Log Hydraulic Conductivity Research Articles

Stochastic analysis of spatial variability in unconfined groundwater flow

In this paper, spatial variability in steady one-dimensional unconfined groundwater flow in heterogeneous formations is investigated. An approach to deriving the variance of the hydraulic head is developed using the nonlinear filter theory. The nonlinear governing equation describing the one-dimensional unconfined groundwater flow is decomposed into three linear partial differential equations using the perturbation method. The linear and quadratic frequency response functions are obtained from the first- and second-order perturbation equations using the spectral method. Furthermore, under the assumption of the exponential covariance function of log hydraulic conductivity, the analytical solutions of both the spectrum and the variance of the hydraulic head produced from the linear system are derived. The results show that the variance derived herein is less than that of Gelhar (1977). The reason is that the log transmissivity is linearized in Gelhar’s work. In addition, the analytical solutions of both the spectrum and the variance of the hydraulic head produced from the quadratic system are derived as well. It is found that the correlation scale and the trend in mean of log hydraulic conductivity are important to the dimensionless variance ratio.

A Comparative Study on Uncertainty Quantification for Flow in Randomly Heterogeneous Media Using Monte Carlo Simulations and Conventional and KL-Based Moment-Equation Approaches

Geological formations are ubiquitously heterogeneous, and the equations that govern flow and transport in such formations can be treated as stochastic partial differential equations. The Monte Carlo method is a straightforward approach for simulating flow in heterogeneous porous media; an alternative based on the moment-equation approach has been developed in the last two decades to reduce the high computational expense required by the Monte Carlo method. However, the computational cost of the moment-equation approach is still high. For example, to solve head covariance up to first order in terms of $\sigma_Y^2$, the variance of log hydraulic conductivity Y = ln Ks, it is required to solve sets of linear algebraic equations with N unknowns for 2N times (N being the number of grid nodes). The cost is even higher if higher-order approximations are needed. Zhang and Lu [J. Comput. Phys., 194 (2004), pp. 773--794] developed a new approach to evaluate high-order moments (fourth order for mean head in terms of $\sigma_Y$, and third order for head variances in terms of $\sigma_Y^2$) of flow quantities based on the combination of Karhunen--Loeve decomposition and perturbation methods. In this study, we systematically investigate the computational efficiency and solution accuracy of three approaches: Monte Carlo simulations, the conventional moment-equation (CME) approach, and the moment-equation approach based on Karhunen--Loeve decomposition (KLME). It is evident that the computational cost for the KLME approach is significantly lower than those required by the Monte Carlo and CME approaches. More importantly, while the computational costs (in terms of the number of times for solving linear algebraic equations with N unknowns) for the CME approach depend on the number of grid nodes, the cost for the KLME approach is independent of the number of grid nodes. This makes it possible to apply the KLME method to solve more realistic large-scale flow problems.

Uncertainty for solute movement in bounded heterogeneous porous media

The main focus of the present study is to assess the impact of finite flow domain size on solute travel time statistics for a nonreactive conservative solute in a bounded three-dimensional statistically isotropic porous medium. This is accomplished by introducing a modified covariance function of log hydraulic conductivity, accounting for a finite sample size, into the evaluation of solute travel time statistics within the framework of stochastic theory and the spectral perturbation techniques. This study finds that the first two moments of solute travel time for advection-dominated solute movement increase with the size of flow domain.

An efficient, high-order perturbation approach for flow in random porous media via Karhunen–Loève and polynomial expansions

