Integration of Tracer Test Data to Refine Geostatistical Hydraulic Conductivity Fields Using Sequential Self-Calibration Method

On the basis of local measurements of hydraulic conductivity,geostatistical methods have been found to be useful in heterogeneity characterization of a hydraulic conductivity field on a regional scale. However,the methods are not suited to directly integrate dynamic production data,such as,hydraulic head and solute concentration,into the study of conductivity distribution. These data,which record the flow and transport processes in the medium,are closely related to the spatial distribution of hydraulic conductivity. In this study,a three-dimensional gradient-based inverse method-the sequential self-calibration (SSC) method-is developed to calibrate a hydraulic conductivity field,initially generated by a geostatistical simulation method,conditioned on tracer test results. The SSC method can honor both local hydraulic conductivity measurements and tracer test data. The mismatch between the simulated hydraulic conductivity field and the reference true one,measured by its mean square error (MSE),is reduced through the SSC conditional study. In comparison with the unconditional results,the SSC conditional study creates the mean breakthrough curve much closer to the reference true curve,and significantly reduces the prediction uncertainty of the solute transport in the observed locations. Further,the reduction of uncertainty is spatially dependent,which indicates that good locations,geological structure,and boundary conditions will affect the efficiency of the SSC study results.     

Groundwater contaminant source identification based on QS-ILUES

When groundwater pollution occurs,to come up with an efficient remediation plan,it is particularly important to collect information of contaminant source(location and source strength)and hydraulic conductivity field of the site accurately and quickly.However,the information can not be obtained by direct observation,and can only be derived from limited measurement data.Data assimilation of observations such as head and concentration is often used to estimate parameters of contaminant source.As for hydraulic conductivity field,especially for complex non-Gaussian field,it can be directly estimated by geostatistics method based on limited hard data,while the accuracy is often not high.Better estimation of hydraulic conductivity can be achieved by solving inverse groundwater problem.Therefore,in this study,the multi-point geostatistics method Quick Sampling(QS)is proposed and introduced for the first time and combined with the iterative local updating ensemble smoother(ILUES)to develop a new data assimilation framework QS-ILUES.It helps to solve the contaminant source parameters and non-Gaussian hydraulic conductivity field simultaneously by assimilating hydraulic head and pollutant concentration data.While the pilot points are utilized to reduce the dimension of hydraulic conductivity field,the influence of pilot points’layout and the ensemble size of ILUES algorithm on the inverse simulation results are further explored.     

Control Function Measures for Hydrodynamic Problems

This short paper shows that different choices made for control functions of information in hydrodynamic flow problems can have significant implications for interpretations of the system. Using as a simple illustration the case of a steady-state, one-dimensional flow model with no internal sources or sinks and with the hydraulic conductivity depending on a single parameter and the distance from the origin, it is shown that, even when a continuous, error-free head data set is provided, statements about the uniqueness or not of the inverse solution are conditioned on the choice of the control function. Care has to be exercised in obtaining physically meaningful results and, depending on the model assumptions and the data available, there may not be acceptable models. It is also shown that there may be more than one model behavior that is acceptable. The results have implications for the hydrodynamic upscaling problem for flow in permeable media, for ensemble averaging methods, and for parameter determination for deterministic models of permeable flow.     

Modeling hydrologic responses of the Zwalm catchment using the REW approach: propagation of uncertainty in soil properties to model output

The research presented in this paper focuses on the application of a newly developed physically based watershed modeling approach, which is called representative elementary watershed approach. The study stressed the effects of uncertainty of input parameters on the watershed responses (i.e., simulated discharges). The approach was applied to the Zwalm catchment, which is an agriculture-dominated watershed with a drainage area of 114 km² located in East Flanders, Belgium. Uncertainty analysis of the model parameters is limited to the saturated hydraulic conductivity because of its high influence on the watershed hydrologic behavior and availability of the data. The assessment of output uncertainty is performed using the Monte Carlo method. The ensemble statistical watershed responses and their uncertainties are calculated and compared with measurements. The results show that the measured discharges fall within the 95% confidence interval of the modeled discharge. This provides the uncertainty bounds of the discharges that account for the uncertainty in saturated hydraulic conductivity. The methodology can be extended to address other uncertain parameters as far as the probability density function of the parameter is defined.     

