A mesh-independent finite element formulation for modeling crack growth in saturated porous media based on an enriched-FEM technique

In this paper, the crack growth simulation is presented in saturated porous media using the extended finite element method. The mass balance equation of fluid phase and the momentum balance of bulk and fluid phases are employed to obtain the fully coupled set of equations in the framework of \(u{-}p\) formulation. The fluid flow within the fracture is modeled using the Darcy law, in which the fracture permeability is assumed according to the well-known cubic law. The spatial discritization is performed using the extended finite element method, the time domain discritization is performed based on the generalized Newmark scheme, and the non-linear system of equations is solved using the Newton–Raphson iterative procedure. In the context of the X-FEM, the discontinuity in the displacement field is modeled by enhancing the standard piecewise polynomial basis with the Heaviside and crack-tip asymptotic functions, and the discontinuity in the fluid flow normal to the fracture is modeled by enhancing the pressure approximation field with the modified level-set function, which is commonly used for weak discontinuities. Two alternative computational algorithms are employed to compute the interfacial forces due to fluid pressure exerted on the fracture faces based on a ‘partitioned solution algorithm’ and a ‘time-dependent constant pressure algorithm’ that are mostly applicable to impermeable media, and the results are compared with the coupling X-FEM model. Finally, several benchmark problems are solved numerically to illustrate the performance of the X-FEM method for hydraulic fracture propagation in saturated porous media.     

Hygrothermomechanical evaluation of porous media under finite deformation. Part I - finite element formulations

The coupled thermomechanical responses of fluid-saturated porous continua subjected to finite deformation are investigated. Field equations governing the transient response of the media are derived from a continuum thermodynamics mixture theory based on mass balance, momentum balance and energy balance laws as well as the Clausius-Duhem inequality. Finite element procedures for the two-dimensional response, employing updated Lagrangian formulations for the solid skeleton deformation and the weak formulations for fluid and thermal transport equations, are implemented in a fully implicit form. Temperature-dependent mechanical properties for the non-linear solid matrix, characterized by Perzyna's viscoplastic model, are assumed. An iterative scheme based on the full Newton-Raphson method is presented for simultaneously solving the coupled non-linear equations.     

Numerical modelling of pressurized fracture evolution in concrete using zero-thickness interface elements

The development of cracks due to the effect of fluid pressure is a problem that concerns many areas of engineering, ranging from structural to geotechnical or petroleum. Within the context of the Finite Element Method, the authors have recently proposed a formulation for the coupled hydro-mechanical behaviour of zero-thickness interface elements. This formulation has been verified for pre-existing discontinuities (e.g. natural joints, faults in rock). In this paper, the above formulation, complemented with an appropriate fracture mechanics-based constitutive model, is applied to developing cracks in plain concrete. The numerical results are compared with a series of wedge splitting tests available in the literature, performed on concrete specimens under the influence of fluid pressure at the notch and along the propagating crack. A good agreement is obtained in terms of wedge-splitting force vs. crack mouth opening displacement (CMOD), crack and fluid fronts vs. CMOD, and fluid pressure along the crack vs. time. The influence of splitting rate and input fluid pressure is also systematically analyzed.     

高温下混凝土中热-湿-气-力学耦合过程数值模拟

对高温下混凝土中热-湿-气-力学耦合过程分析提出了一个多孔多相介质的非混溶-混溶两级数学模型。数学模型基于控制干空气、湿份及基质溶解物的质量守恒、混凝土介质混合体的动量守恒和焓(能量)守恒的耦合偏微分方程组。给出了模型的控制方程、状态方程与所采用的本构定律。发展了相应的非线性耦合问题的有限元数值分析过程,以数值模拟火灾和热辐射等热荷载作用下的热-湿-气-力学耦合行为,并进而分析所发生的破坏现象。数值结果例题显示所发展的数学模型和数值方法在重现高温下混凝土中热-湿-气-力学耦合行为的有效性。     

A two‐scale approach for fluid flow in fractured porous media

A two‐scale numerical model is developed for fluid flow in fractured, deforming porous media. At the microscale the flow in the cavity of a fracture is modelled as a viscous fluid. From the micromechanics of the flow in the cavity, coupling equations are derived for the momentum and the mass couplings to the equations for a fluid‐saturated porous medium, which are assumed to hold on the macroscopic scale. The finite element equations are derived for this two‐scale approach and integrated over time. By exploiting the partition‐of‐unity property of the finite element shape functions, the position and direction of the fractures is independent from the underlying discretization. The resulting discrete equations are non‐linear due to the non‐linearity of the coupling terms. A consistent linearization is given for use within a Newton–Raphson iterative procedure. Finally, examples are given to show the versatility and the efficiency of the approach, and show that faults in a deforming porous medium can have a significant effect on the local as well as on the overall flow and deformation patterns. Copyright © 2006 John Wiley & Sons, Ltd.     

