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.     

A Statistical Correlation Between Permeability,Porosity, Tortuosity and Conductance

A statistical evaluation of 13,000 numerical simulations of random porous structures is used to establish a correlation between permeability, porosity, tortuosity and conductance. The random structures are generated with variable porosities, and parameters such as the permeability and the tortuosity are determined directly from the structures. It is shown that the prevalent definition of tortuosity, as the ratio of length of the real flow path to the projected path in the overall flow direction, does not correlate with permeability in the general case. Also, the correlation between the conductance of the medium, as an indicator of the accessible cross section of a flow path and permeability is no more reliable than the permeability–porosity correlation. However, if the definition of tortuosity is corrected using the cross-sectional variations, the resulting parameter (i.e., the minimum-corrected tortuosity) has a reliable correlation with permeability and can be used to estimate permeability with an acceptable error for most of the simulations of the random porous structures. The feasibility of extending the conclusions from 2-dimensional to 3-dimensional configurations and the numerical percolation thresholds for random structures are also discussed.     

Application of the Carman–Kozeny Correlation to a High‐Porosity and Anisotropic Consolidated Medium: The Compressed Expanded Natural Graphite

The Carman–Kozeny correlation is applied to a medium which is consolidated, highly porous and anisotropic: the expanded then compressed natural graphite. The effective textural properties (i.e. the mean pore diameter, porosity and tortuosity) have been measured by a mercury porosimeter and a heterogeneous diffusion cell. The texture and the permeability (according to the Darcy's law) measured for the two main directions of these orthotropic porous media change over a very wide range depending on their apparent mass densities. Experimental data show that only a part of the total porosity participates in the gas flow in steady state conditions.     

Permeability of Pore Networks Under Compaction

The characteristic pore length fixes the scale of permeability of a porous medium. For pore networks undergoing strong random compaction, this length becomes singular at transition porosities, revealing a change in the microstructure of the porespace controlling the transport. Nodal balances and lattice Boltzmann simulations of flow in pore networks under compaction show that the scaling between permeability and porosity changes near the transition porosities. Simulation results are compared with experimental permeability data from transparent two-dimensional micromodels of networks decorated with the same pore size distribution. Permeability?Cporosity data of media undergoing smooth compaction is well described by a single power law. Under strong compaction, however, the scaling between permeability and porosity is possible by traits only, the scaling exponent changes notably at given transition porosities. These behaviors are common to a wealth of permeability?Cporosity data thus far unexplained.     

Predicting the Capillary Imbibition of Porous Rocks from Microstructure

The kinetics of capillary imbibition into porous rocks is studied experimentally and theoretically. The Washburn law is modified by introducing various corrections relating to the microstructure of the rocks, such as tortuosity, pore shape (obtained experimentally), and applying the effective medium approximation (EMA) in order to calculate the effective radius that defines the hydraulic conductance and the topology of the capillary imbibition. The application of the EMA shows that capillary imbibition is mainly produced in 1-D, and the pore structure is constituted by different pore throats in series, linked by chamber pores. The capillary process has been discussed as a function of their petrography and pore structure. Our study of the Washburn equation and the addition of correction factors for the pore structure allows a very accurate prediction of the weight rate.     

Mathematical modelling of flow through consolidated isotropic porous media

A new mathematical model is proposed for time-independent laminar flow through a rigid isotropic and consolidated porous medium of spatially varying porosity. The model is based upon volumetric averaging concepts. Explicit assumptions regarding the mean geometric properties of the porous microstructure lead to a relationship between tortuosity and porosity. Microscopic inertial effects are introduced through consideration of flow development within the pores. A momentum transport equation is derived in terms of the fluid properties, the porous medium porosity and a characteristic length of the microstructure. In the limiting cases of porosity unity and zero, the model yields the required Navier-Stokes equation for free flow and no flow in a solid, respectively.     

A resistance model for flow through porous media

A new model for resistance of flow through granular porous media is developed based on the average hydraulic radius model and the contracting–expanding channel model. This model is expressed as a function of tortuosity, porosity, ratio of pore diameter to throat diameter, diameter of particles, and fluid properties. The two empirical constants, 150 and 1.75, in the Ergun equation are replaced by two expressions, which are explicitly related to the pore geometry. Every parameter in the proposed model has clear physical meaning. The proposed model is shown to be more fundamental and reasonable than the Ergum equation. The model predictions are in good agreement with the existing experimental data.     

