Log Hydraulic Conductivity Research Articles

Heat and Solute Tracers: How Do They Compare in Heterogeneous Aquifers?

A comparison of groundwater velocity in heterogeneous aquifers estimated from hydraulic methods, heat and solute tracers was made using numerical simulations. Aquifer heterogeneity was described by geostatistical properties of the Borden, Cape Cod, North Bay, and MADE aquifers. Both heat and solute tracers displayed little systematic under- or over-estimation in velocity relative to a hydraulic control. The worst cases were under-estimates of 6.63% for solute and 2.13% for the heat tracer. Both under- and over-estimation of velocity from the heat tracer relative to the solute tracer occurred. Differences between the estimates from the tracer methods increased as the mean velocity decreased, owing to differences in rates of molecular diffusion and thermal conduction. The variance in estimated velocity using all methods increased as the variance in log-hydraulic conductivity (K) and correlation length scales increased. The variance in velocity for each scenario was remarkably small when compared to σ2 ln(K) for all methods tested. The largest variability identified was for the solute tracer where 95% of velocity estimates ranged by a factor of 19 in simulations where 95% of the K values varied by almost four orders of magnitude. For the same K-fields, this range was a factor of 11 for the heat tracer. The variance in estimated velocity was always lowest when using heat as a tracer. The study results suggest that a solute tracer will provide more understanding about the variance in velocity caused by aquifer heterogeneity and a heat tracer provides a better approximation of the mean velocity.

Effective retardation factor for transport of reactive solutes in highly heterogeneous porous formations

[1] The effective retardation factor of highly heterogeneous porous formations is investigated using an analytical derivation and numerical simulations based on the self-consistent approximation and the multi-indicator conductivity model, used in the past for the effective hydraulic conductivity estimation. Both positive and negative correlation between the logarithm of hydraulic conductivity K and the logarithm of sorption distribution coefficient Kd are investigated, including the effects of random noise. The arithmetic average of the random retardation factor is found to represent accurately the effective retardation factor of isotropic and anisotropic formations regardless of correlation between ln Kd and ln K, and for variance of ln K as large as 8.

Dynamic interactions of groundwater flow and soil deformation in randomly heterogeneous porous media

Summary The interaction of groundwater flow and soil deformation within porous media is a complicated process. Water movement or stress change in porous media result in both deformation and alteration in pore water pressure. In this study, the dynamic interactions of flow and deformation in randomly heterogeneous porous media are analyzed using the stochastic first-order second-moment (FOSM) method. Various flow boundaries are considered, including sudden starts and sudden stops of discharge and recharge. The stochastic model is first verified with the results from Monte Carlo simulations with a constant discharge. The patterns of mean, variance, and covariance solutions obtained using the FOSM method are the same as those obtained using Monte Carlo simulations. The results show that the mean behaviors of water pressure variation and deformation are well explained by the concept of effective stress. The second-moment solutions show monotonic increases in the variances of displacement and changes in pore water pressure with distance away from the prescribed boundary. The cross-covariance of log hydraulic conductivity and change in pore water pressure is initially negative and then becomes positive, whereas that of change in pore water pressure and displacement is initially positive and then becomes negative soon after discharge starts at the boundary. The initial behavior is opposite to that in the later period. This reversal of cross-covariance is attributed initially to stress and later to water flow, and it extends from the boundary and then diminishes quickly with time. The Rhade effect occurs for the mean change in pore water pressure in areas away from the discharge boundary when discharge suddenly stops. The reversal of cross-covariance is also found at various depths immediately after the stop of flow at the boundary for both discharge and recharge conditions due to the dynamic interactions of flow and deformation. The results also show that the strength of the dynamic interactions between groundwater flow and soil stress increases with decreasing mean hydraulic conductivity and average stiffness of randomly heterogeneous porous media. Neglecting the dynamic interactions of flow and deformation leads to misinterpretation of aquifer properties and increases estimation uncertainty.