A single-domain dual-boundary-element formulation incorporating a cohesive zone model for elastostatic cracks

A cracked elastostatic structure is artificially divided into subdomains of simpler topology such that the well-developed classic dual integral equations can be applied appropriately to each domain. Applying the continuity and equilibrium conditions along artificial boundaries and properties of the integral kernels a single-domain dual-boundary-integral equation formulation is derived for a cracked elastic structure. A cohesive zone model is used to model the crack tip processes and is coupled with the single-domain dual-boundary-integral equation formulation; the resulting nonlinear equations are solved using the iterative method of successive-over-relaxation. The constitutive law used for a crack includes three parts: a law relating cohesive force to crack displacement difference when a crack is opening, a characterization of tangential interaction between crack surfaces when the crack surfaces are in contact, and a maximum principal stress criterion of crack advance. Incorporation of local unloading effect of the cohesive zone material has enabled a simulation of fracture with initial damage, partial development of the failure process zone at structural instability and multiple crack interaction. Some of the features of the method are demonstrated by considering three examples. The first problem is a single-edge-cracked specimen that exhibits a snap-back instability. The second example is the development of wing cracks from an angled crack under compression. The last example demonstrates the capability to consider mixed-mode crack growth and interaction of cracks. Thus, the problem of crack growth has been reduced to the determination of the cohesive model for the fracture process. This revised version was published online in July 2006 with corrections to the Cover Date.     

饱和弹性裂隙介质的混合物理论

本文首先用损伤力学的方法,按孔隙的配置及几何结构,分别定义了含各向异性分布裂隙的固体介质的二阶连续法向裂纹张量和切向裂纹张量。然后,在裂隙内充满流体时,对组分速度、组分偏应力等混合物理论的基本变量进行了各向异性修正。并用混合物理论,建立了饱和裂隙介质中各组分的质量和动量平衡方程。最后,在仅考虑裂纹的单一张开度时,针对线弹性骨架材料,得到了由不可压缩材料构成的各组分的动力学控制方程。     

On macroscopic equations governing multiphase flow with diffusion and chemical reactions in porous media

Macroscopic balance equations for components, momentum and energy are established for a multiphase flow with diffusion, chemical reactions, heat transfer and exchanges of components between phases in a porous medium. These equations are established separately for each fluid phase, for the solid part of the medium, and for interfaces, by starting from the corresponding equations valid at the pore level and taking their mean value around each point. Then macroscopic entropy balance equations are derived. The entropy source density shows clearly the generalized fluxes and forces which appear in the problem, and suggests how to choose phenomenological equations. A simple example illustrating the method is given in the last paragraph, for a single phase flow with heat transfer in a porous medium. One obtains a generalized form of Darcy's equation. Rigorous conditions along the interfaces and contact lines in a multiphasic medium are given in Appendix.     

基于VOF/PLIC界面追踪方法的RTM工艺充模过程数值模拟

针对基于Darcy定律的树脂传递模塑(RTM)工艺的充模过程数值模拟的局限性,将纤维预制体内的充填流动作为两相流(树脂相和空气相)处理,在动量方程中考虑了惯性项和粘性项,采用有限体积方法(FVM)离散控制方程,并与VOF/PLIC界面追踪方法相结合,发展了求解树脂在纤维预制体内非稳态流动问题的数值模拟方法.在此基础上开发了RTM工艺的充模过程数值模拟程序,其算例的数值模拟结果与解析解或实验结果吻合良好,验证了此数值模拟方法的有效性和可靠性.     

A method for 3-D hydraulic fracturing simulation

We present a method for the simulation of 3-D hydraulic fracturing in fully saturated porous media. The discrete fracture(s) is driven by the fluid pressure. A cohesive fracture model is adopted where the fracture follows the face of the elements around the fracture tip which is closest to the normal direction of the maximum principal stress at the fracture tip. No predetermined fracture path is needed. This requires continuous updating of the mesh around the crack tip to take into account the evolving geometry. The updating of the mesh is obtained by means of an efficient mesh generator based on Delaunay tessellation. The governing equations are written in the framework of porous media mechanics theory and are solved numerically in a fully coupled manner. An examples dealing with a concrete dam is shown.     

Modeling of cohesive crack growth in partially saturated porous media; a study on the permeability of cohesive fracture

Modeling the water flow in cohesive fracture is a fundamental issue in the crack growth simulation of cracked concrete gravity dams and hydraulic fracture problems. In this paper, a mathematical model is presented for the analysis of fracture propagation in the semi-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 modified crack permeability is applied in fracture propagation based on the data obtained from experimental results to implement the roughness of fracture walls.     

