Pore Network Modelling of Electrical Resistivity and Capillary Pressure Characteristics

In order to model petrophysical properties of hydrocarbon reservoir rocks, the underlying physics occurring in realistic rock pore structures must be captured. Experimental evidence showing variations of wetting occurring within a pore, and existence of the so-called 'non-Archie' behaviour, has led to numerical models using pore shapes with crevices (for example, square, elliptic, star-like shapes, etc.). This paper presents theoretical derivations and simulation results of a new pore space network model for the prediction of petrophysical properties of reservoir rocks. The effects of key pore geometrical factors such as pore shape, pore size distribution and pore co-ordination number (pore connectivity) have been incorporated into the theoretical model. In particular, the model is used to investigate the effects of wettability and saturation history on electrical resistivity and capillary pressure characteristics. The petrophysical characteristics were simulated for reservoir rock samples. The use of the more realistic grain boundary pore (GBP) shape allows simulation of the generic behaviour of sandstone rocks, with various wetting scenarios. The predictions are in close agreement with electrical resistivity and capillary pressure characteristics observed in experiments.     

Equivalence Between Semi-empirical and Population-Balance Foam Models

Effective material parameters for diffusion and elastic deformation are calculated for porous materials using a continuum theory-based superposition procedure. The theory that is limited to two-dimensional cases, requires that the pores are sufficiently sparse. The method leads to simple manual calculations that can be performed by, e.g. hospital staff at clinical diagnoses of bone deceases that involve increasing levels of porosity. An advantage is that the result relates to the bone material permeability and stiffness instead of merely pore densities. The procedure uses precalculated pore shape factors and exact size scaling. The remaining calculations do not require any knowledge of the underlying field methods that are used to compute the shape factors. The paper establishes the upper limit for the pore densities that are sufficiently sparse. A cross section of bovine bone is taken as an example. The superposition procedure is evaluated against a full scale finite element calculation. The study compares the pore induced change of the diffusion coefficient and elastic modulus. The predictions differ between superposition and full scale calculations with 0.3% points when pore contribution to the diffusion constant is 3–7%, and 0.7% points when the pore contribution to the modulus of elasticity is 4.5–5%. It is uncertain if the error is in the superposition method, which is exact for small pore densities, while the full scale finite model is not.     

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.     

饱和多孔介质化学-热-弹性模型及应用

本文基于热局部非平衡(LTNE)条件和加权平均温度概念,并假设孔隙流体由溶质和溶剂两组元组成,对页岩（饱和多孔介质）,推导给出了一种LTNE条件下的化学-热-弹性模型,同时讨论了耦合方程组的解耦求解问题．作为模型的应用,考虑无限大平面含一圆形孔的情况,研究了冷/热对流以及溶质摩尔分数突变边界条件下圆孔附近的孔隙压力和化-热应力问题,用Laplace变换得到了平面轴对称情况下有关力学变量的表达式．数值分析了圆孔边界上冷/热对流的Biot数和溶质摩尔分数改变量对圆孔附近孔隙压力和化-热应力的影响．结果表明：在Biot数为中等值(1~5)范围内,LTNE效应是非常明显的;化学作用对孔隙压力和固相应力的影响不可忽视．     

Pore Network Analysis of Resistivity Index for Water-Wet Porous Media

Pore network analysis is used to investigate the effects of microscopic parameters of the pore structure such as pore geometry, pore-size distribution, pore space topology and fractal roughness porosity on resistivity index curves of strongly water-wet porous media. The pore structure is represented by a three-dimensional network of lamellar capillary tubes with fractal roughness features along their pore-walls. Oil-water drainage (conventional porous plate method) is simulated with a bond percolation-and-fractal roughness model without trapping of wetting fluid. The resistivity index, saturation exponent and capillary pressure are expressed as approximate functions of the pore network parameters by adopting some simplifying assumptions and using effective medium approximation, universal scaling laws of percolation theory and fractal geometry. Some new phenomenological models of resistivity index curves of porous media are derived. Finally, the eventual changes of resistivity index caused by the permanent entrapment of wetting fluid in the pore network are also studied.Resistivity index and saturation exponent are decreasing functions of the degree of correlation between pore volume and pore size as well as the width of the pore size distribution, whereas they are independent on the mean pore size. At low water saturations, the saturation exponent decreases or increases for pore systems of low or high fractal roughness porosity respectively, and obtains finite values only when the wetting fluid is not trapped in the pore network. The dependence of saturation exponent on water saturation weakens for strong correlation between pore volume and pore size, high network connectivity, medium pore-wall roughness porosity and medium width of the pore size distribution. The resistivity index can be described succesfully by generalized 3-parameter power functions of water saturation where the parameter values are related closely with the geometrical, topological and fractal properties of the pore structure.     

