A one-dimensional approximate analytical model, which preserves the main features of soil-crop-atmospheric hydrodynamics, has been suggested for plant roots of low soil-root conductivity ratio (SRCR). The proposed approach involves physically based concepts, such as mass balance equation, Darcy’s law, and related water uptake and plant transpiration functions. Two main assumptions have been made to derive the analytical solution: (1) gravitational flow is adopted and (2) the uniform soil moisture distribution within the root water activity zone is supposed. The mass balance equation in its integral form is solved by the method of characteristics. This leads to the two functional equations for soil pressure head and root potential, which can be solved simultaneously by using common software. The model has been further verified against the numerical one. The model represents a reasonable compromise between the complicated mechanism of unsaturated water flow with root water uptake (RWU) and still insufficient knowledge of the soil-plant-atmospheric continuum. It is able to account for temporal fluctuations in root activity zone and provides a relatively simple algorithm for investigation of RWU-mechanism. Besides the theoretical and applicative importance, this flow model yields water and velocity distributions within soil profile, and, thereby, constitutes a preliminary step toward solution of contaminant transport problems in vadose zone.

Transmissivity of aquifers is spatially variable in the plane. Hoeksema & Kitanidis (1985) have analysed field data of tens of aquifers and found that the log-transmissivity, regarded as a normal stationary random function, is characterized by integral scales of the order of kilometres to tens of kilometres. We examine the impact of transmissivity heterogeneity upon steady flow toward a pumping well, for realistic values of the ratio between the integral scale and the well radius of 2000 or larger. Our main finding is that in the zone of influence of the well, of radius up to 2000 well radii, the aquifer practically behaves like a homogeneous one, with the constant transmissivity pertaining to the location of the well. Flow can be modelled, therefore, as one-dimensional (in the radial coordinate) and randomness is parametric. Thus uncertainty can be practically eliminated by conditioning the head on transmissivity measured at the well. We conclude that flow and transport in the zone adjacent to the well are impacted by the three-dimensional local variability of the hydraulic conductivity, but not by the large-scale transmissivity heterogeneity.

Solute transport is investigated in a heterogeneous aquifer for combined natural-gradient and well flows. The heterogeneity is associated with the spatially varying hydraulic conductivity K(x, y, z), which is modelled as a log-normal stationary-random function. As such, the conductivity distribution is characterized by four parameters: the arithmetic mean K-A, the variance sigma(2)(Y) (Y=lnK), the horizontal integral scale I of the axisymmetric log-conductivity autocorrelation and the anisotropy ratio e=I-v/I (I-v is the vertical integral scale). The well fully penetrates an aquifer of constant thickness B and has given constant discharge QB, while the background aquifer flow is driven by an uniform mean head-gradient, - J. Therefore, for a medium of homogeneous conductivity K-A, the steady-state capture zone has a width 2L = Q/(K-A vertical bar J vertical bar) far from the well (herein the term capture zone is used to refer both to the zone from Which water is captured by a pumping well and the zone that captures fluid from an injecting well).The main aim is to determine the mean concentration as a function of time in fluid recovered by a pumping well or in a control volume of the aquifer that captures fluid from an injecting well. Relatively simple solutions to these complex problems are achieved by adopting a few assumptions: a thick aquifer 2 B >> I, of large horizontal extent (so that boundary effects may be neglected), weak heterogeneity sigma(2)(Y)< 1, a highly anisotropic formation e < 0.2 and neglect of pore-scale dispersion. Transport is analyzed to the first order in sigma(2)(Y) in terms of the travel time of particles moving from or towards the well along the steady streamlines within the capture zone. Travel-time mean and variance to any point are computed by two quadratures for an exponential log-conductivity two-point covariance. Spreading is reflected by the 2 variance value, which increases with sigma(2)(Y) and I/L.For illustration, the procedure is applied to two particular cases. In the first one, a well continuously injects water at constant concentration. The mean concentration as function of time for different values of the controlling parameters sigma(2)(Y) and I/L is determined within control volumes surrounding the well or inpiezometers. In the second case, a solute plume, initially occupying a finite volume Omega(0), is drawn towards a pumping well. The expected solute recovery by the well as a function of time is determined in terms of the previous controlling parameters as well as the location and extent of Omega(0).The methodology is tested against a full three-dimensional simulation of a multi-well forced-gradient flow field test ([Lemke, L., W.B. II, Abriola, L., Goovaerts, P., 2004. Matching solute breakthrough with deterministic and stochastic aquifer models. Ground Water 42], SGS simulations). Although the flow and transport conditions are more complex than the ones pertinent to capture zones in uniform background flow, it was found that after proper adaptation the methodology led to results for the breakthrough curve in good agreement with a full three-dimensional simulation of flow and transport. (c) 2006 Elsevier B.V. All rights reserved.

