An evaluation of methodologies for the generation of stochastic hydraulic conductivity fields in highly heterogeneous aquifers

 Stochastic techniques, such as Monte Carlo experiments, are more and more frequently used for the study of flow and transport in heterogeneous aquifers. When the aquifer is composed of distinct hydrofacies, a common way to model heterogeneity is to first generate equally-possible hydrofacies fields, and then convert these hydrofacies fields into hydraulic conductivity (K) fields by assigning a single K value to each facies. This technique assumes relative homogeneity of K within each facies but may not be appropriate for the most conductive facies that often exhibits substantial variability. In this paper, we assessed the impacts of assigning multiple random K, rather than a uniform K value, to the highly conductive facies on the results of a flow and transport model. A set of fifty stochastic hydrofacies maps depicting an environment similar to the Snake River Plain aquifer (SRPA) in south-east Idaho were generated. Simulations demonstrated that a uniform K value, if carefully chosen, can reasonably reproduce the specific discharges and early particle arrival times produced by multiple K values. Yet, the results obtained with a uniform K value are dramatically less variable than those obtained with multiple K values. It is therefore concluded that stochastic simulations with uniform K assigned to the most conductive and variable facies do not necessarily portray the entire uncertainty in the analyses.     

Statistic inversion of multi-zone transition probability models for aquifer characterization in alluvial fans

Understanding the heterogeneity arising from the complex architecture of sedimentary sequences in alluvial fans is challenging. This paper develops a statistical inverse framework in a multi-zone transition probability approach for characterizing the heterogeneity in alluvial fans. An analytical solution of the transition probability matrix is used to define the statistical relationships among different hydrofacies and their mean lengths, integral scales, and volumetric proportions. A statistical inversion is conducted to identify the multi-zone transition probability models and estimate the optimal statistical parameters using the modified Gauss–Newton–Levenberg–Marquardt method. The Jacobian matrix is computed by the sensitivity equation method, which results in an accurate inverse solution with quantification of parameter uncertainty. We use the Chaobai River alluvial fan in the Beijing Plain, China, as an example for elucidating the methodology of alluvial fan characterization. The alluvial fan is divided into three sediment zones. In each zone, the explicit mathematical formulations of the transition probability models are constructed with optimized different integral scales and volumetric proportions. The hydrofacies distributions in the three zones are simulated sequentially by the multi-zone transition probability-based indicator simulations. The result of this study provides the heterogeneous structure of the alluvial fan for further study of flow and transport simulations.     

Accounting for Aquifer Heterogeneity from Geological Data to Management Tools

A nested workflow of multiple‐point geostatistics (MPG) and sequential Gaussian simulation (SGS) was tested on a study area of 6 km² located about 20 km northwest of Quebec City, Canada. In order to assess its geological and hydrogeological parameter heterogeneity and to provide tools to evaluate uncertainties in aquifer management, direct and indirect field measurements are used as inputs in the geostatistical simulations to reproduce large and small‐scale heterogeneities. To do so, the lithological information is first associated to equivalent hydrogeological facies (hydrofacies) according to hydraulic properties measured at several wells. Then, heterogeneous hydrofacies (HF) realizations are generated using a prior geological model as training image (TI) with the MPG algorithm. The hydraulic conductivity (K) heterogeneity modeling within each HF is finally computed using SGS algorithm. Different K models are integrated in a finite‐element hydrogeological model to calculate multiple transport simulations. Different scenarios exhibit variations in mass transport path and dispersion associated with the large‐ and small‐scale heterogeneity respectively. Three‐dimensional maps showing the probability of overpassing different thresholds are presented as examples of management tools.     

Stochastic analysis of immiscible displacement of the fluids with arbitrary viscosities and its dependence on support scale of hydrological data

