Geostatistical analysis and conditional simulation for estimating the spatial variability of hydraulic conductivity in the Choushui River alluvial fan,Taiwan

This work evaluated the spatial variability and distribution of heterogeneous hydraulic conductivity (K) in the Choushui River alluvial fan in Taiwan, using ordinary kriging (OK) and mean and individual sequential Gaussian simulations (SGS). A baseline flow model constructed by upscaling parameters was inversely calibrated to determine the pumping and recharge rates. Simulated heads using different K realizations were then compared with historically measured heads. A global/local simulated error between simulated and measured heads was analysed to assess the different spatial variabilities of various estimated K distributions. The results of a MODFLOW simulation indicate that the OK realization had the smallest sum of absolute mean simulation errors (SAMSE) and the SGS realizations preserved the spatial variability of the measured K fields. Moreover, the SAMSE increases as the spatial variability of the K field increases. The OK realization yields small local simulation errors in the measured K field of moderate magnitude, whereas the SGS realizations have small local simulation errors in the measured K fields, with high and low values. The OK realization of K can be applied to perform a deterministic inverse calibration. The mean SGS method is suggested for constructing a K field when the application focuses on extreme values of estimated parameters and small calibration errors, such as in a simulation of contaminant transport in heterogeneous aquifers. The individual SGS realization is useful in stochastically assessing the spatial uncertainty of highly heterogeneous aquifers. Copyright © 2004 John Wiley & Sons, Ltd.     

Hydraulic Tomography: 3D Hydraulic Conductivity,Fracture Network,and Connectivity in Mudstone

We present the first demonstration of hydraulic tomography (HT) to estimate the three-dimensional (3D) hydraulic conductivity (K) distribution of a fractured aquifer at high-resolution field scale (HRFS), including the fracture network and connectivity through it. We invert drawdown data collected from packer-isolated borehole intervals during 42 pumping tests in a wellfield at the former Naval Air Warfare Center, West Trenton, New Jersey, in the Newark Basin. Five additional tests were reserved for a quality check of HT results. We used an equivalent porous medium forward model and geostatistical inversion to estimate 3D K at high resolution (K blocks <1 m³), using no strict assumptions about K variability or fracture statistics. The resulting 3D K estimate ranges from approximately 0.1 (highest-K fractures) to approximately 10⁻¹³ m/s (unfractured mudstone). Important estimated features include: (1) a highly fractured zone (HFZ) consisting of a sequence of high-K bedding-plane fractures; (2) a low-K zone that disrupts the HFZ; (3) several secondary fractures of limited extent; and (4) regions of very low-K rock matrix. The 3D K estimate explains complex drawdown behavior observed in the field. Drawdown tracing and particle tracking simulations reveal a 3D fracture network within the estimated K distribution, and connectivity routes through the network. Model fit is best in the shallower part of the wellfield, with high density of observations and tests. The capabilities of HT demonstrated for 3D fractured aquifer characterization at HRFS may support improved in situ remediation for contaminant source zones, and applications in mining, repository assessment, or geotechnical engineering.     

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.     

Monitoring of drawdown pattern during pumping in an unconfined physical aquifer model

In this study, we attempted to analyse a drawdown pattern around a pumping well in an unconfined sandy gravelly aquifer constructed in a laboratory tank by means of both experimental and numerical modelling of groundwater flow. The physical model consisted of recharge, aquifer and discharge zones. Permeability and specific yield of the aquifer material were determined by Dupuit approximation under steady‐state flow and stepwise gravitational drainage of groundwater, respectively. The drawdown of water table in pumping and neighbouring observation wells was monitored to investigate the effect of no‐flow boundary on the drawdown pattern during pumping for three different boundary conditions: (i) no recharge and no discharge with four no‐flow boundaries (Case 1); (ii) no recharge and reservoir with three no‐flow boundaries (Case 2); (iii) recharge and discharge with two no‐flow boundaries (Case 3). Based on the aquifer parameters, numerical modelling was also performed to compare the simulated drawdown with that observed. Results showed that a large difference existed between the simulated drawdown and that observed in wells for all cases. The reason for the difference could be explained by the formation of a curvilinear type water table between wells rather than a linear one due to a delayed response of water table in the capillary fringe. This phenomenon was also investigated from a mass balance study on the pumping volume. The curvilinear type of water table was further evidenced by measurement of water contents at several positions in the aquifer between wells using time domain reflectometry (TDR). This indicates that the existing groundwater flow model applicable to an unconfined aquifer lacks the capacity to describe a slow response of water table in the aquifer and care should be taken in the interpretation of water table formation in the aquifer during pumping. Copyright © 2001 John Wiley & Sons, Ltd.     

