First-order indicators for the estimation of discrete fractures in porous media

Faults and geological barriers can drastically affect the flow patterns in porous media. Such fractures can be modelled as interfaces that interact with the surrounding matrix. We propose a new technique for the estimation of the location and hydrogeological properties of a small number of large fractures in a porous medium from given distributed pressure or flow data. At each iteration, the algorithm builds a short list of candidates by comparing fracture indicators. These indicators quantify at the first order the decrease of a data misfit function; they are cheap to compute. Then, the best candidate is picked up by minimization of the objective function for each candidate. Optimally driven by the fit to the data, the approach has the great advantage of not requiring remeshing, nor shape derivation. The stability of the algorithm is shown on a series of numerical examples representative of typical situations.     

多孔介质冻结过程中温度变化的数值模拟

多数对多孔介质冻结过程的研究是建立模型借助于计算机技术给定数值解。笔者在借鉴现有数值模型的基础上，考虑多孔介质的内部特点，采用容积平均法建立数学模型，对无限大平板状多孔介质在第三类边界条件下发生的一维冻结过程中温度变化进行分析并加以实验验证。     

A computational model for fracturing porous media

This paper presents a new computational model for simulating a fracturing process in a porous medium using the finite element method. Two independent numerical techniques are used for describing this process. The partition of unity method is used for describing the fracturing process, and the double porosity model is used for describing the resulting fluid flow. A key feature of the model is the coupling of these two independent numerical techniques, which provide the means for a better simulation of the involved physical and mechanical processes. The paper focuses on the numerical formulation of the model. The capability of the model is illustrated by means of numerical examples, which examine the behaviour of a 1D porous medium under different boundary conditions. The numerical results show that the very complicated physical and mechanical processes of the fracturing porous media can be simulated properly and efficiently. Copyright © 2006 John Wiley & Sons, Ltd.     

Homogenized free surface flow in porous media for wet‐out processing

This paper presents a novel porous media model for homogenized free surface flow, representing wet‐out composites processing. The model is derived from concepts of homogenization applied to a compressible two‐phase flow, accounting for capillary effects and the concept of relative permeability. Based on mass balance considerations, we obtain a nonlinear set of equations of convection‐diffusion type involving the mixture (fluid) pressure and the degree of saturation as primary fields. A staggered Galerkin finite element approach is employed to decouple the solution. Moreover, the streamline upwind/Petrov‐Galerkin technique is applied to attenuate the oscillations in the saturation solutions. The model accuracy and convergence of the finite element solutions are demonstrated through 1‐dimensional and 2‐dimensional examples, representing resin transfer molding flow processes.     

Analysis of vapor condensation in porous media

Abstract

The present work analyzes the problem of vapor condensation in porous media. Specific consideration is given to a transient‐state one‐dimensional formulation, representing a porous slab exposed to the saturated vapor from one side and the cold plate on the other side. When condensation occurs, three zones will exist inside the porous medium. Near the cold plate, a liquid zone is expected. Adjacent to this zone and extending into the medium will be a two‐phase zone which is dominated by capillary and vapor phase transport. Ahead of this zone the medium will remain saturated with vapor phase. Predictions from the three‐zone model used in this study have been compared with the two‐zone model which neglects the effect of surface tension forces. The results show that the two‐zone model can only predict the condensation phenomena in low permeability media. Capillary pressure or surface tension effects become significant and cannot be neglected during the condensation of vapor in high permeability media.     

Complex three‐dimensional microstructural permeability prediction of porous fibrous media with and without compaction

Much of the work in permeability prediction has so far been done with respect to the in‐plane properties. Guided by Darcy's law, the in‐plane permeabilities (i.e. Kxx and Kyy) have been well characterized by researchers both experimentally and, to some extent, analytically and numerically. Work on transverse or through thickness permeability, however, has been sparse. Owing to the fact that the limited length scale in the through thickness direction of most fibre preforms makes transverse permeability a difficult value to measure experimentally, the objective of the present development is to study the feasibility of applying the methodology proposed by the authors in a previous work for in‐plane permeability prediction to the estimation of transverse, as well as the in‐plane, permeability of a typical 3D woven fibre preform. The additional objective of this work is to present a preliminary study on the effects of fibre mat compression on the fibre preform permeability. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical simulation of one‐dimensional flows through porous media with shock waves

This work studies an unsaturated flow of a Newtonian fluid through a rigid porous matrix, using a mixture theory approach in its modelling. The mixture consists of three overlapping continuous constituents: a solid (porous medium), a liquid (Newtonian fluid) and an inert gas (to account for the mixture compressibility). A set of two nonlinear partial differential equations describes the problem, which is approximated by means of a Glimm's scheme, combined with an operator splitting technique. Copyright © 2001 John Wiley & Sons, Ltd.     

