Spatial averaging of hydraulic conductivity in three-dimensional heterogeneous porous media

Numerical models of groundwater flow require the assignment of hydraulic conductivities to large grid blocks discretizing the flow domain; however, conductivity data is usually available only at the much smaller scale of core samples. This paper describes a geostatistical model for hydraulic conductivity at both the core or point scale and that of grid blocks. Conductivity at the block scale is obtained empirically as a spatial power-average of point scale values. Assuming a multivariate Gaussian model for point log-conductivity, expressions are derived for the ensemble mean and variance of block conductivity. The expression for the ensemble mean of block scale conductivity is found to be similar to an expression for the ensemble effective conductivity of an infinite field derived analytically by earlier authors. Here, block conductivities obtained by power averaging are compared with effective conductivities obtained from a numerical flow model and are found to be in excellent agreement for a suitably chosen averaging exponent. This agreement deteriorates gradually as the log variance of conductivity increases beyond 2. For arbitrary flow field geometry and anisotropic conductivity covariances, the averaging exponent can be calibrated by recourse to numerical flow experiments. For cubic fields and an isotropic spatial covariance, the averaging exponent is found to be 1/3. In this particular case, it was found that flow field discretization at the block scale through local averaging of point conductivities gave similar results to those obtained directly using a point scale discretization of the flow field.     

含水层渗透系数空间变异性对地下水数值模拟的影响

地下水数值模拟是目前定量研究地下水水量和水质的重要手段。使用基于随机理论的MonteCarlo方法来进行地下水数值模拟。这种方法能较好地考虑水文地质参数的空间变异性。主要将MonteCarlo方法和确定性模型模拟方法的模拟结果在渗透系数场、水头场、速度场和浓度场等方面进行了比较。结果表明:在模拟三维非均质含水层中的溶质运移问题时,充分考虑了含水层渗透系数空间变异性的MonteCarlo法比确定性方法更为有效,模拟精度提高了很多,且对模拟误差及误差来源有合理的数学解释。     

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.     

Methods for determining the in situ hydraulic conductivity of shallow aquitards – an overview

Shallow clay-rich aquitards limit groundwater recharge to underlying aquifers, but they also protect the aquifers from contamination. The bulk hydraulic conductivity of such shallow aquitards can range from less than 1 mm/year to more than 100 m/year and may be much greater than the hydraulic conductivity of small intact samples of the aquitard material. This enhanced hydraulic conductivity diminishes the qualities of the aquitards for the protection of underlying aquifers but allows a higher rate of recharge. For aquifers that are overlain by aquitards, management and protection of groundwater resources may be critically dependent on reliable determinations of aquitard permeability. A variety of methods for determining bulk hydraulic conductivities of shallow clay aquitards is available; each has drawbacks and advantages, and each is based on simplifying assumptions. These methods include slug tests, pumping tests, response of the aquitard to mechanical loading, and analysis of natural pore-pressure fluctuations. Several of the commonly used methods require an independent measurement of specific storage. Laboratory methods for determining specific storage are probably not representative of in situ conditions and may lead to overestimation of aquitard permeability. Much of the theory developed to date depends on the assumption that horizontal displacement of the solid material is negligible, and this may not be a valid assumption for highly deformable media such as clay aquitards. However, with judicious selection of the most suitable methods for a particular site, good test design, careful instrumentation, and respect for the underlying assumptions, reliable determinations of aquitard permeability can be obtained. Electronic Publication     

河床水力传导度及其各向异性的测定

采用直接测定法观测了黑河中游段河床水力传导度及其各向异性,结果表明:河床水力传导度不仅存在较强的各向异性,而且存在空间尺度上的变异性.河床中心位置在垂直、水平和θ=30°方向的平均水力传导度分别为0.45、22.49和1.71 m/d,河床边分别为5.95、29.69、16.80 m/d.在同一测点,水力传导度随着与水平方向的夹角增大呈幂函数曲线下降.试验结果表明河床边是河水的主要渗漏区,并且以侧渗为主.     