Uncertainty Estimation in One‐Dimensional Heat Transport Model for Heterogeneous Porous Medium

In many practical applications, the rates for ground water recharge and discharge are determined based on the analytical solution developed by Bredehoeft and Papadopulos (1965) to the one-dimensional steady-state heat transport equation. Groundwater flow processes are affected by the heterogeneity of subsurface systems; yet, the details of which cannot be anticipated precisely. There exists a great deal of uncertainty (variability) associated with the application of Bredehoeft and Papadopulos' solution (1965) to the field-scale heat transport problems. However, the quantification of uncertainty involved in such application has so far not been addressed, which is the objective of this wok. In addition, the influence of the statistical properties of log hydraulic conductivity field on the variability in temperature field in a heterogeneous aquifer is also investigated. The results of the analysis demonstrate that the variability (or uncertainty) in the temperature field increases with the correlation scale of the log hydraulic conductivity covariance function and the variability of temperature field also depends positively on the position.

Human Health Risk Assessment of CO2 Leakage into Overlying Aquifers Using a Stochastic, Geochemical Reactive Transport Approach

Increased human health risk associated with groundwater contamination from potential carbon dioxide (CO2) leakage into a potable aquifer is predicted by conducting a joint uncertainty and variability (JUV) risk assessment. The approach presented here explicitly incorporates heterogeneous flow and geochemical reactive transport in an efficient manner and is used to evaluate how differences in representation of subsurface physical heterogeneity and geochemical reactions change the calculated risk for the same hypothetical aquifer scenario where a CO2 leak induces increased lead (Pb(2+)) concentrations through dissolution of galena (PbS). A nested Monte Carlo approach was used to take Pb(2+) concentrations at a well from an ensemble of numerical reactive transport simulations (uncertainty) and sample within a population of potentially exposed individuals (variability) to calculate risk as a function of both uncertainty and variability. Pb(2+) concentrations at the well were determined with numerical reactive transport simulation ensembles using a streamline technique in a heterogeneous 3D aquifer. Three ensembles with variances of log hydraulic conductivity (σ(2)lnK) of 1, 3.61, and 16 were simulated. Under the conditions simulated, calculated risk is shown to be a function of the strength of subsurface heterogeneity, σ(2)lnK and the choice between calculating Pb(2+) concentrations in groundwater using equilibrium with galena and kinetic mineral reaction rates. Calculated risk increased with an increase in σ(2)lnK of 1 to 3.61, but decreased when σ(2)lnK was increased from 3.61 to 16 for all but the highest percentiles of uncertainty. Using a Pb(2+) concentration in equilibrium with galena under CO2 leakage conditions (PCO2 = 30 bar) resulted in lower estimated risk than the simulations where Pb(2+) concentrations were calculated using kinetic mass transfer reaction rates for galena dissolution and precipitation. This study highlights the importance of understanding both hydrologic and geochemical conditions when numerical simulations are used to perform quantitative risk calculations.

A Stochastic Finite Element Heterogeneous Multiscale Method for Seepage Field in Heterogeneous Ground

The finite element heterogeneous multiscale method (FEHM) combined with stochastic collocation method (SCM) called SHMFE is applied to studying the seepage field of naturally heterogeneous multiscale subsurface formations. Kinds of stochastic finite element (SFEM) are mainly computational techniques for the class of problems. But those methods do not report the multiscale nature of the properties of subsurface formations. When the random permeability field is heterogeneous in fine scale comparing to study domain, the simulation by the classic SFEM is not a trivial task. The SHMFE can efficiently solve the problems. In the method, Karhunen-Loµeve (KL) decomposition is used to represent the log hydraulic conductivity Y = lnKεin fine scale. The SCM which couples the generalized polynomial chaos is used to make the problem determined, and then the FEHM method is used to solve it. Sparse grid stochastic collocation method is used when KL expansion has many random variables. The numerical examples demonstrate that the SHMFE approach can efficiently simulate the flow in naturally multiscale heterogeneous subsurface formations with relatively lower computational cost comparing with the SFEM methods.