In this study, we attempt to obtain higher-order solutions of the means and (co)variances of hydraulic head for saturated flow in randomly heterogeneous porous media on the basis of the combination of Karhunen–Loève decomposition, polynomial expansion, and perturbation methods. We first decompose the log hydraulic conductivity Y=ln K s as an infinite series on the basis of a set of orthogonal Gaussian standard random variables { ξ i }. The coefficients of the series are related to eigenvalues and eigenfunctions of the covariance function of log hydraulic conductivity. We then write head as an infinite series whose terms h ( n) represent head of nth order in terms of σ Y , the standard deviation of Y, and derive a set of recursive equations for h ( n) . We then decompose h ( n) with polynomial expansions in terms of the products of n Gaussian random variables. The coefficients in these series are determined by substituting decompositions of Y and h ( n) into those recursive equations. We solve the mean head up to fourth-order in σ Y and the head variances up to third-order in σ Y 2. We conduct Monte Carlo (MC) simulation and compare MC results against approximations of different orders from the moment-equation approach based on Karhunen–Loève decomposition (KLME). We also explore the validity of the KLME approach for different degrees of medium variability and various correlation scales. It is evident that the KLME approach with higher-order corrections is superior to the conventional first-order approximations and is computationally more efficient than the Monte Carlo simulation.

Conditioning mean steady state flow on hydraulic head and conductivity through geostatistical inversion

Nonlocal moment equations allow one to render deterministically optimum predictions of flow in randomly heterogeneous media and to assess predictive uncertainty conditional on measured values of medium properties. We present a geostatistical inverse algorithm for steady-state flow that makes it possible to further condition such predictions and assessments on measured values of hydraulic head (and/or flux). Our algorithm is based on recursive finite-element approximations of exact first and second conditional moment equations. Hydraulic conductivity is parameterized via universal kriging based on unknown values at pilot points and (optionally) measured values at other discrete locations. Optimum unbiased inverse estimates of natural log hydraulic conductivity, head and flux are obtained by minimizing a residual criterion using the Levenberg-Marquardt algorithm. We illustrate the method for superimposed mean uniform and convergent flows in a bounded two-dimensional domain. Our examples illustrate how conductivity and head data act separately or jointly to reduce parameter estimation errors and model predictive uncertainty.

The impact of finite flow domain size on the statistical properties of the flow system in heterogeneous porous media

A modified covariance function of log hydraulic conductivity, accounting for a finite field size, is derived to assess the impact of finite flow domain size on the statistical properties of the flow system in heterogeneous porous media, such as the uncertainty of a large-scale mean model (variation of head fluctuations) and the effective hydraulic conductivity. This development is accomplished by applying a perturbation approximation and spectral representation to a steady three-dimensional flow field of finite extent. Closed-form expressions in a statistically isotropic porous medium indicate that the finite flow domain size has a direct effect on the head variance and the effective hydraulic conductivity. The comparison of the presented formulation for effective hydraulic conductivity is made with existing results.

Numerical studies of the transport behavior of a passive solute in a two-dimensional incompressible random flow field

We study the transport behavior of a passive scalar in a two-dimensional (2D) time-independent Gaussian random velocity field by efficient and highly accurate numerical simulations. The model under consideration has been used in order to gain basic understanding of transport processes in incompressible flow through heterogeneous porous media. The velocity field is derived from the linearized solution of the Darcy equation with a Gauss-distributed log-hydraulic conductivity. The transport of a passive scalar is studied by a high precision random-walk method, which allows for a systematic nonperturbative study of the ensemble and effective dispersion coefficients. The conclusive numerical results validate the range of applicability of the perturbation theory and the consistency of nonperturbative approaches to the transport problem in a random medium. Furthermore, we observe closed streamlines in incompressible 2D Gaussian random fields, which restricts the direct applicability of the simulation method for transport in heterogeneous porous media, and questions the results of similar studies that do not observe this phenomenon.

Mean travel time of conservative solutes in randomly heterogeneous unbounded domains under mean uniform flow

We derive a closed‐form expression for mean travel time of a conservative solute migrating under uniform in the mean flow conditions within an infinite stationary field with simple exponential correlation of the natural logarithm of hydraulic conductivity. Our expression is developed from a consistent second‐order expansion in σY (standard deviation of the log hydraulic conductivity) of the equations for moments of travel time and trajectories of conservative solutes in two‐dimensional randomly nonuniform flows of Guadagnini et al. [2001]. As such, it is nominally valid for moderately heterogeneous fields, with σY2 < 1. Its validity for larger heterogeneity degrees is tested against numerical Monte Carlo simulations. Our results clarify the nonlinear effect in the mean travel time with respect to distance that has been observed numerically (and modeled empirically) in the literature.