Evaluation of Pumping Induced Flow in Observation Wells During Aquifer Testing

The vertical variation of drawdown around pumping wells generates an induced flow in the observation wells. A set of governing equations is presented to couple the drawdown variation and the vertical flux distribution in observation wells. A numerical example is performed to justify the governing equations and to verify the solution methods used by the simulation software WT. The example analyzes the effect of skin loss, wellbore storage, and vertical segmentation on the drawdown and induced flow in observation well during pumping. The evaluation of the Fairborn pumping test involves a vertically homogeneous and anisotropic water table aquifer, uniform well‐face drawdown conditions in the pumping well and simulation of the drawdown evolution in the observation well with and without the effect of induced flow. The computer calibrations resulted in small differences between the measured and simulated drawdown curves.     

On the relationship between indicators of geostatistical,flow and transport connectivity

Connectivity of high conductivity (K) paths is important because it can lead to channeling, i.e. flow along preferential paths, which can reduce travel times very significantly. Nevertheless, limited effort has been devoted to defining the concept quantitatively. We propose and evaluate nine indicators of connectivity. Three account for the presence of flow connectivity, that is, the flow rate increase caused by preferential flow paths. Two account for the presence of transport connectivity, that is, the existence of fast paths allowing early solute arrival. The remaining four are statistical indicators based on two- and multiple-point statistics. We test these indicators on heterogeneous conductivity fields with different visual connectivity. The indicators of flow connectivity and one of the transport connectivity indicators succeed in identifying the increased presence of connected high-K features. The two-point statistical indicators fail to do so. The directional multi-point statistical indicator performs better. None of the statistical indicators correlate with the flow and transport indicators. We find only weak dependence between the flow and transport indicators. Our results suggest that transport connectivity is much less sensitive to barriers, which control flow connectivity. Instead, transport connectivity appears to be controlled by the existence of narrow, possibly discontinuous high-K paths. Therefore, we conclude that connectivity is a process-dependent concept.     

Three-dimensional geostatistical inversion of flowmeter and pumping test data

We jointly invert field data of flowmeter and multiple pumping tests in fully screened wells to estimate hydraulic conductivity using a geostatistical method. We use the steady-state drawdowns of pumping tests and the discharge profiles of flowmeter tests as our data in the inference. The discharge profiles need not be converted to absolute hydraulic conductivities. Consequently, we do not need measurements of depth-averaged hydraulic conductivity at well locations. The flowmeter profiles contain information about relative vertical distributions of hydraulic conductivity, while drawdown measurements of pumping tests provide information about horizontal fluctuation of the depth-averaged hydraulic conductivity. We apply the method to data obtained at the Krauthausen test site of the Forschungszentrum Jülich, Germany. The resulting estimate of our joint three-dimensional (3D) geostatistical inversion shows an improved 3D structure in comparison to the inversion of pumping test data only.     

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).     

New graphical methods for estimating aquifer hydraulic parameters using pumping tests with exponentially decreasing rates

Actual pumping tests may involve continuously decreasing rates over a certain period of time, and the hydraulic conductivity (K) and specific storage (S_s) of the tested confined aquifer cannot be interpreted from the classical constant‐rate test model. In this study, we revisit the aquifer drawdown characteristics of a pumping test with an exponentially decreasing rate using the dimensionless analytical solution for such a variable‐rate model. The drawdown may decrease with time for a short period of time at intermediate pumping times for such pumping tests. A larger ratio of initial to final pumping rate and a smaller radial distance of the observation well will enhance the decreasing feature. A larger decay constant results in an earlier decrease, but it weakens the extent of such a decrease. Based on the proposed dimensionless transformation, we have proposed two graphical methods for estimating K and S_s of the tested aquifer. The first is a new type curve method that does not employ the well function as commonly done in standard type curve analysis. Another is a new analytic method that takes advantage of the decreasing features of aquifer drawdown during the intermediate pumping stage. We have demonstrated the applicability and robustness of the two new graphical methods for aquifer characterization through a synthetic pumping test.     