A stabilized volume‐averaging finite element method for flow in porous media and binary alloy solidification processes

A stabilized equal‐order velocity–pressure finite element algorithm is presented for the analysis of flow in porous media and in the solidification of binary alloys. The adopted governing macroscopic conservation equations of momentum, energy and species transport are derived from their microscopic counterparts using the volume‐averaging method. The analysis is performed in a single domain with a fixed numerical grid. The fluid flow scheme developed includes SUPG (streamline‐upwind/Petrov–Galerkin), PSPG (pressure stabilizing/Petrov–Galerkin) and DSPG (Darcy stabilizing/Petrov–Galerkin) stabilization terms in a variable porosity medium. For the energy and species equations a classical SUPG‐based finite element method is employed. The developed algorithms were tested extensively with bilinear elements and were shown to perform stably and with nearly quadratic convergence in high Rayleigh number flows in varying porosity media. Examples are shown in natural and double diffusive convection in porous media and in the directional solidification of a binary‐alloy. Copyright © 2004 John Wiley & Sons, Ltd.     

泡沫型多孔介质中非达西流动特性的研究

本文通过将泡沫材料复杂结构进行几何简化，形成立方框架结构，利用简化的等效单元流道，提出一种数理模型，它既考虑固体网架表面对流体的摩擦阻力作用，也考虑网架迎着流向的迎风阻力（形状阻力），得到了在多孔介质中非达西流的压力降与流速的关系式。并用简化的框架结构导出了预估泡沫材料渗透率ｋ和Ｆ－Ｗ关系式中流速平方项的系数Ｆ。另外，还通过专用的实验装置测定了四种不同孔径泡沫陶瓷的ｋ和Ｆ。结果表明，根据所给模型理论预测的结果与实验值一致性较好，在泡沫型多孔材料的应用中能够提供流动特性的相关数据。     

Gate elements at injection locations in numerical simulations of flow through porous media: applications to mold filling

Mold filling in polymer and composite processing is usually modelled as a special case of Darcy flow in porous media. The flow pattern and the time necessary to fill the mold depend on the ‘gate’ locations where resin is injected into the closed mold. In composite manufacturing, these are commonly outlets of small tubes transporting resin from a reservoir and their diameters are several orders of magnitude smaller than the mold dimensions. Similar size issue is also encountered in other applications of flow through porous media, such as oil and water pumping and drilling. Traditionally, these inlets are modelled by pressure or flow rate boundary condition as applied at a node of the finite element mesh that represents the injection gate. The omission of the inlet radius in the model results in a mathematical singularity as the mesh gets refined. The computed pressure or flow field depends on the mesh size and does not converge to the accurate solution, as the finite element mesh is refined. It is possible to deal with this phenomenon by modelling the inlet geometry more accurately but this approach is inefficient, as it requires additional degrees of freedom and, above all, significantly complicates the modelling process if the inlet location is not fixed a priori. This paper presents a more efficient alternate solution. It uses special ‘gate’ elements embedded in the mesh around the injection locations. Instead of adjusting the geometrical modelling of the injection location, the adjacent elements use modified shape functions to accurately model pressure field in the neighbourhood of small radial inlet. The proper pressure field shape‐functions for ‘gate’ elements based on linear finite elements are derived. The implementation in an existing mold filling simulation and how the ‘gate elements’ are automatically selected is described. An example to demonstrate the use of ‘gate’ elements and convergence towards the accurate solution with mesh refinement is presented. Copyright © 2004 John Wiley & Sons, Ltd.     

Finite element modelling of saturated porous media at finite strains under dynamic conditions with compressible constituents

Two finite element formulations are proposed to analyse the dynamic conditions of saturated porous media at large strains with compressible solid and fluid constituents. Unlike similar works published in the literature, the proposed formulations are based on a recently proposed hyperelastic framework in which the compressibility of the solid and fluid constituents is fully taken into account when geometrical non‐linear effects are relevant on both micro‐ and macroscales. The first formulation leads to a three‐field finite element method (FEM), which is suitable for analysing high‐frequency dynamic problems, whereas the second is a simplification of the first, leading to a two‐field FEM, in which some inertial effects of the pore fluid are disregarded, hence the second formulation is suitable for studying low‐frequency problems. A fully Lagrangian approach is considered, hence all terms are expressed with reference to the material setting; the balance equations for the pore fluid are also expressed in terms of the chemical potential and the mass flux of the pore fluid in order to take the compressibility of the fluid into account. To improve the numerical response in the case of wave propagation, a discontinuous Galerkin FEM in the time domain is applied to the three‐field formulation. The results are compared with analytical and semi‐analytical solutions, highlighting the different effects of the discontinuous Galerkin method on the longitudinal waves of the first and second kind. Copyright © 2010 John Wiley & Sons, Ltd.     

