Study of desaturation and resaturation in brittle rock with anisotropic damage

This paper deals with numerical modelling of anisotropic damage induced by desaturation and resaturation processes in a brittle rock. This study is conducted in the framework of geological barrier safety analysis for deep disposal of nuclear waste. A non-linear poroelastic model coupled with anisotropic damage is proposed for constitutive modelling of unsaturated rock. A fully coupled FEM method is used for modelling of hydromechanical coupling problems. Instantaneous phase change without dissipation between water liquid phase and vapour is included. Parametric studies are performed to investigate influences of main factors involved in such processes. Rock damage induced by excavation, desaturation and resaturation is evaluated. Finally, we analyse the importance of taking into account the correlation between induced damage and rock permeability.     

Dynamic Green's functions for an anisotropic poroelastic half-space

The dynamic responses of an anisotropic poroelastic half-space under an internal point load and fluid source are investigated in the frequency domain in this paper. By virtue of Fourier transform and Stroh formalism, the three-dimensional (3D) general solutions of the anisotropic Biot's coupling dynamics equations are derived in the frequency domain. Considering the two surface conditions, permeable and impermeable, the analytical solutions for displacement fields and pore pressure in half-space under a point source (point load or a fluid source) are obtained. When the material properties are isotropic, the numerical results of the poroelastic half-space are in excellent agreement with the existing analytical solutions. For anisotropic half-space cases, numerical results show the strong dependence of the dynamic Green's functions on the material properties.     

Numerical Modelling of the Anisotropic Mechanical Behaviour of Opalinus Clay at the Laboratory-Scale Using FEM/DEM

The Opalinus Clay (OPA) is an argillaceous rock formation selected to host a deep geologic repository for high-level nuclear waste in Switzerland. It has been shown that the excavation damaged zone (EDZ) in this formation is heavily affected by the anisotropic mechanical response of the material related to the presence of bedding planes. In this context, the purpose of this study is twofold: (i) to illustrate the new developments that have been introduced into the combined finite-discrete element method (FEM/DEM) to model layered materials and (ii) to demonstrate the effectiveness of this new modelling approach in simulating the short-term mechanical response of OPA at the laboratory-scale. A transversely isotropic elastic constitutive law is implemented to account for the anisotropic elastic modulus, while a procedure to incorporate a distribution of preferentially oriented defects is devised to capture the anisotropic strength. Laboratory results of indirect tensile tests and uniaxial compression tests are used to calibrate the numerical model. Emergent strength and deformation properties, together with the simulated damage mechanisms, are shown to be in strong agreement with experimental observations. Subsequently, the calibrated model is validated by investigating the effect of confinement and the influence of the loading angle with respect to the specimen anisotropy. Simulated fracture patterns are discussed in the context of the theory of brittle rock failure and analyzed with reference to the EDZ formation mechanisms observed at the Mont Terri Underground Research Laboratory.     

Stochastic Representation of Sedimentary Geology

Discrete fracture network representations of discontinuities in rock masses have been shown to be useful in capturing heterogeneity in rock mass properties. Providing computational efficiency in the resulting simulations and analyses is attained, these fracture representations can be combined with structural modelling and sampling algorithms. Multiple fracture network realisations can be generated and the resulting rock mass properties interrogated. Statistical analyses based on fracture connectivity, block size distribution and slope stability can be performed and provide results defined in terms of confidence intervals. For sedimentary geology consisting of dense bedding, equivalent medium continuum methods have traditionally been used in preference to discrete fracture representations due to the large numbers of structures involved and resulting computational complexity. In this paper, it is shown that stochastic representation of these layers can be employed. An analytical solution to accommodate bedding given an assumed block size distribution has been derived. Using this formulation, polyhedral modelling has been used to investigate the influence of bedding on block formation and block size distributions using field data. It is shown that the analysis is both computationally efficient and can capture truncation of size distribution by such layers without numerical methods.     

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.     