Hydrodynamic Characterization of Nickel Metal Foam,Part 2: Effects of Pore Structure and Permeability

In this article, the structural characterization of chemical vapor deposition (CVD) nickel metal foam is presented. Scanning electron microscope and post image processing were used to carefully analyze the surface of the nickel metal foams. Data on foam unit cell, ligament thickness, projected pore diameter, and averaged porosity was obtained. Unit cell and projected pore diameters of CVD nickel metal foam possess Gaussian-like distribution. Characteristics of pore structure and its effect on permeability in Darcian flow regime were analyzed. The relations between the permeability, pore size, and porosity are presented. Present and previous data are compared with these relations. Measurement results indicate that the permeability or the viscous conductivity of the CVD processed metal foam is affected not only by the pore size, and porosity but also by the ligament structure.     

An Analysis of Three-Phase Pore Occupancies and Relative Permeabilities in Porous Media with Variable Wettability

In three-phase flow, the macroscopic constitutive relations of capillary pressure and relative permeability as functions of saturation depend in a complex manner on the underlying pore occupancies. These three-phase pore occupancies depend in turn on the interfacial tensions, the pore sizes and the degree of wettability of the pores, as characterised by the cosines of the oil–water contact angles. In this work, a quasi-probabilistic approach is developed to determine three-phase pore occupancies in media where the degree of wettability varies from pore to pore. Given a set of fluid and rock properties, a simple but novel graphical representation is given of the sizes and oil–water contact angles underlying three-phase occupancies for every allowed combination of capillary pressures. The actual phase occupancies are then computed using the contact angle probability density function. Since a completely accessible porous medium is studied, saturations, capillary pressures, and relative permeabilities are uniquely related to the pore occupancies. In empirical models of three-phase relative permeability it is of central importance whether a phase relative permeability depends only on its own saturation and how this relates to the corresponding two-phase relative permeability (if at all). The new graphical representation of pore sizes and wettabilities clearly distinguishes all three-phase pore occupancies with respect to these saturation-dependencies. Different types of saturation-dependencies may occur, which are shown to appear in ternary saturation diagrams of iso-relative permeability curves as well, thus guiding empirical approaches. However, for many saturation combinations three-phase and two-phase relative permeabilities can not be linked. In view of the latter, the present model has been used to demonstrate an approach for three-phase flow modelling on the basis of the underlying pore-scale processes, in which three-phase relative permeabilities are computed only along the actual flow paths. This process-based approach is used to predict an efficient strategy for oil recovery by simultaneous water-alternating-gas (SWAG) injection.     

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.      

基于粗粒化分子动力学的自由水与水化硅酸钙孔隙水冻结模拟

水化硅酸钙是水泥基材料的主要水化产物,其孔隙内的水分是影响水泥基材料抗冻性的主要因素。本文基于粗粒化分子动力学方法研究水化硅酸钙孔隙水的冻结机制,针对水的粗粒化P₄粒子和水化硅酸钙胶体颗粒,建立了水化硅酸钙孔隙水的冻结模型。根据此模型计算了不同孔径孔隙水冰点,分析了水泥基材料孔径孔隙在冻融破坏中的危害程度;模拟得到了水化硅酸钙孔隙内水的冻结分布特征和密度分布特征。研究工作表明,本文建立的模型有效提高了分子动力学模拟水化硅酸钙孔隙水冻结问题的规模,为后续进行水泥基材料的冻融破坏分析提供了研究基础。     

Characterization of porous media - pore level

Pore size and pore size distribution are defined, 1-D pore structure models, 2-D and 3-D network models of pore structure are reviewed. Simulation of capillary pressure and relative permeability curves with the help of network models of pore structure is discussed. Pore structure determination from serial sections is outlined. Phase immobilization technique is reviewed.     