Contaminant transport in the upper layers of soil during the multiple cycles of short infiltration and prolonged redistribution is investigated. Analytical solutions for the two typical problems encountered in agricultural engineering are derived. The first problem considers the penetration of fertilizer initially applied at the soil surface. The second one is the propagation of contaminant injected with the applied water. Explicit analytical expressions for the solute concentration are obtained under assumptions of one-dimensional gravitational flow and advective solute transport under equilibrium conditions. The properties of the solute penetration are analyzed for the case of contaminant initially applied at the soil surface and picked up by the water flow during ten infiltration–redistribution cycles.

In modeling flow toward wells it is mathematically convenient to replace the well by a singularity line along its axis. In the case of homogeneous aquifers, confined flow, and fully penetrating wells the strength of the sources is constant, and the common condition of constant head on the well is satisfied exactly. It also leads to constant flux on the well envelope. In the case of aquifers of spatially variable conductivity the constant head boundary condition can be satisfied accurately, if the well radius is much smaller than the log conductivity horizontal integral scale, by selecting a variable source strength that is proportional to the local conductivity along the well [ Indelman et al., 1996 ; Fiori et al., 1998 ; Indelman, 2003a , 2003b ]. In this case the equivalent conductivity tends to the arithmetic mean near the well. It changes with distance, approaching from above the effective conductivity in uniform horizontal mean flow far from the well. Selecting inadvertently a constant source strength violates the condition of constant well head. The equivalent conductivity tends to the harmonic mean near the well and approaches from below the effective one in uniform mean flow. The equivalent conductivity is derived for both types of sources for stratified formations as well as for aquifers of a three‐dimensional heterogeneous structure. The implications on interpretation of head measurements by the two models is illustrated by a few examples.

A stochastic methodology is used to investigate transport of a conservative solute by steady flow to a fully penetrating pumping well in a three‐dimensional heterogeneous formation. The spatially variable hydraulic conductivity is modeled as a stationary random function of anisotropic two‐point covariance. Although mean transport is radial, heterogeneity produces three‐dimensional velocity fluctuations which are shown to affect solute recovery and the capture zone distribution. Expressions accurate to the first order in log conductivity variance σY2 are derived for travel time variance, capture zone extent, and solute recovery under the following simplifying assumptions: (1) the vertical integral scale of conductivity is much less than the horizontal integral scale, (2) the depth of the formation is much greater than the vertical integral scale of conductivity, (3) the well bore diameter is much smaller than the horizontal scale of conductivity, (4) longitudinal and lateral local dispersion are negligible while the effect of transverse dispersion is examined. Travel time variance στ2 for transport in convergent flow in a three‐dimensional aquifer is shown to be greater than στ2 in either constant mean flow in a three‐dimensional formation or radially convergent flow in a heterogeneous two‐dimensional medium. Transverse dispersion between layers in a sedimentary formation is found to increase mixing between flow paths leading to a marginal reduction in travel time variance. Because of negative correlation between mass flux and travel time, solute retrieval at the well occurs earlier from a constant flux solute source than from an instantaneous plume. The relationship between initial plume length and relative plume spreading is illustrated for simple plume geometries.