Simultaneous Integration of Pressure, Water Cut,1 and 4-D Seismic Data In Geostatistical Reservoir Modeling

A geostatistically-based inverse technique, the sequential-self calibration (SSC) method, is used to update reservoir models so that they match observed pressure, water cut and time-lapse water saturation derived from 4-D seismic. Within the SSC, a steady-state genetic algorithm (GA) is applied to search the optimal master point locations, as well as the associated optimal permeability perturbations at the master locations. GA provides significant flexibility for SSC to parameterize master point locations, as well as to integrate different types of dynamic data because it does not require sensitivity coefficients. We show that the coupled SSC/GA method is very robust. Integrating dynamic data can significantly improve the characterization of reservoir heterogeneity with reduced uncertainty. Particularly, it can efficiently identify important large-scale spatial variation patterns (e.g., well connectivity, near well averages, high flow channels and low flow barriers) embedded in the reservoir heterogeneity. Using dynamic data, however, could be difficult to reproduce the permeability values on the cell-by-cell basis for the entire model. This reveals the important evidence that dynamic data carry information about large-scale spatial variation features, while they may be not sufficient to resolve the individual local values for the entire model. Through multiple realization analysis, the large-scale spatial features carried by the dynamic data can be extracted and represented by the ensemble mean model. Furthermore, the region informed by the dynamic data can be identified as the area with significant reduced variances in the ensemble variance model. Within this region, the cell-by-cell correlation between the true and updated permeability values can be significantly improved by integrating the dynamic data.     

Improved hydrograph prediction through subsurface characterization: conditional stochastic hillslope simulations

Subsurface heterogeneity is one of the largest sources of uncertainty associated with saturated hydraulic conductivity. Recent work has demonstrated that uncertainty in hydraulic conductivity can impart significant uncertainty in runoff generation processes and surface-water flow. Here, the role of site characterization in reducing hydrograph prediction bias and uncertainty is demonstrated. A fully integrated hydrologic model is used to conduct two sets of stochastic, transient simulation experiments comprising different overland flow mechanisms: Dunne and Hortonian. Conditioning hydraulic conductivity fields using values drawn from a simulated synthetic control case are shown to reduce both mean bias and variance in an ensemble of conditional hydrograph predictions when compared with the control case. The ensemble simulations show a greater reduction in uncertainty in the hydrographs for Hortonian flow. The conditional simulations predict surface ponding and surface pressure distributions with reduced mean error and reduced root mean square error compared with unconditional simulations. Uncertainty reduction in Hortonian and Dunne flow cases demonstrates different temporal signals, with more substantial reduction achieved for Hortonian flow.     

An Application of Bayesian Inverse Methods to Vertical Deconvolution of Hydraulic Conductivity in a Heterogeneous Aquifer at Oak Ridge National Laboratory

A Bayesian inverse method is applied to two electromagnetic flowmeter tests conducted in fractured weathered shale at Oak Ridge National Laboratory. Traditional deconvolution of flowmeter tests is also performed using a deterministic first-difference approach; furthermore, ordinary kriging was applied on the first-difference results to provide an additional method yielding the best estimate and confidence intervals. Depth-averaged bulk hydraulic conductivity information was available from previous testing. The three methods deconvolute the vertical profile of lateral hydraulic conductivity. A linear generalized covariance function combined with a zoning approach was used to describe structure. Nonnegativity was enforced by using a power transformation. Data screening prior to calculations was critical to obtaining reasonable results, and the quantified uncertainty estimates obtained by the inverse method led to the discovery of questionable data at the end of the process. The best estimates obtained using the inverse method and kriging compared favorably with first-difference confirmatory calculations, and all three methods were consistent with the geology at the site.     

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。     

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.     

