Conditional Latin Hypercube Simulation of (Log)Gaussian Random Fields

In earth and environmental sciences applications, uncertainty analysis regarding the outputs of models whose parameters are spatially varying (or spatially distributed) is often performed in a Monte Carlo framework. In this context, alternative realizations of the spatial distribution of model inputs, typically conditioned to reproduce attribute values at locations where measurements are obtained, are generated via geostatistical simulation using simple random (SR) sampling. The environmental model under consideration is then evaluated using each of these realizations as a plausible input, in order to construct a distribution of plausible model outputs for uncertainty analysis purposes. In hydrogeological investigations, for example, conditional simulations of saturated hydraulic conductivity are used as input to physically-based simulators of flow and transport to evaluate the associated uncertainty in the spatial distribution of solute concentration. Realistic uncertainty analysis via SR sampling, however, requires a large number of simulated attribute realizations for the model inputs in order to yield a representative distribution of model outputs; this often hinders the application of uncertainty analysis due to the computational expense of evaluating complex environmental models. Stratified sampling methods, including variants of Latin hypercube sampling, constitute more efficient sampling aternatives, often resulting in a more representative distribution of model outputs (e.g., solute concentration) with fewer model input realizations (e.g., hydraulic conductivity), thus reducing the computational cost of uncertainty analysis. The application of stratified and Latin hypercube sampling in a geostatistical simulation context, however, is not widespread, and, apart from a few exceptions, has been limited to the unconditional simulation case. This paper proposes methodological modifications for adopting existing methods for stratified sampling (including Latin hypercube sampling), employed to date in an unconditional geostatistical simulation context, for the purpose of efficient conditional simulation of Gaussian random fields. The proposed conditional simulation methods are compared to traditional geostatistical simulation, based on SR sampling, in the context of a hydrogeological flow and transport model via a synthetic case study. The results indicate that stratified sampling methods (including Latin hypercube sampling) are more efficient than SR, overall reproducing to a similar extent statistics of the conductivity (and subsequently concentration) fields, yet with smaller sampling variability. These findings suggest that the proposed efficient conditional sampling methods could contribute to the wider application of uncertainty analysis in spatially distributed environmental models using geostatistical simulation.     

Application of multiple-point geostatistics on modelling groundwater flow and transport in a cross-bedded aquifer (Belgium)

Sedimentological processes often result in complex three-dimensional subsurface heterogeneity of hydrogeological parameter values. Variogram-based stochastic approaches are often not able to describe heterogeneity in such complex geological environments. This work shows how multiple-point geostatistics can be applied in a realistic hydrogeological application to determine the impact of complex geological heterogeneity on groundwater flow and transport. The approach is applied to a real aquifer in Belgium that exhibits a complex sedimentary heterogeneity and anisotropy. A training image is constructed based on geological and hydrogeological field data. Multiple-point statistics are borrowed from this training image to simulate hydrofacies occurrence, while intrafacies permeability variability is simulated using conventional variogram-based geostatistical methods. The simulated hydraulic conductivity realizations are used as input to a groundwater flow and transport model to investigate the effect of small-scale sedimentary heterogeneity on contaminant plume migration. Results show that small-scale sedimentary heterogeneity has a significant effect on contaminant transport in the studied aquifer. The uncertainty on the spatial facies distribution and intrafacies hydraulic conductivity distribution results in a significant uncertainty on the calculated concentration distribution. Comparison with standard variogram-based techniques shows that multiple-point geostatistics allow better reproduction of irregularly shaped low-permeability clay drapes that influence solute transport.     

