Computing Petrophysical Properties in Porous Rocks Using a Boundary-Element Technique

An efficient numerical technique has been used to compute the deformation of pores of arbitrary shape embedded in a homogeneous elastic solid under the influence of applied stresses. The scheme is based on the boundary-element method, where single linear elements are used to generate solutions that satisfy prescribed boundary conditions. These solutions can be employed to describe the behavior of elastic moduli and other petrophysical properties in porous rocks. The numerical algorithm allows computation of the stress field induced by the pores in the solid. In this way, the effect of the interactions between pores caused by stress concentrations can be studied from a quantitative point of view. To test the algorithm, some interesting results are compared with existing models, for special cases available in the literature. Also, a model for the compressibility and porosity of sedimentary rocks, as a function of applied hydrostatic stress, was generated by mixing some realistic pore geometries generated with the numerical algorithm. Results were in good agreement with data obtained from selected samples of sandstones.     

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

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

非饱和土中热−湿−盐耦合作用的稳态分析

以多孔介质理论为基础,研究了稳态条件下非饱和土中温度?水分?盐分多场耦合问题。考虑非饱和土的孔隙被液态水、溶解的盐分、水蒸气和干燥气体等填充,在质量和能量守恒的基础上获得了非饱和土中水分、气体、盐分的质量守恒方程以及能量守恒方程。考虑一维稳态问题,选取温度、孔隙气压、孔隙水压和盐溶液浓度以及它们的导数作为状态变量,得到了问题的状态方程组。在给定的边界条件下,采用打靶法求解了该强耦合的非线性变系数微分方程组,通过与已有的试验结果相比较,验证了模型的有效性。基于数值算例,参数分析了含水率、温度边界、孔隙率等条件对非饱和土中温度场、水分场和盐分场分布的影响规律。     

A semianalytical solution for one-dimensional transient wave propagation in a saturated single-layer porous medium with a fluid surface layer

One-dimensional transient wave propagation in a saturated single-layer porous medium with a fluid surface layer is studied in this paper. An analytical solution for a special case with a dynamic permeability coefficient k_f → ∞ and a semianalytical solution for a general case with an arbitrary dynamic permeability coefficient are presented. The eigenfunction expansion and precise time step integration methods are employed. The solution is presented in series form, and thus, the long-term dynamic responses of saturated porous media with small permeability coefficients can be easily computed. We first transform the nonhomogeneous boundary conditions into homogeneous boundary conditions, and then we obtain the eigenvalues and orthogonal eigenfunctions of the fluid–solid system. Finally, the solutions in the time domain are developed. As the model is one dimensional, geometric attenuation is absent, and only the attenuation in the saturated porous medium is considered. We can apply this model to analyse the influences of different seabed types on the propagation of acoustic waves in the fluid layer, which is very important in ocean acoustics and ocean seismic. This solution can also be employed to validate the accuracies of various numerical methods.     

Numerical Simulations of Free Convection about a Vertical Flat Plate Embedded in a Porous Medium

We consider the free convection about a vertical flat plate embedded in a porous medium within the framework of a boundary layer approximation. In some cases, similarity solutions arise in the modelization of such phenomena, allowing a reduction of the dimension of the problem. We suggest two complementary and rather simple numerical methods to compute such solutions. When possible, the numerical analysis for the two discretizations is performed and convergence results are given. Numerical experiments with a physical model are presented to confirm the efficiency of both approaches.     

SILENT BOUNDARY CONDITIONS FOR WAVE PROPAGATION IN SATURATED POROUS MEDIA

Wave propagation both in one- and in two-dimensional saturated elastic porous media is analysed by means of a two-field finite element model with silent boundaries. An extension of the elastic ‘multidirectional’ transmitting boundary to two-phase media is developed to simulate the silent boundary condition. The theoretical assessment and the numerical formulation of the first-order silent boundary technique is presented in detail. Some examples are used to demonstrate the reliability of the first-order method, especially for problems with plane and axisymmetric waves having various angles of incidence. Finally, the wave propagation along a pile shaft is presented, to simulate a common non-destructive dynamic pile test.     

Three-Dimensional Modeling of Mass Transfer in Porous Media Using the Mixed Hybrid Finite Elements and the Random-Walk Methods

A three-dimensional (3D) mass transport numerical model is presented. The code is based on a particle tracking technique: the random-walk method, which is based on the analogy between the advection–dispersion equation and the Fokker–Planck equation. The velocity field is calculated by the mixed hybrid finite element formulation of the flow equation. A new efficient method is developed to handle the dissimilarity between Fokker–Planck equation and advection–dispersion equation to avoid accumulation of particles in low dispersive regions. A comparison made on a layered aquifer example between this method and other algorithms commonly used, shows the efficiency of the new method. The code is validated by a simulation of a 3D tracer transport experiment performed on a laboratory model. It represents a heterogeneous aquifer of about 6-m length, 1-m width, and 1-m depth. The porous medium is made of three different sorts of sand. Sodium chloride is used as a tracer. Comparisons between simulated and measured values, with and without the presented method, also proves the accuracy of the new algorithm.     