A particle mechanics approach for the dynamic strength model of the jointed rock mass considering the joint orientation

In nature, there exist several forms of anisotropy in rock masses due to the presence of bedding planes, joints, and weak layers. It is well understood that the anisotropic properties of jointed rock masses significantly affect the stability of surface and underground excavations. However, these critical anisotropic characteristics are often ignored in existing uniaxial dynamic failure criteria. This study investigates the effect of a pre-existing persistent joint on the rate-dependent mechanical behaviours of a rock mass using a particle mechanics approach, namely, bonded particle model (BPM), to realistically replicate the mechanical response of the rock mass. Firstly, in order to capture the rate-dependent response of the jointed rock mass, the BPM model is validated using published experimental data. Then, a dynamic strength model is proposed based on the Jaeger criterion and simulation results. To further investigate the dynamic behaviours, the dynamic uniaxial compressive strength (UCS) for anisotropic rock masses with various joint orientations is investigated by subjecting the BPM models to uniaxial compression numerical tests with various strain rate. The proposed dynamic strength model is validated based on numerical simulation results. Finally, the fragmentation characteristics of the jointed rock masses are analysed, which demonstrate that the failure mode affects the dynamic UCS. This is further confirmed by the analysis of the orientations of microscopic cracks generated by the compression loading.     

The effect of the nanogranular nature of shale on their poroelastic behavior

Natural composite materials are highly heterogeneous porous materials, with porosities that manifest themselves at scales much below the macroscale of engineering applications. A typical example is shale, the transverse isotropic sealing formation of most hydrocarbon bearing reservoirs. By means of a closed loop approach of microporomechanics modeling, calibration and validation of elastic properties at multiple length scales of shale, we show that the nanogranular nature of this highly heterogeneous material translates into a unique poroelastic signature. The self-consistent scaling of the porous clay stiffness with the clay packing density minimizes the anisotropy of the Biot pore pressure coefficients; whereas the intrinsic anisotropy of the elementary particle translates into a pronounced anisotropy of the Skempton coefficients. This new microporoelasticity model depends only on two shale-specific material parameters which neatly summarize clay mineralogy and bulk density, and which makes the model most appealing for quantitative geomechanics, geophysics and exploitation engineering applications.     

An anisotropic damage–plasticity model for saturated quasi‐brittle materials

This article is devoted to numerical modeling of anisotropic damage and plasticity in saturated quasi‐brittle materials such as rocks and concrete. The damaged materials are represented by an isotropic poroelastic matrix containing a number of families of microcracks. Based on previous works, a discrete thermodynamic approach is proposed. Each family of microcracks exhibits frictional sliding along crack surfaces as well as crack propagation. The frictional sliding is described by a Coulomb–Mohr‐type plastic criterion by taking into account the effect of fluid pressure through a generalized effective stress concept. The damage evolution is entirely controlled by and coupled with the frictional sliding. The effective elastic properties as well as Biot's coefficients of cracked porous materials are determined as functions of induced damage. The inelastic deformation due to frictional sliding is also taken into account. The procedure for the identification of the model's parameters is presented. The proposed model is finally applied to study both mechanical and poromechanical responses of a typical porous brittle rock in drained and undrained compression tests as well as in interstitial pressure controlled tests. The main features of material behaviors are well reproduced by the model. Copyright © 2012 John Wiley & Sons, Ltd.     

Evolution of anisotropy with saturation and its implications for the elastoplastic responses of clay rocks

Many clay rocks have distinct bedding planes. Experimental studies have shown that their mechanical properties evolve with the degree of saturation (DOS), often with higher stiffness and strength after drying. For transversely isotropic rocks, the effects of saturation can differ between the bed-normal (BN) and bed-parallel (BP) directions, which gives rise to saturation-dependent stiffness and strength anisotropy. Accurate prediction of the mechanical behavior of clay rocks under partially saturated conditions requires numerical models that can capture the evolving elastic and plastic anisotropy with DOS. In this study, we present an anisotropy framework for coupled solid deformation-fluid flow in unsaturated elastoplastic media. We incorporate saturation-dependent strength anisotropy into an anisotropic modified Cam-Clay (MCC) model and consider the evolving anisotropy in both the elastic and plastic responses. The model was calibrated using experimental data from triaxial tests to demonstrate its capability in capturing strength anisotropy at various levels of saturation. Through numerical simulations, we demonstrate the role of evolving stiffness and strength anisotropy in the mechanical behavior of clay rocks. Plane strain simulations of triaxial compression tests were also conducted to demonstrate the impacts of material anisotropy and DOS on the mechanical and fluid flow responses.     

