On the Ergodicity Hypothesis in Heterogeneous Formations

Application of the ergodicity hypothesis to the stochastic subsurface hydrology has been checked by investigating the hydraulic conductivity field. The relative variance of the spatial average of conductivity, which is denoted as R, and the error index E.I. = , were employed to justify the uncertainties and errors of using the ergodicity hypothesis. Six factors influence R: autocorrelation function; variance of logconductivity; spatial correlation scales of logconductivity; domain sizes; anisotropy; and dimensionality of the problem. Closed-form analytical solutions of R for the linear autocorrelation function were derived and the numerical integration of R for the exponential autocorrelation function for one-, two-, and three-dimensional problems calculated. It is easy to fulfill the ergodicity hypothesis under these conditions: weak heterogeneity and large ratio of domain size vs. correlation length (L/I). The uncertainties and errors of using this hypothesis increase rapidly when the variances of logconductivity increase and/or the ratios of L/I decrease. The ergodicity hypothesis has less error when applied to a problem with higher dimensionality.     

Equivalent Permeability and Simulation of Two-Phase Flow in Heterogeneous Porous Media

The problem of calculating equivalent grid block permeability tensors for heterogeneous porous media is addressed. The homogenization method used involves solving Darcy's equation subject to linear boundary conditions with flux conservation in subregions of the reservoir and can be readily applied to unstructured grids. The resulting equivalent permeability tensor is stable as defined relative to G-convergence. It is proposed to use both conforming and mixed finite elements to solve the local problems and compute approximations from above and below of the equivalent permeability, respectively. Comparisons with results obtained using periodic, pressure and no-flux boundary conditions and the renormalization method are presented. A series of numerical examples demonstrates the effectiveness of the methodology for two-phase flow in heterogeneous reservoirs.     

Equivalent Permeability and Simulation of Two-Phase Flow in Heterogeneous Porous Media

The problem of calculating equivalent grid block permeability tensors for heterogeneous porous media is addressed. The homogenization method used involves solving Darcy's equation subject to linear boundary conditions with flux conservation in subregions of the reservoir and can be readily applied to unstructured grids. The resulting equivalent permeability tensor is stable as defined relative to G-convergence. It is proposed to use both conforming and mixed finite elements to solve the local problems and compute approximations from above and below of the equivalent permeability, respectively. Comparisons with results obtained using periodic, pressure and no-flux boundary conditions and the renormalization method are presented. A series of numerical examples demonstrates the effectiveness of the methodology for two-phase flow in heterogeneous reservoirs.     

土壤非饱和带中的优先流

优先流是近年来研究土壤水运动所提出的术语。在某些条件下,水分在非饱和带中的运移不是简单的一维垂直向下流动,而是越过大部分土壤的体积沿着优先途径流动。优先流通常分为三类:大孔隙流,漏斗流,指流。本文分别介绍了这三类流动形式,并回顾了有关优先流的理论和实验的研究成果。     

论宁国组和胡乐组

<正> 宁国组和胡乐组系许杰(1934)研究皖南宁国县胡乐地区奥陶系时建立的二个地层单位宁国页岩和胡乐页岩沿革而来。宁国页岩下部为“各种颜色之泥质页岩(下部作绿色,中部暗兰色,顶部棕黄色),厚约一百一十米”,产 Didymograptus hirundo,D. abnormis等对笔石、四笔石、叶笔石、雕笔石化石。上部“为淡黄色、柔软之泥质页岩,计厚三十一米”,笔石丰富,有 Amplexograptus confertus, Didymograptus acutidus,D. ellesae,     

降水入渗补给过程中优先流的确定

优先流是降水、灌溉水等入渗补给地下水的主要形式之一, 流速快, 流动路径复杂, 难以定量描述.针对优先流难以定量描述的问题, 以郑州地中渗透仪观测资料为基础, 探讨了新乡亚砂土等试筒降水入渗过程及其中的优先流补给量比例.根据土壤的水力性质、气候等资料建立不存在优先流的数值模拟模型来刻画降水入渗补给过程, 通过模拟获得的地下水入渗补给量与实测地下水入渗补给量的历时曲线, 将大于模拟值的实测值视为优先流的量及确定其在总补给量中所占的比例.结果表明, 优先流占总补给量的比例在10%~80%之间; 随着土壤粘性增加, 优先流所占比例呈增加趋势; 随地下水位埋深的增大, 优先流所占比例呈逐渐下降趋势.      

