Effective Diffusivities of Point-Like Molecules in Isotropic Porous Media by Monte Carlo Simulation

Monte Carlo simulations of point-like molecules in random and structured media are used to determine and characterize the effective diffusion coefficients of the molecules in the media. Simulations were carried out in 2D and 3D media. Monte Carlo simulation results in 2D and 3D media are compared with those obtained by analytical techniques. Simulation results indicate that for the structured, isotropic media the effective diffusivities can be characterized according to percolation thresholds in addition to porosity. The effective diffusivities in two isotropic media with the same porosity but different percolation thresholds can differ significantly. The effects of dimensionality on the effective diffusivities can also be significant. It is shown that in general the effective diffusion coefficients obtained from 2D simulation are not a good approximation to those of 3D, especially when the percolation thresholds of the 2D media and the 3D media are very different.     

Fractal porous media III: Transversal Stokes flow through random and Sierpinski carpets

The transversal Stokes flow of a Newtonian fluid through random and Sierpinski carpets is numerically calculated and the transversal permeability derived. In random carpets derived from site percolation, the average macroscopic permeability varies as (- _c)^3/2, close to the critical porosity _c. This exponent is found to be slightly different from the conductivity exponent. Results for Sierpinski carpets are presented up to the fourth generation. The Carman equation is not verified in these two model porous media.     

Fractal porous media IV: Three-dimensional stokes flow through random media and regular fractals

The three-dimensional Stokes flow of a Newtonian fluid through random and/or fractal media is numerically determined. The permeability of these media is derived. Results relative to these structures are presented and discussed. The validity of the Carman equation and of a simple scaling argument is questioned.     

Stress-Dependent Porosity and Permeability of Porous Rocks Represented by a Mechanistic Elastic Cylindrical Pore-Shell Model

The stress dependency of the porosity and permeability of porous rocks is described theoretically by representing the preferential flow paths in heterogeneous porous rocks by a bundle of tortuous cylindrical elastic tubes. A Lamé-type equation is applied to relate the radial displacement of the internal wall of the cylindrical elastic tubes and the porosity to the variation of the pore fluid pressure. The variation of the permeability of porous rocks by effective stress is determined by incorporating the radial displacement of the internal wall of the cylindrical elastic tubes into the Kozeny–Carman relationship. The fully analytical solutions of the mechanistic elastic pore-shell model developed by combining the Lamé and Kozeny–Carman equations are shown to lead to very accurate correlations of the stress dependency of both the porosity and the permeability of porous rocks.

     

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.     

Solution of nonlinear wave equation of elastic rod

The longitudinal oscillation of a nonlinear elastic rod with lateral inertia are studied. A nonlinear wave equation is derived. The equation is solved by the method of full approximation.     

Stochastic averaging of n-dimensional non-linear dynamical systems subject to non-Gaussian wide-band random excitations

The averaged generalized Fokker-Planck-Kolmogorov (GFPK) equation for response of n-dimensional (n-d) non-linear dynamical systems to non-Gaussian wide-band stationary random excitation is derived from the standard form of equation of motion. The explicit expressions for coefficients of the fourth-order approximation of the averaged GFPK equation are given in series form. Conditions for convergences of these series are pointed out. The averaged GFPK equation is then reduced to that for 1-d dynamical systems derived by Stratonovich and compared with the closed form of GFPK equation for n-d dynamical systems subject to Poisson white noise derived by Di Paola and Falsone. Finally, this averaged GFPK equation is further reduced to that for quasi linear system subject to non-Gaussian wide-band stationary random excitation. Stationary probability density for quasi linear system subject to filtered Poisson white noise is obtained. Theoretical results for an example are confirmed by using Monte-Carlo simulation for different parameter values.     

Pore-Scale Simulation of Shear Thinning Fluid Flow Using Lattice Boltzmann Method

The present work attempts to identify the roles of flow and geometric variables on the scaling factor which is a necessary parameter for modeling the apparent viscosity of non-Newtonian fluid in porous media. While idealizing the porous media microstructure as arrays of circular and square cylinders, the present study uses multi-relaxation time lattice Boltzmann method to conduct pore-scale simulation of shear thinning non-Newtonian fluid flow. Variation in the size and inclusion ratio of the solid cylinders generates wide range of porous media with varying porosity and permeability. The present study also used stochastic reconstruction technique to generate realistic, random porous microstructures. For each case, pore-scale fluid flow simulation enables the calculation of equivalent viscosity based on the computed shear rate within the pores. It is observed that the scaling factor has strong dependence on porosity, permeability, tortuosity and the percolation threshold, while approaching the maximum value at the percolation threshold porosity. The present investigation quantifies and proposes meaningful correlations between the scaling factor and the macroscopic properties of the porous media.     

