Stochastic Modeling of Variably Saturated Transient Flow in Fractal Porous Media

This work presents the application of a Monte Carlo simulation method to perform an statistical analysis of transient variably saturated flow in an hypothetical random porous media. For each realization of the stochastic soil parameters entering as coefficients in Richards' flow equation, the pressure head and the flow field are computed using a mixed finite element procedure for the spatial discretization combined with a backward Euler and a modified Picard iteration in time. The hybridization of the mixed method provides a novel way for evaluating hydraulic conductivity on interelement boundaries. The proposed methodology can handle both large variability and fractal structure in the hydraulic parameters. The saturated conductivity K _s and the shape parameter _vg in the van Genuchten model are treated as stochastic fractal functions known as fractional Brownian motion (fBm) or fractional Gaussian noise (fGn). The statistical moments of the pressure head, water content, and flow components are obtained by averaging realizations of the fractal parameters in Monte Carlo fashion. A numerical example showing the application of the proposed methodology to characterize groundwater flow in highly heterogeneous soils is presented.     

A Corrected and Generalized Successive Random Additions Algorithm for Simulating Fractional Levy Motions

Simulation of subsurface heterogeneity is important for modeling subsurface flow and transport processes. Previous studies have indicated that subsurface property variations can often be characterized by fractional Brownian motion (fBm) or (truncated) fractional Levy motion (fLm). Because Levy-stable distributions have many novel and often unfamiliar properties, studies on generating fLm distributions are rare in the literature. In this study, we generalize a relatively simple and computationally efficient successive random additions (SRA) algorithm, originally developed for generating Gaussian fractals, to simulate fLm distributions. We also propose an additional important step in response to continued observations that the traditional SRA algorithm often generates fractal distributions having poor scaling and correlation properties. Finally, the generalized and modified SRA algorithm is validated through numerical tests.     

多孔介质中溶质有效扩散系数预测的分形模型

依据分形理论和方法,探索溶质在多孔介质中的有效扩散系数的替代预测方法。在多孔介质溶质扩散的弯曲毛细管束模型的基础上,以分形维数作为介质的基本几何特性参数,建立了多孔介质中溶质扩散的分形毛细管束模型,推导出了溶质有效扩散系数与介质孔隙度之间的幂定律关系式,幂指数是介质孔隙分维和表面分维的函数,反映了介质孔隙体积的层次分布与孔隙通道曲折程度对扩散的影响。对粘性土的分形维数测定数据和有效扩散系数试验测定数据的分析表明,利用该关系式预测多孔介质中溶质的有效扩散系数是较为准确可靠的。     

分形多孔介质孔隙微结构参数与渗透率的分维关系

根据分形几何理论的基本概念,就无序分形多孔介质孔隙率φ和渗透率K与多孔介质结构分数维数Df的关系进行了推导,利用Sierpinski固相分形体(Solid mass fractal)与孔相分形体(Pore mass fractal)概念对分形多孔介质微结构特征、孔隙累积数量-尺寸分布和孔隙率φ等参数及其物理关系给予了详细论述,定量地分析和讨论了基于不同模型的渗透率-分形维数关系与它们的差异.     

多尺度多孔介质有效气体输运参数的分形特征

非常规油气资源的孔隙结构及其连通性非常复杂,其孔隙尺度从毫米到纳米跨越多个量级.多孔介质中气体的输运过程不仅依赖于介质的多尺度微观结构特征,还依赖于气体的相关属性.气体在多尺度多孔介质中的输运过程包括无滑流、滑脱流和过渡流,涉及分子扩散和努森扩散等多种机制,因此很难用唯一的连续介质理论来描述气体的输运特征.大量的数据表明真实多孔介质中的内部孔隙具有分形标度特征,因此采用分形几何表征多尺度多孔介质的孔隙结构,引入孔隙分形维数和迂曲度分形维数定量表征多孔介质的微结构和弯曲流道,建立多尺度多孔介质气体输运过程的细观模型;推导了多尺度多孔介质中气体的有效渗透率和有效扩散系数,并讨论了多尺度多孔介质微结构参数和气体属性对于气体等效输运特性的定量影响.该研究不仅可以丰富渗流理论,且有利于深入理解非常规油气藏的产出机制.      

