Transient Unsaturated Flow Research Articles

The impact of soil moisture availability on forest growth indices for variably layered coarse‐textured soils

ABSTRACTThe reestablishment of productive forests over mining waste and overburden is a primary reclamation goal in oil sands mining in Northern Alberta, Canada. Soil water conditions in coarse‐textured soils can be limiting to forest growth. The objective of this study was to evaluate the effect that textural variability may have on plant‐available water and concomitant forest productivity on coarse‐textured reclamation soils. The ecophysiological and biogeochemical processes model, Biome‐BGC (Thornton et al., Agricultural and Forest Meteorology 113: 185–222, 2002), was employed to simulate forest dynamics. The water flow sub‐model in Biome‐BGC was replaced by a field‐validated physically based formulation for transient unsaturated water flow. The modified model was assessed using validated physiological parameters, and model predictions were compared with measurements of aboveground biomass dynamics for jack pine (Pinus banksiana Lamb), white spruce [Picea glauca (Moench) Voss], and trembling aspen (Populus tremuloides Michx.). The modified Biome‐BGC model was then used to evaluate the response of leaf area index and net primary production to available water holding capacity on texturally variable, coarse‐textured soils. The results indicate that textural variability could increase the available water holding capacity within a 1‐m profile of coarse‐textured soil by 8 to 16 mm. This enhanced available water holding capacity could increase forest leaf area index by 0·3 to 0·8 and net primary production by 14–30% depending on the specific soil texture and tree species. Copyright © 2012 John Wiley & Sons, Ltd.

A Transient Water Release and Imbibitions Method for Rapidly Measuring Wetting and Drying Soil Water Retention and Hydraulic Conductivity Functions

Abstract A transient water release and imbibitions (TWRI) method for measuring the soil water retention curve (SWRC) and hydraulic conductivity function (HCF) under drying and wetting states is presented. The intellectual merit of the TWRI method is its integration of physical and numerical experiments: it employs the simple and reliable measurement of transient water content by electronic balance to record the signature of transient unsaturated flow in soil sample, and it takes advantage of the robust inverse modeling capability to simulate the physical process. The TWRI method; therefore, has two integrated components: testing and modeling. The testing consists of water release upon two-steps increase in matric suction and water imbibitions upon one-step decrease in matric suction. Each process lasts sufficient time in order to obtain transient water content. The data is then used as an objective function in an inverse numerical modeling process to obtain the unsaturated hydrologic parameters that fully define the SWRC and HCF of the soil. A novel feature of the TWRI method is its capability to measure SWRC and HCF under wetting state. The apparatus can accommodate both undisturbed and remolded samples. The testing time required for completing a full drying and wetting loop is approximately one week for most soils. Validation of the technique has been performed both experimentally and numerically. Comparisons between data obtained with the TWRI method and the independent Tempe cell method verify the reliability, applicability, and accuracy of the TWRI method. Results from three different soils are presented to illustrate the procedure and performance of the TWRI method for different soils from fine sand to clayey silt. The TWRI method provides a fast, accurate, and simple testing tool for obtaining SWRC and HCF of various types of soils under both wetting and drying states with high range of matric suction several orders of magnitude above the air-entry pressure of the ceramic stone used in the experimental setup.

A Numerical Model of Transient Unsaturated Flow Under Centrifugation Based on Mass Balance

Numerical modeling of unsaturated flow in porous media under centrifugation is studied. A precise and numerically efficient approximation is presented for the mathematical model, based on Richards’ nonlinear and degenerate equation expressed in terms of effective saturation using the Van Genuchten–Mualem ansatz. The main difference with other methods is the utilization of a nonlocal condition based on mass balance. The method is suitable for determination of soil parameters, including the saturated conductivity, via the solution of an inverse problem in an iterative way. First, the fully saturated sample is centrifugated with a free outflow boundary during some time interval. Next, the output boundary is sealed and the sample is centrifugated for a prescribed time interval, or up to the creation of an equilibrium. Finally, the centrifugation is continued with a free outflow boundary. This procedure can be repeated to increase the information to drive the inverse problem. The application of the present method requires only non-intrusive, cheap measurements: rotational momentum and/or gravitational center of the sample, and optionally, the amount of expelled water.

