Unsaturated Conductivity Research Articles

Characterization of the reaction and transport properties of porous rhyolite and its application to the quantitative understanding of the chemical weathering rate

The chemical weathering of porous rhyolite from Kozushima, Japan was studied, and the factors controlling the weathering rate, including advection, diffusion and dissolution, were characterized using laboratory experiments. The results were used to analyze the reaction and transport rates, and the relationship between the weathering rates obtained in the laboratory and field was evaluated. A flow-through dissolution experiment using a rhyolite core demonstrated that the dissolution rate R (molcm−2s−1) in deionized water depends on time t (days) according to the relationship R=2.9×10−16t−0.65. The dissolution rate was also dependent on the concentration of dissolved Si according to the relationship R∝(1−Ωn), where Ω=c/ceq (ceq=230μmolL−1) and n=1.0 or 0.33. The hydraulic conductivity and diffusion coefficients were measured under water-saturated and unsaturated conditions. The saturated hydraulic conductivity was 10 times greater than the unsaturated conductivity, and the saturated effective diffusion coefficient of dissolved Si was 5 times greater than the unsaturated diffusion coefficient. Previous studies on the weathering of the rhyolites from Kozushima have shown that the field dissolution rate over 52,000years was 3×10−19–6×10−19molSicm−2s−1. These values are 12–1000 times lower than the dissolution rates obtained in the laboratory over a reaction time of 277days. Reactive transport analysis demonstrated that the effect of Ω on the dissolution rate is enhanced as the water saturation decreases. The rate-decreasing effect of Ω was large in the early stages of weathering and then gradually decreased with time. The 52,000-year weathering period is inferred to have occurred following a decrease in the dissolution rate to a value close to the “field rate” during an early weathering stage and subsequent further decrease in the rate owing to the effect of Ω.

Modeling Soil Water Movement under Low Head Ponding and Gravity Infiltration Using Data Determined with Different Methods

The aim of this work is to analyze differences in experimental and calculated model study of the processes of movement of soil water under low head ponding and gravity infiltration. In the forecast of the water regime two kinds of hydrological information were used – experimental and pedotransfer functions (PTFs). The experimental information: the hydraulic conductivities (HC) in saturated conditions were determined in the field conditions, unsaturated conductivity function was calculated (van Genuchten-Mualem model) using HC and water retention curves (WRC), WRC were investigated in laboratory condition with the help of capillarimeters, sand-kaolin boxes and by the centrifuge method. PTFs included: regional PTF, based on the data of the original database for the soil hydrological properties of the study area relying on the soil density and soil organic content, PTF based on particle size distribution, based on field capacity and wilting point as a predictors, and PTFs based on physico-mechanical properties, specifically Atterberg limits. The spatial distribution of moisture indicates that after infiltration under low head ponding, the soil water content variation is rather high (at depths of 50 and 60cm the quartiles were about 2-3 and 6-12%), but in the case of gravity infiltration the quartiles were about 2 and 5%. This points to the presence of preferential flows. The best forecast of the real-mode soil water regime gave experimental hydrological information: no systematic errors in the high min-max range of modeling errors. Among PTFs the modeling results were placed in the following order: regional PTFs work better (>) than PTFs based on the field capacity and wilting point >PTFs based on Atterberg limits> PTF based on particle size distribution. The statistical analysis of the modeling errors by Williams-Klute nonparametric statistical criteria is evidence that models describe best gravity filtration.