Cross-Properties Relations in 3D Percolation Networks: II. Network Permeability

An efficient method to estimate the absolute permeability of three-dimensional percolation networks was proposed. It uses a Kozeny–Carman relationship in the form of a scaling law to relate the network permeability to its hydraulic characteristic length. This characteristic length was determined at the network percolation threshold using a three-dimensional extension of the Hoshen–Kopelman algorithm. For developing the scaling laws, the network permeability was calculated by solving the Kirchoff’s law for all sample spanning clusters that had been identified by the three-dimensional version of the Hoshen–Kopelman algorithm. The method was tested with simple cubic site-bond network models with and without spatial correlations. The universality of the exponents in the scaling laws were also investigated. It was shown that, once the scaling law has been derived, the permeability value can be estimated 3–9 times faster using the present method.     

Sharp approximation to a filtration problem by maximum principle

Consider an incompressible fluid, filtrating through a saturated cylindrical porous layer with rectangular cross-section. A steady pressure gradient, parallel to the axis of the layer, drives a one-directional stationary non recirculating flow when the Darcy law has to include inertial and viscous corrections. This is the case, for instance, when the porosity of the medium or the seeping flow rate are not very small. The resulting nonlinear problem belongs to a class of equations which was proved to have positive solutions. It also satisfies a comparison principle from which approximations from above and from below are derived for the steady flow. The estimate from above is the flat profile which solves the Darcy-Forchheimer equation, which does not take account of viscous effects, and the approximation is excellent when the modified Darcy number is small, under the additional condition that the Forchheimer coefficient be small also. The flow still solves the problem when gradient forces, orthogonal to the axis of the layer, are also present.     

Statistical characteristics of the motion of a fluid particle in a medium with random porosity

In the study of flow of a neutral admixture in a porous medium, it is most often assumed in the stochastic formulation that the porosity is constant and a determinate quantity, and the velocity is a random function [1–4]. The velocity distribution is usually regarded as known. Flow in a porous medium with random porosity has been studied to a far lesser extent. We note [5], which studies the averaged equations obtained within the framework of the correlation approximation. We consider the model problem of one-dimensional motion of a fluid particle (position of the front for flow of a neutral admixture in a porous medium) in a medium with random porosity. For a particular form of random porosity field, expressions are obtained for the one- and two-point densities of the distribution of the position of the particle. A study is made of the dependences of the first four moments and the correlation function of the position of the particle as functions of the time. It is shown that for large values of the time the motion of the particle is asymptotically similar to Brownian motion. It is shown by means of numerical modeling that the results obtained transfer to the case of an arbitrary random porosity field. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 59–65, November–December, 1986.     

Effective Diffusivity Tensors of Point-Like Molecules in Anisotropic Porous Media by Monte Carlo Simulation

Monte Carlo simulations of random walks in anisotropic structured media are performed to determine the dependence of effective diffusivities on geometrical properties. The anisotropic media used in this study are periodic systems, which are generated by extending primitive, face-centered, and body-centered unit cells indefinitely in all axial directions. Results of simulations compare well with published experimental data and the calculations by the volume averaging method. In addition, these results suggest that if the 2D media with percolation thresholds subtantially differ from those of 3D, 2D approximations of 3D media are not satisfactory. When percolation thresholds are the same, the effective diffusivity tensors depend solely on the porosity. This fact has been suggested for isotropic media and it seems to hold for anisotropic media.     

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.     