Lakes on fractal surfaces: A null hypothesis for lake-rich landscapes

A class of stochastic processes known as fractional Brownian motion (fBm) provides strikingly realistic simulations of certain types of terrain, particularly those which appear to be unmodified by geomorphological and geological processes. In addition to their less serious applications in video games and science fiction movies, fractal terrain simulations have proven useful in a number of areas of spatial analysis. For example, they can provide sample data sets for testing the efficiency of data structures and algorithms designed for topographic applications. Previous work has shown that stream networks simulated on fBm surfaces show the same deviations from accepted theories of channel network topology as do real stream networks, implying that such deviations originate in the geometrical constraints of packing channels onto surfaces, rather than from geological or other environmental controls. In effect, this work demonstrates the usefulness of fBm as a null hypothesis for terrain. One difficulty, however, stems from the abundant pits which occur in the simulations, because peaks and pits are equally likely. Flooding of pits on fBm surfaces was simulated to obtain lakes. Lake-rich stream networks were extracted and represented with a suitable integer code. The relative frequencies of various network topologies and groups of topologies were compared to known characteristics of channel networks on real lake-rich landscapes. Lake-string topologies are significantly less abundant than in glaciated landscapes. Lake areas show good fits to hyperbolic distributions, but lake in-degrees do not fit the negative binomial model. fBm surfaces are appropriate null hypotheses of scale-free, lake-rich landscapes.This paper was presented at Emerging Concepts, MGUS-87 Conference, Redwood City, California, 13–15 April 1987.     

多孔介质非饱和导水率预测的分形模型

多孔介质非饱和导水率是地下水污染预测与评价的重要参数。根据分形几何的基本原理和方法,推导出了与Campbell经验公式在形式上完全一致的多孔介质非饱和导水率的预测公式。公式中的幂指数为介质孔隙分维和随机行走分维的函数,分别体现了多孔介质的静态性质与动态性质对其中水分运动的影响,但静态性质的影响是主要的,即导水率主要受多孔介质的结构控制。根据文献中报道的大量数据,利用笔者推导的预测公式计算得到的幂指数的统计值与试验测定的幂指数的统计值基本一致,说明推导的理论公式预测多孔介质非饱和导水率是较为可靠的。     

Modelling Two-Phase Incompressible Flow in Porous Media Using Mixed Hybrid and Discontinuous Finite Elements

In this paper, we present a numerical model for simulating two-phase (oil–water and air–water) incompressible and immiscible flow in porous media. The mathematical model which is based on a fractional flow formulation is formed of two nonlinear partial differential equations: a mean pressure equation and a water saturation equation. These two equations can be solved in a sequential manner. Two numerical methods are used to discretize the equations of the two-phase flow model: mixed hybrid finite elements are used to treat the pressure equation, h-based Richards' equation and the diffusion term in the saturation equation, the advection term in the saturation equation is treated with the discontinuous finite elements. We propose a better way to calculate the nonlinear coefficients contained in our equations on each element of the discretized domain. In heterogeneous porous media, the saturation becomes discontinuous at the interface between two porous media. We show in this paper how to use the capillary pressure–saturation relationship in order to handle the saturation jump in the mixed hybrid finite element method. The two-phase flow simulator is verified against analytical solutions for some flow problems treated by other authors.     

