Application of a Bayesian approach to stochastic delineation of capture zones

This paper presents a Bayesian Monte Carlo method for evaluating the uncertainty in the delineation of well capture zones and its application to a wellfield in a heterogeneous, multiaquifer system. In the method presented, Bayes' rule is used to update prior distributions for the unknown parameters of the stochastic model for the hydraulic conductivity, and to calculate probability-based weights for parameter realizations using head residuals. These weights are then assigned to the corresponding capture zones obtained using forward particle tracking. Statistical analysis of the set of weighted protection zones results in a probability distribution for the capture zones. The suitability of the Bayesian stochastic method for a multilayered system is investigated, using the wellfield Het Rot at Nieuwrode, Belgium, located in a three-layered aquifer system, as an example. The hydraulic conductivity of the production aquifer is modeled as a spatially correlated random function with uncertain parameters. The aquitard and overlying unconfined aquifer are assigned random, homogeneous conductivities. The stochastic results are compared with deterministic capture zones obtained with a calibrated model for the area. The predictions of the stochastic approach are more conservative and indicate that parameter uncertainty should be taken into account in the delineation of well capture zones.     

Time-related capture zones for radial flow in two dimensional randomly heterogeneous media

We consider the effect of randomly heterogeneous hydraulic conductivity on the spatial location of time-related capture zones (isochrones) for a non-reactive tracer in the steady-state radial flow field due to a pumping well in a confined aquifer. A Monte Carlo (MC) procedure is used in conjunction with FFT-based spectral methods. The log hydraulic conductivity field is assumed to be Gaussian and stationary, with isotropic exponential correlation. Various degrees of domain heterogeneity are considered and stability and accuracy of the MC procedure is examined. The location of an isochrone becomes uncertain due to heterogeneity, and it is strongly influenced by hydraulic conductivity variance. The probability that a particle released at a point in the aquifer is pumped by the well within a given time is identified. We propose a new expression for the probabilistic spatial distribution of isochrones, which is formally similar to the analytical solution for a uniform medium and takes into account the effects of heterogeneity.     

Impact of complex aquifer geometry on groundwater storage in high‐elevation meadows of the Sierra Nevada Mountains,CA

Recent research has indicated that Sierra Nevada meadows are hydrologically more complex than previously considered. Improved understanding of the effects of aquifer parameters and climate change on water resources in and downstream of meadows is critically needed to effectively manage mountain meadows for ecosystem services and watershed contributions. This research investigates the roles of bedrock geometry, saturated hydraulic conductivity, and meadow gradient in affecting groundwater storage dynamics and surface‐water outflows in site‐scale high‐elevation meadows. Under current and projected lower snowpack conditions, we modeled groundwater flow in representative high‐elevation meadows considering 2 conceptual aquifer thickness models: uniform and variable thickness. Spatially, variable aquifer thicknesses interpreted from bedrock depths (0–28 m) were identified from a high‐resolution ground‐penetrating radar survey conducted at Tuolumne Meadows, CA. Our interpreted bedrock surface indicated several buried U‐shaped valleys including a buried ridge that separates 2 U‐shaped valleys. Groundwater flow simulations show that an increase in meadow gradient and hydraulic conductivity led to a decrease in seasonal storage and an increase in surface‐water outflow. However, models with varying bedrock geometries change the magnitude and timing of these processes. Uniform thickness models overestimated storage at the model edges and resulted in higher projected volumes of water being released to streams earlier than previously observed.     

Stochastic ground water flow simulation with a fracture zone continuum model

A method is presented for incorporating the hydraulic effects of vertical fracture zones into two-dimensional cell-based continuum models of ground water flow and particle tracking. High hydraulic conductivity features are used in the model to represent fracture zones. For fracture zones that are not coincident with model rows or columns, an adjustment is required for the hydraulic conductivity value entered into the model cells to compensate for the longer flowpath through the model grid. A similar adjustment is also required for simulated travel times through model cells. A travel time error of less than 8% can occur for particles moving through fractures with certain orientations. The fracture zone continuum model uses stochastically generated fracture zone networks and Monte Carlo analysis to quantify uncertainties with simulated advective travel times. An approach is also presented for converting an equivalent continuum model into a fracture zone continuum model by establishing the contribution of matrix block transmissivity to the bulk transmissivity of the aquifer. The methods are used for a case study in west-central Florida to quantify advective travel times from a potential wetland rehydration site to a municipal supply wellfield. Uncertainties in advective travel times are assumed to result from the presence of vertical fracture zones, commonly observed on aerial photographs as photolineaments.     

