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.     

Thermo-hydro-mechanical modeling of fracturing porous media with two-phase fluid flow using X-FEM technique

In this paper, a fully coupled thermo-hydro-mechanical model is presented for two-phase fluid flow and heat transfer in fractured/fracturing porous media using the extended finite element method. In the fractured porous medium, the traction, heat, and mass transfer between the fracture space and the surrounding media are coupled. The wetting and nonwetting fluid phases are water and gas, which are assumed to be immiscible, and no phase-change is considered. The system of coupled equations consists of the linear momentum balance of solid phase, wetting and nonwetting fluid continuities, and thermal energy conservation. The main variables used to solve the system of equations are solid phase displacement, wetting fluid pressure, capillary pressure, and temperature. The fracture is assumed to impose the strong discontinuity in the displacement field and weak discontinuities in the fluid pressure, capillary pressure, and temperature fields. The mode I fracture propagation is employed using a cohesive fracture model. Finally, several numerical examples are solved to illustrate the capability of the proposed computational algorithm. It is shown that the effect of thermal expansion on the effective stress can influence the rate of fracture propagation and the injection pressure in hydraulic fracturing process. Moreover, the effect of thermal loading is investigated properly on fracture opening and fluids flow in unsaturated porous media, and the convective heat transfer within the fracture is captured successfully. It is shown how the proposed computational model is capable of modeling the fully coupled thermal fracture propagation in unsaturated porous media.     

Numerical modeling of hydraulic fracture propagation,closure and reopening using XFEM with application to in‐situ stress estimation

In this paper, a fully coupled model is developed for numerical modeling of hydraulic fracturing in partially saturated weak porous formations using the extended finite element method, which provides an effective means to simulate the coupled hydro‐mechanical processes occurring during hydraulic fracturing. The developed model is for short fractures where plane strain assumptions are valid. The propagation of the hydraulic fracture is governed by the cohesive crack model, which accounts for crack closure and reopening. The developed model allows for fluid flow within the open part of the crack and crack face contact resulting from fracture closure. To prevent the unphysical crack face interpenetration during the closing mode, the crack face contact or self‐contact condition is enforced using the penalty method. Along the open part of the crack, the leakage flux through the crack faces is obtained directly as a part of the solution without introducing any simplifying assumption. If the crack undergoes the closing mode, zero leakage flux condition is imposed along the contact zone. An application of the developed model is shown in numerical modeling of pump‐in/shut‐in test. It is illustrated that the developed model is able to capture the salient features bottomhole pressure/time records exhibit and can extract the confining stress perpendicular to the direction of the hydraulic fracture propagation from the fracture closure pressure. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Calculating the internodal transmissibilities using finite analytic method and its application for multi‐phase flow in heterogeneous porous media

In the traditional numerical reservoir simulations, the internodal transmissibility is usually defined as the harmonic mean of the permeabilities of the adjacent grids. This definition underestimates the phase flux and the speed of the saturation front, especially for the strong heterogeneous case. In this article, the internodal transmissibility is recalculated according to the nodal analytic solution. The redefined internodal transmissibility can be used directly to calculate the multiphase flow in the numerical reservoir simulations. Numerical examples show that, compared to the traditional numerical methods, the proposed scheme makes the convergences much faster as the refinement parameter increases, and the accuracy is independent of the heterogeneity. Copyright © 2016 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.     

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.     

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.     

Numerical study of two‐phase fluid distributions in fractured porous media

Two‐phase fluid distributions in fractured porous media were studied using a single‐component multiphase (SCMP) lattice Boltzmann method (LBM), which was selected among three commonly used numerical approaches through a comparison against the available results of micro X‐ray computed tomography. The influence of the initial configuration and the periodic boundary conditions in the SCMP LBM for the fluid distribution analysis were investigated as well. It was revealed that regular porous media are sensitive to the initial distribution, whereas irregular porous media are insensitive. Moreover, to eliminate the influence of boundaries, the model's buffer size of an SCMP LBM simulation was suggested to be taken as approximately 12.5 times the average particle size. Then, the two‐phase fluid distribution of a porous medium was numerically studied using the SCMP LBM. Both detailed distribution patterns and macroscopic morphology parameters were reasonably well captured. Finally, the two‐phase fluid distributions in a fractured porous media were investigated. The influence of the degree of saturation, fracture length, and fracture width on the fluid distributions and migration was explored. Copyright © 2015 John Wiley & Sons, Ltd.     