1-, 2-, and 3-dimensional effective conductivity of aquifers

Starting with a stochastic differential equation with random coefficients describing steady-state flow, the effective hydraulic conductivity of 1-, 2-, and 3-dimensional aquifers is derived. The natural logarithm of hydraulic conductivity (lnK) is assumed to be heterogeneous, with a spatial trend, and isotropic. The effective conductivity relates the mean specific discharge in an aquifer to the mean hydraulic gradient, thus its importance in predicting Darcian discharge when field data represent mean or average values of conductivity or hydraulic head. Effective conductivity results are presented in exact form in terms of elementary functions after the introduction of special sets of coordinate transformations in two and three dimensions. It was determined that in one, two, and three dimensions, for the type of aquifer heterogeneity considered, the effective hydraulic conductivity depends on: (i) the angle between the gradient of the trend of lnK and the mean hydraulic gradient (which is zero in the one-dimensional situation); (2) (inversely) on the product of the magnitude of the trend gradient of lnK, b, and the correlation scale of lnK, and (3) (proportionally) on the variance of lnK, _f ² . The productb plays a central role in the stability of the results for effective hydraulic conductivity.     

磁共振探测估算含水层渗透系数的原位试验研究

通过一个完整的磁共振探测（MRS）、钻探、抽水试验过程,开展了MRS估算含水层渗透系数的原位试验研究。在分析MRS估算渗透系数准确性的基础上,系统剖析了引起准确性差异的主要因素,并对进一步提高估算效果提出了建议。结果显示：（1）采用现阶段常用的参数设置,与抽水试验计算值相比,MRS估算渗透系数的差值为抽水试验计算值的17.59%。（2）MRS推断的含水层顶板、底板埋深与钻孔揭示的信息相比,差值分别为4.11m、1.03m,表明其对估算渗透系数准确性的影响较小。（3）CP值是影响估算准确性的重要因素,其值为10-9数量级符合大多数地层的特点。另外,通过在已知渗透系数的钻孔附近进行MRS,从而获取CP参考值,应用该值估算的渗透系数准确性高于常用的参数设置。（4）指数a、b设置方面,应用Seevers公式（a=1、b=2）的估算效果优于Kenyon公式（a=4、b=2）。本成果有助于提高MRS估算渗透系数方法在野外条件下的适用性。     

Evaluation of heterogeneity and field-scale groundwater flow regime in a leaky till aquitard

Recent work in southern Ontario, Canada, demonstrates anomalously high vertical groundwater flow velocities (>1 m/year) through a thick (as much as 60 m), sandy silt till aquitard (Northern till), previously assumed to be of very low permeability (hydraulic conductivity <10^–10 m/s). Rapid recharge is attributed to the presence of fractures and sedimentary heterogeneities within the till, but the field-scale flow regime is poorly understood. This study identifies the nature of physical groundwater pathways through the till and provides estimates of the associated groundwater fluxes. The aquitard groundwater flow system is characterized by integrating details of the outcrop and subsurface sedimentary characteristics of the till with field-based hydrogeologic investigation and numerical modeling. Outcrop and subsurface data identify a composite internal aquitard stratigraphy consisting of tabular till beds (till elements) separated by laterally continuous sheet-like sands and gravels (interbeds) and boulder pavements. Individual till elements contain sedimentary heterogeneities, including discontinuous sand and gravel lenses, vertical sand dikes, and zones of horizontal and vertical fractures. Hydrogeologic field investigations indicate a three-layer aquitard flow system, consisting of upper and lower zones of more hydraulically active and heterogeneous till separated by a middle unit of relatively lower hydraulic conductivity. Groundwater pathways and fluxes in the till were evaluated using a two-dimensional aquitard/aquifer flow model which indicates a step-wise flow mechanism whereby groundwater moves alternately downward along vertical pathways (fractures, sedimentary dikes) and laterally along horizontal sand interbeds within the till. This model is consistent with observed hydraulic-head and isotope profiles, and the presence of tritiated pore waters at various depths throughout the till. Simulations suggest that a bulk aquitard vertical hydraulic conductivity on the order of 1×10^–9 m/s is required to reproduce observed hydraulic-head and tritium profiles. Electronic Publication     

Digital simulation of the scale effect in hydraulic conductivity

 Measured hydraulic conductivity increases with the scale of testing, but the reason for this increase is not clear. Possibly, high-conductivity heterogeneities are more effective in raising hydraulic conductivity over the regional scale than at the local scale. Alternatively, borehole skin and storage effects, among others, can systematically bias the results of small-scale tests; thus, the increase may simply be an artifact of the test method. Radial-flow tests were simulated at various scales in digital models of fractured double-porosity media. The mean hydraulic conductivity increases until the radius of influence in the test exceeds the fracture spacing. Therefore, under radial flow, hydraulic conductivity is indeed dependent on measurement scale. The increase in hydraulic conductivity with scale is a natural consequence of heterogeneity. Over short distances, water converging toward a borehole must generally flow across heterogeneities. Therefore, small-scale tests tend to measure a weighted harmonic mean of the hydraulic-conductivity field. Over a large area, however, flow is primarily along high-conductivity heterogeneities. Therefore, large-scale tests approach a weighted arithmetic mean where high-conductivity heterogeneities have a greater influence. Received, April 1997 Revised, January 1998 Accepted, December 1997     

