Pore Space Microstructure Evolution of Regular Sphere Packings Undergoing Compaction and Cementation

We examine the pore space structure evolution of ordered uniform sphere packs: simple cubic (SC), body centered cubic (BCC), and face centered cubic (FCC), undergoing simple diagenetic processes that reduce their pore spaces. Focus is on the occurrence of pore space microstructure changes or transitions, which are followed through their characteristic or critical pore lengths (l _c). For almost all the cubic packings undergoing either compaction or cementation there are no singularities in l _c. This is a consequence of having a single pore shape controlling flow at all stages of the process. However, this is not so for the BCC packing under cementation, for which l _c is non-monotonic exhibiting a kink at ${\phi \approx 0.1452}$ , the porosity at which the pore shape controlling flow switches to a different form and position. These results for uniform compaction/cementation complement our previous works on pore networks under random shrinkage. Kinks in l _c as porosity decreases signal pore space microstructure transitions that anticipate sudden changes in the permeability?Cporosity relation as porosity decreases. The consequences are great; clearly l _c is not a constant unless the diagenetic process is mild. A l _c function of compaction/cementation advancement should be used above a transition and a different l _c function below. For the sphere packs here, once the diagenetic process has reduced the pore space substantially, a l _c function of compaction/cementation advancement is mandatory if we are to capture all significant flow features.     

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.     

Estimation of Shale Intrinsic Permeability with Process-Based Pore Network Modeling Approach

A multi-scale pore network model is developed for shale with the process-based method (PBM). The pore network comprises three types of sub-networks: the \(\upmu \)m-scale sub-network, the nm-scale pore sub-network in organic matter (OM) particles and the nm-scale pore sub-network in clay aggregates. Process-based simulations mimic shale-forming geological processes and generate a \(\upmu \)m-scale sub-network which connects interparticle pores, OM particles and clay aggregates. The nm-scale pore sub-networks in OM and clay are extracted from monodisperse sphere packing. Nm-scale throats in OM and clay are simplified to be cylindrical and cuboid-shaped, respectively. The nm-scale pore sub-networks are inserted into selected OM particles and clay aggregates in the \(\upmu \)m-scale sub-network to form an integrated multi-scale pore network. No-slip permeability is evaluated on multi-scale pore networks. Permeability calculations verify that shales permeability keeps decreasing when nm-scale pores and throats replace \(\upmu \)m-scale pores. Soft shales may have higher porosity but similar range of permeability with hard shales. Small compaction leads to higher permeability when nm-scale pores dominate a pore network. Nm-scale pore networks with higher interconnectivity contribute to higher permeability. Under constant shale porosity, the shale matrix with cuboid-shaped nm-scale throats has lower no-slip permeability than that with cylindrical throats. Different from previous reconstruction processes, the new reconstruction process first considers the porous OM and clay distribution with PBM. The influence of geological processes on the multi-scale pore networks is also first analyzed for shale. Moreover, this study considers the effect of OM porosities and different pore morphologies in OM and clay on shale permeability.     

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.     

Petrophysical and Capillary Properties of Compacted Salt

This paper reports experimental results that demonstrate petrophysical and capillary characteristics of compacted salt. The measured data include porosity, gas permeability, pore size distribution, specific surface area, and gas-brine breakthrough and capillary pressure. Salt samples employed in the experiments were prepared by compacting sodium chloride granulates at high stresses for several hours. They represent an intermediate consolidation stage of crushed salt under in-situ conditions. The porosity and permeability of compacted salt showed similar trends to those expected in backfilled regions of waste repositories excavated in salt rock. The correlation between the measured porosity and permeability seems to be independent of the compaction parameters for the range examined in this study. The correlation also shows a different behaviour from that of rock salt. The data of all petrophysical properties show that the pore structure of compacted salt can be better characterized by fracture permeability models rather than capillary bundle ones. Simple creep tests, conducted on the fully-brine-saturated compacted salt samples, yielded similar strain rates to those obtained by a steady-state mechanical model developed from the tests on fully brine-saturated granular salt. A modified procedure is proposed for the evaluation of restored-state capillary pressure data influenced by the material creep. The characteristic parameters for the capillary behaviour of compacted salt are determined by matching the Brooks-Corey and van Genuchten models with the measured data. The Leverett functions determined with different methods agree well.     

