Frictionless contact formulation for dynamic analysis of nonlinear saturated porous media based on the mortar method

A finite element algorithm for frictionless contact problems in a two‐phase saturated porous medium, considering finite deformation and inertia effects, has been formulated and implemented in a finite element programme. The mechanical behaviour of the saturated porous medium is predicted using mixture theory, which models the dynamic advection of fluids through a fully saturated porous solid matrix. The resulting mixed formulation predicts all field variables including the solid displacement, pore fluid pressure and Darcy velocity of the pore fluid. The contact constraints arising from the requirement for continuity of the contact traction, as well as the fluid flow across the contact interface, are enforced using a penalty approach that is regularised with an augmented Lagrangian method. The contact formulation is based on a mortar segment‐to‐segment scheme that allows the interpolation functions of the contact elements to be of order N. The main thrust of this paper is therefore how to deal with contact interfaces in problems that involve both dynamics and consolidation and possibly large deformations of porous media. The numerical algorithm is first verified using several illustrative examples. This algorithm is then employed to solve a pipe‐seabed interaction problem, involving large deformations and dynamic effects, and the results of the analysis are also compared with those obtained using a node‐to‐segment contact algorithm. The results of this study indicate that the proposed method is able to solve the highly nonlinear problem of dynamic soil–structure interaction when coupled with pore water pressures and Darcy velocity. Copyright © 2015 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.     

On the causes of pressure oscillations in low‐permeable and low‐compressible porous media

Nonphysical pressure oscillations are observed in finite element calculations of Biot's poroelastic equations in low‐permeable media. These pressure oscillations may be understood as a failure of compatibility between the finite element spaces, rather than elastic locking. We present evidence to support this view by comparing and contrasting the pressure oscillations in low‐permeable porous media with those in low‐compressible porous media. As a consequence, it is possible to use established families of stable mixed elements as candidates for choosing finite element spaces for Biot's equations. Copyright © 2011 John Wiley & Sons, Ltd.     

One‐dimensional dynamics of saturated incompressible porous media: analytical solutions and influence of inertia terms

The complexity of formulations for the hydromechanical coupled mechanics of porous media is typically minimised by simplifying assumptions such as neglecting the effect of inertia terms. For example, three formulations commonly employed to model practical problems are classified as fully dynamic, simplified dynamic and quasi‐static. Thus, depending on the porous media conditions, each formulation will have advantages and limitations. This paper presents a comprehensive analysis of these limitations when solving one‐dimensional fully saturated porous media problems in addition to a new solution that considers a more general loading situation. A phase diagram is developed to assist on the selection of which formulation is more appropriate and convenient regarding particular cases of porosity and hydraulic conductivity values. Non‐dimensional formulations are proposed to achieve this goal. Results using the analytical solutions are compared against numerical values obtained with the finite element method, and the effect of porosity is investigated. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical stability for modelling of dynamic two‐phase interaction

Dynamic two‐phase interaction of soil can be modelled by a displacement‐based, two‐phase formulation. The finite element method together with a semi‐implicit Euler–Cromer time‐stepping scheme renders a discrete equation that can be solved by recursion. By experience, it is found that the CFL stability condition for undrained wave propagation is not sufficient for the considered two‐phase formulation to be numerically stable at low values of permeability. Because the stability analysis of the two‐phase formulation is onerous, an analysis is performed on a simplified two‐phase formulation that is derived by assuming an incompressible pore fluid. The deformation of saturated porous media is now captured in a single, second‐order partial differential equation, where the energy dissipation associated with the flow of the fluid relative to the soil skeleton is represented by a damping term. The paper focuses on the different options to discretize the damping term and its effect on the stability criterion. Based on the eigenvalue analyses of a single element, it is observed that in addition to the CFL stability condition, the influence of the permeability must be included. This paper introduces a permeability‐dependent stability criterion. The findings are illustrated and validated with an example for the dynamic response of a sand deposit. Copyright © 2015 John Wiley & Sons, Ltd.     

