Modeling permeability in imperfectly layered porous media. II. A two-dimensional application of block-scale permeability

In the previous paper (Zijl and Stam, 1992), a theory has been developed to calculate the nine components of the three-dimensional intrinsic permeability tensor on the scale of a grid-block from a local-scale, predominantly layered subsurface. The resulting block-scale expressions can be written as a perturbation series of which the first term, or zeroth-order solution, coincides with the conventionally applied arithmetic and harmonic averages over the layers of the subsurface. The derived expressions permit the calculation of the diagonal and off-diagonal terms of the permeability tensor. In the present paper, these expressions will be applied in some numerical examples. Two basic two-dimensional hypothetical permeability distributions are adopted, and the various terms of the theoretical expressions are calculated. The results will be used to derive guidelines to discern the situations where higher order solutions can be neglected, and where conventional harmonic and arithmetic averages give a good estimate of the permeability on grid-block scale.     

Upscaling of Hooke''s Law for Imperfectly Layered Rocks

In this paper, we consider the upscaling of Hooke's law and its parameters on the fine scale, to a similar law with upscaled parameters on a larger scale. It is assumed that the fine scale material properties of the rock are imperfectly layered. In the governing equations, the deviations from perfect layering introduce a small parameter that can be used in perturbation series expansions for the stress, the strain, and the displacement. In the approximation of order zero the upscaled compliance matrix contains the well-known Backus parameters; this approximation holds exactly for a perfect layering. However, many natural rock types are imperfectly layered and in that case the approximation of order zero may not be sufficiently accurate. Therefore, we consider also the first order corrections. The derivation and results are presented both for the most general case and for the much simpler case in which the fine scale Poisson ratio may be assumed constant. From thermodynamic principles, it follows that the compliance tensor is symmetric on the fine scale. However, it is shown that the argument for symmetry cannot be extended to upscaled rigidities. One of the most important conclusions is that upscaled compliance tensors are nonsymmetric when there are trends in the deviations from perfect layering.     

Upscaling of Hooke's Law for Imperfectly Layered Rocks

In this paper, we consider the upscaling of Hooke's law and its parameters on the fine scale, to a similar law with upscaled parameters on a larger scale. It is assumed that the fine scale material properties of the rock are imperfectly layered. In the governing equations, the deviations from perfect layering introduce a small parameter that can be used in perturbation series expansions for the stress, the strain, and the displacement. In the approximation of order zero the upscaled compliance matrix contains the well-known Backus parameters; this approximation holds exactly for a perfect layering. However, many natural rock types are imperfectly layered and in that case the approximation of order zero may not be sufficiently accurate. Therefore, we consider also the first order corrections. The derivation and results are presented both for the most general case and for the much simpler case in which the fine scale Poisson ratio may be assumed constant. From thermodynamic principles, it follows that the compliance tensor is symmetric on the fine scale. However, it is shown that the argument for symmetry cannot be extended to upscaled rigidities. One of the most important conclusions is that upscaled compliance tensors are nonsymmetric when there are trends in the deviations from perfect layering.     

Free convection along a vertical wall in a porous medium with periodic permeability variation

This communication investigates the effect of permeability variation on free convection flow and heat transfer in a porous medium bounded by a vertical porous wall. A transverse periodic variation in permeability is considered to study the effect on heat transfer rate and skin friction. The problem becomes three-dimensional due to the variation in permeability. Analytical expressions for velocity, temperature, skin friction and rate of heat transfer are obtained using the perturbation technique. Effects of permeability and suction parameters on skin friction and heat transfer is studied and it is found that the increase in permeability creates a higher heat transfer rate and greater skin friction. However, the increase in suction parameter causes the skin friction values to decrease. The results obtained will be useful in the design of steam displacement processes in oil recovery and various geothermal systems.     

