Log Hydraulic Conductivity Field Research Articles

Modeling groundwater flow with random hydraulic conductivity using radial basis function partition of unity method

Simulating groundwater flows in heterogeneous aquifers is one of the most widely studied problems. The heterogeneity is modeled through random hydraulic conductivity fields log-normally distributed. In this paper, we aim to generate the realization of the log-normal hydraulic conductivity by summing up a finite number of random periodic modes with the Kraichnan algorithm. To address Neumann conditions, we propose a change of variables that results in Dirichlet boundary conditions along all boundaries of the computational domain. Then, we suggest using the radial basis function partition of unity method (RBFPUM) and RBFPUM with QR factorization (RBFPUM-QR) to analyze groundwater flow in the heterogeneous porous medium. We evaluate the performance of the proposed methods by increasing the number of random modes and the variance of the log-hydraulic conductivity fields using Gaussian and exponential correlation. In addition, we verify the effectiveness of the presented approaches in one and two dimensions using the method of manufactured solutions. Furthermore, we provide statistical inferences through Monte Carlo simulations. Numerical results demonstrate that RBFPUM-QR performs well in handling higher heterogeneity in both single realizations and Monte Carlo simulations.

Inferring geostatistical properties of hydraulic conductivity fields from saline tracer tests and equivalent electrical conductivity time-series

We use Approximate Bayesian Computation and the Kullback–Leibler divergence measure to quantify to what extent horizontal and vertical equivalent electrical conductivity time-series observed during tracer tests constrain the 2-D geostatistical parameters of multivariate Gaussian log-hydraulic conductivity fields. Considering a perfect and known relationship between salinity and electrical conductivity at the point scale, we find that the horizontal equivalent electrical conductivity time-series best constrain the geostatistical properties. The variance, controlling the spreading rate of the solute, is the best constrained geostatistical parameter, followed by the integral scales in the vertical direction. We find that horizontally layered models with moderate to high variance have the best resolved parameters. Since the salinity field at the averaging scale (e.g., the model resolution in tomograms) is typically non-ergodic, our results serve as a starting point for quantifying uncertainty due to small-scale heterogeneity in laboratory-experiments, tomographic results and hydrogeophysical inversions involving DC data.

Tidal Effect on Groundwater Fluctuations and Saltwater Intrusion in Coastal Heterogeneous Aquifers. (Dept. C. ( Irrigation And Hydraulics ))

Groundwater table and saltwater wedge in coastal areas are commonly affected by tidal waves fluctuations. In this research, for the first time, a two-dimensional numerical model in plan is developed for simulating both the tidal fluctuation of the beach groundwater table and movement of interface between saltwater and freshwater in both coastal confined and unconfined heterogeneous aquifers. The sharp interface philosophy is adopted to simulate dynamics of freshwater flow. Tidal waves are simulated by simple harmonic motions in time. The solution is based on an approach of separating large and small time scales motions of groundwater fluctuations. The proposed model consists of two sub-models. The first one applied Finite Element Method (FEM) to simulate the hydraulic responses of the aquifer in large time scale and it is coupled with Finite Difference Method (FDM), in the second sub-model, to simulate the corresponding ones in small time scale. The model is verified against analytical solutions presented by other previous researchers in case of homogenous aquifer. To simulate the aquifer heterogeneity, unconditional random log-hydraulic conductivity fields are generated to present different hydraulic conductivity realizations for the aquifer under consideration. The application is carried out on both the hypothetical confined aquifer and the real Quaternary unconfined aquifer in El Arish-Rafah area, Egypt. Results show that both the groundwater fluctuation and saltwater intrusion due to tidal waves are affected by the aquifers heterogeneity and the tidal waves effects in confined aquifers are more significant than in unconfined aquifers.

Numerical benchmark study for flow in highly heterogeneous aquifers

This article presents numerical investigations on accuracy and convergence properties of several numerical approaches for simulating steady state flows in heterogeneous aquifers. Finite difference, finite element, discontinuous Galerkin, spectral, and random walk methods are tested on two-dimensional benchmark flow problems. Realizations of log-normal hydraulic conductivity fields are generated by Kraichnan algorithms in closed form as finite sums of random periodic modes, which allow direct code verification by comparisons with manufactured reference solutions. The quality of the methods is assessed for increasing number of random modes and for increasing variance of the log-hydraulic conductivity fields with Gaussian and exponential correlation. Experimental orders of convergence are calculated from successive refinements of the grid. The numerical methods are further validated by comparisons between statistical inferences obtained from Monte Carlo ensembles of numerical solutions and theoretical first-order perturbation results. It is found that while for Gaussian correlation of the log-conductivity field all the methods perform well, in the exponential case their accuracy deteriorates and, for large variance and number of modes, the benchmark problems are practically not solvable with reasonably large computing resources, for all the methods considered in this study.