Ground water/surface water interaction in a fractured rock aquifer

In a recent field study of ground water/surface water interaction between a bedrock stream and an underlying fractured rock aquifer, it was determined that the majority of ground water discharge occurred through sparsely located vertical fractures. In this paper, the dominant mechanisms governing ground water/surface water exchange in such an environment are investigated using a numerical model. The study was conducted using several conceptual models based on the field study results. Although the field results provided the motivation for the modeling study, it was not intended to match modeling and field results directly. In addition, the extent of capture zones for discharging or recharging fractures was explored. The results of this study are intended to provide a better understanding of contaminant migration in the vicinity of bedrock streams. Based on the numerical results, the rate of ground water discharge (or recharge) was found to depend on the aperture size of the discharging feature, and on the distribution of hydraulic head with depth within the fracture network. It was determined that the extent of both the capture zone and reverse capture zone for an individual fracture can be extremely large, and will be determined by the height of the stream stage, the fracture apertures of the network, and the hydraulic-head distribution within the network. Because both the stream stage and the hydraulic-head distribution are transient, the size of the capture zone and/or the reverse capture zone for an individual fracture may change significantly over time. As a result, the migration path for contaminants within the fracture network and between the surface and subsurface will also vary significantly with time.     

Performance Assessments of a Novel Well Design for Reducing Exposure to Bedrock‐Derived Arsenic

Arsenic in groundwater is a serious problem in New England, particularly for domestic well owners drawing water from bedrock aquifers. The overlying glacial aquifer generally has waters with low arsenic concentrations but is less used because of frequent loss of well water during dry periods and the vulnerability to surface‐sourced bacterial contamination. An alternative, novel design for shallow wells in glacial aquifers is intended to draw water primarily from unconsolidated glacial deposits, while being resistant to drought conditions and surface contamination. Its use could greatly reduce exposure to arsenic through drinking water for domestic use. Hypothetical numerical models were used to investigate the potential hydraulic performance of the new well design in reducing arsenic exposure. The aquifer system was divided into two parts, an upper section representing the glacial sediments and a lower section representing the bedrock. The location of the well, recharge conditions, and hydraulic properties were systematically varied in a series of simulations and the potential for arsenic contamination was quantified by analyzing groundwater flow paths to the well. The greatest risk of arsenic contamination occurred when the hydraulic conductivity of the bedrock aquifer was high, or where there was upward flow from the bedrock aquifer because of the position of the well in the flow system.     

Water Table Uncertainties due to Uncertainties in Structure and Properties of an Unconfined Aquifer

We consider two sources of geology‐related uncertainty in making predictions of the steady‐state water table elevation for an unconfined aquifer. That is the uncertainty in the depth to base of the aquifer and in the hydraulic conductivity distribution within the aquifer. Stochastic approaches to hydrological modeling commonly use geostatistical techniques to account for hydraulic conductivity uncertainty within the aquifer. In the absence of well data allowing derivation of a relationship between geophysical and hydrological parameters, the use of geophysical data is often limited to constraining the structural boundaries. If we recover the base of an unconfined aquifer from an analysis of geophysical data, then the associated uncertainties are a consequence of the geophysical inversion process. In this study, we illustrate this by quantifying water table uncertainties for the unconfined aquifer formed by the paleochannel network around the Kintyre Uranium deposit in Western Australia. The focus of the Bayesian parametric bootstrap approach employed for the inversion of the available airborne electromagnetic data is the recovery of the base of the paleochannel network and the associated uncertainties. This allows us to then quantify the associated influences on the water table in a conceptualized groundwater usage scenario and compare the resulting uncertainties with uncertainties due to an uncertain hydraulic conductivity distribution within the aquifer. Our modeling shows that neither uncertainties in the depth to the base of the aquifer nor hydraulic conductivity uncertainties alone can capture the patterns of uncertainty in the water table that emerge when the two are combined.     

Effects of Improper Characterization of Aquifer Thickness on Estimates of Hydraulic Conductivity from Pumping Tests

Pumping test data for surficial aquifers are commonly analyzed under the assumption that the base of the aquifer corresponds to the bottom of the test wells (i.e., the aquifer is truncated). This practice can lead to inaccurate hydraulic conductivity estimates, resulting from the use of low saturated thickness values with transmissivity estimates, and not accounting for the effects of partially penetrating wells. Theoretical time-drawdown data were generated at an observation well in a hypothetical unconfined aquifer for various values of saturated thickness and were analyzed by standard curve-matching techniques. The base of the aquifer was assumed to be the bottom of the pumping and observation wells. The overestimation of horizontal hydraulic conductivity was found to be directly proportional to the error in assumed saturated thickness, and to the (actual) ratio of vertical to horizontal hydraulic conductivity (K_v/K_h). Inaccurately high estimates of hydraulic conductivity obtained by aquifer truncation can lead to overestimates of ground water velocity and contaminant plume spreading, narrow capture zone configuration estimates, and overestimates of available ground water resources.     

