An Algorithm for 3D Pore Space Reconstruction from a 2D Image Using Sequential Simulation and Gradual Deformation with the Probability Perturbation Sampler

The construction of a faithful 3D pore space model of a porous medium that could reproduce the macroscopic behavior of that medium is of great interest in various fields including medicine, material science, hydrology and petroleum engineering. A computationally efficient algorithm is developed that uses the probability perturbation method and sequential multiple-point statistics simulations to generate 3D stochastic and equiprobable representations of random porous media when only a 2D thin section image is available. By employing the probability perturbation method as a gradual deformation technique, the pore patterns of a single 2D image are deformed to generate a series of 2D stochastically simulated images. The 3D pore structure is then generated by simply stacking the 2D-simulated images. The quality of the 3D reconstruction is critically dependent on the rate of deformation and a simple general procedure for choosing this parameter is presented. Various criteria such as porosity, two-point auto-correlation function, multiple-point connectivity function, local percolation probability, absolute permeability obtained by lattice-Boltzmann method (LBM), formation factor and two-phase relative permeability calculations are used to validate the results. The method is tested on two random porous solids; Berea Sandstone and synthetic Silica, for which directly measured 3D micro-CT images are available. The stochastically reconstructed 3D pore space preserves the low- and high-order spatial statistics, the macroscopic flow properties and the microstructure of the 3D micro-CT images.     

Grain Shape Effects on Permeability,Formation Factor,and Capillary Pressure from Pore-Scale Modeling

We invoke pore-scale models to evaluate grain shape effects on petrophysical properties of three-dimensional (3D) images from micro-CT scans and consolidated grain packs. Four sets of grain-packs are constructed on the basis of a new sedimentary algorithm with the following shapes: exact angular grain shapes identified from micro-CT scans, ellipsoids fitted to angular grains, and spheres with volume and surface-to-volume ratio equal to original angular grains on a grain-by-grain basis. Subsequently, a geometry-based cementation algorithm implements pore space alteration due to diagenesis. Eight micro-CT scans and 144 grain-pack images with $500 \times 500 \times 500$ voxels (the resolution units of 3D images) are analyzed in this study. Absolute permeability, formation factor, and capillary pressure are calculated for each 3D image using numerical methods and compared to available core measurements. Angular grain packs give rise to the best agreement with experimental measurements. Cement volume and its spatial distribution in the pore space significantly affect all calculated petrophysical properties. Available empirical permeability correlations for non-spherical grains underestimate permeability between 30 and 70 % for the analyzed samples. Kozeny–Carman’s predictions agree with modeled permeability for spherical grain packs but overestimate permeability for micro-CT images and non-spherical grain packs when volume-based radii are used to calculate the average grain size in a pack. We identify surface-to-volume ratio and grain shape as fundamental physical parameters that control fluid distribution and flow in porous media for equivalent porosity samples.     

On the Permeability of Fractal Tube Bundles

The permeability of a porous medium is strongly affected by its local geometry and connectivity, the size distribution of the solid inclusions, and the pores available for flow. Since direct measurements of the permeability are time consuming and require experiments that are not always possible, the reliable theoretical assessment of the permeability based on the medium structural characteristics alone is of importance. When the porosity approaches unity, the permeability?Cporosity relationships represented by the Kozeny?CCarman equations and Archie??s law predict that permeability tends to infinity and thus they yield unrealistic results if specific area of the porous media does not tend to zero. The aim of this article is the evaluation of the relationships between porosity and permeability for a set of fractal models with porosity approaching unity and a finite permeability. It is shown that the tube bundles generated by finite iterations of the corresponding geometric fractals can be used to model porous media where the permeability?Cporosity relationships are derived analytically. Several examples of the tube bundles are constructed, and the relevance of the derived permeability?Cporosity relationships is discussed in connection with the permeability measurements of highly porous metal foams reported in the literature.     

