Cross-Properties Relations in 3D Percolation Networks: I. Network Characteristic Length Determination

This work developed efficient methods to determine the hydraulic characteristic length of percolation networks. The proposed methods are based on the determination of the percolation threshold using the Hoshen–Kopelman algorithm extended to 3D networks. The methods were tested with simple cubic site-bond network models with and without spatial correlations. It was shown that the proposed methods outperformed the algorithms based on the simulation of the porosimetry curve.*Author for correspondence: Fax +55-21-2562-8300; e-mail: paulo@peq.coppe.ufrj.br     

幂律流体在多孔介质中径向渗流的分形模型

基于分形理论及毛细管模型,本文研究了非牛顿幂律流体在各向同性多孔介质中径向流动问题,推导了幂律流体径向有效渗透率的分形解析表达式．研究结果表明,幂律流体径向有效无量纲渗透率模型和Chang and Yortsos’s模型吻合很好;同时还得出幂律流体径向有效渗透率随孔隙率、幂指数的增加而增加,随迂曲度分形维数的增加而减少．     

三维非均匀不稳定渗流方程的二维不均匀粗化算法

在无源汇条件下,根据流过某一个横截面的流体流量等于流过这一横截面内所有精细网格的流体流量之和这一特点提出了粗化网格等效渗透率的计算方法。在粗化区内,利用直接解法求解二维渗流方程,再用这些解合成粗化网格的三维合成解,并由合成解计算粗化网格的等效渗透率。根据精度的要求采用了不均匀网格粗化,在流体流速大的区域采用精细网格。利用所得等效渗透率计算了粗化网格的某三维非均匀不稳定渗流场的压降解,结果表明三维非均匀不稳定渗流方程的二维不均匀粗化解非常逼近采用精细网格的解,但计算的速度比采用精细网格提高了80倍。     

Determination of Permeability Distributions Using NMR Velocity Imaging

This study develops a methodology for determining the absolute permeability distribution in a porous media sample using velocity data obtained from NMR imaging experiments. An objective function describing the discrepancy between observed and simulated data is reduced by iteratively updating the permeability. This parameter estimation scheme is based on an iterative method which uses optimal control theory to refine the estimates. Although this theory is developed for both isotropic and anisotropic porous media, the permeability reconstructions examined in this paper are restricted to the isotropic case. Synthetic data are used to investigate the impact of varying the noise in the experimental data, the degree of parameterization, the relative weighting of the regularization term in the objective function, and the amount and type of data required to obtain a satisfactory permeability reconstruction. These synthetic data are extracted from the solution of numerical experiments that have utilized an assumed permeability distribution. The methodology is also applied to data gathered in laboratory experiments for water flow in a sandstone sample.     

Percolation Approach to the Anisotropy Factor in an Unsaturated Porous Medium

The anisotropy factor is defined as the ratio of the effective (macroscopic) conductivities parallel to the bedding plane and perpendicular to it. The anisotropy factor A(p,a) is a function of both the saturation degree, p, in the void space of the disordered medium and the anisotropy parameter, a, that characterizes the ratio of the local conductivities parallel and normal to the bedding plane. There are two opposite behaviors of the anisotropy factor as a function of the saturation degree described in literature. One presents the anisotropy factor as a curve with a maximum and the other as a curve with a minimum. The main result of calculating the conductivities of a uniaxial percolation anisotropic model is that at the saturation threshold value, p_c, A(p_c, a) = 1, wherefrom it increases at a >1 (or decreases at a < 1) with saturation. An extension of the computed results below the threshold is also proposed.     

Permeability Tensor for Heterogeneous Porous Medium of Fibre Type

A theoretical model which allows us to determine the permeability of a fibrous porous medium is proposed. Fibres are assumed to be parallel and nonuniform in space and material with a low volume fraction of fibres is considered. The model includes two geometric parameters: the diameter of fibres and the diameter of caverns or fissures inside the bundle of fibres. The tensor of permeability of the porous medium is determined based upon a generalized cell model. The components of permeability tensor depend on two parameters which are determined using experimental data and least-squares approximation. The influence of the geometric parameters on components of permeability tensor is discussed.     