Effect of spatial variation in hydraulic conductivity on groundwater flow by alternate solution techniques

The behaviour of an aquifer in which there is spatial variation in hydraulic conductivity is simulated by means of a Monte Carlo procedure. The lognormal function is used to model the distribution of conductivities. Thus variations in hydraulic head and specific discharge are studied. As an alternate technique a finite-difference technique is adopted for comparative purposes. The standard deviation of two-dimensional flows is related to the standard deviation of conductivity. Hence, an empirical probability distribution of flows is obtained by specific values of the variance in conductivities.     

Stream water infiltration, bank storage, and storage zone changes due to stream-stage fluctuations

During a flood period, stream-stage increases induce infiltration of stream water into an aquifer; subsequent declines in stream stage cause a reverse motion of the infiltrated water. This paper presents the results of the water exchange rate between a stream and aquifer, the storage volume of the infiltrated stream water in the surrounding aquifer (bank storage), and the storage zone. The storage zone is the part of aquifer where groundwater is replaced by stream water during the flood. MODFLOW was used to simulate stream–aquifer interactions and to quantify rates of stream infiltration and return flow. MODPATH was used to trace the pathlines of the infiltrated stream water and to determine the size of the storage zone. Simulations were focused on the analyses of the effects of the stream-stage fluctuation, aquifer properties, the hydraulic conductivity of streambed sediments, regional hydraulic gradients, and recharge and evapotranspiration (ET) rates on stream–aquifer interactions. Generally, for a given stream–aquifer system, larger flow rates result from larger stream-stage fluctuations; larger storage volumes and storage zones are produced by larger and longer-lasting fluctuations. For a given stream-stage hydrograph, a lower-permeable streambed, an aquitard, or an anisotropic aquifer of low vertical hydraulic conductivity can significantly reduce the rate of infiltration and limit the size of the storage zone. The bank storage solely caused by the stage fluctuation differs slightly between gaining and losing streams. Short-term rainfall recharge and ET loss in the shallow groundwater slightly influence on the flow rate, but their effects on bank storage in a larger area for a longer period can be considerable.     

A numerical investigation of bedrock groundwater recharge and exfiltration on soil mantled hillslopes

Here we use Richards Equation models of variably saturated soil and bedrock groundwater flow to investigate first-order patterns of the coupling between soil and bedrock flow systems. We utilize a Monte Carlo sensitivity analysis to identify important hillslope parameters controlling bedrock recharge and then model the transient response of bedrock and soil flow to seasonal precipitation. Our results suggest that hillslopes can be divided into three conceptual zones of groundwater interaction, (a) the zone of lateral unsaturated soil moisture accumulation (upper portion of hillslope), (b) the zone of soil saturation and bedrock recharge (middle of hillslope) and (c) the zone of saturated-soil lateral flow and bedrock groundwater exfiltration (bottom of hillslope). Zones of groundwater interaction expand upslope during periods of precipitation and drain downslope during dry periods. The amount of water partitioned to the bedrock groundwater system a can be predicted by the ratio of bedrock to soil saturated hydraulic conductivity across a variety of hillslope configurations. Our modelled processes are qualitatively consistent with observations of shallow subsurface saturation and groundwater fluctuation on hillslopes studied in our two experimental watersheds and support a conceptual model of tightly coupled shallow and deep subsurface circulation where groundwater recharge and discharge continuously stores and releases water from longer residence time storage.     

A stochastic optimization framework for the restoration of an over-exploited aquifer