Stochastic study on groundwater flow and solute transport in a porous medium with multi-scale heterogeneity

In this study, a numerical moment method (NMM) is applied to study groundwater flow and solute transport in a multiple-scale heterogeneous formation. The formation is composed of various materials and conductivity distribution within each material is heterogeneous. The distribution of materials in the study domain is characterized by an indicator function and the conductivity field within each material is assumed to be statistically stationary. Based on this assumption, the covariance function of log-hydraulic conductivity in the composite field is analytically derived and expressed in terms of the covariance of the indicator function and the statistics of log conductivity in every material. The NMM is used to investigate the effects of various uncertain parameters on flow and transport predictions in two case studies. It is shown from the case studies that the two-scale stochastic processes will both significantly influence the flow and transport predictions, especially for the variances of hydraulic head and solute fluxes. The case studies also show that the NMM can be used to study flow and transport in complex subsurface environments. In comparison with Monte Carlo simulation, NMM results are consistent with those obtained by Monte Carlo simulation method even when the total variance of log conductivity is larger than 1.

Temporal behavior of a solute cloud in a heterogeneous porous medium 3. Numerical simulations

The article presents systematic numerical simulations of the temporal behavior of a passive solute in a saturated three‐dimensional heterogeneous medium. The groundwater flow is derived from the linearized solution of the Darcy equation with Gauss‐distributed log hydraulic conductivity. The transport of a passive solute is studied by a random‐walk method, which allows for a systematic study of the temporal behavior of the effective and ensemble dispersion coefficients. The numerical results are compared to the second‐order perturbation theory expressions given in two companion papers [Dentz et al., 2000a, 2000b] and to nonperturbative results which follow from Corrsin's conjecture. The low‐order perturbation theory is intrinsically based on the assumption of small heterogeneity, while Corrsin's conjecture does not take into account certain contributions due to higher‐order terms of the perturbation series. The simulations yield, for the first time, systematic quantitative information on the validity and the limitations of these analytic approximations. For increasing heterogeneities, considerable deviations from the theoretically predicted transport behavior are observed.

3D inverse modelling of groundwater flow at a fractured site using a stochastic continuum model with multiple statistical populations

3D groundwater flow at the fractured site of Aspo (Sweden) is simulated. The aim was to characterise the site as adequately as possible and to provide measures on the uncertainty of the estimates. A stochastic continuum model is used to simulate both groundwater flow in the major fracture planes and in the background. However, the positions of the major fracture planes are deterministically incorporated in the model and the statistical distribution of the hydraulic conductivity is modelled by the concept of multiple statistical populations; each fracture plane is an independent statistical population. Multiple equally likely realisations are built that are conditioned to geological information on the positions of the major fracture planes, hydraulic conductivity data, steady state head data and head responses to six different interference tests. The experimental information could be reproduced closely. The results of the conditioning are analysed in terms of ensemble averaged average fracture plane conductivities, the ensemble variance of average fracture plane conductivities and the statistical distribution of the hydraulic conductivity in the fracture planes. These results are evaluated after each conditioning stage. It is found that conditioning to hydraulic head data results in an increase of the hydraulic conductivity variance while the statistical distribution of log hydraulic conductivity, initially Gaussian, becomes more skewed for many of the fracture planes in most of the realisations.

