Dual models of pore spaces

We present a model for pore spaces that consists of two parts related by duality: (1) a decomposition of an open polyhedral pore space into open contractible pore bodies separated by relatively open interfaces and (2) a pore network that is homotopy equivalent to the pore space. The dual model is unique and free of parameters, but it relies on regularity conditions for the pore space. We show how to approximate any pore space by the interior of a polyhedral complex such that the regularity conditions are fulfilled. Thus, we are able to calculate the dual model from synthetic porous media and images of real porous media. The pore bodies are unions of relatively open Delaunay cells with respect to the corners of the pore boundary, and the pore network consists of certain at most two-dimensional (2D) Voronoi cells with respect to the corners of the pore boundary. The pore network describes the neighborhood relations between the pore bodies. In particular, any relatively open 2D Delaunay face f separating two pore bodies has a unique (relatively open) dual network edge. In our model, f is a pore throat only if it is hit by its dual network edge. Thus, as opposed to widespread intuition, any pore throat is convex, and adjacent pore bodies are not necessarily separated by pore throats. Due to the duality between the pore network and the decomposition of the pore space into pore bodies it is straightforward to store the geometrical properties of the pore bodies [pore throats] as attributes of the dual network vertices [edges]. Such an attributed network is used to perform 2D drainage simulations. The results agree very well with those from a pore-morphology based modeling approach performed directly on the digital image of a porous medium. Contractibility of the pore bodies and homotopy equivalence of the pore space and the pore network is proven using discrete Morse theory and the nerve theorem from combinatorial topology.     

Numerical analysis of liquefaction of porous seabed around pipeline fixed in space under seismic loading

Liquefaction of seabed under seismic loading is one of the main points that govern the overall stability of submarine pipeline. However, most previous investigations concerned only with free seabed and searched for seismic accumulative excess pore pressure by solving Terzaghi's consolidation equation containing pore pressure source term. It is not able to introduce two-dimensional structures such as submarine pipelines in one-dimensional problem, and it is also not able to obtain the distribution of seismic accumulative excess pore pressure in seabed around submarine pipelines by such a way. In this study, a FEM numerical analysis method for determining the liquefaction of sandy seabed around a buried pipeline under seismic loading is presented. The empirical mode of dynamic increase of pore pressure under undrained shearing induced by seismic loading is incorporated with two-dimensional dynamic consolidation equation and a numerical procedure based on FEM is developed to assess the accumulative excess pore pressure. By numerical computations, the accumulative process of pore pressure and liquefaction potential of seabed soil during seismic loading is evaluated. From a series of numerical computations based on the presented model with various parameters, the effects of soil characteristic parameters and pipeline geometry on seismic accumulative excess pore pressure around submarine pipeline and along the depth of seabed are explored in detail.     

不同循环荷载作用下软黏土动力特性对比试验研究

研究交通循环荷载作用下路基软黏土的长期沉降和动力力学性质对路基设计具有重要意义。本文通过GDS动三轴实验,研究(不排水条件下)振动波形、排水条件以及动应力比三因素对于软黏土动应变和动孔压的影响。试验结果显示:排水条件对饱和软黏土的动应变和动孔压影响最大,在部分排水条件下动孔压逐渐消散,动应变迅速发展。振动波形对软黏土动应变和动孔压影响较大,单向纯压半正弦波作用下软黏土的动应变和动孔压较容易达到最大值。在较少的振次内动应力比对孔压影响较大,但在归一化的孔压模型中,动应力比对孔压影响较小。通过以上分析,本文建立包含循环振次和纯压因素的孔压增长模型。     

