Random walk on graphs: An application to the double diffusivity model

Preliminary but interesting and definite results are given on the application of graph theory concepts (random walk on graphs) to the double diffusivity theory proposed by Aifantis in the late 70s to model transport in media with high diffusivity paths such as metal polycrystals with a continuous distribution of grain boundaries possessing much higher diffusivity than the bulk, as well as in nanopolycrystals for which it has been shown recently that the double diffusivity model fits experimental observations. The new information provided by employing the graph theory tool is concerned with certain restrictions and relations that the phenomenological coefficients, entering in the coupled partial differential equations of double diffusivity, should satisfy depending on the topology and related details of the graph model adopted.     

Performance of slotted pores in particle manufacture using rotating membrane emulsification

This paper addresses the use of different slotted pores in rotating membrane emulsification technology. Pores of square and rectangular shapes were studied to understand the effect of aspect ratio （1-3.5） and their orientation on oil droplet formation. Increasing the membrane rotation speed decreased the droplet size, and the oil droplets produced were more uniform using slotted pores as compared to circular geometry. At a given rotation speed, the droplet size was mainly determined by the pore size and the fluid velocity of oil through the pore （pore fluid velocity）. The ratio of droplet diameter to the equivalent diameter of the slotted pore increased with the pore fluid velocity. At a given pore fluid velocity and rotation speed, pore orientation significantly influences the droplet formation rate： horizontally disposed pores （with their longer side perpendicular to the membrane axis） generate droplets at double the rate of vertically disposed pores. This work indicates practical benefits in the use of slotted membranes over conventional methods.     

Performance of slotted pores in particle manufacture using rotating membrane emulsification

This paper addresses the use of different slotted pores in rotating membrane emulsification technology.Pores of square and rectangular shapes were studied to understand the effect of aspect ratio (1-3.5) and their orientation on oil droplet formation.Increasing the membrane rotation speed decreased the droplet size,and the oil droplets produced were more uniform using slotted pores as compared to circular geometry.At a given rotation speed,the droplet size was mainly determined by the pore size and the fluid velocity of oil through the pore (pore fluid velocity).The ratio of droplet diameter to the equivalent diameter of the slotted pore increased with the pore fluid velocity.At a given pore fluid velocity and rotation speed,pore orientation significantly influences the droplet formation rate: horizontally disposed pores (with their longer side perpendicular to the membrane axis) generate droplets at double the rate of vertically disposed pores.This work indicates practical benefits in the use of slotted membranes over conventional methods.     

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.     

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.     

A characterization of the coupled evolution of grain fabric and pore space using complex networks: Pore connectivity and optimized flows in the presence of shear bands

A framework for the multiscale characterization of the coupled evolution of the solid grain fabric and its associated pore space in dense granular media is developed. In this framework, a pseudo-dual graph transformation of the grain contact network produces a graph of pores which can be readily interpreted as a pore space network. Survivability, a new metric succinctly summarizing the connectivity of the solid grain and pore space networks, measures material robustness. The size distribution and the connectivity of pores can be characterized quantitatively through various network properties. Assortativity characterizes the pore space with respect to the parity of the number of particles enclosing the pore. Multiscale clusters of odd parity versus even parity contact cycles alternate spatially along the shear band: these represent, respectively, local jamming and unjamming regions that continually switch positions in time throughout the failure regime. Optimal paths, established using network shortest paths in favor of large pores, provide clues on preferential paths for interstitial matter transport. In systems with higher rolling resistance at contacts, less tortuous shortest paths thread through larger pores in shear bands. Notably the structural patterns uncovered in the pore space suggest that more robust models of interstitial pore flow through deforming granular systems require a proper consideration of the evolution of in situ shear band and fracture patterns – not just globally, but also inside these localized failure zones.     

Diffusion in pore fractals: A review of linear response models

A major aspect of describing transport in heterogeneous media has been that of relating effective diffusivities to the topological properties of the medium. While such effective transport coefficients may be useful for mass fractals or under steady state conditions, they are not adequate under transient conditions for self-similar pore fractal media. In porous formations without scale, diffusion is anomalous with the mean-squared displacement of a particle proportional to time raised to a fractional exponent less than unity. The objective of this review is to investigate the nature of the laws of diffusion in fractal media using the framework of linear response theory of nonequilibrium statistical mechanics. A Langevin/Fokker-Planck approach reveals that the particle diffusivity depends on its age defined as the time spent by the particle since its entry into the medium. An analysis via generalized hydrodynamics describes fractal diffusion with a frequency and wave number dependent diffusivity.     