Variability of volume strain in bounded heterogeneous media

The deformation of the solid matrix affects the fluid pore pressure and flow by altering the pore volume. Such interaction in turn affects the storage of groundwater in the void space. Obviously, this subject is of interest in groundwater hydrology. This paper describes an investigation of the effect of aquifer heterogeneity on the variability of the fluid pressure head and solid's volume strain, where the assumption of a constant vertical total stress leads to a relatively simple relationship between changes in solid's volume strain and fluid pressure head. To solve the problem analytically, focus is placed on the one-dimensional models. It is found from our closed-form solutions that the variance and correlation length of the log hydraulic conductivity are important in increasing the variability of pressure head and solid's volume strain. It is hoped that our findings will provide a basic framework for understanding and quantifying field-scale volume strain processes and be useful in stimulating further research in this area. Copyright © 2011 John Wiley & Sons, Ltd.

Stochastic analysis of field-scale heat advection in heterogeneous aquifers

Abstract. Owing to the analogy between the solute and heat transport processes, it can be expected that the rate of growth of the spatial second moments of the heat flux in a heterogeneous aquifer over relatively large space scales is greater than that predicted by applying the classical heat transport model. The motivation of stochastic analysis of heat transport at the field scale is therefore to quantify the enhanced growth of the field-scale second moments caused by the spatially varying specific discharge field. Within the framework of stochastic theory, an effective advection-dispersion equation containing effective parameters (namely, the macrodispersion coefficients) is developed to model the mean temperature field. The rate of growth of the field-scale spatial second moments of the mean temperature field in the principal coordinate directions is described by the macrodispersion coefficient. The variance of the temperature field is also developed to characterize the reliability to be anticipated in applying the mean heat transport model. It is found that the heterogeneity of the medium and the correlation length of the log hydraulic conductivity are important in enhancing the field-scale heat advection, while the effective thermal conductivity plays the role in reducing the field-scale heat advection.

Operational real‐time modeling with ensemble Kalman filter of variably saturated subsurface flow including stream‐aquifer interaction and parameter updating

Urban groundwater is frequently contaminated, and the exact location of the pollution spots is often unknown. Intelligent monitoring of the temporal variations in groundwater flow in such an area assists in selectively extracting groundwater of drinking water quality. Here an example from the city of Zurich (Switzerland) is shown. The monitoring strategy consists of using the ensemble Kalman filter (EnKF) for optimally combining online observations and online models for the real‐time characterization of groundwater flow. We conducted numerical simulation experiments for the period January 2004 to December 2007 with a 3‐D finite element model for variably saturated groundwater flow. It was found that the daily assimilation of piezometric head data with EnKF results in a better characterization of piezometric heads than does a model which is inversely calibrated with historical data but not updated in real time. The positive impact of model updating with observations can still be observed 10 days after the update. These simulations also suggest that parameters (hydraulic conductivity and leakage) are successfully updated: 1 and 10 day piezometric head predictions are better with than without updating of parameters. Additional experiments with a synthetic model for the same site, in which the only difference is that certain parameter values are selected as the unknown “true” conditions, show that EnKF also successfully updates unknown parameters. However, this is only the case if spatially distributed hydraulic conductivities and leakage coefficients are jointly updated and if a damping parameter is used. The mean absolute error of estimated log leakage coefficients decreased by up to 63%; for log hydraulic conductivity a decrease of up to 27% was observed. From January 2009 the method has been operational at the Water Works Zurich and showed a remarkable performance until present (October 2010).

Characteristics and controlling factors of dispersion in bounded heterogeneous porous media