Modelling Two-Phase Incompressible Flow in Porous Media Using Mixed Hybrid and Discontinuous Finite Elements

In this paper, we present a numerical model for simulating two-phase (oil–water and air–water) incompressible and immiscible flow in porous media. The mathematical model which is based on a fractional flow formulation is formed of two nonlinear partial differential equations: a mean pressure equation and a water saturation equation. These two equations can be solved in a sequential manner. Two numerical methods are used to discretize the equations of the two-phase flow model: mixed hybrid finite elements are used to treat the pressure equation, h-based Richards' equation and the diffusion term in the saturation equation, the advection term in the saturation equation is treated with the discontinuous finite elements. We propose a better way to calculate the nonlinear coefficients contained in our equations on each element of the discretized domain. In heterogeneous porous media, the saturation becomes discontinuous at the interface between two porous media. We show in this paper how to use the capillary pressure–saturation relationship in order to handle the saturation jump in the mixed hybrid finite element method. The two-phase flow simulator is verified against analytical solutions for some flow problems treated by other authors.     

A Comparative Study of Numercial Solution of PDE inGeosciences

A number of phenomena and processes in geosciences can be summarized by second order partial differential equations. The major numerical methods for their solution include the classical finite difference method and the finite element method newly developed in the last two or three decades. Since 1977 the author has proved that for the Laplace and Poisson equations, these two methods are identical and are different only in the process of formulation. For transient problems, such as heat conduction in the earth and the groundwater and oil-gas unsteady flow in porous media, there are some differences in resulting linear algebraic euqations. In general, two methods give similar results, but when the time step is decreased to some extent, the resulting algebraic equation will be consistent with the anti-heat conduction equation rather than the original heat conduction equation. This is the reason why unrealistic potentials are produced by the finite element method. Such a problem can be overcome by using the     

A space-time discontinuous Galerkin method applied to single-phase flow in porous media

A space-time discontinuous Galerkin finite element method is proposed and applied to a convection-dominant single-phase flow problem in porous media. The numerical scheme is based on a coupled space-time finite element discretization allowing for discontinuous approximations in space and in time. The continuities on the element interfaces are weakly enforced by the flux treatments, so that no extra penalty factor has to be determined. The resulting space-time formulation possesses the advantage of capturing the steep concentration front with sharp gradients efficiently. The stability and reliability of the proposed approach is demonstrated by numerical experiments. The author is grateful to the DFG (German Science Foundation—Deutsche Forschungsgemeinschaft) for the financial support under the grant number Di 430/4-2.     

Solid–liquid–air coupling in multiphase porous media

This paper addresses various issues concerning the modelling of solid–liquid–air coupling in multiphase porous media with an application to unsaturated soils. General considerations based on thermodynamics permit the derivation and discussion of the general form of field equations; two cases are considered: a three phase porous material with solid, liquid and gas, and a two phase porous material with solid, liquid and empty space. Emphasis is placed on the presentation of differences in the formulation and on the role of the gas phase. The finite element method is used for the discrete approximation of the partial differential equations governing the problem. The two formulations are then analysed with respect to a documented drainage experiment carried out by the authors. The merits and shortcomings of the two approaches are shown. Copyright © 2003 John Wiley & Sons, Ltd.     

流体饱和两相多孔介质动力反应计算分析

基于流体饱和两相多孔介质的弹性波动方程组,运用显式逐步积分格式与局部透射人工边界相结合的时域显式有限元方法对该波动方程组进行求解,对两相介质在输入地震波作用下的弹性动力反应进行计算和分析;对在是否考虑孔隙流体渗流的两种情况下计算得到的两相介质弹性动力反应结果的差异进行对比研究,从而揭示孔隙流体渗流对两相介质动力反应性质的影响。计算结果表明：两相介质弹性动力反应时程的波形与入射地震波的波形相同,且弹性动力反应的峰值出现的时刻对应于入射地震波的峰值出现的时刻;孔隙流体的渗流将对两相介质的弹性动力反应性质产生显著的影响。数值计算同时表明,时域显式有限元方法是进行流体饱和两相多孔介质弹性动力反应计算分析的一种有效的方法。     