선행강우를 고려한 인제지역 불포화 화강풍화토 사면의 안정성 특성

실제 사면 안정성을 평가하기 위해서는 불포화 지반정수를 반영함과 동시에 여러 인자(강우 특성, 지형학적 특성 등)를 고려해야 한다. 또한 사면안정성에 큰 영향을 미치는 선행강우를 고려해야 함에도 불구하고 사면안정해석 시 거의 반영이 되지 않고 있는 실정이다. 따라서 선행강우를 고려한 사면안정해석을 위해 기존의 현장계측치에 근거하여 초기 모관흡수력의 변화를 바탕으로 선행강우를 고려하였다. 선행강우를 고려한 실제 사면의 안정성을 분석하기 위해 급경사지 붕괴다발지역인 인제지역의 불포화 지반정수(함수특성곡선, 불포화투수곡선, 불포화 강도특성)를 산정하였다. 강우강도 특성과 인제지역 불포화 지반특성을 반영하여 선행강우가 불포화 사면의 안전율 변화에 미치는 영향을 분석하였을 뿐만 아니라, 선행강우(초기 모관흡수력)에 따른 파괴시점 및 파괴시점까지의 누적강우량을 비교 분석 하였다. 초기 모관흡수력이 높을수록 안전율이 급격히 감소하는 시점과 파괴시점이 늦어진다. 사면안정해석 시 습윤 함수특성곡선을 사용할 때가 건조 함수특성곡선을 사용했을 때보다 더 빠른 시점에 파괴가 유발된다. Unsaturated geotechnical properties and other factors(rainfall characteristic, geomorphology, etc) have to be considered in order to evaluate the stability of real slope. Even though antecedent rainfall is also an important factor on the slope stability, it is rarely considered in analysis of slope stability. In this paper slope stability analysis for unsaturated weathered granite soil in Inje was conducted considering antecedent rainfall. Effect of antecedent rainfall was considered in analysis by using initial matric suction, which was induced from field measurement. Unsaturated geotechnical properties(soil-water characteristic curve, unsaturated conductivity, unsaturated shear strength) of weathered granite soil in Inje were also investigated. Effects of antecedent rainfall on variation of factor of safety(FS) of slope, failure time and cumulative rainfall were evaluated. For a slope with high initial matric suction, FS slowly decreased and failure time delayed than a slope with low initial matric suction. Numerical analysis also indicated that a slope with wetting SWCC more rapidly approached to failure than a slope with drying SWCC.

Pedotransfer functions to determine water conducting macroporosity in South African soils

Macropores play an important role in the rapid transport of water, solutes and pollutants through the soil. Transport through these pores (>0.5 mm) is dominated by gravitational forces (i.e. matrix forces have low impact) resulting in flow rates orders of magnitude higher than rates that would be predicted, posing problems for modelling and understanding water and solute transport through soils. This study aimed to quantify the water conducting macroporosity (WCM) in a range of soils in South Africa and to develop three pedotransfer functions (PTFs) able to predict WCM. Saturated (K(s)) and unsaturated (K₃₀) conductivities were measured in situ on 120 soil profiles using double ring and tension infiltrometers methods. Differences between K(s) and K₃₀ in conjunction with Poiseuille's law and the capillary rise equation were used to calculate WCM. The first two multiple regression functions made use of all available soil properties influencing WCM using a 'best model' and 'backward' analysis approach respectively. The third model used only easily observable soil properties to predict the WCM. The functions were validated using a double-cross method. Results are encouraging with R² values of 0.78, 0.74 and 0.69 for functions 1, 2 and 3 respectively.

A Note on the Apparent Conductivity of Stratified Porous Media in Unsaturated Steady Flow Above a Water Table

We aim at deriving the apparent unsaturated conductivity (AUC) K (ap), defined as the ratio between the mean flux and the mean head gradient in a stratified vadose zone above the water table. This is achieved for steady flow generated by a constant infiltrating flux applied at the soil surface. By adopting the first-order approximation in the two parameters of the conductivity curve, and under a few additional simplifying assumptions, we were able to analytically compute K (ap). It is shown that this latter varies between K (ap) ≡ K H (the harmonic mean) at the water table, and K (ap) ≡ K ef (the effective conductivity in gravitational flow) far above the water table. Profiles of the AUC are illustrated, and the impact of parameters values is discussed.