In this study, we present first‐order (in terms of the log conductivity variance) analytical solutions to displacement covariances Xij and macrodispersion coefficients Dij for transport of conservative solutes in two‐dimensional, bounded heterogeneous porous media. These solutions are presented as infinity series and are explicit functions of the statistics of the log hydraulic conductivity. Using numerical examples, the convergence of these solutions in terms of the number of terms in truncated finite summations is first investigated, and the accuracy of these solutions is examined by comparing them with results from Monte Carlo simulations and the semianalytical solutions of Osnes (1998). Through several examples, the general features of time‐dependent Xij and Dij are discussed. Unlike in unbounded domains, the longitudinal macrodispersivity D11 for bounded domains does not approach an asymptotic value at large dimensionless time but instead increases quickly near the downstream constant head boundary. In addition, D11 for bounded domains is always larger than that in unbounded domains, and accordingly, the transverse macrodispersivity D22 in bounded domains is smaller than that in unbounded domains. These differences stem from the lateral no‐flow boundaries in our bounded domain models. We also investigated the effect of domain sizes on Xij and Dij. Our study shows that both Xij and Dij depend not only on the dimensionless size of the domain but also on its aspect ratio. The dependence of Xij and Dij on the particles' initial location has also been investigated at detail. Our results indicate that while this dependence is very small for X11 and D11, the release location has a significant impact on both X22 and D22. Our solutions to both displacement covariances and macrodispersivity are compared against those derived from tracer test data at the Borden site. The comparison shows that our solutions are quite close to observed data, indicating that they may be applicable to predict solute transport at the field scale.

Hydraulic conductivity changes of a low-plasticity till subjected to freeze–thaw cycles

This paper examines the influence of several freeze–thaw cycles on the hydraulic conductivity changes of a glacial till from Péribonka in Quebec, Canada. Samples reconstituted by consolidation of a slurry and by compaction at dry of optimum conditions were subjected to closed freezing and thaw as well as constant-head permeability tests. For saturated consolidated samples, the relationship between the logarithm of hydraulic conductivity and void ratio was linear for both unfrozen and thawed states. At a given void ratio, the increase of the hydraulic conductivity of the thawed soil was larger for compacted and unsaturated samples than for saturated consolidated samples. Actual freeze–thaw-induced hydraulic conductivity changes, however, may be offset by a decrease in void ratio during thaw consolidation. A general framework considering void ratio, water content, stress and hydraulic conductivity relationships is presented for Péribonka glacial till.

Spatial variability of the poroelastic response of the heterogeneous medium

In this work, a stochastic methodology is applied to analyze the variability of the poroelastic response of the heterogeneous medium at the field scale. To solve the problem analytically, we restrict our attention to the one-dimensional models, where fluid flow as well as deformation occurs in one direction only under a constant applied stress. Assuming statistic homogeneity, the closed-form solutions that describe the variability of fluid pressure head, and a solid's strain and displacement are developed using a spectral approach based on Fourier–Stieltjes representations for the perturbed quantities. The influence of the correlation length of the log hydraulic conductivity on these results is investigated. It is found that the variances of the solid's strain and displacement increase with the correlation length of the log hydraulic conductivity, while the correlation length of the log hydraulic conductivity plays the role in reducing the variability of the specific discharge.

A stochastic collocation based Kalman filter for data assimilation

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

Wavelet analysis of characteristic length scales and orientation of two-dimensional heterogeneous porous media

The global wavelet energy spectrum (GWES) has been used on one-dimensional transects of porous media to identify the characteristic length scale of a material property. The characteristic length scale is a measure of the distance over which property values are correlated. We extend the wavelet analysis technique to two-dimensional stationary porous media for identifying both the characteristic length scales and orientation of heterogeneity. We theoretically develop and evaluate the GWES for isotropic and anisotropic random fields using the two-dimensional Mexican hat wavelet. The relationship between the wavelet scale and the characteristic length scale of a two-dimensional random field is investigated for different covariance structures. The ability of the GWES to identify multiple characteristic length scales and orientations in a random field with nested covariance structures is also investigated. We use the technique to identify the characteristic length scales of a log hydraulic conductivity field used in a laboratory experiment.

