APPLICATION OF THE PRECONDITIONED CONJUGATE GRADIENTS METHOD IN THE SIMULATION OF RELATIVE PERMEABILITY PROPERTIES OF POROUS MEDIA

A model that is after the determination of the relative permeability characteristics of porous media is presented. It is part of a general model that deals with the simulation of capillary phenomena and immiscible fluid flow behaviour in porous media. The relative permeability characteristics in a water wet porous medium are simulated with the use of three-dimensional (3-D) network models of pore structure with pore body and pore throat size distributions. The major assumption involved is that a cubic network of pore bodies connected by pore throats with pore body and pore throat size distribution respectively, is a realistic representation of the pore structure of a porous medium. The physical laws that apply in real media are simulated in the network analysis, and the computed results are compared with experimental findings. A new algorithm was developed that leads to the solution of a large set of linear equations, with a sparse and positive definite coefficient matrix. Results obtained with the application of the Preconditioned Conjugate Gradients method and numerical aspects of the simulation are presented and discussed. Comparison of this method with other available numerical methods is also made. It is concluded that the Preconditioned Conjugate Gradients method is advantageous for large networks with regards to time of solution, convergence, and accuracy. The validity of the algorithm is tested against other methods in the literature.     

用三维网络模型模拟多孔介质的吸着回路

在简单立方网络框架的基础上构造了具有规整和随机特性的三维网络模型,并引入了周期边界条件.采用Weibull分布作为孔尺寸参数的分布函数,模拟了不同配位数和孔尺寸范围下氮在多孔介质中的等温吸附和脱附过程.着重考察了多孔介质网络模型的几何性质和拓扑性质对吸着等温回路的影响.结果发现,影响吸着等温线回路形状的主要因素是孔喉的半径和网络的配位数.     

Dynamic modeling of drainage through three-dimensional porous materials

The interplay of viscous, gravity and capillary forces determines the flow behavior of two or more phases through porous materials. In this study, a rule-based dynamic network model is developed to simulate two-phase flow in three-dimensional porous media. A cubic network analog of porous medium is used with cubic bodies and square cross-section throats. The rules for phase movement and redistribution are devised to honor the imbibition and drainage physics at pore scale. These rules are based on the pressure field within the porous medium that is solved for by applying mass conservation at each node. The pressure field governs the movement and flow rates of the fluids within the porous medium. Film flow has been incorporated in a novel way. A pseudo-percolation model is proposed for low but non-zero capillary number (ratio of viscous to capillary forces). The model is used to study primary drainage with constant inlet flow rate and constant inlet pressure boundary conditions. Non-wetting phase front dynamics, apparent wetting residuals (S_wr), and relative permeability are computed as a function of capillary number (N_ca), viscosity ratio (M), and pore-throat size distribution. The simulation results are compared with experimental results from the literature. Depending upon the flow rate and viscosity ratio, the displacement front shows three distinct flow patterns—stable, viscous fingering and capillary fingering. Capillary desaturation curves (S_wr vs. N_ca) depend on the viscosity ratio. It is shown that at high flow rates (or high N_ca), relative permeability assumes a linear dependence upon saturation. Pseudo-static capillary pressure curve is also estimated (by using an invasion percolation model) and is compared with the dynamic capillary pressure obtained from the model.     

Pore-Scale Simulation of Drying of a Porous Media Saturated with a Sucrose Solution

