An overview on nonlinear porous flow in low permeability porous media

This paper gives an overview on nonlinear porous flow in low permeability porous media, reveals the microscopic mechanisms of flows, and clarifies properties of porous flow fluids. It shows that, deviating from Darcy's linear law, the porous flow characteristics obey a nonlinear law in a low-permeability porous medium, and the viscosity of the porous flow fluid and the permeability values of water and oil are not constants. Based on these characters, a new porous flow model, which can better describe low permeability reservoir, is established. This model can describe various patterns of porous flow, as Darcy's linear law does. All the parameters involved in the model, having definite physical meanings, can be obtained directly from the experiments.     

Exponential stability of stochastic generalized porous media equations with jump

Stochastic generalized porous media equation with jump is considered. The aim is to show the moment exponential stability and the almost certain exponential stability of the stochastic equation.     

Efficient stochastic finite element methods for flow in heterogeneous porous media. Part 2: Random lognormal permeability

Efficient and robust iterative methods are developed for solving the linear systems of equations arising from stochastic finite element methods for single phase fluid flow in porous media. Permeability is assumed to vary randomly in space according to some given correlation function. In the companion paper, herein referred to as Part 1, permeability was approximated using a truncated Karhunen-Loève expansion (KLE). The stochastic variability of permeability is modeled using lognormal random fields and the truncated KLE is projected onto a polynomial chaos basis. This results in a stochastic nonlinear problem since the random fields are represented using polynomial chaos containing terms that are generally nonlinear in the random variables. Symmetric block Gauss-Seidel used as a preconditioner for CG is shown to be efficient and robust for stochastic finite element method.     

Determination of permeability tensors for two-phase flow in homogeneous porous media: Theory

In this paper we continue previous studies of the closure problem for two-phase flow in homogeneous porous media, and we show how the closure problem can be transformed to a pair of Stokes-like boundary-value problems in terms of pressures that have units of length and velocities that have units of length squared. These are essentially geometrical boundary value problems that are used to calculate the four permeability tensors that appear in the volume averaged Stokes' equations. To determine the geometry associated with the closure problem, one needs to solve the physical problem; however, the closure problem can be solved using the same algorithm used to solve the physical problem, thus the entire procedure can be accomplished with a single numerical code.Nomenclature a a vector that maps V onto , m^-1. - A a tensor that maps V onto . - A area of the - interface contained within the macroscopic region, m². - A area of the -phase entrances and exits contained within the macroscopic region, m². - A area of the - interface contained within the averaging volume, m². - A area of the -phase entrances and exits contained within the averaging volume, m². - Bo Bond number (= (=(–)g²/). - Ca capillary number (= v/). - g gravitational acceleration, m/s². - H mean curvature, m^-1. - I unit tensor. - permeability tensor for the -phase, m². - viscous drag tensor that maps V onto V. - ^* dominant permeability tensor that maps onto v , m². - ^* coupling permeability tensor that maps onto v , m². - characteristic length scale for the -phase, m. - l characteristic length scale representing both and , m. - L characteristic length scale for volume averaged quantities, m. - n unit normal vector directed from the -phase toward the -phase. - n unit normal vector representing both n and n . - n unit normal vector representing both n and n . - P pressure in the -phase, N/m². - p superficial average pressure in the -phase, N/m². - p intrinsic average pressure in the -phase, N/m². - p –p , spatial deviation pressure for the -phase, N/m². - r ₀ radius of the averaging volume, m. - r position vector, m. - t time, s. - v fluid velocity in the -phase, m/s. - v superficial average velocity in the -phase, m/s. - v intrinsic average velocity in the -phase, m/s. - v –v , spatial deviation velocity in the -phase, m/s. - V volume of the -phase contained within the averaging volmue, m³. - averaging volume, m³. Greek Symbols V /, volume fraction of the -phase. - viscosity of the -phase, Ns/m². - density of the -phase, kg/m³. - surface tension, N/m. - (v +v ^T ), viscous stress tensor for the -phase, N/m².     

A LGA model for fluid flow in heterogeneous porous media

A lattice gas automaton (LGA) model is proposed to simulate fluid flow in heterogeneous porous media. Permeability fields are created by distributing scatterers (solids, grains) within the fluid flow field. These scatterers act as obstacles to flow. The loss in momentum of the fluid is directly related to the permeability of the lattice gas model. It is shown that by varying the probability of occurrence of solid nodes, the permeability of the porous medium can be changed over several orders of magnitude. To simulate fluid flow in heterogeneous permeability fields, isotropic, anisotropic, random, and correlated permeability fields are generated. The lattice gas model developed here is then used to obtain the effective permeability as well as the local fluid flow field. The method presented here can be used to simulate fluid flow in arbitrarily complex heterogeneous porous media.     