Mortar Upscaling for Multiphase Flow in Porous Media

In mortar space upscaling methods, a reservoir is decomposed into a series of subdomains (blocks) in which independently constructed numerical grids and possibly different physical models and discretization techniques can be employed in each block. Physically meaningful matching conditions are imposed on block interfaces in a numerically stable and accurate way using mortar finite element spaces. Coarse mortar grids and fine subdomain grids provide two-scale approximations. In the resulting effective solution flow is computed in subdomains on the fine scale while fluxes are matched on the coarse scale. In addition the flexibility to vary adaptively the number of interface degrees of freedom leads to more accurate multiscale approximations. This methodology has been implemented in the Center for Subsurface Modeling's multiphysics multiblock simulator IPARS (Integrated Parallel Accurate reservoir Simulator). Computational experiments demonstrate that this approach is scalable in parallel and it can be applied to non-matching grids across the interface, multinumerics and multiphysics models, and mortar adaptivity. Moreover unlike most upscaling approaches the underlying systems can be treated fully implicitly.     

Modeling Hysteresis in Porous Media Flow via Relaxation

Two-phase flow in a porous medium can be modeled, using Darcy's law, in terms of the relative permeability functions of the two fluids (say, oil and water). The relative permeabilities generally depend not only on the fluid saturations but also on the direction in which the saturations are changing. During water injection, for example, the relative oil permeability k _ro falls gradually until a threshold is reached, at which stage the k _ro begins to decrease sharply. The latter stage is termed imbibition. If oil is subsequently injected, then k _ro does not recover along the imbibition path, but rather increases only gradually until another threshold is reached, whereupon it rises sharply. This second stage is called drainage, and the type of flow that occurs between the imbibition and drainage stages is called scanning flow. Changes in permeability during scanning flow are approximately reversible, whereas changes during drainage and imbibition are irreversible. Thus there is hysteresis, or memory, exhibited by the two-phase flow in the porous medium. In this work, we describe two models of permeability hysteresis. Common to both models is that the scanning flow regime is modeled with a family of curves along which the flow is reversible. In the Scanning Hysteresis Model (SHM), the scanning curves are bounded by two curves, the drainage and imbibition curves, where the flow can only occur in a specific direction. The SHM is a heuristic model consistent with experiments, but it does not have a nice mathematical specification. For instance, the algorithm for constructing solutions of Riemann problems involves several ad hoc assumptions. The Scanning Hysteresis Model with Relaxation (SHMR) augments the SHM by (a) allowing the scanning flow to extend beyond the drainage and imbibition curves and (b) treating these two curves merely as attractors of states outside the scanning region. The attraction, or relaxation, occurs on a time scale that corresponds to the redistribution of phases within the pores of the medium driven by capillary forces. By means of a formal Chapman–Enskog expansion, we show that the SHM with additional viscosity arises from the SHMR in the limit of vanishing relaxation time, provided that the diffusion associated with capillarity exceeds that induced by relaxation. Moreover, through a rigorous study of traveling waves in the SHMR, we show that the shock waves used to solve Riemann problems in the SHM are precisely those that have diffusive profiles. Thus the analysis of the SHMR justifies the SHM model. Simulations based on a simple numerical method for the simulation of flow with hysteresis confirm our analysis.     

Numerical Homogenization of Two-Phase Flow in Porous Media

Homogenization has proved its effectiveness as a method of upscaling for linear problems, as they occur in single-phase porous media flow for arbitrary heterogeneous rocks. Here we extend the classical homogenization approach to nonlinear problems by considering incompressible, immiscible two-phase porous media flow. The extensions have been based on the principle of preservation of form, stating that the mathematical form of the fine-scale equations should be preserved as much as possible on the coarse scale. This principle leads to the required extensions, while making the physics underlying homogenization transparent. The method is process-independent in a way that coarse-scale results obtained for a particular reservoir can be used in any simulation, irrespective of the scenario that is simulated. Homogenization is based on steady-state flow equations with periodic boundary conditions for the capillary pressure. The resulting equations are solved numerically by two complementary finite element methods. This makes it possible to assess a posteriori error bounds.     

崇陵流域土石山区坡面优先流发育路径研究

大孔隙优先流对山坡产汇流过程有重要影响。为摸清太行山土石山坡不同坡位点(坡顶、坡中上、坡中、坡中下、坡脚)的大孔隙优先流发育路径规律,以崇陵流域的典型山坡为研究对象,采用亮蓝染色剂开展了野外双环入渗染色剂示踪试验,并从水平方向与垂直方向对比分析大孔隙优先流发育路径。结果表明:(1)从坡顶到坡脚,垂直方向优先流发育减弱,水平方向优先流增强。(2)坡中以下,大孔隙优先流水平发育明显;而坡中以上,垂直方向优先流发育明显,水平方向大孔隙优先流鲜有发育。(3)崇陵流域土山区坡面表层深度20 cm以上很少出现水平方向的优先流侧向补给,为垂直向下的活塞式下渗方式。20 cm以下开始出现水平方向的大孔隙优先流,30～70 cm为优先流发育显著区。以上结论可以为基于优先流的山坡产汇流模拟提供参考。     

