Rapoport-leas two-phase flow model for anisotropic porous media

The Rapoport-Leas mathematical model of two-phase flow is generalized to include the case of anisotropic porous media. The formula for the capillary pressure, which specifies the relationship between the phase pressures, contains a scalar function of a vector argument. In order to determine the scalar function, the capillary pressure tensor and the tensor inverse to the tensor of characteristic linear dimensions are introduced. The capillary pressure is determined by the contraction of the second-rank tensors with a unit vector collinear to the phase pressure gradients, also assumed to be collinear. It is shown that the saturation function introduced for isotropic porous media (Leverett function) can be generalized to include anisotropic media and is now determined by a fourth-rank tensor. Generalized expressions for the Leverett and relative phase permeability functions are given for orthotropic and transversely isotropic media with account for the hysteresis of the phase permeabilities and capillary pressure.     

Flow and capacity tensor characteristics of anisotropic porous media: Theory and experiment

Tensors determining and describing the main flow and capacity characteristics of anisotropic media are introduced. A system of laboratorymeasurements for the determination of the tensors of areal porosity, permeability, limiting gradients, and characteristic linear dimensions is considered. The pressure dependences of permeability are generalized to take into account the medium anisotropy. The results of an experiment for the determination of the permeability and limiting (initial) gradient tensors in band sandstone with orthotropic flow characteristics are presented. The measurements were carried out on four cores: two along the principal stratification directions, one perpendicular to the stratification, and one inclined at an angle of 45° to the stratification plane. The latter (fourth) specimen was taken for reference: to test the tensor nature of the introduced mathematical objects and formulas. The good agreement of the theoretical and experimental results makes it possible to recommend for engineering calculations both the formulas proposed and the developed method of laboratory investigation of the flow characteristics of anisotropic reservoirs.     

Anisotropy effects in two-phase flows through porous media

Effects manifested in two-phase flows through anisotropic porous reservoirs with monoclinic and triclinic characteristics are analyzed. It is shown that in two-phase flows through media with monoclinic and triclinic symmetries of flow characteristics the position of the principal axes of the phase permeability tensors depends on the saturation and does not coincide with the position of the principal axes of the absolute permeability tensor in single-phase flows and that going over from single-to two-phase flow may lead to a change in the symmetry group of the flow characteristics. A general representation of the phase permeability tensor components is presented and formulas are given for the diagonal and nondiagonal components of the relative phase permeabilities, which are universal and can be used for anisotropic media with any type of anisotropy (symmetry) of flow characteristics. A complex of laboratory tests for finding the nondiagonal components of the phase and relative phase permeability tensors is discussed.     

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.     

Determining equations of two-phase flows through anisotropic porous media

A new interpretation of the concept of relative phase permeability is given. Relative phase permeabilities are represented in the form of fourth-rank tensors. It is shown that in the case of anisotropic porous media functions depending not only on the saturation but also on the anisotropy parameters represented in the form of ratios of the principal values of the absolute permeability coefficient tensor correspond to the classical representation of the relative phase permeabilities. For a two-phase flow in anisotropic porous media with orthotropic and transversely-isotropic symmetry a generalized two-term Darcy’s law is analyzed. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 87–94, March–April, 1998. The work was carried out with support from the Russian Foundation for Fundamental Research (project No. 96-01-00623).     

Practical Considerations for Selection of Representative Elementary Volumes for Fluid Permeability in Fibrous Porous Media

We investigate the lower bound of the area of a square-shaped representative elementary volume (REV) for the permeability tensor for transverse Stokes flow through randomly packed, parallel, and monodisperse cylinders. The investigation is significant to flow models using small calculation regions for fibrous porous media, such as modeling defect formation during directional solidification in the mushy zone of dendritic alloys. Using 90 ensembles of 1,000 domains, where each ensemble comprises domains with the same number and size of cylinders, we develop correlations between the permeability tensor invariants and macroscopic features of the domain. We find that for ensembles of domains with fewer than 200 cylinders, the eigenvectors of the permeability tensors exhibit preferential alignment with the domain axes, demonstrating that the estimated permeability is significantly affected by the periodic boundary conditions for these cases. Our results also suggest that the anisotropy of the permeability tensor may not be insignificant even for large sampling volumes. These results provide a practical lower bound for the calculation volumes used in permeability simulations in fibrous porous media, and also suggest that modelers should consider using an anisotropic tensor for small calculation volumes if phenomena such as channeling are important.     