Stochastic analysis is commonly used to address uncertainty in the modeling of flow and transport in porous media. In the stochastic approach, the properties of porous media are treated as random functions with statistics obtained from field measurements. Several studies indicate that hydrological properties depend on the scale of measurements or support scales, but most stochastic analysis does not address the effects of support scale on stochastic predictions of subsurface processes. In this work we propose a new approach to study the scale dependence of stochastic predictions. We present a stochastic analysis of immiscible fluid–fluid displacement in randomly heterogeneous porous media. While existing solutions are applicable only to systems in which the viscosity of one phase is negligible compare with the viscosity of the other (water–air systems for example), our solutions can be applied to the immiscible displacement of fluids having arbitrarily viscosities such as NAPL–water and water–oil. Treating intrinsic permeability as a random field with statistics dependant on the permeability support scale (scale of measurements) we obtained, for one-dimensional systems, analytical solutions for the first moments characterizing unbiased predictions (estimates) of system variables, such as the pressure and fluid–fluid interface position, and we also obtained second moments, which characterize the uncertainties associated with such predictions. Next we obtained empirically scale dependent exponential correlation function of the intrinsic permeability that allowed us to study solutions of stochastic equations as a function of the support scale. We found that the first and second moments converge to asymptotic values as the support scale decreases. In our examples, the statistical moments reached asymptotic values for support scale that were approximately 1/10000 of the flow domain size. We show that analytical moment solutions compare well with the results of Monte Carlo simulations for moderately heterogeneous porous media, and that they can be used to study the effects of heterogeneity on the dynamics and stability of immiscible flow.     

Patterns and dynamics of river–aquifer exchange with variably-saturated flow using a fully-coupled model

The parallel physically-based surface–subsurface model PARFLOW was used to investigate the spatial patterns and temporal dynamics of river–aquifer exchange in a heterogeneous alluvial river–aquifer system with deep water table. Aquifer heterogeneity at two scales was incorporated into the model. The architecture of the alluvial hydrofacies was represented based on conditioned geostatistical indicator simulations. Subscale variability of hydraulic conductivities (K) within hydrofacies bodies was created with a parallel Gaussian simulation. The effects of subscale heterogeneity were investigated in a Monte Carlo framework. Dynamics and patterns of river–aquifer exchange were simulated for a 30-day flow event. Simulation results show the rapid formation of saturated connections between the river channel and the deep water table at preferential flow zones that are characterized by high conductivity hydrofacies. Where the river intersects low conductivity hydrofacies shallow perched saturated zones immediately below the river form, but seepage to the deep water table remains unsaturated and seepage rates are low. Preferential flow zones, although only taking up around 50% of the river channel, account for more than 98% of total seepage. Groundwater recharge is most efficiently realized through these zones. Subscale variability of K_sat slightly increased seepage volumes, but did not change the general seepage patterns (preferential flow zones versus perched zones). Overall it is concluded that typical alluvial heterogeneity (hydrofacies architecture) is an important control of river–aquifer exchange in rivers overlying deep water tables. Simulated patterns and dynamics are in line with field observations and results from previous modeling studies using simpler models. Alluvial heterogeneity results in distinct patterns and dynamics of river–aquifer exchange with implications for groundwater recharge and the management of riparian zones (e.g. river channel-floodplain connectivity via saturated zones).     

A Model-Averaging Method for Assessing Groundwater Conceptual Model Uncertainty

This study evaluates alternative groundwater models with different recharge and geologic components at the northern Yucca Flat area of the Death Valley Regional Flow System (DVRFS), USA. Recharge over the DVRFS has been estimated using five methods, and five geological interpretations are available at the northern Yucca Flat area. Combining the recharge and geological components together with additional modeling components that represent other hydrogeological conditions yields a total of 25 groundwater flow models. As all the models are plausible given available data and information, evaluating model uncertainty becomes inevitable. On the other hand, hydraulic parameters (e.g., hydraulic conductivity) are uncertain in each model, giving rise to parametric uncertainty. Propagation of the uncertainty in the models and model parameters through groundwater modeling causes predictive uncertainty in model predictions (e.g., hydraulic head and flow). Parametric uncertainty within each model is assessed using Monte Carlo simulation, and model uncertainty is evaluated using the model averaging method. Two model-averaging techniques (on the basis of information criteria and GLUE) are discussed. This study shows that contribution of model uncertainty to predictive uncertainty is significantly larger than that of parametric uncertainty. For the recharge and geological components, uncertainty in the geological interpretations has more significant effect on model predictions than uncertainty in the recharge estimates. In addition, weighted residuals vary more for the different geological models than for different recharge models. Most of the calibrated observations are not important for discriminating between the alternative models, because their weighted residuals vary only slightly from one model to another.     