渗透系数空间变异性研究

水文地质参数的空间变异性是随机理论研究的基础,而渗透系数是最为重要的水文地质参数。国外有关渗透参数空间变异性的研究工作已开展很多,但渗透系数究竟服从什么分布目前尚无确切答案。利用Borden含水层试验数据,对渗透系数的空间变异性进行探讨,结果表明若处理方法得当,渗透系数应服从对数正态分布。同时,还对今后野外开展含水层渗透系数空间变异性试验研究提出几点建议。     

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。     

土壤饱和导水率空间预测的不确定性分析

当土壤转换函数应用于土壤水力性质估计时,对于预测值的不确定性往往容易被忽视。为了有针对性地提出减少这种不确定性的方法和措施,提高土壤转换函数的实际应用能力,以两种现有的土壤转换函数(Vereecken和HYPRES模型)为例,将其应用于山东省平度市土壤饱和导水率的空间预测,并利用拉丁超立方抽样(LHS)方法对预测结果的不确定性进行了分析。结果表明,饱和导水率空间预测的不确定性主要来源于土壤基本性质的空间插值误差和土壤转换函数自身的预测误差。当Vereecken模型应用于饱和导水率空间预测时,预测结果的不确定性主要由土壤基本性质空间插值误差所决定,土壤转换函数预测误差的影响较小,而HYPRES模型则是受二者的双重影响。     

The hydraulic conductivity field and groundwater flow in the unconfined aquifer system of the Keta Strip,Ghana

This study investigates the hydraulic conductivity field and the groundwater flow pattern as predicted by a calibrated steady state groundwater flow model for the Keta Strip, southeastern Ghana. The hydraulic conductivity field is an important parameter in evaluating aquifer properties in space, and in general basin-wide groundwater resources evaluation and management. This study finds that the general hydraulic conductivity of the unconsolidated unconfined aquifer system of the Keta Strip ranges between 2 m/d and 20 m/d, with an average of 15 m/d. The spatial variation in horizontal hydraulic conductivity appears to take the trend in the variations in the nature of the material in space. Calibrated groundwater recharge suggests that 6.9–34% of annual precipitation recharges the shallow aquifer system. This amount of recharge is significant and suggests high fortunes in terms of groundwater resources development for agriculture and industrial activities in the area. A spatial distribution of groundwater recharge from precipitation is presented in this study. The spatial pattern appears to take the form of the distribution in horizontal hydraulic conductivity, and suggests that the vertical hydraulic conductivity takes the same pattern of spatial variation as the horizontal hydraulic conductivity. This is consistent with observations in other areas. The resulting groundwater flow is dominated by local flow systems as the unconfined system is quite shallow. A general northeast – southwest flow pattern has been observed in the study area.     

地下水污染强度及渗透系数场的反演识别研究

在制定地下水污染修复方案时,污染源参数和渗透系数场是最重要的地下水数值模型参数,但前人研究多集中于单一类型参数的识别。文章中采用地下水污染物运移模型（MT3DMS）和数据同化方法（迭代局部更新集合平滑器,ILUES）构成地下水污染源识别的求解框架,并利用Karhunen-Loève展开技术实现渗透系数场的参数降维,最后通过同化水头与浓度数据实现地下水污染源强和渗透系数场的联合反演。结果表明：（1）ILUES算法能精确识别污染源参数和渗透系数场,并且具有很高的普适性;（2）精确表征渗透系数在空间上呈现出的非均质性,是预测污染物迁移路径、反演污染强度的关键;（3）ILUES算法参数影响着反演效果,综合考虑计算效率和计算精度等,可以得到算例的最佳样本集合大小（Ne=4000）和ILUES算法最佳参数组合（局部临近样本集合占比α=0.4,相对权重b=4）。但在实际工程案例中,如果对精度的要求不是过高,经验组合（α=0.1,b=1）更值得推荐。研究结果对于区域地下水资源调查、评价和管理等工作具有较强的实践意义,并可为后期地下水污染预测及地下水监测井网优化提供技术支撑。     