On the flow of fluids through inhomogeneous porous media due to high pressure gradients

Most porous solids are inhomogeneous and anisotropic, and the flows of fluids taking place through such porous solids may show features very different from that of flow through a porous medium with constant porosity and permeability. In this short paper we allow for the possibility that the medium is inhomogeneous and that the viscosity and drag are dependent on the pressure (there is considerable experimental evidence to support the fact that the viscosity of a fluid depends on the pressure). We then investigate the flow through a rectangular slab for two different permeability distributions, considering both the generalized Darcy and Brinkman models. We observe that the solutions using the Darcy and Brinkman models could be drastically different or practically identical, depending on the inhomogeneity, that is, the permeability and hence the Darcy number.     

多孔材料孔隙尺寸对渗透系数影响的数值模拟

采用有限元方法数值模拟了多孔材料的孔隙尺寸与等效渗透系数之间的非线性关系.有限元模型中的固体骨架和孔隙根据孔隙率的大小随机生成,模型中的材料参数和单元属性用ANSYS中的APDL参数化语言赋值.根据有限元随机模拟断面的流量分布和稳态渗流问题的达西定律,计算在不同孔隙尺寸的等效渗透系数,研究等效渗透系数与孔隙尺寸之间的关系.计算结果表明,在孔隙率不变的情况下,等效渗透系数与孔隙尺寸的平方成正比,该结论与经验公式相一致.而孔隙尺寸不变的条件下,随着孔隙率的增加等效渗透系数近似呈线性增加.     

Numerical Approach of the Permeability of a Macroporous Bioceramic with Interconnected Spherical Pores

Manufacturing a hybrid bone substitute requires a dynamic culture of the cells preliminarily seeded in a scaffold through a flow of physiological fluid. The velocity, pressure, and the distribution of fluid flow in this kind of macroporous medium are the important keys. Because of the difficulties in determining these parameters by experiment, a numerical approach has been chosen. One of the primary step of this study consists in the determination of permeability K. In this article, two types of structure of macroporous bioceramics are concerned. One is the interconnected pore spheres arranged either simple cubic, body-centered cubic or face-centered cubic systems. The other is the interconnected pore spheres randomly arranged. Based on Darcy??s law, the permeability K was calculated for many cases (type, porosity) by simulating the fluid flow through a small representative volume. These results are compared with some previous models such as Ergun, Carman?CKozeny, Rumpf?CGupte, and Du Plessis. The limits of Darcy??s law and the above-mentioned models have been determined using numerical simulation. The result showed that the porous media with spherical interconnected pores of BCC systems can be used to replace a complex random system in a range of porosity from 0.71 to 0.76 (i.e., porosity of our scaffolds). This assumption is validated for a pressure gradient lower the 1,000?Pa m^?C1 and a simple polynomial relation linking permeability and porosity (0.71?C0.76) has been established.     

Network Model of Flow,Transport and Biofilm Effects in Porous Media

In this paper, we develop a network model to determine porosity and permeability changes in a porous medium as a result of changes in the amount of biomass. The biomass is in the form of biofilms. Biofilms form when certain types of bacteria reproduce, bond to surfaces, and produce extracellular polymer (EPS) filaments that link together the bacteria. The pore spaces are modeled as a system of interconnected pipes in two and three dimensions. The radii of the pipes are given by a lognormal probability distribution. Volumetric flow rates through each of the pipes, and through the medium, are determined by solving a linear system of equations, with a symmetric and positive definite matrix. Transport through the medium is modeled by upwind, explicit finite difference approximations in the individual pipes. Methods for handling the boundary conditions between pipes and for visualizing the results of numerical simulations are developed. Increases in biomass, as a result of transport and reaction, decrease the pipe radii, which decreases the permeability of the medium. Relationships between biomass accumulation and permeability and porosity reduction are presented.     