Pattern-based conditional simulation with a raster path: a few techniques to make it more efficient

Mass transport is known to depend on heterogeneity in geological formations. This entails geological bodies with complex geometries. The major interest of multiple-point simulation is its ability to reproduce such geological features through the use of a training image. The idea behind the training image is to describe a geological concept with the expected geological architecture. Its structural content is then used to infer multiple-point statistics. This yields a database with a variety of possible patterns or events. In this paper, we present a hybrid algorithm combining geostatistical multiplepoint and texture synthesis techniques for simulating geological reservoir models constrained to hard data. The proposed algorithm is a two steps process, involving first analysis with the building of an organized database from the training image content, and second synthesis with the simulation of a realization. Various tests are performed to investigate the potential of the algorithm in terms of computation time and ability to properly reproduce the shapes and connectivity features of the objects represented in the training image. We also propose a few improvements to make the algorithm more efficient. Last, six examples are presented based upon different kinds of training images depicting large-scale channelized and fractured media as well as fine-scale porous media.     

From Grain to Floodplain: Evaluating heterogeneity of floodplain hydrostatigraphy using sedimentology,geophysics, and remote sensing

Stratigraphy is a fundamental component of floodplain heterogeneity and hydraulic conductivity and connectivity of alluvial aquifers, which affect hydrologic processes such as groundwater flow and hyporheic exchange. Watershed-scale hydrological models commonly simplify the sedimentology and stratigraphy of floodplains, neglecting natural floodplain heterogeneity and anisotropy. This study, conducted in the upper reach of the East River in the East River Basin, Colorado, USA, combines point-, meander-, and floodplain-scale data to determine key features of alluvial aquifers important for estimating hydrologic processes. We compare stratigraphy of two meanders with disparate geometries to explore floodplain heterogeneity and connectivity controls on flow and transport. Meander shape, orientation, and internal stratigraphy affected residence time estimates of laterally exchanged hyporheic water. Although the two meanders share a sediment source, vegetation, and climate, their divergent river migration histories resulted in contrasting meander hydrofacies. In turn, the extent and orientation of these elements controlled the effective hydraulic conductivity and, ultimately, estimates of groundwater transport and hyporheic residence times. Additionally, the meanders’ orientation relative to the valley gradient impacted the hydraulic gradient across the meanders—a key control of groundwater velocity. Lastly, we combine our field data with remotely sensed data and introduce a potential approach to estimate key hydrostratigraphic packages across floodplains. Prospective applications include contaminant transport studies, hyporheic models, and watershed models. © 2019 John Wiley & Sons, Ltd.     

Analysis of subsurface contaminant migration and remediation using high performance computing

Highly resolved simulations of groundwater flow, chemical migration and contaminant recovery processes are used to test the applicability of stochastic models of flow and transport in a typical field setting. A simulation domain encompassing a portion of the upper saturated aquifer materials beneath the Lawrence Livermore National Laboratory was developed to hierarchically represent known hydrostratigraphic units and more detailed stochastic representations of geologic heterogeneity within them. Within each unit, Gaussian random field models were used to represent hydraulic conductivity variation, as parameterized from well test data and geologic interpretation of spatial variability. Groundwater flow, transport and remedial extraction of two hypothetical contaminants were made in six different statistical realizations of the system. The effective flow and transport behavior observed in the simulations compared reasonably with the predictions of stochastic theories based upon the Gaussian models, even though more exacting comparisons were prevented by inherent nonidealities of the geologic model and flow system. More importantly, however, biases and limitations in the hydraulic data appear to have reduced the applicability of the Gaussian representations and clouded the utility of the simulations and effective behavior based upon them. This suggests a need for better and unbiased methods for delineating the spatial distribution and structure of geologic materials and hydraulic properties in field systems. High performance computing can be of critical importance in these endeavors, especially with respect to resolving transport processes within highly variable media.©1998 Elsevier Science Limited. All rights reserved     

An Open,Object-Based Framework for Generating Anisotropy in Sedimentary Subsurface Models