A LGA model for dispersion in heterogeneous porous media

The lattice gas automaton (LGA) model proposed in the previous paper is applied to the problem of simulating dispersion and mixing in heterogeneous porous media. We demonstrate here that tracer breakthrough profiles and longitudinal dispersion coefficients can be computed for heterogeneous porous media.     

A model for solute transport following an abrupt pressure impact in saturated porous media

A mathematical model is developed of an abrupt pressure impact applied to a compressible fluid with solute, flowing through saturated porous media. Nondimensional forms of the macroscopic balance equations of the solute mass and of the fluid mass and momentum lead to dominant forms of these equations. Following the onset of the pressure change, we focus on a sequence of the first two time intervals at which we obtain reduced forms of the balance equations. At the very first time period, pressure is proven to be distributed uniformly within the affected domain, while solute remains unaffected. During the second time period, the momentum balance equation for the fluid conforms to a wave form, while the solute mass balance equation conforms to an equation of advective transport. Fluid's nonlinear wave equation together with its mass balance equation, are separately solved for pressure and velocity. These are then used for the solution of solute's advective transport equation. The 1-D case, conforms to a pressure wave equation, for the solution of fluid's pressure and velocity. A 1-D analytical solution of the transport problem, associates these pressure and velocity with an exponential power which governs solute's motion along its path line.     

Self-similar approximants of the permeability in heterogeneous porous media from moment equation expansions

We use a mathematical technique, the self-similar functional renormalization, to construct formulas for the average conductivity that apply for large heterogeneity, based on perturbative expansions in powers of a small parameter, usually the log-variance of the local conductivity. Using perturbation expansions up to third order and fourth order in obtained from the moment equation approach, we construct the general functional dependence of the scalar hydraulic conductivity in the regime where is of order 1 and larger than 1. Comparison with available numerical simulations show that the proposed method provides reasonable improvements over available expansions.     

流体饱和多孔介质黏弹性动力人工边界

基于Biot流体饱和多孔介质本构方程,分别考察具有辐射阻尼性质的外行柱面波和球 面波在圆柱面和球面人工边界上引起的法向、切向应力的表达式. 在应力表达形式上,固相 介质和孔隙流体的法向和切向应力都是由两项组成,它们分别与质点的位移和速度成正比, 因此,可在人工边界的法向和切向设置连续分布的并联弹簧------黏滞阻尼器,用来模拟人工边 界以外的无限域介质对来自有限计算域的外行波动的能量吸收作用,从而形成了流体饱和多 孔介质的黏弹性动力人工边界. 流体饱和多孔介质的黏弹性动力人工边界可方便地与大型通 用软件结合,用于分析饱和土中复杂的结构-地基动力相互作用问题. 算例表明流体饱和多 孔介质黏弹性动力人工边界具有较好的精度和稳定性.     

基于蒙特卡罗随机有限元方法的随机多孔介质内流体自然对流不确定性研究

为分析孔隙率不确定性对多孔介质方腔内自然对流换热的影响,发展了一种基于KL（Karhunen-Loeve展开）-蒙特卡罗随机有限元算法的随机多孔介质内自然对流不确定性分析数理模型及有限元数值模拟程序框架。通过K-L展开及基于拉丁抽样法生成多孔介质孔隙率随机实现,并耦合多孔介质自然对流有限元程序,进行随机多孔介质内自然对流传热数值模拟,得出了多孔介质内流场与温度场平均值与标准偏差,并分析了孔隙率不确定性条件下Da数对Nu数的影响。结果表明,孔隙率不确定性对多孔介质方腔内自然对流有重要影响。随机多孔介质内流场及温度场与确定性条件下的流场及温度场存在一定偏差,Nu数标准偏差随着Da的增大先增大后减小。     