Towards Improving the Efficiency of Bayesian Model Averaging Analysis for Flow in Porous Media via the Probabilistic Collocation Method

The characterization of flow in subsurface porous media is associated with high uncertainty. To better quantify the uncertainty of groundwater systems, it is necessary to consider the model uncertainty. Multi-model uncertainty analysis can be performed in the Bayesian model averaging (BMA) framework. However, the BMA analysis via Monte Carlo method is time consuming because it requires many forward model evaluations. A computationally efficient BMA analysis framework is proposed by using the probabilistic collocation method to construct a response surface model, where the log hydraulic conductivity field and hydraulic head are expanded into polynomials through Karhunen–Loeve and polynomial chaos methods. A synthetic test is designed to validate the proposed response surface analysis method. The results show that the posterior model weight and the key statistics in BMA framework can be accurately estimated. The relative errors of mean and total variance in the BMA analysis results are just approximately 0.013% and 1.18%, but the proposed method can be 16 times more computationally efficient than the traditional BMA method.

Theoretical analysis of non‐Gaussian heterogeneity effects on subsurface flow and transport

AbstractMuch of the stochastic groundwater literature is devoted to the analysis of flow and transport in Gaussian or multi‐Gaussian log hydraulic conductivity (or transmissivity) fields, (x being a position vector), characterized by one or (less frequently) a multiplicity of spatial correlation scales. Yet Y and many other variables and their (spatial or temporal) increments, ΔY, are known to be generally non‐Gaussian. One common manifestation of non‐Gaussianity is that whereas frequency distributions of Y often exhibit mild peaks and light tails, those of increments are generally symmetric with peaks that grow sharper, and tails that become heavier, as separation scale or lag between pairs of Y values decreases. A statistical model that captures these disparate, scale‐dependent distributions of Y and in a unified and consistent manner has been recently proposed by us. This new “generalized sub‐Gaussian (GSG)” model has the form where is (generally, but not necessarily) a multiscale Gaussian random field and is a nonnegative subordinator independent of G. The purpose of this paper is to explore analytically, in an elementary manner, lead‐order effects that non‐Gaussian heterogeneity described by the GSG model have on the stochastic description of flow and transport. Recognizing that perturbation expansion of hydraulic conductivity diverges when Y is sub‐Gaussian, we render the expansion convergent by truncating Y's domain of definition. We then demonstrate theoretically and illustrate by way of numerical examples that, as the domain of truncation expands, (a) the variance of truncated Y (denoted by Yt) approaches that of Y and (b) the pdf (and thereby moments) of Yt increments approach those of Y increments and, as a consequence, the variogram of approaches that of . This in turn guarantees that perturbing to second order in (the standard deviation of Yt) yields results which approach those we obtain upon perturbing to second order in even as the corresponding series diverges. Our analysis is rendered mathematically tractable by considering mean‐uniform steady state flow in an unbounded, two‐dimensional domain of mildly heterogeneous Y with a single‐scale function G having an isotropic exponential covariance. Results consist of expressions for (a) lead‐order autocovariance and cross‐covariance functions of hydraulic head, velocity, and advective particle displacement and (b) analogues of preasymptotic as well as asymptotic Fickian dispersion coefficients. We compare these theoretically and graphically with corresponding expressions developed in the literature for Gaussian Y. We find the former to differ from the latter by a factor k = ( denoting ensemble expectation) and the GSG covariance of longitudinal velocity to contain an additional nugget term depending on this same factor. In the limit as Y becomes Gaussian, k reduces to one and the nugget term drops out.