Using Multiple-Point Geostatistics for Tracer Test Modeling in a Clay-Drape Environment with Spatially Variable Conductivity and Sorption Coefficient

This study investigates the effect of fine-scale clay drapes on tracer transport. A tracer test was performed in a sandbar deposit consisting of cross-bedded sandy units intercalated with many fine-scale clay drapes. The heterogeneous spatial distribution of the clay drapes causes a spatially variable hydraulic conductivity and sorption coefficient. A fluorescent tracer (sodium naphthionate) was injected in two injection wells and ground water was sampled and analyzed from five pumping wells. To determine (1) whether the fine-scale clay drapes have a significant effect on the measured concentrations and (2) whether application of multiple-point geostatistics can improve interpretation of tracer tests in media with complex geological heterogeneity, this tracer test is analyzed with a local three-dimensional ground-water flow and transport model in which fine-scale sedimentary heterogeneity is modeled using multiple-point geostatistics. To reduce memory needs and calculation time for the multiple-point geostatistical simulation step, this study uses the technique of direct multiple-point geostatistical simulation of edge properties. Instead of simulating pixel values, model cell edge properties indicating the presence of irregularly shaped surfaces are simulated using multiple-point geostatistical simulations. Results of a sensitivity analysis show under which conditions clay drapes have a significant effect on the concentration distribution. Calibration of the model against measured concentrations from the tracer tests reduces the uncertainty on the clay-drape parameters. The calibrated model shows which features of the breakthrough curves can be attributed to the geological heterogeneity of the aquifer and which features are caused by other processes.     

Using Sequential Self-Calibration Method to Identify Conductivity Distribution: Conditioning on Tracer Test data

An iterative inverse method, the sequential self-calibration method, is developed for mapping spatial distribution of a hydraulic conductivity field by conditioning on nonreactive tracer breakthrough curves. A streamline-based, semi-analytical simulator is adopted to simulate solute transport in a heterogeneous aquifer. The simulation is used as the forward modeling step. In this study, the hydraulic conductivity is assumed to be a deterministic or random variable. Within the framework of the streamline-based simulator, the efficient semi-analytical method is used to calculate sensitivity coefficients of the solute concentration with respect to the hydraulic conductivity variation. The calculated sensitivities account for spatial correlations between the solute concentration and parameters. The performance of the inverse method is assessed by two synthetic tracer tests conducted in an aquifer with a distinct spatial pattern of heterogeneity. The study results indicate that the developed iterative inverse method is able to identify and reproduce the large-scale heterogeneity pattern of the aquifer given appropriate observation wells in these synthetic cases.     

基于局域化EnKF同时同化地下水水流和溶质运移数据的渗透系数场识别研究

局域化改进集合卡尔曼滤波（EnKF）可以克服EnKF方法在使用小集合时,对参数识别精度较低的缺陷,其能同化 地下水位观测数据有效识别渗透系数场。实际工作中,溶质运移数据也较容易获得。崔凯鹏（2013）尝试增加溶质运移 数据以改进只同化水流数据对渗透系数的估计结果,但是精度提高有限。本文在其基础上修改模型,进一步增加溶质注 入井,探究同时同化水头和溶质运移数据,对渗透系数场识别效果,之后对比了局域化EnKF与非局域化EnKF参数识别结 果,并分析了溶质影响范围与参数识别的关系。结果表明：同时同化溶质运移和水头资料,比同化单一种类观测数据识别 的渗透系数精度更高;相同实现数目下,局域化EnKF比EnKF对渗透系数场的估计结果与真实场更为接近;仅考虑溶质影 响范围内的渗透系数,同化水头数据在最后时刻参数识别结果好于同化溶质运移数据参数识别结果,但差别不大。     

基于不同地质统计方法的渗透系数场对污染物运移的影响

渗透系数场的空间变异性是影响污染物运移结果的决定因素,而地质统计方法是解决渗透系数空间变异性的主要技术手段。本文利用野外场地实测数据,采用普通克里格法和指示克里格法、顺序高斯模拟法和顺序指示模拟法四种地质统计方法,插值估测和模拟再现随机渗透系数场,进而对比研究四种渗透系数场对大尺度污染物运移的影响。研究结果表明,污染羽的质心位置（一阶矩）主要由渗透系数的平均值来决定;污染羽在空间上的展布范围（二阶矩）主要受渗透系数空间变异方差的影响;条件模拟克服了估计法的平滑效果,较好地再现真实曲线的波动性,渗透系数（ lnK）估计方差与污染羽空间二阶矩随着条件模拟次数的增加而减小,并且顺序指示模拟程度更加明显。     