Numerical manifold method for dynamic nonlinear analysis of saturated porous media

It is well known that the Babuska–Brezzi stability criterion or the Zienkiewicz–Taylor patch test precludes the use of the finite elements with the same low order of interpolation for displacement and pore pressure in the nearly incompressible and undrained cases, unless some stabilization techniques are introduced for dynamic analysis of saturated porous medium where coupling occurs between the displacement of solid skeleton and pore pressure. The numerical manifold method (NMM), where the interpolation of displacement and pressure can be determined independently in an element for the solution of u–p formulation, is derived based on triangular mesh for the requirement of high accurate calculations from practical applications in the dynamic analysis of saturated porous materials. The matrices of equilibrium equations for the second‐order displacement and the first‐order pressure manifold method are given in detail for program coding. By close comparison with widely used finite element method, the NMM presents good stability for the coupling problems, particularly in the nearly incompressible and undrained cases. Numerical examples are given to illustrate the validity and stability of the manifold element developed. Copyright © 2006 John Wiley & Sons, Ltd.     

Analytical solutions of a finite two‐dimensional fluid‐saturated poroelastic medium with compressible constituents

An analytical solution is proposed for transient flow and deformation coupling of a fluid‐saturated poroelastic medium within a finite two‐dimensional (2‐D) rectangular domain. In this study, the porous medium is assumed to be isotropic, homogeneous, and compressible. In addition, the point sink can be located at an arbitrary position in the porous medium. The fluid–solid interaction in porous media is governed by the general Biot's consolidation theory. The method of integral transforms is applied in the analytical formulation of closed‐form solutions. The proposed analytical solution is then verified against both exact and numerical results. The analytical solution is first simplified and validated by comparison with an existing exact solution for the uncoupled problem. Then, a case study for pumping from a confined aquifer is performed. The consistency between the numerical solution and the analytical solution confirms the accuracy and reliability of the analytical solution presented in this paper. The proposed analytical solution can help us to obtain in‐depth insights into time‐dependent mechanical behavior due to fluid withdrawal within finite 2‐D porous media. Moreover, it can also be of great significance to calibrate numerical solutions in plane strain poroelasticity and to formulate relevant industry norms and standards. Copyright © 2014 John Wiley & Sons, Ltd.     

Some numerical issues using element‐free Galerkin mesh‐less method for coupled hydro‐mechanical problems

A new formulation of the element‐free Galerkin (EFG) method is developed for solving coupled hydro‐mechanical problems. The numerical approach is based on solving the two governing partial differential equations of equilibrium and continuity of pore water simultaneously. Spatial variables in the weak form, i.e. displacement increment and pore water pressure increment, are discretized using the same EFG shape functions. An incremental constrained Galerkin weak form is used to create the discrete system equations and a fully implicit scheme is used for discretization in the time domain. Implementation of essential boundary conditions is based on a penalty method. Numerical stability of the developed formulation is examined in order to achieve appropriate accuracy of the EFG solution for coupled hydro‐mechanical problems. Examples are studied and compared with closed‐form or finite element method solutions to demonstrate the validity of the developed model and its capabilities. The results indicate that the EFG method is capable of handling coupled problems in saturated porous media and can predict well both the soil deformation and variation of pore water pressure over time. Some guidelines are proposed to guarantee the accuracy of the EFG solution for coupled hydro‐mechanical problems. Copyright © 2008 John Wiley & Sons, Ltd.     

