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     

Seasonal and event-scale variations in solute chemistry for four Sierra Nevada catchments

Hydrobiogeochemical processes controlling stream water chemistry were examined in four small (<5 km²) catchments having contrasting bedrock lithologies in the western Sierra Nevada foothills of California. The Mediterranean climate with its cool/wet and hot/dry cycle produces strong seasonal patterns in hydrological, biological and geochemical processes. Stream water solutes fall into three general groups according to seasonal fluctuation in concentration: strong, rainy season minimum–dry season maximum (Cl⁻, SO₄²⁻, base cations); weak, rainy season minimum–dry season maximum (Si); and rainy season maximum–dry season minimum (NO₃⁻ and K⁺). Solute dynamics in soil solutions and stream water suggest that mixing of drainage waters from bedrock and soil sources regulate stream water solute concentrations. Patterns are further altered by the leaching of solutes accumulated in the soil over the summer period of desiccation and the temporal discoupling of nutrient cycles that occurs due to differences in the timing between vegetation growth (late spring) and leaching (early winter). Solute concentrations are remarkably similar between watersheds with varying bedrock types, with the exception of nitrate, sulfate and bicarbonate. Three watersheds have nitrogen-bearing metasedimentary bedrock that contributes to elevated nitrate concentrations in stream waters. Watersheds whose bedrock includes mineralized veins of sulfide and carbonate minerals similarly have greater sulfate and bicarbonate concentrations in stream water. Hydrobiogeochemical processes are highly dynamic at the seasonal and storm-event temporal scales and spatially complex at the watershed scale making management of stream water chemical composition, such as nitrate concentrations, very challenging.     

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.     

Eulerian spatial moments for solute transport in three-dimensional heterogeneous,dual-permeability media

A Eulerian analytical method is developed for nonreactive solute transport in heterogeneous, dual-permeability media where the hydraulic conductivities in fracture and matrix domains are both assumed to be stochastic processes. The analytical solution for the mean concentration is given explicitly in Fourier and Laplace transforms. Instead of using the fast fourier transform method to numerically invert the solution to real space (Hu et al., 2002), we apply the general relationship between spatial moments and concentration (Naff, 1990; Hu et al., 1997) to obtain the analytical solutions for the spatial moments up to the second for a pulse input of the solute. Owing to its accuracy and efficiency, the analytical method can be used to check the semi-analytical and Monte Carlo numerical methods before they are applied to more complicated studies. The analytical method can be also used during screening studies to identify the most significant transport parameters for further analysis. In this study, the analytical results have been compared with those obtained from the semi-analytical method (Hu et al., 2002) and the comparison shows that the semi-analytical method is robust. It is clearly shown from the analytical solution that the three factors, local dispersion, conductivity variation in each domain and velocity convection flow difference in the two domains, play different roles on the solute plume spreading in longitudinal and transverse directions. The calculation results also indicate that when the log-conductivity variance in matrix is 10 times less than its counterpart in fractures, it will hardly influence the solute transport, whether the conductivity field is matrix is treated as a homogeneous or random field.     

An efficient probabilistic finite element method for stochastic groundwater flow

We present an efficient numerical method for solving stochastic porous media flow problems. Single-phase flow with a random conductivity field is considered in a standard first-order perturbation expansion framework. The numerical scheme, based on finite element techniques, is computationally more efficient than traditional approaches because one can work with a much coarser finite element mesh. This is achieved by avoiding the common finite element representation of the conductivity field. Computations with the random conductivity field only arise in integrals of the log conductivity covariance function. The method is demonstrated in several two- and three-dimensional flow situations and compared to analytical solutions and Monte Carlo simulations. Provided that the integrals involving the covariance of the log conductivity are computed by higher-order Gaussian quadrature rules, excellent results can be obtained with characteristic element sizes equal to about five correlation lengths of the log conductivity field. Investigations of the validity of the proposed first-order method are performed by comparing nonlinear Monte Carlo results with linear solutions. In box-shaped domains the log conductivity standard deviation σ_Y may be as large as 1.5, while the head variance is considerably influenced by nonlinear effects as σ_Y approaches unity in more general domains.     