Effect of Bentonite on Compacted Clay Landfill Barriers

Compacted clay barriers (liner and cap) are one of the most important components of municipal waste landfills. On-site soils are generally used to construct the clay barriers as long as they can be compacted to standard specifications, including hydraulic conductivity. Wherever the available on-site soils are not suitable to be used for constructing clay barriers, soils amended with bentonite are commonly used. This article presents the results of an experimental study conducted on compacted clay-bentonite mixtures to develop data on the effects of bentonite on engineering properties of compacted clay-bentonite mixtures. Clay-bentonite mixtures with bentonite contents of up to 7% were tested to determine consistency using Atterberg limits, moisture-density relationships using standard Proctor compaction tests, shear strength using unconfined compression tests, hydraulic conductivity using triaxial flexible-wall hydraulic conductivity tests, and consolidation properties using one-dimensional consolidation tests. Unconfined compression tests were also performed with 11% bentonite content.The laboratory test results indicated that liquid limit, plastic limit, and plasticity index increased linearly with increased amount of bentonite. The addition of bentonite resulted in decreased maximum dry unit weight but the optimum moisture content increased slightly. Unconfined compression strength of compacted clay-bentonite mixtures increased linearly with an increase in the amount of bentonite. Hydraulic conductivity of compacted clay-bentonite mixtures decreased nonlinearly with increased amount of bentonite, but a linear relationship was observed between logarithm of hydraulic conductivity and bentonite content. The compression index increased slightly from 0 to 3% bentonite content but increased nonlinearly beyond 3% bentonite content, whereas the swelling index of clay-bentonite mixtures has been observed to increase approximately linearly with increase in the amount of bentonite.

Analysis of variance method for the equivalent conductivity of rectangular blocks

We use an analysis of variance (ANOVA) decomposition of the log hydraulic conductivity F(x) = ln K(X) as the sum of an average value F¯ plus main effects and interactions of various orders to evaluate the equivalent conductivity of rectangular blocks in D‐dimensional space. Some of the ANOVA components are dealt with exactly. The effect of the other components is approximated through calibration to the effective conductivity Keff under ergodic conditions. Our analysis applies to both isotropic and anisotropic lognormal K fields. We evaluate the theoretical findings through analytical and numerical comparisons with exact results and previously proposed approximations. We find that the ANOVA formula is nearly unbiased, although its performance is generally inferior to numerical renormalization. Advantages of the ANOVA formula are that it is less computationally demanding and explicitly shows the dependence of the block conductivity Kb on the various components of fluctuation of F inside the block. Such dependence is more complex than previously assumed. Considering the ANOVA formula to be unbiased, we derive the bias of the geometric mean of the Cardwell‐Parsons bounds for three‐dimensional rectangular blocks.

How well are hydraulic conductivity variations approximated by additive stable processes?

Analysis of the higher statistical moments of a hydraulic conductivity ( K) and an intrinsic permeability ( k) data set leads to the conclusion that the increments of the data and the logs of the data are not governed by Levy-stable or Gaussian distributions. The distribution tails appear to display a Pareto-like power-law decay (i.e. Prob(| x|> s|)∝s −α as s→∞), with α values of ≈2.5 and 3.6, respectively. Unlike the calculations of Liu and Molz (1997), α values were largely independent of lag size. These results suggest that the Levy model does not fit the tail behavior of the data well, even prior to the need for truncation in order to keep the statistical moments of simulated K distributions from becoming unrealistically large. It is suggested also that the fractional diffusion equation, based on an underlying Levy motion rather than the usual Brownian motion, might be better justified if porosity variations, as well as K variations, were considered. For the past 10 years, hydrologists and petroleum scientists have explored the use of non-stationary stochastic processes with stationary increments as models for log hydraulic conductivity distributions — the so-called scaling fractal models. Initially, Gaussian processes were used based on fractional Brownian motion. Later, a non-Gaussian model, fractional Levy motion, was suggested as a more realistic alternative. Even more recently, Levy multi-fractals have been proposed as direct models for K variations. In the on-going effort to arrive at the most practical and realistic model for K or log( K) increments, the present communication attempts to develop useful information by presenting a careful analysis of the tail behavior of K and log( K) data sets. It is concluded that although more realistic than the Gaussian model, the Levy model over-estimates the variability inherent in the two carefully measured data sets that were studied.