An expression for the permeability of anisotropic granular media

There are many expressions proposed for the permeability of isotropic media based on flow channel and pore size distribution concepts, but there are no such expressions for anisotropic media. In this paper an expression for the permeability of an anisotropic medium is proposed, which has been verified in the laboratory. The mechanism behind fluid flow through soil was investigated using microscopic computer simulations to propose an expression for macroscopic permeability. The soil was assumed to be a spatially periodic porous medium, and the Navier-Stokes equation was solved using the FEM with appropriate boundary conditions for several different arrangements of the porous medium. The basic variables influencing flow through soil at the microscopic level were identified as specific surface area, void ratio, particle shape, material heterogeneity and the arrangement of particles in a porous medium. A sensitivity analysis was carried out to obtain an expression for the permeability in terms of the above variables. The corresponding macroscopic variables for the above microscopic variables are average specific surface area, average void ratio, anisotropy, tortuosity due to material heterogeneity, and the arrangement of particles respectively. An expression for the directional permeability is proposed in terms of these variables for the most common occurrence of particles in a porous medium. For the verification of the proposed equation, the permeability values of a fine-grained sand were measured at different void ratios and were compared with those predicted by the proposed equation. The results show that the predicted permeability values from the proposed equation are very close to the measured values.     

Stochastic representation for anisotropic permeability tensor random fields

In this paper, we introduce a novel stochastic model for the permeability tensor associated with stationary random porous media. In the light of recent works on mesoscale modeling of permeability, we first discuss the physical interpretation of the permeability tensor randomness. Subsequently, we propose a nonparametric prior probabilistic model for non‐Gaussian permeability tensor random fields, making use of the information theory and a maximum entropy procedure, and provide a physical interpretation of the model parameters. Finally, we demonstrate the capability of the considered class of random fields to generate higher levels of statistical fluctuations for selected stochastic principal permeabilities. This unique flexibility offered by the parameterization of the model opens up many new possibilities for both forward simulations (e.g. for uncertainty propagation in predictive simulations) and stochastic inverse problem solving. Copyright © 2011 John Wiley & Sons, Ltd.     

非均匀多孔介质等效渗透率的普适表达式

多孔介质渗流是普遍的物理过程,涉及地下工程、地热开采、环境工程等各行各业,尤其是工程建设,常面临防渗问题。由于地质条件的复杂性,工程区域地层受到成岩、压实、风化、生物作用等各种影响,故渗流性质复杂,常需要对建设区域的渗流状况进行数值模拟,从而为工程的设计施工提供决策依据。数值仿真结果依赖于对地层介质关键参数的选取,但目前工程多将其视为均匀介质处理,对于介质的非均匀特性考虑较少。文章旨在研究非均质多孔介质渗透率空间分布与等效渗透率的关系。基于连续介质假定、达西定律以及非均匀多孔介质渗透率空间分布函数,建立一维到三维的达西渗流问题模型,通过求解偏微分方程和理论推导,得到基于渗透率空间分布函数的等效渗透率理论表达式,并与有限元计算的数值解进行对比分析,结果表明理论值和数值解误差很小,证明等效渗透率的表达式的合理性。利用该成果可通过多点局部渗透率的测定构建渗透率空间分布函数,从而对整体渗流区域的渗透性质进行快速计算和评估,从而简化异常复杂的工程地质模型以减少计算量需求,对于工程仿真的快速计算和结果评估有重要意义。     

复杂形体重力场、梯度及磁场计算方法

在改进均匀多面体重力场正演公式基础上, 利用二阶张量的坐标变换实现对多面体重力场梯度的求解, 推导了新的多面体重力梯度和磁场的正演公式, 给出了新的统一的均匀多面体重力场、梯度及磁场正演表达式形式, 并用理论模型进行了检验.同时, 应用新的多面体重力场梯度正演公式推导出新的长方体重力场垂直梯度理论表达式.本文给出的均匀多面体重力场、梯度及磁场正演表达式形式统一, 重磁场联合正演中可相互利用其计算过程中的结果, 避免重复计算以提高正演计算效率.      

Modeling the 3D crack network and anisotropic permeability of saturated cracked soil

Cracks on a natural soil surface provide preferential pathways for water infiltration and contaminant solute transport. Before the mechanical property and permeability of cracked soil can be studied, a crack network model must first be established. Based on statistical analysis of crack geometry from field observations, a new method of representing a 3D crack network was developed. A horizontal plane of a crack network was derived using the Voronoi diagram. Each vertical crack was idealized as an inverted triangular prism. The 3D permeability tensor was determined by modeling the water flow through the crack network. A computer program was developed to generate a 3D crack network automatically and compute the permeability tensor of cracked soil at different depths. The model was verified by comparing the measured permeability and computed permeability of cracked soil. The results showed that the computed permeability was consistent with the measured permeability.     