Stochastic analysis of unsaturated transport in soils with fractal log-conductivity distribution

Within the framework of stochastic theory and the spectral perturbation techniques, three-dimensional dispersion in partially saturated soils with fractal log hydraulic conductivity distribution is analyzed. Our analysis is focused on the impact of fractal dimension of log hydraulic conductivity distribution, local dispersivity, and unsaturated flow parameters, such as the soil poresize distribution parameter and the moisture distribution parameter, on the spreading behavior of solute plume and the concentration variance. Approximate analytical solutions to the stochastic partial differential equations are derived for the variance of asymptotic solute concentration and asymptotic macrodispersivities.     

A field‐scale infiltration model accounting for spatial heterogeneity of rainfall and soil saturated hydraulic conductivity

This study first explores the role of spatial heterogeneity, in both the saturated hydraulic conductivity K_s and rainfall intensity r, on the integrated hydrological response of a natural slope. On this basis, a mathematical model for estimating the expected areal‐average infiltration is then formulated. Both K_s and r are considered as random variables with assessed probability density functions. The model relies upon a semi‐analytical component, which describes the directly infiltrated rainfall, and an empirical component, which accounts further for the infiltration of surface water running downslope into pervious soils (the run‐on effect). Monte Carlo simulations over a clay loam soil and a sandy loam soil were performed for constructing the ensemble averages of field‐scale infiltration used for model validation. The model produced very accurate estimates of the expected field‐scale infiltration rate, as well as of the outflow generated by significant rainfall events. Furthermore, the two model components were found to interact appropriately for different weights of the two infiltration mechanisms involved. Copyright © 2005 John Wiley & Sons, Ltd.     

FIELD ESTIMATION OF MACROPORE FUNCTIONING AND SURFACE HYDRAULIC CONDUCTIVITY IN A FEN PEAT

A limitation of existing models of water and solute movement in fen peats is that they fail to represent processes in the unsaturated zone. This limitation is largely due to a lack of data on the hydraulic properties of unsaturated peat, in particular the relationship between hydraulic conductivity (K) and pressure head (ψ). A tension infiltrometer was used to measure K(ψ) of a fen peat in Somerset, England. It was found that macropores could be important in water and solute movement in this soil type. It was also found that (i) variability of K in this peat was less than that reported for other peats and mineral soils, and (ii) the K data were better described by a log-normal distribution than a normal distribution in accord with findings from other peat and mineral soils. Recommendations on improving the understanding of water and solute movement in the unsaturated zone of this soil type are made. © 1997 by John Wiley & Sons, Ltd.     

Analytical solutions for transport of decaying solutes in rivers with transient storage

Analytical solutions are presented for solute transport in rivers including the effects of transient storage and first order decay. The solute transport model considers an advection–dispersion equation for transport in the main channel linked to a first order mass exchange between the main channel and the transient storage zones. In case of a conservative tracer, it is shown that different analytical solutions presented in the literature are mathematically identical. For non-conservative solutes, first order decay reactions are considered with different reaction rate coefficients in the main river channel and in the dead zones. New analytical solutions are presented for different boundary conditions, i.e. instantaneous injection in an infinite river reach, and variable concentration time series input in a semi-infinite river reach. The correctness and accuracy of the analytical solutions is verified by comparison with the OTIS numerical model. The results of analytical and numerical approaches compare favourably and small differences can be attributed to the influence of boundary conditions. It is concluded that the presented analytical solutions for solute transport in rivers with transient storage and solute decay are accurate and correct, and can be usefully applied for analyses of tracer experiments and transport characteristics in rivers with mass exchange in dead zones.     

Probabilistic risk assessment of contaminant transport in groundwater and vapour intrusion following remediation of a contaminant source

Community-scale simulations were performed to investigate the risk to groundwater and indoor air receptors downgradient of a contaminated site following the remediation of a long-term source. Six suites of Monte Carlo simulations were performed using a numerical model that accounted for groundwater flow, reactive solute transport, soil gas flow, and vapour intrusion in buildings. The model was applied to a three-dimensional, community-scale (250 m × 1000 m × 14 m) domain containing heterogeneous, spatially correlated distributions of the hydraulic conductivity, fraction of organic carbon, and biodegradation rate constant, which were varied between realizations. Analysis considered results from both individual realizations as well as the suite of Monte Carlo simulations expressed through several novel, integrated parameters, such as the probability of exceeding a regulatory standard in either groundwater or indoor air. Results showed that exceedance probabilities varied considerably with the consideration of biodegradation in the saturated zone, and were less sensitive to changes in the variance of hydraulic conductivity or the incorporation of heterogeneous distributions of organic carbon at this spatial scale. A sharp gradient in exceedance probability existed at the lateral edges of the plumes due to variability in lateral dispersion, which defined a narrow region of exceedance uncertainty. Differences in exceedance probability between realizations (i.e., due to heterogeneity uncertainty) were similar to differences attributed to changes in the variance of hydraulic conductivity or fraction of organic carbon. Simulated clean-up times, defined by reaching an acceptable exceedance probability, were found to be on the order of decades to centuries in these community-scale domains. Results also showed that the choice of the acceptable exceedance probability level (e.g., 1 vs. 5 %) would likely affect clean up times on the order of decades. Moreover, in the scenarios examined here, the risk of exceeding indoor air standards was greater than that of exceeding groundwater standards at all times and places. Overall, simulations of coupled transport processes combined with novel spatial and temporal quantification metrics for Monte Carlo analyses, provide practical tools for assessing risk in wider communities when considering site remediation.     