Melting and refreezing of porous media

During severe nuclear reactor accidents similar to Three-Mile Island, the fuel rods can fragment and thus convert the reactor core into a large rubble bed composed primarily of UO₂ and ZrO₂ particles. In the present study a one-dimensional model is developed for the melting and refreezing of such a bed. The analysis includes mass conservation equations for the species of interest (UO₂ and ZrO₂); a momentum equation that represents a balance among drag, capillary and gravity forces; an energy equation that incorporates the effects of convection by the melt, radiation and conduction through the bed and internal heat generation; and a UO₂---ZrO₂ phase diagram. A few key results are that (1) capillary forces are only important in beds composed of particles smaller than a few millimeters in diameter and in such beds, melt relocates both upward and downward until it freezes, forming crusted regions above and below the melt zone; (2) as melt flows downward and freezes, a flow blockage forms near the bottom of the bed and the location of this blockage is determined by the bottom thermal boundary layer thickness; (3) the maximum thickness of the lower crust increases linearly with the height of the bed; and (4) deviations from initially uniform composition profiles occur because ZrO₂ is preferentially melted and these deviations decrease as the initial ZrO₂ concentration is increased.     

The thermogravitational effect in porous media: A modelling approach

The thermogravitational effect may induce large concentration contrasts, particularly in porous media. This phenomenon arises from a coupling of the Soret effect and convection currents in a temperature field. The present study of this phenomenon is motivated by the safety assessment of nuclear waste repositories, which are sources of thermal energy. Here, we present a modelling approach of laboratory experiments carried out at the University of Toulouse. The results of this model, though more adequate than the analytical solution to account for the influence of permeability, remain far from the experimental ones. In conclusion, it appears that the research must now focus on both a comprehensive phenomenology of the transport processes and experiments with new dimensional constraints.Notation C concentration of the solute - c mass fraction of the solute - D molecular diffusion coefficient - D coefficient of thermodiffusion - F ₁ external forces on the solute - F ₂ external forces on the solvent - g gravity - J _q heat flux - J _l solute mass flux - J _x horizontal mass flux of the solute - k intrinsic permeability - k _L longitudinal permeability - k _T transverse permeability - T ₀ reference temperature - U Darcy velocity - thermal expansion coefficient of the mixture - thermal conductivity - porosity of the medium - dynamic viscosity of the mixture - _T ⁰ dynamic viscosity of pure water atT - _1T dynamic viscosity of 1 atT - _2T dynamic viscosity of 2 atT - ₁ chemical potential of the solute - ₂ chemical potential of the solvent - specific mass of the fluid - ₀ specific mass of pure water atT ₀     

ONSET CONDITION OF STRAIN LOCALIZATION IN MATRIX OF SATURATED POROUS MEDIA

Introduction Strainlocalizationofgeomaterialsisoneofmostpopularfailuretypesinnature,which canbeshowedaslandslidesandmudflowsinmountainousareasunderincessantorheavy raining,especiallythevegetationisseverelydamagedbywoodsharvest;pipingeffect,a typeoflocalfa…     

Determination of permeability in fibrous porous media using the lattice Boltzmann method with application to PEM fuel cells

The lattice Boltzmann method (LBM) is used to simulate the flow through an idealized proton exchange membrane fuel cell (PEMFC) porous transport layer (PTL) geometry generated using a Monte Carlo method. Using the calculated flow field, Darcy's law is applied and the permeability is calculated. This process is applied in both through‐ and in‐plane directions of the paper as both of these permeability values are important in computational fluid dynamics models of PEMFCs. It is shown that the LBM can be used to determine permeability in a random porous media by solving the flow in the microstructure of the material. The permeability in the through‐ and in‐plane directions is shown to be different and the anisotropic nature of the geometry creates anisotropic permeability. It is also found that fiber arrangement plays a large role in the permeability of the PTL. New correlations are presented for in‐ and though‐plane permeabilities of fibrous porous media with (0.6<ε<0.8). Copyright © 2008 John Wiley & Sons, Ltd.     

Hydraulic radius and transport in reconstructed model three-dimensional porous media

Methods for reconstructing three-dimensional porous media from two-dimensional cross sections are evaluated in terms of the transport properties of the reconstructed systems. Two-dimensional slices are selected at random from model three-dimensional microstructures, based on penetrable spheres, and processed to create a reconstructed representation of the original system. Permeability, conductivity, and a critial pore diameter are computed for the original and reconstructed microstructures to assess the validity of the reconstruction technique. A surface curvature algorithm is utilized to further modify the reconstructed systems by matching the hydraulic radius of the reconstructed three-dimensional system to that of the two-dimensional slice. While having only minor effects on conductivity, this modification significantly improves the agreement between permeabilities and critical diameters of the original and reconstructed systems for porosities in the range of 25–40%. For lower porosities, critical pore diameter is unaffected by the curvature modification so that little improvement between original and reconstructed permeabilities is obtained by matching hydraulic radii.     