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.     

页岩有机质纳米孔隙气体吸附与流动规律研究

页岩储层孔隙结构复杂, 气体赋存方式多样. 有机质孔隙形状对受限空间气体吸附和流动规律的影响尚不明确, 导致难以准确认识页岩气藏气体渗流机理. 为解决该问题, 本文首先采用巨正则蒙特卡洛方法模拟气体在不同形状有机质孔隙(圆形孔隙、狭长孔隙、三角形孔隙、方形孔隙)内吸附过程, 发现不同形状孔隙内吸附规律符合朗格缪尔单层吸附规律, 分析了绝对吸附量、过剩吸附浓量、气体吸附参数随孔隙尺寸、压力的变化, 研究了孔隙形状对气体吸附的影响. 在明确不同形状有机质孔隙内气体热力学吸附规律基础上, 建立不同形状有机质孔隙内吸附气表面扩散数学模型和考虑滑脱效应的自由气流动数学模型, 结合分子吸附模拟结果研究了不同孔隙形状、孔隙尺寸有机质孔隙内吸附气流动与自由气流动对气体渗透率的贡献. 结果表明, 狭长孔隙内最大吸附浓度和朗格缪尔压力最高, 吸附气表面扩散能力最弱. 孔隙半径5 nm以上时, 吸附气表面扩散对气体渗透率影响可忽略. 本文研究揭示了页岩气藏实际生产过程中有机质孔隙形状对页岩气吸附和流动能力的影响机制.      

Application of Fractional Differential Equations for Modeling the Anomalous Diffusion of Contaminant from Fracture into Porous Rock Matrix with Bordering Alteration Zone

Solute diffusion from a fracture into a porous rock with an altered zone bordering the fracture is modeled by a system of two diffusion equations (one for the altered zone and another for the intact porous matrix) with different coefficients of effective diffusivity. Since experimental studies of diffusion into rock samples with altered zones indicate that mathematical models of diffusion based on Fick’s law do not adequately describe the concentration field in a sample, fractional order diffusion equations are chosen in this study for modeling the anomalous mass transport in the rocks. In the case of significantly higher porosity of the altered zone (e.g., this is typical for carbonates) the effective diffusivity here can be much higher than the effective diffusivity of non-altered rocks. By introducing a small parameter that is the ratio of effective diffusivities in the non-altered and altered regions and applying the technique of perturbations, approximate analytical solutions for concentrations in the altered zone bordering the fracture and in the intact surrounding rocks are obtained. Based on these solutions, different regimes of diffusion into the rocks with different physical properties are modeled and analyzed. It is shown that, using experimentally obtained data, the orders of the fractional derivatives in the differential equations can be readily calibrated for the every specific rock.     

基于规则网络的碳酸盐岩多尺度网络模型构建方法研究

碳酸盐岩油藏非均质性强,孔隙大小变化可达好几个数量级,描述碳酸盐岩油藏多尺度孔隙特征具有重要意义.本文首先基于三维规则网络模型建立了不同物理尺寸的溶洞网络、大孔隙网络和微孔隙网络;然后提出一种耦合算法,以溶洞网络为基础,通过添加适当比例的大孔隙和微孔隙,构建出碳酸盐岩多尺度网络模型;最后对比分析了各网络模型的几何性质、拓扑性质和绝对渗透率.结果表明,碳酸盐岩多尺度网络模型能够同时描述不同尺度孔隙的几何和拓扑特征;且相比各单一尺度的孔隙网络模型,多尺度网络模型有着较高的绝对渗透率,这是由于各尺度孔隙之间的相互连通极大地提高了网络的整体连通性和流动能力,为碳酸盐岩油藏微观渗流模拟提供了重要的研究平台.     

