mesh‐generating tools for fractured media, this weak formulation has been implemented in a numerical code and applied to some typical problems of hydromechanical coupling in fractured porous media. It is shown that in this way, the FEM that has proved its efficiency to model hydromechanical phenomena in porous media is extended with all its performances (calculation time, couplings, and nonlinearities) to fractured porous media. Copyright © 2015 John Wiley & Sons, Ltd.     

多孔介质非饱和导水率预测的分形模型

多孔介质非饱和导水率是地下水污染预测与评价的重要参数。根据分形几何的基本原理和方法,推导出了与Campbell经验公式在形式上完全一致的多孔介质非饱和导水率的预测公式。公式中的幂指数为介质孔隙分维和随机行走分维的函数,分别体现了多孔介质的静态性质与动态性质对其中水分运动的影响,但静态性质的影响是主要的,即导水率主要受多孔介质的结构控制。根据文献中报道的大量数据,利用笔者推导的预测公式计算得到的幂指数的统计值与试验测定的幂指数的统计值基本一致,说明推导的理论公式预测多孔介质非饱和导水率是较为可靠的。     

Modeling of dynamic cohesive fracture propagation in porous saturated media

In this paper, a mathematical model is presented for the analysis of dynamic fracture propagation in the saturated porous media. The solid behavior incorporates a discrete cohesive fracture model, coupled with the flow in porous media through the fracture network. The double‐nodded zero‐thickness cohesive interface element is employed for the mixed mode fracture behavior in tension and contact behavior in compression. The crack is automatically detected and propagated perpendicular to the maximum effective stress. The spatial discretization is continuously updated during the crack propagation. Numerical examples from the hydraulic fracturing test and the concrete gravity dam show the capability of the model to simulate dynamic fracture propagation. The comparison is performed between the quasi‐static and fully dynamic solutions, and the performance of two analyses is investigated on the values of crack length and crack mouth opening. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

A three‐dimensional staggered finite element approach for random parametric modeling of thermo‐hygral coupled phenomena in porous media

The aim of this paper is to present a three‐dimensional (3D) finite element modeling of heat and mass transfer phenomena in partially saturated open porous media with random fields of material properties. Randomness leads to transfer processes within the porous medium that naturally need a full 3D modeling for any quantitative assessment of these processes. Nevertheless, the counterpart of 3D modeling is a significant increase in computations cost. Therefore, a staggered solution strategy is adopted which permits to solve the equations sequentially. This appropriate partitioning reduces the size of the discretized problem to be solved at each time step. It is based on a specific iterative algorithm to account for the interaction between all the transfer processes. Accordingly, a suitable linearization of mass convective boundary conditions, consistent with the staggered algorithm, is also derived. After some validation tests, the 3D numerical model is used for studying the drying process of a cementitious material with regard to its intrinsic permeability randomness. Copyright © 2011 John Wiley & Sons, Ltd.     

A mixed solution for two-dimensional unsteady flow in fractured porous media

A mixed finite element–boundary element solution for the analysis of two-dimensional flow in porous media composed of rock blocks and discrete fractures is described. The rock blocks are modelled implicitly by using boundary elements whereas finite elements are adopted to model the discrete fractures. The computational procedure has been implemented in a hybrid code which has been validated first by comparing the numerical results with the closed-form solution for flow in a porous aquifer intercepted by a vertical fracture only. Then, a more complex problem has been solved where a pervious, homogeneous and isotropic matrix containing a net of fractures is considered. The results obtained are shown to describe satisfactorily the main features of the flow problem under study. © 1997 by John Wiley & Sons, Ltd.     

Thermo‐hydro‐mechanical modeling of damage in unsaturated porous media: Theoretical framework and numerical study of the EDZ

The damage model presented in this article (named ‘THHMD’ model) is dedicated to non‐isothermal unsaturated porous media. It is formulated by means of three independent strain state variables, which are the thermodynamic conjugates of net stress, suction and thermal stress. The damage variable is a second‐order tensor. Stress/strain relationships are derived from Helmholtz free energy, which is assumed to be the sum of damaged elastic potentials and ‘crack‐closure energies’. Damage is assumed to grow with tensile strains due to net stress, with pore shrinkage due to suction and with thermal dilatation. Specific conductivities are introduced to account for the effects of cracking on the intensification and on the orientation of liquid water and vapor flows. These conductivities depend on damage and internal length parameters. The mechanical aspects of the THHMD model are validated by comparing the results of a triaxial compression test with experimental measurements found in the literature. Parametric studies of damage are performed on three different heating problems related to nuclear waste disposals. Several types of loading and boundary conditions are investigated. The thermal damage potential is thoroughly studied. The THHMD model is expected to be a useful tool in the assessment of the Excavation Damaged Zone, especially in the vicinity of nuclear waste repositories. Copyright © 2011 John Wiley & Sons, Ltd.     

