Flow through anisotropic networks

This paper presents an analysis of Hagen-Poiseulle flow through plane random anisotropic networks of interconnected channels. Macroscopic permeability tensor of the network is expressed in terms of statistico-geometrical characteristics like the degree of anisotropy in channel orientations, average co-ordination number of the network and first two moments of channel length distribution. Analytical results are illustrated and verified using numerical analysis of flow in a simulated random network. The emphasis of the paper is on the effects of anisotropy on distributions of flow rates in channels. It is shown that, due to anisotropy the maximum flow rate generally occurs in channels that are not aligned along the direction of the macroscopic pressure gradient.     

Statistical aspects of flow in a random network of channels

This paper presents a detailed statistical analysis of Hagen-Poiseuille flow in plane random isotropic networks of interconnected channels. The emphasis is on statistico-geometrical features of networks that affect macroscopic permeability. It is shown that permeability of a network depends on its average co-ordination number, the first two moments of the channel length distribution and other explicitly identifiable geometrical features. Distributions of flow rates in channels and average flow rates are established by minimization of the rate of energy dissipation. Theoretical developments are interpreted in the context of classical statistical mechanics. Analytical results are illustrated and verified using numerical analysis of flow in a simulated random network.     

裂隙各向异性介质中的NMO速度

推导了各向异性介质中由弹性系数表示的方位动校NMO速度的具体表达式,表明各向异性介质中方位NMO速度程椭圆形状,并分别对具水平对称轴的横向各向同性介质(HTI)、正交介质和单斜各向异性介质及在不同的裂隙填充物的性质下方位NMO速度进行了计算,结果表明裂隙的存在对NMO速度的影响不仅与裂隙密度有关,还取决于裂隙填充物的性质.同时,研究表明对于裂隙型单斜各向异性介质,其方位NMO速度椭圆轴向并不象HTI介质和正交介质中的那样与自然坐标系的坐标轴一致,而是发生了一定角度的偏离,其大小与裂隙填充物的性质、两组裂隙密度的比值及裂隙间的夹角等因素有关,研究结果为进一步区分裂隙介质的类型及裂隙填充物的性质提供依据.     

Generation of three dimensional flow fields for statistically anisotropic heterogeneous porous media

A methodology for generating three dimensional (3D) flow fields for statistically anisotropic heterogeneous porous media is presented and demonstrated. The simulated flow fields are shown to exhibit the input spatial correlation structure and observe mass continuity. Sample flow fields are presented in the form of cross sectional slices of the 3D formation. These cross sections demonstrate visually the characteristics of subsurface flow. The method was found to be faster than traditional techniques in terms of its computational requirements. Given this method, it is possible to generate the large number of realizations of a velocity field necessary to compute high order statistics in transport problems.     

Ray-centred coordinate systems in anisotropic media

Whereas the ray-centred coordinates for isotropic media by Popov and Pšenčík are uniquely defined by the selection of the basis vectors at one point along the ray, there is considerable freedom in selecting the ray-centred coordinates for anisotropic media. We describe the properties common to all ray-centred coordinate systems for anisotropic media and general conditions, which may be imposed on the basis vectors. We then discuss six different particular choices of ray-centred coordinates in an anisotropic medium. This overview may be useful in choosing the ray-centred coordinates best suited for a particular application. The equations are derived for a general homogeneous Hamiltonian of an arbitrary degree and are thus applicable both to the anisotropic-ray-theory rays and anisotropic common S-wave rays.     

各向异性介质中的弹性阻抗及其反演

地震反演已成为油藏描述中的重要组成部分.绝大多数的常规地震反演是叠后地震数据体上进行,很少考虑各向异性存在的情况.随着勘探开发的发展,地震各向异性和叠前地震波阻抗反演引起了人们极大关注.本文在各向同性介质中弹性阻抗研究基础上,推导出了各向异性介质中的弹性阻抗方程,提出了地震各向异性介质中用弹性阻抗进行储层参数描述的技术路线和框架,并对反演过程中存在的问题进行了有益探讨.     

Velocities of elastic waves andv P/v S ratios in dry and water-saturated rock samples at high pressure

Elastic wave velocities and the ratiov _P/v _S were studied for dry and initially saturated samples of carbonate and crystalline rocks at pressures to 2 kbar. In initially saturated samplesv _P increases in crystalline rock, whereas in sedimentary rock it may either increase or decrease with increasing pressure. Under the same conditionsv _S remains approximately constant in crystalline rocks and decreases in sedimentary samples. The ratiov _P/v _S as a function of pressure increases in dry rocks and decreases in saturated ones. Saturated samples always have higherv _P/v _S ratios than dry ones. It appears that the geometry of the pore space influences the acoustic properties of saturated crystalline rocks.     