This work presents results of Monte Carlo simulations of isothermal drying of a nonhygroscopic porous media initially saturated with a sugar solution. The porous media is represented by a two-dimensional network of cubic pores connected by throats with a given radius distribution. The considered network had just one open side (the three other sides were sealed) from which water evaporation occurred. Water evaporation, hydraulic flow, and diffusivity of sucrose in water are considered in the physical model. It was considered that drying occurred under isothermal conditions (low drying rates) and that the capillary forces surpass the viscous forces, as in invasion percolation. It was also considered that water evaporation inside the network of pores and throats causes solution concentration, which remains at the corners, allowing hydraulic connection throughout the whole network. At each simulation step, a single meniscus moves through a particular pore segment with the higher displacing force. As drying progresses, air replaces the solution. Determination of the mechanism prevailing at any given drying stage requires calculation of evaporation. In other words, each step of the simulation involves finding the solution to three systems of equations: the vapor pressure field in the vapor phase, the pressure field in the liquid phase, and the solutes' concentration in the liquid phase. Herein, we report results of drying curves calculated as a function of the sucrose and water saturation and of the distribution of liquid, sucrose, and vapor as drying advances. The results presented in this work showed that network models are a powerful tool for investigating the influence of the main mechanisms controlling drying at its different stages; that is, from liquid saturation condition to very low saturation (end of drying). Despite the applied simplifications, the model can capture the main aspects of drying of liquids and solutions present in porous media.     

Effect of the coordination number of the pore-network on the transport and deposition of particles in porous media

Effect of the coordination number of the pore-network structure on the transport and deposition of colloidal particles in porous media was investigated. Applying with the orthogonal collocation principle and the pseudo-spectral method based on the Chebyshev polynomial, the complete solutions of the population balance equations describing the deposited particles, the suspended particles and the unblocked pore throats of porous media were obtained for the simple square network and the capillary tube network. It is shown that the network structure of porous media may significantly affect filtration rate, and therefore the effluent concentration and pressure drop. The filter with a capillary tube network structure owns higher filter coefficient than that of the filter with a simple square network structure. Theoretical predictions for different influent concentrations of particles, different flow rates and the case of blocking occurred in the filter bed are in agreement with the available experimental data.     

NETmix®, a new type of static mixer: Modeling,simulation, macromixing,and micromixing characterization

NETmix® is a new technology for static mixing based on a network of chambers connected by channels. The NETmix® model is the basis of a flow simulator coupled with chemical reaction used to characterize macro and micromixing in structured porous media. The chambers are modeled as perfectly mixing zones and the channels as plug flow perfect segregation zones. A segregation parameter is introduced as the ratio between the channels volume and the whole network volume. Different kinetics and reactants injection schemes can be implemented. Results show that the number of rows in the flow direction and the segregation parameter control both macro and micromixing, but the degree of micromixing is also controlled by the reactants injection scheme. The NETmix® model enables the systematic study of micromixing and macromixing for different network structures and reaction schemes, enabling the design of network structures to ensure the desired yield and selectivity. © 2009 American Institute of Chemical Engineers AIChE J, 2009     

A PORE NETWORK MODEL FOR THE CALCULATION OF NON-DARCY FLOW COEFFICIENTS IN FLUID FLOW THROUGH POROUS MEDIA

A computational method was developed to determine the correlation between permeability and the Forchheimer (non-Darcy) coefficient and to investigate numerically the factors that affect this correlation. The method is based on the construction of a 2-D random pore network. The porous medium is represented as a network of cylindrical pipes with randomly generated size, orientation, and connectivity. Probability density functions are used to characterize the orientation, length, diameter, and connectivity properties of the pipes that form the pore space. In this article the development and validation of the computational method are discussed. The computational procedure provides an expected value for the medium properties and a variation around this expected value, which is a measure of the uncertainty associated with the calculation of porous media properties. It was found that the porous medium structure has a strong effect on the flow properties. Specifically, the splitting of the pores is mostly responsible for non-Darcy flow behavior. Examination of the behavior of the medium properties under compaction shows that there is a unifying correlation for the behavior of the permeability. However, each type of porous medium exhibits different non-Darcy flow behavior under compaction. Finally, a statistical model, which correlates the structure characteristics of the porous medium to the measured properties, is presented.     