Geostatistical simulation of random fields with bivariate isofactorial distributions by adding mosaic models

This work deals with the geostatistical simulation of a family of stationary random field models with bivariate isofactorial distributions. Such models are defined as the sum of independent random fields with mosaic-type bivariate distributions and infinitely divisible univariate distributions. For practical applications, dead leaf tessellations are used since they provide a wide range of models and allow conditioning the realizations to a set of data via an iterative procedure (simulated annealing). The model parameters can be determined by comparing the data variogram and madogram, and enable to control the spatial connectivity of the extreme values in the realizations. An illustration to a forest dataset is presented, for which a negative binomial model is used to characterize the distribution of coniferous trees over a wooded area.     

Stochastic inverse method for estimation of geostatistical representation of hydrogeologic stratigraphy using borehole logs and pressure observations

An approach is presented for identifying statistical characteristics of stratigraphies from borehole and hydraulic data. The approach employs a Markov-chain based geostatistical framework in a stochastic inversion. Borehole data provide information on the stratigraphy while pressure and flux data provide information on the hydraulic performance of the medium. The use of Markov-chain geostatistics as opposed to covariance-based geostatistics can provide a more easily interpreted model geologically and geometrically. The approach hinges on the use of mean facies lengths (negative inverse auto-transition rates) and mean transition lengths (inverse cross-transition rates) as adjustable parameters in the stochastic inversion. Along with an unconstrained Markov-chain model, simplifying constraints to the Markov-chain model, including (1) proportionally-random and (2) symmetric spatial correlations, are evaluated in the stochastic inversion. Sensitivity analyses indicate that the simplifying constraints can facilitate the inversion at the cost of spatial correlation model generality. Inverse analyses demonstrate the feasibility of this approach, indicating that despite some low parameter sensitivities, all adjustable parameters do converge for a sufficient number of ensemble realizations towards their “true” values. This paper extends the approach presented in Harp et al. (doi:, 2008) to (1) statistically characterize the hydraulic response of a geostatistical model, thereby incorporating an uncertainty analysis directly in the inverse method, (2) demonstrate that a gradient-based optimization strategy is sufficient, thereby providing relative computational efficiency compared to global optimization strategies, (3) demonstrate that the approach can be extended to a 3-D analysis, and (4) introduce the use of mean facies lengths and mean transition lengths as adjustable parameters in a geostatistical inversion, thereby allowing the approach to be extended to greater than two category Markov-chain models.     

Flow depletion in a small stream caused by ground water abstraction from wells

The interaction between a gaining stream and a water-table aquifer is studied at an outwash plain. The aquifer is hydraulically well connected to the stream. Pumping tests were carried out in 1997 and 1998 in two wells 60 m from the stream, screening different depths of the aquifer. Drawdown was measured on both sides of the stream. Hydraulic head, drawdown, and stream depletion data were analyzed using numerical flow models. Similar models were fitted to each of two different data sets: Model A was fitted to steady-state hydraulic head and streamflow gain data not influenced by pumping; and model B was fitted to drawdown data measured during the 1998 pumping test. Each calibrated model closely fits its calibration data; however, predictions were biased if model A was used to predict the calibration data of model B, and vice versa. To further test the models, they were used to predict streamflow depletion during the two pumping tests as well as the drawdown during the 1997 test. Neither of these data were used for calibration. Model A predicted the measured depletions fairly accurately during both tests, whereas the predicted drawdowns in 1997 were significantly larger than actually measured. Contrary to this, the 1997 drawdowns predicted by model B were nearly unbiased; the predicted depletions deviate significantly from the measured depletions in 1997, but they compare well with the observations in 1998. Thus, although field work and analyses were extensive and done carefully to develop a ground water flow model that could predict both drawdown and streamflow depletion, the model predictions are biased. Analyses indicate that the deviations between model and data may be because of error in the models' representations of either the release of water from storage or of the hydrology in the riparian zone.     

