Spatial averaging of hydraulic conductivity under radial flow conditions

Steady-state radial flow in three-dimensional heterogeneous media is investigated using a geostatistical approach. The goal of the study is to develop a model of the relationship between corescale hydraulic conductivities measured at the wellbore and the conductivity of the surrounding drainage region as measured by a larger scale flow experiment such as a pump test. Conductivity at the point or core-scale is modeled as a stationary and multivariate lognormal spatial random function. Conductivity of the drainage region is obtained by a weighted nonlinear spatial average over the point-scale values within. This empirical spatial averaging process is shown to yield excellent approximations of true effective drainage region conductivities calculated using a numerical flow model. The geostatistical model for point-scale conductivity and the spatial averaging process are used to determine the first and second order ensemble moments of drainage region conductivity. In particular, an expression is derived for the conditional expectation of drainage region conductivity given point-scale values measured at the wellbore. The results are illustrated in a case study of a well from a sandstone oil reservoir where both core and transient-test conductivity data from the same interval are available for comparison.     

含水介质各向异性对渗透系数空间变异性统计的影响

针对目前渗透系数实测样本假定满足各向同性的局限,以Borden含水层试验场实测数据为例,通过反证法进行相关数据分析,总结了含水介质各向异性对渗透系数空间变异性统计的影响,并指明渗透系数各向同性假设的适用条件以及不合理之处。同时相应地给出了研究尺度下渗透系数场能否采用平稳随机场描述的判定依据。最后为今后进一步开展渗透系数空间变异性研究提出几点建议。     

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。     

Stochastic groundwater flow analysis in the presence of trends in heterogeneous hydraulic conductivity fields

Due to changes in lithostatic pressure, differential fracturing across bedding planes and irregularities in depositional environments, hydraulic conductivity exhibits heterogeneities and trends at various spatial scales. Using spectral theory, we have examined the effect of trends in hydraulic conductivity on (1) the solution of the mean equation for hydraulic head, (2) the covariance of hydraulic head, (3) the cross-covariances of hydraulic head and log-hydraulic conductivity perturbations and their gradients, and (4) the effective hydraulic conductivity. It is shown that the field of hydraulic head is sensitive to the presence of trends in ways that cannot be predicted by the classical analysis based on stationary hydraulic conductivity fields. The controlling variables for the second moments of hydraulic head are the mean hydraulic gradient, the correlation scale of log-hydraulic conductivity and its variance, and the slope of the trend in log-hydraulic conductivity. The mean hydraulic gradient introduces complications in the analysis since it is, in general, spatially variable. In this respect, our results are approximate, yet indicative of the true role of spatially variable patterns of log-hydraulic conductivity on groundwater flow systems.     

准饱和多孔介质中圈闭气体对渗透系数的影响

含圈闭气体的地下水流称为准饱和流,准饱和流中的圈闭气体对含水层渗透系数有重大影响。通过柱试验开展了粉砂、细砂、中砂和粗砂4种介质圈闭气体饱和度与准饱和渗透系数关系的研究。结果表明:圈闭气体饱和度明显受介质的粒径影响,在细粒介质中圈闭气体饱和度明显较大;4种介质圈闭气体饱和度在0~15%范围内,准饱和渗透系数与完全饱和相比减少了32.82%~56.38%,且准饱和渗透系数与圈闭气体饱和度之间可表达为一个负线性相关的经验公式;该公式与Faybishenko公式等效,但形式简单,参数较少,使用方便;准饱和渗透系数的变化规律可概化为圈闭气体占据了原有的有效孔隙,造成原有效孔隙度减少,从而使渗透系数减小。利用该理论,Kozeny-Carman方程能较准确地描述准饱和渗透系数的变化规律,而基于哈根-泊肃叶方程的渗透系数公式则存在较大误差,不适用于描述准饱和渗透系数;试验结果证明了室内测定饱和渗透系数时排除圈闭气体的必要性。     

Unsteady source flow in weakly heterogeneous porous media

Average nonuniform flows in heterogeneous formations are modeled with the aid of the nonlocal effective Darcy's law. The mean head for flow toward source of instantaneous discharge in a heterogeneous medium of given statistics represents the fundamental solution of the average flow equation and is called the Mean Green Function (MGF). The general representation of the MGF is obtained for weakly heterogeneous formations as a functional of the logconductivity correlation function. For Gaussian logconductivity correlation, the MGF is derived in terms of one quadrature in time t and it is analyzed for isotropic media of any dimensionality d and for 3D axisymmetric formations. The MGF is further applied to determining the mean head distribution for flow driven by a continuous source of constant discharge. The large time asymptotic of the mean head is analyzed in details.     