Porous Medium Modeling of Air-Cooled Condensers

This article presents a porous media transport approach to model the performance of an air-cooled condenser. The finned tube bundles in the condenser are represented by a porous matrix, which is defined by its porosity, permeability, and the form drag coefficient. The porosity is equal to the tube bundle volumetric void fraction and the permeability is calculated by using the Karman–Cozney correlation. The drag coefficient is found to be a function of the porosity, with little sensitivity to the way this porosity is achieved, i.e., with different fin size or spacing. The functional form was established by analyzing a relatively wide range of tube bundle size and topologies. For each individual tube bundle configuration, the drag coefficient was selected by trial and error so as to make the pressure drop from the porous medium approach match the pressure drop calculated by the heat exchanger design software ASPEN B-JAC. The latter is a well-established commercial heat exchanger design program that calculates the pressure drop by using empirical formulae based on the tube bundle properties. A close correlation is found between the form drag coefficient and the porosity with the drag coefficient decreasing with increasing porosity. A second order polynomial is found to be adequate to represent this relationship. Heat transfer and second law (of thermodynamics) performance of the system has also been investigated. The volume-averaged thermal energy equation is able to accurately predict the hot spots. It has also been observed that the average dimensionless wall temperature is a parabolic function of the form drag coefficient. The results are found to be in good agreement with those available in the open literature.     

On the Construction of an Experimentally Based Set of Equations to Describe Cocurrent and Countercurrent, Two-Phase Flow of Immiscible Fluids Through Porous Media

Generalized flow equations developed for two-phase flow through porous media contain a second term that enables proper account to be taken of capillary coupling between the two flowing phases. In this study, a partition concept, together with a novel capillary pressure equation for countercurrent flow, have been introduced into Kalaydjian’s generalized flow equations to construct modified flow equations which enable a better understanding of the role of capillary coupling in horizontal, two-phase flow. With the help of these equations it is demonstrated that the reduced flux observed in countercurrent flow, as compared to cocurrent flow, can be explained by the reduction in the driving force per unit volume which comes about because of capillary coupling. Also, it is shown experimentally that, because fluids flow through a void space reduced in magnitude due to the presence of immobile irreducible and residual saturations, the capillary coupling parameter should be defined in terms of a reduced porosity, rather than in terms of porosity. Moreover, it is shown statistically that the countercurrent relative permeability curve is proportional to the cocurrent relative permeability curve, the constant of proportionality being the capillary coupling parameter. Finally it is suggested that one can eliminate the need to determine experimentally countercurrent relative permeability curves by making use of an equation constructed for predicting the magnitude of the capillary coupling parameter.     

Surface porosity and permeability of porous media with a periodic microstructure

Various ways of determining the surface porosity, the relation between the porosity and the surface porosity and the representation of the permeability in terms of the characteristics of the microstructure of the porous medium are analyzed with reference to model porous media with a periodic microstructure. It is shown that it is necessary to distinguish between the geometric (scalar) and physical (tensor) suface porosities and that the geometric surface porosity, the physical surface porosity and the porosity are different characteristics of the porous medium.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 79–85, January–February, 1995.     

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.     

Geometric and Hydrodynamic Characteristics of Three-dimensional Saturated Prefractal Porous Media Determined with Lattice Boltzmann Modeling

Fractal and prefractal geometric models have substantial potential for contributing to the analysis of flow and transport in porous media such as soils and reservoir rocks. In this study, geometric and hydrodynamic parameters of saturated 3D mass and pore–solid prefractal porous media were characterized using the lattice Boltzmann model (LBM). The percolation thresholds of the 3D prefractal porous media were inversely correlated with the fraction of micro-pore clusters and estimated as 0.36 and 0.30 for mass and pore–solid prefractal porous media, respectively. The intrinsic permeability and the dispersivity of the 3D pore–solid prefractals were larger than those of the 3D mass prefractals, presumably because of the occurrence of larger solid and pore cluster sizes in the former. The intrinsic permeability and dispersivity of both types of structure increased with increasing porosity, indicating a positive relationship between permeability and dispersivity, which is at odds with laboratory data and current theory. This discrepancy may be related to limitations of the convection dispersion equation at the relatively high porosity values employed in the present study.     

