Percolation Cluster Statistics and Conductivity Semi-Variograms

Cluster statistics of percolation theory have been shown to generate expressions for the distribution of hydraulic conductivity values in accord with field studies. Percolation theory yields directly the smallest possible generalized resistance value, R _c, for which a continuous path through an infinite heterogeneous system can avoid all larger resistances. R _c, defines an infinite system hydraulic conductivity. Cluster statistics generate the number of clusters of resistors of a given size with a given R, for which a continuous path through the cluster can avoid resistances larger than R. The probability that a volume of size x ³ falls on a particular cluster gives the probability that volume has a characteristic resistance, R. Determining the semi-variogram of the hydraulic conductivity is now elementary; it is necessary only to determine whether translation h of the center of the volume x ³ removes it from the cluster in question. If the cluster is larger than (x+h)³, then, on the average, the same cluster resistance R will control K. Otherwise, the value of K at x+h will be uncorrelated with its value at x. The condition is then expressed as an integral related to the one, which gives the distribution of K. Then an integral over the derived distribution of K gives the variogram. Results obtained are that the variogram should be similar to either the exponential or Gaussian forms typically in use, if K is a power law function of random variables (as in Poiseuilles Law), or more closely related to the spherical approximation if K is an exponential function of random variables.     

Electrical Conductivity and Percolation Aspects of Statistically Homogeneous Porous Media

A method of 3-D stochastic reconstruction of porous media based on statistical information extracted from 2-D sections is evaluated with reference to the steady transport of electric current. Model microstructures conforming to measured and simulated pore space autocorrelation functions are generated and the formation factor is systematically determined by random walk simulation as a function of porosity and correlation length. Computed formation factors are found to depend on correlation length only for small values of this parameter. This finding is explained by considering the general percolation behavior of a statistically homogeneous system. For porosities lower than about 0.2, the dependence of formation factor on porosity shows marked deviations from Archie's law. This behavior results from the relatively high pore space percolation threshold (0.09) of the simulated media and suggests a limitation to the applicability of the method to low porosity media. It is additionally demonstrated that the distribution of secondary porosity at a larger scale can be simulated using stochastic methods. Computations of the formation factor are performed for model media with a matrix-vuggy structure as a function of the amount and spatial distribution of vuggy porosity and matrix conductivity. These results are shown to be consistent with limited available experimental data for carbonate rocks.     

A Percolation Study of Wettability Effect on the Electrical Properties of Reservior Rocks

Measurements of the electrical resistivity of oil reservoirs are commonly used to estimate other properties of reservoirs, such as porosity and hydrocarbon reserves. However, the interpretation of the measurements is based on empirical correlations, because the underlying mechanisms that control the electrical properties of oil bearing rocks have not been well understood. In this paper, we employ percolation concepts to investigate the effect of wettability on the electrical conductivity of a reservoir formation. A three-dimensional simple cubic network is used to represent an ideal reservoir formation, for which the effect of the wettability can be isolated from the others. The phase distribution in the network is analyzed for different flow processes, and the conductivity is then estimated using a power law approximation of the percolation quantities.To whom correspondence should be addressed.The proposed conceptual model predicts the generic behavior of reservoir resistivities of different wettabilities. It demonstrates that the resistivity index depends on saturation history and wettability. For strongly oil-wet systems, significant hysteresis is expected, while there is little hysteresis for strongly water-wet systems, and some hysteresis is also expected for intermediate wet systems. One of the interesting results from this study is that for intermediate wet systems, Archie's saturation exponent is between 1.9 and 3.0.Chemical Engineering Department, Technical University of Denmark, DK-2800 Lyngby, Denmark.     

Capillary fingering: Percolation and fractal dimension

We present experimental and theoretical results concerning immiscible displacements (drainage) in 2-dimensional permeable media. When capillary forces are predominant, the injected fluid presents very thin fingers and the Representative Elementary Volume concept cannot be used for describing the partial saturations. The purpose of this paper is to show how this classical concept can be replaced by a statistical approach based on fractal geometry.Communication presented at the International Symposium on the Stochastic Approach to Subsurface Flow, Montvillargenne 4–7 June 1985.     

