Advances in Water Resources 47 (2012) 31–42

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Advances in Water Resources

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Numerical analysis of solute transport in variably saturated bimodalheterogeneous formations with mobile–immobile-porosity

David RussoDepartment of Environmental Physics and Irrigation Institute of Soils, Water and Environmental Sciences, Agricultural Research Organization, The Volcani Center,Bet Dagan 50250, Israel

a r t i c l e i n f o a b s t r a c t

Article history:Received 3 October 2011Received in revised form 3 May 2012Accepted 10 May 2012Available online 3 July 2012

Keywords:Spatial heterogeneitySolute transportUnsaturated flow

0309-1708/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.advwatres.2012.05.017

E-mail address: vwrosd@agri.gov.il

Considering flow and transport in three-dimensional, variably saturated, composite bimodal heteroge-neous formations, the main purpose of this study was to extend the previous analyses [37], restrictedto the one-region case in which the entire water-filled pore space is mobile, to the two-region case inwhich part of the water-filled pore space of each of the sub-soils of the composite formation is stagnant,and to investigate the effect of the interaction between the mobile and the immobile regions on solutetransport in these formations. Following Russo [37], formations with fine- and coarse-textured embeddedsoils (FTES- and CTES-formations, respectively), were considered in the analyses. Main results of the pres-ent study suggest that mass exchange between the two regions masks features of the transport that existin bimodal, one-region flow domains, related to characteristics of the unsaturated hydraulic conductivityin variably saturated bimodal, heterogeneous formations. In particular, the crossover behavior (i.e., thatunder relatively wet conditions, solute spread is larger in the FTES-formations than in the CTES-forma-tions, while the opposite occurs under relatively dry conditions) characterizing one-region, bimodal flowdomains disappears in two-region, bimodal flow domains. The latter attributes to the transfer of massfrom the mobile region to the immobile region and the extension of the capture zone for the solute par-ticles associated with the fine-textured embedded soil to lower water saturations. Consequently, for bothsteady state- and transient-flows, as water saturation decreases, the response of the composite forma-tions is essentially independent of the texture of the embedded soil.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Soil properties relevant to solute transport such as the hydraulicconductivity of near-surface geologic formations often exhibit aconsiderable spatial heterogeneity ([4,24,39,15,38,42,19], amongothers) that generally is irregular. Based on experimental evidence(e.g., [39,6,47,38]), it is generally assumed that the heterogeneousformation may be viewed as a single population whose propertiesfollow a unimodal distribution and a two-point spatial covariancewith a single, finite length-scale. This assumption may besupported on theoretical grounds, based on the concept of the exis-tence of a discrete hierarchy of length-scales of heterogeneity [9],with disparity between scales. Still, a more general approach is re-quired for the situation in which the disparity between differentlength-scales is not large enough to neglect the variability on onescale when considering the other.

There are two alternative approaches to deal with situations inwhich the spatial variability of the formation properties is charac-terized by a relatively complex correlation structure. The first ap-proach (e.g., [12,31,21,33,34,36,37]), adopted also in the present

ll rights reserved.

study, replaces the spatial arrangement of distinct soil materialswith a single, composite material whose properties are multimodalbut statistically homogeneous. The second approach, the randomdomain decomposition (RDD) model [53–55] identifies the shapeof the soil inclusions and their three-dimensional arrangementprobabilistically and results in non-stationary statistics. The firstapproach, which applies naturally to cases in which the soil mate-rials are not compactly grouped in clusters, has few advantagesover the RDD model due to its simplicity [53]. Unlike the RDDmodel, however, the first approach may not be appropriate whenthe contrast between mean conductivities of the composite forma-tion exceeds few orders of magnitude.

