Unsaturated flows in soils with self-similar hydraulic conductivity distribution

In this article, we are concerned with the statistics of steady unsaturated flow in soils with a fractal hydraulic conductivity distribution. It is assumed that the spatial distribution of log hydraulic conductivity can be described as an isotropic stochastic fractal process. The impact of the fractal dimension of this process, the soil pore-size distribution parameter, and the characteristic length scale on the variances of tension head and the effective conductivity is investigated. Results are obtained for one-dimensional and three-dimensional flows. Our results indicate that the tension head variance is scale-dependent for fractal distribution of hydraulic conductivity. Both tension head variance and effective hydraulic conductivity depend strongly on the fractal dimension. The soil pore-size distribution parameter is important in reducing the variability of the unsaturated hydraulic conductivity and of the fluxes.     

A concise parameterization of the hydraulic conductivity of unsaturated soils

The parameter n in the well-known expression for hydraulic conductivity K=K₀S_eⁿ (where K₀ is its value at satiation and S_e the effective saturation) is determined as a function of the exponent in the power form of the soil–water retention relationship. The result is validated with an extensive experimental database comprising some 43 soils, collected by Mualem.     

On the effective hydraulic conductivity in mean vertical unsaturated steady flows

Water flow in partially saturated heterogeneous porous formations is modelled by regarding the hydraulic parameters as stationary random space functions (RSFs). As a consequence, the flow variables are also RSFs, and we aim to develop a procedure to derive the effective hydraulic conductivity (EHC). The methodology relies on a perturbation approach which regards the variances of the hydraulic parameters as small quantities. By using the Gardner’s [Gardner WR. Some steady state solutions of unsaturated moisture flow equations with application to evaporation from a water table. Soil Sci 1958;85:228–32] two-parameters (K_s, α) model for the local unsaturated conductivity, we obtain the EHC for any dimensionality d of the flow domain, and arbitrary correlation functions of the input RSFs. Unlike previous studies [e.g. Yeh T-CJ, Gelhar J, Gutjahr A. Stochastic analysis of unsaturated flow in heterogenous soils. 1. Statistically isotropic media. Water Resour Res 1985;21;447–56, Yeh T-CJ, Gelhar J, Gutjahr A. Stochastic analysis of unsaturated flow in heterogenous soils. 2. Statistically anisotropic media with variable α. Water Resour Res 1985:21:457–64], the EHC is represented here as product between the local scale conductivity valid for a domain of mean parameters, and a correction function κ¹ which depends on the medium heterogeneity structure and the mean pressure head.Generally, the correction function κ¹ is expressed by d-fold quadrature. These quadratures are further reduced after adopting specific (i.e. exponential and Gaussian) structure for the (cross) correlation functions involved in the computation of κ¹. We have also focused on some particular formation structures which are relevant for the applications, and permit simplification of the computational aspect, as well.We investigate effects of the heterogeneity formation properties as well as the mean head on the structure of κ¹. Overall, results suggest that, given the formation statistics, the impact of the heterogeneity upon κ¹ is enhanced as the medium becomes drier. This is particularly so when the variability of the fluctuation of Y = ln K_s is small compared with that of ζ = ln α. Conversely, when the heterogeneity of Y is prevalent upon that of ζ, κ¹ is influenced solely by the anisotropic structure of the formation unless the horizontal correlation scales are much greater than the vertical ones.     

A partial explanation of the dependency of hydraulic conductivity on positive pore water pressure in peat soils

Part of the relationship between positive pore water pressures and hydraulic conductivity in peat soils may be explained by accumulations of methane bubbles. We show how compression and expansion of gas bubbles with changes in pore water pressure could cause changes in hydraulic conductivity and thus help to explain some observations of dependency of hydraulic conductivity in peats on pore water pressure. Consideration is also given to the effect on hydraulic conductivities of methane gas going into solution with increase in pore water pressure.     

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.     

On the dependence of the saturated hydraulic conductivity upon the effective porosity through a power law model at different scales

We study the scale dependence of the saturated hydraulic conductivity K_s through the effective porosity n_e by means of a newly developed power‐law model (PLM) which allows to use simultaneously measurements at different scales. The model is expressed as product between a single PLM (capturing the impact of the dominating scale) and a characteristic function κ^? accounting for the correction because of the other scale(s). The simple (closed form) expression of the κ^?‐function enables one to easily identify the scales which are relevant for K_s. The proposed model is then applied to a set of real data taken at the experimental site of Montalto Uffugo (Italy), and we show that in this case two (i.e. laboratory and field) scales appear to be the main ones. The implications toward an important application (solute transport) in Hydrology are finally discussed. Copyright © 2016 John Wiley & Sons, Ltd.     