How the Rhizosphere May Favor Water Availability to Roots

Recent studies have shown that rhizosphere hydraulic properties may differ from those of the bulk soil. Specifically, mucilage at the root–soil interface may increase the rhizosphere water holding capacity and hydraulic conductivity during drying. The goal of this study was to point out the implications of such altered rhizosphere hydraulic properties for soil–plant water relations. We addressed this problem through modeling based on a steady‐rate approach. We calculated the water flow toward a single root assuming that the rhizosphere and bulk soil were two concentric cylinders having different hydraulic properties. Based on our previous experimental results, we assumed that the rhizosphere had higher water holding capacity and unsaturated conductivity than the bulk soil. The results showed that the water potential gradients in the rhizosphere were much smaller than in the bulk soil. The consequence is that the rhizosphere attenuated and delayed the drop in water potential in the vicinity of the root surface when the soil dried. This led to increased water availability to plants, as well as to higher effective conductivity under unsaturated conditions. The reasons were two: (i) thanks to the high unsaturated conductivity of the rhizosphere, the radius of water uptake was extended from the root to the rhizosphere surface; and (ii) thanks to the high soil water capacity of the rhizosphere, the water depletion in the bulk soil was compensated by water depletion in the rhizosphere. We conclude that under the assumed conditions, the rhizosphere works as an optimal hydraulic conductor and as a reservoir of water that can be taken up when water in the bulk soil becomes limiting.

Direct Measurement of Unsaturated Conductivity including Hydraulic Nonequilibrium and Hysteresis

The unsaturated hydraulic conductivity function is the dominant material property for modeling soil water dynamics. Because it is difficult to measure directly, it is often derived from the water retention characteristic combined with a geometric model of the pore space. In this study, we developed an automated, simple multistep flux (MSF) experiment to directly measure unsaturated conductivities, K(ψm), at a number of water potentials, ψm, using the experimental setup of classical multistep outflow (MSO) experiments. In contrast to the MSO experiment, the MSF experiment measures the conductivity directly at a spatially constant water potential assuming macroscopically homogeneous materials. Additionally, the proposed method reveals the hysteresis of K(ψm) with respect to increasing and decreasing water potentials as well as the temporal dynamics of K(ψm) during transient‐flow conditions. This temporal behavior is explained by the dynamics of fluid configurations at the pore scale during drainage and imbibition leading to hydraulic nonequilibrium. It may provoke a systematic underestimation of hydraulic conductivity using inverse optimization of K(ψm) based on classical MSO experiments. The new approach will improve the determination of K(ψm) and it provides an experimental tool to quantify the effects of hydraulic nonequilibrium under transient conditions.

Combined Brooks-Corey/Burdine and van Genuchten/Mualem Closed-Form Model for Improving Prediction of Unsaturated Conductivity

The performance of the widely used conventional closed-form models of Brooks and Corey and van Genuchten is restricted to soils characterized by a specific shape form of the water retention curve. Hence, the van Genuchten model more accurately describes “S”-shaped retention curves characterizing finer-textured soils, whereas the Brooks-Corey model is much better adapted for “J”-shaped retention curves characterizing relatively coarse-textured soils. In this work, a new closed-form soil hydraulic model is proposed. The suggested continuous-form function accurately describes soil retention curves irrespective of their specific shape form. New algebraic expressions based on Mualem’s statistical model and another new model that is a combination of the Mualem and Burdine theories were derived for the prediction of the unsaturated hydraulic conductivity function. Comparisons of ten soils known from international bibliography were performed. It is concluded that the proposed water retention curve expression as w...

Influence of Unsaturated Soil Properties Uncertainty on Moisture Flow Modeling

The quality of a numerical modeling solution of moisture flow through unsaturated soil, in part, depends on properly described unsaturated soil properties. The variability of the Soil Water Characteristic Curve, SWCC, is attributed to hysteresis and reproducibility of measurement. Because the unsaturated conductivity function is rarely directly measured, the variability of the unsaturated soil hydraulic conductivity function is attributed to the uncertainty associated with the estimation of this parameter with currently available fitting functions, and hence a range of reasonable variation was considered. One-dimensional modeling of expansive soil under dry initial conditions (suction of 1,500 kPa) was performed; both potential evaporation and infiltration boundary conditions were considered. It was found that small variations in the unsaturated soil hydraulic conductivity function result in significantly different modeling outputs, as expected, while substential variation in SWCC alone (assuming the same unsaturated soil hydraulic conductivity for all SWCCs) produced almost identical soil response in terms of soil suction when the slope of the SWCC is similiar. Thus, proper characterization of the slope of the SWCC is important to proper suction profile determination.