Three-Dimensional Conjugate Gradient Inversion of Magnetotelluric Impedance Tensor Data

We developed a three-dimensional(3D) conjugate gradient inversion algorithm for in-verting magnetotelluric impedance tensor measurements.In order to show the importance of including diagonal components of magnetotelluric impedance tensor in 3D inversion,synthetic data were inverted using the 3D conjugate gradient inversion,and the inversion results were compared and analyzed.The results from the 3D inversion of synthetic data indicate that both the off-diagonal and the diagonal components are required in in...     

Implementation of a Locally Conservative Numerical Subgrid Upscaling Scheme for Two-Phase Darcy Flow

We present a locally mass conservative scheme for the approximation of two-phase flow in a porous medium that allows us to obtain detailed fine scale solutions on relatively coarse meshes. The permeability is assumed to be resolvable on a fine numerical grid, but limits on computational power require that computations be performed on a coarse grid. We define a two-scale mixed finite element space and resulting method, and describe in detail the solution algorithm. It involves a coarse scale operator coupled to a subgrid scale operator localized in space to each coarse grid element. An influence function (numerical Greens function) technique allows us to solve these subgrid scale problems independently of the coarse grid approximation. The coarse grid problem is modified to take into account the subgrid scale solution and solved as a large linear system of equations posed over a coarse grid. Finally, the coarse scale solution is corrected on the subgrid scale, providing a fine grid representation of the solution. Numerical examples are presented, which show that near-well behavior and even extremely heterogeneous permeability barriers and streaks are upscaled well by the technique.     

Determination of permeability in anisotropic rock-masses from integral samples

Summary Determination of Permeability in Anisotropic Rock-Masses From Integral SamplesA method is presented which makes it possible to characterize the permeability of a rock mass as an anisotropic magnitude — i. e. to determine its permeability tensor — from a characterization of its fracturing by means of integral samples.For the purpose, a theory is developed by means of which the permeability tensor can be calculated from the attitudes and openings of the fractures and — if infillings are present — also from their coefficient of permeability. All these magnitudes are determined in integral samples, it being assumed that the sampled fractures are continuous and plane, and have the same characteristics as the section of the fractures present in the samples. Possible deviations with respect to this assumption are taken into account by means of correcting factors derived from the results of pressure tests in situ. The permeability tensor of a rock mass at a point can be determined from a single integral sample, provided this is representative of the fracturing.Results of the application of the method are presented, which show it to look very promising.With 16 Figures     

Effect of compressible parameters on vertical vibration of an elastic pile in multilayered poroelastic media

This paper mainly investigates the influences of compressible parameters on the vertical vibration of a pile embedded in layered poroelastic soil media. The pile is treated as a 1D elastic bar by the finite element method, and fundamental solutions for the layered poroelastic soils due to a vertical dynamic load are obtained by the analytical layer element method. Based on the compatibility conditions, the pile-soil dynamic interaction problem is solved. The numerical scheme has been compiled into a Fortran program for numerical calculation. Influences of the pile-soil stiffness ratio, compressible parameters, vibration frequency and the soil stratification are discussed.     

Permeability tensors for sedimentary structures

Accurate modeling of fluid flow through sedimentary units is of great importance in assessing the performance of both hydrocarbon reservoirs and aquifers. Most sedimentary rocks display structure from the mm or cm scale upwards. Flow simulation should therefore begin with grid blocks of this size in order to calculate effective permeabilities for larger structures. In this paper, we investigate several flow models for sandstones, and examine their impact on the calculation of effective permeability for single phase flow. Crossflow arises in some structures, in which case it may be necessary to use a tensor representation of the effective permeability. We establish conditions under which tensors are required, e.g., in crossbedded structures with a high bedding angle, high permeability contrast, and laminae of comparable thickness. Cases where the off-diagonal terms can be neglected, such as in symmetrical systems, are also illustrated. We indicate how the method of calculating tensor permeabilities may be extended to model multiphase flow in sedimentary structures.     