单层非饱和多孔介质一维瞬态响应半解析解

基于Zienkiewicz提出的非饱和多孔介质波动理论,考虑两相流体和固体颗粒的压缩性以及惯性、黏滞和机械耦合作用,采用半解析的方法获得了一类典型边界条件下单层非饱和多孔介质一维瞬态响应解。首先推导出无量纲化后以位移表示的控制方程,并将其写成矩阵形式;然后,将边界条件齐次化,求解控制方程所对应的特征值问题,得到了满足齐次边界条件的特征值和相对应的特征函数。根据变异系数法并利用特征函数的正交性,得到了一系列仅黏滞耦合的关于时间的二阶常微分方程及相应的初始条件。在此基础上,运用精细时程积分法给出了常微分方程组的数值解。最后,通过若干算例验证了结果的正确性并探讨了单层非饱和多孔介质一维瞬态动力响应的特点。该方法可推广应用于其他典型的边界条件。     

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.     

Three‐dimensional transient thermo‐hydro‐mechanical fundamental solutions of unsaturated soils

The governing differential equations of unsaturated soils considering the thermo‐poro‐mechanical behaviour consist of equilibrium, moisture air and heat transfer equations. In this paper at first, following some necessary simplifications, the thermal three‐dimensional fundamental solution for an unsaturated deformable porous medium with linear elastic behaviour in Laplace transform domain is presented. Subsequently, the closed‐form time domain fundamental solutions are derived by analytical inversion of the Laplace transform domain solutions. Then a set of numerical results are presented, which demonstrate the accuracies and some salient features of the derived analytical transient fundamental solutions. Finally, the closed‐form time domain fundamental solution will be verified mathematically by comparison with the previously introduced corresponding fundamental solution. Copyright © 2009 John Wiley & Sons, Ltd.     

采用复合单元法模拟裂隙多孔介质变饱和流动

采用复合单元法建立了模拟裂隙多孔介质变饱和流动的数值模型。该模型具有以下特点:裂隙不需要离散成特定单元,而是根据几何位置插入到孔隙基质单元中形成复合单元;在复合单元中,分别建立裂隙流和孔隙基质流的计算方程,二者通过裂隙-基质界面产生联系并整合成复合单元方程;复合单元方程具有和常规有限单元方程相同的格式,因此,可以使用常规有限单元方程的求解技术。采用欠松弛迭代、集中质量矩阵以及自适应时步调节等技术,开发了裂隙多孔介质变饱和流动计算程序。通过模拟一维干土入渗和复杂裂隙含水层内的流动问题,验证了该模型的合理性和适用性。模拟结果为进一步认识非饱和裂隙含水层地下水流动特性提供了理论依据。     

Dynamics of unsaturated poroelastic solids at finite strain

We derive the governing equations for the dynamic response of unsaturated poroelastic solids at finite strain. We obtain simplified governing equations from the complete coupled formulation by neglecting the material time derivative of the relative velocities and the advection terms of the pore fluids relative to the solid skeleton, leading to a so‐called u^s ? p^w ? p^a formulation. We impose the weak forms of the momentum and mass balance equations at the current configuration and implement the framework numerically using a mixed finite element formulation. We verify the proposed method through comparison with analytical solutions and experiments of quasi‐static processes. We use a neo‐Hookean hyperelastic constitutive model for the solid matrix and demonstrate, through numerical examples, the impact of large deformation on the dynamic response of unsaturated poroelastic solids under a variety of loading conditions. Copyright © 2011 John Wiley & Sons, Ltd.     

复杂水力路径下非饱和流动的数值分析

多孔介质中的流动问题,与孔隙介质的特性,含水量状态以及含水量的变化历史密切相关。基于毛细循环滞回理论模型,考虑含水量变化历史对土水特征关系的影响,在开发的U-DYSAC2有限元程序中进行了相应的数值实施。在试验给定的初边值条件下进行了非饱和渗流模拟分析,并将模拟结果与实测数据比较,表明在压力边界条件反复变化下,考虑滞回效应能获得更接近实测的结果,证实该模型在模拟各种循环变化条件下非饱和土渗流初边值问题的适用性与必要性。对入渗重分布反复变化条件下非饱和土柱流动的数值模拟表明,考虑滞回与不考虑滞回条件下,含水量、孔隙水压力和湿峰的迁移的预测在入渗后的重分布过程差异较大。考虑滞回效应时,土柱上部的脱湿速率、下部的吸湿速率比不考虑滞回时要低。从而证实了非饱和多孔介质中的土水状态依赖于含水量变化,而且强烈依赖于土体的水力路径变化。因此,循环边界条件变化下,毛细滞回效应在非饱和渗流模拟中的影响显著,必须加以考虑。     