Land-use effects on flood generation – considering soil hydraulic measurements in modelling

Abstract. The investigation in the catchment of the Mulde (51°0'55" N, 13°15'54" E Saxony, Germany) researches the effect of afforestation measures on the soil hydraulic properties. The concept of a "false chronosequence" was used to quantify the time-dependent dynamical character of the forest impact. Four adjacent plots were identified at a test location with comparable pedological start conditions and a set of tree stands of different age: (1) arable field (initial state); (2) 6-year-old afforestation; (3) 50-year-old afforestation; (4) ancient natural forest ("target" stocking). Water retention curves and unsaturated conductivities were analysed in the lab. In the field, the undisturbed infiltration capacities were measured quantitatively (hood infiltrometer) and qualitatively (brilliant blue tracer). Pronounced differences between all 4 plots were detected. The afforestation causes an increased infiltration and soil water retention potential. Especially the topsoil layers showed a distinct increase in conductivity and portion of coarse/middle pores. The influence of these changes on rainfall-runoff calculations at the test location was analysed in this study.

Stochastic analysis of unsaturated flow with probabilistic collocation method

In this study, we present an efficient approach, called the probabilistic collocation method (PCM), for uncertainty analysis of flow in unsaturated zones, in which the constitutive relationship between the pressure head and the unsaturated conductivity is assumed to follow the van Genuchten‐Mualem model. Spatial variability of soil parameters leads to uncertainty in predicting flow behaviors. The aim is to quantify the uncertainty associated with flow quantities such as the pressure head and the effective saturation. In the proposed approach, input random fields, i.e., the soil parameters, are represented via the Karhunen‐Loeve expansion, and the flow quantities are expressed by polynomial chaos expansions (PCEs). The coefficients in the PCEs are determined by solving the equations for a set of carefully selected collocation points in the probability space. To illustrate this approach, we use two‐dimensional examples with different input variances and correlation scales and under steady state and transient conditions. We also demonstrate how to deal with multiple‐input random parameters. To validate the PCM, we compare the resulting mean and variance of the flow quantities with those from Monte Carlo (MC) simulations. The comparison reveals that the PCM can accurately estimate the flow statistics with a much smaller computational effort than the MC.

Use of soil moisture dynamics and patterns at different spatio-temporal scales for the investigation of subsurface flow processes

Abstract. Spatial patterns as well as temporal dynamics of soil moisture have a major influence on runoff generation. The investigation of these dynamics and patterns can thus yield valuable information on hydrological processes, especially in data scarce or previously ungauged catchments. The combination of spatially scarce but temporally high resolution soil moisture profiles with episodic and thus temporally scarce moisture profiles at additional locations provides information on spatial as well as temporal patterns of soil moisture at the hillslope transect scale. This approach is better suited to difficult terrain (dense forest, steep slopes) than geophysical techniques and at the same time less cost-intensive than a high resolution grid of continuously measuring sensors. Rainfall simulation experiments with dye tracers while continuously monitoring soil moisture response allows for visualization of flow processes in the unsaturated zone at these locations. Data was analyzed at different spacio-temporal scales using various graphical methods, such as space-time colour maps (for the event and plot scale) and binary indicator maps (for the long-term and hillslope scale). Annual dynamics of soil moisture and decimeter-scale variability were also investigated. The proposed approach proved to be successful in the investigation of flow processes in the unsaturated zone and showed the importance of preferential flow in the Malalcahuello Catchment, a data-scarce catchment in the Andes of Southern Chile. Fast response times of stream flow indicate that preferential flow observed at the plot scale might also be of importance at the hillslope or catchment scale. Flow patterns were highly variable in space but persistent in time. The most likely explanation for preferential flow in this catchment is a combination of hydrophobicity, small scale heterogeneity in rainfall due to redistribution in the canopy and strong gradients in unsaturated conductivities leading to self-reinforcing flow paths.

