Stochastic analysis of effective rate constant for heterogeneous reactions

A probability density function (pdf) formulation is applied to a heterogeneous chemical reaction involving an aqueous solution reacting with a solid phase in a batch. This system is described by a stochastic differential equation with multiplicative noise. Both linear and nonlinear kinetic rate laws are considered. An effective rate constant for the mean field approximation describing the change in mean concentration with time is derived. The effective rate constant decreases with increasing time eventually approaching zero as the system approaches equilibrium. This behavior suggests that a possible explanation for the observed discrepancy between laboratory measured rate constants on uniform grain sizes and field measurements may in part be caused by the heterogeneous distribution of grain sizes in natural systems. This work was supported in part by the US Department of Energy under the DOE/BES Program in the Applied Mathematical Sciences, Contract KC-07-01-01, and the Environmental Management Science Program, Office of Biological and Environmental Research. This work made use of shared facilities supported by SAHRA (Sustainability of Semi-Arid Hydrology and Riparian Areas) under the STC Program of the National Science Foundation under agreement EAR-9876800. Los Alamos National Laboratory is operated by the University of California for the US Department of Energy under contact W-7405-ENG-36.     

Numerical method of moments for solute transport in a nonstationary flow field conditioned on hydraulic conductivity and head measurements

The unconditional stochastic studies on groundwater flow and solute transport in a nonstationary conductivity field show that the standard deviations of the hydraulic head and solute flux are very large in comparison with their mean values (Zhang et al. in Water Resour Res 36:2107–2120, 2000; Wu et al. in J Hydrol 275:208–228, 2003; Hu et al. in Adv Water Resour 26:513–531, 2003). In this study, we develop a numerical method of moments conditioning on measurements of hydraulic conductivity and head to reduce the variances of the head and the solute flux. A Lagrangian perturbation method is applied to develop the framework for solute transport in a nonstationary flow field. Since analytically derived moments equations are too complicated to solve analytically, a numerical finite difference method is implemented to obtain the solutions. Instead of using an unconditional conductivity field as an input to calculate groundwater velocity, we combine a geostatistical method and a method of moment for flow to conditionally simulate the distributions of head and velocity based on the measurements of hydraulic conductivity and head at some points. The developed theory is applied in several case studies to investigate the influences of the measurements of hydraulic conductivity and/or the hydraulic head on the variances of the predictive head and the solute flux in nonstationary flow fields. The study results show that the conditional calculation will significantly reduce the head variance. Since the hydraulic head measurement points are treated as the interior boundary (Dirichlet boundary) conditions, conditioning on both the hydraulic conductivity and the head measurements is much better than conditioning only on conductivity measurements for reduction of head variance. However, for solute flux, variance reduction by the conditional study is not so significant.     

Analytical solutions to steady state unsaturated flow in layered,randomly heterogeneous soils via Kirchhoff transformation

In this study, we derive analytical solutions of the first two moments (mean and variance) of pressure head for one-dimensional steady state unsaturated flow in a randomly heterogeneous layered soil column under random boundary conditions. We first linearize the steady state unsaturated flow equations by Kirchhoff transformation and solve the moments of the transformed variable up to second order in terms of σ_Y and σ_β, the standard deviations of log hydraulic conductivity Y=ln(K_s) and of the log pore size distribution parameter β=ln(α). In addition, we also give solutions for the mean and variance of the unsaturated hydraulic conductivity. The analytical solutions of moment equations are validated via Monte Carlo simulations.     

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

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

Using data assimilation method to calibrate a heterogeneous conductivity field conditioning on transient flow test data

A data assimilation method is developed to calibrate a heterogeneous hydraulic conductivity field conditioning on transient pumping test data. The ensemble Kalman filter (EnKF) approach is used to update model parameters such as hydraulic conductivity and model variables such as hydraulic head using available data. A synthetical two-dimensional flow case is used to assess the capability of the EnKF method to calibrate a heterogeneous conductivity field by assimilating transient flow data from observation wells under different hydraulic boundary conditions. The study results indicate that the EnKF method will significantly improve the estimation of the hydraulic conductivity field by assimilating continuous hydraulic head measurements and the hydraulic boundary condition will significantly affect the simulation results. For our cases, after a few data assimilation steps, the assimilated conductivity field with four Neumann boundaries matches the real field well while the assimilated conductivity field with mixed Dirichlet and Neumann boundaries does not. We found in our cases that the ensemble size should be 300 or larger for the numerical simulation. The number and the locations of the observation wells will significantly affect the hydraulic conductivity field calibration.     