Flow of viscous fluids through a porous deformable matrix

A self-contained account of mixture theory is presented as a framework for describing the flow of fluids, liquids and gases, through a porous deformable matrix, incorporating both mechanical and thermal effects. The theory comprises the conservation laws of mass, momentum and energy for each constituent and the mixture properties which describe the interactions between constituents. Mass transfer between constituents which arises during phase change and chemical reactions influences both conservation laws and mixture properties. An analysis of discontinuity conditions at a singular surface is presented, which would be needed, for example, to describe an advancing phase-change front. Details are presented for the flow of viscous fluids through a thermoelastic matrix undergoing infinitesimal deformation, a common model for underground reservoirs. The interactions of immiscible and miscible fluids are discussed. An essential ingredient is the relation between partial physical variables defined as mean values over mixture elements, and intrinsic variables defined with respect to the constituent elements.     

含泥质低孔渗各向异性黏弹性介质中的波频散和衰减研究

本文定义了各向异性黏弹性参数修正因子,并将其引入到黏弹性模型中以体现泥质含量对黏弹性机制的影响,同时将波传播过程中孔隙介质骨架黏弹性力学机制与两种孔隙流体流动力学机制（Biot流动和喷射流动机制）有机地统一起来处理,从而给出了描述含泥质低孔渗孔隙各向异性介质中波传播规律的黏弹性Biot/squirt (BISQ)模型.数值计算结果表明,入射波的方位角、各向异性渗透率以及泥质含量等对含流体复杂孔隙介质中波频散和衰减的影响具有显著的方位各向异性特征,在低频范围内（地震波勘探频率）黏弹性力学机制对波传播能量的衰减起主导作用.     

Spatial derivatives and perturbation derivatives of amplitude in isotropic and anisotropic media

Explicit equations for the spatial derivatives and perturbation derivatives of amplitude in both isotropic and anisotropic media are derived. The spatial and perturbation derivatives of the logarithm of amplitude can be calculated by numerical quadratures along the rays. The spatial derivatives of amplitude may be useful in calculating the higher-order terms in the ray series, in calculating the higher-order amplitude coefficients of Gaussian beams, in estimating the accuracy of zero-order approximations of both the ray method and Gaussian beams, in estimating the accuracy of the paraxial approximation of individual Gaussian beams, or in estimating the accuracy of the asymptotic summation of paraxial Gaussian beams. The perturbation derivatives of amplitude may be useful in perturbation expansions from elastic to viscoelastic media and in estimating the accuracy of the common-ray approximations of the amplitude in the coupling ray theory.     

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).     

岩石物理的理论模拟和数值实验新方法

当代物理实验科学和计算机科学技术的巨大进步为岩石物理的理论和实验研究开辟了新的领域.在细胞自动机基础上发展起来的格子气自动机和格子玻尔滋曼方法就是这种技术进步的产物.本文讨论了格子气自动机的发展及其应用于研究多孔岩石微观孔隙结构对宏观导电特性影响的新方法. 我们开发的模拟多孔岩石导电特性的数值实验新方法，提供了一种可深入到从微观孔隙结构这个尺度上来考察具有复杂孔隙结构的，孔隙度小于10%的低孔、低渗岩石的宏观导电特性及其影响因素.研究表明:Archie 系数m,n并不是传统上的意义，研究各种不同条件下F-、I-Sw关系，结果显示非阿尔奇现象与孔隙介质中孔隙、骨架和流体的混合导电网络有关，得出新的饱和度计算公式.同时还研究了泥质含量和分布、电阻率各向异性对宏观导电特性的影响，提出适用于电阻率各向异性地层的F-、I-Sw的关系.本文最后探讨了应用格子气自动机和格子玻尔滋曼方法进行岩心渗流和核磁共振数值实验的新方法及其发展前景.     

A new upscaling method for fractured porous media

We present a method to determine equivalent permeability of fractured porous media. Inspired by the previous flow-based upscaling methods, we use a multi-boundary integration approach to compute flow rates within fractures. We apply a recently developed multi-point flux approximation Finite Volume method for discrete fracture model simulation. The method is verified by upscaling an arbitrarily oriented fracture which is crossing a Cartesian grid. We demonstrate the method by applying it to a long fracture, a fracture network and the fracture network with different matrix permeabilities. The equivalent permeability tensors of a long fracture crossing Cartesian grids are symmetric, and have identical values. The application to the fracture network case with increasing matrix permeabilities shows that the matrix permeability influences more the diagonal terms of the equivalent permeability tensor than the off-diagonal terms, but the off-diagonal terms remain important to correctly assess the flow field.     