土壤水力特性分形特征的研究进展

从土壤结构特征的描述、土壤水力特性的研究等方面,综合评述了80年代以来国内外应用分形理论描述土壤水力特性参数及分形特征的研究进展,并探讨了其发展方向和前景。     

考虑扩散路径的宾汉姆流体渗透注浆机制

浆液在多孔介质中的扩散路径对渗透扩散范围和注浆效果具有非常重要的影响。采用理论分析,以分形特征与宾汉姆流体在多孔介质中的渗流运动方程为基础,揭示了考虑扩散路径的宾汉姆流体渗透注浆机制,并利用团队前期开展的渗透注浆试验对其进行了验证。分析了多孔介质孔隙率、宾汉姆水泥浆液水灰比、多孔介质渗透系数、注浆压力、地下水压力等对扩散半径的影响变化规律。同时,基于Comsol Multiphysics平台,采用计算机编程技术二次开发得到了考虑扩散路径的宾汉姆流体渗透注浆机制的渗透注浆三维数值模拟程序,并以此开展了宾汉姆水泥浆液在多孔介质中渗透扩散形态效果的数值模拟。研究结果表明：与不考虑扩散路径的宾汉姆流体渗透注浆球形扩散公式获得的扩散半径理论计算值相比,采用考虑扩散路径的宾汉姆流体渗透注浆机制得到的扩散半径理论计算值更接近试验值。该研究成果可为实践注浆工程提供一定的理论支撑。     

分形技术在探地雷达资料处理解释中的应用

将分形理论应用到探地雷达 (GPR)资料处理解释中 ,以提高探地雷达剖面的分辨率。通过建立相应的GPR的fBm模型 ,采用逐次随机增加方法 (SuccessiveRandomAdditions)对GPR数据进行细分。实验分析表明 ,分形处理是一种提取GPR高分辨率信息的非常有效的方法 ,其效果要比常规处理 (如线性滤波等 )明显 ;最后用分形技术对实际探地雷达检测剖面进行了处理 ,取得了较好效果     

Flexible spectral methods for the generation of random fields with power-law semivariograms

Random field generators serve as a tool to model heterogeneous media for applications in hydrocarbon recovery and groundwater flow. Random fields with a power-law variogram structure, also termed fractional Brownian motion (fBm) fields, are of interest to study scale dependent heterogeneity effects on one-phase and two-phase flow. We show that such fields generated by the spectral method and the Inverse Fast Fourier Transform (IFFT) have an incorrect variogram structure and variance. To illustrate this we derive the prefactor of the fBm spectral density function, which is required to generate the fBm fields. We propose a new method to generate fBm fields that introduces weighting functions into the spectral method. It leads to a flexible and efficient algorithm. The flexibility permits an optimal choice of summation points (that is points in frequency space at which the weighting function is calculated) specific for the autocovariance structure of the field. As an illustration of the method, comparisons between estimated and expected statistics of fields with an exponential variogram and of fBm fields are presented. For power-law semivariograms, the proposed spectral method with a cylindrical distribution of the summation points gives optimal results.     

Numerical simulation of fully saturated porous materials

Fully coupled, porous solid–fluid formulation, implementation and related modeling and simulation issues are presented in this work. To this end, coupled dynamic field equations with u?p?U formulation are used to simulate pore fluid and soil skeleton (elastic–plastic porous solid) responses. Present formulation allows, among other features, for water accelerations to be taken into account. This proves to be useful in modeling dynamic interaction of media of different stiffnesses (as in soil–foundation–structure interaction). Fluid compressibility is also explicitly taken into account, thus allowing excursions into modeling of limited cases of non‐saturated porous media. In addition to these features, present formulation and implementation models in a realistic way the physical damping, which dissipates energy. In particular, the velocity proportional damping is appropriately modeled and simulated by taking into account the interaction of pore fluid and solid skeleton. Similarly, the displacement proportional damping is physically modeled through elastic–plastic processes in soil skeleton. An advanced material model for sand is used in present work and is discussed at some length. Also explored in this paper are the verification and validation issues related to fully coupled modeling and simulations of porous media. Illustrative examples describing the dynamical behavior of porous media (saturated soils) are presented. The verified and validated methods and material models are used to predict the behavior of level and sloping grounds subjected to seismic shaking. Copyright © 2008 John Wiley & Sons, Ltd.     

水力传导度空间变异性的分形模拟研究进展

水力传导度是描述孔隙介质物理特性的重要参数,水力传导度的空间变异性直接影响到水分与溶质在介质中的运移状况。由于基于随机理论的方法难于描述具有多重变异尺度的水力传导度的空间变异性,使得基于分形理论的方法得到了较快发展和应用。详细介绍并评述了分形理论和方法的基本特征及研究进展,水力传导度的空间变异分形与弥散尺度效应的关系及其对溶质运移的影响。     

Smooth particle hydrodynamics simulations of low Reynolds number flows through porous media