Comparison of conductivity averaging methods for one-dimensional unsaturated flow in layered soils

One of the important factors influencing the accuracy of the numerical solution of 1D unsaturated flow equation (Richards’ equation) is the averaging method applied to compute hydraulic conductivity between two adjacent nodes of the computational grid. A number of averaging schemes have been proposed in the literature for homogeneous soil, including arithmetic, geometric, upstream and integrated means, as well as more sophisticated approaches, based on the local solution of steady state flow between the neighboring nodes (Darcian means). Another group of methods have been developed for the case when a material interface is present between the nodes. They range from simple arithmetic averaging to more complex schemes using the pressure- and flux-continuity conditions at the interface. In this paper we compare several averaging schemes for a number of steady and unsteady flow problems in layered soils. The first group of methods is applied in the framework of the vertex-centered approach to spatial discretization, where the nodes are placed at the material interfaces, while the second group is used with the cell-centered approach, where the material interfaces are located between computational nodes. The resulting numerical schemes are evaluated in terms of accuracy and computational time. It is shown that the averaging schemes based on Darcian mean principle [19] used in the framework of either vertex-centered or cell-centered approach compare favorably to other methods for a range of test cases.     

A dripper-TDR method for in situ determination of hydraulic conductivity and chemical transport properties of surface soils

Field determined hydraulic and chemical transport properties can be useful for the protection of groundwater resources from land-applied chemicals. Most field methods to determine flow and transport parameters are either time or energy consuming and/or they provide a single measurement for a given time period. In this study, we present a dripper-TDR field method that allows measurement of hydraulic conductivity and chemical transport parameters at multiple field locations within a short time period. Specifically, the dripper-TDR determines saturated hydraulic conductivity (K_s), macroscopic capillary length (λ_c), immobile water fraction (θ_im/θ), mass exchange coefficient (α) and dispersion coefficient (D_m). Multiple dripper lines were positioned over five crop rows in a field. Background and step solutions were applied through drippers to determine surface hydraulic conductivity parameters at 44 locations and surface transport properties at 38 locations. The hydraulic conductivity parameters (K_s, λ_c) were determined by application of three discharge rates from the drippers and measurements of the resultant steady-state flux densities at the soil surface beneath each dripper. Time domain reflectometry (TDR) was used to measure the bulk electrical conductivity of the soil during steady infiltration of a salt solution. Breakthrough curves (BTCs) for all sites were determined from the TDR measurements. The K_s and λ_c values were found to be lognormally distributed with average values of 31.4 cm h⁻¹ and 6.0 cm, respectively. BTC analysis produced chemical properties, θ_im/θ, α, and D_m with average values of 0.23, 0.0036 h⁻¹, and 1220 cm² h⁻¹, respectively. The estimated values of the flow and transport parameters were found to be within the ranges of values reported by previous studies conducted at nearby field locations. The dripper TDR method is a rapid and useful technique for in situ measurements of hydraulic conductivity and solute transport properties. The measurements reported in this study give clear evidence to the occurrence of non-equilibrium water and chemical movement in surface soil. The method allows for quantification of non-equilibrium model parameters and preferential flow. Quantifying the parameters is a necessary step toward determining the influences of surface properties on infiltration, runoff, and vadose zone transport.     

Influence of coarse soils with high hydraulic conductivity on the applicability of the SCS-CN method

The impact of the existence of coarse soils with high hydraulic conductivity on the applicability of the Soil Conservation Service curve number (SCS-CN) method is investigated in a heterogeneous catchment in the Zagros mountain region, southwest Iran. The findings indicate that curve numbers (CN) decline steadily as rainfall depths increase, showing no evidence of reaching stable CN values. This is attributed to a small low permeability area in the catchment, although most parts of the region are covered by high to very high permeability soils. Although this behaviour has been attributed mainly to forested areas by some researchers, it is shown here that it can be observed in mountain regions with coarse soils. The results also reveal that a linear rainfall–runoff formula of the form Q = CP provides more accurate results than the SCS-CN method, and the runoff coefficient (C) can be taken as equal to the low-permeability area fraction, meaning that these parts can be considered to be the main runoff-producing areas.     

Spatial dependence of the hydraulic conductivity in a well‐type configuration at the mesoscale

The spatial distribution of the hydraulic conductivity κ is modelled by a power law, and we present a methodological approach to quantify the exponent (crowding index) of such a law as detected within a well‐type flow configuration. Based upon the outcome of several pumping tests conducted into a caisson (mesoscale), we identify the crowding index as function of the volumetric flow rate. Hence, we develop a simple (although approximated) procedure to assess whether the spatial distribution of κ can be characterized by a power law. We demonstrate that, even at the mesoscale, the conductivity κ can not be regarded as a formation's property (nonlocality), in agreement with the recent developments on the theory of flows into radial configurations.     