Colloid transport with wetting fronts: Interactive effects of solution surface tension and ionic strength

Transport of colloids with transient wetting fronts represents an important mechanism of contaminant migration in the vadose zone. The work presented here used steady-state saturated and transient unsaturated flow columns to evaluate the transport of a fluorescent latex microsphere (980 nm in diameter) with capillary wetting fronts of different solution surface tensions and ionic strengths. The saturated transport experiments demonstrated that decreasing solution surface tension and ionic strength decreased colloid deposition at the solid-liquid interface and increased colloid recovery in the column effluent. The effect of solution surface tension on colloid transport and deposition was greater at lower ionic strength, suggesting an interaction between these two factors. Under transient unsaturated flow conditions, the number of colloids retained in sand decreased exponentially with travel distance through the porous media. However, lowering the solution surface tension and ionic strength resulted in a more even distribution of colloids along the column. The measured zeta potentials of colloids in different solutions suggest that both lowering surface tension and ionic strength would enhance the electrostatic repulsion between colloid and sand. The experimental results revealed that the effects are nonlinear, implying the possible existence of critical threshold values, beyond which the effects were not significant. In addition, colloid migration slowed down as solution surface tension decreased due to reduction of capillary forces that drove liquid movement.

Colloid transport and remobilization in porous media during infiltration and drainage

Colloids are potential vectors of many contaminants in porous media. Understanding colloid transport is critical for assessing the migration of contaminants (e.g., pathogens) in the vadose zone. In this study, a series of column experiments were conducted to investigate the coupled effects of flow velocity, water content, and solution ionic strength on transport and remobilization of a model colloid (montmorillonite clay) in a model porous medium (Accusand) during transient unsaturated flow and steady-state saturated flow. The unsaturated transport experiments included a series of infiltration and drainage pulses (e.g., infiltration with colloids, followed by drainage of colloid suspensions, followed by infiltration with a colloid-free solution and drainage of the solution). Saturated flow experiments included only the infiltration of the colloid and colloid-free solutions. Tests were repeated for a variety of solution ionic strengths. Results showed that colloid transport was more sensitive to changes in solution ionic strength at low infiltration rates, and the effect of infiltration rate was more significant at high ionic strength. As a result, increased flow velocities and water content, resulting from high infiltration rates, enhanced colloid transport and remobilization under ionic strength conditions (e.g., 100 mM) that would otherwise lead to strong colloid retention. This observation conceptually suggests that chemical threshold values for preventing colloid movement in porous media might be larger for transient flow conditions than for uniform flow conditions. In addition, drainage was found to induce remobilization of the retained colloids, suggesting transport of colloids even after termination of injection. Overall, the study experimentally highlights the complicated interdependence of the effects of water content, flow velocity, and solution chemistry on colloid transport and remobilization.

Modeling Transient Evaporation from Descending Shallow Groundwater Table Based on Brooks–Corey Retention Function

In arid and semi-arid regions a large amount of rainfall and irrigation water that enters into the soil is lost through soil surface via evaporation. In such regions, there are some areas with shallow groundwater table, evaporating huge amounts of water and accumulating salts at the soil surface. Thus, the evaporation phenomenon not only is responsible for water loss but also is a major reason for soil salinization. The objective of this study was to develop and verify an analytical model for one dimensional transient unsaturated upward flow from water table to soil surface. Consequently, an analytical solution was developed based on the Richards equation with initial and boundary conditions governing evaporation phenomenon. The parametric Brooks and Corey retention function was used to describe water status in the vadose zone. Based on the proposed model, the saccumulative evaporation is estimated as function of water table drawdown and soil retention parameters. To collect the data required for model verification, nine large lysimeters were constructed and packed with sandy loam, silty clay loam, and silty clay soil textures. The results indicated reasonable agreements between the experimental data and those predicted with the proposed model. Although the overall predicted results were well resemble the real conditions, there were some underestimations for a certain period. This can be attributed to evaporation from side gap of columns, upward flow due to vapor phase movement of moisture, and the collapse of macropores resulting from soil packing.

Simultaneous transport of water and solutes under transient unsaturated flow conditions — A case study

The imbalance between incoming and outgoing salt causes salinization of soils and sub-soils that result in increasing the salinity of stream-flows and agriculture land. This salinization is a serious environmental hazard particularly in semi-arid and arid lands. In order to estimate the magnitude of the hazard posed by salinity, it is important to understand and identify the processes that control salt movement from the soil surface through the root zone to the ground water and stream flows. In the present study, Malaprabha sub-basin (up to dam site) has been selected which has two distinct climatic zones, sub-humid (upstream of Khanapur) and semi-arid region (downstream of Khanapur). In the upstream, both surface and ground waters are used for irrigation, whereas in the downstream mostly groundwater is used. Both soils and ground waters are more saline in downstream parts of the study area. In this study we characterized the soil salinity and groundwater quality in both areas. An attempt is also made to model the distribution of potassium concentration in the soil profile in response to varying irrigation conditions using the SWIM (Soil-Water Infiltration and Movement) model. Fair agreement was obtained between predicted and measured results indicating the applicability of the model.