Laws of viscoplastic fluid flow in anisotropic porous and fractured media

In invariant tensor form, the laws of viscoplastic fluid flow are formulated for capillary and fractured media with a periodic microstructure that has orthotropic and transversely isotropic symmetry in the flow properties. An analysis of the laws of viscoplastic fluid flow in transversely isotropic and orthotropic porous and fractured media shows that in formulating the equations it is necessary to distinguish between the permeability tensor and the limiting gradient tensor, which may differ in the symmetry of the flow characteristics, and that the flow law is multivariant and admits one-, two-, and three-dimensional flows.     

Calculating the Anisotropic Permeability of Porous Media Using the Lattice Boltzmann Method and X-ray Computed Tomography

A lattice Boltzmann (LB) method is developed in this article in a combination with X-ray computed tomography to simulate fluid flow at pore scale in order to calculate the anisotropic permeability of porous media. The binary 3D structures of porous materials were acquired by X-ray computed tomography at a resolution of a few microns, and the reconstructed 3D porous structures were then combined with the LB model to calculate their permeability tensor based on the simulated velocity field at pore scale. The flow is driven by pressure gradients imposed in different directions. Two porous media, one gas diffusion porous layer used in fuel cells industry and glass beads, were simulated. For both media, we investigated the relationship between their anisotropic permeability and porosity. The results indicate that the LB model is efficient to simulate pore-scale flow in porous media, and capable of giving a good estimate of the anisotropic permeability for both media. The calculated permeability is in good agreement with the measured date; the relationship between the permeability and porosity for the two media is well described by the Kozeny–Carman equation. For the gas diffusion layer, the simulated results showed that its permeability in one direction could be one order of magnitude higher than those in other two directions. The simulation was based on the single-relaxation time LB model, and we showed that by properly choosing the relaxation time, it could give similar results to those obtained using the multiple-relaxation time (MRT) LB method, but with only one third of the computational costs of MRTLB model.     

Analytical expressions for odd-order anisotropic tensor dimension

According to the symmetries of the matter, the number of coefficients needed to define a tensorial relation varies. It is well known that in linear elasticity the number of generic coefficients varies from 21, for a complete anisotropic material, to 2, in case of isotropy. In a previous contribution, we provided analytical expressions that give the number of generic anisotropic coefficients in any anisotropic system for an even-order tensor. In the present note, we aim at extending the previous results to the case of odd-order tensors. As an illustration, the dimension of any anisotropic system for third-order piezoelectricity tensors and of the fifth-order coupling tensors of Mindlin's strain-gradient elasticity are determined.     

Calculation of strain histories in Protean coordinate systems

Integral equations for the relative deformation gradient tensors are solved to give analytical expressions which involve velocities and velocity gradients along streamlines. For some Protean coordinate systems, metric tensors are presented, and deformation gradients and strain histories are calculated. The results are tested for two types of flow: rotational shearing flow and extensional flow. They are found to give the existing exact relations for the Finger strain tensor.     

A Potential Formulation of Non-Linear Models of Flow through Anisotropic Porous Media

A theoretical analysis, based on the search for a normal dissipation potential, is performed in order to generalize the empirical non-Darcy one-dimensional flow models to 3-D flows through anisotropic porous media. In an abstract framework, it is proven that a large number of heuristic non-linear equations governing the multidimensional flow through isotropic porous media can be derived starting from a potential strictly related to the mechanical power dissipated by the fluid. Such a formulation allows to define, for the tensor permeability case, a wide class of filtration models according to the Onsager's generalized theory of dissipative mechanical systems. A consistent generalization to anisotropic permeability case of polynomial flow models is proposed. Both primal and dual mixed variational formulations associated to the proposed quadratic and incomplete cubic flow models are introduced and discussed.     

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.     

Dependency of Tortuosity and Permeability of Porous Media on Directional Distribution of Pore Voids

Single-phase fluid flow in porous media is usually direction dependent owing to the tortuosity associated with the internal structures of materials that exhibit inherent anisotropy. This article presents an approach to determine the tortuosity and permeability of porous materials using a structural measure quantifying the anisotropic distribution of pore voids. The approach uses a volume averaging method through which the macroscopic tortuosity tensor is related to both the average porosity and the directional distribution of pore spaces. The permeability tensor is derived from the macroscopic momentum balance equation of fluid in a porous medium and expressed as a function of the tortuosity tensor and the internal structure of the material. The analytical results generally agree with experimental data in the literature.     

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.     

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.     