The effect of random recharge on uniform steady free-surface flow in heterogeneous porous formations

The effect of parametric uncertainty in recharge rate and spatial variability of hydraulic conductivity upon free-surface flow is investigated in a stochastic framework. We examine the three-dimensional free-surface gravitational flow problem for sloped mean uniform flow in a randomly heterogeneous porous medium under the influence of random recharge. We develop analytic solutions for the variance of free-surface position, head, and specific discharge on the free surface. Additionally, we obtain semi-analytic solutions for the statistical moments of head and specific discharge beneath the free-surface. Statistical moments are derived using a first-order approximation and then compared with their parallel in an unbounded medium. The effect of recharge mean and variability on the statistical moments is analyzed. Results can be applied to more complex flows, slowly varying in the mean.     

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 Asp? (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.     

Near‐saturated surface soil hydraulic properties under different land uses in the St Denis National Wildlife Area,Saskatchewan, Canada

Surface soil hydraulic properties are key factors controlling the partition of rainfall and snowmelt into runoff and soil water storage, and their knowledge is needed for sound land management. The objective of this study was to evaluate the effects of three land uses (native grass, brome grass and cultivated) on surface soil hydraulic properties under near‐saturated conditions at the St Denis National Wildlife Area, Saskatchewan, Canada. For each land use, water infiltration rates were measured using double‐ring and tension infiltrometers at ?0·3, ?0·7, ?1·5 and ?2·2 kPa pressure heads. Macroporosity and unsaturated hydraulic properties of the surface soil were estimated. Mean field‐saturated hydraulic conductivity (K_fs), unsaturated hydraulic conductivity at ?0·3 kPa pressure head, inverse capillary length scale (α) and water‐conducting macroporosity were compared for different land uses. These parameters of the native grass and brome grass sites were significantly (p < 0·1) higher than that of the cultivated sites. At the ?0·3 kPa pressure head, hydraulic conductivity of grasslands was two to three times greater than that of cultivated lands. Values of α were about two times and values of K_fs about four times greater in grasslands than in cultivated fields. Water‐conducting macroporosity of grasslands and cultivated fields were 0·04% and 0·01% of the total soil volume, respectively. Over 90% of the total water flux at ?0·06 kPa pressure head was transmitted through pores > 1·36 × 10^?4 m in diameter in the three land uses. Land use modified near‐saturated hydraulic properties of surface soil and consequently may alter the water balance of the area by changing the amount of surface runoff and soil water storage. Copyright © 2004 John Wiley & Sons, Ltd.     

Flow in unsaturated random porous media,nonlinear numerical analysis and comparison to analytical stochastic models

This work presents a rigorous numerical validation of analytical stochastic models of steady state unsaturated flow in heterogeneous porous media. It also provides a crucial link between stochastic theory based on simplifying assumptions and empirical field and simulation evidence of variably saturated flow in actual or realistic hypothetical heterogeneous porous media. Statistical properties of unsaturated hydraulic conductivity, soil water tension, and soil water flux in heterogeneous soils are investigated through high resolution Monte Carlo simulations of a wide range of steady state flow problems in a quasi-unbounded domain. In agreement with assumptions in analytical stochastic models of unsaturated flow, hydraulic conductivity and soil water tension are found to be lognormally and normally distributed, respectively. In contrast, simulations indicate that in moderate to strong variable conductivity fields, longitudinal flux is highly skewed. Transverse flux distributions are leptokurtic. the moments of the probability distributions obtained from Monte Carlo simulations are compared to modified first-order analytical models. Under moderate to strong heterogeneous soil flux conditions (σ²_y≥1), analytical solutions overestimate variability in soil water tension by up to 40% as soil heterogeneity increases, and underestimate variability of both flux components by up to a factor 5. Theoretically predicted model (cross-)covariance agree well with the numerical sample (cross-)covarianaces. Statistical moments are shown to be consistent with observed physical characteristics of unsaturated flow in heterogeneous soils.©1998 Elsevier Science Limited. All rights reserved     

Steady flow patterns in saturated and unsaturated,isotropic soils

A comprehensive analysis of steady flow patterns in saturated and unsaturated, possibly heterogeneous, isotropic soils is presented. It is shown that, at any point, the divergence of the unit tangent vector field to the streamlines is equal to the directional derivative along the streamlines of the orthogonal cross-sectional area of an infinitesimal stream tube divided by that area and also equal to the mean curvature of the surface of constant total head. Expressions are derived for the distribution of the flux, the water content, the velocity, the hydraulic conductivity, the total head, and the pressure head along a stream line or an infinitesimal, stream tube. Among the results is a simpler derivation, further interpretation, and extension of earlier work on calculating the hydraulic conductivity distribution from detailed measurements of the total head distribution in combination with measurements of the hydraulic conductivity at a few locations. In the last section, the jumps of various quantities at an interface are discussed.     