Intercomparison of 3D pore-scale flow and solute transport simulation methods

Multiple numerical approaches have been developed to simulate porous media fluid flow and solute transport at the pore scale. These include 1) methods that explicitly model the three-dimensional geometry of pore spaces and 2) methods that conceptualize the pore space as a topologically consistent set of stylized pore bodies and pore throats. In previous work we validated a model of the first type, using computational fluid dynamics (CFD) codes employing a standard finite volume method (FVM), against magnetic resonance velocimetry (MRV) measurements of pore-scale velocities. Here we expand that validation to include additional models of the first type based on the lattice Boltzmann method (LBM) and smoothed particle hydrodynamics (SPH), as well as a model of the second type, a pore-network model (PNM). The PNM approach used in the current study was recently improved and demonstrated to accurately simulate solute transport in a two-dimensional experiment. While the PNM approach is computationally much less demanding than direct numerical simulation methods, the effect of conceptualizing complex three-dimensional pore geometries on solute transport in the manner of PNMs has not been fully determined. We apply all four approaches (FVM-based CFD, LBM, SPH and PNM) to simulate pore-scale velocity distributions and (for capable codes) nonreactive solute transport, and intercompare the model results. Comparisons are drawn both in terms of macroscopic variables (e.g., permeability, solute breakthrough curves) and microscopic variables (e.g., local velocities and concentrations). Generally good agreement was achieved among the various approaches, but some differences were observed depending on the model context. The intercomparison work was challenging because of variable capabilities of the codes, and inspired some code enhancements to allow consistent comparison of flow and transport simulations across the full suite of methods. This study provides support for confidence in a variety of pore-scale modeling methods and motivates further development and application of pore-scale simulation methods.     

Application of GA in optimization of pore network models generated by multi-cellular growth algorithms

In pore network modeling, the void space of a rock sample is represented at the microscopic scale by a network of pores connected by throats. Construction of a reasonable representation of the geometry and topology of the pore space will lead to a reliable prediction of the properties of porous media. Recently, the theory of multi-cellular growth (or L-systems) has been used as a flexible tool for generation of pore network models which do not require any special information such as 2D SEM or 3D pore space images. In general, the networks generated by this method are irregular pore network models which are inherently closer to the complicated nature of the porous media rather than regular lattice networks. In this approach, the construction process is controlled only by the production rules that govern the development process of the network. In this study, genetic algorithm has been used to obtain the optimum values of the uncertain parameters of these production rules to build an appropriate irregular lattice network capable of the prediction of both static and hydraulic information of the target porous medium.     

Characteristics of petroleum accumulation in syncline of the Songliao basin and discussion on its accumulation mechanism

The relation between oil and water in reservoirs with low and ultra-low permeability is very complicated. Gravitational separation of oil and water is not obvious. Normal reservoirs are located in depression and structural high spot, oil and water transitions are located in their middle. Stagnation is the key fact of oil-forming reservoir in the axis of a syncline based on the research of oil, gas and water migration manner, dynamics and non-Darcy flow in the Songliao basin. In low and ultra-low permeable reservoir, gas and water migrate easily through pore throats because their molecules are generally smaller than the pore throats; but the minimum diameter of oil droplets is larger than pore throats and they must be deformed to go through. Thus, gas and water migrate in advance of oil, and oil droplets remain behind. Pressure differential and the buoyancy force in a syncline reservoir are a main fluid driving force; and capillary force is the main resistance to flow. When the dynamics force is less than resistance, oil is immobile. When the buoyancy force is less than the capillary force, a gravitational separation of oil and water does not occur. The reservoir in the mature source rock of a syncline area with the low and ul- tra-low permeability belongs to an unconventional petroleum reservoir.     

Dissolution of residual non-aqueous phase liquids in porous media: pore-scale mechanisms and mass transfer rates

Experiments designed to elucidate the pore-scale mechanisms of the dissolution of a residual non-aqueous phase liquid (NAPL), trapped in the form of ganglia within a porous medium, are discussed. These experiments were conducted using transparent glass micromodels with controlled pore geometry, so that the evolution of the size and shape of individual NAPL ganglia and, hence, the pore-scale mass transfer rates and mass transfer coefficients could be determined by image analysis. The micromodel design permitted reasonably accurate control of the pore water velocity, so that the mass transfer coefficients could be correlated in terms of a local (pore-scale) Peclet number. A simple mathematical model, incorporating convection and diffusion in a slit geometry was developed and used successfully to predict the observed mass transfer rates. For the case of non-wetting NAPL ganglia, water flow through the corners in the pore walls was seen to control the rate of NAPL dissolution, as recently postulated by Dillard and Blunt [Water Resour. Res. 36 (2000) 439–454]. Break-up of doublet non-wetting phase ganglia into singlet ganglia by snap-off in pore throats was also observed, confirming the interplay between capillarity and mass transfer. Additionally, the effect of wettability on dissolution mass transfer was demonstrated. Under conditions of preferential NAPL wettability, mass transfer from NAPL films covering the solid surfaces was seen to control the dissolution process. Supply of NAPL from the trapped ganglia to these films by capillary flow along pore corners was observed to result in a sequence of pore drainage events that increase the interfacial area for mass transfer. These observations provide new experimental evidence for the role of capillarity, wettability and corner flow on NAPL ganglia dissolution.     