The effect of anisotropic permeability on free convective boundary layer flow in porous media

The effect of an anisotropic permeability on thermal boundary layer flow in porous media is studied. The convective flow is induced by a vertical, uniformly heated surface embedded in a fluid-saturated medium. A leading-order boundary layer theory is presented. It is shown that the thickness of the resulting boundary layer flow is different from that obtained in an isotropic porous medium. In general, an anisotropic permeability induces a fluid drift in the spanwise direction, the strength of which depends on the precise nature of the anisotropy. Conditions are found which determine whether or not the boundary layer flow is three-dimensional.     

A formulation of anisotropic continuum elastoplasticity at finite strains. Part I: Modelling

A constitutive model for anisotropic elastoplasticity at finite strains is developed together with its numerical implementation. An anisotropic elastic constitutive law is described in an invariant setting by use of structural tensors and the elastic strain measure C_e. The elastic strain tensor as well as the structural tensors are assumed to be invariant in relation to superimposed rigid body rotations. An anisotropic Hill-type yield criterion, described by a non-symmetric Eshelby-like stress tensor and further structural tensors, is developed, where use is made of representation theorems for functions with non-symmetric arguments. The model also considers non-linear isotropic hardening. Explicit results for the specific case of orthotropic anisotropy are given. The associative flow rule is employed and the features of the inelastic flow rule are discussed in full. It is shown that the classical definition of the plastic material spin is meaningless in conjunction with the present formulation. Instead, the study motivates an alternative definition, which is based on the demand that such a quantity must be dissipation-free, as the plastic material spin is in the case of isotropy. Equivalent spatial formulations are presented too. The full numerical treatment is considered in Part II.     

Flows of constant stretch history for polymeric materials with power-law distributions of relaxation times

It is herein shown that for separable integral constitutive equations with power-law distributions of relaxation times, the streamlines in creeping flow are independent of flow rate.For planar flows of constant stretch history, the stress tensor is the sum of three terms, one proportional to the rate-of-deformation tensor, one to the square of this tensor, and the other to the Jaumann derivative of the rate-of-deformation tensor. The three tensors are the same as occur in the Criminale-Ericksen-Filbey Equation, but the coefficients of these tensors depend not only on the second invariant of the strain rate, but also on another invariant which is a measure of flow strength. With the power-law distribution of relaxation times, each coefficient is equal to the second invariant of the strain rate tensor raised to a power, times a function that depends only on strength of the flow. Axisymmetric flows of constant stretch history are more complicated than the planar flows, because three instead of two nonzero normal components appear in the velocity gradient tensor. For homogeneous axisymmetric flows of constant stretch history, the stress tensor is given by the sum of the same three terms. The coefficients of these terms again depend on the flow strength parameter, but in general the dependences are not the same as in planar flow.     

Permeability coefficient tensor in the Kozeny-Carman capillary model

The representation of the permeability coefficient tensor for capillary models of porous media displaying isotropic and anisotropic flow properties is considered. The representation proposed is compared with the Kozeny-Carman formula. It is shown that in general, as distinct from the widely accepted representation of the Carman constant in the form of a product of the form factor and tortuosity, this constant is equal to a combination of three coefficients, namely the form factor, the tortuosity, and the structure coefficient. The presence of the latter is due to the fact that in periodic capillary models the porosity is not equal to the surface porosity. It is shown that the Carman constant, the form factor, and the structure coefficient are not universal and their intervals of variation are calculated. The results obtained make it possible to explain and interpret numerous experimental data on the determination of the Carman constant in various porous media.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 96–104, July–August, 1996.     

The evolution of Hooke's law due to texture development in FCC polycrystals

Initially isotropic aggregates of crystalline grains show a texture-induced anisotropy of both their inelastic and elastic behavior when submitted to large inelastic deformations. The latter, however, is normally neglected, although experiments as well as numerical simulations clearly show a strong alteration of the elastic properties for certain materials. The main purpose of the work is to formulate a phenomenological model for the evolution of the elastic properties of cubic crystal aggregates. The effective elastic properties are determined by orientation averages of the local elasticity tensors. Arithmetic, geometric, and harmonic averages are compared. It can be shown that for cubic crystal aggregates all of these averages depend on the same irreducible fourth-order tensor, which represents the purely anisotropic portion of the effective elasticity tensor. Coupled equations for the flow rule and the evolution of the anisotropic part of the elasticity tensor are formulated. The flow rule is based on an anisotropic norm of the stress deviator defined by means of the elastic anisotropy. In the evolution equation for the anisotropic part of the elasticity tensor the direction of the rate of change depends only on the inelastic rate of deformation. The evolution equation is derived according to the theory of isotropic tensor functions. The transition from an elastically isotropic initial state to a (path-dependent) final anisotropic state is discussed for polycrystalline copper. The predictions of the model are compared with micro–macro simulations based on the Taylor–Lin model and experimental data.