Application of the finite difference heterogeneous multiscale method to the Richards' equation

This paper extends the finite difference heterogeneous multiscale method (FDHMM) to simulate transient unsaturated water flow problems in random porous media. The numerical method is based on the use of two different schemes for the original equation, at different grid levels which allows numerical results at a lower cost than solving the original equations. The main feature of FDHMM is that the necessary data for the macroscopic model are supplied by solving the microscopic model on a sparse spatial domain. The generated code is verified by applying the linearization model of the Richards' equation. Considering two different constitutive relationships, this method is applied to several test examples with different soil textures and boundary conditions. Both the Dirichlet and the periodic boundary conditions are considered for solving the local microscopic model when the water flow in heterogeneous unsaturated soils is simulated by FDHMM. The numerical experiments demonstrate that FDHMM can effectively simulate the transient unsaturated water flow in the specific soils. The numerical experiments also demonstrate that FDHMM can achieve accurate global mass balance and is a globally convergent algorithm, and the spatial correlation length of random coefficients under the specific standard deviation has relatively little influence on the accuracy of the method.

Reactive transport in unsaturated soil: Comprehensive modelling of the dynamic spatial and temporal mass balance of water and chemical components

By implementing the moisture-based form of Richards’ equation into the geochemical modelling framework PHREEQC, a generic tool for the simulation of one-dimensional flow and solute transport in the vadose zone undergoing complex geochemical reactions was developed. A second-order, cell-centred, explicit finite difference scheme was employed for the numerical solution of the partial differential equations of flow and transport. In this scheme, the charge-balanced soil solution is treated as an assembly of elements, where changes in water and solute contents result from fluxes of elements across cell boundaries. Therefore, water flow is considered in terms of oxygen and hydrogen transport. The direct implementation into the geochemical framework provides access to the full set of reactions available in PHREEQC, giving capabilities beyond existing software for unsaturated flow and reaction. Possible reactions include complex aqueous speciation, cation exchange, equilibrium phase dissolution and precipitation, formation of solid solutions, redox reactions, gas phase exchange, surface adsorption considering electrostatics and kinetic reactions with user-defined rate equations, among others. Geochemical reactions were coupled to transport processes by non-iterative sequential operator splitting. The scheme is currently limited to cases where changes in physical fluid properties and hydraulic flow characteristics due to geochemical reactions are negligible. Results from extensive code verification with analytical and accurate numerical solutions as well as HYDRUS-1D show the excellent performance of the scheme for a variety of hydraulic models including the Brooks and Corey model and the van Genuchten model. High accuracy was gained by the use of integrated diffusivities in the finite difference formulation. The integration of complex geochemical reactions was verified with HP1 by simulating the infiltration of a hyperalkaline solution into a clay soil involving aqueous speciation, equilibrium and kinetic phase dissolution/precipitation and cation exchange reactions. The novel capability of the scheme to account for the influence of geochemical reactions on water contents was demonstrated by comparing reactive and non-reactive transport under transient flow conditions. The simulation of surface complexation according to the diffuse double layer model with an explicit calculation of the ion composition in the diffuse layer in transient unsaturated flow conditions, as shown, represents an additional advance in vadose zone modelling.

Virus Transport during Infiltration of a Wetting Front into Initially Unsaturated Sand Columns

We investigated the effect of different flow conditions on the transport of bacteriophage phiX174 in Memphis aquifer sand. Virus transport associated with a wetting front moving into an initially unsaturated horizontal sand column was experimentally compared with that observed under steady-state saturated vertical flow. Results obtained by sectioning the sand columns showthattotal (retained and free) resident virus concentrations decreased approximately exponentially with the travel distance. The rate of decline was similar under both transient unsaturated flow and steady-state saturated flow conditions. Total resident virus concentrations near the inlet were an order of magnitude greater than the virus concentration of the influent solution in both experiments, indicating continuous virus sorption during flow through this zone. Virus retardation was quantified using the ratio of the centroids of the relative saturation and virus concentration versus relative distance functions. The mean retardation factors were 6.43 (coefficient of variation, CV = 14.4%) and 8.22 (CV = 8.22%) for the transient unsaturated and steady-state saturated flow experiments, respectively. Attest indicated no significant difference between these values at P < 0.05. Air-water and air-water-solid interfaces are thought to enhance virus inactivation and sorption to solid particles. The similar retardation factors obtained may be attributable to the reduced presence of these interfaces in the two flow systems investigated as compared to steady-state unsaturated flow experiments in which these interfaces occur throughout the entire column.