Horizontal magnetic dipole over an inhomogeneous earth model with exponential variation of conductivity

The paper outlines the formulation of the problem of calculating the electromagnetic field components due to a horizontal magnetic dipole placed over a multilayered earth model with one of the layers having exponential variation of conductivity with depth. Analytical solutions and numerical computations are performed for three-layered earth models possessing the described conductivity variation in the transition layer. It is assumed that the conduction currents dominate the displacement currents. Results presented here show the influence of the transition layer thickness and the conductivity contrast between the top and the bottom layers on electric and magnetic field components. The results show a characteristic dependence on the conductivity inhomogeneity.     

Evaluation of streambed hydraulic conductivity heterogeneity in an urban watershed

It has long been understood that streambed hydraulic conductivity plays an important role in surface-subsurface solute exchange. Using a portable falling head permeameter in situ, we estimated the horizontal hydraulic conductivity, K, of the near-surface streambed sediments at a total of 85 locations encompassing two depth intervals: 7.5–10 and 10–12.5 cm. The measurements were conducted in an 80 m reach of Indian Creek, a small urban stream in Philadelphia, PA, USA. We found that the ln K data within each sediment layer were Gaussian, but the combined data set was not. The results indicated that while the mean hydraulic conductivity decreased with depth, horizontal heterogeneity (e.g. the variance) increased with depth. This strong contrast between layers suggests that they should be treated as separated entities in modeling studies. Variogram analyses across the stream suggested symmetry with respect to the thalweg in the upper layer and fractality in the lower layer. The variograms along the streams suggested that the K data are random.     

Estimation of hydraulic conductivity in an alluvial system using temperatures

Well water temperatures are often collected simultaneously with water levels; however, temperature data are generally considered only as a water quality parameter and are not utilized as an environmental tracer. In this paper, water levels and seasonal temperatures are used to estimate hydraulic conductivities in a stream-aquifer system. To demonstrate this method, temperatures and water levels are analyzed from six observation wells along an example study site, the Russian River in Sonoma County, California. The range in seasonal ground water temperatures in these wells varied from <0.2 degrees C in two wells to approximately 8 degrees C in the other four wells from June to October 2000. The temperature probes in the six wells are located at depths between 3.5 and 7.1 m relative to the river channel. Hydraulic conductivities are estimated by matching simulated ground water temperatures to the observed ground water temperatures. An anisotropy of 5 (horizontal to vertical hydraulic conductivity) generally gives the best fit to the observed temperatures. Estimated conductivities vary over an order of magnitude in the six locations analyzed. In some locations, a change in the observed temperature profile occurred during the study, most likely due to deposition of fine-grained sediment and organic matter plugging the streambed. A reasonable fit to this change in the temperature profile is obtained by decreasing the hydraulic conductivity in the simulations. This study demonstrates that seasonal ground water temperatures monitored in observation wells provide an effective means of estimating hydraulic conductivities in alluvial aquifers.     

A Linking Test to reduce the number of hydraulic parameters necessary to simulate groundwater recharge in unsaturated soils

In environmental studies, numerical simulation models are valuable tools for testing hypothesis about systems functioning and to perform sensitivity studies under scenarios of land use or climate changes. The simulations depend upon parameters which are not always measurable quantities and must be calibrated against observations, using for instance inverse modelling. Due to the scarcity of these observations, it has been found that parameter sets allowing a good matching between simulated and measured quantities are often non-unique, leading to the problem of equifinality. This can lead to non-physical values, erroneous fluxes and misleading sensitivity analysis. Therefore, a simple but robust inverse method coined the Linking Test is presented to determine if the parameters are linked. Linked parameters are then sub-divided into classes according to their impact on water fluxes. The Linking Test establishes the causes of non-uniqueness of parameter sets and the feasibility of the inverse modelling.     