Inferring the relation between seismic slowness and hydraulic conductivity in heterogeneous aquifers

Cross‐well seismic tomography can be used to develop high‐resolution seismic slowness (1/velocity) estimates along planes through aquifers. Unfortunately, the relation between seismic slowness and hydraulic conductivity is poorly understood, resulting in poor characterization of hydraulic properties from seismic data. This relation is generally developed from laboratory measurements, but slowness values measured with very high frequencies in the lab are often poorly correlated with lower frequency cross‐well and surface seismic slowness values. To address this problem, we developed an approach to infer the relation between slowness and hydraulic conductivity using field scale geophysical and hydrogeologic measurements. We first develop an a priori relation between the conductivity measurements and the cross‐well slowness estimates. Multiple three‐dimensional slowness realizations, conditioned on the cross‐well estimates, are then generated and remapped into log conductivity fields using the a priori slowness to log conductivity relation. We simulate groundwater flow and tracer transport through these conductivity fields and calculate the residuals between measured and simulated concentration arrival time quantiles and drawdown. The slope and intercept of the relation between slowness and log hydraulic conductivity and the dispersivity are then estimated for each slowness realization to minimize the sum of these squared residuals. We demonstrate this approach for the Kesterson aquifer, California, where seismic tomography provided valuable information about aquifer properties. The groundwater flow and tracer transport simulations, through the estimated conductivity fields, yield reasonable fits to the observed tracer concentration histories for two multiple‐well tracer tests (one of which was not used in the inversion) and to the measured drawdown. This approach provides estimates of seismic slowness and hydraulic conductivity, and information about the relation between slowness and log conductivity for a field site.

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.

Estimation of spatially variable residual nonaqueous phase liquid saturations in nonuniform flow fields using partitioning tracer data

Estimates of spatially variable residual NAPL saturations SN are obtained in heterogeneous porous media using first temporal moments of breakthrough curves (BTCs) obtained from multilevel samplers during in situ partitioning tracer tests. An approach is adopted in which the distribution of the log NAPL/water volumetric ratio (Y = ln [SN/(1 − SN∥]) and log hydraulic conductivity (F = ln K) are treated as spatially correlated random fields. A nonlinear Gauss‐Newton search technique is used to identify the spatial distribution of Y that minimizes the weighted sum of the deviation of the temporal moment predictions from their measured values and the deviation of the estimate of Y from its prior estimate obtained from the temporal moments of extraction well BTCs. Sensitivities required for the algorithm are obtained using a coupled flow and transport adjoint sensitivity method. In addition to obtaining optimal estimates for the spatial distribution of Y, the method also provides the estimation error covariance. The estimation error covariance can be used to evaluate the information that may be obtained from alternate pumping and monitoring configurations for tracer tests designed to detect NAPL in the subsurface. To this end, we tested the method using two different NAPL distributions (one with a random spatially correlated field and a second that was a block of NAPL) and three different pumping configurations (a double five‐spot pattern, an inverted double five‐spot pattern, and a line‐drive pattern). The results show that measured temporal moments are more sensitive to Y in the double five‐spot and inverted double five‐spot patterns, and estimates produced in these configurations are slightly superior to those produced in the line‐drive pattern.

Role of boundary conditions in convergence and nonlocality of solutions to stochastic flow problems in bounded domains