An enriched FEM technique for modeling hydraulically driven cohesive fracture propagation in impermeable media with frictional natural faults: Numerical and experimental investigations

In this paper, an enriched finite element technique is presented to simulate the mechanism of interaction between the hydraulic fracturing and frictional natural fault in impermeable media. The technique allows modeling the discontinuities independent of the finite element mesh by introducing additional DOFs. The coupled equilibrium and flow continuity equations are solved using a staggered Newton solution strategy, and an algorithm is proposed on the basis of fixed‐point iteration concept to impose the flow condition at the hydro‐fracture mouth. The cohesive crack model is employed to introduce the nonlinear fracturing process occurring ahead of the hydro‐fracture tip. Frictional contact is modeled along the natural fault using the penalty method within the framework of plasticity theory of friction. Moreover, an experimental investigation is carried out to perform the hydraulic fracturing experimental test in fractured media under plane strain condition. The results of several numerical and experimental simulations are presented to verify the accuracy and robustness of the proposed computational algorithm as well as to investigate the mechanisms of interaction between the hydraulically driven fracture and frictional natural fault. Copyright © 2015 John Wiley & Sons, Ltd.     

On finite dynamic equations for fluid-saturated porous media

Summary The paper concerns the relations between two principally different approaches to the formulation of momentum balance equations in porous media theories, namely, the dynamic approach similar to Biot's theory and the modern approach as a result of mixture theories extended by the concept of volume fractions. In particular, it is shown that both approaches necessarily lead to the same type of momentum balance equations and furthermore contain, in a certain sense, within the geometrically linear case, the well-known classical equations of Biot's theory.     

The nonlinear fracture behavior of an arbitrarily oriented dielectric crack in piezoelectric materials

Summary. This paper presents a comprehensive study on the plane problem of an arbitrarily oriented crack in a piezoelectric medium. Using a dielectric crack model, the electric boundary condition along the crack surfaces is assumed to be governed by the opening displacement of the crack. The formulation of this nonlinear problem is based on the use of Fourier transform and solving the resulting nonlinear singular integral equations. Multiple deformation modes are observed according to different geometric and loading conditions. The effects of the crack orientation and the applied loads upon the fracture behavior of cracked piezoelectric materials are studied. The relation between the current crack model and the commonly used permeable and impermeable models is discussed.     

Governing equations for unsaturated flow through woven fiber mats. Part 1. Isothermal flows

Correct modeling of resin flow in liquid composite molding (LCM) processes is important for accurate simulation of the mold-filling process. Recent experiments indicate that the physics of resin flow in woven (also stitched or braided) fiber mats is very different from the flow in random fiber mats. The dual length-scale porous media created by the former leads to the formation of a sink term in the equation of continuity; such an equation in combination with the Darcy's law successfully replicate the drooping inlet pressure history, and the region of partial saturation behind the flow-front, for the woven mats. In this paper, the mathematically rigorous volume averaging method is adapted to derive the averaged form of mass and momentum balance equations for unsaturated flow in LCM. The two phases used in the volume averaging method are the dense bundle of fibers called tows, and the surrounding gap present in the woven fiber mats. Averaging the mass balance equation yields a macroscopic equation of continuity which is similar to the conventional continuity equation for a single-phase flow except for a negative sink term on the right-hand side of the equation. This sink term is due to the delayed impregnation of fiber tows and is equal to the rate of liquid absorbed per unit volume. Similar averaging of the momentum balance equation is accomplished for the dual-scale porous medium. During the averaging process, the dynamic interaction of the gap flow with the tow walls is lumped together as the drag force. A representation theorem and dimensional analysis are used to replace this drag force with a linear function of an average of the relative velocity of the gap fluid with respect to the tow matrix for both the isotropic and anisotropic media. Averaging of the shear stress term of the Navier–Stokes equation gives rise to a new quantity named the interfacial kinetic effects tensor which includes the effects of liquid absorption by the tows, and the presence of slip velocity on their surface. Though the gradient of the tensor contributes a finite force in the final momentum balance equation, a scaling analysis leads to its rejection in the fibrous dual-scale porous medium if the permeability of flow through the gaps is small. For such a porous medium, the momentum equation reduces to the Darcy's law for single-phase flow.     