A monotone nonlinear finite volume method for diffusion equations and multiphase flows

We present a new nonlinear monotone finite volume method for diffusion equation and its application to two-phase flow model. We consider full anisotropic discontinuous diffusion or permeability tensors on conformal polyhedral meshes. The approximation of the diffusive flux uses the nonlinear two-point stencil which provides the conventional seven-point stencil for the discrete diffusion operator on cubic meshes. We show that the quality of the discrete flux in a reservoir simulator has great effect on the front behavior and the water breakthrough time. We compare two two-point flux approximations (TPFA), the proposed nonlinear TPFA and the conventional linear TPFA, and multipoint flux approximation (MPFA). The new nonlinear scheme has a number of important advantages over the traditional linear discretizations. Compared to the linear TPFA, the nonlinear TPFA demonstrates low sensitivity to grid distortions and provides appropriate approximation in case of full anisotropic permeability tensor. For nonorthogonal grids or full anisotropic permeability tensors, the conventional linear TPFA provides no approximation, while the nonlinear flux is still first-order accurate. The computational work for the new method is higher than the one for the conventional TPFA, yet it is rather competitive. Compared to MPFA, the new scheme provides sparser algebraic systems and thus is less computational expensive. Moreover, it is monotone which means that the discrete solution preserves the nonnegativity of the differential solution.     

Determining the equivalent permeability tensor for fractured rock masses using a stochastic REV approach: Method and application to the field data from Sellafield,UK

A numerical procedure to determine the equivalent permeability tensor of a fractured rock is presented, using a stochastic REV (Representative Elementary Volume) concept that uses multiple realizations of stochastic DFN (Discrete Fracture Network) models. Ten square DFN models are generated using the Monte Carlo simulations of the fracture system based on the data obtained from a site characterization program at Sellafield, Cumbria, UK. Smaller models with varying sizes of from 0.25 m×0.25 m to 10 m×10 m are extracted from the generated DFN models and are used as two-dimensional geometrical models for calculation of equivalent permeability tensor. The DFN models are also rotated in 30º intervals to evaluate the tensor characteristics of calculated directional permeability. Results show that the variance of the calculated permeability values decreases significantly as the side lengths of the DFN models increase, which justifies the existence of a REV. The REV side length found in this analysis is about 5 m and 8 m with 20% and 10% acceptable variations, respectively. The calculated directional permeability values at the REV size have tensor characteristic that is confirmed by a close approximation of an ellipse in a polar plot of the reciprocal of square roots of the directional permeability.

低渗透岩芯水驱油试验相似理论

以相似理论为基础,考虑低渗透多孔介质非达西渗流特性,研究了油水两相渗流模拟试验理论。应用方程分析法,推导出油水两相渗流控制方程的无量纲形式,获得了低渗透岩芯水驱油模拟试验的相似准则。采用隐式求解压力、显式求解饱和度的方法（IMPES）对无量纲控制方程进行求解,得到包含相似准数的数值模拟器。通过敏感性分析,得到了每个相似准数对试验结果的影响程度,并通过实际低渗透岩芯的水驱油试验结果进行了验证。结果表明,敏感因子大的相似准数对试验结果影响较大,反之影响则较小。因此,在设计物理模拟试验时,当无法满足模型和原型所有相似准数相等时,应该优先满足敏感因子较大的相似准数相等。     

The behavior of a multilayered porous thermo-elastic medium with anisotropic thermal diffusivity and permeability

With the aid of integral transform techniques, this paper presents an extended precise integration solution for thermal consolidation problems of a multilayered porous thermo-elastic medium with anisotropic thermal diffusivity and permeability due to a heat source. From the fundamental governing equations, ordinary differential equations are derived by employing Laplace–Hankel transforms. By applying the extended precise integration method, equations in the transformed domain can be solved, and the actual solutions are further obtained by adopting a numerical inverse transformation. The accuracy and feasibility of the proposed theory is demonstrated by contrastive analysis with existing studies. Finally, several examples are carried out to investigate the influence of heat source’s type, axial distance, burial depth of heat source, ratio of thermo-permeability, permeability anisotropy, thermal diffusivity anisotropy and stratification on the thermal consolidation process.     

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.