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.     

Influence of pore size and geometry on peat unsaturated hydraulic conductivity computed from 3D computed tomography image analysis

In organic soils, hydraulic conductivity is related to the degree of decomposition and soil compression, which reduce the effective pore diameter and consequently restrict water flow. This study investigates how the size distribution and geometry of air‐filled pores control the unsaturated hydraulic conductivity of peat soils using high‐resolution (45 µm) three‐dimensional (3D) X‐ray computed tomography (CT) and digital image processing of four peat sub‐samples from varying depths under a constant soil water pressure head. Pore structure and configuration in peat were found to be irregular, with volume and cross‐sectional area showing fractal behaviour that suggests pores having smaller values of the fractal dimension in deeper, more decomposed peat, have higher tortuosity and lower connectivity, which influences hydraulic conductivity. The image analysis showed that the large reduction of unsaturated hydraulic conductivity with depth is essentially controlled by air‐filled pore hydraulic radius, tortuosity, air‐filled pore density and the fractal dimension due to degree of decomposition and compression of the organic matter. The comparisons between unsaturated hydraulic conductivity computed from the air‐filled pore size and geometric distribution showed satisfactory agreement with direct measurements using the permeameter method. This understanding is important in characterizing peat properties and its heterogeneity for monitoring the progress of complex flow processes at the field scale in peatlands. Copyright © 2010 John Wiley & Sons, Ltd.     

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].     

Impact of fractal characteristics on evaporation and infiltration in unsaturated heterogeneous soils

ABSTRACT

In this study, an approach is developed to investigate the impact of fractal characteristics of unsaturated soil between the water table and land surface on the steady-state evaporation and infiltration across a heterogeneous landscape. The soil domain is conceptualized as a collection of stream tubes of soils and the particle diameters in various stream tubes follow a fractal distribution. The saturated hydraulic conductivity of each stream tube is related to the representative particle diameter in the tube. The effective specific discharge is then integrated from the specific discharge for each stream tube and the fractal distribution. The effective evaporation and infiltration in unsaturated soils increase with the fractal dimension. The ratio of minimum over maximum diameters does not significantly affect the specific discharge in the fractal soil. The specific discharge in unsaturated fractal soils calculated by using the simple average particle diameter mostly over-predicts the actual effective specific discharge.     

Simultaneous scaling of soil water retention and unsaturated hydraulic conductivity functions assuming lognormal pore-size distribution

Using simultaneous scaling, soil spatial variability of hydraulic functions can be described from a single set of scaling factors. The conventional scaling approach is based on empirical curve fitting, without paying much attention to the physical significance of the scaling factors. In this study, the concept of simultaneous scaling of the soil water retention and unsaturated hydraulic conductivity functions is applied to a physically based scaling theory. In this approach, it is assumed that soils are characterized by a lognormal pore-size distribution, which leads directly to lognormally distributed scaling factors. To test this concept, a total of 143 undisturbed soil samples were collected from two soil depths (25 and 50 cm), with each depth divided into two subsets based on the median soil capillary pressure head value, as determined from the lognormal pore-size distribution assumption. Moreover, the theory was compared with the conventional simultaneous scaling method. Both the conventional and physically based simultaneous scaling method performed equally well for all four subsets, as determined from the reduction in weighted root mean squared residual (WRMSR) values after scaling. We showed that the theoretical interpretation of the lognormal scaling factor distribution was applicable to simultaneous scaling of soil hydraulic functions.     