Free Flow at the Interface of Porous Surfaces: A Generalization of the Taylor Brush Configuration

A solution to the problem of shallow laminar water flow above a porous surface is essential when modeling phenomena such as erosion, resuspension, and mass transfer between the porous media and the flow above it. Previous studies proposed theoretical, experimental, and numerical insight with no single general solution to the problem. Many studies have used the Brinkman equation, while others showed that it does not represent the actual interface flow conditions. In this paper we show that the interface macroscopic velocity can be accurately modeled by introducing a modification to the Brinkman equation. A moving average approach was proved to be successful when choosing the correct representative elementary volume and comparing the macroscopic solution with the average microscopic flow. As the size of the representative elementary volume was found to be equal to the product of the square root of the permeability and an exponential function of the porosity, a general solution is now available for any brush configuration. Given the properties of the porous media (porosity and permeability), the flow height and its driving force, a complete macroscopic solution of the interface flow is obtained.     

多孔介质非线性渗流问题的摄动解

考虑变形多孔介质渗透参数(渗透率和孔隙度)与孔隙压力呈负指数变化的特点，建立了多孔介质渗流问题的数学模型，采用积分变换方法求出了一维非线性渗流问题的摄动解，并对常数渗透参数和指数渗透参数的渗流问题进行对比分析，计算结果表明：两者之间的差别较大，且渗透参数的变化对于流体渗流中后期过程有着重要的影响，但对渗流早期影响不大，这对于定量研究工程中非线性渗流问题模型参数的相对重要性提供了可靠的理论依据。     

Measurement of Void Size Correlation in Inhomogeneous Porous Media

In previous works, we have described a void space reconstruction method based on non-wetting fluid intrusion, wetting fluid drainage, and image analysis data. The method has been applied to a wide range of substances, including sandstone, compressed and sintered powders, paper substrates and coatings, soil and fibrous mats. We have also demonstrated in a previous work that the spatial correlation of similarly sized voids within inhomogeneous porous media has a huge effect on permeability. We therefore describe a method of measuring such correlation, suitable for use in our void space reconstructions. The method involves a cubic spline smoothing of a variogram of the void sizes in a binary image of the porous medium. It has been successfully tested on an artificially correlated void network, comprising two sintered glass discs of different void size ranges. Stereological effects, caused by the off-centre sectioning of voids, can interfere with the variogram features. Our method is sh own to be insensitive to artificially generated stereological interference. The method is also applied to sandstone samples.     

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.     

Time-Dependent Diffusion and Surface-Enhanced Relaxation in Stochastic Replicas of Porous Rock

Understanding the connection between pore structure and NMR behavior of fluid-saturated porous rock is essential in interpreting the results of NMR measurements in the field or laboratory and in establishing correlations between NMR parameters and petrophysical properties. In this paper we use random-walk simulation to study NMR relaxation and time-dependent diffusion in 3D stochastic replicas of real porous media. The microstructures are generated using low-order statistical information (porosity, void–void autocorrelation function) obtained from 2D images of thepore space. Pore size distributions obtained directly by a 3D pore space partitioning method and indirectly by inversion of NMR relaxation data are compared for the first time. For surface relaxation conditions typical of reservoir rock, diffusional coupling between pores of different size is observed to cause considerable deviations between the two distributions. Nevertheless, the pore space correlation length and the size of surface asperity are mirrored in the NMR relaxation data for the media studied. This observation is used to explain the performance of NMR-based permeability correlations. Additionally, the early time behavior of the time-dependent diffusion coefficient is shown to reflect the average pore surface-to-volume ratio. For sufficiently high values of the self-diffusion coefficient, the tortuosity of the pore space is also recovered from the long-time behavior of the time-dependent diffusion coefficient, even in the presence of surface relaxation. Finally, the simulations expose key limitations of the stochastic reconstruction method, and allow suggestions for future development to be made.     