Filtration in a Porous Granular Medium: 1. Simulation of Pore-Scale Particle Deposition and Clogging

This paper presents a numerical model for simulating the pore-scale transport and infiltration of dilute suspensions of particles in a granular porous medium under the action of hydrodynamic and gravitational forces. The formulation solves the Stokes’ flow equations for an incompressible fluid using a fixed grid, multigrid finite difference method and an embedded boundary technique for modeling particle–fluid coupling. The analyses simulate a constant flux of the fluid suspension through a cylindrical model pore. Randomly generated particles are collected within the model pore, initially through contact and attachment at the grain surface (pore wall) and later through mounding close to the pore inlet. Simple correlations have been derived from extensive numerical simulations in order to estimate the volume of filtered particles that accumulate in the pore and the differential pressure needed to maintain a constant flux through the pore. The results show that particle collection efficiency is correlated with the Stokes’ settling velocity and indirectly through the attachment probability with the particle–grain surface roughness. The differential pressure is correlated directly with the maximum mound height and indirectly with particle size and settling velocity that affect mound packing density. Simple modification factors are introduced to account for pore length and dip angle. These parameters are used to characterize pore-scale infiltration processes within larger scale network models of particle transport in granular porous media in a companion paper. Articlenote: Currently at GZA GeoEnvironmental Inc., 1 Edgewater Drive, Norwood, MA 02062, U.S.A.     

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.     

Plastic potential and yield function of porous materials with aligned and randomly oriented spheroidal voids

Based on an energy approach, the plastic potential and yield function of a porous material containing either aligned or randomly oriented spheroidal voids are developed at a given porosity and pore shape. The theory is applicable to both elastically compressible and incompressible matrix and, it is proved that, in the incompressible case, the theory with spherical and aligned spheroidal voids also coincides with Ponte Castaneda's bounds of the Hashin-Shtrikman and Willis types, respectively. Comparison is also made between the present theory and those of Gurson and Tvergaard, with a result giving strong overall support of this new development. For the influence of pore shape, the yield function and therefore the stress-strain curve of the isotropic porous material are found to be stiffest when the voids are spherical, and those associated with other pore shapes all fall below these values, the weakest one being caused by the disc-shaped voids. The transversely isotropic nature of the yield function and stress-strain curves of a porous material containing aligned pores are also demonstrated as a function of porosity and pore shape, and it is further substantiated with a comparison with an exact, local analysis when the void shape becomes cylindrical.     

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.     

受移动简谐力作用的多孔弹性半平面问题

研究了匀速移动的振动荷载作用下半无限多孔饱和固体中产生的应力和孔隙水压力．应用Fourier变换求解该问题的控制偏微分方程,考虑了荷载的移动速度及振动频率对多孔饱和固体中应力与孔隙水压力的影响,并与相应的弹性介质的解答进行了比较．结果显示多孔饱和半平面中应力和孔隙水压力随荷载的移动速度与振动频率的增加而增大,多孔饱和固体在移动荷载下的动力响应与相应的单相弹性固体的动力响应有较大的差别。     

不同孔隙率及孔径泡沫铝的力学与吸能特性研究

对三种孔径两种孔隙率共六种泡沫铝试件进行了静态压缩试验,发现对于中孔隙率材料,孔隙率对材料力学性能和吸能性能仍有明显影响:孔隙率越大,其吸能能力越强,但是屈服极限越小.孔径对材料吸能能力也有一定影响.同时,试验中还发现,对于孔隙率为55%的材料,当孔径≤1mm时,泡沫铝材料的孔径对其力学性能及吸能性能影响甚小.     

Comparison of the time-temperature superposition of the relaxation modulus and time-to-break of preswollen gels

This paper describes the work carried out to investigate the pore pressures occurring in secondary consolidation. A theoretical approach and an experimental technique was developed in order to conduct the study. By considering compression to occur only due to water leaving the soil it was possible to derive an expression for the dissipation of pore pressure in the secondary phase. By further simplified assumptions which are based on experimental observations, the above general solution was reduced to a simple formula which predicted the observed behaviour of pore water pressures during secondary consolidation.     

超临界CO_2作用后煤微观结构与渗透率变化规律实验研究

在宏观超临界CO_2增透实验基础上进行微观成像实验,提取煤微观孔隙特征,自编Matlab程序,计算孔隙率,得到煤各微区孔隙率和渗透率矩阵,绘制孔隙率和渗透率等值线图。结果表明:增透实验前后煤微观结构差别显著,经超临界CO_2作用后,煤微观孔隙充分发育,孔隙的数量、尺寸明显增加,孔隙率是增透前的9.11倍;随着孔隙压力的增大,煤中粒间孔隙数量增多,孔隙之间的连通性增加,孔隙率呈指数增大的趋势,煤体各微区孔隙率等值线密集程度增加;煤体的渗透率随着孔隙率的增加呈正指数递增的趋势,且随着孔隙压力的增加,渗透率呈指数递增的变化规律,渗透率等值线的密集程度增加,宏微观实验结果是一致的,随着注入超临界CO_2孔隙压力的增加,煤微观孔隙结构的发育程度提高,为煤层气的运移提供更多的通道,有效提高了煤体的渗透性。