Variational principles and finite element simulations for thermo-elastic consolidation

In this paper, a general variational principle for the initial boundary value problem of quasi-static thermoelastic consolidation is developed by assuming infinitesimal deformation and an incompressible fluid flowing through a linearly elastic solid. By manipulating the coupling operators, an extended form of the variational pronciple is derved. The associated finite element formulation based on this principle is presented and numerical applications for plane strain thermo-elastic consolidation are revealed.     

饱和多孔介质中水力-力学-传质耦合过程的混合有限元法

对饱和多孔介质提出了一个含溶混污染物输运（传质）过程的混合元方法,其中污染物输运过程数学模型包含了对流、机械逸散、分子弥散和吸附等机制。固相位移、应变和有效应力,孔隙水压力、压力空间梯度和Darcy速度,污染物浓度、浓度空间梯度和浓度流量在单元内均为独立变量分别插值。基于胡海昌-Washizu三变量广义变分原理,结合可以滤掉虚假振荡的特征线方法,推导出饱和土中水力-力学-传质耦合问题控制方程的单元弱形式,并导出了混合元计算公式。数值模拟证明了所提出的方法可以提供与传统4点积分方案同样精度,同时能够提高计算效率。     

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.     

Numerical performance of some finite element schemes for analysis of seepage in porous elastic media

Several finite element schemes for analysis of seepage in porous elastic media, based on different spatial and temporal discretization, were implemented in computer programs. Their numerical performance is evaluated by comparison with the exact solution for Terzaghi's problem of one-dimensional consolidation.     

A finite element method for localized erosion in porous media with applications to backward piping in levees

This paper presents a computational method able to effectively model both the simultaneous processes typically observed in backward erosion piping, ie, the pipe tip propagation and the conduit cross-section enlargement. The numerical method is based on the novel formulation of a problem of localized erosion along a line propagating in a multidimensional porous medium. In this line, a conduit with evolving transverse size is embedded, which conveys a multiphase flow. The two systems, porous medium and pipe, are bridged by exchange terms of multiphase fluid mass and by a shared fluid pressure field. On the contrary, different fields are considered to describe flows, which are assumed as Darcian in the porous medium and turbulent in the conduit. These two flows drive pipe propagation and enlargement, respectively, as modeled by means of proper erosion kinetic laws. The corresponding numerical formulation is based on the combination between one- and multidimensional finite elements, to model the erosion conduit and the porous medium, respectively. Several simulations are proposed to demonstrate the ability of the proposed approach in reproducing available experimental data of real-scale tests on levees. Our results point out the crucial role played by the combined influence of pipe propagation and enlargement, as well as of three-dimensional (3D) effects. We also assess the mesh independence of the proposed numerical solution, particularly as concerns the calculated pipe propagation history.     

Combining the modified discrete element method with the virtual element method for fracturing of porous media

Simulation of fracturing processes in porous rocks can be divided into two main branches: (i) modeling the rock as a continuum enhanced with special features to account for fractures or (ii) modeling the rock by a discrete (or discontinuous) approach that describes the material directly as a collection of separate blocks or particles, e.g., as in the discrete element method (DEM). In the modified discrete element (MDEM) method, the effective forces between virtual particles are modified so that they reproduce the discretization of a first-order finite element method (FEM) for linear elasticity. This provides an expression of the virtual forces in terms of general Hook’s macro-parameters. Previously, MDEM has been formulated through an analogy with linear elements for FEM. We show the connection between MDEM and the virtual element method (VEM), which is a generalization of FEM to polyhedral grids. Unlike standard FEM, which computes strain-states in a reference space, MDEM and VEM compute stress-states directly in real space. This connection leads us to a new derivation of the MDEM method. Moreover, it enables a direct coupling between (M)DEM and domains modeled by a grid made of polyhedral cells. Thus, this approach makes it possible to combine fine-scale (M)DEM behavior near the fracturing region with linear elasticity on complex reservoir grids in the far-field region without regridding. To demonstrate the simulation of hydraulic fracturing, the coupled (M)DEM-VEM method is implemented using the Matlab Reservoir Simulation Toolbox (MRST) and linked to an industry-standard reservoir simulator. Similar approaches have been presented previously using standard FEM, but due to the similarities in the approaches of VEM and MDEM, our work provides a more uniform approach and extends these previous works to general polyhedral grids for the non-fracturing domain.     