Mixing and Mass Transfer in Multicontact Miscible Displacements

In this work, we investigate the accuracy of some physical models that are frequently used to describe and interpret dispersive mixing and mass transfer in compositional reservoir simulation. We have designed a quaternary analog fluid system (alcohol?Cwater?Chydrocarbon) that mimics the phase behavior of CO₂-hydrocarbon mixtures at high pressure and temperature. A porous medium was designed using PolyTetraFlouroEthylene (PTFE) materials to ensure that the analog oil acts as the wetting phase, and the properties of the porous medium were characterized in terms of porosity, permeability and dispersivity. Relative permeability and interfacial tension (IFT) measurements were also performed to delineate interactions between the fluid system and the porous medium. The effluent concentrations from two-component first-contact miscible (FCM) displacement experiments exhibit a tailing behavior that is attributed to imperfect sweep of the porous medium: A feature that is not captured by normal dispersion models. To represent this behavior in displacement calculations, we use dual-porosity (DP) models including mass transfer between flowing and stagnant porosities. Two 4-component two-phase displacement experiments were performed at near-miscible and multicontact miscible (MCM) conditions and the effluent concentrations were interpreted by numerical calculations. We demonstrate that the accuracy of our displacement calculations relative to the experimental observations is sensitive to the selected models for dispersive mixing, mass transfer between flowing and stagnant porosities, and IFT scaling of relative permeability functions. We also demonstrate that numerical calculations substantially agree with the experimental observations for some physical models with limited need for model parameter adjustment. The combined experimental and modeling effort presented in this work identifies and explores the impact of a set of physical mechanisms (dispersion and mass transfer) that must be upscaled adequately for field-scale displacement calculations in DP systems.     

3D Morphology and Permeability of Highly Porous Cellulosic Fibrous Material

Fibrous materials are structures whose complexity depends on the size and geometry of the fibres and on their arrangement induced by the manufacturing process. We interest here in the permeability for air of wood-based fibrous materials with high porosity on which experimental measurements are difficult to be implemented without structural modifications. In this study, we show the usefulness of 3D X-ray tomography imaging in both experimental and numerical permeability measurements. Image analysis tools derived from mathematical morphology are used to access quantitative structural parameters of the interconnected pores (porosities, pore size distributions and correlation lengths), as well as needed information on the experimental samples (cross-section area and length). A specific experimental setup and methodology is proposed to assess the permeability derived from Darcy’s law. Results are then compared to direct numerical simulation of Stokes flow carried out on 3D representative volumes of the fibrous materials.     

Ramp loading in Russian doll poroelasticity

Aporoelastic model for porous materials with a nested pore space structure is developed to represent the interstitial fluid flow in bone tissue. The nested porosity model is applied to the problem of determining the exchange of pore fluid between the vascular porosity (PV) and the lacunar-canalicular porosity (PLC) in bone tissue in a ramp loading in the case where the fluid and solid constituents are assumed to be compressible. The compressibility assumption is appropriate for hard tissues while the incompressibility assumption is appropriate for soft tissues. The influence of blood pressure in the PV is included in the analysis. A formula for the fluid that moves between the two porosities is developed. The analysis showed the coupling of the two porosities and their influence on each other and concluded that the PV pore pressure has an influence less than 3% on the PLC pore pressure while the absence of the PV pore pressure will affect the fluid exchange between the PV and PLC by less than 6% (the blood pressure range is 40-60 mmHg). Also the analysis has shown that the draining time of the PLC is inversely proportional to its permeability. The significance of the result is basic to the understanding of interstitial flow in bone tissue that, in turn, is basic to understanding of nutrient transport from the vasculature to the bone cells buried in the bone tissue and to the process of mechanotransduction by these cells.     

Prediction of Porosity and Permeability of Caved Zone in Longwall Gobs

The porosity and permeability of the caved zone (gob) in a longwall operation impact many ventilation and methane control related issues, such as air leakage into the gob, the onset of spontaneous combustion, methane and air flow patterns in the gob, and the interaction of gob gas ventholes with the mining environment. Despite its importance, the gob is typically inaccessible for performing direct measurements of porosity and permeability. Thus, there has always been debate on the likely values of porosity and permeability of the caved zone and how these values can be predicted. This study demonstrates a predictive approach that combines fractal scaling in porous medium with principles of fluid flow. The approach allows the calculation of porosity and permeability from the size distribution of broken rock material in the gob, which can be determined from image analyzes of gob material using the theories on a completely fragmented porous medium. The virtual fragmented fractal porous medium so generated is exposed to various uniaxial stresses to simulate gob compaction and porosity and permeability changes during this process. The results suggest that the gob porosity and permeability values can be predicted by this approach and the presented models are capable to produce values close to values documented by other researchers.     

Experimental study of air natural convection on metallic foam-sintered plate

The natural convection on metallic foam-sintered plate at different inclination angles was experimentally studied. Seven copper foam samples with different pore densities (10–40 pore per inch), porosities (0.90–0.95), and aspect ratios (the ratio of foam thickness to sample length, 0.1–0.5) were measured at inclination angles of 0° (vertical orientation), 15°, 30°, 45°, 60°, 75°, 90° (horizontal orientation). The heat conduction and natural convection inside the foam both contributed to the total heat transfer. Although, the form and viscous drag, which are influenced by permeability and viscous friction in the thermal boundary layer respectively, tend to suppress the natural convection, the heat transfer was finally enhanced by the foam sintered surface due to large surface area extension. Optimum inclination range 60–75° corresponding to maximum average Nu number was found in the heat flux range of 600–1800 W/m². The sintered foam surface with lower porosity and pore density was recommended for heat transfer enhancement. Particularly, the sample with porosity 0.9, pore density of 10 PPI, aspect ratio of 0.5 offered the highest average Nu number among the studied samples. An empirical correlation for modified Nusselt number at isoflux boundary condition considering the foam morphology parameter and inclination angle was proposed within deviation ±15% between the correlation and the experimental data.     