A linear dynamic model for a saturated porous medium

A linear isothermal dynamic model for a porous medium saturated by a Newtonian fluid is developed in the paper. In contrast to the mixture theory, the assumption of phase separation is avoided by introducing a single constitutive energy function for the porous medium. An important advantage of the proposed model is it can account for the couplings between the solid skeleton and the pore fluid. The mass and momentum balance equations are obtained according to the generalized mixture theory. Constitutive relations for the stress, the pore pressure are derived from the total free energy accounting for inter-phase interaction. In order to describe the momentum interaction between the fluid and the solid, a frequency independent Biot-type drag force model is introduced. A temporal variable porosity model with relaxation accounting for additional attenuation is introduced for the first time. The details of parameter estimation are discussed in the paper. It is demonstrated that all the material parameters in our model can be estimated from directly measurable phenomenological parameters. In terms of the equations of motion in the frequency domain, the wave velocities and the attenuations for the two P waves and one S wave are calculated. The influences of the porosity relaxation coefficient on the velocities and attenuation coefficients of the three waves of the porous medium are discussed in a numerical example.     

Thermal stretching in two-phase porous media: Physical basis for Maxwell model

An alternate yet general form of the classical effective thermal conductivity model (Maxwell model) for two-phase porous materials is presented, serving an explicit thermo-physical basis. It is demonstrated that the reduced effective thermal conductivity of the porous media due to non-conducting pore inclusions is caused by the mechanism of thermal stretching, which is a combination of reduced effective heat flow area and elongated heat transfer distance (thermal tortuosity).     

Modelling of solute transport in a mild heterogeneous porous medium using stochastic finite element method: Effects of random source conditions

Randomness in the source condition other than the heterogeneity in the system parameters can also be a major source of uncertainty in the concentration field. Hence, a more general form of the problem formulation is necessary to consider randomness in both source condition and system parameters. When the source varies with time, the unsteady problem, can be solved using the unit response function. In the case of random system parameters, the response function becomes a random function and depends on the randomness in the system parameters. In the present study, the source is modelled as a random discrete process with either a fixed interval or a random interval (the Poisson process). In this study, an attempt is made to assess the relative effects of various types of source uncertainties on the probabilistic behaviour of the concentration in a porous medium while the system parameters are also modelled as random fields. Analytical expressions of mean and covariance of concentration due to random discrete source are derived in terms of mean and covariance of unit response function. The probabilistic behaviour of the random response function is obtained by using a perturbation‐based stochastic finite element method (SFEM), which performs well for mild heterogeneity. The proposed method is applied for analysing both the 1‐D as well as the 3‐D solute transport problems. The results obtained with SFEM are compared with the Monte Carlo simulation for 1‐D problems. Copyright © 2007 John Wiley & Sons, Ltd.     

Microstructural effects on the permeability of periodic fibrous porous media

An analytical–numerical approach is presented for computing the macroscopic permeability of fibrous porous media taking into account their microstructure. A finite element (FE) based model for viscous, incompressible flow through a regular array of cylinders/fibers is employed for predicting the permeability associated with this type of media. High resolution data, obtained from our simulations, are utilized for validating the commonly used semi-analytical models of drag relations from which the permeability is often derived. The effect of porosity, or volume fraction, on the macroscopic permeability is studied. Also microstructure parameters like particle shape, orientation and unit cell staggered angle are varied. The results are compared with the Carman–Kozeny (CK) equation and the Kozeny factor (often assumed to be constant) dependence on the microstructural parameters is reported and used as an attempt to predict a closed form relation for the permeability in a variety of structures, shapes and wide range of porosities.     

Mathematical model of two-phase fluid nonlinear flow in low-permeability porous media with applications

IntroductionItisasuccessfulexampleinadevelopmentstoryofscienceandtechnologyformechanicsoffluidsinporousmediatocombinewithengineeringtechnology .Fieldsinfluencedbythemechanicsinvolveddevelopmentofoil_gasandgroundwaterresources,controlonseawaterintrusionandsubsidenceandgeologichazards,geotechnicalengineeringandbioengineering ,andairlineindustry[1～ 7].Aproblemonnonlinearflowinlow_permeabilityporousmediaisbutonlyabasiconeindifferentkindsofengineeringfields,butalsooneoffrontlineresearchfieldsofmod…