多尺度非均质多孔介质中溶质运移的蒙特卡罗模拟

探讨了将蒙特卡罗(Monte Carlo)方法应用于多尺度非均质含水层中溶质运移模拟的方法。所研究的含水层由两种具有不同渗透系数统计特征的多孔介质所组成,每一种多孔介质是非均质的,且其渗透系数场符合平稳假设,而整个模拟区的渗透系数是非平稳的。Monte Carlo方法要求参数是平稳的,因此,分别对两种多孔介质产生若干随机渗透系数场后,用两种方法进行组合,并进行溶质运移的模拟计算。通过对计算结果的分析,综合考虑计算精度、计算时间等因素,得出了处理多尺度非均质多孔介质中溶质运移问题的较好方法。     

Sensitivity Analysis of a Combined Groundwater Flow and Solute Transport Model Using Local-Grid Refinement: A Case Study

Combining groundwater flow models with solute transport models represents a common challenge in groundwater resources assessments and contaminant transport modeling. Groundwater flow models are usually constructed at somewhat larger scales (involving a coarser discretization) to include natural boundary conditions. They are commonly calibrated using observed groundwater levels and flows (if available). The groundwater solute transport models may be constructed at a smaller scale with finer discretization than the flow models in order to accurately delineate the solute source and the modeled target, to capture any heterogeneity that may affect contaminant migration, and to minimize numerical dispersion while still maintaining a reasonable computing time. The solution that is explored here is based on defining a finer grid subdomain within a larger coarser domain. The local-grid refinement (LGR) implemented in the Modular 3D finite-difference ground-water flow model (MODFLOW) code has such a provision to simulate groundwater flow in two nested grids: a higher-resolution sub-grid within a coarse grid. Under the premise that the interface between both models was well defined, a comprehensive sensitivity and uncertainty analysis was performed whereby the effect of a parameter perturbation in a coarser-grid model on transport predictions using a higher-resolution grid was quantified. This approach was tested for a groundwater flow and solute transport analysis in support of a safety evaluation of the future Belgian near-surface radioactive waste disposal facility. Our reference coarse-grid groundwater flow model was coupled with a smaller fine sub-grid model in two different ways. While the reference flow model was calibrated using observed groundwater levels at a scale commensurate with that of the coarse-grid model, the fine sub-grid model was used to run a solute transport simulation quantifying concentrations in a hypothetical well nearby the disposal facility. When LGR coupling was compared to a one-way coupling, LGR was found to provide a smoother flow solution resulting in a more CPU-efficient transport solution. Parameter sensitivities performed with the groundwater flow model resulted in sensitivities at the head observation locations. These sensitivities identified the recharge as the most sensitive parameter, with the hydraulic conductivity of the upper aquifer as the second most sensitive parameter in regard to calculated groundwater heads. Based on one-percent sensitivity maps, the spatial distribution of the observations with the highest sensitivities is slightly different for the upper aquifer hydraulic conductivity than for recharge. Sensitivity analyses were further performed to assess the prediction scaled sensitivities for hypothetical contaminant concentrations using the combined groundwater flow and solute transport models. Including all pertinent parameters into the sensitivity analysis identified the hydraulic conductivity of the upper aquifer as the most sensitive parameter with regard to the prediction of contaminant concentrations.     

溶质入渗土工合成衬垫的化学-渗透特性研究

溶质在土工合成衬垫（GCL）长期入渗过程中具有明显的化学-渗透特性。室内渗透试验表明：阳离子之间的置换效应对GCL衬垫渗透系数影响较大,10 mM的CaCl2溶液使渗透系数上升至2.5×10-11 m/s,而30 mM的CaCl2溶液使渗透系数上升至5.6×10-11 m/s。渗透液体浓度的增加缩短了溶质穿透GCL的时间,且预饱和处理试剂对GCL渗透系数的变化影响较大,采用蒸馏水作为预饱和试剂处理GCL衬垫对其渗透系数的影响明显小于CaCl2溶液;建立了考虑膜效应和离子交换效应条件下溶质运移耦合动力学模型,并对GCL穿透试验过程中溶质浓度的变化进行了预测,仿真计算结果表明,10 mM和30 mM两种CaCl2溶液渗透条件下,Ca2+浓度变化的试验结果和计算结果均相吻合,验证了耦合动力学模型的可靠性;从Ca2+浓度及流量穿透曲线分布可知,化学-渗透效应可有效地延缓溶质的迁移速度。随着溶质浓度的降低,阻滞作用更显著。因此,在分析GCL衬垫中溶质入渗特征时,必须考虑化学-渗透效应的影响。     