The spatial distribution of hydraulic properties in the subsurface controls groundwater flow and solute transport. However, many approaches to modeling these distributions do not produce geologically realistic results and/or do not model the anisotropy of hydraulic conductivity caused by bedding structures in sedimentary deposits. We have developed a flexible object-based package for simulating hydraulic properties in the subsurface—the Hydrogeological Virtual Realities (HyVR) simulation package. This implements a hierarchical modeling framework that takes into account geological rules about stratigraphic bounding surfaces and the geometry of specific sedimentary structures to generate realistic aquifer models, including full hydraulic-conductivity tensors. The HyVR simulation package can create outputs suitable for standard groundwater modeling tools (e.g., MODFLOW), is written in Python, an open-source programming language, and is openly available at an online repository. This paper presents an overview of the underlying modeling principles and computational methods, as well as an example simulation based on the Macrodispersion Experiment site in Columbus, Mississippi. Our simulation package can currently simulate porous media that mimic geological conceptual models in fluvial depositional environments, and that include fine-scale heterogeneity in distributed hydraulic parameter fields. The simulation results allow qualitative geological conceptual models to be converted into digital subsurface models that can be used in quantitative numerical flow-and-transport simulations, with the aim of improving our understanding of the influence of geological realism on groundwater flow and solute transport.     

Delineating alluvial aquifer heterogeneity using resistivity and GPR data

Conceptual geological models based on geophysical data can elucidate aquifer architecture and heterogeneity at meter and smaller scales, which can lead to better predictions of preferential flow pathways. The macrodispersion experiment (MADE) site, with >2000 measurements of hydraulic conductivity obtained and three tracer tests conducted, serves as an ideal natural laboratory for examining relationships between subsurface flow characteristics and geophysical attributes in fluvial aquifers. The spatial variation of hydraulic conductivity measurements indicates a large degree of site heterogeneity. To evaluate the usefulness of geophysical methods for better delineating fluvial aquifer heterogeneities and distribution of preferential flow paths, a surface grid of two-dimensional ground penetrating radar (GPR) and direct current (DC) resistivity data were collected. A geological model was developed from these data that delineate four stratigraphic units with distinct electrical and radar properties including (from top to bottom) (1) a meandering fluvial system (MFS); (2) a braided fluvial system (BFS); (3) fine-grained sands; and (4) a clay-rich interval. A paleochannel, inferred by other authors to affect flow, was mapped in the MFS with both DC resistivity and GPR data. The channel is 2 to 4 m deep and, based on resistivity values, is predominantly filled with clay and silt. Comparing previously collected hydraulic conductivity measurements and tracer-plume migration patterns to the geological model indicates that flow primarily occurs in the BFS and that the channel mapped in the MFS has no influence on plume migration patterns.     

Estimation of the 3D correlation structure of an alluvial aquifer from surface-based multi-frequency ground-penetrating radar reflection data

Knowledge about the stochastic nature of heterogeneity in subsurface hydraulic properties is critical for aquifer characterization and the corresponding prediction of groundwater flow and contaminant transport. Whereas the vertical correlation structure of the heterogeneity is often well constrained by borehole information, the lateral correlation structure is generally unknown because the spacing between boreholes is too large to allow for its meaningful inference. There is, however, evidence to suggest that information on the lateral correlation structure may be extracted from the correlation statistics of the subsurface reflectivity structure imaged by surface-based ground-penetrating radar measurements. To date, case studies involving this approach have been limited to 2D profiles acquired at a single antenna centre frequency in areas with limited complementary information. As a result, the practical reliability of this methodology has been difficult to assess. Here, we extend previous work to 3D and consider reflection ground-penetrating radar data acquired using two antenna centre frequencies at the extensively explored and well-constrained Boise Hydrogeophysical Research Site. We find that the results obtained using the two ground-penetrating radar frequencies are consistent with each other, as well as with information from a number of other studies at the Boise Hydrogeophysical Research Site. In addition, contrary to previous 2D work, our results indicate that the surface-based reflection ground-penetrating radar data are not only sensitive to the aspect ratio of the underlying heterogeneity, but also, albeit to a lesser extent, to the so-called Hurst number, which is a key parameter characterizing the local variability of the fine-scale structure.     