Variations in hydraulic conductivity with scale of measurement during aquifer tests in heterogeneous, porous carbonate rocks

 Previous studies have shown that hydraulic conductivity of an aquifer seems to increase as the portion of the aquifer tested increases. To date, such studies have all relied on different methods to determine hydraulic conductivity at each scale of interest, which raises the possibility that the observed increase in hydraulic conductivity is due to the measurement method, not to the scale. This study analyzes hydraulic conductivity with respect to scale during individual aquifer tests in porous, heterogeneous carbonate rocks in southeastern Wisconsin, USA. Results from this study indicate that hydraulic conductivity generally increases during an individual test as the volume of aquifer impacted increases, and the rate of this increase is the same as the rate of increase determined by using different measurement methods. Thus, scale dependence of hydraulic conductivity during single tests does not depend on the method of measurement. This conclusion is supported by 22 of 26 aquifer tests conducted in porous-flow-dominated carbonate units within the aquifer. Instead, scale dependency is probably caused by heterogeneities within the aquifer, a conclusion supported by digital simulation. All of the observed types of hydraulic-conductivity variations with scale during individual aquifer tests can be explained by a conceptual model of a simple heterogeneous aquifer composed of high-conductivity zones within a low-conductivity matrix. Received, January 1997 Revised, August 1997, November 1997 Accepted, November 1997     

裂隙岩体渗透系数确定方法综述

总结了近年来裂隙岩体渗透系数的确定方法,主要有现场水力试验法、裂隙测量法和离散裂隙网络渗流数值试验法.每种方法都有自己的适用性和测量尺度.裂隙岩体渗透系数存在尺度效应,针对不同尺度的研究对象,应尽量选取与渗流模型网格剖分尺度匹配的测量方法.裂隙的延伸具有方向性,测试点的布设应合理科学,不同测量方法需要结合起来才能得到裂隙岩体真实的渗透系数.     

基于GA-SVR的渗透系数参数反演方法

渗透系数参数反演的本质是优化问题求解,遗传算法是一种基于自然选择和群体遗传机理的新的全局优化求解方法,可以较好地用于求解诸如渗透系数参数反演等复杂非线性组合优化问题。基于结构风险最小化原理的支持向量机具有逼近复杂非线性系统、较强的学习泛化能力,可以用来计算渗透系数参数反演过程中的测点水头值。实验表明,基于遗传算法-支持向量回归机的地下水渗透系统参数反演拟合效果良好,能大大提升区间搜索效率,避免出现局部最优解,其参数识别精度符合实际应用要求。     

高围压下砂土的渗透特性试验研究

利用高压三轴渗透试验系统,对某矿区深部的粗砂和细砂进行了高围压条件下的渗透试验。研究了同一水力梯度下渗透系数与围压的关系;同一围压下渗透系数与渗透水力梯度的关系;同一水力梯度下,围压逐级加载的渗透系数与一次加载的渗透系数之间的差别。结果表明,粗砂和细砂在同一水力梯度下渗透系数均随围压的增大而逐渐减小,在同一围压下,渗透系数会随渗透水力梯度的增大而逐渐增大;同一水力梯度下,围压逐级加载下的渗透系数明显小于一次加载条件下的渗透系数。根据围压与渗透系数的关系拟合出了两种砂样渗透系数与围压关系的数学表达式。为探究高围压下渗透系数变化的原因,研究了砂样试验过程中的体积变化和试验前后的粒度成分变化。结果表明,围压的施加过程伴随着试样的体积减小,相应的孔隙度减小,渗透系数降低;高围压条件下,砂土颗粒被挤碎成细颗粒,使得砂土的细粒含量增多,孔隙度减小,导致了砂土渗透系数的降低。     

Inverse modelling of groundwater flow around a large-scale underground cavern system considering the excavation-induced hydraulic conductivity variation

This study proposed an inverse modelling procedure for evaluating the anisotropic hydraulic conductivity and its variation induced by excavation in fractured rocks by integrating a strain-dependent hydraulic conductivity model. The time-series measurements of both hydraulic head and discharge were used to construct the objective function for improving the reliability, which was solved with a combined method of orthogonal design, transient groundwater flow modelling, artificial neural network and genetic algorithm-based optimization for reducing the computational cost. The proposed methodology proves its effectiveness by successful inverse modelling of the groundwater flow around the underground caverns at the Jinping-I Hydropower Station.