In situ ozonation of anthracene in unsaturated porous media

Soil column experiments were conducted to investigate the effect of ozonation duration, contaminant content, particle size, moisture content, OH radical scavenger and soil organic matter on the removal of anthracene by in situ ozonation. In the whole study, the gas flow rate was 100 mL/min and concentration of gaseous ozone was 40 mg/L. The removal efficiency increased with the elapsed time, but the removal rate decreased in the range of 0–90 min. As anthracene content in sand decreased from 50 to 10 mg/kg, the removal efficiency increased from 42.1% to 62.0%, and ozone passed through soil column more rapidly. However, the ozone effectiveness reduced when anthracene content dropped. Small particle size provides a large interfacial area, which led to the high removal efficiency and long ozone breakthrough time in the column. The profile of residual anthracene in soil column varied more greatly at smaller particle size. The removal efficiency reduced when the moisture content rose from 0% to 9.1%. The ozone breakthrough time also decreased with the increasing moisture content. The presence of sodium bicarbonate or humic acid reduced the removal efficiency to some extent. GC–MS was employed in this study to determine 9,10-anthraquinone as the main ozonation product.     

Density difference driven instabilities in porous media

In the present work, we analyze density driven instabilities in miscible fluids in a random packing of monosized glass spheres (porous medium). The aim of this work is to detect heterogeneities in flow velocity using tracer dispersion because of the sensitivity of this technique. The influence of non-Newtonian rheological properties on the displacement front between two fluids of different densities is particularly studied. We compared the width of the displaced fronts in the case of Newtonian fluids with the same density contrast. Our analysis allows us to find a characteristic gravity number G (ratio between hydrostatic and viscous pressure gradients). A same threshold value of G for Newtonian and non Newtonian solutions is directly related to the structural properties of the porous medium. Received: 30 March 2000     

The effect of solid conversion on travelling combustion waves in porous media

A system of nonlinear partial differential equations, which describe the combustion of a gas passing through a porous medium, is examined. This model provides a bridge between recent porous-medium combustion studies and more classical combustion models. In particular, the effect of solid conversion on the downstream temperature is determined for travelling-wave solutions to the system. In cases for which the solid matrix is a perfect catalyst, solely enhancing the exothermic chemical reaction, with zero mass exchange, it is proved that the downstream temperature must always exceed the upstream temperature. However, when the solid is allowed to react with the gas, travelling-wave solutions for which the up- and down-stream temperatures are equal may be realised. A key result of this work is then the investigation of physical processes and related parameter ranges that give rise to travelling wave solutions with equal up- and down-stream temperatures. Specifically, two critical wavespeeds c_i with 0 < c₁ < c₂ are identified. For c < c₁ the downstream temperature always exceeds that upstream, whilst for c > c₂ no bounded travelling-wave solutions exist. Behaviour in the region c₁ < c < c₂ is new: here solutions having equal up- and down-stream temperatures are realised. The two factors upon which this result depends are the inclusion into the model of solid conversion effects and the distinction between solid and gas temperatures. Thus the inclusion of a new physical process, that of heat storage in the solid varying as the reaction proceeds, allows the existence of a new type of travelling-wave solution. Further phenomena studied include the appearance of nonunique solutions (which are of a type that may be related to ignition processes) and degenerate solutions which terminate precisely when all the solid and gaseous fuel is used up. Parameter regimes for the existence of these various types of solutions are found and the stability of such solutions discussed.     