Mean field calculation of effective permeability based on fractal pore space

Effective permeability for porous rocks is calculated using mean field theory. We make two simplifying assumptions about the internal conductances in a network representation of the porous rock: (i) Pore space is characterized by a uniform fractal scaling; (ii) the internal conductances depend only on the characteristic pore sizes. Within these approximations, it is possible to derive a simple probability density for the internal conductances which is used for calculating effective permeability. Good agreement between calculations and experimental data of permeability vs. porosity is achieved.     

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.     

Porosity–Permeability Relations for Evolving Pore Space: A Review with a Focus on (Bio-)geochemically Altered Porous Media

Reactive transport processes in a porous medium will often both cause changes to the pore structure, via precipitation and dissolution of biomass or minerals, and be affected by these changes, via changes to the material’s porosity and permeability. An understanding of the pore structure morphology and the changes to flow parameters during these processes is critical when modeling reactive transport. Commonly applied porosity–permeability relations in simulation models on the REV scale use a power-law relation, often with slight modifications, to describe such features; they are often used for modeling the effects of mineral precipitation and/or dissolution on permeability. To predict the reduction in permeability due to biomass growth, many different and often rather complex relations have been developed and published by a variety of authors. Some authors use exponential or simplified Kozeny–Carman relations. However, many of these relations do not lead to fundamentally different predictions of permeability alteration when compared to a simple power-law relation with a suitable exponent. Exceptions to this general trend are only few of the porosity–permeability relations developed for biomass clogging; these consider a residual permeability even when the pore space is completely filled with biomass. Other exceptions are relations that consider a critical porosity at which the porous medium becomes impermeable; this is often used when modeling the effect of mineral precipitation. This review first defines the scale on which porosity–permeability relations are typically used and aims at explaining why these relations are not unique. It shows the variety of existing approaches and concludes with their essential features.     

Generic strength model for dry jointed rock masses

A new nonlinear thermo-mechanical model for heavily jointed rock masses is presented. The model describes poroelasticity, shear-enhanced compaction and brittle–ductile transition in dry porous rocks. The key input parameters of the model, such as elastic moduli, tensile and compressive strength are expressed as functions of the reference porosity of the rock. These functions are based on empirical data for limestones and sandstones and assume that the medium is isotropic. The effect of joints is modeled by scaling down the key model parameters. The scaling rules are found with the help of explicit numerical modeling of randomly jointed 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.     

Comparison between oxide-reduced and water-atomized copper powders used in making sintered wicks of heat pipe

Oxide-reduced copper powder can be produced efficiently at low cost. The volume shrinkage, porosity, maximum pore size, permeability and thermal conductivity of wicks sintered from two oxide-reduced （OR） powders were compared with one from water-atomized （WA） powder. The green specimens were sintered at temperatures from 800 to 1000 ℃ in a tube furnace under a reduction stream of 10% hydrogen and 90% argon. The results show that the property variations of OR - 100 and WA wicks due to porosity changes have a similar tendency and range. Nine hundred degree celsius is a recommended sintering temperature for producing ideal wicks for use in heat pipes. A smaller maximum pore size can be obtained by increasing the green density.2007 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V.     

高温高压气体在低含水率介质中迁移的数值模拟

针对气液两相非等温渗流模型高度非线性的特点,发展了适宜的数值离散方法。根据相态转换准则和控制方程的性质,采用最低饱和度法简化算法。空间离散方面,使用有限体积法;时间离散方面,设计了一套包含合理求解顺序的Picard迭代法,解决了方程组强耦合的问题。利用上述数值方法对高温高压气体的迁移行为进行数值模拟,证明了气体在低含水率介质和等效孔隙度的干燥介质内的运动基本一致,并分析了空腔内的气液相态转变过程。在此基础上,研究了多孔介质孔隙度和渗透率对气体压强演化和示踪气体迁移的影响。研究表明,孔隙度越小（相同渗透率）、渗透率越高（相同孔隙度）,示踪气体的迁移距离越远,并给出了估算不同孔隙度和渗透率下迁移距离的半经验公式。     

A Statistical Correlation Between Permeability,Porosity, Tortuosity and Conductance

A statistical evaluation of 13,000 numerical simulations of random porous structures is used to establish a correlation between permeability, porosity, tortuosity and conductance. The random structures are generated with variable porosities, and parameters such as the permeability and the tortuosity are determined directly from the structures. It is shown that the prevalent definition of tortuosity, as the ratio of length of the real flow path to the projected path in the overall flow direction, does not correlate with permeability in the general case. Also, the correlation between the conductance of the medium, as an indicator of the accessible cross section of a flow path and permeability is no more reliable than the permeability–porosity correlation. However, if the definition of tortuosity is corrected using the cross-sectional variations, the resulting parameter (i.e., the minimum-corrected tortuosity) has a reliable correlation with permeability and can be used to estimate permeability with an acceptable error for most of the simulations of the random porous structures. The feasibility of extending the conclusions from 2-dimensional to 3-dimensional configurations and the numerical percolation thresholds for random structures are also discussed.