Homogenized constitutive laws for rocks with elastoplastic fractures

This work deals with the development of a constitutive law for fractured rocks. Fractures are considered as penny‐shaped inclusions, whose constitutive law is deduced from an interface law and a regularization procedure. Such a method is applied to linear and non‐linear interface behaviours in order to reproduce effects such as an increase in stiffness during fracture closing, dilatancy or asperity surface degradation. Then, considering the fractured rock as a composite material, we use a Mori–Tanaka method to estimate the homogenized properties of the rock. Numerical experiments illustrate the interest of the proposed homogenization procedure. Copyright © 2007 John Wiley & Sons, Ltd.     

Elastic Solution for Deep Tunnels. Application to Excavation Damage Zone and Rockbolt Support

Summary  A new formulation is presented for deep circular tunnels in rock with cylindrical anisotropy. The formulation is an exact solution since it satisfies equilibrium, strain compatibility, and the anisotropic constitutive model. Complete solutions have been found for two scenarios: tunnel with excavation damage zone, and tunnel with rockbolt support. The solution is based on the assumption of a deep, circular tunnel in a medium with two homogeneous zones: an inner zone surrounding the tunnel, which is either isotropic or anisotropic, and an outer zone, for the remainder of the medium, which is isotropic. Plane strain conditions, elastic response of rock, rockbolts and support, and simultaneous excavation and support installation are also assumed. For tunnels surrounded by an excavation damage zone with reduced rock properties, the tangential stresses and the radial deformations at the tunnel wall are very sensitive to both the magnitude of stiffness reduction of the damaged rock and the size of the damaged zone. The effect of the rockbolts on the rock is approximated by treating the rockbolt-rock composite as a material with cylindrical anisotropy with stiffnesses related to the properties of the rock and rockbolts, and spacing of the rockbolts. Comparisons between the analytical solution and a numerical method show small differences and provide confidence in the approach suggested.     

Linear coupled analysis of desiccation shrinkage in a double‐layer partially saturated medium: semi‐explicit solutions

Solutions are presented for the problem of isothermal dessiccation shrinkage in a double‐layer porous partially saturated medium. The rheological model taken into account is linear poroelastic. Hence the analysis is mainly focused on hydromechanical coupling effects and contrasts of mechanical and hydraulic properties between two materials: a low thickness skin comprised between the outer boundary and the reference porous material. Three one‐dimensional ideal structures are taken into account: a wall of finite thickness (cartesian geometry), a thick cylinder and a thick sphere. The solution of the time‐dependent problem is arrived at by applying Laplace transforms to the field variables. Exact solutions are obtained in Laplace transform space using Mathematica^© to solve the field equations whilst taking into account the continuity equations at the interface and the boundary conditions. The Talbot's modified algorithm has been performed to invert the Laplace transform solutions. A bibliographical and numerical study shows that this method is remarkably precise, stable and close to the analytical inversion. Results are presented using poroelastic data representative of a concrete material and involve a strong coupling effect between hydraulical and mechanical behaviours. A first approach elastic modelling of degradation process have been presented using a thin outer layer. Apart from emphasising the semi‐explicit solution utility due to accurate speed calculation, this paper deals with more complex problems than those which can be solved using purely analytical solutions. Copyright © 2001 John Wiley & Sons, Ltd.     