Multi-scale Analysis of Gas Transport Mechanisms in Kerogen

Quantification of natural gas transport in organic-rich shale is important in predicting natural gas production. However, laboratory measurements are challenging due to tight nature of the rock and include large uncertainties. The emphasis of this work is to understand mass transport mechanisms inside the organic nanoporous material known as kerogen under subsurface conditions and describe its permeability. This requires a multi-scale theoretical approach that includes flow measurements in model nanocapillaries and within their network. Molecular dynamics simulation results of steady-state supercritical methane flow in single-wall carbon nanotube are presented in this article. A transition from convection to molecular diffusion is observed. The simulation results show that the adsorbed methane molecules are mobile and contribute a significant portion to the total mass flux in nanocapillaries with diameter \({<}\)10 nm. They experience cluster diffusion that is dependent on the applied pressure drop across the capillary. A modified Hagen–Poiseuille equation is proposed considering the convective–diffusive nature of the overall transport in nanocapillary. The molecular-level study of steady-state transport is extended to a simple network of interconnected nanocapillaries representing kerogen. The modified Hagen–Poiseuille equation leads to a representative elementary volume of the model kerogen. The estimated permeability of the volume is sensitive to compressed and adsorbed fluids density ratio and to surface properties of the nanocapillary walls, indicating that fluid–wall interactions driven by molecular forces could be significant during the large-scale transport within shale. A modified Kozeny–Carman correlation is proposed, relating kerogen porosity and tortuosity to the permeability.     

Measurement of Sediment Retention in a Sandy Tidal Flat Based on Pressure Drop Model

Sediment can either play an important role in subsurface environments as a food source for bacteria or deteriorate the subsurface environments by its retention. Thus, understanding sediment retention is useful for designing the management of subsurface environments. The pressure drop model derived from the Kozeny–Carman model is experimentally verified by the seepage flow in sand beds. It was found that the water head in the sand bed under steady-state flow and variations of the water head corresponding to changes in the boundary water head could be reproduced by the pressure drop model. As the porosity of the sand bed is taken into account in the pressure drop model, the sediment retention can be predicted from variations of the porosity. Experimental results showed that the water head in the sand bed varied due to sediment retention. This ensured that variances in the porosity of the sand bed could be predicted, leading to the investigation onto sediment retention. A method based on the pressure drop model is proposed to measure temporal variations of the water head in a sandy tidal flat and river water head. From field experiments, the temporal variations of the water head in the tidal flat could be predicted when the porosity of the tidal flat was used. Conversely, it is expected that sediment retention in the tidal flat can be predicted based on the variations of the porosity, if the water head in the tidal flat is observed temporally.     

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.     

Principles of the micromechanics of material damage. 1. Long-term damage

The principles of the theory of long-term damage based on the mechanics of stochastically inhomogeneous media are set out. The process of damage is modeled as randomly dispersed micropores resulting from the destruction of microvolumes. A failure criterion for a single microvolume is associated with its long-term strength dependent on the relationship of the time to brittle failure and the difference between the equivalent stress and the Huber-von Mises failure stress, which is assumed to be a random function of coordinates. The stochastic elasticity equations for porous media are used to determine the effective moduli and the stress-strain state of microdamaged materials. The porosity balance equation is derived in finite-time and differential-time forms for given macrostresses or macrostrains and arbitrary time using the properties of the distribution function and the ergodicity of the random field of short-term strength as well as the dependence of the time to brittle failure on the stress state and the short-term strength. The macrostress-macrostrain relationship and the porosity balance equation describe the coupled processes of deformation and long-term damage __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 2, pp. 108–121, February 2007. For the centenary of the birth of G. N. Savin.     

Stability of a spherical cavity in a sound field

The linear theory of the stability of the spherical shape of a cavity and the stability of its radial oscillations in a sound field are discussed. An equation is derived for the amplitudes of the spherical harmonics with allowance for surface tension, viscosity, and compressibility of the surrounding liquid in the Herring-Flynn approximation. The radial pulsation stability is analyzed in the same approximation. The equations derived in the article are subjected to numerical analysis.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 109–114, November–December, 1973.     

Tortuosity based on Anisotropic Diffusion Process in Structured Plate-like Obstacles by Monte Carlo Simulation

The diffusion of fluids in porous media, composed of regularly aligned plate-like obstacles, was studied by Monte Carlo simulation. The diffusion coefficients and all diagonal components of the diffusion tensor were estimated for these media. The calculated tortuosities were modeled as a function of porosity by using the Koponen’s equation related to percolation threshold. These results indicated that a media with a homogeneous porosity has a heterogeneous tortuosity, is affected by the alignments of the plate-like obstacles. Furthermore, the calculation results were compared with the experimental results for fixed perpendicular plates of Comiti and Renaud as a function of porosity. The results for tortuosity compared well for porosity larger than 0.86.