Network model evaluation of permeability and spatial correlation in a real random sphere packing

In principle, network models can replicate exactly the microstructure of porous media. In practice, however, network models have been constructed using various assumptions concerning pore structure. This paper presents a network model of a real, disordered porous medium that invokes no assumptions regarding pore structure. The calculated permeability of the model agrees well with measured permeabilities, providing a new and more rigorous confirmation of the validity of the network approach. Several assumptions commonly used in constructing network models are found to be invalid for a random packing of equal spheres. In addition, the model permits quantification of the effect of pore-scale correlation (departure from randomness) upon permeability. The effect is comparable to reported discrepancies between measured permeabilities and predictions of other network models. The implications of this finding are twofold. First, a key assumption of several theories of transport in porous media, namely that pore dimensions are randomly distributed upon a network, may be invalid for real porous systems. Second, efforts both to model and to measure pore-scale correlations could yield more accurate predictions of permeability.     

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.     

The effect of pore-structure on hysteresis in relative permeability and capillary pressure: Pore-level modeling

The effect of pore-structure upon two-phase relative permeability and capillary pressure of strongly-wetting systems at low capillary number is simulated. A pore-level model consisting of a network of pore-bodies interconnected by pore-throats is used to calculate scanning loops of hysteresis between primary drainage, imbibition and secondary drainage. The pore-body to pore-throat aspect ratio strongly influences the pattern of hysteresis. Changes in the patterns of hysteresis often attributed to consolidation can be understood in terms of changes in aspect ratio. Correlation between the sizes of neighboring pore-throats affects the shape of the relative permeability curves, while the width and shape of the pore-size distribution have only a minor influence.     

A Two-Dimensional Network Model to Simulate Permeability Decrease Under Hydrodynamic Effect of Particle Release and Capture

A model was developed to simulate permeability decrease induced by hydrodynamic effects when injecting a fluid in a reservoir with respect to particle release and capture mechanisms and the parameters of the fluid–rock system. The kinetics of particle release and capture were integrated after computing the initial permeability of the porous medium with a square lattice of a two–dimensional network model. The rate of particle release is related to the difference between a microscopic velocity of the fluid and a critical velocity. The permeability decrease shows a direct link to the reduction of pore throat radii by three mechanisms of particle capture: straining and particle accumulation through direct interception or diffusion. Comparison between the simulations and the experimental results shows that the model reproduces the physics of the permeability decrease phenomenon, although the values are overestimated. The difference between the two sets of results can be explained by the fact that the simulations are realized at constant pressure whereas the experiments are realized at constant flow rate, and that re–entrainment of the trapped particles was not taken into account in the model.     

Experimental investigation of characteristic length scale in periodic heterogeneous porous media

An experimental investigation of scale-dependent dispersion in periodic heterogeneous porous media was conducted. Models with two-, three- and four-layer periodic heterogeneities were constructed to investigate the effect of heterogeneity size on the scale-dependence of dispersion. Longitudinal dispersion coefficients were determined as a function of column length by measuring the breakthrough of a continuous injection of potassium chloride tracer solution. Chloride ion concentration was monitored by recording the millivolt potential of silver/silver chloride electrodes placed at intervals along the length of the column. In all three models, dispersion appeared to be scale dependent up to a distance of approximately 20–30 times the size of the repeated heterogeneity group (hydraulic unit). Because all three models suggested a similar dependence, it was concluded that a medium with periodic heterogeneity may likely be characterized by the scale of its hydraulic unit.     