Two-phase flow in porous media

Two-phase flow in porous media depends on many factors, such as displacement vs steady two-phase flow, saturation, wettability conditions, wetting fluid vs non-wetting fluid is displacing, the capillary number, interfacial tension, viscosity ratio, pressure gradient, uniformly wetted vs mixed-wet pore surface, uniform vs distributed pore throats, small vs large pores, well-connected pores vs pores connected by small throats, etc. These parameters determine how the two fluids are distributed in the pores, e.g. whether they flow in seperate channels or side-by-side in the same channels, either with both fluids being continous or only one fluid being continous and the other discontinuous. In displacement, the capillary number and the viscosity ratio determine whether the displacement front is sharp, or if there is either capillary or viscous fingering.     

LIQUID-LIQUID RELATIVE PERMEABILITY: NETWORK MODELS AND EXPERIMENTS

Two different but related two-dimensional network models are used to elucidate the concept of relative permeability in simultaneous liquid-liquid flow in porous media. The first model is designed for monosized sphere packs while the other, consisting of spherical pore chambers interconnected by capillaries of variable length and radius, is intended for more general porous media. The effects of various operating conditions and physical factors—pressure gradient, wettability, surface tension, throat and pore size distributions, imbibition or drainage—on relative permeability are studied. Experimental data are also presented to confirm various aspects of the theory.     

CAPILLARY NETWORK MODELS FOR TRANSPORT IN PACKED BEDS: CONSIDERATIONS OF PORE ASPECT RATIO

The conventional analysts for the estimation of the tortuosity factor for transport in porous media is modified here to account for the effect of pore aspect ratio. Structural models of the porous medium are also constructed for calculating the aspect ratio as a function of porosity. Comparison of the model predictions with the extensive data of Currie (1960) for the effective diffusivity of hydrogen in packed beds shows good agreement with a network model of randomly oriented intersecting pores for porosities upto about 50 percent, which is the region of practical interest. The predictions based on this network model are also found to be in better agreement with the data of Currie than earlier expressions developed Tor unconsolidated and grainy media.     

A study on environment‐friendly polymer gel for water shut‐off treatments in low‐temperature reservoirs

In this study, the zirconium acetate crosslinked gel systems are studied owing to their environment‐friendly and gelation performance in low‐temperature reservoirs through rheological measurements, environmental scanning electron microscopy, and scanning electron microscopy. The effects of various parameters on the gelation properties, stability, and microstructure in bottle test and porous media were addressed. With the increase of concentrations and temperature, gelation time is reduced and gel strength is increased. In addition, the gel systems show salt tolerance and shearing resistance. The environment‐friendly gel systems have a high stability in both injection water and formation water. A three‐dimensional network structure was formed in the gel and confirmed by environmental scanning electron microscopy. The three‐dimensional gel network was also formed in porous media, which bridges across the pore throats and reduced the water permeability in the formation. This study suggests that environment‐friendly polymer gels can be used for water shut‐off treatments in low‐temperature reservoirs. © 2013 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2014 , 131, 40154.     

多孔介质干燥孔道网络模型的研究进展

建立在物质微观传输基础上的孔道网络干燥理论，通过完全离散化的方法在孔道等级上对干燥过程进行研究，描述了多孔介质内部结构参数对干燥过程的影响。介绍了建立孔道网络模型的原理和方法，阐述了基于单元体上孔道网络研究的内容及目的，综述了基于产品等级上孔道网络研究的最新进展，阐明了孔道网络模型方法对干燥理论研究的重要意义。指出，进一步提高网络模型中孔道的拓扑等价性、形状的不规则性及尺寸的相关性，探索网络构建新方法以及增加孔道网络信息量，是孔道网络干燥理论的主要发展方向，并应加强同分形、渗流理论的进一步结合。     

曲折因子与多孔介质微观结构的定性关系

采用三维网络模型对多孔介质中的两组分气体扩散过程进行了计算机模拟 .在考虑了多孔介质微观结构的前提下 ,分析了外界因素和多孔介质结构对曲折因子的影响 .结果表明 ,曲折因子仅与多孔介质的孔道连通性———网络配位数密切相关 ,而与孔道的几何尺寸、扩散温度、扩散物种及浓度无关 .经过Wicke -Kallenbach定态扩散实验验证了这一定性结论的可靠性 .     