ABSTRACT

This study investigates the impact of hydraulic conductivity uncertainty on the sustainable management of the aquifer of Lake Karla, Greece, using the stochastic optimization approach. The lack of surface water resources in combination with the sharp increase in irrigation needs in the basin over the last 30 years have led to an unprecedented degradation of the aquifer. In addition, the lack of data regarding hydraulic conductivity in a heterogeneous aquifer leads to hydrogeologic uncertainty. This uncertainty has to be taken into consideration when developing the optimization procedure in order to achieve the aquifer’s sustainable management. Multiple Monte Carlo realizations of this spatially-distributed parameter are generated and groundwater flow is simulated for each one of them. The main goal of the sustainable management of the ‘depleted’ aquifer of Lake Karla is two-fold: to determine the optimum volume of renewable groundwater that can be extracted, while, at the same time, restoring its water table to a historic high level. A stochastic optimization problem is therefore formulated, based on the application of the optimization method for each of the aquifer’s multiple stochastic realizations in a future period. In order to carry out this stochastic optimization procedure, a modelling system consisting of a series of interlinked models was developed. The results show that the proposed stochastic optimization framework can be a very useful tool for estimating the impact of hydraulic conductivity uncertainty on the management strategies of a depleted aquifer restoration. They also prove that the optimization process is affected more by hydraulic conductivity uncertainty than the simulation process.

Editor Z.W. Kundzewicz; Guest editor S. Weijs     

Polynomial chaos expansion for global sensitivity analysis applied to a model of radionuclide migration in a randomly heterogeneous aquifer

We perform global sensitivity analysis (GSA) through polynomial chaos expansion (PCE) on a contaminant transport model for the assessment of radionuclide concentration at a given control location in a heterogeneous aquifer, following a release from a near surface repository of radioactive waste. The aquifer hydraulic conductivity is modeled as a stationary stochastic process in space. We examine the uncertainty in the first two (ensemble) moments of the peak concentration, as a consequence of incomplete knowledge of (a) the parameters characterizing the variogram of hydraulic conductivity, (b) the partition coefficient associated with the migrating radionuclide, and (c) dispersivity parameters at the scale of interest. These quantities are treated as random variables and a variance-based GSA is performed in a numerical Monte Carlo framework. This entails solving groundwater flow and transport processes within an ensemble of hydraulic conductivity realizations generated upon sampling the space of the considered random variables. The Sobol indices are adopted as sensitivity measures to provide an estimate of the role of uncertain parameters on the (ensemble) target moments. Calculation of the indices is performed by employing PCE as a surrogate model of the migration process to reduce the computational burden. We show that the proposed methodology (a) allows identifying the influence of uncertain parameters on key statistical moments of the peak concentration (b) enables extending the number of Monte Carlo iterations to attain convergence of the (ensemble) target moments, and (c) leads to considerable saving of computational time while keeping acceptable accuracy.     

Groundwater-supported evapotranspiration within glaciated watersheds under conditions of climate change

This paper analyzes the effects of geology and geomorphology on surface-water/-groundwater interactions, evapotranspiration, and recharge under conditions of long-term climatic change. Our analysis uses hydrologic data from the glaciated Crow Wing watershed in central Minnesota, USA, combined with a hydrologic model of transient coupled unsaturated/saturated flow (HYDRAT2D). Analysis of historical water-table (1970–1993) and lake-level (1924–2002) records indicates that larger amplitude and longer period fluctuations occur within the upland portions of watersheds due to the response of the aquifer system to relatively short-term climatic fluctuations. Under drought conditions, lake and water-table levels fell by as much as 2–4 m in the uplands but by 1 m in the lowlands. The same pattern can be seen on millennial time scales. Analysis of Holocene lake-core records indicates that Moody Lake, located near the outlet of the Crow Wing watershed, fell by as much as 4 m between about 4400 and 7000 yr BP. During the same time, water levels in Lake Mina, located near the upland watershed divide, fell by about 15 m. Reconstructed Holocene climate as represented by HYDRAT2D gives somewhat larger drops (6 and 24 m for Moody Lake and Lake Mina, respectively). The discrepancy is probably due to the effect of three-dimensional flow. A sensitivity analysis was also carried out to study how aquifer hydraulic conductivity and land-surface topography can influence water-table fluctuations, wetlands formation, and evapotranspiration. The models were run by recycling a wet year (1985, 87 cm annual precipitation) over a 10-year period followed by 20 years of drier and warmer climate (1976, 38 cm precipitation). Model results indicated that groundwater-supported evapotranspiration accounted for as much as 12% (10 cm) of evapotranspiration. The aquifers of highest hydraulic conductivity had the least amount of groundwater-supported evapotranspiration owing to a deep water table. Recharge was even more sensitive to aquifer hydraulic conductivity, especially in the lowland regions. These findings have important implications for paleoclimatic studies, because the hydrologic response of a surface-water body will vary across the watershed to a given climate signal.     