基于Delaunay剖分的二维非规则重力建模及重力计算

针对计算截面为任意形状的非均匀密度二度体的重力问题,提出了二维情况下的非规则重力建模方法,即利用二维Delaunay剖分方法,将其截面分割为若干三角形,将该二度体分解成为若干三角棱柱的组合;通过求取变密度的三角棱柱体的重力异常,并将其累加,最后得出了截面为任意形状的水平无限延伸的二度体的重力计算公式.以千米桥油田的一条实测重力剖面为例,用该方法进行了反演计算,并将反演结果与常规方法做了分析比较,进行了相应的分析和讨论.     

Frequency domain expressions for surface and borehole gravity potential due to two- and three-dimensional mass models

Frequency domain expressions for surface and borehole gravity potential due to selected two- and three-dimensional mass models with linear variation of density are derived using conjugate complex variables formulation. The mass models include elliptical bodies, vertical and horizontal polygonal prisms, polyhedral bodies and bodies with rotational symmetry. Such analytic expressions are essential for development of interpretative theory for surface and borehole gravity data in the Fourier domain.     

Rock physics model of tight oil siltstone for seismic prediction of brittleness

Tight oil siltstones are rocks with complex structure at pore scale and are characterized by low porosity and low permeability at macroscale. The production of tight oil siltstone reservoirs can be increased by hydraulic fracturing. For optimal fracking results, it is desirable to map the ability to fracture based on seismic data prior to fracturing. Brittleness is currently thought to be a key parameter for evaluating the ability to fracture. To link seismic information to the brittleness distribution, a rock physics model is required. Currently, there exists no commonly accepted rock physics model for tight oil siltstones. Based on the observed correlation between porosity and mineral composition and known microstructure of tight oil siltstone in Daqing oilfield of Songliao basin, we develop a rock physics model by combining the Voigt–Reuss–Hill average, self-consistent approximation and differential effective medium theory. This rock physics model allows us to explore the dependence of the brittleness on porosity, mineral composition, microcrack volume fraction and microcrack aspect ratio. The results show that, as quartz content increases and feldspar content decreases, Young's modulus tends to increase and Poisson ratio decreases. This is taken as a signature of higher brittleness. Using well log data and seismic inversion results, we demonstrate the versatility of the rock physics template for brittleness prediction.     

Delaunay三角剖分算法在地震烈度速报中的应用——以天津市强震台网为例

利用计算图形学中的Delaunay三角剖分算法及其对偶结构的Voronoi图算法,构建天津市强震台网的Delaunay三角网及Voronoi图.结果表明,Voronoi图可以较直观地显示台网的布局疏密,并为等震曲线绘制的合理性提供判断依据.     

Computer simulation of flow through a lattice flow-cell model

For single-phase flow through a network model of a porous medium, we report (1) solutions of the Navier–Stokes equation for the flow, (2) micro-particle imaging velocimetry (PIV) measurements of local flow velocity vectors in the “pores throats” and “pore bodies,” and (3) comparisons of the computed and measured velocity vectors. A “two-dimensional” network of cylindrical pores and parallelepiped connecting throats was constructed and used for the measurements. All pore bodies had the same dimensions, but three-different (square cross-section) pore-throat sizes were randomly distributed throughout the network. An unstructured computational grid for flow through an identical network was developed and used to compute the local pressure gradients and flow vectors for several different (macroscopic) flow rates. Numerical solution results were compared with the experimental data, and good agreement was found. Cross-over from Darcy flow to inertial flow was observed in the computational results, and the permeability and inertia coefficients of the network were estimated. The development of inertial flow was seen as a “two-step” process: (1) recirculation zones appeared in more and more pore bodies as the flow rate was increased, and (2) the strengths of individual recirculation zones increased with flow rate. Because each pore-throat and pore-body dimension is known, in this approach an experimental (and/or computed) local Reynolds number is known for every location in the porous medium at which the velocity has been measured (and/or computed).     