Effect of mineral reactions on the hydraulic properties of unsaturated soils: Model development and application

The selective radius shift model was used to relate changes in mineral volume due to precipitation/dissolution reactions to changes in hydraulic properties affecting flow in porous media. The model accounts for (i) precipitation/dissolution taking place only in the water-filled part of the pore space and further that (ii) the amount of mineral precipitation/dissolution within a pore depends on the local pore volume. The pore bundle concept was used to connect pore-scale changes to macroscopic soil hydraulic properties. Precipitation/dissolution induces changes in the pore radii of water-filled pores and, consequently, in the effective porosity. In a time step of the numerical model, mineral reactions lead to a discontinuous pore-size distribution because only the water-filled pores are affected. The pore-size distribution is converted back to a soil moisture characteristic function to which a new water retention curve is fitted under physically plausible constraints. The model equations were derived for the commonly used van Genuchten/Mualem hydraulic properties. Together with a mixed-form solution of Richards’ equation for aqueous phase flow, the model was implemented into the geochemical modelling framework PHREEQC, thereby making available PHREEQC’s comprehensive geochemical reactions. Example applications include kinetic halite dissolution and calcite precipitation as a consequence of cation exchange. These applications showed marked changes in the soil’s hydraulic properties due to mineral precipitation/dissolution and the dependency of these changes on water contents. The simulations also revealed the strong influence of the degree of saturation on the development of the saturated hydraulic conductivity through its quadratic dependency on the van Genuchten parameter α. Furthermore, it was shown that the unsaturated hydraulic conductivity at fixed reduced water content can even increase during precipitation due to changes in the pore-size distribution.     

考虑非饱和特性的黄土湿陷性与微观结构分析

利用扫描电子显微镜测试技术对3个场地16个土样的微观结构进行观测,并使用图像处理软件对微观图像进行处理、对土样孔隙的几何特征参数和分维数进行了提取,土样孔隙分布分维数为1.816~1.936。利用分形几何学原理建立非饱和土的孔隙分布函数,对天然湿度下黄土中水分分布进行分析,运用回归分析的方法对孔隙的分维数、非饱和孔隙孔隙率和湿陷性的关系进行了分析。结果表明：孔隙分维数越大,孔隙结构越复杂;天然湿度下处于非饱和状态的黄土孔隙孔径均大于40μm,黄土的湿陷系数随着孔隙分维数、非饱和孔隙孔隙率的增大而增大,非饱和孔隙是造成黄土湿陷的主要原因。     

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.     

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.     

Influence of heterogeneity on unsaturated hydraulic properties (2) – percentage and shape of heterogeneity

In subsurface porous media, the soil water retention curve (WRC) and unsaturated hydraulic conductivity curve (UHC) are two important soil hydraulic property curves. Spatial heterogeneity is ubiquitous in nature, which may significantly affect soil hydraulic property curves. The main theme of this paper is to investigate how spatial heterogeneities, including their arrangements and amounts in soil flumes, affect soil hydraulic property curves. This paper uses a two‐dimensional variably saturated flow and solute transport finite element model to simulate variations of pressure and moisture content in soil flumes under a constant head boundary condition. To investigate the behavior of soil hydraulic property curves owing to variations of heterogeneities and their arrangements as well, cases with different proportions of heterogeneities are carried out. A quantitative evaluation of parameter variations in the van Genuchten model (VG model) resulting from heterogeneity is presented. Results show that the soil hydraulic properties are strongly affected by variations of heterogeneities and their arrangements. If the pressure head remains at a specific value, the soil moisture increases when heterogeneities increase in the soil flumes. On the other hand, the unsaturated hydraulic conductivity decreases when heterogeneities increase in the soil flumes under a constant pressure head. Moreover, results reveal that parameters estimated from both WRC and UHC also are affected by shapes of heterogeneity; this indicates that the parameters obtained from the WRC are not suitable for predicting the UHC of different shapes in heterogeneous media. Copyright © 2011 John Wiley & Sons, Ltd.     

The exponential decline in saturated hydraulic conductivity with depth: a novel method for exploring its effect on water flow paths and transit time distribution