多孔介质内流体流动的格子Boltzmann模拟

采用格子Boltzmann方法模拟多孔介质内的流动过程,通过预测渗透率,比较了单松弛模型、多松弛模型和熵格子模型在多孔介质计算中的优劣,为研究多松弛模型中各自由参数的影响,选择了12种组合进行模拟,此外,还将大涡模拟与格子Boltzmann方法相结合模拟了多孔介质内高Reynolds数下的流动及流型的转变。结果表明：单松弛模型和熵格子模型预测的渗透率随黏度逐渐增大,而多松弛模型得到的结果随黏度变化很小,另外,多松弛模型中不同松弛参数的组合对结果有较大的影响,通过比较推荐了模拟多孔介质时的最佳组合,计算结果与经验公式吻合较好。大涡模拟与多松弛模型结合较好地预测了多孔介质内流型的转变,Reynolds数越大,多孔介质内的涡越多,并且变大。     

The effective diffusivities in porous media with and without nonlinear reactions

The question of whether effective diffusivities in porous materials under reactive and nonreactive conditions are equal is addressed. Previous studies had considered the problem with first-order reactions. We study the issue with two nonlinear reactions—a second-order reaction and one governed by the Michaelis-Menten kinetics. Pore network and continuum models of porous media are utilized to estimate the effective diffusivities under reactive and nonreactive conditions. We show that the two effective diffusivities are significantly different. The difference is due to the heterogeneities of the porous material, and the fluctuations that they cause in the spatially varying local concentrations and diffusivities, and can be as large as a few orders of magnitude. Theoretical analysis of diffusion and reactions in porous media is also presented that supports the results of the simulations. In particular, it is shown that the results of pore network simulations cannot be fitted to the classical continuum equation of diffusion and reaction, and that a more complex continuum equation should be used for this purpose.     

Network model for straining dominated particle entrapment in porous media

A network model has been developed to study and describe formation damage resulting from particle entrapment in porous media by straining or size exclusion. Unlike the previous network models, this model considers the simultaneous entry of a number of particles into the network, as well as the effects of fluid flow on the particle transport path. A systematic study has been carried out on the flow and entrapment of monodispersed particles as well as particles with a size distribution through different networks. The effects of various parameters such as network size, particle size distribution and pore size distribution on the extent of formation damage, manifested by permeability reduction have been discussed in this paper. The model has also been used to determine the degree of prefiltration required to prevent damage to injection wells during water flooding. The model predictions show good agreement with experimental data for several different runs. A single parameter is used to match the exact number of pore volumes required to produce damage to the porous media. This parameter was found to be constant for the two different sandstones studied and for different concentrations of particles in the suspension. The simulation was also performed using the “random walk model” (which does not account for the fluid flow effects on particle flow) for purposes of comparison. The permeability responses predicted by this random walk model show trends that are significantly different from those observed experimentally. The network model developed in this paper has wide application in water flooding and matrix acidizing operations where diverting agents are used.     

Influence of the boundary conditions on capillary flow dynamics and liquid distribution in a porous medium