Coupled formulation and algorithms for the simulation of non‐planar three‐dimensional hydraulic fractures using the generalized finite element method

This paper presents a coupled hydro‐mechanical formulation for the simulation of non‐planar three‐dimensional hydraulic fractures. Deformation in the rock is modeled using linear elasticity, and the lubrication theory is adopted for the fluid flow in the fracture. The governing equations of the fluid flow and elasticity and the subsequent discretization are fully coupled. A Generalized/eXtended Finite Element Method (G/XFEM) is adopted for the discretization of the coupled system of equations. A Newton–Raphson method is used to solve the resulting system of nonlinear equations. A discretization strategy for the fluid flow problem on non‐planar three‐dimensional surfaces and a computationally efficient strategy for handling time integration combined with mesh adaptivity are also presented. Several three‐dimensional numerical verification examples are solved. The examples illustrate the generality and accuracy of the proposed coupled formulation and discretization strategies. Copyright © 2015 John Wiley & Sons, Ltd.     

Weak discontinuity in porous media: an enriched EFG method for fully coupled layered porous media

In this paper, a series of multimaterial benchmark problems in saturated and partially saturated two‐phase and three‐phase deforming porous media are addressed. To solve the process of fluid flow in partially saturated porous media, a fully coupled three‐phase formulation is developed on the basis of available experimental relations for updating saturation and permeabilities during the analysis. The well‐known element free Galerkin mesh‐free method is adopted. The partition of unity property of MLS shape functions allows for the field variables to be extrinsically enriched by appropriate functions that introduce existing discontinuities in the solution field. Enrichment of the main unknowns including solid displacement, water phase pressure, and gas phase pressure are accounted for, and a suitable enrichment strategy for different discontinuity types are discussed. In the case of weak discontinuity, the enrichment technique previously used by Krongauz and Belytschko [Int. J. Numer. Meth. Engng., 1998; 41:1215–1233] is selected. As these functions possess discontinuity in their first derivatives, they can be used for modeling material interfaces, generating only minor oscillations in derivative fields (strain and pressure gradients for multiphase porous media), as opposed to unenriched and constrained mesh‐free methods. Different problems of multimaterial poro‐elasticity including fully saturated, partially saturated one, and two‐phase flows under the assumption of fully coupled extended formulation of Biot are examined. As a further development, problems involved with both material interface and impermeable discontinuities, where no fluid exchange is permitted across the discontinuity, are considered and numerically discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

A finite element algorithm for parameter identification of material models for fluid saturated porous media

In this contribution an algorithm for parameter identification of geometrically linear Terzaghi–Biot‐type fluid‐saturated porous media is proposed, in which non‐uniform distributions of the state variables such as stresses, strains and fluid pore pressure are taken into account. To this end a least‐squares functional consisting of experimental data and simulated data is minimized, whereby the latter are obtained with the finite element method. This strategy allows parameter identification based on in situ experiments. In order to improve the efficiency of the minimization process, a gradient‐based optimization algorithm is applied, and therefore the corresponding sensitivity analysis for the coupled two‐phase problem is described in a systematic manner. For illustrative purpose, the performance of the algorithm is demonstrated for a slope stability problem, in which a quadratic Drucker–Prager plasticity model for the solid and a linear Darcy law for the fluid are combined. Copyright © 2001 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.     

Modelling of coupled fluid‐mechanical problems in fractured geological media using enriched finite elements

Geological environments, such as petroleum reservoirs, normally exhibit physical discontinuities, for example, fractures and faults. Because of the reduced thickness of these discontinuities, finite element formulations with strong discontinuity have been applied to the numerical modelling of geological environments. Until now, two relevant characteristics of petroleum reservoirs have not been addressed by these formulations. The first is the pore pressure jump in the direction normal to a discontinuity in a fluid‐mechanical coupling condition, which is present primarily in sealing faults owing to the contrast of permeability with the porous medium. The absence of this jump can affect the prediction of the deformability of a physical discontinuity. Furthermore, reservoir models frequently use coarse meshes. Thus, the method used to evaluate the pore pressure in the discontinuity may exhibit a strong dependence relative to the mesh refinement. Based on these characteristics, in this study, a formulation of an enriched finite element for application to coupled fluid‐mechanical problems with pre‐existing physical discontinuities saturated by a single fluid is presented. The formulation employs discontinuous interpolation functions and enables the reproduction of jumps of displacement and pore pressure associated with a discontinuity inside the element without the need to discretise it. An approximation to estimate the pore pressure in the discontinuity was developed, one which seeks to minimise the influence of refinement. The element's response is verified by comparison with a one‐dimensional analytical solution and simple examples that are simulated using commercial software. Copyright © 2015 John Wiley & Sons, Ltd.     