Generalized Coarse Graining Procedures for Flow in Porous Media

This paper focuses on heterogeneous soil conductivities and on the impact their resolution has on a solution of the piezometric head equation: owing to spatial variations of the conductivity, the flow properties at larger scales differ from those found for experiments performed at smaller scales. The method of coarse graining is proposed in order to upscale the piezometric head equation on arbitrary intermediate scales. At intermediate scales large scale fluctuations of the conductivities are resolved, whereas small scale fluctuations are smoothed by a partialy spatial filtering procedure. The filtering procedure is performed in Fourier space with the aid of a low-frequency cut-off function. We derive the partially upscaled head equations. In these equations, the impact of the small scale variability is modeled by scale dependent effective conductivities which are determined by additional differential equations. Explicit results for the scale dependent conductivity values are presented in lowest order perturbation theory. The perturbation theory contributions are summed up with using a renormalisation group analysis yielding explicit results for the effective conductivity in isotropic media. Therefore, the results are also valid for highly heterogeneous media. The results are compared with numerical simulations performed by Dykaar and Kitanidis (1992). The method of coarse graining combined by a renormalisation group analysis offers a tool to derive exact and explicit expressions for resolution dependent conductivity values. It is, e.g., relevant for the interpretation of measurement data on different scales and for reduction of grid-block resolution in numerical modeling. This revised version was published online in July 2006 with corrections to the Cover Date.     

数值模拟方法验证光透法监测流体饱和度的有效性:以非均质孔隙介质中的水/气两相流为例

光透法作为一种安全经济的非侵入监测方法广泛应用于室内砂箱多相流实验中,但该方法在局部空间点上结果的可 靠性尚缺乏验证。鉴于当前难以采用实验手段直接进行局部监测点取样验证光透法结果,本文以非均质孔隙介质中的水/气 两相流为例,基于砂箱实验开展了数值模拟研究以进行论证。研究表明：（1） 模拟结果中气体在非均质含水层中的迁移规 律、展布形态和饱和度分布总体上再现了光透法的测量结果。（2） 数值模拟和光透法得到的6个局部点饱和度数据在各时 刻均拟合较好。同时选取12个时刻共5316个监测点的饱和度数据进行分析,其决定系数达0.94,这表明模拟结果从统计 角度较好地验证了光透法在局部点监测上的有效性,两者间的差异主要由人工填砂导致的介质“不均匀”引起。（3） 气体 总量方面,实测排水值、模拟结果与光透法计算结果均拟合较好。因此数值模拟结果系统地验证了光透法监测流体饱和度 的有效性。     

非均质土壤中二维非饱和土壤水分运动的随机分析

以一水平向具有空间变异而垂直向为均质的二维非饱和流动区域中的均匀入渗问题为例,应用Monte Carlo随机模拟方法分析了土壤水分变量的随机统计特性及其一、二阶矩的时空分布规律。在随机分析过程中,将所研究的流动区域土壤水力特性的空间变异性以随空间位置变化的标定参数δ(x)表示,并将标定参数δ(x)视为一维随机空间函数的实现,应用随机生成模型来生成参数δ(x)的随机样本。通过随机模拟分析得到:垂直方向上负压水头方差与平均负压水头近似呈一线性关系;随入渗时间的延长,不同深度处表征负压水头空间变异结构的自相关函数趋于一稳定结构。     

非均质含水层中渗流与溶质运移研究进展简

Natural aquifer heterogeneity controls the groundwater flow and solute transport, and how to accurately quantify the flow and solute transport in heterogeneous aquifers has received wide attention by many scholars, and has become a hot research topic in earth science. Theoretically, a systematic review is given by the following aspect: flow and solute transport model, moment analysis, multi scale analysis. The resolved and remained issues for scale conversion in hydrogeology research are pointed out. Secondly, recent advances of heterogeneous field test, uncertainty and velocity connectivity are analyzed. Finally, the geophysical inversion of aquifer heterogeneity, stochastic theory and development of stochastic simulation software, scale conversion and uncertainty of velocity connectivity, and the relationship between heterogeneity and hydrogeological condition on the major four aspects of the future research direction is pointed out.     

Comparison of Numerical Formulations for Two-phase Flow in Porous Media

Numerical approximation based on different forms of the governing partial differential equation can lead to significantly different results for two-phase flow in porous media. Selecting the proper primary variables is a critical step in efficiently modeling the highly nonlinear problem of multiphase subsurface flow. A comparison of various forms of numerical approximations for two-phase flow equations is performed in this work. Three forms of equations including the pressure-based, mixed pressure–saturation and modified pressure–saturation are examined. Each of these three highly nonlinear formulations is approximated using finite difference method and is linearized using both Picard and Newton–Raphson linearization approaches. Model simulations for several test cases demonstrate that pressure based form provides better results compared to the pressure–saturation approach in terms of CPU_time and the number of iterations. The modification of pressure–saturation approach improves accuracy of the results. Also it is shown that the Newton–Raphson linearization approach performed better in comparison to the Picard iteration linearization approach with the exception for in the pressure–saturation form.