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.     

Embedded discontinuity finite element modeling of fluid flow in fractured porous media

In many geomaterials, particularly rocks and clays, permeability is greatly enhanced by the presence of fractures. Fracture sets create an overall permeability that is anisotropic, enhanced in the directions of the fractures. In modeling the fractures via a finite element method, for example, meshing around these fractures can become quite difficult and result in computationally intensive systems. In this article, we develop a relatively simple method for including the fractures within the elements. Flow through the bulk medium is assumed to be governed by Darcy’s law, and the flow on the fracture by a generalization of the law. This model is embedded in a finite element framework, with the fractures passing through the elements. In this formulation, elements with fractures are given an enhanced permeability in the direction of the fractures. With these enhancements, the material essentially becomes anisotropically more permeable in the direction of fracture sets.     

On the use of enriched finite element method to model subsurface features in porous media flow problems

In this paper, a new enrichment scheme is proposed to model fractures and other conduits in porous media flow problems. Inserting this scheme into a partition of unity based method results in a new numerical method that does not require the mesh to honor the specific geometry of these subsurface features. The new scheme involves a specially designed integration procedure and enrichment functions, which can capture effects of local heterogeneity introduced by subsurface features on the pressure solution. The new method is also capable of modeling fractures with low as well as high conductivity. Another feature of the proposed scheme is that, even though two enrichment functions are used to model the permeability change at the two rock/fracture interfaces of a fracture, only one element partition is made for numerical integration. To demonstrate the accuracy and effectiveness of the proposed approach, production problems for wells that were stimulated or completed by longitudinal fracture, transverse fractures, and perforations are studied.     

A novel finite element method for seepage analysis

A novel finite element method has been proposed in this paper for the solution of seepage problems economically and accurately. In this method the governing equation and the prescribed boundary conditions are transformed so that they refer to a suitable logarithmically condensed ‘image’ space; the physical problem domain is also mapped into the image space. The transformed equation is then solved in the image space using standard finite elements, subject to the transformed boundary conditions. Because physical space is logarithmically condensed in the image space, the proposed method is capable of dealing with large or very large aspect ratio seepage problems economically and accurately. The validity of the method has been demonstrated by means of a number of examples including anisotropy and non-linearity. In all cases an excellent degree of accuracy was achieved, efficiently and economically.     

A porous media finite element approach for soil instability including the second-order work criterion

This paper deals with the hydromechanical modelling of the initiation of failure in soils with particular reference to landslides. To this end, localized and diffused failure modes are simulated with a finite element model for coupled elasto-plastic variably saturated porous geomaterials, in which the material point instability is detected with the second-order work criterion based on Hill’s sufficient condition of stability. Three different expressions of the criterion are presented, in which the second-order work is expressed in terms of generalized effective stress, of total stress and thirdly by taking into account the hydraulic energy contribution for partially saturated materials. The above-mentioned computational framework has been applied to study two initial boundary value problems: shear failure of a plane strain compression test of globally undrained water-saturated dense sand (where cavitation occurs at strain localization) and isochoric grain matter, and the onset of a flowslide from southern Italy due to rainfall (Sarno-Quindici events, May 5–6 1998). It is shown that the second-order work criterion applied at the material point level detects the local material instability and gives a good spatial indication of the extent of the potentially unstable domains in both the localized and diffused failure mechanisms of the cases analyzed, is able to capture the instability induced by cavitation of the liquid water and gives results according to the time evolution of plastic strains and displacement rate.     

A mapping finite element method for stress calculation in two-dimensions

A mapping finite element method is proposed for the solution of elastic problems in two-dimensions. This method, based on the logarithmic condensation of physical space, is found to give significantly greater accuracy than when the problem is solved in the physical plane using the standard FEM. Logarithmic condensation of space also permits the solution of large aspect-ratio problems of this type, accurately and economically.     