The transport of instantaneously injected conservative solute through a well in the formation of random conductivity is analyzed. The solute is advected by the recharging well flow with the uniform background gradient. The longitudinal trajectory variance is derived for the central mean streamline. It is shown that the solute is spread as in a radial flow at small travel distances and as in a uniform flow far from the well. Closed form expressions of the longitudinal trajectory variance and macrodispersivity are derived for the case of small scale heterogeneity. It is shown that the macrodispersivity is bounded between the asymptotic macrodispersivities pertinent to the well and uniform flows.

Transport of reactive solute in unsaturated soils under an infiltration-redistribution cycle is investigated. The study is based on the model of vertical flow and transport in the unsaturated zone proposed by Indelman et al. [J. Contam. Hydrol. 32 (1998) 77], and generalizes it by accounting for linear nonequilibrium kinetics. An exact analytical solution is derived for an irreversible desorption reaction. The transport of solute obeying linear kinetics is modeled by assuming equilibrium during the redistribution stage. The model which accounts for nonequilibrium during the infiltration and assumes equilibrium at the redistribution stage is termed partial equilibrium infiltration-redistribution model (PEIRM). It allows to derive approximate closed form solutions for transport in one-dimensional homogeneous soils. These solutions are further applied to computing the field-scale concentration by adopting the Dagan and Bresler [Soil Sci. Soc. Am. J. 43 (1979) 461] column model. The effect of soil heterogeneity on the solute spread is investigated by modeling the hydraulic saturated conductivity as a random function of horizontal coordinates. The quality of the PEIRM is illustrated by calculating the critical values of the Damköhler number which provide the achievable accuracy in estimating the solute mass in the mobile phase. The distinguishing feature of transport during the infiltration-redistribution cycle as compared to that of infiltration only is the finite depth of solute penetration. For irreversible desorption, the maximum solute penetration W/theta(r) is determined by the amount of applied water W and the residual water content theta(r). For sorption-desorption kinetics, the maximum depth of penetration z(r)(e, infinity ) also depends on the ratio between the rate of application and the column-saturated conductivity. It is shown that z(r)(e, infinity ) is bounded between the depths W/(theta(r)+K(d)) and W/theta(r) corresponding to the maximum solute penetration for equilibrium transport and for irreversible desorption, respectively. This feature of solute penetration explains the unusual phenomena of plume contraction after an initial period of spreading [Lessoff, S.C., Indelman, P., Dagan, G., 2002. Solute transport in infiltration-redistribution cycles in heterogeneous soils. In Raats, P.A.C., Smiles, D.,Warrick, A.W. (Eds), Environmental Mechanics: Water, Mass and Energy Transport in the Biosphere. American Geophysical Union, pp. 133-144]. Unlike transport under equilibrium conditions, when the solute is completely concentrated at the front, the solute under nonequilibrium conditions is spread out behind the front. Heterogeneity leads to additional spreading of the plume.

The problem of averaging transient flows by sources of a given head boundary condition in heterogeneous formations of random conductivity is investigated. The study generalizes the recently developed mathematical model of average transient nonuniform flow [Indelman, 1996;Tartakovsky and Neuman, 1998a]. The latter allows calculating the mean head for sources of flux boundary condition and, as such, is not applicable to modeling common well flows. To account for the head condition, the source term in the local flow model is modeled by a random function proportional to the hydraulic conductivity. The local flow equations are further averaged in order to determine the mean flow variables. It is shown that the effective conductivity cannot be defined for arbitrary initial head distributions. This precludes deriving the mathematical model of an average transient flow with the head boundary condition. However, the average flow equations are derived for the important particular case of a uniformly distributed initial head. The effective conductivity tensor is defined for an arbitrary heterogeneous formation and analyzed in detail for isotropic media. It is shown that at the initial stage of the transient flow, the effective conductivity is larger than the conductivity arithmetic mean. The fundamental solution of the average flow equations (mean Green function), corresponding to the mean head distribution due to the instantaneous injection of the unit mean amount of water through a point source, is derived at first order in the conductivity variance and is compared with the mean head for the flux boundary condition.