Numerical Studies of Three-dimensional Stochastic Darcy’s Equation and Stochastic Advection-Diffusion-Dispersion Equation

Solute transport in randomly heterogeneous porous media is commonly described by stochastic flow and advection-dispersion equations with a random hydraulic conductivity field. The statistical distribution of conductivity of engineered and naturally occurring porous material can vary, depending on its origin. We describe solutions of a three-dimensional stochastic advection-dispersion equation using a probabilistic collocation method (PCM) on sparse grids for several distributions of hydraulic conductivity. Three random distributions of log hydraulic conductivity are considered: uniform, Gaussian, and truncated Gaussian (beta). Log hydraulic conductivity is represented by a Karhunen-Loeve (K-L) decomposition as a second-order random process with an exponential covariance function. The convergence of PCM has been demonstrated. It appears that the accuracy in both the mean and the standard deviation of PCM solutions can be improved by using the Jacobi-chaos representing the truncated Gaussian distribution rather than the Hermite-chaos for the Gaussian distribution. The effect of type of distribution and parameters such as the variance and correlation length of log hydraulic conductivity and dispersion coefficient on leading moments of the advection velocity and solute concentration was investigated.

Numerical evaluation of multicomponent cation exchange reactive transport in physically and geochemically heterogeneous porous media

Experimental evidence and stochastic studies strongly show that the transport of reactive solutes in porous media is significantly influenced by heterogeneities in hydraulic conductivity, porosity, and sorption parameters. In this paper, we present Monte Carlo numerical simulations of multicomponent reactive transport involving competitive cation exchange reactions in a two-dimensional vertical physically and geochemically heterogeneous medium. Log hydraulic conductivity, log K, and log cation exchange capacity (log CEC) are assumed to be random Gaussian functions with spherical semivariograms. Random realizations of log K and log CEC are used as input data for the numerical simulation of multicomponent reactive transport with CORE2D, a general purpose reactive transport code. Longitudinal features of the fronts of reactive and conservative species are computed from the temporal and spatial moments of depth-averaged concentrations. Monte Carlo simulations show that: (1) the displacement of reactive fronts increases with increasing variance of log K, while it decreases with the variance of log CEC; (2) second-order spatial moments increase with increasing variances of log K and log CEC; (3) uncertainties in the mean arrival time are largest (smallest) for negatively (positively) correlated log K and Log CEC; (4) cations undergoing competitive cation exchange exhibit different apparent velocities and retardation factors due to both physical and geochemical heterogeneities; and (5) the correlation between log K and log CEC affects significantly apparent cation retardation factors in heterogeneous aquifers.

Laplace-Transform Finite Element Solution of Nonlocal and Localized Stochastic Moment Equations of Transport

Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1238-1255) developed exact first and second nonlocal moment equations for advective-dispersive transport in finite, randomly heterogeneous geologic media. The velocity and concentration in these equations are generally nonstationary due to trends in heterogeneity, conditioning on site data and the influence of forcing terms. Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1399-1418) solved the Laplace transformed versions of these equations recursively to second order in the standard deviation σY of (natural) log hydraulic conductivity, and iteratively to higher-order, by finite elements followed by numerical inversion of the Laplace transform. They did the same for a space-localized version of the mean transport equation. Here we recount briefly their theory and algorithms; compare the numerical performance of the Laplace-transform finite element scheme with that of a high-accuracy ULTIMATE-QUICKEST algorithm coupled with an alternating split operator approach; and review some computational results due to Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1399-1418) to shed light on the accuracy and computational efficiency of their recursive and iterative solutions in comparison to conditional Monte Carlo simulations in two spatial dimensions. AMS subject classifications: 65Z05, 65Z05, 68U20, 76S05