Multivariate grid-free geostatistical simulation with point or block scale secondary data

A novel grid-free geostatistical simulation method (GFS) allows representing coregionalized variables as an analytical function of the coordinates of the simulation locations. Simulation on unstructured grids, regridding and refinement of available realizations of natural phenomena including, but not limited to, environmental systems are possible with GFS in a consistent manner. The unconditional realizations are generated by utilizing the linear model of coregionalization and Fourier series-based decomposition of the covariance function. The conditioning to data is performed by kriging. The data can be measured at scattered point-scale locations or sampled at a block scale. Secondary data are usually used in conjunction with primary data for the improved modeling. Satellite imaging is an example of exhaustively sampled secondary data. Improvements and recommendations are made to the implementation of GFS to properly assimilate secondary exhaustive data sets in a grid-free manner. Intrinsic cokriging (ICK) is utilized to reduce computational time and preserve the overall quality of the simulation. To further reduce the computational cost of ICK, a block matrix inversion is implemented in the calculation of the kriging weights. A projection approach to ICK is proposed to avoid artifacts in the realizations around the edges of the exhaustive data region when the data do not cover the entire modeling domain. The point-scale block value representation of the block-scale data is developed as an alternative to block cokriging to integrate block-scale data into realizations within the GFS framework. Several case studies support the proposed enhancements.     

How to Find Aquifer Statistics Utilizing Pumping Tests? Two Field Studies Using welltestpy

We present a workflow to estimate geostatistical aquifer parameters from pumping test data using the Python package welltestpy . The procedure of pumping test analysis is exemplified for two data sets from the Horkheimer Insel site and from the Lauswiesen site, Germany. The analysis is based on a semi-analytical drawdown solution from the upscaling approach Radial Coarse Graining, which enables to infer log-transmissivity variance and horizontal correlation length, beside mean transmissivity, and storativity, from pumping test data. We estimate these parameters of aquifer heterogeneity from type-curve analysis and determine their sensitivity. This procedure, implemented in welltestpy , is a template for analyzing any pumping test. It goes beyond the possibilities of standard methods, for example, based on Theis' equation, which are limited to mean transmissivity and storativity. A sensitivity study showed the impact of observation well positions on the parameter estimation quality. The insights of this study help to optimize future test setups for geostatistical aquifer analysis and provides guidance for investigating pumping tests with regard to aquifer statistics using the open-source software package welltestpy .     

Detecting Drawdowns Masked by Environmental Stresses with Water‐Level Models

Detecting and quantifying small drawdown at observation wells distant from the pumping well greatly expands the characterized aquifer volume. However, this detection is often obscured by water level fluctuations such as barometric and tidal effects. A reliable analytical approach for distinguishing drawdown from nonpumping water‐level fluctuations is presented and tested here. Drawdown is distinguished by analytically simulating all pumping and nonpumping water‐level stresses simultaneously during the period of record. Pumping signals are generated with Theis models, where the pumping schedule is translated into water‐level change with the Theis solution. This approach closely matched drawdowns simulated with a complex three‐dimensional, hypothetical model and reasonably estimated drawdowns from an aquifer test conducted in a complex hydrogeologic system. Pumping‐induced changes generated with a numerical model and analytical Theis model agreed (RMS as low as 0.007 m) in cases where pumping signals traveled more than 1 km across confining units and fault structures. Maximum drawdowns of about 0.05 m were analytically estimated from field investigations where environmental fluctuations approached 0.2 m during the analysis period.     