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     

模拟裂隙多孔介质中变饱和渗流的广义等效连续体方法

描述了一种计算裂隙多孔介质中变饱和渗流的广义等效连续体方法。这种方法忽略裂隙的毛细作用,设定一个与某孔隙饱和度相对应的综合饱和度极限值,并假定：（1）如果裂隙多孔介质的综合饱和度小于该极限值,水只在孔隙中存在并流动,而裂隙中则没有水的流动;（2）如果综合饱和度等于或大于该极限值,水将进入裂隙,并在裂隙内运动。分析比较了等效连续体模型的不同计算方法,并给出了一个模拟裂隙岩体中变饱和渗流与传热耦合问题的应用算例。结果表明,所述方法具有一般性,可以有效地模拟裂隙多孔介质中变饱和渗流的基本特征。     

A fully coupled hydro‐thermo‐poro‐mechanical model for black oil reservoir simulation

A fully coupled formulation of a hydro‐thermo‐poro‐mechanical model for a three‐phase black oil reservoir model is presented. The model is based upon the approach proposed by one of the authors which fully couples geomechanical effects to multiphase flow. Their work is extended here to include non‐isothermal effects. The gas phase contribution to the energy equation has been neglected based on a set of assumptions. The coupled formulation given herein differs in several ways when compared to the earlier work and an attempt is made to link the flow based formulation and mixture theory. The Finite Element Method is employed for the numerical treatment and essential algorithmic implementation is discussed. Numerical examples are presented to provide further understanding of the current methodology. Copyright © 2001 John Wiley & Sons, Ltd.     

A three‐phase model for unsaturated soils

A comprehensive framework to define the constitutive behaviour of unsaturated soils is developed within the theory of mixtures applied to three‐phase porous media. Each of the three phases is endowed with its own strain and stress. Elastic and elastic–plastic constitutive equations are developed. Particular emphasis is laid on the interactions between the phases both in the elastic and plastic regimes. Nevertheless, the clear structure of the constitutive equations requires a minimal number of material parameters. Their identification is provided: in particular, it incorporates directly the soil–water characteristic curve. Crucial to the formulation is an appropriate definition of the effective stress. The coupled influence of this effective stress and of suction makes it possible to describe qualitatively many of the characteristic features observed in experiments, e.g. for normally consolidated soils, a plastic behaviour up to air entry followed by an elastic behaviour at increasing suctions, and, on the way back, an elastic behaviour, unless compression is applied in which case plastic collapse occurs. Copyright © 2000 John Wiley & Sons, Ltd.     

A comparison of variational formats for porous media subjected to dynamic loading

This paper presents a comparison of two variational formats for fully saturated porous media subjected to dynamic loading, whereby the general situation of relative fluid acceleration is considered: (1) the classical three‐field ( u , p, w )‐format and (2) a novel two‐field ( u , p)‐format, where the seepage velocity w is a spatially ‘local’ field whose treatment resembles that of internal variables in material models. The limited numerical comparison shows that the ( u , p)‐format competes well with the ( u , p, w )‐format. Indeed, it is consistent with the general acceleration modeling in the full range of permeabilities. Moreover, in the low permeability regime (where the magnitude of w is insignificant), the new format reflects the situation pertinent to ‘added‐mass’ and is more efficient than the classical ( u , p, w )‐format. Finally, the ( u , p)‐formatcan conveniently be implemented in existing FE‐codes based on the ‘added mass’ formulation. Copyright © 2010 John Wiley & Sons, Ltd.     

A coupled poroplastic damage model accounting for cracking effects on both hydraulic and mechanical properties of unsaturated media

Damage induced by microcracking affects not only the mechanical behaviour of geomaterials but also their hydraulic properties. Evaluating these impacts is important for many engineering applications, such as the safety assessment of radioactive waste disposal facilities. This paper presents a new constitutive model accounting simultaneously for the impact of damage on hydraulic and mechanical properties of unsaturated poroplastic geomaterials. The hydro‐mechanical coupling is formulated by means of the thermodynamic framework for partially saturated media, extended by taking into account isotropic damage and plasticity. State and complementary laws are governed by the so‐called plastic effective stress and equivalent pore pressure. Assuming a bimodal pore size distribution for cracked porous media, the hydraulic part (water retention curve and hydraulic conductivity) is modelled using phenomenological functions of damage variable. The participation of damage on both mechanical and hydraulic part enables this model to describe bilateral couplings between them. This coupled model is then validated against a number of experimental data obtained from Callovo‐Oxfordian argillite, which is the possible host rock for a radioactive waste disposal in France. Parametric studies are also carried out to check the consistency and to better demonstrate the bilateral couplings in the model. Copyright © 2015 John Wiley & Sons, Ltd.