Flux in Porous Media with Memory: Models and Experiments

The classic constitutive equation relating fluid flux to a gradient in potential (pressure head plus gravitational energy) through a porous medium was discovered by Darcy in the mid 1800s. This law states that the flux is proportional to the pressure gradient. However, the passage of the fluid through the porous matrix may cause a local variation of the permeability. For example, the flow may perturb the porous formation by causing particle migration resulting in pore clogging or chemically reacting with the medium to enlarge the pores or diminish the size of the pores. In order to adequately represent these phenomena, we modify the constitutive equations by introducing a memory formalism operating on both the pressure gradient–flux and the pressure–density variations. The memory formalism is then represented with fractional order derivatives. We perform a number of laboratory experiments in uniformly packed columns where a constant pressure is applied on the lower boundary. Both homogeneous and heterogeneous media of different characteristic particle size dimension were employed. The low value assumed by the memory parameters, and in particular by the fractional order, demonstrates that memory is largely influencing the experiments. The data and theory show how mechanical compaction can decrease permeability, and consequently flux.     

Multiscale, Multiphysics Network Modeling of Shale Matrix Gas Flows

We present a pore network model to determine the permeability of shale gas matrix. Contrary to the conventional reservoirs, where permeability is only a function of topology and morphology of the pores, the permeability in shale depends on pressure as well. In addition to traditional viscous flow of Hagen–Poiseuille or Darcy type, we included slip flow and Knudsen diffusion in our network model to simulate gas flow in shale systems that contain pores on both micrometer and nanometer scales. This is the first network model in 3D that combines pores with nanometer and micrometer sizes with different flow physics mechanisms on both scales. Our results showed that estimated apparent permeability is significantly higher when the additional physical phenomena are considered, especially at lower pressures and in networks where nanopores dominate. We performed sensitivity analyses on three different network models with equal porosity; constant cross-section model (CCM), enlarged cross-section model (ECM) and shrunk length model (SLM). For the porous systems with variable pore sizes, the apparent permeability is highly dependent on the fraction of nanopores and the pores’ connectivity. The overall permeability in each model decreased as the fraction of nanopores increased.     

Non-Darcy double-diffusive natural convection in axisymmetric fluid saturated porous cavities

Double-diffusive natural convection in a fluid saturated porous medium has been investigated using the finite element method. A generalised porous medium model is used to study both Darcy and non-Darcy flow regimes in an axisymmetric cavity. Results indicate that the Darcy number should be a separate parameter to understand flow characteristics in non-Darcy regime. The influence of porosity on heat and mass transfer is significant and the transport rates may differ by 25% or more, at higher Darcy and Rayleigh numbers. When compared with the Darcy and other specialised models of Brinkman and Forchheimer, the present generalised model predicts the least heat and mass transfer rates. It is also observed that an increase in radius ratio leads to higher Nusselt and Sherwood numbers along the inner wall.     

Law of nonlinear flow in saturated clays and radial consolidation

It was derived that micro-scale amount level of average pore radius of clay changed from 0.01 to 0.1 micron by an equivalent concept of flow in porous media.There is good agreement between the derived results and test ones.Results of experiments show that flow in micro-scale pore of saturated clays follows law of nonlinear flow.Theoretical analyses demonstrate that an interaction of solid-liquid interfaces varies inversely with permeability or porous radius.The interaction is an important reason why nonlinear flow in saturated clays occurs.An exact mathematical model was presented for nonlinear flow in micro-scaie pore of saturated clays.Dimension and physical meanings of parameters of it are definite.A new law of nonlinear flow in saturated clays was established.It can describe characteristics of flow curve of the whole process of the nonlinear flow from low hydraulic gradient to high one.Darcy law is a special case of the new law.A math- ematical model was presented for consolidation of nonlinear flow in radius direction in saturated clays with constant rate based on the new law of nonlinear flow.Equations of average mass conservation and moving boundary,and formula of excess pore pressure distribution and average degree of consolidation for nonlinear flow in saturated clay were derived by using an idea of viscous boundary layer,a method of steady state in stead of transient state and a method of integral of an equation.Laws of excess pore pressure distribution and changes of average degree of consolidation with time were obtained.Re- suits show that velocity of moving boundary decreases because of the nonlinear flow in saturated clay.The results can provide geology engineering and geotechnical engineering of saturated clay with new scientific bases.Calculations of average degree of consolidation of the Darcy flow are a special case of that of the nonlinear flow.     