A stochastic collocation based Kalman filter for data assimilation

In this paper, a stochastic collocation-based Kalman filter (SCKF) is developed to estimate the hydraulic conductivity from direct and indirect measurements. It combines the advantages of the ensemble Kalman filter (EnKF) for dynamic data assimilation and the polynomial chaos expansion (PCE) for efficient uncertainty quantification. In this approach, the random log hydraulic conductivity field is first parameterized by the Karhunen–Loeve (KL) expansion and the hydraulic pressure is expressed by the PCE. The coefficients of PCE are solved with a collocation technique. Realizations are constructed by choosing collocation point sets in the random space. The stochastic collocation method is non-intrusive in that such realizations are solved forward in time via an existing deterministic solver independently as in the Monte Carlo method. The needed entries of the state covariance matrix are approximated with the coefficients of PCE, which can be recovered from the collocation results. The system states are updated by updating the PCE coefficients. A 2D heterogeneous flow example is used to demonstrate the applicability of the SCKF with respect to different factors, such as initial guess, variance, correlation length, and the number of observations. The results are compared with those from the EnKF method. It is shown that the SCKF is computationally more efficient than the EnKF under certain conditions. Each approach has its own advantages and limitations. The performance of the SCKF decreases with larger variance, smaller correlation ratio, and fewer observations. Hence, the choice between the two methods is problem dependent. As a non-intrusive method, the SCKF can be easily extended to multiphase flow problems.     

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.     

基于NSMC方法的某核工程场地裂隙地下水数值模拟不确定分析

裂隙介质渗透结构表现为高度的非均质性与各项异性。为了科学有效地预测某核工程场地裂隙地下水的流动规律,揭示裂隙岩体地下水的渗流特性,笔者等采用Pilot Point调参方法与null space Monte Carlo方法（NSMC）,开展了裂隙岩体渗透结构的不确定性分析研究,构建了符合实际水文地质条件的多个渗流数值模型集合。结果表明：该方法获得的各个实现地下水位模拟结果能够与实际观测数据较好吻合,可反映工程场地裂隙地下水动力特征与流动趋势;各个实现的参数化渗透结构在空间上存在一定的差异性,但整体变化趋势是保持一致的,渗透参数的不确定性表现为在实测数据分布区域相对较低,钻孔空白区域相对较高;该方法可以弥补单一、确定性模拟结果在表征裂隙介质渗透结构方面的局限性,有效地降低模型参数的不确定性与随机性。此方法对进一步提升裂隙岩体渗流模拟精度与预测能力,深化裂隙地下水迁移规律的认识具有重要的意义。     

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。     

Modeling approaches and strategies for data-scarce aquifers: example of the Dar es Salaam aquifer in Tanzania

Management of groundwater resources can be improved by using groundwater models to perform risk analyses and to improve development strategies, but a lack of extensive basic data often limits the implementation of sophisticated models. Dar es Salaam in Tanzania is an example of a city where increasing groundwater use in a Pleistocene aquifer is causing groundwater-related problems such as saline intrusion along the coastline, lowering of water-table levels, and contamination of pumping wells. The lack of a water-level monitoring network introduces a problem for basic data collection and model calibration and validation. As a replacement, local water-supply wells were used for measuring groundwater depth, and well-top heights were estimated from a regional digital elevation model to recalculate water depths to hydraulic heads. These were used to draw a regional piezometric map. Hydraulic parameters were estimated from short-time pumping tests in the local wells, but variation in hydraulic conductivity was attributed to uncertainty in well characteristics (information often unavailable) and not to aquifer heterogeneity. A MODFLOW model was calibrated with a homogeneous hydraulic conductivity field and a sensitivity analysis between the conductivity and aquifer recharge showed that average annual recharge will likely be in the range 80–100 mm/year.     

Aquifer hydraulic conductivity prediction via coupling model of MCMC-ANN

Grain-size distribution data,as a substitute for measuring hydraulic conductivity(K),has often been used to get K value indirectly.With grain-size distribution data of 150 sets of samples being input data,this study combined the Artificial Neural Network technology(ANN)and Markov Chain Monte Carlo method(MCMC),which replaced the Monte Carlo method(MC)of Generalized Likelihood Uncertainty Estimation(GLUE),to establish the GLUE-ANN model for hydraulic conductivity prediction and uncertainty analysis.By means of applying the GLUE-ANN model to a typical piedmont region and central region of North China Plain,and being compared with actually measured values of hydraulic conductivity,the relative error ranges are between 1.55%and 23.53%and between 14.08%and 27.22%respectively,the accuracy of which can meet the requirements of groundwater resources assessment.The global best parameter gained through posterior distribution test indicates that the GLUEANN model,which has satisfying sampling efficiency and optimization capability,is able to reasonably reflect the uncertainty of hydrogeological parameters.Furthermore,the influence of stochastic observation error(SOE)in grain-size analysis upon prediction of hydraulic conductivity was discussed,and it is believed that the influence can not be neglected.     