Transverse mixing in three-dimensional nonstationary anisotropic heterogeneous porous media

Groundwater plumes originating from continuously emitting sources are typically controlled by transverse mixing between the plume and reactants in the ambient solution. In two-dimensional domains, heterogeneity causes only weak enhancement of transverse mixing in steady-state flows. In three-dimensional domains, more complex flow patterns are possible because streamlines can twist. In particular, spatially varying orientation of anisotropy can cause steady-state groundwater whirls. We analyze steady-state solute transport in three-dimensional locally isotropic heterogeneous porous media with blockwise anisotropic correlation structure, in which the principal directions of anisotropy differ from block to block. For this purpose, we propose a transport scheme that relies on advective transport along streamlines and transverse-dispersive mass exchange between them based on Voronoi tessellation. We compare flow and transport results obtained for a nonstationary anisotropic log-hydraulic conductivity field to an equivalent stationary field with identical mean, variance, and two-point correlation function disregarding the nonstationarity. The nonstationary anisotropic field is affected by mean secondary motion and causes neighboring streamlines to strongly diverge, which can be quantified by the two-particle semivariogram of lateral advective displacements. An equivalent kinematic descriptor of the flow field is the advective folding of plumes, which is more relevant as precursor of mixing than stretching. The separation of neighboring streamlines enhances transverse mixing when considering local dispersion. We quantify mixing by the flux-related dilution index, which is substantially larger for the nonstationary anisotropic conductivity field than for the stationary one. We conclude that nonstationary anisotropy in the correlation structure has a significant impact on transverse plume deformation and mixing. In natural sediments, contaminant plumes most likely mix more effectively in the transverse directions than predicted by models that neglect the nonstationarity of anisotropy.

Multimodel Bayesian analysis of groundwater data worth

AbstractWe explore the way in which uncertain descriptions of aquifer heterogeneity and groundwater flow impact one's ability to assess the worth of collecting additional data. We do so on the basis of Maximum Likelihood Bayesian Model Averaging (MLBMA) by accounting jointly for uncertainties in geostatistical and flow model structures and parameter (hydraulic conductivity) as well as system state (hydraulic head) estimates, given uncertain measurements of one or both variables. Previous description of our approach was limited to geostatistical models based solely on hydraulic conductivity data. Here we implement the approach on a synthetic example of steady state flow in a two‐dimensional random log hydraulic conductivity field with and without recharge by embedding an inverse stochastic moment solution of groundwater flow in MLBMA. A moment‐equations‐based geostatistical inversion method is utilized to circumvent the need for computationally expensive numerical Monte Carlo simulations. The approach is compatible with either deterministic or stochastic flow models and consistent with modern statistical methods of parameter estimation, admitting but not requiring prior information about the parameters. It allows but does not require approximating lead predictive statistical moments of system states by linearization while updating model posterior probabilities and parameter estimates on the basis of potential new data both before and after such data are actually collected.

Conditional Simulation of Flow in Heterogeneous Porous Media with the Probabilistic Collocation Method

AbstractA stochastic approach to conditional simulation of flow in randomly heterogeneous media is proposed with the combination of the Karhunen-Loeve expansion and the probabilistic collocation method (PCM). The conditional log hydraulic conductivity field is represented with the Karhunen-Loeve expansion, in terms of some deterministic functions and a set of independent Gaussian random variables. The propagation of uncertainty in the flow simulations is carried out through the PCM, which relies on the efficient polynomial chaos expansion used to represent the flow responses such as the hydraulic head. With the PCM, existing flow simulators can be employed for uncertainty quantification of flow in heterogeneous porous media when direct measurements of hydraulic conductivity are taken into consideration. With illustration of several numerical examples of groundwater flow, this study reveals that the proposed approach is able to accurately quantify uncertainty of the flow responses conditioning on hydraulic conductivity data, while the computational efforts are significantly reduced in comparison to the Monte Carlo simulations.

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.

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.

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].