Based on the first-order, Lagrangian-stochastic analysis of va-dose-zone transport [34,35], Russo [37] invesigated the effect ofthe embedded soil’s texture and the mean pressure head (i.e., meanwater saturation), on solute transport in steady-state flows in theseformations. Two distinct variably saturated composite formationsconsisting of relatively low-conductive, fine-textured embeddedsoil with appreciable capillary forces, and high-conductive,coarse-textured embedded soil with relatively weak capillaryforces (will be termed hereafter as FTES- and CTES-formations,respectively), were considered in the analyses. The main results

32 D. Russo / Advances in Water Resources 47 (2012) 31–42

of the first-order analyses, confirmed by three-dimensional numer-ical simulations for more realistic conditions [37], suggested thatfeatures of solute transport in variably saturated, heterogeneousbimodal FTES and CTES formations exhibit a crossover behaviorattributed to the concave nature of the unsaturated hydraulic con-ductivity in these formations.

This crossover behavior means that under relatively wet condi-tions solute spread is larger in the FTES-formations than in theCTES-formations, while the opposite occurs when the formationis relatively dry. Furthermore, the results of the numerical simula-tions [37] suggested that also under transient, non-monotonousflows, the difference between the responses of the FTES-formationsand the CTES-formations decreases substantially, similar to the sit-uation in steady state flows associated with intermediate watersaturations corresponding to the mean pressure head at whichthe crossover occurs.

The analyses of Russo [37], which, considered bimodal forma-tions associated with relatively small volume fraction of theembedded soil, are restricted to limited-contrasts situations; fur-thermore, the analyses focused on the case in which the entirewater-filled pore space is mobile, will be termed hereafter as theone-region case. Near-surface formations, however, may exhibitcomplex-structured features such as clay soils comprised of aggre-gates of small-diameter particles [56,18,14], or sandy soils inwhich the individual sand particles comprising the formation havenon-zero porosity [2].

In these circumstances, the water-filled pores between the soilaggregates and/or the sand particles are viewed as channelsthrough which relatively rapid water flow and solute transportmay take place. In turn, the soil aggregates and/or the sand parti-cles are typically viewed as regions within which the water-filled-pore-space is essentially stagnant and may exchange solutewith the mobile water region by a rate-limiting diffusion process(e.g., [7,52,28,51,40]). The latter case will be termed hereafter asthe two-region case.

Field-scale investigations of solute transport (e.g., the transportof bromacil in transient, vadose zone flow [43] and the transport ofbromide in steady state, groundwater flow [20], suggest that thetransport in these spatially heterogeneous sites is better quantifiedby the two-region, mobile-immobile transport model, than by theclassical, one-region, convection dispersion equation model. Theresults of these studies support the significance of the rate-limitingmass transfer between the mobile and the immobile regions occur-ring in heterogeneous formations.

Considering composite bimodal heterogeneous formations, thepresent study focuses on the case wherein each of the sub-soilsof the composite formation has secondary porosity features (e.g.,microporosity) that may give rise to mobile-immobile behavior.Specifically, the main purpose of this study is to extend the previ-ous analyses of flow and transport in variably saturated, one-region, bimodal heterogeneous formations [37] to the case inwhich part of the water-filled pore space of each of the sub-soilsof the composite formation is stagnant, and to investigate theeffect of the interaction between the mobile and the immobileregions on solute transport in these formations.

The study will be carried out through a series of detailednumerical analyses of flow and transport in a hypothetical, yetrealistic, three-dimensional (3-D) variably saturated, two-region,composite, bimodal heterogeneous flow domain. The approachadopted in the present study, viewed as a ‘‘numerical experi-ment’’, is an efficient tool for studying processes’ mechanismand evaluating the flow system’s response to plausible scenarios.At the price of reduced generality it circumvents most of thestringent assumptions of analytical studies, and, facilitates analy-sis of simplified, yet realistic situations at a fraction of the cost ofphysical experiments

2. Governing partial differential equations

A Cartesian coordinate system (x1, x2, x3, where x1 is directedvertically downwards) which coincides with the principal axesassociated with the principal components of the hydraulic conduc-tivity tensor, is considered here. Taking into account waterextraction by plant roots, the Richards equation that governs flowin a rigid, variably saturated 3-D flow domain is:

@h@t¼X3

i¼1

@

@xiKii

@w@xi

� �� @K11

@x1� Sw ð1Þ

where t is time, w = w(x, t) is the pressure head, h = h(x, t) is thevolumetric water content, Kii = Kii(w,x), i = 1,2,3, are the principalcomponents of taken as a symmetrical tensor of rank two withzero off-diagonal components and Sw = Sw(x, t) is a sink term,representing water uptake by plant roots, given [25,5] as:

Swðx; tÞ ¼ �Reðx; tÞKðw; xÞ½wrðtÞ � wðx; tÞ � pðx; tÞ� ð2Þ

where Re(x, t) is the root effectiveness function, wr is the total pres-sure head at the root-soil interface and p is the osmotic pressurehead of the soil solution.

Neglecting solute uptake by plant roots, the equation governingtwo-region, mobile-immobile transport of a passive solute (tracer)in a variably saturated 3-D flow system is:

@ðhmcmÞ@t

þ @ðhimcimÞ@t

¼X3

i¼1

X3

j¼1

@

@xihmDij

@cm

@xj

� ��X3

i¼1

@ðuihmcmÞ@xi

ð3aÞ

@ðhimcimÞ@t

¼ cðcm � cimÞ ð3bÞ

where hm(x, t) = h(x, t) � him(x) and him(x) are the mobile and theimmobile water contents, respectively; cm(x, t) and cim(x, t) are theresident solute concentrations (expressed as mass per unit volumeof the soil solution) in the mobile and immobile regions, respec-tively; ui (i = 1,2,3) are components of the Eulerian velocity vectorand Dij (i,j = 1,2,3) are components of the pore-scale dispersiontensor given [3] as:

Dij ¼ dijðkT juj þ DmÞ þ ðkL � kTÞuiuj=juj ð4aÞ

where kL and kT are the longitudinal and the transverse pore-scaledispersivities; dij is the Kronecker delta (i.e., dij = 1, if i = j, anddij = 0 if i – j); |u|=(u1

2 + u22 + u3

2)1/2 and Dm is the effective molec-ular diffusion coefficient in the mobile region, given (Millington andQuirk, 1961) as

Dm ¼ D0 h10=3m =h2

s

� �ð4bÞ

where D0 is the molecular diffusion coefficient in water, hs = hs(x) isthe saturated volumetric water content, and c = c(x) is the first-order mass transfer coefficient representing solute diffusionbetween the mobile and the immobile regions, calculated for spher-ical particles [48] as:

c ¼ 15himDim=r2a ð5Þ

where ra is the average radius of the soil particles, him(x) is theimmobile water content and Dim = Dim(x) is the effective solute dif-fusion coefficient in the immobile region (given by (4b) with him

replacing hm and Dim replacing Dm).Regarding the transport equation (3), it should be emphasized

(e.g., [43]) that at the small c limit, c?0, there is no mass transferto the immobile region, i.e., cim = 0 and (3) reduces to the one-region convection–dispersion equation (CDE) with c = cm andh = hm. At the large c limit, c?1, physical equilibrium between

Table 2Estimatesa of the linear cross-correlation coefficients between the soil parameters ofthe van Genuchten [51] model.

Soil parameter logKs logaVG N hs hr

logKs 1.000logaVG 0.501 1.000n –0.200 0.745 1.000hs –0.501 �0.841 0.561 1.000hr –0.700 �0.272 0.740 0.058 1.000

a Adopted from Russo et al. [42].

D. Russo / Advances in Water Resources 47 (2012) 31–42 33

the two regions is reached, i.e., cm = cim and (3) reduces to the one-region CDE with c = cm = cim and h = hm + him.

3. The numerical simulations

The approach adopted here is the ‘single realization’ approach(e.g., [1,32,27,43–46,41,49]. The approach is focused on space-averaged quantities derived from the local, random, three-dimensional flow-controlled attributes associated with a singlerealization of the spatially-heterogeneous formation properties.The procedure is feasible if the extent of the averaging domain ismuch larger than that of the correlation length-scale of the forma-tion heterogeneity [10]. If this requirement is fulfilled, the flowdomain may be considered as ergodic and the statistics of theflow-controlled attributes can be derived from the single realiza-tion statistics.