Calculating the internodal transmissibilities using finite analytic method and its application for multi‐phase flow in heterogeneous porous media

In the traditional numerical reservoir simulations, the internodal transmissibility is usually defined as the harmonic mean of the permeabilities of the adjacent grids. This definition underestimates the phase flux and the speed of the saturation front, especially for the strong heterogeneous case. In this article, the internodal transmissibility is recalculated according to the nodal analytic solution. The redefined internodal transmissibility can be used directly to calculate the multiphase flow in the numerical reservoir simulations. Numerical examples show that, compared to the traditional numerical methods, the proposed scheme makes the convergences much faster as the refinement parameter increases, and the accuracy is independent of the heterogeneity. Copyright © 2016 John Wiley & Sons, Ltd.     

高水力梯度条件下颗粒堆积型多孔介质渗流规律试验研究

煤矿开采面临的水文地质条件越来越复杂,尤其是遭遇承压含水层的水压力越来越大,突水灾害发生时必然会带来高水力梯度引起的破碎岩体突水通道内高速非线性渗流问题。据此,研制高水力梯度（最大600）条件下堆积型多孔介质中高速非线性渗流试验装置,采用堆积型钢球模拟破碎岩体,对粒径为1、2、3、4、5、6 mm共6种光滑钢球分别开展了一维均质圆柱渗流试验。试验结果表明：对于由1～6 mm钢球堆积而成的孔隙率为0.44～0.45的多孔介质,当水力梯度大于145时,通过分析水力梯度-平均流速（J-v）曲线和水力梯度-雷诺数（J-Re）关系曲线,将流动状态划分为3个模式：线性层流、非线性层流、紊流,并获得了从线性层流过渡到非线性层流的临界流速为0.23～0.78 cm/s、临界水力梯度为3～8;从层流到紊流转捩的临界流速为1.6～4.8 cm/s、临界水力梯度为90～145。从小粒径多孔介质到大粒径多孔介质的渗流过程中,临界流速越来越大,而临界水力梯度逐渐减小。 渗透率与粒径的平方、非达西流影响系数与粒径的倒数均呈线性正相关,非达西流影响系数随着渗透率的增加呈指数减小。该研究对多孔介质非线性渗流的理论研究以及实际工程中高承压含水层突涌水问题有重要借鉴意义。     

准饱和多孔介质中地下水驱替速率、圈闭气体饱和度和准饱和渗透系数的关系

圈闭有不连续、非流动和非混溶气体的含水层,被称为准饱和含水层。圈闭气体的存在会降低含水层的渗透性能,但准饱和渗透系数、圈闭气体饱和度和驱替速率之间的关系还没有完全被理解。本文通过室内砂性介质的驱替试验开展了圈闭气体饱和度、准饱和渗透系数和驱替速率三者相互关系的研究。研究表明,虽然总体上圈闭气体饱和度随驱替速率的增大而减小,但是在高驱替速率和低驱替速率分别出现了圈闭气体饱和度不随驱替速率变化的等值区间,该“台阶”型曲线可以采用与van Genuchten曲线类似的数学模型来表征,而砂性介质的准饱和渗透系数与圈闭气体饱和度之间呈简单的线性负相关关系。这两个经验模型说明可以用驱替速率定量估算准饱和渗透系数,该关系为利用驱替速率估算准饱和渗透系数提供了理论依据。上述准饱和水力学性质为深入理解准饱和地下水水流提供了依据。另外,驱替速率与圈闭气体饱和度的关系还表明,实验室测量完全饱和渗透系数时,应以大于初始毛细上升速率的速率进行饱和,以避免介质中的空气被水圈闭。     

Dynamic capillary effects in heterogeneous porous media

In standard multi-phase flow models on porous media, a capillary pressure saturation relationship developed under static conditions is assumed. Recent experiments have shown that this static relationship cannot explain dynamic effects as seen for example in outflow experiments. In this paper, we use a static capillary pressure model and a dynamic capillary pressure model based on the concept of Hassanizadeh and Gray and examine the behavior with respect to material interfaces. We introduce a new numerical scheme for the one-dimensional case using a Lagrange multiplier approach and develop a suitable interface condition. The behavior at the interface is discussed and verified by various numerical simulations.     

A pore-scale numerical model for flow through porous media

A pore-scale numerical model based on Smoothed Particle Hydrodynamics (SPH) is described for modelling fluid flow phenomena in porous media. Originally developed for astrophysics applications, SPH is extended to model incompressible flows of low Reynolds number as encountered in groundwater flow systems. In this paper, an overview of SPH is provided and the required modifications for modelling flow through porous media are described, including treatment of viscosity, equation of state, and no-slip boundary conditions. The performance of the model is demonstrated for two-dimensional flow through idealized porous media composed of spatially periodic square and hexagonal arrays of cylinders. The results are in close agreement with solutions obtained using the finite element method and published solutions in the literature. Copyright © 1999 John Wiley & Sons, Ltd.     