Numerical study of boundary conditions for solute transport through a porous medium

A transition region may be defined as a region of rapid change in medium properties about the interface between two porous media or at the interface between a porous medium and a reservoir. Modelling the transition region between different porous media can assist in the selection of the most appropriate boundary conditions for the standard advection–dispersion equation (ADE). An advantage of modelling the transition region is that it removes the need for explicitly defining boundary conditions, though boundary conditions may be recovered as limiting cases. As the width of a transition region is reduced, the solution of the transition region model (TR model) becomes equivalent to the solution of the standard ADE model with correct boundary conditions. In this paper numerical simulations using the TR model are employed to select the most appropriate boundary conditions for the standard ADE under a variety of configurations and conditions. It is shown that at the inlet boundary between a reservoir and porous medium, continuity of solute mass flux should be used as the boundary condition. At the boundary interface between two porous media both continuity of solute concentration and solute mass flux should be used. Finally, in a finite porous medium where the solute is allowed to advect freely from the exit point, both continuity of solute concentration and solute mass flux should be used as the outlet boundary condition. The findings made here are discussed with reference to a detailed review of previous relevant theoretical and experimental observations. Copyright © 2001 John Wiley & Sons, Ltd.     

A Particle Tracking Model for Non-Fickian Subsurface Diffusion

In this paper, a new particle tracking technique is described which can simulate non-Fickian diffusion within porous media. The technique employs fractional Brownian motions (fBms), a generalization of regular Brownian motion. These random fractal functions allow both super- and subdiffusive particle paths to be produced and hence non-Fickian diffusion of the resulting panicle clouds can be modeled. In recent years, fBm trace functions have been used by many authors to reproduce self-affine random fields to simulate various porous media properties. In contrast, a method is detailed herein which uses self-similar spatial fBm trajectories to simulate directly non-Fickian behavior of the particle clouds. Although fractal trajectories have been previously suggested as the basis for possible methods of modeling non-Fickian diffusion, the authors believe that this paper contains the first algorithm to be presented which does not require an a priori knowledge of the end condition of the random walk and, more importantly, allows both a definable scaling exponent and (fractal) diffusion coefficient to be specified. The resulting non-Fickian diffusion using the new algorithm is illustrated and some applications are discussed. The purpose of this paper is to bring the potential usefulness of fBm trajectories in simulating non-Fickian processes within homogeneous media to the attention of numerical modelers active in the simulation of subsurface diffusive processes. The method has a particular environmental application in the simulation of the non-Fickian dispersion of groundwater contaminants through porous media.     

Explicit solutions for the instantaneous undrained contraction of hollow cylinders and spheres in porous elastoplastic medium

In this article we present closed‐form solutions for the undrained variations in stress, pore pressure, deformation and displacement inside hollow cylinders and hollow spheres subjected to uniform mechanical pressure instantaneously applied to their external and internal boundary surfaces. The material is assumed to be a saturated porous medium obeying a Mohr–Coulomb model failure criterion, exhibiting dilatant plastic deformation according to a non‐associated flow rule which accounts for isotropically strain hardening or softening. The instantaneous response of a porous medium submitted to an instantaneous loading is undrained, i.e. without any fluid mass exchange. The short‐term equilibrium problem to be solved is now formally identical to a problem of elastoplasticity where the constitutive equations involve the undrained elastic moduli and particular equivalent plastic parameters. The response of the model is presented (i) for extension and compression undrained triaxial tests, and (ii) for unloading problems of hollow cylinders and spheres through the use of appropriately developed closed‐form solutions. Numerical results are presented for a plastic clay stone with strain hardening and an argilite with strain softening. The effects of plastic dilation, of the strain softening law and also of geometry of the cavity on the behaviour of the porous medium have been underlined. Analytical solutions provide valuable benchmarks enabling various numerical methods in undrained conditions with a finite boundary to be verified. Copyright © 2002 John Wiley & Sons, Ltd.     