Percolation losses in paddy fields with a dynamic soil structure: model development and applications

The hydraulic characteristics of the plough pan of paddy fields provide continuous ponding conditions during the growing season and control the water use efficiency in wet rice production. Its saturated hydraulic conductivity K_s, however, exhibits a large spatiotemporal variability as a consequence of a highly dynamic soil structure involving temporary shrinkage cracks. Water flow through the earthen bunds surrounding the fields further contributes to the uncertainty in water flux calculations. The objective of this study was to develop a simple deterministic model with stochastic elements (‘PADDY‐FLUX’) for depiction of deep percolation, and to assess the effect of different water management scenarios on percolation in two channel command areas. Darcy's law is used as the fundamental equation for water flow calculations with the ponding water depth h as a time‐dependent variable. Flux uncertainty is estimated by a Monte‐Carlo‐type implementation. K_s is treated as a random variable of a bimodal probability density function (PDF), which is the weighted sum of two Gaussian PDFs (accounting for a matrix and a preferential flow domain). The weighing factor α is a function of h, reflecting an increasing risk for preferential flow situations after desiccation and the development of shrinkage cracks. Under‐bund percolation is calculated using transfer functions. The results demonstrate that percolation losses increase in the following order: continuous soil saturation < continuous flooding (CF) < mid‐season drainage and intermittent irrigation (MD + II) < mid‐season drainage and continuous flooding. The bunds contribute up to 54 and 17% to total fluxes under CF and MD + II, respectively. Preferential water fluxes are responsible for the major part of water losses as soon as desiccation causes the formation of shrinkage cracks. As a conclusion, continuous soil saturation should be promoted as the least water‐intensive irrigation regime, while intermittent irrigation is recommended only in case that irreversible shrinkage cracks have already developed. Copyright © 2010 John Wiley & Sons, Ltd.     

Estimation of soil hydraulic properties and their uncertainty through the Beerkan infiltration experiment

The Beerkan method based on in situ single‐ring water infiltration experiments along with the relevant specific Beerkan estimation of soil transfer parameters (BEST) algorithm is attractive for simple soil hydraulic characterization. However, the BEST algorithm may lead to erroneous or null values for the saturated hydraulic conductivity and sorptivity especially when there are only few infiltration data points under the transient flow state, either for sandy soil or soils in wet conditions. This study developed an alternative algorithm for analysis of the Beerkan infiltration experiment referred to as BEST‐generalized likelihood uncertainty estimation (GLUE). The proposed method estimates the scale parameters of van Genuchten water retention and Brooks–Corey hydraulic conductivity functions through the GLUE methodology. The GLUE method is a Bayesian Monte Carlo parameter estimation technique that makes use of a likelihood function to measure the goodness‐of‐fit between modelled and observed data. The results showed that using a combination of three different likelihood measurements based on observed transient flow, steady‐state flow and experimental steady‐state infiltration rate made the BEST‐GLUE procedure capable of performing an efficient inverse analysis of Beerkan infiltration experiments. Therefore, it is more applicable for a wider range of soils with contrasting texture, structure, and initial and saturated water content. Copyright © 2015 John Wiley & Sons, Ltd.     