Steady flow through a heterogeneous porous medium in a bounded domain is investigated using a recursive perturbation scheme. The effect of boundary conditions on a two‐dimensional flow with arbitrary variation of the mean flow (allowing large gradients) is investigated using analytical expressions for the head and velocity covariance functions. Boundary conditions were decomposed into deterministic and stochastic components. Two flow cases with the same zero‐order and different first‐order boundary conditions were analyzed. Boundary conditions are deterministic for the first case and random for the second. Significant differences between the two cases indicate random (or absence of randomness) processes must be modeled at boundaries. First‐order solutions for the head and velocity covariance functions for a bounded rectangular domain are derived. The resulting integral kernels involve Greens functions, and they are evaluated by numerical integration. Boundary conditions for higher‐order problems influence the absolute value and shape of the kernels (and therefore of the head and velocity variance and covariance functions). It is found that the validity of the perturbation scheme is dependent on the magnitude of the kernels and not only on the condition σf2 ≪ 1, where σf is the variance of the log hydraulic conductivity, assumed Gaussian and weakly homogeneous in space. The type of boundary conditions affects the values of the kernels and therefore determine the convergence limits for the problem. Milder head gradients also allow larger values of σf2. Nonlocality is also contingent on boundary type. Weakly homogeneous log‐fluctuating conductivity fields give rise to head and velocity covariances which are not weakly homogeneous. The inhomogeneous fields are obtained from a linear filter of the solution to the problem without stochasticity. Higher‐gradient regions induce higher head and velocity variances. In the presence of space‐varying gradients, nonlocal effects are most important away from the boundaries lpar;“center of the domain”). Small local head gradients result in small head and velocity gradients, and therefore where observations are made in stagnation regions, data should be analyzed taking into account this effect. The results may be used to interpret experimental data for columns or data taken where conditions do not fit the average uniform flow assumption and when processes at the boundaries influence the flow and therefore the mixing of contaminants.

A multiscale approach for estimating solute travel time distributions

This paper uses a multiscale statistical framework to estimate groundwater travel times and to derive conditional travel time probability densities. In the applications of interest here travel time uncertainties depend primarily on uncertainties in hydraulic conductivity. These uncertainties can be reduced if the travel times are conditioned on scattered measurements of hydraulic conductivity and/or hydraulic head. In our approach the spatially discretized log hydraulic conductivity is modeled as a multiscale stochastic process, where each scale describes the process at a different spatial resolution. Related dependent variables such as hydraulic head and travel time are approximated by discrete linear functions of the log conductivity. The linearization makes it possible to incorporate these variables into efficient multiscale estimation and conditional simulation algorithms. We illustrate the application of these algorithms by considering two options for estimating travel time densities: (1) a Monte Carlo technique which only requires linearization of the groundwater flow equation and (2) a Gaussian approximation which also requires linearization of Darcy's law and an implicit particle tracking equation. Both options provide reasonable estimates of the travel time probability density in a synthetic experiment if the underlying log hydraulic conductivity variance is small (0.5). When this variance is increased (to 5.0), the Monte Carlo result is still quite good but the Gaussian approximation is unsatisfactory. The multiscale Monte Carlo option is a very competitive approach for estimating travel time since it provides accurate results over a wide range of conditions and it is more computationally efficient than competing alternatives.

A Full‐Bayesian Approach to parameter inference from tracer travel time moments and investigation of scale effects at the Cape Cod Experimental Site

A method for inverting the travel time moments of solutes in heterogeneous aquifers is presented and is based on peak concentration arrival times as measured at various samplers in an aquifer. The approach combines a Lagrangian [Rubin and Dagan, 1992] solute transport framework with full‐Bayesian hydrogeological parameter inference. In the full‐Bayesian approach the noise values in the observed data are treated as hyperparameters, and their effects are removed by marginalization. The prior probability density functions (pdfs) for the model parameters (horizontal integral scale, velocity, and log K variance) and noise values are represented by prior pdfs developed from minimum relative entropy considerations. Analysis of the Cape Cod (Massachusetts) field experiment is presented. Inverse results for the hydraulic parameters indicate an expected value for the velocity, variance of log hydraulic conductivity, and horizontal integral scale of 0.42 m/d, 0.26, and 3.0 m, respectively. While these results are consistent with various direct‐field determinations, the importance of the findings is in the reduction of confidence range about the various expected values. On selected control planes we compare observed travel time frequency histograms with the theoretical pdf, conditioned on the observed travel time moments. We observe a positive skew in the travel time pdf which tends to decrease as the travel time distance grows. We also test the hypothesis that there is no scale dependence of the integral scale λ with the scale of the experiment at Cape Cod. We adopt two strategies. The first strategy is to use subsets of the full data set and then to see if the resulting parameter fits are different as we use different data from control planes at expanding distances from the source. The second approach is from the viewpoint of entropy concentration. No increase in integral scale with distance is inferred from either approach over the range of the Cape Cod tracer experiment.