裂缝诱导双相HTI介质模型及其弹性波传播方程

将Biot双相介质理论与Gurevich裂缝各向异性理论相结合,建立了能够同时考虑实际裂缝性储层孔隙性和各向异性的裂缝诱导双相HTI介质模型。从本构方程、动力学方程和动力学达西定律出发,推导出了裂缝诱导双相HTI介质中弹性波传播的一阶速度-应力方程,并针对方程的刚性问题,给出了利用显式二阶时间积分法数值求解该方程时所需要满足的稳定性条件。该方程能够定量地给出双相HTI介质的波场特征与裂缝参数、背景孔隙介质参数之间的关系,描述弹性波在这种介质中的传播机理。     

A coupled time integration algorithm for discontinuous deformation analysis using the numerical manifold method

To improve the computational efficiency of the numerical manifold method for discontinuous deformation simulations, a spatial-domain coupled explicit-implicit time integration algorithm is proposed. A subdomain partition algorithm based on a super manifold element is developed for the numerical manifold method to simulate dynamic motions of blocky rock mass. In different subdomains, explicit or implicit time integration method is employed respectively based on its contact and motion status. These subdomains interact through assembling the corresponding explicit or implicit time integration-based matrices of different rock blocks. The computational efficiency of the discontinuity system under dynamic loading is improved by partially diagonalizing the global matrices. Two verification examples of a sliding block along an inclined plane under a horizontal acceleration excitation and a multiblock system acted on by dynamic forces are studied to examine the accuracy of the proposed numerical method, respectively. A highly fractured rock mass situated on an inclined slope subjected to seismic excitations is then studied to show the computational efficiency of the developed algorithm. The simulated results are in good agreement with those from the versions using purely implicit or explicit time integration algorithm for the numerical manifold method. The computational efficiency is shown to be higher using the proposed algorithm, which demonstrates its potential for application in dynamic analysis of highly fractured rock masses.     

含结构面岩体试样单轴强度与变形特征

结构面是岩体区别于岩石材料的一大特征,其产状、迹长、密度等参数对岩体的力学性质有着重要影响。本文利用FLAC3D对含结构面岩体试样的单轴压缩特性进行了较为系统的数值模拟研究。文中建立了含不同组贯通性结构面的岩体试样模型和含不同倾角及迹长的非贯通结构面岩体试样模型,对每个试样进行单轴压缩试验的数值模拟,结构体和结构面均采用Mohr-Coulomb剪切和拉伸破坏准则。模拟中用编制的伺服控制程序通过调节加载速度,控制试样内最大不平衡力,研究含结构面试样单轴压缩情况下的变形、强度及破坏方式等特征。模拟结果显示,含1-3组贯通性结构面试样呈现各向异性特征,而含4组贯通性结构面试件呈现各向同性特征。随着贯通性结构面数量的增多,同尺寸试件的变形强度参数劣化。含单组非贯通性结构面试件,其单轴压缩模拟试验的应力-应变曲线峰值后出现应力降。基于Mohr-Coulomb抗剪强度准则和损伤理论所得的解析解与数值模拟结果所得的非贯通性结构面试件的单轴压缩强度不符,说明用抗剪强度准则与损伤理论刻画非贯通结构面试样的强度并不合理。随着非贯通性结构面贯通率的增大,试件的变形、强度参数劣化。含单组结构面试件的破坏方式可分为结构面控制破坏,结构面部分控制破坏和结构面不控制破坏3种类型,而随着结构面组数的增多,结构面控制试样破坏的概率增加。     

Micromechanics approach to poroelastic behavior of a jointed rock

The formulation of macroscopic poroelastic behavior of a jointed rock is investigated within the framework of a micro–macro approach. The joints are modeled as interfaces, and their behavior is modeled by means of generalized poroelastic state equations. Starting from Hill's lemma extended for a jointed medium and extending the concept of strain concentration to relate the joint displacement jump to macroscopic strain, the overall poroelastic constitutive equations for the jointed rock are formulated. The analysis emphasizes the main differences and similarities of the resulting behavior with respect to that characterizing ordinary porous media. It is shown that, unlike ordinary porous media, conditions on the poroelastic parameters of joints are required for the macroscopic drained stiffness to entirely define the poroelastic behavior. This is achieved, for instance, if the joint network is characterized by a unique Biot coefficient. Extension of the analysis to non‐linear poroelasticity is also outlined. Finally, the theoretical formulation is applied to two particular cases of jointed rock for which explicit expressions of the overall poroelastic parameters are derived. Copyright © 2011 John Wiley & Sons, Ltd.     