The properties of the solute spread in a uniform-radial divergent flow are analysed. Analytical expressions are derived for the central trajectory variance. It is shown that the solute is spread as in a radial flow at small travel distances and as in a uniform flow far from the well. For microscale heterogeneity, the macrodispersivity along the central trajectory grows with distance from the well from alpha(11) = alpha(11)((u))/3 to alpha(11) = alpha(11)((u)) where alpha(11)((u)) is the asymptotic macrodispersivity pertinent to the uniform flow.

We consider a general model of transient flow in media of random conductivity and storativity. The flow is driven by the spatially distributed source function φ(x, t) and the initial head distribution h0(x). The function φ models sources and wells and can be deterministic, random or a sum of both. The deterministic source function corresponds to singularities of deterministic strength, whereas the random φ models the head boundary condition. In the latter case, φ is shown to be proportional to the hydraulic conductivity. The aim of the study is to analyze the feasibility of averaging the flow equations and of developing the mathematical model of average flow (AFM) without solving problems in detail. It is shown that the problem of averaging is reduced to deriving two constitutive equations. The first equation, the effective Darcy's law (EDL) stems from averaging Darcy's law at local scale. The second one is related to the medium ability to store a fluid and expresses the correlation between the storativity and head in terms of the mean head. Both relationships are required to be completely determined by the medium structure (conductivity and storativity statistical properties) and independent of the flow configuration (functions φ and h0). We show that if one of the constitutive equations exists, the same is true respective to the second. This reduces the problem of averaging to the classic one of deriving the EDL. For steady flows the EDL is shown to exist for flows driven by sources (wells) of either deterministic flux or head boundary conditions. No EDL can be derived if both types of sources are present in the flow domain. For unsteady flows the EDL does not exist if the initial head correlates with the medium properties. For uncorrelated initial head distribution, its random residual (due to the measurement errors and scarcity of the data) has no impact on the EDL and is immaterial. For deterministic h0, the only case for which the EDL exists is the flow by sources of deterministic discharge. For sources of given head boundary condition the EDL can be derived only for uniform initial head distribution. For all other cases, the EDL does not exist. The results of the study are not limited by usually adopted assumptions of weak heterogeneity and of stationarity of the formation random properties.

Penetration of reactive solute into a soil during a cycle of water infiltration and redistribution is investigated by deriving analytical closed form solutions for fluid flux, moisture content and contaminant concentration. The solution is developed for gravitational flow and advective transport and is applied to two scenarios of solute applications encountered in the applications: a finite pulse of solute dissolved in irrigation water and an instantaneous pulse broadcasted onto the soil surface. Through comparison to simulations of Richards' flow, capillary suction is shown to have contrasting effects on the upper and lower boundaries of the fluid pulse, speeding penetration of the wetting front and reducing the rate of drying. This leads to agreement between the analytical and numerical solutions for typical field and experimental conditions. The analytical solution is further incorporated into a stochastic column model of flow and transport to compute mean solute concentration in a heterogeneous field. An unusual phenomenon of plume contraction is observed at long times of solute propagation during the drying stage. The mean concentration profiles match those of the Monte-Carlo simulations for capillary length scales typical of sandy soils.

Reactive transport properties under unsteady unsaturated flow conditions are estimated from field-measured transport data. The data is from a recent field experiment (Dror et al. 1999a,b) in which herbicides (Bromacil, Atrazine, and Terbuthylazine) applied at the soil surface were transported by a few cycles of intermittent irrigation, and concentrations were measured to a depth of 1.2 m by gas chromatography. An analytic stochastic model of flow and reactive transport accounting for multiple infiltration-redistribution cycles is applied in an inverse mode to identify soil and chemical transport properties and measurement uncertainty. Transport, measured at the end of water redistribution, is more sensitive to residual water in the field and less sensitive to the hydraulic conductivity and maximum saturated moisture content. Sorption in the field is found to be less than laboratory-based predictions. The degradation-rate coefficients agree with laboratory measurements.