Sensitivity analysis applied to finite element method model for coupled multiphase system

Today multiphysics problems applied to various fields of engineering have become increasingly important. Among these, in the areas of civil, environmental and nuclear engineering, the problems related to the behaviour of porous media under extreme conditions in terms of temperature and/or pressure are particularly relevant. The mathematical models used to solve these problems have an increasing complexity leading to increase of computing times. This problem can be solved by using more effective numerical algorithms, or by trying to reduce the complexity of these models. This can be achieved by using a sensitivity analysis to determine the influence of model parameters on the solution. In this paper, the sensitivity analysis of a mathematical/numerical model for the analysis of concrete as multiphase porous medium exposed to high temperatures is presented. This may lead to a reduction of the number of the model parameters, indicating what parameters should be determined in an accurate way and what can be neglected or found directly from the literature. Moreover, the identification parameters influence may allow to proceeding to a simplification of the mathematical model (i.e. model reduction). The technique adopted in this paper to performing the sensitivity analysis is based on the automatic differentiation (AD), which allowed to develop an efficient tool for the computation of the sensitivity coefficients. The results of the application of AD technique have been compared with the results of the more standard finite difference method, showing the superiority of the AD in terms of numerical accuracy and execution times. From the results of the sensitivity analysis, it follows that a drastic simplification of the model for thermo‐chemo‐hygro‐mechanical behaviour of concrete at high temperature, is not possible. Therefore, it is necessary to use different model reduction techniques in order to obtain a simplified version of the model that can be used at industrial level. Copyright © 2012 John Wiley & Sons, Ltd.     

THE SETTLEMENT OF A RIGID CIRCULAR FOUNDATION RESTING ON A HALF-SPACE EXHIBITING A NEAR SURFACE ELASTIC NON-HOMOGENEITY

The present paper examines the elastostatic problem pertaining to the axisymmetric loading of a rigid circular foundation resting on the surface of a non-homogeneous elastic half-space. The non-homogeneity corresponds to a depth variation in the linear elastic shear modulus according to the exponential form G(z)=G₁+G₂e^-ζz. The equations of elasticity governing this type of non-homogeneity are solved by employing a Hankel transform technique. The mixed boundary value problem associated with the indentation of the half-space by the rigid circular foundation is reduced to a Fredholm integral equation which is solved via a numerical technique. The numerical results presented in the paper illustrate the influence of the near-surface elastic non-homogeneity on the settlement of the foundation.     

Efficient solution algorithms for multiaxial probabilistic elasto‐plastic constitutive simulations of soils

A Fokker‐Planck‐Kolmogorov (FPK) equation approach has recently been developed to probabilistically solve any elastic‐plastic constitutive equation with uncertain material parameters by transforming the nonlinear, stochastic constitutive rate equation into a linear, deterministic partial differential equation (PDE) and thereby simplifying the numerical solution process. For an uniaxial problem, conventional numerical techniques, such as the finite difference or finite element methods, may be used to solve the resulting univariate FPK PDE. However, for a multiaxial problem, an efficient algorithm is necessary for tractability of the numerical solution of the multivariate FPK PDE. In this paper, computationally efficient algorithms, based on a Fourier spectral approach, are presented for solving FPK PDEs in (stress) space and (pseudo) time, having space‐independent but time‐dependent coefficients and both space‐ and time‐dependent coefficients, that commonly arise in probabilistic elasto‐plasticity. The algorithms are illustrated by probabilistically simulating 2 common laboratory constitutive experiments in geotechnical engineering, namely, the unconfined compression test and the unconsolidated undrained triaxial compression test.     

Automatic calibration of soil–water characteristic curves using genetic algorithms

An automatic technique for the determination of the coefficients of models for soil–water characteristic curves (SWCC) or water retention curves (WRC) is presented. The technique is based on optimisation using genetic algorithms, in which the error between predictions and experimental data is minimised by varying the model parameters. The method is powerful and reasonably efficient in finding the best parameters. Four models are analysed including one accounting for hysteresis behaviour. Details of a simple genetic algorithm (SGA) and its complete application are explained. To account for the hysteresis of the SWCC, the models are programmed in a rate form, in which numerical integration is employed to advance the state variables. One advantage of the optimisation presented is that the best curves averaging both the drying and wetting paths are obtained when hysteresis is present.     

An ELLAM Approximation for Highly Compressible Multicomponent Flows in Porous Media

We develop an ELLAM-MFEM approximation to the strongly coupled systems of time-dependent nonlinear partial differential equations (PDEs) and constraining equations, which describe fully miscible, highly compressible, multicomponent flows through heterogeneous and compressible porous media with singular sources and sinks. An Eulerian–Lagrangian localized adjoint method (ELLAM) is presented to solve the transport equations for concentrations. A mixed finite element method (MFEM) is used to solve the pressure PDE for the pressure and Darcy velocity simultaneously, which generates accurate fluid velocities and minimizes the numerical difficulties occurring in standard methods caused by differentiation of the pressure and then multiplication by rough coefficients. The ELLAM-MFEM solution technique symmetrizes and stabilizes the governing transport PDEs and greatly reduces nonphysical oscillation and/or excessive numerical dispersion present in many large-scale simulators. Computational experiments show that the ELLAM-MFEM solution technique can generate stable and physically reasonable numerical simulations even if coarse spatial grids and very large time steps are used.     