Simplified modelling of areal average infiltration at the hillslope scale

Two models for estimating expected areal‐average infiltration rate, ī, at the hillslope scale are presented. The first relies upon the condition of a negligible infiltration of surface water running downslope (run‐on process) into a previous heterogeneous soil. It is an adapted version of an earlier semi‐analytical model. The second incorporates the run‐on process and is based on a lumped approach that uses an effective saturated hydraulic conductivity. This latter was parameterized in terms of the main characteristics of rainfall and soil. Both the models were tested by comparison with the results carried out by Monte‐Carlo simulations over different soil types. It was found that the first model simulated ī with maximum errors in magnitude typically less than 10%. The second model provided similar errors in the total volume of overland flow, and the rising limb of the hydrograph experienced a distortion. Lastly, satisfactory results were obtained by comparing the model without run‐on with an empirical approach particularly accurate for fine‐textured soils. Copyright © 2002 John Wiley & Sons, Ltd.     

Identification of optimal soil hydraulic functions and parameters for predicting soil moisture

Abstract

The accuracy of six combined methods formed by three commonly-used soil hydraulic functions and two methods to determine soil hydraulic parameters based on a soil hydraulic parameter look-up table and soil pedotransfer functions was examined for simulating soil moisture. A novel data analysis and modelling approach was used that eliminated the effects of evapotranspiration so that specific sources of error among the six combined methods could be identified and quantified. By comparing simulated and observed soil moisture at six sites of the USDA Soil Climate Analysis Network, we identified the optimal soil hydraulic functions and parameters for predicting soil moisture. Through sensitivity tests, we also showed that adjusting only the soil saturated hydraulic conductivity, K_s , is insufficient for representing important effects of macropores on soil hydraulic conductivity. Our analysis illustrates that, in general, soil hydraulic conductivity is less sensitive to K_s than to the soil pore-size distribution parameter.

Editor D. Koutsoyiannis; Associate editor D. Hughes

Citation Pan, F., McKane, R.B. and Stieglitz, M., 2012. Identification of optimal soil hydraulic functions and parameters for predicting soil moisture. Hydrological Sciences Journal, 57 (4), 723–737.     

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.     

Functional evaluation of a simplified scaling method for assessing the spatial variability of soil hydraulic properties at the hillslope scale

Abstract

Mapping soil hydraulic parameters with traditional scaling methods that use laboratory-determined hydraulic characteristics (the LAB method) is not always feasible as it involves expensive, time-consuming and sophisticated measurements on soil samples collected in several locations of the study area. An alternative scaling method (the AP method) has been recently proposed to indirectly retrieve the soil hydraulic properties following the Arya-Paris physico-empirical pedotransfer function, which makes use of particle-size distribution and bulk density values. In this synthetic study we verify the performance of the AP method from a functional perspective, by evaluating the differences in the simulated soil water budget through a Monte Carlo approach. Notwithstanding that the AP method can provide soil hydraulic property patterns with faster experimental procedures and minor costs, we observe significant bias in the predicted spatially-averaged soil water budget due to a poor parametric calibration of the AP method and an imprecise identification of the spatial correlation structure of the AP-estimated scaling factors.

Citation Nasta, P., Romano, N., and Chirico, G.B., 2013. Functional evaluation of a simplified scaling method for assessing the spatial variability of soil hydraulic properties at the hillslope scale. Hydrological Sciences Journal, 58 (5), 1059–1071.     

Bridging the gap between numerical solutions of travel time distributions and analytical storage selection functions

Recent advancements in analytical solutions to quantify water and solute travel time distributions (TTDs) and the related StorAge Selection (SAS) functions synthesize catchment complexity into a simplified, lumped representation. Although these analytical approaches are efficient in application, they require rarely available long‐term and high‐frequency hydrochemical data for parameter estimation. Alternatively, integrated hydrologic models coupled to Lagrangian particle‐tracking approaches can directly simulate age under different catchment geometries and complexity, but at a greater computational expense. Here, we bridge the two approaches, using a physically based model to explore the uncertainty in the estimation of the SAS function shape. In particular, we study the influence of subsurface heterogeneity, interactions between distinct flow domains (i.e., the vadose zone and saturated groundwater), diversity of flow pathways, and recharge rate on the shape of TTDs and the SAS functions. We use an integrated hydrology model, ParFlow, linked with a particle‐tracking model, SLIM, to compute transient residence times (or ages) at every cell in the domain, facilitating a direct characterization of the SAS function. Steady‐state results reveal that the SAS function shape shows a wide range of variation with respect to the variability in the structure of subsurface heterogeneity. Ensembles of spatially correlated realizations of hydraulic conductivity indicate that the SAS functions in the saturated groundwater have an overall weak tendency toward sampling younger ages, whereas the vadose zone gives a strong preference for older ages. We further show that the influence of recharge rate on the TTD is tightly dependent on the variability of subsurface hydraulic conductivity.