A consistent u‐p formulation for porous media with hysteresis

This paper presents a continuum formulation based on the theory of porous media for the mechanics of liquid unsaturated porous media. The hysteresis of the liquid retention model is carefully modelled, including the derivation of the corresponding consistent tangent moduli. The quadratic convergence of Newton's method for solving the highly nonlinear system with an implicit finite element code is demonstrated. A u‐p formulation is proposed where the time discretisation is carried out prior to the space discretisation. In this way, the derivation of all consistent moduli is fairly straightforward. Time integration is approximated with the Theta and Newmark's methods, and hence the fully coupled nonlinear dynamics of porous media is considered. It is shown that the liquid retention model requires also the consistent second‐order derivative for quadratic convergence. Some predictive simulations are presented illustrating the capabilities of the formulation, in particular to the modelling of complex porous media behaviour. Copyright © 2014 John Wiley & Sons, Ltd.     

Computational coupled large-deformation periporomechanics for dynamic failure and fracturing in variably saturated porous media

The large-deformation mechanics and multiphysics of continuous or fracturing partially saturated porous media under static and dynamic loads are significant in engineering and science. This article is devoted to a computational coupled large-deformation periporomechanics paradigm assuming passive air pressure for modeling dynamic failure and fracturing in variably saturated porous media. The coupled governing equations for bulk and fracture material points are formulated in the current/deformed configuration through the updated Lagrangian–Eulerian framework. It is assumed that the horizon of a mixed material point remains spherical and its neighbor points are determined in the current configuration. As a significant contribution, the mixed interface/phreatic material points near the phreatic line are explicitly considered for modeling the transition from partial to full saturation (vice versa) through the mixed peridynamic state concept. We have formulated the coupled constitutive correspondence principle and stabilization scheme in the updated Lagrangian–Eulerian framework for bulk and interface points. We numerically implement the coupled large deformation periporomechanics through a fully implicit fractional-step algorithm in time and a hybrid updated Lagrangian–Eulerian meshfree method in space. Numerical examples are presented to validate the implemented stabilized computational coupled large-deformation periporomechanics and demonstrate its efficacy and robustness in modeling dynamic failure and fracturing in variably saturated porous media.     

Finite analytic method for 2D steady fluid flows in heterogeneous porous media with unstructured grids

The finite analytic method (FAM) is developed to solve the 2D steady fluid flows in heterogeneous porous media with full tensor permeability on unstructured grids. The proposed FAM is constructed based upon the power‐law analytic nodal solution in the angular domain with arbitrary shape. When approaching the grid node joining the subdomains, 3 different flow patterns may exist: power‐law flow, linear flow, or the stagnant flow. Based on the nodal analytic solution, the triangle‐based FAM is proposed. Numerical examples show that the proposed numerical scheme makes the convergences much quickly than the traditional methods, typically the weighted harmonic mean method under the cell refinement. In practical applications, the grid refinement parameter n = 2 or n = 3 is recommended, and the relative error of the calculated equivalent permeability will below 4% independent of the heterogeneity. In contrast, when using the traditional numerical scheme the refinement ratio for the grid cell needs to increase dramatically to get an accurate result, especially for strong heterogeneous porous medium.     

Elastic wave propagation in unsaturated porous media using hybrid‐Trefftz stress elements

The stress model of the hybrid‐Trefftz finite element is formulated for the analysis of elastodynamic problems defined on unsaturated porous media. The supporting mathematical model is the theory of mixtures with interfaces and considers the full coupling between the solid, fluid and gas phases, including the effect of seepage acceleration. Hybrid‐Trefftz stress elements use the free‐field regular solutions of the homogeneous Navier (or Beltrami) equation to construct the approximation of the generalized stresses in the domain of the element. The influence of non‐homogeneous terms in the Navier equation is modelled using solutions of the corresponding static problem. The resulting elements are highly convergent under p‐refinement and robust to both low and high excitation frequencies, as the trial functions embody relevant physical information on the modelled phenomenon. Copyright © 2013 John Wiley & Sons, Ltd.     

FDM and FEM solutions to linear dynamics of porous media: stabilised,monolithic and fractional schemes

A number of methods have been developed for solving the dynamics of saturated porous media. However, most solutions are based on the finite element method, and only a few employ finite differences (FDM). One problem with the FDM is the difficulty in fulfilling the inf‐sup (Lady?enskaja‐Babu?ka‐Brezzi) condition. This paper explores solutions with the FDM, including the development of new schemes aiming at stabilised formulations. The efficiency, accuracy and stability of several FDM and finite element method algorithms are thoroughly investigated as well. A combination of primary variables from the theory of porous media is considered, including the so‐called up and uvp formulations. Six numerical schemes are produced and quantitatively studied. Simulations of 1D and 2D wave propagation problems are performed in order to reveal the advantages and drawbacks of all schemes. Copyright © 2016 John Wiley & Sons, Ltd.