Macroscale Properties of Porous Media from a Network Model of Biofilm Processes

We demonstrate how a network model can predict porosity and permeability changes in a porous medium as a result of biofilm buildup in the pore spaces. A biofilm consists of bacteria and extracellular polymeric substances (EPS) bonded together and attached to a surface. In this case, the surface consists of the walls of the porous medium, which we model as a random network of pipes.Our model contains five species. Four of these are bacteria and EPS in both fluid and adsorbed phases. The fifth species is nutrient, which we assume to reside in the fluid phase only. Bacteria and EPS transfer between the adsorbed and fluid phases through adsorption and erosion or sloughing. The adsorbed species influence the effective radii of the pipes in the network, which affect the porosity and permeability.We develop a technique for integrating the coupled system of ordinary and partial differential equations that govern transport of these species in the network. We examine ensemble averages of simulations using different arrays of pipe radii having identical statistics. These averages show how different rate parameters in the biofilm transport processes affect the concentration and permeability profiles.     

Effect of yield stress on the flow of a Casson fluid in a homogeneous porous medium bounded by a circular tube

The effect of yield stress on the flow characteristics of a Casson fluid in a homogeneous porous medium bounded by a circular tube is investigated by employing the Brinkman model to account for the Darcy resistance offered by the porous medium. The non-linear coupled implicit system of differential equations governing the flow is first transformed into suitable integral equations and are solved numerically. Analytical solution is obtained for a Newtonian fluid in the case of constant permeability, and the numerical solution is verified with that of the analytic solution. The effect of yield stress of the fluid and permeability of the porous medium on shear stress and velocity distributions, plug flow radius and flow rate are examined. The minimum pressure gradient required to start the flow is found to be independent of the permeability of the porous medium and is equal to the yield stress of the fluid.     

A Geometrical Pore Model for Estimating the Microscopical Pore Geometry of Soil with Infiltration Measurements

This paper presents a simple geometrical pore model designed to relate characteristic pore radii of the porous network of soils with macroscopic infiltration parameters. The model composed of a stack of spherical hollow elements is described with two radii values: the pore access radius and the actual pore radius. The model was compared to cylindrical pore models and its mathematical consistency was assessed. Soil sorptivity S and the second parameter A of the Philip infiltration equation (1957), have been determined by numerically simulated infiltration. A diagram and an empirical relation have been set in order to relate the pore access and pore radii to the infiltration parameters S and A. The consistency of the model was validated by comparing the predicted sorptivity and hydraulic conductivity values, with the widely used unsaturated soil hydraulic functions (van Genuchten, 1980). The model showed good agreement with experimental infiltration data, and it is therefore concluded that the use of a model with two radii improves the relation between microscopic pore size and macroscopic infiltration parameters.     

Effective Correlation of Apparent Gas Permeability in Tight Porous Media

Gaseous flow regimes through tight porous media are described by rigorous application of a unified Hagen–Poiseuille-type equation. Proper implementation is accomplished based on the realization of the preferential flow paths in porous media as a bundle of tortuous capillary tubes. Improved formulations and methodology presented here are shown to provide accurate and meaningful correlations of data considering the effect of the characteristic parameters of porous media including intrinsic permeability, porosity, and tortuosity on the apparent gas permeability, rarefaction coefficient, and Klinkenberg gas slippage factor.     

Factorization of Transport Coefficients in Macroporous Media

We prove the fundamental theorem about factorization of the phenomenological coefficients for transport in macroporous media. By factorization we mean the representation of the transport coefficients as products of geometric parameters of the porous medium and the parameters characteristic of the multicomponent fluid saturating the porous space. The two permeabilities of the porous medium, the convective and the diffusional ones, are separated. A similarity between the diffusional permeability and the porosity–tortuosity factor of the Kozeny–Carman theory is demonstrated. We do not make any specific assumption about stochastic or deterministic structure of the porous medium. The fluxes in fluid on the pore level are described by general relations of the non-equilibrium thermodynamics.