Influence of antecedent soil moisture on hydraulic conductivity in a series of texture‐contrast soils

Antecedent soil moisture significantly influenced the hydraulic conductivity of the A1, A2e and B21 horizons in a series of strong texture‐contrast soils. Tension infiltration at six supply potentials demonstrated that in the A1 horizon, hydraulic conductivity was significantly lower in the ‘wet’ treatment than in the ‘dry’ treatment. However in the A2e horizon, micropore and mesopore hydraulic conductivity was lower in the ‘dry’ treatment than the ‘wet’ treatment, which was attributed to the precipitation of soluble amorphous silica. In the B21 horizon, desiccation of vertic clays resulted in the formation of shrinkage cracks which significantly increased near‐saturated hydraulic conductivity and prevented the development of subsurface lateral flow in the ‘dry’ treatment. In the ‘wet’ treatment, the difference between the hydraulic conductivity of the A1 and B21 horizons was reduced; however, lateral flow still occurred in the A1 horizon due to difficulty displacing existing soil water further down the soil profile. Results demonstrate the need to account for temporal variation in soil porosity and hydraulic conductivity in soil‐water model conceptualisation and parameterisation. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

On estimating the hydraulic properties of soil,Part 2. A new empirical equation for estimating hydraulic conductivity for sands

A new empirical equation to estimate hydraulic conductivity is proposed, based on a large set of measured data for hydraulic properties of soil. The equation is simpler and more accurate than the series-parallel model. Under conditions of insufficient data, the new equation provides a good estimation of hydraulic conductivity for sands. For the same class of soils, another empirical equation is proposed to estimate the power N in the Averjanov-Irmay function.     

A diagnosis of sub-surface water table dynamics in low hydraulic conductivity soils in the sugar cane fields of Pongola,South Africa

Water and land are the two natural resources restraining crop production in South Africa. With the increasing demand for food, emphasis has shifted from the sole reliance on rain fed crop production, to irrigation. The deterioration in irrigation water quality from surface water sources is, however, posing a big challenge to the sustainability of irrigated crop production. This is because more water is required for leaching, resulting in shallow water tables in agricultural lands. The installation of well designed subsurface drainage systems alone is not enough; the provision of timely maintenance is also necessary. In this study, the extent and severity of problems as a consequence of shallow water tables and their possible causes were investigated at three sugarcane fields in Pongola, South Africa, having low hydraulic conductivity soils. Also investigated were soil salinity levels and the temporal variation in the salinity of the irrigation water. A water table map of a 32 ha sugarcane field was generated, using observed water table depth (WTD) data from 36 piezometers monitored from September 2011 to February 2012. Out of the total 32 ha under cultivation, 12% was found to be affected by shallow WTDs of less than the 1.0 m design WTD. The inability of natural drainage to cope with subsurface drainage needs and the poor maintenance of subsurface drainage systems contributed to the shallow water tables in the area. Furthermore, the currently adopted drainage design criteria also proved unsatisfactory with mean observed water table depth and drainage discharge (DD) of 20% and 50%, respectively, less than their respective design levels. The salinity of the irrigation water was, on average, 32% higher than threshold tolerance level of sugarcane. The root zone soil salinity levels at the three study sites were greater than the 1.7 dS m⁻¹ threshold for sugar cane. The subsurface drainage design criteria adopted at the site needs to be revisited by ensuring that the slope of the land is taken into consideration in the drainage design in addition to adhering to a recommended maintenance schedule.     

Quantification of the hydraulic diffusivity of a bentonite-sand mixture using the water head decrease measured upon sudden flow interruption

This paper presents a new modelling approach to quantify the hydraulic diffusivity of low-permeability unconsolidated porous media under confined saturated-flow conditions in the laboratory. The derived analytical solution for the transient variation of the hydraulic head after flow interruption was applied to experimental data obtained from continuous measurements of the water pressure at two locations in the soil column. Three soil samples made of a mixture of natural bentonite (at different mass fractions) and medium sand were studied during a series of stepwise constant flow rates of water. The numerical results well fit the experimentally measured decrease of the dimensionless hydraulic head. The study shows that the increase of the mass fraction of bentonite in the soil sample from 10 to 30% is accompanied by a strong decrease of the hydraulic diffusivity from 2.4 × 10⁻² to 1.1 × 10⁻³ m² s⁻¹, which is clearly due to the decrease of the hydraulic conductivity of the soil sample. The specific storages obtained for each of the three samples are in the same order of magnitude and seem to decrease with the increase of mass fraction of bentonite. However, they clearly reflect the predominant portion of the compressibility of the porous medium compared with that of water. Compared with reported literature values for clayey soils, the specific storage values in this study are slightly higher, varying within the range of 2 × 10⁻³ to 8.1 × 10⁻³ m⁻¹._. The experimental results also give insight into the limitations of the modelling approach. In the case of low-permeability soils (K < 2 × 10⁻⁶ ms⁻¹) and steady-flow conditions with low Reynolds numbers, for example, Re < 0.003, it is recommended to choose a time duration for flow interruption between subsequent flow rate steps of longer than 5 s. For high-permeability porous media, to increase the precision of the quantified hydraulic diffusivity, it might be useful to select a measuring frequency significantly higher than 1 Hz.