Stochastic analysis of groundwater flow in aquifers with leakage

 Stochastic analysis of one- and two-dimensional flow through a shallow semi-confined aquifer with spatially variable hydraulic conductivity K represented by a stationary (statistically homogeneous) random process is carried out by using the spectral technique. The hydraulic head covariance functions for flows in a semi-confined aquifer bounded by a leaky layer above and an impervious stratum below are derived by assuming that the randomness forcing the head variation to originate from the hydraulic conductivity field of the aquifer. The head covariance functions are studied using two convenient forms of the logarithmic hydraulic conductivity process. The results demonstrate the significant reduction in the head variances and covariances due to the presence of a leaky layer. The hydraulic head correlation distance is also reduced greatly due to the presence of the leaky layer.     

Analytical Interpretation of Slug Test in a Vertical Cutoff Wall

An analysis method for slug tests performed in a partially penetrating well within a vertical cutoff wall is presented. A steady‐state shape factor for evaluating hydraulic conductivity of the material within the wall was derived by applying the method of images to the previously developed analytical solution of Zlotnik et al. (2010) for an infinite aquifer. Two distinct boundary conditions were considered: constant‐head boundary for the case of direct contact between the wall and the aquifer, and no‐flux boundary representing an impermeable filter cake on the sides of the wall. The constant‐head and no‐flux boundary conditions yield significantly higher and lower shape factors, respectively, than those for the infinite aquifer. Consequently the conventional line‐fitting method for slug test analysis would yield an inaccurate estimate of the hydraulic conductivity of a vertical cutoff wall.     

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.This work is supported in part by NSF/ITR Grant EAR-0110289. The first author was additionally supported by scholarships from CONACYT and Instituto de Investigaciones Electricas of Mexico. Additional support was provided by the European Commission under Contract EVK1-CT-1999-00041 (W-SAHaRA-Stochastic Analysis of Well Head Protection and Risk Assessment).     

Usefulness of flux measurements during hydraulic tomographic survey for mapping hydraulic conductivity distribution in a fractured medium

Using the first-order analysis, we investigate the spatial cross-correlation between hydraulic conductivity variation and specific discharge (flux) as well as its components measured in a borehole under steady-state flow conditions during cross-hole pumping tests in heterogeneous aquifers. These spatial correlation patterns are found to be quite different from that between the hydraulic conductivity variation and the hydraulic head measurement in the same borehole. This finding suggests that a specific discharge measurement carries non-redundant information about the spatial distribution of heterogeneity, even this measurement is collected from the same location where the head measurement is taken. As such, specific discharge observations should be included in the analysis of hydraulic tomography to increase the resolution of estimated aquifer heterogeneity. Using numerical experiments, we demonstrate the effectiveness of the joint interpretation of both hydraulic heads and fluxes for mapping fracture distributions in a hypothetic geologic medium.     

Electrokinetic Migration of Nitrate Through Heterogeneous Granular Porous Media

This study investigates and quantifies the influence of physical heterogeneity in granular porous media, represented by materials with different hydraulic conductivity, on the migration of nitrate, used as an amendment to enhance bioremediation, under an electric field. Laboratory experiments were conducted in a bench‐scale test cell under a low applied direct current using glass bead and clay mixes and synthetic groundwater to represent ideal conditions. The experiments included bromide tracer tests in homogeneous settings to deduce controls on electrokinetic transport of inorganic solutes in the different materials, and comparison of nitrate migration under homogeneous and heterogeneous scenarios. The results indicate that physical heterogeneity of subsurface materials, represented by a contrast between a higher‐hydraulic conductivity and lower‐hydraulic conductivity material normal to the direction of the applied electric field exerts the following controls on nitrate migration: (1) a spatial change in nitrate migration rate due to changes in effective ionic mobility and subsequent accumulation of nitrate at the interface between these materials; and (2) a spatial change in the voltage gradient distribution across the hydraulic conductivity contrast, due to the inverse relationship with effective ionic mobility. These factors will contribute to higher mass transport of nitrate through low hydraulic conductivity zones in heterogeneous porous media, relative to homogeneous host materials. Overall electrokinetic migration of amendments such as nitrate can be increased in heterogeneous granular porous media to enhance the in situ bioremediation of organic contaminants present in low hydraulic conductivity zones.     