The strong vertical gradient in soil and subsoil saturated hydraulic conductivity is characteristic feature of the hydrology of catchments. Despite the potential importance of these strong gradients, they have proven difficult to model using robust physically based schemes. This has hampered the testing of hypotheses about the implications of such vertical gradients for subsurface flow paths, residence times and transit time distribution. Here we present a general semi‐analytical solution for the simulation of 2D steady‐state saturated‐unsaturated flow in hillslopes with saturated hydraulic conductivity that declines exponentially with depth. The grid‐free solution satisfies mass balance exactly over the entire saturated and unsaturated zones. The new method provides continuous solutions for head, flow and velocity in both saturated and unsaturated zones without any interpolation process as is common in discrete numerical schemes. This solution efficiently generates flow pathlines and transit time distributions in hillslopes with the assumption of depth‐varying saturated hydraulic conductivity. The model outputs reveal the pronounced effect that changing the strength of the exponential decline in saturated hydraulic conductivity has on the flow pathlines, residence time and transit time distribution. This new steady‐state model may be useful to others for posing hypotheses about how different depth functions for hydraulic conductivity influence catchment hydrological response. Copyright © 2016 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.     

Consistent parameter constraints for soil hydraulic functions

Parameters of functions to describe soil hydraulic properties are derived from measurements by means of parameter estimation. Of crucial importance here is the choice of correct constraints in the parameter space. Often, the parameters are mere shape parameters without physical meaning, giving flexibility to the model. A fundamental requirement is that the hydraulic functions are monotonic: the retention function and the conductivity function can only decrease as the capillary suction increases. A stricter physical requirement for the conductivity function is that its decrease with respect to saturation is at least linear. This linear decrease would occur if all pores of a capillary bundle had an equal radius. In the first part of this contribution, we derive constraints for the so-called tortuosity parameter of the Mualem conductivity model, which allow highest possible flexibility on one hand and guarantee physical consistency on the other hand. In combination with the retention functions of Brooks and Corey, van Genuchten, or Durner, such a constraint can be expressed as a function of the pore-size distribution parameters. In the second part, we show that a common modification of retention models, which is applied to reach zero water content at finite suction, can lead to the physically unrealistic case of increasing water content with increasing suction. We propose a solution for this problem by slightly modifying these models and introducing a correct parameter constraint.     

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     

A Simple Field Method for Assessing Near‐Surface Saturated Hydraulic Conductivity

This technical note describes an effective and inexpensive field technique for measuring the saturated hydraulic conductivity of both undisturbed cores and repacked soil samples. The method requires no specialized equipment; everything that is required can be obtained in a hardware store. The method is a straightforward field implementation of the widely used falling‐head laboratory analysis directly derived from Darcy's law. As such, it sidesteps the need for empirical assumptions about soil texture and the relationship between saturated and unsaturated flow components which many permeameter‐based methods rely upon. The method is shown to produce results that are consistent with K values obtained elsewhere in the same homogeneous sand formation. Furthermore, the proposed method is useful for measuring hydraulic conductivity in drill cuttings obtained from direct push or auguring drill techniques, which cannot be done with any other field method. The range of hydraulic conductivity values that this test is appropriate for is on the order of 1E ? 7 m/s to 1E ? 3 m/s.     

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.     

Influence of heterogeneity on unsaturated hydraulic properties: (1) local heterogeneity and scale effect

Spatial heterogeneity is ubiquitous in nature, which may significantly affect the soil hydraulic property curves. The models of a closed‐form functional relationship of soil hydraulic property curves (e.g. VG model or exponential model) are valid at point or local scale based on a point‐scale hydrological process, but how do scale effects of heterogeneity have an influence on the parameters of these models when the models are used in a larger scale process? This paper uses a two‐dimensional variably saturated flow and solute transport finite element model (VSAFT2) to simulate variations of pressure and moisture content in the soil flume under a constant head boundary condition. By changing different numerical simulation block sizes, a quantitative evaluation of parameter variations in the VG model, resulting from the scale effects, is presented. Results show that the parameters of soil hydraulic properties are independent of scale in homogeneous media. Parameters of α and n in homogeneous media, which are estimated by using the unsaturated hydraulic conductivity curve (UHC) or the soil water retention curve (WRC), are identical. Variations of local heterogeneities strongly affect the soil hydraulic properties, and the scale affects the results of the parameter estimations when numerical experiments are conducted. Furthermore, the discrepancy of each curve becomes considerable when moisture content becomes closer to a dry situation. Parameters estimated by UHC are totally different from the ones estimated by WRC. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

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.