In this paper, a three‐dimensional smooth particle hydrodynamics (SPH) simulator for modeling grain scale fluid flow in porous media is presented. The versatility of the SPH method has driven its use in increasingly complex areas of flow analysis, including the characterization of flow through permeable rock for both groundwater and petroleum reservoir research. SPH provides the means to model complex multi‐phase flows through such media; however, acceptance of the methodology has been hampered by the apparent lack of actual verification within the literature, particulary in the three‐dimensional case. In this paper, the accuracy of SPH is addressed via a comparison to the previously recognized benchmarks of authors such as Sangani and Acrivos (Int. J. Multiphase Flow 1982; 8 (4): 343–360), Zick and Homsy (J. Fluid Mech. 1982; 115 :13–26) and Larson and Higdon (Phys. Fluids A 1989; 1 (1):38–46) for the well‐defined classical problems of flow through idealized two‐ and three‐dimensional porous media. The accuracy of results for such low Reynolds number flows is highly dependent on the implementation of no‐slip boundary conditions. A new, robust and numerically efficient, method for implementing such boundaries in SPH is presented. Simulation results for friction coefficient and permeability are shown to agree well with the available benchmarks. Copyright © 2010 John Wiley & Sons, Ltd.     

离散介质中地震波传播的本构问题

实际地球介质是相当复杂的.基于均匀的、连续介质模型建立的弹性波动理论可能导致对地球结构地震响应的不当解释, 有时可能是错误解释.由于没有更接近实际地球介质模型的波动理论, 许多有用的地震信息因得不到合理解释, 被作为噪音处理了.从等效介质角度来看, 储层介质可以划分为弹性区、粘性区、空白区(空白区是由孔隙、裂隙、结构面产生的介质性质弱化区) 组成的离散介质.在局部平均思想的指导下, 利用区间内聚定理建立起离散(储层) 介质的本构关系, 为建立更接近实际地球介质模型的波动理论提供了力学基础.      

Precipitation of calcium carbonate and calcium phosphate under diffusion controlled mixing

Multi-component mineral precipitation in porous, subsurface environments is challenging to simulate or engineer when in situ reactant mixing is controlled by diffusion. In contrast to well-mixed systems, the conditions that favor mineral precipitation in porous media are distributed along chemical gradients, which evolve spatially due to concurrent mineral precipitation and modification of solute transport in the media. The resulting physical and chemical characteristics of a mixing/precipitation zone are a consequence of coupling between transport and chemical processes, and the distinctive properties of individual chemical systems. We examined the spatial distribution of precipitates formed in “double diffusion” columns for two chemical systems, calcium carbonate and calcium phosphate. Polyacrylamide hydrogel was used as a low permeability, high porosity medium to maximize diffusive mixing and minimize pressure- and density-driven flow between reactant solutions. In the calcium phosphate system, multiple, visually dense and narrow bands of precipitates were observed that were reminiscent of previously reported Liesegang patterns. In the calcium carbonate system, wider precipitation zones characterized by more sparse distributions of precipitates and a more open channel structure were observed. In both cases, formation of precipitates inhibited, but did not necessarily eliminate, continued transport and mixing of the reactants. A reactive transport model with fully implicit coupling between diffusion, chemical speciation and precipitation kinetics, but where explicit details of nucleation processes were neglected, was able to qualitatively simulate properties of the precipitation zones. The results help to illustrate how changes in the physical properties of a precipitation zone depend on coupling between diffusion-controlled reactant mixing and chemistry-specific details of precipitation kinetics.     

Heterogeneous growth of cordierite in low P/T Tsukuba metamorphic rocks from central Japan