Tomography-Based Characterization and Optimization Of Fluid Flow Through Porous Media

Numerical simulations to characterize fluid flow through porous media have been carried out using tomography-derived real geometry data that has been manipulated using digital image processing techniques to obtain a wide range of porosities. Two kinds of porous media have been analyzed: (a) a reticulated porous ceramic (RPC) foam and (b) a packed bed of CaCO₃ particles. The porosity of the media is varied via morphological operations between 0.727 and 0.913 in case of the RPC and between 0.329 and 0.824 in case of the packed bed. A mesh generator based on the pore space indicator function is then used to generate unstructured tetrahedral grids from the processed tomography data. Fluid flow simulations are carried out for Reynolds numbers ranging from 0.1 to 200 and the results are used to determine the permeability and the Dupuit?CForchheimer coefficient in each case. The results are then compared with existing analytical models and the applicability of the models is examined. In the RPC case, the Happel?CBrenner (parallel-flow) model predicts the permeability with a normalized root mean square error (NRMSE) of 11.8 % across the porosity range and Modified Ergun (Macdonald et. al) model predicts the Dupuit?CForchheimer coefficient within a NRMSE of 13.5 %. In the packed-bed case, the Brinkman drag model predicts the permeability within a NRMSE of 8.26 % across the porosity range and the Modified Ergun model predicts the Dupuit?CForchheimer coefficient within an NRMSE of 5.94 %. For each material, an adjusted Kozeny constant is determined. For the RPC, the Kozeny constant is evaluated at 7.73 and for the CaCO₃ packed bed, it is found to be 6.10, leading to predictions of the permeability with an NRMSE of 4.16 and 3.37 %, respectively.     

Effect of a nonuniform distribution of voids on the plastic response of voided materials: a computational and statistical analysis

This study investigates the overall and local response of porous media composed of a perfectly plastic matrix weakened by stress-free voids. Attention is focused on the specific role played by porosity fluctuations inside a representative volume element. To this end, numerical simulations using the Fast Fourier Transform (FFT) are performed on different classes of microstructure corresponding to different spatial distributions of voids. Three types of microstructures are investigated: random microstructures with no void clustering, microstructures with a connected cluster of voids and microstructures with disconnected void clusters. These numerical simulations show that the porosity fluctuations can have a strong effect on the overall yield surface of porous materials. Random microstructures without clusters and microstructures with a connected cluster are the hardest and the softest configurations, respectively, whereas microstructures with disconnected clusters lead to intermediate responses. At a more local scale, the salient feature of the fields is the tendency for the strain fields to concentrate in specific bands. Finally, an image analysis tool is proposed for the statistical characterization of the porosity distribution. It relies on the distribution of the ‘distance function’, the width of which increases when clusters are present. An additional connectedness analysis allows us to discriminate between clustered microstructures.     

Tomographic Study of Internal Erosion of Particle Flows in Porous Media

In particle-laden flows through porous media, porosity and permeability are significantly affected by the deposition and erosion of particles. Experiments show that the permeability evolution of a porous medium with respect to a particle suspension is not smooth, but rather exhibits significant jumps followed by longer periods of continuous permeability decrease. Their origin seems to be related to internal flow path reorganization by avalanches of deposited material due to erosion inside the porous medium. We apply neutron tomography to resolve the spatiotemporal evolution of the pore space during clogging and unclogging to prove the hypothesis of flow path reorganization behind the permeability jumps. This mechanistic understanding of clogging phenomena is relevant for a number of applications from oil production to filters or suffosion as the mechanisms behind sinkhole formation.     

Foam Mobility in Heterogeneous Porous Media

It is shown experimentally that in situ generation of foam is an effective method for achieving gas mobility control and diverting injected fluid to low permeability strata within heterogeneous porous media. The experimental system is composed of a 0.395 porosity, 5.35 µm² synthetic sandstone and a 0.244 porosity, 0.686 µm² natural sandstone. The cores are arranged in parallel and communicate through common injection and production conditions. Nitrogen is the gas phase and alpha-olefin sulfonate (AOS 1416) in brine is the foamer. Three types of experiments were conducted. First, gas alone was injected into the system after presaturation with the foamer solution. Second, gas and foamer solution were coinjected at an overall gas fraction of 90% into cores presaturated with surfactant. Each core accepted a portion of the injected gas and liquid according to the mobility within the core. Lastly, gas and foamer solution were coinjected into the individual, isolated porous media in order to establish baseline behavior. The results are striking. It is possible to achieve total diversion of gas injection to the low permeability medium in some cases. The results also confirm previous predictions that foamed gas can be more mobile in lower permeability porous media.     

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.