The Motion of Long Bubbles in a Network of Tubes

The motion and interaction of discrete bubbles in porous materials is studied numerically using a network model. The goal is to extend analytical results for the motion of bubbles through a single straight tube to a more realistic geometry for porous materials, modeled here as a planar network of straight tubes of different radii. The problem is characterized by two dimensionless parameters, the capillary number (Ca) and the volume fraction of bubbles (); results are characterized by determining the effective permeability of the network and the mean residence time of bubbles in the material. The simulations indicate that at low volume fraction most of the bubbles follow a limited number of high-flow pathways through the network. In this case the predictions of our simulations can be approximated by a simple analytical model. Bubbles interact with each other because their presence changes the local resistance to flow in individual tubes. As increases, interactions between individual bubbles become important resulting in a wider range of residence times in the porous material.     

3D simulation of radionuclide transport in porous media

We present an efficient and easily implementable finite volume method simulating radionuclide transport through highly heterogeneous grounds in three space dimensions. The numerical concentration of the transported chemicals are proved to remain nonnegative and stable. Then, we run a realistic test case in which some radioactive iodine I¹²⁹ particles are released from a leak in an underground nuclear waste disposal site. The question of whether the radionuclide invades the underground and reach the ground surface is investigated. Because of the 3D nature of the problem, a particular emphasis is made on the control of CPU time. Copyright © 2009 John Wiley & Sons, Ltd.     

横向三维化学反应流动数值模拟

对一个三维化学反应流程序作了改进,使之适应于模拟带横向碘蒸汽注入的COIL(化学氧碘激光)喷管流动。其中,主副气流的入流边界条件的改进是基本的,重要的。这里,入流边界附近的气体流动被视为一维均熵流动,用双曲方程的特征理论,分析以流量作变量的EULER方程的特征关系,用特征方法数值求解简化的特征方程,从而确定亚声速(跨声速)流动边界点的入流密度或速度。对相关文献提供的算例的计算,佐证了程序改进的有效性。对氮气作载气的条件也作了模拟,得出了与氦气条件不同的流动特征与增益分布特征。     

Determination of permeability tensors for two-phase flow in homogeneous porous media: Theory

In this paper we continue previous studies of the closure problem for two-phase flow in homogeneous porous media, and we show how the closure problem can be transformed to a pair of Stokes-like boundary-value problems in terms of pressures that have units of length and velocities that have units of length squared. These are essentially geometrical boundary value problems that are used to calculate the four permeability tensors that appear in the volume averaged Stokes' equations. To determine the geometry associated with the closure problem, one needs to solve the physical problem; however, the closure problem can be solved using the same algorithm used to solve the physical problem, thus the entire procedure can be accomplished with a single numerical code.Nomenclature a a vector that maps V onto , m^-1. - A a tensor that maps V onto . - A area of the - interface contained within the macroscopic region, m². - A area of the -phase entrances and exits contained within the macroscopic region, m². - A area of the - interface contained within the averaging volume, m². - A area of the -phase entrances and exits contained within the averaging volume, m². - Bo Bond number (= (=(–)g²/). - Ca capillary number (= v/). - g gravitational acceleration, m/s². - H mean curvature, m^-1. - I unit tensor. - permeability tensor for the -phase, m². - viscous drag tensor that maps V onto V. - ^* dominant permeability tensor that maps onto v , m². - ^* coupling permeability tensor that maps onto v , m². - characteristic length scale for the -phase, m. - l characteristic length scale representing both and , m. - L characteristic length scale for volume averaged quantities, m. - n unit normal vector directed from the -phase toward the -phase. - n unit normal vector representing both n and n . - n unit normal vector representing both n and n . - P pressure in the -phase, N/m². - p superficial average pressure in the -phase, N/m². - p intrinsic average pressure in the -phase, N/m². - p –p , spatial deviation pressure for the -phase, N/m². - r ₀ radius of the averaging volume, m. - r position vector, m. - t time, s. - v fluid velocity in the -phase, m/s. - v superficial average velocity in the -phase, m/s. - v intrinsic average velocity in the -phase, m/s. - v –v , spatial deviation velocity in the -phase, m/s. - V volume of the -phase contained within the averaging volmue, m³. - averaging volume, m³. Greek Symbols V /, volume fraction of the -phase. - viscosity of the -phase, Ns/m². - density of the -phase, kg/m³. - surface tension, N/m. - (v +v ^T ), viscous stress tensor for the -phase, N/m².     