Using the 3-D simulation model of Russo et al. [43,44], a seriesof numerical simulations of flow and transport in a 3-D, variablysaturated heterogeneous, bimodal flow domain was conducted.The domain setup for the simulations is similar to the one consid-ered by Russo et al. [45]; the flow domain extends over distanceL1 = 5 m in the vertical x1 axis and over distances L2 = 8 m andL3 = 8 m in the horizontal x2 and x3 axes, respectively. The two dis-tinct composite FTES- and CTES-formations analyzed by Russo[37], are also considered in the present study.

In both formations, the volume fraction of the embedded soilwas relatively small (P⁄ = 0.1) and for each of the sub-domains ofthe bimodal formation, the van Genuchten [50] (VG) parametricexpressions were adopted for the local description of the K(w;x)and the h(w;x) relationships. The soil parameters of the VG model,each viewed as a realization of a second-order, stationary randomspace function (RSF), include the saturated conductivity, Ks, thesaturated, hs, and the residual, hr, volumetric water contents andthe parameters a and n, related to the soil pore size distribution.

The immobile water content, him(x), viewed also as a realizationof a second-order RSF, was selected here as the volumetric water

content, h(w,x), at the so-called ‘‘field-capacity’’, i.e., the watercontent of the near surface soil profile at which drainage becomes

negligible, hFC(x), often defined as h(w,x) corresponding to the pres-sure head, wFC = �3.33 m [16]; the corresponding mean relative

conductivities, <K(hFC(x)/Ks(x)>, are 1.8 � 10�3 and 3.3 � 10�4, forthe FTES- and the CTES-formations, respectively. For both forma-tions, a deterministic pore-scale dispersion tensor (with longitudi-nal dispersivity, kL = 2 � 10�3 m and transverse dispersivity,kT = 1 � 10�4m, Perkins and Johnston [26]), and a deterministicaverage radius of the soil particles, ra = 0.125 mm, were consideredin the simulations. The latter value may be considered as a lowerboundary for the large sand particles and/or soil aggregatescontributing to soil structure and drainable pores [8].

Table 1Mean values and standard deviation (SD) of the soil parameters of the van Genuchtencoefficient,b c, for the background soil and the two different embedded soils.c

Soil parameter Background soil Fine-textured em

Mean SD Mean

Ks (md�1) 1.012 1.313 0.029aVG (m�1) 1.023 0.886 0.170n 2.003 0.203 1.500hs 0.401 0.040 0.477hr 0.100 0.020 0.120him 0.203 0.048 0.427c (d�1) 89.96 98.54 964.5

a Given by h(w,x) corresponding to wFC = �3.33 m.b Calculated from (5) for ra = 0.125 mm.c Ks, a and c are log-normally distributed RSFs; n, hs and hr are normally-distributed R

Mean values and standard deviations of the soil parameters ofthe VG model, the immobile water content, him(x), and the masstransfer coefficient, c(x), (5) are given in Table 1; linear cross-correlation coefficients between the various VG parameters aregiven in Table 2. Note (Table 1) that for the two-domain flow systemconsidered here, the fine-textured embedded soil (which consists ofclay aggregates) is associated with considerably larger immobilewater content and mass transfer coefficient as compared with thecoarse-textured embedded soil (which consists of sand particles).

Following Russo [37], the correlation length-scales of logKs

associated with the background soil (Iyv1 = 0.2 m and Iyh1 = 1 m)and with the embedded soil (Iyv2 = 1 m and Iyh2 = 5 m) wereadopted for the other VG parameters. Note that for the parameter’sstatistics employed in this study the correlation length-scales ofthe composite formations [21,34] are Iyv � 2Iyv1 and Iyh � 2Iyh1.The two-stage procedure [11] used to generate discrete numericalrepresentations of the composite, heterogeneous parameters’ fieldsis described elsewhere [45]. Since multi-Gaussian fields tend to ex-hibit low connectivity (e.g., [13]), the hydraulic properties fields inthe present study may be considered as poorly connected fields.