An analytic model of transient unsaturated infiltration is presented. The model is based on the column conceptualization of flow and transport in unsaturated soil which is expanded here to incorporate repeated infiltration and redistribution stages. The transport of reactive solute is modeled by assuming three mechanisms: advection by gravitational water flow, equilibrium sorption and linear decay. Solutions of the flow and transport equations are derived for multiple infiltration-redistribution cycles and for Dirac and finite pulse solute applications. Expressions are derived for average moisture content and for average concentration regarding the soil saturated conductivity a random value.

The flow of fluids in heterogeneous porous media is modelled by regarding the hydraulic conductivity as a stationary random space function. The flow variables, the pressure head and velocity field are random functions as well and we are interested primarily in calculating their mean values. The latter had been intensively studied in the past for flows uniform in the average. It has been shown that the average Darcy's law, which relates the mean pressure head gradient to the mean velocity, is given by a local linear relationship. As a result, the mean head and velocity satisfy the local flow equations in a fictitious homogeneous medium of effective conductivity. However, recent analysis has shown that for nonuniform flows the effective Darcy's law is determined by a nonlocal relationship of a convolution type. Hence, the average flow equations for the mean head are expressed as a linear integro-differential operator. Due to the linearity of the problem, it is useful to derive the mean head distribution for a flow by a source of unit discharge. This distribution represents a fundamental solution of the average flow equations and is called the mean Green function Gd (x). The mean head Gd(x) is derived here at first order in the logconductivity variance for an arbitrary correlation function ρ(x) and for any dimensionality d of the flow. It is obtained as a product of the solution Gd(0)(x) for source flow in unbounded domain of the mean conductivity KA and the correction Ψ d (x) which depends on the medium heterogeneous structure. The correction Ψ d is evaluated for a few cases of interest.

Transport of conservative and reactive solute by flow from a recharging well, and by flow between a recharging and a pumping well, is analysed. The macrodispersivity for transport in the radial flow is shown to be smaller by a factor of three than in the uniform flow. The breakthrough curves in the pumping well are determined as functions of time and the heterogeneity of the formation.

Average nonuniform flows in heterogeneous formations are modeled with the aid of the nonlocal effective Darcy's law. The mean head for flow toward source of instantaneous discharge in a heterogeneous medium of given statistics represents the fundamental solution of the average flow equation and is called the Mean Green Function (MGF). The general representation of the MGF is obtained for weakly heterogeneous formations as a functional of the logconductivity correlation function. For Gaussian logconductivity correlation, the MGF is derived in terms of one quadrature in time t and it is analyzed for isotropic media of any dimensionality d and for 3D axisymmetric formations. The MGF is further applied to determining the mean head distribution for flow driven by a continuous source of constant discharge. The large time asymptotic of the mean head is analyzed in details.

Fluid flow and solute transport in an anisotropic, heterogeneous porous medium with mean flow normal to a constant‐flux boundary are considered. The statistical moments of flow and transport variables are determined at second order in log conductivity fluctuation, and they are expressed in terms of the log conductivity variance and integral scales, the mean flow velocity, and the distance from the boundary. The variance of the longitudinal and transverse components of velocity as well as hydraulic head variance and the longitudinal macrodispersivity are analyzed for a bounded medium with axisymmetric, Gaussian log conductivity covariance structure. In this case, all of the moments can be solved by means of a single numerical quadrature. The constant flow boundary increases the variability of head and of flow transverse to the mean flow direction and causes a reduction of the macrodispersivity in a zone adjacent to the boundary. Our results should be useful for the design and testing of numerical models and have important implications for surface infiltration of solutes.