Effective Hydraulic Properties Determined from Transient Unsaturated Flow in Anisotropic Soils

An important requirement for accurate modeling of field‐scale unsaturated transport using local‐scale parameters is a method to account for multiscale heterogeneity and the connectivity of the facies that control flow. A series of infiltration tests were conducted at the Hanford Site along a 60‐m‐long transect, instrumented at 1‐m intervals to measure water content (θ) and capillary pressure head (h) to quantify the spatial correlation structure of hydraulic properties and assess connectivity of the facies. Hydraulic parameters including the pore connectivity–tortuosity tensor (Li) were inversely estimated using the Subsurface Transport Over Multiple Phases (STOMP) numerical simulator coupled with the parameter estimation code, UCODE. Results show that six of eight parameters required for a modified van Genuchten–Mualem model could be inversely estimated using θ measured during transient infiltration from a surface line source and approximated prior information. Soils show evidence of saturation‐dependent anisotropy that was well described with the connectivity tensor. Variability of the vertical saturated hydraulic conductivity, Ksv, was larger than the horizontal, Ksh The autocorrelation ranges for Ksh, Ksv, the inverse of the air‐entry value, α, and the horizontal connectivity, Lh, were between 2.4 and 4.6 m, whereas the van Genuchten shape parameter, n, and saturated water content, θs, showed no autocorrelation. Accurate upscaling of hydraulic properties requires the correct assessment of the connectivity of facies.

In Situ Colloid Mobilization in Hanford Sediments under Unsaturated Transient Flow Conditions: Effect of Irrigation Pattern

Colloid transport may facilitate off-site transport of radioactive wastes at the Hanford site, Washington State. In this study, column experiments were conducted to examine the effect of irrigation schedule on releases of in situ colloids from two Hanford sediments during saturated and unsaturated transientflow and its dependence on solution ionic strength, irrigation rate, and sediment texture. Results show that transient flow mobilized more colloids than steady-state flow. The number of short-term hydrological pulses was more important than total irrigation volume for increasing the amount of mobilized colloids. This effect increased with decreasing ionic strength. At an irrigation rate equal to 5% of the saturated hydraulic conductivity, a transient multipulse flow in 100 mM NaNO3 was equivalent to a 50-fold reduction of ionic strength (from 100 mM to 2 mM) with a single-pulse flow in terms of their positive effects on colloid mobilization. Irrigation rate was more important for the initial release of colloids. In addition to water velocity, mechanical straining of colloids was partly responsible for the smaller colloid mobilization in the fine than in the coarse sands, although the fine sand contained much larger concentrations of colloids than the coarse sand.

Monte Carlo simulation of nonlinear reactive contaminant transport in unsaturated porous media

In the current proposed solutions of radioactive waste repositories, the protective function against the radionuclide water-driven transport back to the biosphere is to be provided by an integrated system of engineered and natural geologic barriers. The occurrence of several nonlinear interactions during the radionuclide migration process may render burdensome the classical analytical–numerical approaches. Moreover, the heterogeneity of the barriers’ media forces approximations to the classical analytical–numerical models, thus reducing their fidelity to reality. In an attempt to overcome these difficulties, in the present paper we adopt a Monte Carlo simulation approach, previously developed on the basis of the Kolmogorov–Dmitriev theory of branching stochastic processes. The approach is here extended for describing transport through unsaturated porous media under transient flow conditions and in presence of nonlinear interchange phenomena between the liquid and solid phases. This generalization entails the determination of the functional dependence of the parameters of the proposed transport model from the water content and from the contaminant concentration, which change in space and time during the water infiltration process. The corresponding Monte Carlo simulation approach is verified with respect to a case of nonreactive transport under transient unsaturated flow and to a case of nonlinear reactive transport under stationary saturated flow. Numerical applications regarding linear and nonlinear reactive transport under transient unsaturated flow are reported.