Probability of plume interception using conditional simulation of hydraulic head and inverse modeling

A procedure to estimate the probability of intercepting a contaminant groundwater plume for monitoring network design has been developed and demonstrated. The objective of the procedure is to use all available information in a method that accounts for the heterogeneity of the aquifer and the paucity of data. The major components of the procedure are geostatistical conditional simulation and parameter estimation that are used sequentially to generate flow paths from a suspected contaminant source location to a designated monitoring transect. From the flow paths, a histogram is constructed that represents the spatial probability distribution of plume centerlines. With an independent estimate of the plume width, a relationship between the total cost and the probability of detecting a plume can be made. The method uses geostatistical information from hydraulic head measurements and is conditioned by the data and the physics of groundwater flow. This procedure was developed specifically for the design of monitoring systems at sites where very few, if any, hydraulic conductivity data are available.     

纵向弥散作用与渗透介质非均质性定量关系的模拟研究

在地下水污染模拟预报中,弥散参数是很难确定的一个模型参数。因实验室小尺度弥散规律一般不能用于大尺度弥散过程,而野外示踪试验却耗资大、周期长,限制了其实用性。文中利用随机数值模拟手段、基于随机理论的蒙特卡罗方法及序贯高斯模拟技术来生成渗透系数随机场,并研究渗透系数对数场的方差、相关长度以及变异函数类型在不同尺度上对纵向弥散度的影响,进而建立纵向弥散度与随机分布渗透系数场的方差和相关长度的统计定量关系,并与Gelhar理论计算结果进行比较。数值模拟结果表明,经过一定迁移距离后纵向弥散度与随机分布渗透系数对数场的方差和相关长度具有良好的线性统计关系,与Gelhar理论公式表达的关系类型类似。但对于较大的方差,纵向弥散度模拟结果明显大于Gelhar理论计算值,而对于较大相关长度在迁移距离不很大时,纵向弥散度模拟结果明显小于Gelhar理论计算值。本研究可为野外大尺度地下水污染预报模型中水动力弥散参数的确定提供方法借鉴。     

Analysis of karst hydrodynamics through comparison of dissolved and suspended solids' transport

In karst systems, rain events often result in a decrease of the conductivity (a tracer of dissolved phase transport) and an increase in turbidity (a tracer of suspended solids transport) at wells and springs. This study shows that the comparison of suspended solids and solute transport by the coupled approach of $T\text{–}C$ curves (Turbidity–Conductivity) and autocorrelations gives evidence of the transport processes in the karst network and allows understanding the karst hydrodynamics. To cite this article: D. Valdes et al., C. R. Geoscience 337 (2005).     

Stochastic forecasts of seawater intrusion towards sustainable groundwater management: application to the Korba aquifer (Tunisia)

A stochastic study of long-term forecasts of seawater intrusion with an application to the Korba aquifer (Tunisia) is presented. Firstly, a geostatistical model of the exploitation rates was constructed, based on a multi-linear regression model combining incomplete direct data and exhaustive secondary information. Then, a new method was designed and used to construct a geostatistical model of the hydraulic conductivity field by combining lithological information and data from hydraulic tests. Secondly, the effects of the uncertainties associated with the pumping rates and the hydraulic conductivity field on the 3D density-dependent transient model were analysed separately and then jointly. The forecasts of the impacts of two different management scenarios on seawater intrusion in the year 2048 were performed by means of Monte Carlo simulations, accounting for uncertainties in the input parameters as well as possible changes of the boundary conditions. Combining primary and secondary data allowed maps of pumping rates and the hydraulic conductivity field to be constructed, despite a lack of direct data. The results of the stochastic long-term forecasts showed that, most probably, the Korba aquifer will be subject to important losses in terms of regional groundwater resources.     