JHomogenizer: a computational tool for upscaling permeability for flow in heterogeneous porous media

This paper presents an object-oriented programming approach for the design of numerical homogenization programs, called JHomogenizer. It currently includes five functional modules to compute effective permeability and simple codes for computing solutions for flow in porous media. Examples with graphical output are shown to illustrate some functionalities of the program. A series of numerical examples demonstrates the effectiveness of the methodology for two-phase flow in heterogeneous reservoirs. The software is freely available, and the open architecture of the program facilitates further development and can adapt to suit specific needs easily and quickly.     

Simulations of solute transport in fractured porous media using 2D percolation networks with uncorrelated hydraulic conductivity fields

Two-dimensional percolation networks have been used to model a disordered and fractured porous medium. The advantage of percolation networks is that they allow the flow and transport properties of the system to be systematically studied as a function of the connectivity of the fractures and/or the permeable regions. The aim of this research is to study hydrodynamic dispersion in such networks, and to investigate the behavior of the longitudinal dispersion coefficient D_L with binary and log-normally distributed hydraulic conductivity fields. In particular, the study focuses on the behavior of D_L at the percolation threshold p_c, where the insufficiency of flow field homogenization and the limited number of tortuous paths for flow and transport force D_L to behave anomalously, i.e., to be scale- and time-dependent. The simulations indicate that the D_L population taken over a large number of the network realizations resembles a log-normal distribution, hence indicating that, unlike the hydraulic conductivity, D_L is not a self-averaged property whose variance should tend to zero when the size of the system tends to infinity. In addition, it was found that the power law that characterizes the scale dependence of D_L is contingent upon its computation method. Moreover, D_L is found to have a completely different behavior in networks with low and high connectivities.

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Resumen Simulaciones de transporte de solutos en medio heterogéneo utilizando redes de precolación 2D con campos de conductividad no correlacionados. Se han utilizado redes de percolación en dos dimensiones para modelizar un medio poroso fracturado y desordenado. La ventaja de redes de percolación es que permiten estudiar sistemáticamente las propiedades de flujo y transporte del sistema en función de la conectividad de las fracturas y/o regiones permeables. El objetivo de esta investigación es estudiar la dispersión hidrodinámica en las redes mencionadas, e investigar el comportamiento del coeficiente de dispersión longitudinal DL con campos de conductividad hidráulica que tienen distribución log-normal y binaria. El estudio se enfoca particularmente en el comportamiento de DL cuando se alcanza el threshold de percolación pc, donde la insuficiencia de homogenización del campo de flujo y el número limitado de trayectorias tortuosas de flujo y fuerza de transporte DL se comportan de manera anómala, i.e., son dependientes del tiempo y la escala. Las simulaciones indican que la población DL tomada en un número grande en las redes realizadas se asemeja a una distribución log-normal, indicando por lo tanto que, a diferencia de la conductividad hidráulica, DL no es una propiedad auto-promediable cuya variación tienda a cero cuando el tamaño del sistema tiende al infinito. Se encontró además que la ley potencia que caracteriza la dependencia en escala de DL es dependiente del método de cálculo. Más aún, se encontró que DL tiene un comportamiento completamente distinto en las redes de alta y baja conectividad.

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Résumé Des réseaux de percolation bi-dimensionnels ont été utilisés pour modéliser des milieux poreux fracturés et désordonnés. Lavantage de ces réseaux est quils permettent détudier les propriétés découlement et de transport en fonction de la connexité des fractures et/ou des zones perméables. Lobjectif est dappréhender la dispersion hydrodynamique et dinvestiguer le comportement du coefficient de dispersion longitudinal D_L pour des réseaux ayant une distribution des conductivités hydrauliques binaire et log-normale. En particulier, le comportement de D_L a été étudié au seuil de percolation p_c, là où linhomogénéité des vitesses du fluide et le nombre limité de chemins tortueux disponibles pour lécoulement et le transport entraînent un comportement anormal de D_L, à savoir, une dépendance vis-à-vis des échelles de temps et despace. Les simulations montrent que les populations de D_L définies pour un grand nombre de réseaux ressemblent à des distributions log-normales, indiquant que, contrairement à la conductivité hydraulique, D_L nest pas une propriété dont la variance tend vers zéro lorsque la taille du système tend vers linfini. Il a également été trouvé que les lois de puissance qui caractérisent la dépendance déchelle de D_L découlent directement de la méthode de calcul. Enfin, les simulations engendrent un comportement très différent de D_L dans des réseaux faiblement ou fortement connectés.