Sorption of Isoxaflutole Diketonitrile Degradate (DKN) and Dicamba in Unsaturated Soil

When analyzing the sorption characteristics of weakly sorbing or labile pesticides, batch methods tend to yield a high margin of error attributable to errors in concentration measurement and to degradation, respectively. This study employs a recently developed unsaturated transient flow method to determine the sorption of isoxaflutole's herbicidally active diketonitrile degradate (DKN) and dicamba. A 20-cm acrylic column was packed with soils with varied texture that had been uniformly treated with 14C-labeled chemical. The antecedent solution herbicide in equilibrium with sorbed phase herbicide was displaced by herbicide-free water, which was infiltrated into the column. Sorption coefficients, Kd, were obtained from a plot of total herbicide concentration in the soil versus water content in the region where the antecedent solution accumulated. DKN Kd values were ∼2–3 times (average Kd = 0.71 L kg−1) greater using the unsaturated transient flow method as compared to the batch equilibration method in clay loam (Kd = 0.33 L kg−1), but similar for the two methods in sand (0.12 vs 0.09 L kg−1) soils. Dicamba Kd values were 3 times greater using the unsaturated transient flow method as compared to the batch equilibration method in the clay loam soil (0.38 vs 0.13 L kg−1), however, the Kd values were the same for the two methods in the sand (∼0.06 L kg−1). This demonstrates that to determine sorption coefficients for labile hydrophilic pesticides, an unsaturated transient flow method may be a suitable alternative to the batch method. In fact, it may be better in cases where transport models have overpredicted herbicide leaching when batch sorption coefficients have been used.

Quantifying the Pore Size Spectrum of Macropore‐Type Preferential Pathways under Transient Flow

It is well known that there is a spectrum of pores in a soil profile. The conventional use of a single lumped value of soil hydraulic conductivity to describe a spectrum of hydraulically active pores may have unintentionally impeded the development of field‐scale chemical transport theory and perhaps indirectly hindered the development of management protocols for chemical application and waste disposal. In this study, three sets of four field‐scale tracer mass flux breakthrough patterns measured under transient unsaturated flow conditions were used to evaluate the validity of an indirect method to quantify equivalent pore spectra of macropore‐type preferential flow pathways. Results indicated that there were distinct trends in how pore spectra of macropore‐type preferential flow pathways changed when a soil profile became wetter during a precipitation event. This suggests that the indirect method has predictive value and is perhaps a better alternative to the lumped soil hydraulic conductivity approach in accurately determining the impact of macropore‐type preferential flow pathways on water movement and solute transport under transient unsaturated flow conditions.

Intermittent filtration of bacteria and colloids in porous media

Intermittent filtration through porous media used for water and wastewater treatment can achieve high pathogen and colloid removal efficiencies. To predict the removal of bacteria, the effects of cyclic infiltration and draining events (transient unsaturated flow) were investigated. Using physical micromodels, we visualized the intermittent transport of bacteria and other colloids in unsaturated porous media. Column experiments provided quantitative measurements of the phenomena observed at the pore scale. Tagged Escherichia coli and a conservative tracer (NaI) were introduced in an initial pulse into a 1.5 m sand column. Subsequent hydraulic flushes without tagged bacteria or tracer were repeated every 4 hours for the next 4 days, during which outflow concentrations were monitored. Breakthrough behavior between colloids and dissolved tracer differed significantly, reflecting the differences in transport processes. Advancement of the wetting front remobilized bacteria which were held in thin water films, attached to the air‐water interface (AWI), or entrapped in stagnant pore water between gas bubbles. In contrast, the tracer was only remobilized by diffusion from immobile to mobile water. Remobilization led to successive concentration peaks of bacteria and tracer in the effluent but with significant temporal differences. Observations at the pore‐scale indicated that the colloids were essentially irreversibly attached to the solid‐water interface, which explained to some extent the high removal efficiency of microbes in the porous media. Straining, cluster filtration, cell lysis, protozoa grazing, and bacteriophage parasitism could also contribute to the removal efficiency of bacteria.

Measuring Sorption of Hydrophilic Organic Compounds in Soils by an Unsaturated Transient Flow Method