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

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

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.     

A novel finite element double porosity model for multiphase flow through deformable fractured porous media

Based on the theory of double-porosity, a novel mathematical model for multiphase fluid flow in a deforming fractured reservoir is developed. The present formulation, consisting of both the equilibrium and continuity equations, accounts for the significant influence of coupling between fluid flow and solid deformation, usually ignored in the reservoir simulation literature. A Galerkin-based finite element method is applied to discretize the governing equations both in the space and time domain. Throughout the derived set of equations the solid displacements as well as the fluid pressure values are considered as the primary unknowns and may be used to determine other reservoir parameters such as stresses, saturations, etc. The final set of equations represents a highly non-linear system as the elements of the coefficient matrices are updated during each iteration in terms of the independent variables. The model is employed to solve a field scale example where the results are compared to those of ten other uncoupled models. The results illustrate a significantly different behaviour for the case of a reservoir where the impact of coupling is also considered. © 1997 by John Wiley & Sons, Ltd.     

Boundary element analysis of contaminant transport in fractured porous media

A two-dimensional boundary integral method to analyse the flow of contaminant in fractured media having a two- or three-dimensional orthogonal fracture network is presented. The method assumes that the fractures provide the paths of least resistance for transport of contaminants while the matrix, because of its low permeability, acts as ‘storage blocks’ into which the contaminant diffuses. Laplace transform is used to eliminate the time variable in the governing equation in order to facilitate the formulation of a boundary integral equation in the Laplace transform space. Conventional boundary element techniques are applied to solve for the contaminant concentrations at specified locations in the spatial domain. The concentration in the time domain is then obtained by using an efficient inversion technique developed by Talbot. The method is able to analyse the behaviour of waste repositories which have diminishing concentration due to the mass transport of the contaminant into the surrounding fractured media.     

Hydro‐mechanical modeling of cohesive crack propagation in multiphase porous media using the extended finite element method

In this paper, a numerical model is developed for the fully coupled hydro‐mechanical analysis of deformable, progressively fracturing porous media interacting with the flow of two immiscible, compressible wetting and non‐wetting pore fluids, in which the coupling between various processes is taken into account. The governing equations involving the coupled solid skeleton deformation and two‐phase fluid flow in partially saturated porous media including cohesive cracks are derived within the framework of the generalized Biot theory. The fluid flow within the crack is simulated using the Darcy law in which the permeability variation with porosity because of the cracking of the solid skeleton is accounted. The cohesive crack model is integrated into the numerical modeling by means of which the nonlinear fracture processes occurring along the fracture process zone are simulated. The solid phase displacement, the wetting phase pressure and the capillary pressure are taken as the primary variables of the three‐phase formulation. The other variables are incorporated into the model via the experimentally determined functions, which specify the relationship between the hydraulic properties of the fracturing porous medium, that is saturation, permeability and capillary pressure. The spatial discretization is implemented by employing the extended finite element method, and the time domain discretization is performed using the generalized Newmark scheme to derive the final system of fully coupled nonlinear equations of the hydro‐mechanical problem. It is illustrated that by allowing for the interaction between various processes, that is the solid skeleton deformation, the wetting and the non‐wetting pore fluid flow and the cohesive crack propagation, the effect of the presence of the geomechanical discontinuity can be completely captured. Copyright © 2012 John Wiley & Sons, Ltd.     

An improved accelerated initial stress procedure for elasto-plastic finite element analysis

The most flexible and generally applicable methods for elasto-plastic analysis are those based on an incremental-iterative form of the initial stress approach, but such methods often exhibit slow convergence. The acceleration procedure known as the alpha-constant stiffness method is reconsidered and some modifications are proposed. The principal difference in the present approach lies in the use of a single acceleration parameter, rather than a diagonal matrix of acceleration coefficients. The new scheme shows a significant improvement in numerical stability and converges three times faster than the standard initial stress method. Some practical aspects associated with the method are discussed and a number of applications are presented.