基于核磁共振双截止值的致密砂岩渗透率评价新方法

对比分析致密砂岩岩心在完全含水状态和束缚水状态下的核磁共振T₂谱,明确了致密砂岩孔隙中流体的赋存状态和渗流规律,指出常规核磁共振方法预测渗透率的局限性并提出核磁共振双截止值的概念.基于核磁共振双截止值,将储集空间细分为完全可动、完全束缚、部分可动等三部分,分析不同孔隙组分对渗透率的影响,并应用三组分法建立了核磁共振渗透率表征新模型.研究表明：致密砂岩渗透率与完全可动流体饱和度、部分可动流体T₂几何平均值、核磁孔隙度成正比,与完全束缚流体饱和度成反比.在此基础上,结合完全含水核磁共振T₂谱的二阶差分得到了双截止值的自适应确定方法,可以连续地计算储层双截止值.将该研究成果应用于生产实践,渗透率计算精度有较大的提高.     

Three-dimensional convection in spherical shells

Abstract

In this study, the equations of the three-dimensional convective motion of an infinite Prandtl number fluid are solved in spherical geometry, for Rayleigh numbers up to 15 times the critical number. An iterative method is used to find stationary solutions. The spherical parts of the operators are treated using a Galerkin collocation method while the radial and time dependences are expressed using finite difference methods. A systematic search for stationary solutions has led to eight different stream patterns for a low Rayleigh number (1.28 times the critical number). They can be classified as:

I) Axisymmetrical solutions, analogous to rolls in plane geometry.

II) Solutions which have several ascending plumes within a large area of ascending current, and also several descending plumes within an area of descending current. This type of flow is analogous to bimodal circulation in plane geometry.

III) Solutions characterized by isolated ascending (or descending) plumes separated from each other by a closed polyhedral network of descending (or ascending) currents. This type of circulation is called ‘polygonal’ in analogy with hexagonal circulation in plane geometry.

The behaviour of each of the eight solutions has been studied by increasing the Rayleigh number up to 15 times the critical number. A trend towards transitions from type (I) and type (II) solutions to type (III) solutions is observed. It is inferred that only the “polygonal” solutions are stable for a Rayleigh number greater than 15 times the critical number.     

Using a pore‐scale model to quantify the effect of particle re‐arrangement on pore structure and hydraulic properties

A pore‐scale model based on measured particle size distributions has been used to quantify the changes in pore space geometry of packed soil columns resulting from a dilution in electrolyte concentration from 500 to 1 mmol l^?1 NaCl during leaching. This was applied to examine the effects of particle release and re‐deposition on pore structure and hydraulic properties. Two different soils, an agricultural soil and a mining residue, were investigated with respect to the change in hydraulic properties. The mining residue was much more affected by this process with the water saturated hydraulic conductivity decreasing to 0·4% of the initial value and the air‐entry value changing from 20 to 50 cm. For agricultural soil, there was little detectable shift in the water retention curve but the saturated hydraulic conductivity decreased to 8·5% of the initial value. This was attributed to localized pore clogging (similar to a surface seal) affecting hydraulic conductivity, but not the microscopically measured pore‐size distribution or water retention. We modelled the soil structure at the pore scale to explain the different responses of the two soils to the experimental conditions. The size of the pores was determined as a function of deposited clay particles. The modal pore size of the agricultural soil as indicated by the constant water retention curve was 45 µm and was not affected by the leaching process. In the case of the mining residue, the mode changed from 75 to 45 µm. This reduction of pore size corresponds to an increase of capillary forces that is related to the measured shift of the water retention curve. Copyright © 2007 John Wiley & Sons, Ltd.     