Contribution of the Dusty Gas Model to Permeability/Diffusion Tests on Partially Saturated Clay Rocks

Gas transfer experiments on claystone and numerical simulations have been conducted to enhance the knowledge of gas transport in nuclear waste repositories in the Callovo-Oxfordian clay formation in Bure, France. Laboratory Gas transfer experiments were performed with a specific device dedicated to very low permeability measurement (10^?23 to 10^?20 m²). Experiments were performed on both dry and close to saturation claystone. The Dusty Gas Model, based on multi-component gas transfer equations with Knudsen diffusion, was used to describe the experimental results. The parameters obtained are the effective permeability, the Knudsen diffusion (Klinkenberg effect) and molecular diffusion coefficients and the porosity accessible to gas. Numerical simulations were carried with various boundary conditions and for different gases (helium vs hydrogen) and were compared with experiments to test the reliability of the model parameters and to better understand the mechanisms involved in clays close to saturation. The numerical simulation fitted the experimental data well whereas simpler models cannot describe the complexity of the Knudsen/Klinkenberg effects. Permeabilities lie between 10^?22 and 10^?20 m². Claystones close to saturation have an accessible porosity to gas transfer that is lower than 0.1?C1% of the porosity. Analysis of the Klinkenberg effect suggests that this accessible pore network should be made of 50?C200?nm diameter pores. It represents pore networks accessible at capillary pressure lower than 4?MPa.     

Investigation of Various Pressure Transient Techniques on Permeability Measurement of Unconventional Gas Reservoirs

Chloride ingress into concrete is a major cause for material degradation, such as cracking due to corrosion-induced steel reinforcement expansion. Corresponding transport processes encompass diffusion, convection, and migration, and their mathematical quantification as a function of the concrete composition remains an unrevealed enigma. Approaching the problem step by step, we here concentrate on the diffusivity of cement paste, and how it follows from the microstructural features of the material and from the chloride diffusivity in the capillary pore spaces. For this purpose, we employ advanced self-consistent homogenization theory as recently used for permeability upscaling, based on the resolution of the pore space as pore channels being oriented in all space directions, resulting in a quite compact analytical relation between porosity, pore diffusivity, and the overall diffusivity of the cement paste. This relation is supported by experiments and reconfirms the pivotal role that layered water most probably plays for the reduction of the pore diffusivity, with respect to the diffusivity found under the chemical condition of a bulk solution.     

Fractal and superdiffusive transport and hydrodynamic dispersion in heterogeneous porous media

We review and discuss diffusion and hydrodynamic dispersion in a heterogeneous porous medium. Two types of heterogeneities are considered. One is percolation disorder in which a fraction of the pores do not allow transport to take place at all. In the other type, the permeabilities of various regions of the pore space are fractally distributed with long-range correlations. Both systems give rise to unusual transport in which the mean square displacement <r ²(t)> of a particle grows nonlinearly with time. Depending on the heterogeneities and the mechanism of diffusion and disperison, we may havefractal transport in which <r ²> growsslower than linearly with time, orsuperdiffusive transport in which <r ²> growsfaster than linearly with time. We show that percolation models can give rise to both types of transport with scale-dependent transport coefficients such as diffusivity and dispersion coefficients, which are consistent with many experimental observations.     

Effective Diffusivity of Porous Materials with Microcracks: Self-Similar Mean-Field Homogenization and Pixel Finite Element Simulations

We investigate the influence of distributed microcracks on the overall diffusion properties of a porous material using the self-similar cascade continuum micromechanics model within the framework of mean-field homogenization and computational homogenization of diffusion simulations using a high-resolution pixel finite element method. In addition to isotropic, also anisotropic crack distributions are considered. The comparison of the results from the cascade continuum micromechanics model and the numerical simulations provides a deeper insight into the qualitative transport characteristics such as the influence of the crack density on the complexity and connectivity of crack networks. The analysis shows that the effective diffusivity for a disordered microcrack distribution is independent of the absolute length scale of the cracks. It is observed that the overall effective diffusivity of a microcracked material with the microcracks oriented in the direction of transport is not necessarily higher than that of a material with a random orientation of microcracks, independent of the microcrack density.     