On the geostatistical characterization of hierarchical media

The subsurface consists of porous and fractured materials exhibiting a hierarchical geologic structure, which gives rise to systematic and random spatial and directional variations in hydraulic and transport properties on a multiplicity of scales. Traditional geostatistical moment analysis allows one to infer the spatial covariance structure of such hierarchical, multiscale geologic materials on the basis of numerous measurements on a given support scale across a domain or “window” of a given length scale. The resultant sample variogram often appears to fit a stationary variogram model with constant variance (sill) and integral (spatial correlation) scale. In fact, some authors, who recognize that hierarchical sedimentary architecture and associated log hydraulic conductivity fields tend to be nonstationary, nevertheless associate them with stationary “exponential‐like” transition probabilities and variograms, respectively, the latter being a consequence of the former. We propose that (1) the apparent ability of stationary spatial statistics to characterize the covariance structure of nonstationary hierarchical media is an artifact stemming from the finite size of the windows within which geologic and hydrologic variables are ubiquitously sampled, and (2) the artifact is eliminated upon characterizing the covariance structure of such media with the aid of truncated power variograms, which represent stationary random fields obtained upon sampling a nonstationary fractal over finite windows. To support our opinion, we note that truncated power variograms arise formally when a hierarchical medium is sampled jointly across all geologic categories and scales within a window; cite direct evidence that geostatistical parameters (variance and integral scale) inferred on the basis of traditional variograms vary systematically with support and window scales; demonstrate the ability of truncated power models to capture these variations in terms of a few scaling parameters; show that exponential and truncated power variograms are often difficult to distinguish from each other, which helps explain why hierarchical data may appear to fit the former; note that truncated power models are unique in their ability to represent multiscale random fields having either Gaussian or heavy‐tailed symmetric Levy stable probability distributions; detail the way in which these models allow conditioning the spatial statistics of such fields on multiscale measurements via cokriging; and illustrate these capabilities on multiscale hydraulic data from an unconfined aquifer near Tübingen, Germany.

Assessment of uncertainty associated with the estimation of well catchments by moment equations

Non-local stochastic moment equations are used successfully to analyze groundwater flow in randomly heterogeneous media. Here we present a moment equations-based approach to quantify the uncertainty associated with the estimation of well catchments. Our approach is based on the development of a complete second order formalism which allows obtaining the first statistical moments of the trajectories of conservative solute particles advected in a generally non-uniform groundwater flow. Approximate equations of moments of particles’ trajectories are then derived on the basis of a second order expansion in terms of the standard deviation of the aquifer log hydraulic conductivity. Analytical expressions are then obtained for the predictors of locations of mean stagnation points, together with their associated uncertainties. We implement our approach on heterogeneous media in bounded two-dimensional domains, with and without including the effect of conditioning on hydraulic conductivity information. The impact of domain size, boundary conditions, heterogeneity and non-stationarity of hydraulic conductivity on the prediction of a well catchment is explored. The results are compared against Monte Carlo simulations and semi-analytical solutions available in the literature. The methodology is applicable to both infinite and bounded domains and is free of distributional assumptions (and so applies to both Gaussian and non-Gaussian log hydraulic conductivity fields) and formally includes the effect of conditioning on available information.

Modeling the effects of nonlinear equilibrium sorption on the transport of solute plumes in saturated heterogeneous porous media

Transport of sorbing solutes in 2D steady and heterogeneous flow fields is modeled using a particle tracking random walk technique. The solute is injected as an instantaneous pulse over a finite area. Cases of linear and Freundlich sorption isotherms are considered. Local pore velocity and mechanical dispersion are used to describe the solute transport mechanisms at the local scale. This paper addresses the impact of the degree of heterogeneity and correlation lengths of the log-hydraulic conductivity field as well as negative correlation between the log-hydraulic conductivity field and the log-sorption affinity field on the behavior of the plume of a sorbing chemical. Behavior of the plume is quantified in terms of longitudinal spatial moments: center-of-mass displacement, variance, 95% range, and skewness. The range appears to be a better measure of the spread in the plumes with Freundlich sorption because of plume asymmetry. It has been found that the range varied linearly with the travelled distance, regardless of the sorption isotherm. This linear relationship is important for extrapolation of results to predict behavior beyond simulated times and distances. It was observed that the flow domain heterogeneity slightly enhanced the spreading of nonlinearly sorbing solutes in comparison to that which occurred for the homogeneous flow domain, whereas the spreading enhancement in the case of linear sorption was much more pronounced. In the case of Freundlich sorption, this enhancement led to further deceleration of the solute plume movement as a result of increased retardation coefficients produced by smaller concentrations. It was also observed that, except for plumes with linear sorption, correlation between the hydraulic conductivity and the sorption affinity fields had minimal effect on the spatial moments of solute plumes with nonlinear sorption.