Hydraulic Tomography Offers Improved Imaging of Heterogeneity in Fractured Rocks

Fractured rocks have presented formidable challenges for accurately predicting groundwater flow and contaminant transport. This is mainly due to our difficulty in mapping the fracture‐rock matrix system, their hydraulic properties and connectivity at resolutions that are meaningful for groundwater modeling. Over the last several decades, considerable effort has gone into creating maps of subsurface heterogeneity in hydraulic conductivity (K) and specific storage (S_s) of fractured rocks. Developed methods include kriging, stochastic simulation, stochastic inverse modeling, and hydraulic tomography. In this article, I review the evolution of various heterogeneity mapping approaches and contend that hydraulic tomography, a recently developed aquifer characterization technique for unconsolidated deposits, is also a promising approach in yielding robust maps (or tomograms) of K and S_s heterogeneity for fractured rocks. While hydraulic tomography has recently been shown to be a robust technique, the resolution of the K and S_s tomograms mainly depends on the density of pumping and monitoring locations and the quality of data. The resolution will be improved through the development of new devices for higher density monitoring of pressure responses at discrete intervals in boreholes and potentially through the integration of other data from single‐hole tests, borehole flowmeter profiling, and tracer tests. Other data from temperature and geophysical surveys as well as geological investigations may improve the accuracy of the maps, but more research is needed. Technological advances will undoubtedly lead to more accurate maps. However, more effort should go into evaluating these maps so that one can gain more confidence in their reliability.     

A methodology for quantifying the value of spatial information for dynamic Earth problems

We develop a methodology for assessing the value of information (VOI) from spatial data for groundwater decisions. Two sources of uncertainty are the focus of this VOI methodology: the spatial heterogeneity (how it influences the hydrogeologic response of interest) and the reliability of geophysical data (how they provide information about the spatial heterogeneity). An existing groundwater situation motivates and in turn determines the scope of this research. The objectives of this work are to (1) represent the uncertainty of the dynamic hydrogeologic response due to spatial heterogeneity, (2) provide a quantitative measure for how well a particular information reveals this heterogeneity (the uncertainty of the information) and (3) use both of these to propose a VOI workflow for spatial decisions and spatial data. The uncertainty of the hydraulic response is calculated using many Earth models that are consistent with the a priori geologic information. The information uncertainty is achieved quantitatively through Monte Carlo integration and geostatistical simulation. Two VOI results are calculated which demonstrate that a higher VOI occurs when the geophysical attribute (the data) better discriminates between geological indicators. Although geophysical data can only indirectly measure static properties that may influence the dynamic response, this transferable methodology provides a framework to estimate the value of spatial data given a particular decision scenario.     

Simulating rainfall time-series: how to account for statistical variability at multiple scales?

Daily rainfall is a complex signal exhibiting alternation of dry and wet states, seasonal fluctuations and an irregular behavior at multiple scales that cannot be preserved by stationary stochastic simulation models. In this paper, we try to investigate some of the strategies devoted to preserve these features by comparing two recent algorithms for stochastic rainfall simulation: the first one is the modified Markov model, belonging to the family of Markov-chain based techniques, which introduces non-stationarity in the chain parameters to preserve the long-term behavior of rainfall. The second technique is direct sampling, based on multiple-point statistics, which aims at simulating a complex statistical structure by reproducing the same data patterns found in a training data set. The two techniques are compared by first simulating a synthetic daily rainfall time-series showing a highly irregular alternation of two regimes and then a real rainfall data set. This comparison allows analyzing the efficiency of different elements characterizing the two techniques, such as the application of a variable time dependence, the adaptive kernel smoothing or the use of low-frequency rainfall covariates. The results suggest, under different data availability scenarios, which of these elements are more appropriate to represent the rainfall amount probability distribution at different scales, the annual seasonality, the dry-wet temporal pattern, and the persistence of the rainfall events.     

Inverse sequential simulation: Performance and implementation details

For good groundwater flow and solute transport numerical modeling, it is important to characterize the formation properties. In this paper, we analyze the performance and important implementation details of a new approach for stochastic inverse modeling called inverse sequential simulation (iSS). This approach is capable of characterizing conductivity fields with heterogeneity patterns difficult to capture by standard multiGaussian-based inverse approaches. The method is based on the multivariate sequential simulation principle, but the covariances and cross-covariances used to compute the local conditional probability distributions are computed by simple co-kriging which are derived from an ensemble of conductivity and piezometric head fields, in a similar manner as the experimental covariances are computed in an ensemble Kalman filtering. A sensitivity analysis is performed on a synthetic aquifer regarding the number of members of the ensemble of realizations, the number of conditioning data, the number of piezometers at which piezometric heads are observed, and the number of nodes retained within the search neighborhood at the moment of computing the local conditional probabilities. The results show the importance of having a sufficiently large number of all of the mentioned parameters for the algorithm to characterize properly hydraulic conductivity fields with clear non-multiGaussian features.     