On the definition and derivatives of macroscale energy for the description of multiphase systems

Modeling of flow and transport in environmental systems often involves formulation of conservation equations at spatial scales involving tens to hundreds of pore diameters in porous media or the depth of flow in a channel. Quantities such as density, temperature, internal energy, and velocity may not be uniform over these macroscopic length scales. The external gravitational potential causes gradients in density, pressure, and chemical potential even at equilibrium. Despite these complications, it is important to formulate the thermodynamic analysis of environmental systems at the macroscopic scale. Heretofore, this has been accomplished primarily using the approach of rational thermodynamics whereby the thermodynamic dependence of macroscale internal energy on macroscale variables is hypothesized directly without development of any systematic method for transforming microscale energy dependence from the microscale to the macroscale. However when thermodynamic variables are inhomogeneous at the microscale, the functional dependence of macroscale internal energy on macroscale variables is not a simple extension of the microscale case. In the present work, the relation between the definitions of microscale and macroscale intensive thermodynamic variables is established. Expressions for the material derivatives of macroscale internal energy of phases, interfaces, and common lines are derived from and consistent with their microscopic counterparts by integrating to the macroscale. The forms obtained and the consistency required will be important for use in analyses of systems at scales where microscopic heterogeneities cannot be neglected.     

The effect of microporosity on transport properties in porous media

Sizeable amounts of connected microporosity with various origins can have a profound effect on important petrophysical properties of a porous medium such as (absolute/relative) permeability and capillary pressure relationships. We construct pore-throat networks that incorporate both intergranular porosity and microporosity. The latter originates from two separate mechanisms: partial dissolution of grains and pore fillings (e.g. clay). We then use the reconstructed network models to estimate the medium flow properties. In this work, we develop unique network construction algorithms and simulate capillary pressure–saturation and relative permeability–saturation curves for cases with inhomogeneous distributions of pores and micropores. Furthermore, we provide a modeling framework for variable amounts of cement and connectivity of the intergranular porosity and quantifying the conditions under which microporosity dominates transport properties. In the extreme case of a disconnected inter-granular network due to cementation a range of saturations within which neither fluid phase is capable of flowing emerges. To our knowledge, this is the first flexible pore scale model, from first principles, to successfully approach this behavior observed in tight reservoirs.     

考虑碎石桩排水能力复合地基中孔压长消解析解

由于地震作用时间较短,且碎石桩渗透能力和土体渗透能力相比并不是无限大,因此本文考虑碎石桩排水能力研究了碎石桩桩体材料由地震引起的孔压的长消规律。根据比奥固结理论综合考虑碎石桩的排水能力和相应的初始条件及边界条件,推导出了能够真实反映碎石桩排水减压作用在地震期超孔隙水压力产生、扩散、消散过程中的贡献作用的一般解析解公式。同时讨论了碎石桩渗透能力的不同对抗震液化的影响作用。     

Numerical modeling of formation damage by two-phase particulate transport processes during CO2 injection in deep heterogeneous porous media

Prediction of CO₂ injection performance in deep subsurface porous media relies on the ability of the well to maintain high flow rates of carbon dioxide during several decades typically without fracturing the host formation or damaging the well. Dynamics of solid particulate suspensions in permeable media are recognized as one major factor leading to injection well plugging in sandstones. The invading supercritical liquid-like fluid can contain exogenous fine suspensions or endogenous particles generated in situ by physical and chemical interactions or hydrodynamic release mechanisms. Suspended solids can plug the pores possibly leading to formation damage and permeability reduction in the vicinity of the injector. In this study we developed a finite volume simulator to predict the injectivity decline near CO₂ injection wells and also for production wells in the context of enhanced oil recovery. The numerical model solves a system of two coupled sets of finite volume equations corresponding to the pressure-saturation two-phase flow, and a second subsystem of solute and particle convection-diffusion equations. Particle transport equations are subject to mechanistic rate laws of colloidal, hydrodynamic release from pore surfaces, blocking in pore bodies and pore throats, and interphase particle transfer. The model was validated against available laboratory experiments at the core scale. Example results reveal that lower CO₂ residual saturation and formation porosity enhance CO₂-wet particle mobility and clogging around sinks and production wells. We conclude from more realistic simulations with heterogeneous permeability spanning several orders of magnitude that the control mode of mobilization, capture of particles, and permeability reduction processes strongly depends on the type of permeability distribution and connectivity between injection and production wells.