     

基于连续分形理论的土壤非饱和水力传导度的研究

土壤的孔隙是具有连续分形性质的物理结构,根据土壤孔隙分形结构建立了非饱和水力传导度模型。模型包括综合系数、分形维数和临界体积比3个参数,综合系数为不同土壤基质势对应的非饱和水力传导度与饱和传导度之间的水力联系,与土壤质地有关;分形维数反映土壤孔隙结构对于非饱和水力传导度的作用,土壤不同尺寸孔隙之间的连通性则通过临界体积加以描述。模型具有较为明确的物理解释。将模型应用于5种不同土壤的结果表明,所提出的非饱和水力传导度模型具有较好的模拟效果。     

土壤非饱和导水率模型中参数的敏感性分析

针对邵明安根据再分布过程得出的非饱和导水率模型,对其中参数的敏感性进行分析,通过实测资料计算非饱和导水率对不同参数的敏感度,对比参数对非饱和导水率的影响程度。研究表明采用线性关系表示土壤湿润剖面的平均湿度和湿润锋处湿度之间的关系时,非饱和导水率对参数的敏感性比其他参数高得多。     

Transverse and longitudinal fluid flow modelling in fractured porous media with non‐matching meshes

A new discrete fracture model is introduced to simulate the steady‐state fluid flow in discontinuous porous media. The formulation uses a multi‐layered approach to capture the effect of both longitudinal and transverse permeability of the discontinuities in the pressure distribution. The formulation allows the independent discretisation of mesh and discontinuities, which do not need to conform. Given that the formulation is developed at the element level, no additional degrees of freedom or special integration procedures are required for coupling the non‐conforming meshes. The proposed model is shown to be reliable regardless of the permeability of the discontinuity being higher or lower than the surrounding domain. Four numerical examples of increasing complexity are solved to demonstrate the efficiency and accuracy of the new technique when compared with results available in the literature. Results show that the proposed method can simulate the fluid pressure distribution in fractured porous media. Furthermore, a sensitivity analysis demonstrated the stability regarding the condition number for wide range values of the coupling parameter.     

Theoretical and numerical study of the steady‐state flow through finite fractured porous media

This paper investigates the two‐dimensional flow problem through an anisotropic porous medium containing several intersecting curved fractures. First, the governing equations of steady‐state fluid flow in a fractured porous body are summarized. The flow follows Darcy's law in matrix and Poiseuille's law in fractures. An infinite transversal permeability is considered for the fractures. A multi‐region boundary element method is used to derive a general pressure solution as a function of discharge through the fractures and the pressure and the normal flux on the domain boundary. The obtained solution fully accounts for the interaction and the intersection between fractures. A numerical procedure based on collocation method is presented to compute the unknowns on the boundaries and on the fractures. The numerical solution is validated by comparing with finite element solution or the results obtained for an infinite matrix. Pressure fields in the matrix are illustrated for domains containing several interconnected fractures, and mass balance at the intersection points is also checked. Copyright © 2013 John Wiley & Sons, Ltd.     

利用ERT数据推求非均质多孔介质渗透系数初探

高密度电阻率成像法（ERT）能通过改变间距来对不同尺度上的地质体特征进行描述，近年来广泛地应用于地质、工程等领域。本文从电流场和渗流场的相似性出发来研究电阻率和渗透系数之间的关系，简要介绍了利用ERT技术所得的电阻率数据推求渗透系数的基本原理，并利用张家港大新和塘桥两处的ERT测量数据推求出渗透系数。与两地抽水试验结果和岩性取样分析的结果分别进行对比，结果较为吻合。     

Multiscale finite-volume method for density-driven flow in porous media

The multiscale finite-volume (MSFV) method has been developed to solve multiphase flow problems on large and highly heterogeneous domains efficiently. It employs an auxiliary coarse grid, together with its dual, to define and solve a coarse-scale pressure problem. A set of basis functions, which are local solutions on dual cells, is used to interpolate the coarse-grid pressure and obtain an approximate fine-scale pressure distribution. However, if flow takes place in presence of gravity (or capillarity), the basis functions are not good interpolators. To treat this case correctly, a correction function is added to the basis function interpolated pressure. This function, which is similar to a supplementary basis function independent of the coarse-scale pressure, allows for a very accurate fine-scale approximation. In the coarse-scale pressure equation, it appears as an additional source term and can be regarded as a local correction to the coarse-scale operator: It modifies the fluxes across the coarse-cell interfaces defined by the basis functions. Given the closure assumption that localizes the pressure problem in a dual cell, the derivation of the local problem that defines the correction function is exact, and no additional hypothesis is needed. Therefore, as in the original MSFV method, the only closure approximation is the localization assumption. The numerical experiments performed for density-driven flow problems (counter-current flow and lock exchange) demonstrate excellent agreement between the MSFV solutions and the corresponding fine-scale reference solutions.