After depositing a wetting liquid onto a porous medium surface, and under the influence of the capillary pressure, the liquid is imbibed into the porous medium creating a wetted imprint. The flow within the porous medium does not cease once all the liquid is imbibed but continues as a secondary capillary flow, where the liquid flows from large pores into small pores along the liquid interface. The flow is solved using the capillary network model, and the influence of the boundary condition on the liquid distribution within the porous medium is investigated. The pores at the porous medium boundaries can be defined as open or closed pores, where an open pore is checked for the potential threshold condition for flow to take place. In contrast, the closed pore is defined as a static entity, in which the potential condition for flow to take place is never satisfied. By defining the pores at distinct porous medium boundaries as open or closed, one is able to obtain a very different liquid distribution within the porous medium. The liquid saturation profiles along the principal flow direction, ranging from constant to steadily decreasing, to the profile with a local maximum, are found numerically. It is shown that these saturation profiles are also related to the geometrical dimension that is perpendicular to the flow principal direction, and changing the boundary type from open to closed allows the liquid distribution within the porous medium to be controlled. In addition to the liquid distribution, the influence of the boundary conditions on capillary pressure and relative permeability is investigated, where both parameters are not influenced by variation of the boundary condition types. © 2011 American Institute of Chemical Engineers AIChE J, 2012     

A finite element/control volume approach to mold filling in anisotropic porous media

Mold filling in anisotropic porous media is the governing phenomena in a number of composite manufacturing processes, such as resin transfer molding (RTM) and structural reaction injection molding (SRIM). In this paper we present a numerical simulation to predict the flow of a viscous fluid through a fiber network. The simulation is based on the finite element/control volume method. It can predict the movement of a free surface flow front in a thin shell mold geometry of arbitrary shape and with varying thickness. The flow through the fiber network is modeled using Darcy's law. Different permeabilities may be specified in the principal directions of the preform. The simulation permits the permeabilities to vary in magnitude and direction throughout the medium. Experiments were carried out to measure the characteristic permeabilities of fiber preforms. The results of the simulation are compared with experiments performed in a flat rectangular mold using a Newtonian fluid. A variety of preforms and processing conditions were used to verify the numerical model.     

Evaluation of the capillary pressure curve techniques for determining pore size distribution — a network approach

Porous media are inter-connected networks of void spaces having different shapes and sizes. Attempts at developing analytical expressions describing fluid flow through them them have not been satisfactory to date for the lack of a satisfactory characterization of the structure of void spaces. The classical model, of bundle of tubes, is too simplified a model to be realistic and useful in most real situations. In the present work the flow behaviour in porous media was modelled by a network of inter-connected tubes of different sizes. An attempt was made to evaluate and synthesize various techniques of relating pore-size distribution to the capillary pressure behaviour of a medium. The validity of the “ink-bottle” effect and the techniques of Fatt and Meyer were tested by comparing the pore-size distributions obtained from capillary pressure data with the actual tube-size distribution of the network. It was seen that the existing techniques of obtaining pore-size distribution yield poor results even in the case of highly idealized network representations of porous media. Some modifications for making these techniques more realistic are discussed.The extent of inter-connections between the pores in a network was found to be a very important factor in influencing the shape of the capillary pressure (P_c) curves. This effect has been tested by allowing 2, 6, 10 and 14 inter-connections (on the average) between pores in the network, thus simulating parallel and intersecting tube models.In order to improve the reliability of results, it must be ascertained that there is, indeed, a one-to-one correspondence between pore-size distributions and capillary pressure curves.     

THE INFLUENCE OF POROUS MEDIA ON THE FLOW OF POLYMER MELTS IN CAPILLARIES†

Porous media placed in the entrance of capillaries were found to reduce the pressure drop across the capillaries (?P_c) by a factor of two or three for polystyrene. The reduction in ?P_c was found to be a function of the distance of the porous media from the capillary entrance, the type of porous media, the length of the capillary, and the rheological properties of the polymer melt. No significant reduction in ?P_c was observed for a polymer melt such as polyethyleneterephthalate (PET) which is nearly devoid of memory. The apparent shear rate for the onset of melt fracture was extended by a factor of three when polystyrene passed through the porous media before entering the capillary. No significant difference in die swell values was observed with the use of porous media in the entrance of the capillaries. The mechanism which accounts for these phenomena is believed to be associated with the break up of the entanglement network in the porous medium which temporarily changes the rheological properties of the polymer melt.