Invasion Percolation on Correlated and Elongated Lattices: Implications for the Interpretation of Residual Saturations in Rock Cores

The invasion percolation model is used to investigate the effect of correlated heterogeneity on capillary dominated displacements in porous media. The breakthrough and residual saturations are shown to be strongly influenced by the correlations. Correlated heterogeneity leads to lower residual saturations than those observed in random systems and the scatter commonly observed in laboratory core measurements of the residual saturations can be attributed to the presence of such heterogeneity at the pore scale. Invasion percolation computations on elongated lattices, those with a geometry of L ^d–1 × nL where n denotes the aspect ratio, show that residual saturations for systems with correlated heterogeneity exhibit a strong dependence on aspect ratio. This effect is not considered by experimentalists who advocate the use of long (high aspect ratio) cores in order to overcome end-effects in experiments on shorter cores. A new scaling law is proposed for the residual saturations in elongated systems with correlated heterogeneity, and is confirmed by numerical simulations.     

Equivalent Diffusion in a Thin Film Flow with a Non-One-Dimensional Velocity Field and Anisotropic Diffusion Tensor

For describing the mass transfer processes in channels, Taylor's dispersion theory is widely used. This theory makes it possible, with asymptotic rigor, to replace the complete diffusion (heat conduction) equation with a convective term that depends on the coordinate transverse to the flow by an effective diffusion (dispersion) equation with constant coefficients, averaged over the channel cross-section. In numerous subsequent studies, Taylor's theory was generalized to include more complex situations, and novel algorithms for constructing the dispersion equations were proposed. For thin film flows a theory similar to Taylor's leads to a matrix of dispersion coefficients.In this study, Taylor's theory is extended to film flows with a non-one-dimensional velocity field and anisotropic diffusion tensor. These characteristics also depend to a considerable extent on the spatial coordinates and time. The dispersion equations obtained can be simplified in regions in which the effective diffusion coefficient tensor changes sharply.     

The Onset of Convection in an Anisotropic Porous Layer Using a Thermal Non-Equilibrium Model

The stability of a horizontal fluid saturated anisotropic porous layer heated from below and cooled from above is examined analytically when the solid and fluid phases are not in local thermal equilibrium. Darcy model with anisotropic permeability is employed to describe the flow and a two-field model is used for energy equation each representing the solid and fluid phases separately. The linear stability theory is implemented to compute the critical Rayleigh number and the corresponding wavenumber for the onset of convective motion. The effect of thermal non-equilibrium and anisotropy in both mechanical and thermal properties of the porous medium on the onset of convection is discussed. Besides, asymptotic analysis for both very small and large values of the interphase heat transfer coefficient is also presented. An excellent agreement is found between the exact and asymptotic solutions. Some known results, which correspond to thermal equilibrium and isotropic porous medium, are recovered in limiting cases.     

Coriolis Effect on the Stability of Centrifugally Driven Convection in a Rotating Anisotropic Porous Layer Subjected to Gravity

We investigate natural convection in a fluid saturated rotating anisotropic porous layer subjected to centrifugal gravitational and Coriolis body forces. The Darcy model (including the centrifugal, gravitational and Coriolis terms; and permeability anisotropy effects) and a modified energy equation (including the effects of thermal anisotropy) is used in the current analysis. The linear stability theory is used to evaluate the critical Rayleigh number for the onset of convection in the presence of thermal and mechanical anisotropy. It is shown that the preferred solution comprises roll cells aligned parallel to the vertical z-axis. As a result, it is found that the Coriolis acceleration (or Taylor number) and the gravitational term play no role in the stability of convection.     

An Analytical Study of Free Convective Boundary-Layer Flow in Porous Media: The Effect of Anisotropic Diffusitivity

The effect of an anisotropic thermal diffusivity tensor on the free convective boundary-layer flow in porous media is studied. Convection is induced by a generally inclined, uniformly heated surface embedded in a fluid-saturated medium. A third-order boundary-layer theory is presented in order to obtain accurate information on the effect of anisotropy on the rate of heat transfer into the porous medium. It is shown that the thickness of the resulting leading order boundary-layer flow depends on the precise nature of the anisotropy. On the other hand, the anisotropic diffusivity does not induce a fluid drift in the spanwise direction, a result which is different from that obtained in our earlier study of the effects of an anisotropic permeability. It is found that the second order temperature field does not contribute to the overall rate of heat transfer. Finally, we show that the third-order correction to the leading-order rate of heat transfer is given in terms of an explicit formula.     