Hydraulic Conductivity as Influenced by Stoniness in Degraded Drylands of Chile

Infiltration measurements in arid, stony soils are notoriously variable in visually homogeneous areas, and have been reported to be influenced by embedded stone fragments. This study aimed to identify the influence of rock fragment contents, orientation, and position within a small arid watershed on hydraulic conductivity in northern Chile. Two different measurement techniques were used, a single‐ring infiltrometer with constant ponding head and a tension infiltrometer, which were applied at both an undisturbed field site (44 locations along three transects) and on the disturbed <2‐mm soil fraction from the same locations. Variations in saturated and unsaturated hydraulic conductivities were observed when using different calculation methods, adding to the observed variability. For saturated conditions, only small differences in conductivities were observed between two calculation methods, whereas unsaturated hydraulic conductivities calculated by five different methods showed more important variations. Stone fragment content correlated significantly with both saturated and unsaturated conductivities, probably due to a positive correlation between stone content and coarse lacunar pore space. Slope orientations with higher amounts of stone fragments gave higher infiltration rates, as well as transects with steeper slopes and more, but smaller, rock fragments. When evaluating differences in infiltration rates observed along three transects in the watershed, variability could be mainly attributed to stone fragment content influences.

Biexponential model for water retention characteristics

Water retention characteristic (WRC) is an important soil property used in many hydrologic applications. Although there are numerous models in the literature for its prediction, the models are largely empirical with fundamental derivations based on soil textural pore-spaces. This paper presents a biexponential model for WRC that has its basis on textural and structural soil pore-spaces. The model contains physical parameters related to soil textural and structural pore characteristics and can also navigate through all inflections present in measured WRC. Thus, it adequately represents possible classes of soil pores in WRC. The performance of this model on different soil types from various parts of the world was compared to some of the popular WRC models in the literature. The models tested were those proposed by van Genuchten [van Genuchten, M.T., 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Science Society of America Journal, 44, 892–898], Campbell [Campbell, G.S., 1974. A simple method for determining unsaturated conductivity from moisture retention data. Soil Science, 117, 311–314], Brooks and Corey [Brooks, R.H., Corey, A.T., 1964. Hydraulic Properties of Porous Medium. Hydrology Paper Number 3. Colorado State University, USA.], Gardner [Gardner, W.R., 1958. Some steady state solutions of the unsaturated moisture flow equation with application to evaporation from a water table. Soil Science, 85, 228–232], Groenevelt and Grant [Groenevelt, P.H., Grant, C.D., 2004. A new model for the soil-water retention curve hat solves the problem of residual water contents. European Journal of Soil Science, 55, 479–485]. These models were found inadequate in fitting the whole range of measured WRC, are limited to certain geographic regions of the world, and contain parameters not directly linked to soil physical properties. The biexponential model does not only overcome these limitations but also gives the best fit of WRC. Its model parameters have physical significance and have been shown in this paper to be useful in fitting different soil physical conditions. The model is continuous and continuously differentiable throughout the whole range of WRC and is envisaged to perform equally well in fitting unsaturated hydraulic conductivity.

Comparison of empirical, semi-empirical and physically based models of soil hydraulic functions derived for bi-modal soils