Bayesian Markov chain Monte Carlo inversion for anisotropy of PP- and PS-wave in weakly anisotropic and heterogeneous media

A single set of vertically aligned cracks embedded in a purely isotropic background may be considered as a long-wavelength effective transversely isotropy (HTI) medium with a horizontal symmetry axis. The crack-induced HTI anisotropy can be characterized by the weakly anisotropic parameters introduced by Thomsen. The seismic scattering theory can be utilized for the inversion for the anisotropic parameters in weakly anisotropic and heterogeneous HTI media. Based on the seismic scattering theory, we first derived the linearized PP- and PS-wave reflection coefficients in terms of P- and S-wave impedances, density as well as three anisotropic parameters in HTI media. Then, we proposed a novel Bayesian Markov chain Monte Carlo inversion method of PP- and PS-wave for six elastic and anisotropic parameters directly. Tests on synthetic azimuthal seismic data contaminated by random errors demonstrated that this method appears more accurate, anti-noise and stable owing to the usage of the constrained PS-wave compared with the standards inversion scheme taking only the PP-wave into account.     

利用三维非均匀不稳定渗流场的数值解求解垂直井与水平井的联合开采

用差分的方法给出了三维非均匀不稳定渗流方程的离散方程组，将配斯曼（Peaceman）方程拓展成能够被用于同时求解垂直井与水平井联合开采问题的形式，利用迭代解法求解差分方程和配斯曼方程所组成的方程组，并以实际层状不均匀介质为例计算了其压强的空间分布。     

Harmonic wave propagation through viscoelastic heterogeneous media exhibiting mild stochasticity — II. Applications

The results obtained in Part I (Soil Dynam. Earth. Engng, 1996, 15, 119-27) of this work for modelling harmonic wave propagation through viscoelastic heterogeneous media that exhibit a small random fluctuation of their material properties about mean values are now used here to investigate SH wave propagation in two naturally occurring media, namely sandstone and topsoil. These results are in the form of depth dependent, deterministic mean values and non-isotropic covariances for both the wave speed profile in the medium and for the fundamental solution in terms of displacement due to a unit point source. The results are also compared against conventional Monte Carlo simulations.     

Flow and transport through two-dimensional isotropic media of binary conductivity distribution. Part 1: NUMERICAL methodology and semi-analytical solutions

Flow and transport take place in a heterogeneous medium made up from inclusions of conductivity K submerged in a matrix of conductivity K ₀. We consider two-dimensional isotropic media, with circular inclusions of uniform radii, that are placed at random and without overlap in the matrix. The system is completely characterized by the conductivity contrast =K/K ₀ and by the volume fraction n. The flow is uniform in the mean, of velocity U=const. The derivation of the velocity field is achieved by a numerical method of high accuracy, based on analytical elements. Approximate analytical solutions are derived by a few methods: composite elements, effective medium, dilute systems and first-order approximation in logconductivity variance. The latter was employed by Rubin (1995), while the dilute system approximation was used by Eames and Bush (1999) and Dagan and Lessoff (2001). Transport is solved in a Lagrangean framework, with trajectories determined numerically from the velocity field, by particle tracking. Results for the velocity variance and for the longitudinal macrodispersivity, for a few values of and n, are presented in Part 2.     

Time-related capture zones for radial flow in two dimensional randomly heterogeneous media

We consider the effect of randomly heterogeneous hydraulic conductivity on the spatial location of time-related capture zones (isochrones) for a non-reactive tracer in the steady-state radial flow field due to a pumping well in a confined aquifer. A Monte Carlo (MC) procedure is used in conjunction with FFT-based spectral methods. The log hydraulic conductivity field is assumed to be Gaussian and stationary, with isotropic exponential correlation. Various degrees of domain heterogeneity are considered and stability and accuracy of the MC procedure is examined. The location of an isochrone becomes uncertain due to heterogeneity, and it is strongly influenced by hydraulic conductivity variance. The probability that a particle released at a point in the aquifer is pumped by the well within a given time is identified. We propose a new expression for the probabilistic spatial distribution of isochrones, which is formally similar to the analytical solution for a uniform medium and takes into account the effects of heterogeneity.     

Common-ray tracing and dynamic ray tracing for S waves in a smooth elastic anisotropic medium

Anisotropic common S-wave rays are traced using the averaged Hamiltonian of both S-wave polarizations. They represent very practical reference rays for calculating S waves by means of the coupling ray theory. They eliminate problems with anisotropic-ray-theory ray tracing through some S-wave slowness-surface singularities and also considerably simplify the numerical algorithm of the coupling ray theory for S waves. The equations required for anisotropic-common-ray tracing for S waves in a smooth elastic anisotropic medium, and for corresponding dynamic ray tracing in Cartesian or ray-centred coordinates, are presented. The equations, for the most part generally known, are summarized in a form which represents a complete algorithm suitable for coding and numerical applications.