多尺度有限单元法求解非均质多孔介质中的三维地下水流问题

应用多尺度有限单元法模拟非均质多孔介质中的三维地下水流问题。与传统有限单元法相比，多尺度有限单元法的基函数具有能反映单元内参数变化的优点，所以这种方法能在大尺度上抓住解的小尺度特征获得较精确的解。在介绍多尺度有限单元法求解非均质多孔介质中三维地下水流问题的基本原理之后，对参数水平方向渐变垂直方向突变的非均质多孔介质中的三维地下水流和Borden实验场的三维地下水流分别用多尺度有限单元法和传统等参有限单元法进行了计算，结果表明在模拟高度非均质多孔介质中的三维地下水流问题时，多尺度有限单元法比传统有限单元法有效，既节省计算量又有较高的精度；在模拟非均质性弱的多孔介质中的三维地下水流问题时，多尺度有限单元法虽然也能在大尺度上获得较为精确的解，但效果不明显。     

Theoretical and numerical analyses of chemical‐dissolution front instability in fluid‐saturated porous rocks

The chemical‐dissolution front propagation problem exists ubiquitously in many scientific and engineering fields. To solve this problem, it is necessary to deal with a coupled system between porosity, pore‐fluid pressure and reactive chemical‐species transport in fluid‐saturated porous media. Because there was confusion between the average linear velocity and the Darcy velocity in the previous study, the governing equations and related solutions of the problem are re‐derived to correct this confusion in this paper. Owing to the morphological instability of a chemical‐dissolution front, a numerical procedure, which is a combination of the finite element and finite difference methods, is also proposed to solve this problem. In order to verify the proposed numerical procedure, a set of analytical solutions has been derived for a benchmark problem under a special condition where the ratio of the equilibrium concentration to the solid molar density of the concerned chemical species is very small. Not only can the derived analytical solutions be used to verify any numerical method before it is used to solve this kind of chemical‐dissolution front propagation problem but they can also be used to understand the fundamental mechanisms behind the morphological instability of a chemical‐dissolution front during its propagation within fluid‐saturated porous media. The related numerical examples have demonstrated the usefulness and applicability of the proposed numerical procedure for dealing with the chemical‐dissolution front instability problem within a fluid‐saturated porous medium. Copyright © 2007 John Wiley & Sons, Ltd.     

Variational principles and finite element method for stress analysis of porous media

A number of variational principles are established in this paper for the stress analysis of porous media with compressible constituents. A three-dimensional finite element method is proposed based on the variational principle. The finite element method thus established is applied to the study of the temperature, deformation and flow field associated with the water-flood technique in secondary recovery projects for oil exploration. In the study model, the layout of injection wells and production wells is considered to have a regular pattern where symmetry conditions exist. The injection fluid diffuses slowly into the formation through a vertical crack which is initially generated by an explosion. The problem is analysed by a plane strain formulation with the effects of heat conduction and convection included.     

Parametric variational principle for elastic–plastic consolidation analysis of saturated porous media

Finite element procedures for numerical solution of various engineering problems are often based on variational formulations. In this paper, a parametric variational principle applicable to elastic-plastic coupled field problems in consolidation analysis of saturated porous media is presented. This principle can be used to solve problems where materials are inconsistent with Drucker's postulate of stability, such as in non-associated plasticity flow or softening problems. The finite element formulation was given, and it can be solved by either the conventional method or a parametric quadratic programming method.     

Control‐volume mixed finite element methods

A key ingredient in simulation of flow in porous media is accurate determination of the velocities that drive the flow. Large‐scale irregularities of the geology (faults, fractures, and layers) suggest the use of irregular grids in simulation. This paper presents a control‐volume mixed finite element method that provides a simple, systematic, easily implemented procedure for obtaining accurate velocity approximations on irregular (i.e., distorted logically rectangular) block‐centered quadrilateral grids. The control‐volume formulation of Darcy’s law can be viewed as a discretization into element‐sized “tanks” with imposed pressures at the ends, giving a local discrete Darcy law analogous to the block‐by‐block conservation in the usual mixed discretization of the mass‐conservation equation. Numerical results in two dimensions show second‐order convergence in the velocity, even with discontinuous anisotropic permeability on an irregular grid. The method extends readily to three dimensions.     

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.