Lattice Boltzmann simulation of two-sided lid-driven flow in a staggered cavity

In this article, the lattice Boltzmann method is employed in order to explore incompressible fluid flow inside a two-sided lid-driven staggered cavity. Results of the lattice Boltzmann simulation for antiparallel motion of lids are compared with the data from existing literature. For parallel motion of lids, the characteristics of flow pattern for a variety of Re numbers (50–3200) are presented. An asymmetric steady-state flow pattern for parallel motion of lids is obtained.     

A Technical Note: Theory of Spontaneous Versus Induced Capillary Imbibition

The prevailing current wisdom about how to properly formulate an algorithm to describe spontaneous and coupled spontaneous/induced capillary imbibition into and through the interstices of porous Darcian scale sediments implies, but perhaps incorrectly, that the early empirical relations of Buckley and Leverett (1942) can safely be employed. Here this persistently popular point of view is found to be inadequate since it overlooks taking into full account how common sense suggests that it is surface energy gradient driving forces which are the underlying cause for the occurrences of these processes. Specifically it will be noted that a satisfactory algorithm to model cases where spontaneous capillary imbibition occurs will not be one which ignores the fact that necessarily the local free surface energy is continuously diminished in magnitude until a minimum value together with a maximum entropy condition has been reached. Hence, the conclusion is drawn herein that the Buckley–Leverett formulations alone at best can only be used to describe what is called those induced capillary imbibition processes caused by the action of mechanical energy driving forces.     

On Compressible Materials Capable of Sustaining Axisymmetric Shear Deformations. Part 4: Helical Shear of Anisotropic Hyperelastic Materials

Conditions on the form of the strain energy function in order that homogeneous, compressible and isotropic hyperelastic materials may sustain controllable static, axisymmetric anti-plane shear, azimuthal shear, and helical shear deformations of a hollow, circular cylinder have been explored in several recent papers. Here we study conditions on the strain energy function for homogeneous and compressible, anisotropic hyperelastic materials necessary and sufficient to sustain controllable, axisymmetric helical shear deformations of the tube. Similar results for separate axisymmetric anti-plane shear deformations and rotational shear deformations are then obtained from the principal theorem for helical shear deformations. The three theorems are illustrated for general compressible transversely isotropic materials for which the isotropy axis coincides with the cylinder axis. Previously known necessary and sufficient conditions on the strain energy for compressible and isotropic hyperelastic materials in order that the three classes of axisymmetric shear deformations may be possible follow by specialization of the anisotropic case. It is shown that the required monotonicity condition for the isotropic case is much simpler and less restrictive. Restrictions necessary and sufficient for anti-plane and rotational shear deformations to be possible in compressible hyperelastic materials having a helical axis of transverse isotropy that winds at a constant angle around the tube axis are derived. Results for the previous case and for a circular axis of transverse isotropy are included as degenerate helices. All of the conditions derived here have essentially algebraic structure and are easy to apply. The general rules are applied in several examples for specific strain energy functions of compressible and homogeneous transversely isotropic materials having straight, circular, and helical axes of material symmetry.     

Anisotropic Elastic Materials that Uncouple All Three Displacement Components, and Existence of One-Displacement Green's Function

It was shown in an earlier paper that, under a two-dimensional deformation, there are anisotropic elastic materials for which the antiplane displacement u ₃ and the inplane displacements u ₁, u ₂ are uncoupled but the antiplane stresses σ₃₁, σ₃₂ and the inplane stresses σ₁₁, σ₁₂, σ₂₂ remain coupled. The conditions for this to be possible were derived, but they have a complicated expression. In this paper new and simpler conditions are obtained, and a general anisotropic elastic material that satisfies the conditions is presented. For this material, and for certain monoclinic materials with the symmetry plane at x ₃ = 0, we show that the unnormalized Stroh eigenvectors a _k for k = 1, 2, 3 are all real. The matrix A =[a ₁, a ₂, a ₃] is a unit matrix when the material has a symmetry plane at x ₂ = 0. Thus any one of the u ₁, u ₂, u ₃ can be the only nonzero displacement, and the solution is a one-displacement field. Application to the Green's function due to a line of concentrated force f and a line dislocation with Burgers vector v in the infinite space, the half-space with a rigid boundary, and the infinite space with an elliptic rigid inclusion shows that one can indeed have a one-displacement field u ₁, u ₂ or u ₃. One can also have a two-displacement field polarized on a plane other than the (x ₁, x ₂)-plane. The material that uncouples u ₁, u ₂, u ₃ is not as restrictive as one might have thought. It can be triclinic, monoclinic, orthotropic, tetragonal, transversely isotropic, or cubic. However, it cannot be isotropic. This revised version was published online in August 2006 with corrections to the Cover Date.     