Analytical decoupling of poroelasticity equations for acoustic-wave propagation and attenuation in a porous medium containing two immiscible fluids

Poroelasticity theory has become an effective and accurate approach to analyzing the intricate mechanical behavior of a porous medium containing two immiscible fluids, a system encountered in many subsurface engineering applications. However, the resulting partial differential equations in the theory intrinsically take on a coupled form in the terms pertinent to inertial drag, viscous damping, and applied stress, making it difficult to obtain closed-form, steady-state analytical solutions to boundary-value problems except in special cases. In the present paper, we demonstrate that, for dilatational wave excitations, these partial differential equations can be decoupled analytically into three Helmholtz equations featuring complex-valued, frequency-dependent normal coordinates that correspond physically to three independent modes of dilatational wave motion. The normal coordinates in turn can be expressed in the frequency domain as three different linear combinations of the solid dilatation and the linearized increment of fluid content for each pore fluid, or equivalently, as three different linear combinations of total dilatational stress and two pore fluid pressures. These representations are applicable to strain-controlled and stress-prescribed boundary conditions, respectively. Numerical calculations confirm that the phase speed and attenuation coefficient of the three dilatational waves represented by the Helmholtz equations are exactly identical to those obtained previously by numerical solution of the dispersion relations for dilatational wave excitation of a porous medium containing two immiscible fluids. Thus, dilatational wave motions in unsaturated porous media subject to suitable boundary conditions can now be accurately modeled analytically.     

Dislocation pile-up and cleavage: effects of strain gradient plasticity on micro-crack initiation in ferritic steel

The paper is devoted to the problem of slow crack growth in heterogeneous media. The crack is subjected to arbitrary pressure distribution on the crack surface. The problem relates to construction of the so-called equilibrium crack. For such a crack, stress intensity factors are equal to the material fracture toughness at each point of the crack contour. The crack shape and size depend on spatial distributions of the elastic properties and fracture toughness of the medium, and the type of loading. In the paper, attention is focused on the case of layered elastic media when a planar crack propagates orthogonally to the layers. The problem is reduced to a system of surface integral equations for the crack opening vector and volume integral equations for stresses in the medium. For discretization of these equations, a regular node grid and Gaussian approximating functions are used. For iterative solution of the discretized equations, fast Fourier transform technique is employed. An iteration process is proposed for the construction of the crack shape in the process of crack growth. Examples of crack evolution for various properties of medium and types of loading are presented.     

On a moving dielectric crack in a piezoelectric interface with spatially varying properties

This paper provides a comprehensive theoretical analysis of a finite crack propagating with constant speed along an interface between two dissimilar piezoelectric media under inplane electromechanical loading. The interface is modeled as a graded piezoelectric layer with spatially varying properties (functionally graded piezoelectric materials, i.e., FGPMs). The analytical formulations are developed using Fourier transforms and the resulting singular integral equations are solved with Chebyshev polynomials. Using a dielectric crack model with deformation-dependent electric boundary condition, the dynamic stress intensity factors, electric displacement intensity factor, crack opening displacement (COD) intensity factor, and energy release rate are derived to fully understand this inherent mixed mode dynamic fracture problem. Numerical simulations are made to show the effects of the material mismatch, the thickness of the interfacial layer, the crack position, and the crack speed upon the dynamic fracture behavior. A critical state for the electromechanical loading applied to the medium is identified, which determines whether the traditional impermeable (or permeable) crack model serves as the upper or lower bound for the dielectric model considering the effect of dielectric medium crack filling.     

A two-scale model for fluid flow in an unsaturated porous medium with cohesive cracks

A two-scale model is developed for fluid flow in a deforming, unsaturated and progressively fracturing porous medium. At the microscale, the flow in the cohesive crack is modelled using Darcy’s relation for fluid flow in a porous medium, taking into account changes in the permeability due to the progressive damage evolution inside the cohesive zone. From the micromechanics of the flow in the cavity, identities are derived that couple the local momentum and the mass balances to the governing equations for an unsaturated porous medium, which are assumed to hold on the macroscopic scale. The finite element equations are derived for this two-scale approach and integrated over time. By exploiting the partition-of-unity property of the finite element shape functions, the position and direction of the fractures are independent from the underlying discretization. The resulting discrete equations are nonlinear due to the cohesive crack model and the nonlinearity of the coupling terms. A consistent linearization is given for use within a Newton–Raphson iterative procedure. Finally, examples are given to show the versatility and the efficiency of the approach. The calculations indicate that the evolving cohesive cracks can have a significant influence on the fluid flow and vice versa.     

各向异性双重孔隙介质的应力与油水两相渗流耦合理论模型

针对天然裂缝性油藏的特性,建立了描述双重孔隙介质中油水两相流体流动特性的流固耦合理论模型。该模型不仅考虑了渗透率的各向异性,而且考虑了岩石固体骨架变形的各向异性。渗流方程是依据双重孔隙的概念建立起来的,而固体骨架变形控制方程则是根据Biot 的等温、线性孔隙弹性理论建立起来的。同时,给出了横向各向同性及结构各向异性、固体材料各向同性时的双重孔隙介质的应力与油水两相渗流耦合理论模型。对该模型进行了简化,并将其简化后模型与单相流的各项同性和各向异性双重孔隙介质流固耦合理论模型进行了比较。