Exact solutions for one‐dimensional transient response of fluid‐saturated porous media

Based on the Biot theory, the exact solutions for one‐dimensional transient response of single layer of fluid‐saturated porous media and semi‐infinite media are developed, in which the fluid and solid particles are assumed to be compressible and the inertial, viscous and mechanical couplings are taken into account. First, the control equations in terms of the solid displacement u and a relative displacement w are expressed in matrix form. For problems of single layer under homogeneous boundary conditions, the eigen‐values and the eigen‐functions are obtained by means of the variable separation method, and the displacement vector u is put forward using the searching method. In the case of nonhomogeneous boundary conditions, the boundary conditions are first homogenized, and the displacement field is constructed basing upon the eigen‐functions. Making use of the orthogonality of eigen‐functions, a series of ordinary differential equations with respect to dimensionless time and their corresponding initial conditions are obtained. Those differential equations are solved by the state‐space method, and the series solutions for three typical nonhomogeneous boundary conditions are developed. For semi‐infinite media, the exact solutions in integral form for two kinds of nonhomogeneous boundary conditions are presented by applying the cosine and sine transforms to the basic equations. Finally, three examples are studied to illustrate the validity of the solutions, and to assess the influence of the dynamic permeability coefficient and the fluid inertia to the transient response of porous media. Copyright © 2010 John Wiley & Sons, Ltd.     

Finite element modelling of transport of organic chemicals through soils

Aquifer contamination by organic chemicals in subsurface flow through soils due to leaking underground storage tanks filled with organic fluids is an important groundwater pollution problem. The problem involves transport of a chemical pollutant through soils via flow of three immiscible fluid phases: namely air, water and an organic fluid. In this paper, assuming the air phase is under constant atmospheric pressure, the flow field is described by two coupled equations for the water and the organic fluid flow taking interphase mass transfer into account. The transport equations for the contaminant in all the three phases are derived and assuming partition equilibrium coefficients, a single convective – dispersive mass transport equation is obtained. A finite element formulation corresponding to the coupled differential equations governing flow and mass transport in the three fluid phase porous medium system with constant air phase pressure is presented. Relevant constitutive relationships for fluid conductivities and saturations as function of fluid pressures lead to non-linear material coefficients in the formulation. A general time-integration scheme and iteration by a modified Picard method to handle the non-linear properties are used to solve the resulting finite element equations. Laboratory tests were conducted on a soil column initially saturated with water and displaced by p-cymene (a benzene-derivative hydrocarbon) under constant pressure. The same experimental procedure is simulated by the finite element programme to observe the numerical model behaviour and compare the results with those obtained in the tests. The numerical data agreed well with the observed outflow data, and thus validating the formulation. A hypothetical field case involving leakage of organic fluid in a buried underground storage tank and the subsequent transport of an organic compound (benzene) is analysed and the nature of the plume spread is discussed.     

Ion transport in porous media: derivation of the macroscopic equations using upscaling and properties of the effective coefficients

In this work, we undertake a numerical study of the effective coefficients arising in the upscaling of a system of partial differential equations describing transport of a dilute N-component electrolyte in a Newtonian solvent through a rigid porous medium. The motion is governed by a small static electric field and a small hydrodynamic force, around a nonlinear Poisson–Boltzmann equilibrium with given surface charges of arbitrary size. This approach allows us to calculate the linear response regime in a way initially proposed by O’Brien. The O’Brien linearization requires a fast and accurate solution of the underlying Poisson–Boltzmann equation. We present an analysis of it, with the discussion of the boundary layer appearing as the Debye–Hückel parameter becomes large. Next, we briefly discuss the corresponding two-scale asymptotic expansion and reduce the obtained two-scale equations to a coarse scale model. Our previous rigorous study proves that the homogenized coefficients satisfy Onsager properties, namely they are symmetric positive definite tensors. We illustrate with numerical simulations several characteristic situations and discuss the behavior of the effective coefficients when the Debye–Hückel parameter is large. Simulated qualitative behavior differs significantly from the situation when the surface potential is given (instead of the surface charges). In particular, we observe the Donnan effect (exclusion of co-ions for small pores).     

Solving stiff mass transfers in compositional multiphase flow models: Numerical stability and spurious solutions

This paper points out two numerical problems linked to the resolution of compositional multiphase flow models for porous media with the finite‐volume technique. In particular, we consider fluid mixtures featuring fast mass transfers between the phases, hence stiff. In this context, we show how the computation of mass exchange kinetics can be expensive and that erroneous saturation front locations arise. A numerical splitting method is developed which is proven to be stable with advection‐type time steps, whatever the stiffness of the mass transfer. The erroneous front location problem is illustrated and shown to be intrinsically linked to the numerical diffusion. If we assume that the fluids are in thermodynamical equilibrium, we find that spurious solutions can be avoided by deriving and solving a new uncoupled hyperbolic equation for the saturation. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.