液滴撞击具有浸润性分布的倾斜固壁铺展行为的格子Boltzmann研究

采用基于单组分多相伪势模型的格子Boltzmann方法,模拟了三维液滴撞击左右两侧浸润性不同的倾斜固壁的铺展过程,获得了液滴在壁面两侧的铺展因子、相对铺展宽度、相对高度和液滴运动速度随时间的变化情况,研究了壁面浸润性分布和壁面倾斜角度对液滴铺展过程的影响．结果表明,液滴在倾斜壁面的铺展过程受到重力和表面力的综合作用,重力影响液滴的铺展和沿壁面向下的滑动,壁面浸润性分布影响液滴向壁面亲水侧横向移动．     

Generalized Darcy's Law and the Structure of the Phase and Relative Phase Permeabilities for Two-Phase Flows through Anisotropic Porous Media

For two-phase immiscible fluid flows a generalized Darcy's law is written in invariant tensor form for crystallographic point symmetry groups and anisotropic textures. The representation of the phase permeability coefficient tensors and the structure of the expressions for the relative phase permeabilities are analyzed for all symmetry groups. The relation between the phase and absolute permeability coefficient tensors is specified by a fourth-rank tensor with the external symmetry coinciding with external symmetry of the phase permeability tensors. It is shown that the external symmetry of the phase permeability coefficient tensors can differ from the external symmetry of the absolute permeability tensor. For triclinic and monoclinic symmetry groups it is shown that the phase permeability coefficient tensors may not be coaxial with each other and with the absolute permeability tensor; moreover, the directions of the principal axes of the phase permeability coefficient tensors can depend on the saturation.     

具有填充材料的金属格栅夹层板在高速冲击下动态响应的数值分析

利用大型非线性有限元程序ABAQUS和LS-DYNA,对具有填充材料的金属格栅结构的冲击问题进行数值模拟.研究了不同的填充材料(金属泡沫和陶瓷)分别填充到不同的格栅构型(波纹型、蜂窝型和加强六边形)夹层板后,各类夹层板受到金属泡沫子弹和不锈钢子弹冲击时变形与能量吸收特性,探讨了夹层板上下面层板、支撑格栅及填充材料等各部分的吸能比率.研究结果表明,泡沫填充夹层板在缓冲吸能方面具有优势,陶瓷填充夹层板则在抵抗冲击穿透方面更具有优势,不同构型的夹层板,性能略有不同.     

A Lattice Gas Automaton Simulation of the Nonlinear Diffusion Equation: A Model for Moisture Flow in Unsaturated Porous Media

We investigate a two-dimensional lattice gas automaton (LGA) for simulating the nonlinear diffusion equation in a random heterogeneous structure. The utilility of the LGA for computation of nonlinear diffusion arises from the fact that, the diffusion coefficient in the LGA depends on the local density of fluid particles which statistically determines the collision rate and thus, the mean free path of the particles at the microscopic scale. The LGA may therefore be used as a physical analogue to simulate moisture flow in unsaturated porous media. The capability of the LGA to account for unsaturated flow is tested through a set of numerical experiments simulating one-dimensional infiltration in a simplified semi-infinite homogenous isotropic porous material. Different mechanisms of interactions are used between the fluid and the solid phase to simulate various fluid–solid interfaces. The heterogeneous medium, initially at low density is submitted to a steep density gradient by continuously injecting fluid particles at high concentration and zero velocity along one face of the model. The propagation of the infiltration front is visualized at different time steps through concentration profiles parallel to the applied concentration gradient and the infiltration rate is measured continuously until steady-state flow is reached. The numerical results show close agreement with the classical theory of flow in unsaturated porous media. The cumulative absorption exhibits the expected t ^1/2 dependence. The evolution of the effective diffusion coefficient with the particle concentration is estimated from the measured density profiles for the various porous materials. Depending on the applied fluid–solid interactions, the macroscopic effective diffusivity may vary by more than two orders of magnitude with density.     