Temporal moments in geoelectrical monitoring of salt tracer experiments

Monitoring of salt tracer experiments by electrical resistivity tomography (ERT) has been shown to be a valuable tool for characterizing the hydraulics of an aquifer, but efficient approaches of determining the spatial hydraulic conductivity distribution from ERT data are still missing. Standard inversion of ERT data obtained during salt tracer tests may even lead to estimates of the concentration distribution that are in contradiction to flow and transport of conservative compounds in porous media. In order to avoid nonphysical behavior, we consider the governing equations of groundwater flow, solute transport, and geoelectrics as a coupled system. While the tracer passes through the part of the domain that is sensitive for a particular electrode configuration, the measured electrical potential differences are perturbed. We characterize these perturbations by their temporal moments and relate them to the temporal moments of concentration, which themselves depend on hydraulic conductivity. We present temporal moment‐generating equations leading from the hydraulic conductivity field via heads and velocities to the temporal moments of concentration and electrical potential perturbations. The approach makes use of a linearized version of the Poisson equation. On the basis of this system of coupled steady state equations, we compute the sensitivity of electrical potential perturbations with respect to the log hydraulic conductivity distribution by the continuous adjoint state method for coupled systems. For demonstration, we simulate salt tracer experiments in a virtual quasi‐two‐dimensional sandbox, monitored by ERT. We show that the ratio of the first over the zeroth temporal moment of potential perturbation is less affected by the linearization of the Poisson equation than the zeroth and first moments themselves. Thus, it appears recommendable to use the ratio of first to zeroth moments also as data in inversion. We compare sensitivity patterns resulting from different electrode configurations. The methods of forward simulations and sensitivity calculations presented in this paper can be combined with any inverse kernel to develop a complete inverse model. Altogether, using temporal moments of potential perturbation appears promising for fully coupled hydrogeophysical inversion of ERT surveys during salt tracer tests.

Stochastic analysis of bounded unsaturated flow in heterogeneous aquifers: Spectral/perturbation approach

This paper describes a stochastic analysis of steady state flow in a bounded, partially saturated heterogeneous porous medium subject to distributed infiltration. The presence of boundary conditions leads to non-uniformity in the mean unsaturated flow, which in turn causes non-stationarity in the statistics of velocity fields. Motivated by this, our aim is to investigate the impact of boundary conditions on the behavior of field-scale unsaturated flow. Within the framework of spectral theory based on Fourier–Stieltjes representations for the perturbed quantities, the general expressions for the pressure head variance, variance of log unsaturated hydraulic conductivity and variance of the specific discharge are presented in the wave number domain. Closed-form expressions are developed for the simplified case of statistical isotropy of the log hydraulic conductivity field with a constant soil pore-size distribution parameter. These expressions allow us to investigate the impact of the boundary conditions, namely the vertical infiltration from the soil surface and a prescribed pressure head at a certain depth below the soil surface. It is found that the boundary conditions are critical in predicting uncertainty in bounded unsaturated flow. Our analytical expression for the pressure head variance in a one-dimensional, heterogeneous flow domain, developed using a nonstationary spectral representation approach [Li S-G, McLaughlin D. A nonstationary spectral method for solving stochastic groundwater problems: unconditional analysis. Water Resour Res 1991;27(7):1589–605; Li S-G, McLaughlin D. Using the nonstationary spectral method to analyze flow through heterogeneous trending media. Water Resour Res 1995; 31(3):541–51], is precisely equivalent to the published result of Lu et al. [Lu Z, Zhang D. Analytical solutions to steady state unsaturated flow in layered, randomly heterogeneous soils via Kirchhoff transformation. Adv Water Resour 2004;27:775–84].

An efficient, high-order probabilistic collocation method on sparse grids for three-dimensional flow and solute transport in randomly heterogeneous porous media

In this study, a probabilistic collocation method (PCM) on sparse grids is used to solve stochastic equations describing flow and transport in three-dimensional, saturated, randomly heterogeneous porous media. The Karhunen–Loève decomposition is used to represent log hydraulic conductivity Y = ln K s . The hydraulic head h and average pore-velocity v are obtained by solving the continuity equation coupled with Darcy’s law with random hydraulic conductivity field. The concentration is computed by solving a stochastic advection–dispersion equation with stochastic average pore-velocity v computed from Darcy’s law. The PCM approach is an extension of the generalized polynomial chaos (gPC) that couples gPC with probabilistic collocation. By using sparse grid points in sample space rather than standard grids based on full tensor products, the PCM approach becomes much more efficient when applied to random processes with a large number of random dimensions. Monte Carlo (MC) simulations have also been conducted to verify accuracy of the PCM approach and to demonstrate that the PCM approach is computationally more efficient than MC simulations. The numerical examples demonstrate that the PCM approach on sparse grids can efficiently simulate solute transport in randomly heterogeneous porous media with large variances.