Determination of sorption of hydrophilic, weakly sorbing organic compounds in soil by conventional batch methods using a slurried suspension is often prone to considerable errors because small changes in the solution concentration on equilibration must be accurately determined. This difficulty is exacerbated for compounds susceptible to degradation, which also decreases the solution concentration. The objective of this study was to determine sorption of hydrophilic pesticides by applying an unsaturated transient flow method, which enables determination of sorption at sufficiently small solution to soil ratios. The method makes use of piston-like displacement of the antecedent solution in equilibrium with sorbed phase when pesticide-free water is infiltrated into a soil column spiked with a pesticide. Pesticide sorption and the solution concentration are inferred from a plot of total pesticide content per unit mass of soil vs. water content in a region where the antecedent solution is accumulated. Thus, extraction of solution from relative dry soil is unnecessary. We tested this method for two hydrophilic pesticides, monocrotophos [dimethyl (E)-1-methyl-2-(methyl-carbamoyl) vinyl phosphate] and dichlorvos (2,2-dichlorovinyl dimethyl phosphate). The sorption coefficient, K(d), obtained for monocrotophos was slightly lower than that by batch method (K(d) = 0.10 vs. 0.19 L kg(-1)), whereas for dichlorvos, a compound highly susceptible to degradation, the unsaturated flow method yielded a much smaller K(d) (0.19 vs. 3.22 L kg(-1)). The K(d) values for both compounds were consistent with the observed retardation in the pesticide displacement in the columns. The proposed method is more representative of field conditions and particularly suitable for weakly sorbing organic compounds in soils.

SIMULATING WATER MOVEMENT AND ITS UPTAKE BY PLANT ROOTS IN UNSATURATED ZONES

Existing mathematical models that simulate water movement and contaminant transport in unsaturated zone do not take into account the water uptake by plant crop roots resulting in an error in the prediction of water flux and, thereby, contaminant concentration. Moreover the application of these models is often limited due to the lack of easily accessible and representative soil hydraulic properties, like moisture retention characteristic and unsaturated hydraulic conductivity. In this study, a mathematical model is developed for simulating water movement in unsaturated zones by integrating the one‐dimensional transient unsaturated water flow equation (Richard's equation) with a root water extraction term (sink), and also incorporates pedo‐transfer functions (PTF) for estimating soil hydraulic properties. The governing non‐linear partial differential equation is solved numerically by the implicit finite difference method using Picard's iterative technique, the formulation has been illustrated by a characteristic example.

Determining competitive nitrate and chloride adsorption in an andisol by the unsaturated transient flow method

Adsorption of monovalent electrolyte anions during transport process in Andisols is largely due to the increase in the total anion adsorption rather than through anion exchange with strongly adsorbed native S04 2−. The unsaturated transient flow method has been developed for determining adsorption isotherms for weakly reactive ions without causing excessive desorption of strongly adsorbed native ions. The objective of this study was to extend the method to determine NO3 − adsorption isotherms in an Andisol in the absence or presence of Cl competing for the adsorption sites Kannondai subsoil (Hydric Hapludand), premixed with a Ca(NO3)2 or CaCl2-Ca(NO3)2 solution at different concentrations, was packed into sectionable columns, and one-dimensional water absorption experiments were conducted. Anion adsorption by soil, Q n, and the liquid-phase concentration, C n, prior to the water imbibition were obtained from the plots of the anion content vs. water content in the region beyond the ‘plane of separation,’ where the antecedent solution was accumulated. The values of Qn and C n found in a series of column experiments were then used to construct the isotherms describing the solution composition-dependent adsorption of NO3 − and Cl−. e fitted to Langmuir-type equations, the maximum adsorption for NO3 − (Qmax= 27,1-29.0 mmolc kg−1) was consistently smaller than that for Cl− (Qmax = 455 46.1 mmolc kg−1), while the empirical constant K (= 0.0238 - 0.0274 m3 molc −1) was insensitive to the anion species. The anion content profiles predicted by the inferred adsorption isotherms closely agreed with the measured content profiles.

Gaussian Closure of Transient Unsaturated Flow in Random Soils

The Gaussian closure approximation, previously used by the authors to solve steady state stochastic unsaturated flow problems in randomly heterogeneous soils, is extended here to transient flow. The method avoids linearizing the governing flow equations or the soil constitutive relations. It places no theoretical limit on the variance of constitutive parameters and applies to a broad class of soils with flow properties that scale according to a linearly separable model. Closure is obtained by treating the dimensionless pressure head ψ as a multivariate Gaussian function. It yields a system of coupled nonlinear differential equations for the first and second moments of ψ. We apply the Gaussian closure technique to the problem of transient infiltration into a randomly stratified soil. In each layer, hydraulic conductivity and water content vary exponentially with ψ. Elsewhere we show that application of the technique to other constitutive relations is straightforward. Our solution for the mean and variance of ψ in a one-dimensional layer with random conductivity compares well with Monte Carlo results over a wide range of parameters, provided that the spatial variability of the constitutive exponent is small. The solution provides considerable insight into the behavior of the transient unsaturated stochastic flow problem.