Hybrid time integration and coupled solution methods for nonlinear finite element analysis of partially saturated deformable porous media at small strain

The goal of the paper is to determine the most efficient, yet accurate and stable, finite element nonlinear solution method for analysis of partially saturated deformable porous media at small strain. This involves a comparison between fully implicit, semi‐implicit, and explicit time integration schemes, with monolithically coupled and staggered‐coupled nonlinear solution methods and the hybrid combination thereof. The pore air pressure p_a is assumed atmospheric, that is, p_a=0 at reference pressure. The solid skeleton is assumed to be pressure‐sensitive nonlinear isotropic elastic. Coupled partially saturated ‘consolidation’ in the presence of surface infiltration and traction is simulated for a simple one‐dimensional uniaxial strain example and a more complicated plane strain slope example with gravity loading. Three mixed plane strain quadrilateral elements are considered: (i) Q4P4; (ii) stabilized Q4P4S; and (iii) Q9P4; “Q” refers to the number of solid skeleton displacement nodes, and “P” refers to the number of pore fluid pressure nodes. The verification of the implementation against an analytical solution for partially saturated pore water flow (no solid skeleton deformation) and comparison between the three time integration schemes (fully implicit, semi‐implicit, and explicit) are presented. It is observed that one of the staggered‐coupled semi‐implicit schemes (SIS(b)), combined with the fully implicit monolithically coupled scheme to resolve sharp transients, is the most efficient computationally. Copyright © 2015 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。     

Factor of safety in a partially saturated slope inferred from hydro‐mechanical continuum modeling

Rainfall weakens an earth slope and triggers mass movement. Relevant triggering mechanisms are complex and include reduction of capillary pressure due to increased saturation and frictional drag on the sediment induced by fluid flow. Physics‐based continuum models utilizing modern computational tools are useful for understanding the mechanisms of deformation in partially saturated slopes; however, they do not provide a scalar indicator called ‘factor of safety’ that measures the potential of a given slope for mass movement. In the present work, we employ sequential calculations consisting of a physics‐based finite element model that couples solid deformation with fluid flow to quantify the stress and deformation fields in a steep hillside slope subjected to rainfall infiltration. This is followed by a limit‐equilibrium calculation based on the method of slices that evaluates the desired factor of safety. The field condition investigated is similar to the steep experimental catchment CB1 near Coos Bay, Oregon, which failed as a large debris flow from heavy rainfall. Copyright © 2011 John Wiley & Sons, Ltd.     

An elasto‐plastic three phase model for partially saturated soil for the finite element simulation of compressed air support in tunnelling

This paper presents a fully coupled finite element formulation for partially saturated soil as a triphasic porous material, which has been developed for the simulation of shield tunnelling with heading face support using compressed air. While for many numerical simulations in geotechnics use of a two‐phase soil model is sufficient, the simulation of compressed air support demands the use of a three‐phase model with the consideration of air as a separate phase. A multiphase model for soft soils is developed, in which the individual constituents of the soil—the soil skeleton, the fluid and the gaseous phase—and their interactions are considered. The triphasic model is formulated within the framework of the theory of porous media, based upon balance equations and constitutive relations for the soil constituents and their mixture. An elasto‐plastic, cam–clay type model is extended to partially saturated soil conditions by incorporating capillary pressure according to the Barcelona basic model. The hydraulic properties of the soil are described via DARCY 's law and the soil–water characteristic curve after VAN GENUCHTEN . Water is modelled as an incompressible and air as a compressible phase. The model is validated by means of selected benchmark problems. The applicability of the model to geotechnical problems is demonstrated by results from the simulation of a compressed air intervention in shield tunnelling. Copyright © 2009 John Wiley & Sons, Ltd.     

General coupling extended multiscale FEM for elasto‐plastic consolidation analysis of heterogeneous saturated porous media

This paper presents a general coupling extended multiscale FEM (GCEMs) for solving the coupling problem of elasto‐plastic consolidation of heterogeneous saturated porous media. In the GCEMs, the numerical multiscale base functions for the solid skeleton and fluid phase of the coupling system are all constructed on the basis of the equivalent stiffness matrix of the unit cell, which not only contain the interaction between the solid and fluid phases but also consider the time effect. Furthermore, in order to improve the computational accuracy for two‐dimensional problems, a multi‐node coarse element strategy for the GCEMs is proposed, and a two‐scale iteration algorithm for the elasto‐plastic consolidation analysis is developed. Some one‐dimensional and two‐dimensional homogeneous and heterogeneous numerical examples are carried out to validate the proposed method through the comparison with the coupling multiscale FEM and standard FEM. Numerical results show that the newly developed GCEMs can almost preserve the same convergent property as the standard FEM and also possesses the advantages of high computational efficiency. In addition, the GCEMs can be easily applied to other coupling multifield and multiphase transient problems. Copyright © 2014 John Wiley & Sons, Ltd.