Anisotropy, lacunarity, and upscaled conductivity and its autocovariance in multiscale random fields with truncated power variograms

It has been shown by Di Federico and Neuman [1997, 1998a, b] that observed multiscale behaviors of subsurface fluid flow and transport variables can be explained within the context of a unified stochastic framework, which views hydraulic conductivity as a random fractal characterized by a power variogram. Such a random field is statistically nonhomogeneous but possesses homogeneous spatial increments. Di Federico and Neuman [1997] have demonstrated that the power variogram and associated spectra of a statistically isotropic fractal field can be constructed as a weighted integral from zero to infinity (an infinite hierarchy) of exponential or Gaussian variograms and spectra of mutually uncorrelated fields (modes) that are homogeneous and isotropic. We show in this paper that the same holds true when the field and its constituent modes are statistically anisotropic, provided the ratios between principal integral (spatial correlation) scales are the same for all modes. We then analyze the effect of filtering out (truncating) modes of low, high, and intermediate spatial frequency from this infinite hierarchy in the real and spectral domains. A low‐frequency cutoff renders the truncated hierarchy homogeneous. The integral scales of the lowest‐ and highest‐frequency cutoff modes are related to length scales of the sampling window (domain) and data support (sample volume), respectively. Taking the former to be proportional to the latter renders expressions for the integral scale and variance of the truncated field dependent on window and support scale (in a manner previously shown to be consistent with observations in the isotropic case). It also allows (in principle) bridging across scales at a specific locale, as well as among locales, by adopting either site‐specific or generalized variogram parameters. The introduction of intermediate cutoffs allows us to account, in a straightforward manner, for lacunarity due to gaps in the multiscale hierarchy created by the absence of modes associated with discrete ranges of scales (for example, where textural and structural features are associated with distinct ranges of scale, such as fractures having discrete ranges of trace length and density, which dissect the rock into matrix blocks having corresponding ranges of sizes). We explore mathematically and graphically the effects that anisotropy and lacunarity have on the integral scale, variance, covariance, and spectra of a truncated fractal field. We then develop an expression for the equivalent hydraulic conductivity of a box‐shaped porous block, embedded within a multiscale log hydraulic conductivity field, under mean‐uniform flow. The block is larger than the support scale of the field but is smaller than a surrounding sampling window. Consequently, its equivalent hydraulic conductivity is a random variable whose variance and spatial autocorrelation function, conditioned on a known mean value of support‐scale conductivity across the window, are given explicitly by our multiscale theory.

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.

Time‐Related Capture Zones for Contaminants in Randomly Heterogeneous Formations

AbstractPrediction of solute travel distance based on homogeneous and isotropic hydraulic conductivity may lead to substantial differences in the actual travel distance in a heterogeneous system. The effect of randomly varying hydraulic conductivity on the spatial location of time‐related well capture zones for a nonreactive tracer in a confined aquifer with uniform base flow is considered. A numerical Monte Carlo procedure is used in conjunction with fast Fourier transform‐based spectral methods. The log hydraulic conductivity field is assumed to be Gaussian and stationary, with isotropic exponential autocovariance. Various degrees of domain heterogeneity are considered, and stability and accuracy of the numerical procedure are examined. The total probability that a particle injected at a point in the aquifer is extracted by the well within a given time is identified. The concept of probabilistic isochrone, which is the boundary of a given time‐related capture zone, is introduced. A simple analytical model that extends the deterministic capture‐zone model for uniform media is derived, by taking into account the effect of random variations of hydraulic conductivity. Knowledge of the spatial pattern of probabilistic isochrones allows reconstructing the breakthrough curve at the well for solute injections at selected locations in the aquifer.

Numerical solutions to statistical moment equations of groundwater flow in nonstationary, bounded, heterogeneous media

In this paper we investigate the combined effect of nonstationarity in log hydraulic conductivity and the presence of boundaries on flow in heterogeneous aquifers. We derive general equations governing the statistical moments of hydraulic head for steady state flow by perturbation expansions. Due to their mathematical complexity, we solve the moment equations by the numerical technique of finite differences. The numerical approach has flexibility in handling (moderately) irregular geometry, different boundary conditions, various trends in the mean log hydraulic conductivity, spatial variabilities in the magnitude and direction of mean flow, and different covariance functions, all of which are important factors to consider for real‐world applications. The effect of boundaries on the first two statistical moments involving head is strong and persistent. For example, in the case of stationary log hydraulic conductivity the head variance is always finite in a bounded domain while the head variance may be infinite in an unbounded domain. As in many other stochastic models, the statistical moment equations are derived under the assumption that the variance of log hydraulic conductivity is small. Accounting for a spatially varying, large‐scale trend in the log hydraulic conductivity field reduces the variance of log hydraulic conductivity. Although this makes the conductivity field nonstationary and significantly increases the mathematical complexity in the problem, it justifies the small‐variance assumption for many aquifers.