Multiscale Structures to Describe Porous Media Part II: Transport Properties and Application to Test Materials

A renormalization method for the computation of the transport properties of a porous medium modelled as a multiscale random network is proposed. The method applies to electrical conduction, molecular diffusion, hydraulic transport under low Reynolds number, transport of condensable vapour, in the medium fully or partially saturated by one or two immiscible fluids. For 31 test materials, the method previously exposed by the authors for the reconstitution of the pore structure from the mercury intrusion curve is applied. Then, the intrinsic permeability is computed. The results are in good agreement with the measured permeability.     

Pore-Level Modeling of Gas and Condensate Flow in Two- and Three-Dimensional Pore Networks: Pore Size Distribution Effects on the Relative Permeability of Gas and Condensate

We present a mechanistic model of retrograde condensation processes in two- and three-dimensional capillary tube networks under gravitational forces. Condensate filling-emptying cycles in pore segments and gas connection–isolation cycles are included. With the pore-level distribution of gas and condensate in hand, we determine their corresponding relative permeabilities. Details of pore space and displacement are subsumed in pore conductances. Solving for the pressure field in each phase, we find a single effective conductance for each phase as a function of condensate saturation. Along with the effective conductance for the saturated network, the relative permeability for each phase is calculated. Our model porous media are two- and three-dimensional regular networks of pore segments with distributed size and square cross-section. With a Monte Carlo sampling we find the optimum network size to avoid size effects and then we investigate the effect of network dimensionality and pore size distribution on the relative permeabilities of gas and condensate.     

Achieving Rapid Absorption and Extensive Liquid Uptake Capacity in Porous Structures by Decoupling Capillarity and Permeability: Nanoporous Modified Calcium Carbonate

Porous media with rapid absorption properties are greatly sought after in the fields of super absorbers and catalysts. Natural materials, such as diatomite, or synthetic zeolite feature strongly in industrial reaction processes. Most, or all, of such materials, however, are surface acidic. A novel rapidly absorbing alkaline porous structure, with a high absorption capacity, is presented here. As in the case of diatomite or zeolite, the pigment design incorporates strong capillarity within a highly permeable packed medium. A model is proposed for general use with highly absorbing media that can be proven microscopically to have separate domains of micro- or nano-capillarity embedded within a permeable matrix. The new pigment morphology, based on natural ground calcium carbonate (gcc), exhibits this property using special surface structure modifications. It is contrasted with standard gcc by using consolidated tablet blocks made from a suspension of the pigment and chosen mixtures thereof. The blocks are characterised after drying by mercury porosimetry, and the absorption dynamic of a selected liquid is studied. It is shown that using a self-assembly method of discrete pore structures provides a much faster absorption rate and a liquid capacity for up to 10 times more fluid than a conventional homogeneously distributed pore concept. In such unique discrete network systems, the mercury intrusion curve provides a separable analysis of permeability and capillarity in respect to the inflection point of the cumulative intrusion curve. The discrete decoupled properties each follow the absorption behaviour predicted by previous modelling (Ridgway and Gane, Colloids and Surfaces A: Physicochemical and Engineering Aspects, 206(1–3), 2002). The absorption driving force is shown to be determined by the proportion of fine pores present up to a size equal to a Bosanquet inertially-defined optimum within the timescale of absorption. Combining the wetting force, from the capillarity-controlled fine pore structure, with the experimental flow resistance of the sample, consisting of the assembly of particles, it is possible to predict the trends in absorption dynamic using the pore and throat model Pore-Cor.* Use of this model allows existing materials as well as new synthetic designs to be modelled prior to manufacture. The novel alkaline material is compared with independent absorption data for diatomite and shown to be comparable. *Pore-Cor is a software product name of the Environmental and Fluid Modelling Group, University of Plymouth, Devon PL4 8AA, U.K.