The accelerated flow in soil pores is responsible for a rapid transport of pollutants from the soil surface to deeper layers up to groundwater. The term preferential flow is used for this type of transport. Our study was aimed at the preferential flow realized in the structural porous domain in bi-modal soils. We compared equations describing the soil water retention function h( θ) and unsaturated hydraulic conductivity K( h), eventually K( θ) modified for bi-modal soils, where θ is the soil water content and h is the pressure head. The analytical description of a curve passing experimental data sets of the soil hydraulic function is typical for the empirical equation characterized by fitting parameters only. If the measured data are described by the equation derived by the physical model without using fitting parameters, we speak about a physically based model. There exist several transitional subtypes between empirical and physically based models. They are denoted as semi-empirical, or semi-physical. We tested 3 models of soil water retention function and 3 models of unsaturated conductivity using experimental data sets of sand, silt, silt loam and loam. All used soils are typical by their bi-modality of the soil porous system. The model efficiency was estimated by RMSE (Root mean square error) and by RSE (Relative square error). The semi-empirical equation of the soil water retention function had the lowest values of RMSE and RSE and was qualified as “optimal” for the formal description of the shape of the water retention function. With this equation, the fit of the modelled data to experiments was the closest one. The fitting parameters smoothed the difference between the model and the physical reality of the soil porous media. The physical equation based upon the model of the pore size distribution did not allow exact fitting of the modelled data to the experimental data due to the rigidity and simplicity of the physical model when compared to the real soil porous system. On the other hand, the “optimal” unsaturated conductivity equation was obtained for a model based on the pore size distribution where the additional fitting parameters allowed a good agreement between the model and the reality of the measured data. We denoted this equation as semi-physical.

Validation of Mualem’s Conductivity Model and Prediction of Saturated Permeability from Sorptivity

Advanced modeling of unsaturated water and contaminant transport in concrete requires knowledge of the unsaturated conductivity (permeability) function and the water retention curve. An assumption of simple Fickian diffusion is not sufficient due to the nonlinear water and contaminant fluxes. The Van Genuchten-Mualem conductivity model, widely used in describing hydraulic properties of soils, is validated by predicting experimental moisture content profiles obtained by nuclear magnetic resonance during simple absorption experiments with concrete cylinders. A new tortuosity parameter is suggested. Analytical methods are extended based on the Van Genuchten-Mualem model, to provide a means of estimating the saturated permeability from the much simpler sorptivity experiment. The predicted permeability is similar to the long-term experimental permeability. This suggests that only the very long-term saturated permeability is suitable for describing unsaturated moisture flux.

Upscaling hydraulic conductivity based on the topology of the sub-scale structure

The hydraulic conductivity of heterogeneous porous media depends on the distribution function and the geometry of local conductivities at the smaller scale. There are various approaches to estimate the effective conductivity K eff at the larger scale based on information about the small scale heterogeneity. A critical geometric property in this ‘upscaling’ procedure is the spatial connectivity of the small-scale conductivities. We present an approach based on the Euler-number to quantify the topological properties of heterogeneous conductivity fields, and we derive two key parameters which are used to estimate K eff. The required coefficients for the upscaling formula are obtained by regression based on numerical simulations of various heterogeneous fields. They are found to be generally valid for various different isotropic structures. The effective unsaturated conductivity function K eff ( ψ m) could be predicted satisfactorily. We compare our approach with an alternative based on percolation theory and critical path analysis which yield the same type of topological parameters. An advantage of using the Euler-number in comparison to percolation theory is the fact that it can be obtained from local measurements without the need to analyze the entire structure. We found that for the heterogeneous field used in this study both methods are equivalent.

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.

Closed‐Form Expression for the Multi‐Modal Unsaturated Conductivity Function

In this note we derive a closed‐form expression representing the hydraulic conductivity function for soils with a multi‐modal pore size distribution. By combining the multi‐modal representation of the retention function of Durner with the conductivity representation model of Mualem and following van Genuchten, we derive a simple analytical expression for the conductivity of soils with heterogeneous pore systems. Examples for the representation of bi‐modal and tri‐modal retention and corresponding conductivity curves demonstrate the applicability of the simple analytical expressions. It is concluded that the usefulness of the multi‐modal representation of retention functions is increased by providing an analytical expression to directly calculate unsaturated hydraulic conductivities.

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 κ ∗ which depends on the medium heterogeneity structure and the mean pressure head. Generally, the correction function κ ∗ 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 κ ∗. 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 κ ∗. Overall, results suggest that, given the formation statistics, the impact of the heterogeneity upon κ ∗ 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, κ ∗ is influenced solely by the anisotropic structure of the formation unless the horizontal correlation scales are much greater than the vertical ones.