全拖曳式深海直流电阻率法三维任意各向异性正演模拟

高精度、高效率的深海勘探是当前国家海洋资源勘探开发的研究热点之一,设计观测效率高、勘探成本低的探测方式,开发可精确计算海底复杂电性环境的正演模拟算法有助于推进该研究的进展。本文基于前人的研究经验,引入了全拖曳式深海直流电阻率观测方式,同时考虑到海底沉积环境的电各向异性,开发了基于非结构有限元方法的三维任意各向异性深海直流电阻率正演算法,实现了对海底任意电各向异性情况的仿真模拟。本文算法与层状一维模型半解析解的对比验证了算法的精度。对典型各向异性电性模型的仿真模拟和分析,证明了全拖曳式深海探测方式对海底沉积层和矿产资源电各向异性的高分辨率。在此基础上,总结了电导率各向异性主轴沿笛卡尔坐标旋转时视电阻率分布也发生同方向旋转的视电阻率分布特征,并根据海底地形对各向异性探测的影响进行了简单探讨。     

Poroelastic behaviour of saturated brittle rock with anisotropic damage

A new anisotropic poroelastic damage model is proposed for saturated brittle porous materials. The model is formulated in the framework of the continuum damage mechanics. A second‐rank symmetric tensor is used to characterize material damage due to oriented microcracks. The classic Biot poroelastic theory is then extended to include poroelastic damage coupling. Both the deterioration of elastic properties and poroelastic coefficients is taken into account. A suitable procedure for determination of model parameters from standard laboratory tests is presented. The validity of the model is tested through comparison between numerical predictions and experimental data in various loading conditions. The overall performance of the model is evaluated. The choice of relevant effective stress for the microcrack propagation criterion in saturated cohesive geomaterials is discussed. Copyright © 2000 John Wiley & Sons, Ltd.     

An Integrated Numerical Modelling–Discrete Fracture Network Approach Applied to the Characterisation of Rock Mass Strength of Naturally Fractured Pillars

Naturally fractured mine pillars provide an excellent example of the importance of accurately determining rock mass strength. Failure in slender pillars is predominantly controlled by naturally occurring discontinuities, their influence diminishing with increasing pillar width, with wider pillars failing through a combination of brittle and shearing processes. To accurately simulate this behaviour by numerical modelling, the current analysis incorporates a more realistic representation of the mechanical behaviour of discrete fracture systems. This involves realistic simulation and representation of fracture networks, either as individual entities or as a collective system of fracture sets, or a combination of both. By using an integrated finite element/discrete element–discrete fracture network approach it is possible to study the failure of rock masses in tension and compression, along both existing pre-existing fractures and through intact rock bridges, and incorporating complex kinematic mechanisms. The proposed modelling approach fully captures the anisotropic and inhomogeneous effects of natural jointing and is considered to be more realistic than methods relying solely on continuum or discontinuum representation. The paper concludes with a discussion on the development of synthetic rock mass properties, with the intention of providing a more robust link between rock mass strength and rock mass classification systems.     

An analytical approach for cased borehole stress calculation in general anisotropic formations

We present an explicit analytical approach for calculating the distribution of stress around a circular cased borehole in a general anisotropic formation. This approach is developed based on a pure elastic cased borehole model, which can include multiple cemented concentric casing strings that are surrounded by a homogeneous anisotropic medium. The wall of the innermost casing is under the compression of borehole internal fluid pressure, and the formation can be subjected to arbitrary far‐field stresses. Every casing–cement boundary is assumed to be welded, so as the cement–formation boundary. The derived analytical solution can be applied to an arbitrary wellbore trajectory in an arbitrary anisotropic formation and is also applicable to degenerate isotropic formations. Accuracy and robustness of the analytical solution is validated through comparing its results with those calculated from the classical Kirsch solution for an isotropic example that has identical elastic properties for all materials and from a finite element method for an anisotropic cased borehole example.