Investigation of fine scale structure of turbulent and molecular diffusion in coaxial jets of He/CO2 in air by LDA and Rayleigh scattering

This paper revisits the important issue of differential diffusion and provides new experimental results and subsequent analysis that attempts to quantify the relationship between molecular diffusion, turbulent diffusion and their mutual interference in non-reacting axisymmetric coaxial jets of variable Reynolds number. The reported investigation has been focused on the analysis of molecular diffusion of a He/CO₂ mixture in air by combining line imaging of Rayleigh scattering and laser Doppler anemometry (LDA) to determine length scales associated with differential diffusion and turbulent transport. Line imaging Rayleigh scattering was performed applying the index-matching method with a mixture of two gaseous species having scattering cross-sections respectively lower and higher of that of air and the cross-section of the mixture identical to that of the co-flowing air. Any measured variation in scattering intensity is therefore due to differential diffusion between the two species. Instantaneous and ensemble averaged line profiles of Rayleigh scattering intensity are presented and a characteristic length scale associated with differential diffusion is deduced. Autocorrelation analysis is applied to obtain the characteristic scale of differential diffusion fluctuations and the integral length scales of velocity fluctuations, as measured by LDA. Theoretical information from the literature is used in relating these scales to the molecular and turbulent diffusion coefficients, assuming homogeneous and isotropic turbulence, and the ratio of molecular to turbulent diffusivity is estimated as a function of the Reynolds number. The results confirm that the average contribution of molecular diffusion to the effective diffusivity into the air stream progressively reduces when the turbulence level increases. They also suggest that, at higher Re, the differential diffusion remains significant down to the scalar dissipation length scale, and could influence mixing at the molecular level and thus chemical reactions.     

Heat transfer characteristics in a horizontal swirling fluidized bed

An innovative horizontal swirling fluidized bed (HSFB) with a rectangular baffle in the center of an air distributor and three layers of horizontal secondary air nozzles located at each corner of fluidized bed was developed. Experiments on heat transfer characteristics were conducted in a cold HSFB test model. Heat transfer coefficients between immersed tubes and bed materials in the HSBF were measured with the help of a fast response heat transfer probe. The influences of fluidization velocity, particle size of bed materials, measurement height, probe orientation, and secondary air injection, etc. on heat transfer coefficients were intensively investigated. Test results indicated that heat transfer coefficients increase with fluidization velocity, and reach their maximum values at 1.5-3 times of the minimum fluidization velocity. Heat transfer coefficients are variated along the circumference of the probe, and heat transfer coefficients on the leeward side of the probe are larger than that on the windward side of the probe. Heat transfer coefficients decrease with increasing of measurement height; heat transfer coefficients of the longitudinal probe are larger than that of the transverse probe. The proper secondary air injection and particle size of bed materials can generate a preferred hydrodynamics in the dense zone and enhance heat transfer in a HSFB.     

Porosity dependence of strength in brittle solids

The effect of pore size and pore volume fraction on strength in brittle solids is evaluated. The analysis considers that the strength degradaton of a solid containing a large number of spherical pores is due to a strong effect of porosity on Young's modulus. Each pore is assumed to possess radial or annular flaws emanating from the pore surface whose lengths are considered to be independent of pore size. The effect of stress concentration induced by the presence of the pore is included in the equation for strength through the Young's modulus dependence of porosity originally developed using the concept of crack opening displacement. It is shown that the strength of a solid containing spherical pores is controlled by the pore size, pore volume fracton and the radial (or annular) crack size to pore size ratio. Predicted variation of strength with pore volume fraction is tested against experimental data for glass and polycrystalline alumina.     

厚壁胞体多孔结构接触强度研究

选用H59黄铜为材料,制备了含圆形单孔、双孔、三孔以及四孔的圆柱体试样,采用型号为YE-600的液压式压力试验机对试样进行压缩试验,探讨了布孔方式对厚壁胞体多孔结构接触强度的影响;并用有限元法探讨了多孔结构的破坏机理.结果表明:双孔材料水平布孔方式优于垂直布孔方式;三孔材料斜正三角形布孔方式优于正三角形布孔方式;四孔材料正方形布孔方式优于菱形布孔方式;微孔分布半径的增大和微孔个数的增加均有助于提高厚壁胞体多孔结构的接触强度;裂纹萌生于微孔边缘且离初始接触点最近的地方,它由最大拉伸主应力引起并沿着微孔拉应力集中区进行扩展.