Three-dimensional unsaturated flow in heterogeneous systems and implications on groundwater contamination: A stochastic approach

A stochastic approach for modeling transient unsaturated flow in large-scale spatially variable soils is developed in order to overcome the problem of limited information about the local details of spatial soil variability. It is assumed that local soil properties are realizations of three-dimensional stationary random fields, and a large-scale model representation is derived by averaging the local governing flow equation over the ensemble of realizations of the underlying soil property random fields. The three-dimensionality of the local flow equations and the nonlinear dependence of the local flow output on the local soil properties are considered. The resulting mean representation (structure) is in the form of a partial differential equation in which averaged or effective model parameters occur. These effective model parameters are evalutated using a quasi-linearized fluctuation equation and a spectral representation of stationary processes. The large-scale model structure considers the large-scale effects of soil variability and have relatively few parameters which should be identifiable from a realistic data set. The general stochastic theory is then applied to the case of flow in stratified soil formations, which is of practical importance in applications such as waste disposal control. An important finding of this study is that spatial variability of the hydraulic soil properties produces significant large-scale effects, such as large-scale hysteresis and anisotropy of the effective parameters. These large-scale effects should be considered in field applications such as for predicting the movement of liquid wastes in the unsaturated zone.     

Viscous coupling in a model porous medium geometry: Effect of fluid contact area

We study the effect of fluid contact area on viscous coupling in the parallel flow of immiscible fluids in a porous media geometry. We consider flow on opposite sides of a planar interface, consisting of alternating solid and open (slit) segments. We use the analytical solution of Tio and Sadhal [15] to derive explicit expressions for viscous coupling in terms of the fractional area of contact between the fluids and the viscosity ratio,M. ForM=1, the coefficient matrix obtained is symmetric showing that Onsager's relations are satisfied. In this case, the resulting viscous coupling is typically very small, in agreement with recent experimental results. Lattice gas simulations forM=1 using theBGK model support the theoretical results and show that viscous coupling further diminishes as the wall thickness increases. Assuming the same configuration, analytical results are next derived forM1. The results confirm an existing reciprocity relation between the off-diagonal terms. Viscous coupling remains small.     

Determination of the anisotropic permeability of a carbon cloth gas diffusion layer through X‐ray computer micro‐tomography and single‐phase lattice Boltzmann simulation

An investigation of the anisotropic permeability of a carbon cloth gas diffusion layer (GDL) based on the integration of X‐ray micro‐tomography and lattice Boltzmann (LB) simulation is presented. The method involves the generation of a 3D digital model of a carbon cloth GDL as manufactured using X‐ray shadow images acquired through X‐ray micro‐tomography at a resolution of 1.74 µm. The resulting 3D model is then split into 21 volumes and integrated with a LB single‐phase numerical solver in order to predict three orthogonal permeability tensors when a pressure difference is prescribed in the through‐plane direction. The 21 regions exhibit porosity values in the range of 0.910–0.955, while the average fibre diameter is 4 µm. The results demonstrate that the simulated through‐plane permeability is about four times higher than the in‐plane permeability for the sample imaged and that the corresponding degrees of anisotropy for the two orthogonal off‐principal directions are 0.22 and 0.27. The results reveal that flow channelling can play an important role in gas transport through the GDL structure due to the non‐homogeneous porosity distribution through the material. The simulated results are also applied to generate a parametric coefficient for the Kozeny–Carman (KC) method of determining permeability. The current research reveals that by applying the X‐ray tomography and LB techniques in a complementary manner, there is a strong potential to gain a deeper understanding of the microscopic fluidic phenomenon in representative models of porous fuel cell structures and how this can influence macroscopic transport characteristics which govern fuel cell performance. Copyright © 2010 John Wiley & Sons, Ltd.