Parameter identification in fresh-saltwater flow based on borehole resistivities and freshwater head data

By means of a series of borehole resistivity measurements and a resistivity–salinity relation, a particular salt-freshwater inversion was found under the shore with semi-diurnal tides at the French–Belgian border. These resistivity data provide valuable information about the vertical variation of the saltwater percentage in different boreholes. At different places and depths fluctuations of freshwater heads are observed. A regression modelling procedure is proposed in which the hydraulic parameters for density dependent flow and solute transport can be simultaneously considered with the parameters of the resistivity–salinity relation. The object function comprises resistivity residuals and freshwater head residuals along with saltwater percentage residuals and parameter residuals. First, a synthetic problem is elaborated with this regression modelling procedure. It is followed by the application of the procedure on the observed fresh-saltwater flow problem under the shore. In the synthetic problem the identification of the hydraulic parameters was demonstrated without the inclusion of prior information about these parameters. The resistivity–salinity relation was slightly adjusted in this regression. During the regression modelling of the observations made in one of the shore-normal cross sections, the horizontal and vertical conductivity are identified along with the effective porosity and the longitudinal and transverse dispersivity. The optimal values of the dispersivities are very small. Finally, it is shown that the high waters on the back shore forms the main threat of saltwater enchroachment from the sea side of the dunes and that the isolated fresh-brackish lens under the lower part of the shore before the build up area of De Panne can be explained by overexploitation.     

Validity limits for the van Genuchten–Mualem model and implications for parameter estimation and numerical simulation

The calculation of the relative hydraulic conductivity function based on water retention data is an attractive and widely used approach, since direct measurements of unsaturated conductivities are difficult. We show theoretically under which conditions an air-entry value for water retention data is definitely required when using the statistical approach of Mualem. Moreover we rigorously specify the conditions for which the classical van Genuchten–Mualem model leads to wrong predictions of relative hydraulic conductivity and, hence, an alternative formulation including an air-entry value should be used. Significant consequences are demonstrated for the inverse parameter estimation based on multistep outflow experiments. Furthermore it is shown that the use of a physically correct formulation of the water retention curve including an air-entry value and the derived hydraulic conductivity function influences not only the stability of numerical simulations but also their final results. This is especially grave as simulations with van Genuchten–Mualem parameters are frequently used to compare experiments and simulations and to draw conclusions on the correctness of Richards’ equation.     

The effect of non divergence-free velocity fields on field scale ground water solute transport

Solute plume subjected to field scale hydraulic conductivity heterogeneity shows a large dispersion/macrodispersion, which is the manifestation of existing fields scale heterogeneity on the solute plume. On the other hand, due to the scarcity of hydraulic conductivity measurements at field scale, hydraulic conductivity heterogeneity can only be defined statistically, which makes the hydraulic conductivity a random variable/function. Random hydraulic conductivity as a parameter in flow equation makes the pore flow velocity also random and the ground water solute transport equation is a stochastic differential equation now. In this study, the ensemble average of stochastic ground water solute transport equation is taken by the cumulant expansion method in order to upscale the laboratory scale transport equation to field scale by assuming pore flow velocity is a non stationary, non divergence-free and unsteady random function of space and time. Besides the stochastic explanation of macrodispersion and the velocity correction term obtained by Kavvas and Karakas (J Hydrol 179:321–351, 1996) before a new velocity correction term, which is a function of mean pore flow velocity divergence, is obtained in this study due to strict second order cumulant expansion (without omitting any term after the expansion) performed. The significance of the new velocity correction term is investigated on a one dimensional transport problem driven by a density dependent flow field.     

Tensor Hydraulic Conductivity Estimation for Heterogeneous Aquifers under Unknown Boundary Conditions

A physically based inverse method is developed using hybrid formulation and coordinate transform to simultaneously estimate hydraulic conductivity tensors, steady‐state flow field, and boundary conditions for a confined aquifer under ambient flow or pumping condition. Unlike existing indirect inversion techniques, the physically based method does not require forward simulations to assess model‐data misfits. It imposes continuity of hydraulic head and Darcy fluxes in the model domain while incorporating observations (hydraulic heads, Darcy fluxes, or well rates) at measurement locations. Given sufficient measurements, it yields a well‐posed inverse system of equations that can be solved efficiently with coarse grids and nonlinear optimization. When pumping and injection are active, well rates are used as measurements and flux sampling is not needed. The method is successfully tested on synthetic aquifer problems with regular and irregular geometries, different hydrofacies and flow patterns, and increasing conductivity anisotropy ratios. All problems yield stable inverse solutions under increasing head measurement errors. For a given set of observations, inversion accuracy is strongly affected by the conductivity anisotropy ratio. Conductivity estimation is also affected by flow pattern: within a hydrofacies, when Darcy flux component is very small, the corresponding directional conductivity perpendicular to streamlines becomes less identifiable. Finally, inversion is successful even if the location of aquifer boundaries is unknown. In this case, the inversion domain is defined by the location of the measurements.