Temporal sampling and role of flux measurements for subsurface heterogeneous characterization in groundwater basins using hydraulic tomography

Accurate characterization of heterogeneity in groundwater basins is crucial to the sustainable management of groundwater resources. This study explores the temporal sampling issues and the role of flux measurements in the characterization of heterogeneity in groundwater basins using numerical experiments. The experiments involve a digital basin imitating the groundwater basin of the North China Plain (NCP), where the groundwater exploitation reduction program is ongoing. Using the experiments, we champion that the reduction program could collect groundwater level information induced by operational variations of existing pumping wells at different locations in the basin. Such a dataset could serve as a basin-scale hydraulic tomography (HT) to characterize the basin-scale heterogeneity cost-effectively. Both steady-state and transient-state inversion experiments demonstrate the advantage of HT surveys in characterizing basin-scale heterogeneity over conventional pumping tests at fixed well locations. Additionally, head data at the early, intermediate, and late time from well hydrographs should be selected for the HT analysis to maximize HT's power and save computational costs. When accurate geological zones are incorporated in prior information, flux measurements significantly improve parameter estimates based on conventional pumping tests. However, their effects are less noticeable for long-term HT surveys in such basin-scale aquifers without fissures or fractures. This basin-scale tomographic survey example serves a guide for field data collection and optimization of the analysis of future basin-scale HT.     

Block-effective macrodispersion for numerical simulations of sorbing solute transport in heterogeneous porous formations

Heterogeneity is prevalent in aquifers and has an enormous impact on contaminant transport in groundwater. Numerical simulations are an effective way to deal with heterogeneity directly by assigning different hydraulic property values to each numerical grid block. Because hydraulic properties vary on different scales, but they cannot be sampled exhaustively and the number of numerical grid blocks is limited by computational considerations, the dispersive effects of unmodeled heterogeneity need to be accounted for. Dispersion tensors can be used to model the dispersion caused by unmodeled heterogeneity. The concept of block-effective macrodispersion tensors for modeling the effects of small-scale variability on solute transport introduced by Rubin et al. [Rubin Y, Sun A, Maxwell R, Bellin A. The concept of block-effective macrodispersivity and a unified approach for grid-scale- and plume-scale-dependent transport. J Fluid Mech 1999;395:161–80] is extended in this paper for use with reactive solutes. The tensors are derived for reactive solutes with spatially variable retardation factors and for solutes experiencing spatially uniform rate-limited sorption. The longitudinal block-effective macrodispersion coefficient is largest for perfect negative correlation between the log-hydraulic conductivity and the retardation factor. Because dispersion tensors, as they are usually implemented in numerical simulations, produce symmetric spreading, the applicability of the concept depends on the portion of the plume asymmetry caused by small-scale variability. The presented results show that the concept is applicable for rate-limited sorption for block sizes of one and two integral scales.     

Stochastic environmental risk analysis: an integrated methodology for predicting cancer risk from contaminated groundwater

Stochastic environmental risk assessment considers the effects of numerous biological, chemical, physical, behavioral and physiological processes that involve elements of uncertainty and variability. A methodology for predicting health risks to individuals from contaminated groundwater is presented that incorporates the elements of uncertainty and variability in geological heterogeneity, physiological exposure parameters, and in cancer potency. An idealized groundwater basin is used to perform a parametric sensitivity study to assess the relative impact of (a) geologic uncertainty, (b) behavioral and physiological variability in human exposure and (c) uncertainty in cancer potency on the prediction of increased cancer risk to individuals in a population exposed to contaminants in household water supplied from groundwater. A two-dimensional distribution (or surface) of human health risk was generated as a result of the simulations. Cuts in this surface (fractiles of variability and percentiles of uncertainty) are then used as a measure of relative importance of various model components on total uncertainty and variability. A case study for perchloroethylene or PCE, shows that uncertainty and variability in hydraulic conductivity play an important role in predicting human health risk that is on the same order of influence as uncertainty of cancer potency.