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。     

区域地下水溶质运移随机理论的研究与进展

在总结近年来国内外区域地下水溶质运动研究的基本理论、方法和部分成果的基础上,论述了溶质在大区域运动的主要影响因素为区域介质的空间变异性。首先总结了野外条件下饱和介质和非饱和介质土壤渗透性能的空间变异性结果,由于野外渗透介质严重的空间变异性,研究溶质在野外条件下的运动采用了随机理论方法。基于Lagrange方法和Euler方法,研究结果表明,在渗透系数为对数正态二阶平稳及一阶扰动近似条件下,平均浓度满足对流－弥散方程,方程中宏观弥散度决定于介质渗透性能的统计特征,总结了一系列宏观弥散系数的表达形式,在此基础上,指出了需要进一步研究的问题。     

Efficient Simulation of (Log)Normal Random Fields for Hydrogeological Applications

Two methods for generating representative realizations from Gaussian and lognormal random field models are studied in this paper, with term representative implying realizations efficiently spanning the range of possible attribute values corresponding to the multivariate (log)normal probability distribution. The first method, already established in the geostatistical literature, is multivariate Latin hypercube sampling, a form of stratified random sampling aiming at marginal stratification of simulated values for each variable involved under the constraint of reproducing a known covariance matrix. The second method, scarcely known in the geostatistical literature, is stratified likelihood sampling, in which representative realizations are generated by exploring in a systematic way the structure of the multivariate distribution function itself. The two sampling methods are employed for generating unconditional realizations of saturated hydraulic conductivity in a hydrogeological context via a synthetic case study involving physically-based simulation of flow and transport in a heterogeneous porous medium; their performance is evaluated for different sample sizes (number of realizations) in terms of the reproduction of ensemble statistics of hydraulic conductivity and solute concentration computed from a very large ensemble set generated via simple random sampling. The results show that both Latin hypercube and stratified likelihood sampling are more efficient than simple random sampling, in that overall they can reproduce to a similar extent statistics of the conductivity and concentration fields, yet with smaller sampling variability than the simple random sampling.     

Groundwater contaminant source identification based on iterative local update ensemble smoother

Identification of the location and intensity of groundwater pollution source contributes to the effect of pollution remediation, and is called groundwater contaminant source identifcation. This is a kind of typical groundwater inverse problem, and the solution is usually ill-posed. Especially considering the spatial variability of hydraulic conductivity field, the identification process is more challenging. In this paper, the solution framework of groundwater contaminant source identification is composed with groundwater pollutant transport model (MT3DMS) and a data assimilation method (Iterative local update ensemble smoother, ILUES). In addition, Karhunen-Loève expansion technique is adopted as a PCA method to realize dimension reduction. In practical problems, the geostatistical method is usually used to characterize the hydraulic conductivity feld, and only the contaminant source information is inversely calculated in the identifcation process. In this study, the identification of contaminant source information under Kriging K-field is compared with simultaneous identification of source information and K-field. The results indicate that it is necessary to carry out simultaneous identification under heterogeneous site, and ILUES has good performance in solving high-dimensional parameter inversion problems.     

An empirical probabilistic approach for constraining the uncertainty of long-term solute transport predictions in fractured rock using in situ tracer experiments

Although radionuclide tracer tests have been carried out for over 30  years, the role of tracer tests in radioactive waste repository performance assessment (PA) has been questioned due to the differences between the time scales for tracer tests and PA. The possibility of using in situ tracer tests to constrain PA time scale (over 10,000  years) solute transport has been demonstrated using a systematic “microstructural model” approach to define advective and retentive materials. A series of simulations were conducted of the “TRUE-1” sorbing solute transport experiments in a well-characterized block of fractured granite. These experiments were then used to constrain the uncertainty of long-term transport on the same pathways, using a gradient several orders of magnitude smaller. The comparison of uncertainty of this long-term transport, with and without this conditioning step, provided a measure of the ability of tracer tests to reduce PA time-scale uncertainty. Although this approach for quantifying uncertainty reduction is somewhat empirical, it does indicate the potential usefulness of tracer experiments in reducing the uncertainty of the key PA time-scale transport parameters such as the flow wetted surface, provided that immobile zone properties such as sorption (Kd), porosity and diffusivity are available.