This paper examines the spatial statistics of matrix minerals and complex patterned cordierite porphyroblasts in the low‐pressure, high‐temperature (low P/T) Tsukuba metamorphic rocks from central Japan, using a density correlation function. The cordierite‐producing reaction is sillimanite + biotite + quartz = K‐feldspar + cordierite + water. The density correlation function shows that quartz is distributed randomly. However, the density correlation functions of biotite, plagioclase and K‐feldspar show that their spatial distributions are clearly affected by the formation of cordierite porphyroblasts. These observations suggest that cordierite growth occurred through a selective growth mechanism: quartz adjacent to cordierite has a tendency to prevent the growth of cordierite, whereas other matrix minerals adjacent to cordierite have a tendency to enhance the growth of cordierite. The density correlation functions of complex patterned cordierite porphyroblasts show power‐law behaviour. A selective growth mechanism alone cannot explain the origin of the power‐law behaviour. Comparison of the morphology and fractal dimension of cordierite with two‐dimensional sections from a three‐dimensional diffusion‐limited aggregation (DLA) suggests that the formation of cordierite porphyroblasts can be modelled as a DLA process. DLA is the simple statistical model for the universal fractal pattern developed in a macroscopic diffusion field. Diffusion‐controlled growth interacting with a random field is essential to the formation of a DLA‐like pattern. The selective growth mechanism will provide a random noise for the growth of cordierite due to random distribution of quartz. Therefore, a selective growth mechanism coupled with diffusion‐controlled growth is proposed to explain the power‐law behaviour of the density correlation function of complex patterned cordierite. The results in this paper suggest that not only the growth kinetics but also the spatial distribution of matrix minerals affect the progress of the metamorphic reaction and pattern formation of metamorphic rocks.     

Multi‐species diffusion analysis in porous media under various dry density and temperature conditions: molecular dynamics simulation,homogenization analysis,and finite element method

In porous media, chemical species that dissolve in pore water can be transported via diffusion mechanisms or advective fluxes, close to or far away from where precipitation occurs. In the case of a high‐level radioactive waste disposal system, compacted bentonite is used in a buffer material in an engineering barrier system to minimize the amount of specific nuclides that breach into the surrounding host rock. To minimize breaching, it is very important to understand the transport mechanism of multiple chemical species in porous media. In the following research, we introduced FEM analysis methods using the results of the molecular dynamics simulation and homogenization analysis (MD/HA) method. First, the diffusion coefficients of ions (Cl^?, I^?, and Na⁺) in different water layers of Na‐beidellite were calculated using the MD/HA procedure under various dry density (1.2, 1.6, and 2.0 Mg/m³) and temperature (293, 323, and 363 K) conditions. Next, using FEM analysis that used the MD/HA results as input parameters, the diffusion behaviors of ions in porous media were calculated. The results indicate that the diffusion coefficients of the interlayer water in Na‐beidellite are different from the diffusion coefficients under dry density conditions. Further, the concentration profiles (C_t/C₀) of iodine and chloride are proportional to temperature but inversely proportional to dry density. Copyright © 2014 John Wiley & Sons, Ltd.     

Montecarlo DLA-type simulations of wetting effects in fluid displacement in porous media

In this work, we report diffusion-limited aggregation (DLA)-type Montecarlo computations of a stochastic model of displacement of a viscous fluid by another that preferentially wets a porous medium, for the case when both fluids are immiscible in the absence of buoyancy forces. The model has the aim to simulate cooperative invasion processes found in experiments of immiscible wetting displacement. The model considers the nonlocal effects of the Laplacian pressure field and the capillary forces via hydrodynamic equations in the Darcy regime with a boundary condition for the pressure at the interface. The boundary condition contains two different types of disorder: the capillary term, which constitutes an additive random disorder, and a term containing an effective random surface tension, which couples to a curvature (it constitutes a multiplicative random term that carries nonlocal information of the whole pressure). We generate different displacement patterns for different setting of the parameters of the model. We analyze these patterns by studying the scaling properties of the interface that separate the two fluids and calculating the fractal dimension of the interface. The results show the existence of three distinct regimes of scaling. One regime at the smallest-length scales is due to the multiplicative random disorder together with the nonlocal coupling; it reveals itself in a roughness exponent α ≈ 0.80. Additionally, we find a DLA-type scaling regime with a roughness exponent α ≈ 0.60 at the largest scales and intermediate scaling regime with α ≈ 0.70 corresponding to invasion percolation with trapping. Each regime has definite scaling ranges that depend on the capillary number and the relative wetting tendency of the fluids. The behavior of the fractal dimensions of the interfaces of the aggregates constitutes a further confirmation of the existence of three scaling regimes and the multi-self-affinity of the perimeter of the interface boundaries.