The Effect of Modeled Recharge Distribution on Simulated Groundwater Availability and Capture

Simulating groundwater flow in basin‐fill aquifers of the semiarid southwestern United States commonly requires decisions about how to distribute aquifer recharge. Precipitation can recharge basin‐fill aquifers by direct infiltration and transport through faults and fractures in the high‐elevation areas, by flowing overland through high‐elevation areas to infiltrate at basin‐fill margins along mountain fronts, by flowing overland to infiltrate along ephemeral channels that often traverse basins in the area, or by some combination of these processes. The importance of accurately simulating recharge distributions is a current topic of discussion among hydrologists and water managers in the region, but no comparative study has been performed to analyze the effects of different recharge distributions on groundwater simulations. This study investigates the importance of the distribution of aquifer recharge in simulating regional groundwater flow in basin‐fill aquifers by calibrating a groundwater‐flow model to four different recharge distributions, all with the same total amount of recharge. Similarities are seen in results from steady‐state models for optimized hydraulic conductivity values, fit of simulated to observed hydraulic heads, and composite scaled sensitivities of conductivity parameter zones. Transient simulations with hypothetical storage properties and pumping rates produce similar capture rates and storage change results, but differences are noted in the rate of drawdown at some well locations owing to the differences in optimized hydraulic conductivity. Depending on whether the purpose of the groundwater model is to simulate changes in groundwater levels or changes in storage and capture, the distribution of aquifer recharge may or may not be of primary importance.     

Scale effect and calibration of contaminant transport models

The influence on solute transport of the small-scale spatial variation of aquifer hydraulic conductivity (K) was analyzed by comparing results from fine-grid (2 m by 2 m) simulations of a synthetic heterogeneous aquifer to those from coarse-grid (8 m by 4 m) simulations of an equivalent homogeneous aquifer. Realizations of the K field of the heterogeneous aquifer were generated, using the Monte Carlo approach, from a lognormal distribution with mean log K of 2 (K in m/d) and three levels of log K variance of 0.1, 0.5, and 1.0. Numerical simulation results show that the average standard deviation of point concentrations increased from 1.21 to 5.78 when the value of log K variance was increased from 0.1 to 1.0. The average discrepancy between modeled concentrations (obtained from a coarse-grid deterministic numerical simulation) and the actual mean point concentrations (obtained from fine-grid Monte Carlo numerical simulations) increased from 0.91 to 4.23 with the increase in log K variance. The results from this study illustrate the uncertainty in predictions from contaminant transport models due to their inability to simulate the effects of heterogeneities at scales smaller than the model grid.     

Influence of the amount of available K data on uncertainty about contaminant transport prediction

The effect of increasing the amount of hydraulic conductivity (K) data on the level of uncertainty about contaminant concentration predictions was examined. Contaminant concentrations were predicted using Monte Carlo numerical simulations with progressively more K information, and were compared to reference data obtained from a physical experiment. Increasing the amount of K data used in generating the aquifer's K field improved the measure of confidence in the predicted contaminant concentration. The normalized concentration interval with 95% confidence decreased from 1.29 to 0.75 when the amount of K data increased from 20 to 71. However, the trade-off is the increase in the costs of aquifer investigation. Conducting Monte Carlo numerical simulations may help design an optimal aquifer investigation scheme, so that it can provide enough input data to predict concentrations with the desired level of confidence and at the same time avoid the excessive costs of aquifer investigation.     

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).     

Biased monitoring of fresh water-salt water mixing zone in coastal aquifers

In coastal aquifers, significant vertical hydraulic gradients are formed where fresh water and underlying salt water discharge together upward to the seafloor. Monitoring boreholes may act as "short circuits" along these vertical gradients, connecting between the higher and the lower hydraulic head zones. When a sea tide is introduced, the fluctuations of both the water table and the depth of the mixing zone are also biased due to this effect. This problem is intensified in places of long-screen monitoring boreholes, which are common in many places in the world. For example, all approximately 500 boreholes of the fresh water-salt water mixing zone in the coastal aquifer of Israel are installed with 10 to 50 m long screens. We present field measurements of these fluctuations, along with a three-dimensional numerical model. We find that the in-well fluctuation magnitude of the mixing zone is an order of magnitude larger than that in the porous media of the actual aquifer. The primary parameters that affect the magnitude of this bias are the anisotropy of the aquifer conductivity and the borehole hydraulic parameters. With no sea tide, borehole interference is higher for the anisotropic case because the vertical hydraulic gradients are high. When tides are introduced, the amplitude of the mixing zone fluctuation is higher for the isotropic case because the overall effective hydraulic conductivity is greater than the conductivity in the anisotropic case. In the aquifer, the fresh water-salt water mixing zone fluctuations are dampened, and tens of meters inland from the shoreline, the fluctuations are on the order of few centimeters.