Conservative models: parametric entropy vs. temporal entropy in outcomes

The geologic architecture in aquifer systems affects the behavior of fluid flow and the dispersion of mass. The spatial distribution and connectivity of higher-permeability facies play an important role. Models that represent this geologic structure have reduced entropy in the spatial distribution of permeability relative to models without structure. The literature shows that the stochastic model with the greatest variance in the distribution of predictions (i.e., the most conservative model) will not simply be the model representing maximum disorder in the permeability field. This principle is further explored using the Shannon entropy as a single metric to quantify and compare model parametric spatial disorder to the temporal distribution of mass residence times in model predictions. The principle is most pronounced when geologic structure manifests as preferential-flow pathways through the system via connected high-permeability sediments. As per percolation theory, at certain volume fractions the full connectivity of the high-permeability sediments will not be represented unless the model is three-dimensional. At these volume fractions, two-dimensional models can profoundly underrepresent the entropy in the real, three-dimensional, aquifer system. Thus to be conservative, stochastic models must be three-dimensional and include geologic structure.     

Geologic heterogeneity recognition using discrete wavelet transformation for subsurface flow solute transport simulations

Subsurface flow and solute transport simulations are performed using different scenarios of permeability fields generated from the sequential Gaussian simulation method (SGS), the multiple-point FILTERSIM algorithm and a new multiple-point wavelet-based simulation method (SWS). The SWS method is a multiple-point pattern-based simulation method which uses discrete wavelet transformation for the representation of geologic heterogeneity. For pattern-based simulation, patterns are generated by scanning a training image with a spatial template. The pattern classifications were performed after reducing the dimension of patterns by wavelet decomposition at the suitable scale and by taking only scaling components of wavelet decomposed patterns. The simulation is performed in a sequential manner by finding the best-matched class corresponding to the conditioning data and by randomly sampling a pattern from the best-matched class. The developed method is compared with two other multi-point simulation algorithms, FLTERSIM and SIMPAT. The comparative results revealed that the proposed method is computationally faster than the other two methods while the simulation maps are comparable. Numerical simulations of two flow problems are performed using SGS, SWS and FILTERSIM realizations. The numerical results show a superiority of the SWS method over SGS and FILTERSIM in terms of reproduction of the reference images main features, and agreement with flow and transport results obtained on reference images.     

Some results comparing Monte Carlo simulation and first order Taylor series approximation for steady groundwater flow

The expected head and standard deviation of the head from the first order Taylor series approximation is compared to Monte Carlo simulation, for steady flow in a confined aquifer with transmissivity as a random variable. Emphasis is on the effect of changes in the covariance structure of the transmissivity, and pumping rates, on the errors in the first order Taylor series approximation. The accuracy of the first order Taylor series approximation is found to be particularly sensitive to pumping rates. With significant pumping the approximation is found to under estimate both the expected drawdown and head variance, and the error increases as the pumping rate increases. This can lead to large errors in probability constraints based on moments from the first order Taylor series approximation.     

A Type-Curve Method for the Analysis of Pumping Tests with Piecewise-Linear Pumping Rates

Aquifer hydraulic parameters are commonly inferred from constant-rate pumping tests, while variable pumping rates are frequently encountered in actual field conditions. In this study, we propose a generally applicable dimensionless form of the analytical solution for variable-rate pumping tests in confined aquifers. In particular, we adopt a piecewise-linear fitting of variable pumping rates and propose a new type-curve method for estimating the hydraulic conductivity (K ) and specific storage (S_s ) of the investigated confined aquifer. For each test, a series of type curves, which depend on the variable pumping rates, the location of observation wells and the introduced first dimensionless inflection time, need to be provided for matching the observed drawdown data on a log-log graph. We first demonstrate the applicability and robustness of this method through a synthetic pumping test. Subsequently, we apply this method to analyze drawdown data from four pumping tests conducted within a multilayered aquifer/aquitard system in Wuxi city, Jiangsu Province, China. The parameter estimates are then compared with those reported by PEST. The K and S_s values estimated by the new type-curve method are found to be quite close to PEST-based estimates. Parameter estimation results demonstrate the difference in K and S_s values between observation wells. The difference could be attributed to the spatial heterogeneity in K and S_s . A future research topic may focus on the characterization of K and S_s heterogeneity with the currently available drawdown data from variable-rate pumping tests.