Statistical inverse analysis based on genetic algorithm and principal component analysis: Method and developments using synthetic data

This study concerns the identification of parameters of soil constitutive models from geotechnical measurements by inverse analysis. To deal with the non‐uniqueness of the solution, the inverse analysis is based on a genetic algorithm (GA) optimization process. For a given uncertainty on the measurements, the GA identifies a set of solutions. A statistical method based on a principal component analysis (PCA) is, then, proposed to evaluate the representativeness of this set. It is shown that this representativeness is controlled by the GA population size for which an optimal value can be defined. The PCA also gives a first‐order approximation of the solution set of the inverse problem as an ellipsoid. These developments are first made on a synthetic excavation problem and on a pressuremeter test. Some experimental applications are, then, studied in a companion paper, to show the reliability of the method. Copyright © 2009 John Wiley & Sons, Ltd.     

Hyperfiltration of nickel sulphate solutions through silty sandstone and its effect on hydraulic conductivity

Hyperfiltration (reverse osmosis) and subsequent precipitation of minerals from the hyperfiltrated solution are processes that potentially decrease the hydraulic conductivity of porous media. These processes were demonstrated by hyperfiltrating NiSO₄ solutions through fine-grained sandstone. The mineral precipitates occur in very small (mm sized) layers at the high-pressure side of the samples where they create zones of lowered hydraulic conductivity (2–3 orders of magnitude lower than initial). The total amount of precipitates is very small compared to the dissolved mass which was passed through the membrane. Hyperfiltration-induced precipitates and the resulting lowering of hydraulic conductivities were observed at solute saturations as low as 10%. Nevertheless, at saturations higher than 50% the conductivity reduction strongly increased. Full reversibility of the hydraulic conductivity reduction by extensive re-flushing with water was only obtained at low initial solute saturations (10%). This indicates that precipitated minerals in many pores are susceptible only to very slow, diffusion-controlled re-dissolution.     

Towards a hierarchical parametrization to address prior uncertainty in ensemble-based data assimilation

Ensemble-based methods are becoming popular assisted history matching techniques with a growing number of field applications. These methods use an ensemble of model realizations, typically constructed by means of geostatistics, to represent the prior uncertainty. The performance of the history matching is very dependent on the quality of the initial ensemble. However, there is a significant level of uncertainty in the parameters used to define the geostatistical model. From a Bayesian viewpoint, the uncertainty in the geostatistical modeling can be represented by a hyper-prior in a hierarchical formulation. This paper presents the first steps towards a general parametrization to address the problem of uncertainty in the prior modeling. The proposed parametrization is inspired in Gaussian mixtures, where the uncertainty in the prior mean and prior covariance is accounted by defining weights for combining multiple Gaussian ensembles, which are estimated during the data assimilation. The parametrization was successfully tested in a simple reservoir problem where the orientation of the major anisotropic direction of the permeability field was unknown.     

Estimation of Hydraulic Conductivity and Its Uncertainty from Grain-Size Data Using GLUE and Artificial Neural Networks

Various approaches exist to relate saturated hydraulic conductivity (K _s) to grain-size data. Most methods use a single grain-size parameter and hence omit the information encompassed by the entire grain-size distribution. This study compares two data-driven modelling methods??multiple linear regression and artificial neural networks??that use the entire grain-size distribution data as input for K _s prediction. Besides the predictive capacity of the methods, the uncertainty associated with the model predictions is also evaluated, since such information is important for stochastic groundwater flow and contaminant transport modelling. Artificial neural networks (ANNs) are combined with a generalised likelihood uncertainty estimation (GLUE) approach to predict K _s from grain-size data. The resulting GLUE-ANN hydraulic conductivity predictions and associated uncertainty estimates are compared with those obtained from the multiple linear regression models by a leave-one-out cross-validation. The GLUE-ANN ensemble prediction proved to be slightly better than multiple linear regression. The prediction uncertainty, however, was reduced by half an order of magnitude on average, and decreased at most by an order of magnitude. This demonstrates that the proposed method outperforms classical data-driven modelling techniques. Moreover, a comparison with methods from the literature demonstrates the importance of site-specific calibration. The data set used for this purpose originates mainly from unconsolidated sandy sediments of the Neogene aquifer, northern Belgium. The proposed predictive models are developed for 173 grain-size K _s-pairs. Finally, an application with the optimised models is presented for a borehole lacking K _s data.