Statistics of the resultant logK(w;x) fields associated with theFTES- and the CTES-formations, expressed in the terms of themean, Y, and the variance, r2

y, of logK(w;x), depicted as functionsof w in Fig. 4 of Russo [37], demonstrate the crossover behaviorthat characterizes the hydraulic conductivity in variably saturated,composite, bimodal, heterogeneous flow systems. This crossoverbehavior is in agreement with the results of the first-order analyses[34,36], and is similar to the crossover between the hydraulic con-ductivities of fine- and coarse-textured, unsaturated, homoge-neous soils [30].

Flow and transport simulations were performed here in order toinvesigate the effect of the stagnant water region within soil aggre-gates and/or sand particles on solute transport in 3-D, variablysaturated, two-region, composite, bimodal, heterogeneous forma-tions. Details of the numerical approach used to approximate theequations governing flow and transport in the aforementionedformations are given in Russo et al. [43,44] and Russo [37]. Forthe solute transport, the numerical scheme (similar to the third-or-

[51] model, Ks, a, n, hs, hr, the immobile water content,a him, and the mass transfer

bedded soil Coarse-textured embedded soil

SD Mean SD

0.031 29.33 31.200.135 5.406 4.3040.145 2.996 0.2960.047 0.358 0.0350.024 0.080 0.0160.026 0.085 0.015210.54 2.052 2.681

SFs; relative variability (SD/Mean) is based on Russo and Bouton [38].

34 D. Russo / Advances in Water Resources 47 (2012) 31–42

der, total-variation-diminishing (TVD) scheme implemented in theMT3D code [57]), originally developed to solve the one-region CDE,was modified in order to solve the two-region, mobile–immobiletransport equation (3) subject to the appropriate boundary andinitial conditions [43].

Following Russo [37], the flow and the transport were simu-lated for both steady state and transient, variably saturated flowregimes. The relatively simple steady state flow regime is pertinentto the flow conditions assumed in the first-order stochastic analy-ses of solute transport [34,35] employed by Russo [37]. The steady-state simulations were performed in order to analyze the effect ofthe texture of the embedded soil (relative to that of the back-ground soil) and the mean pressure head (i.e., degree of water sat-uration) on solute spread and breakthrough; they were designed toidentify the existence of a crossover behavior. The more compli-cated transient flow regime is pertinent to realistic situationsinvolving periodic forcing conditions at the soil surface and wateruptake by plant roots. The transient simulations were designed toanalyze the effect of the texture of the embedded soil on solutespread and breakthrough associated with water saturation distri-butions relevant to realistic conditions.

The simulations were performed for a flow domain associatedwith initial water pressure head, wi, and zero solute concentration.The steady state and the transient flows had originated from a con-tinuous infiltration under a predetermined, spatially uniform,time-invariant pressure head, w0, and a spatially-uniform, periodicinflux, respectively, imposed on the entire soil surface located atthe x1 = 0 boundary. No-flow conditions were assumed for the ver-tical planes of the flow domain located at the x2 = 0 and x2 = L2 andthe x3 = 0 and x3 = L3 boundaries, while unit-head gradient was as-sumed for the lower horizontal plane located at the x1 = L1 bound-ary. For the steady-state flows, wi = w0, while for the transientflows, wi = �2 m. A pulse t0 of a tracer with concentration c0 wasinjected into the flow system via a planar source of dimensions(x12 � x02) � (x13 � x03) located at the soil surface, orthogonal tothe direction of the mean flow. No-transport conditions were as-sumed for the horizontal plane located at x1 = 0 boundary, outsidethe planar source, and for the vertical planes located at the x2 = 0and x2 = L2 and x3 = 0 and x3 = L3 boundaries. A zero-gradient-boundary was specified for the lower horizontal plane located atthe x1 = L1 boundary.

Note that under transient flow conditions, during periods be-tween successive irrigation and/or rain events (associated withan appreciable water extraction by the plant roots), water contentmay decrease below hFC. Consequently, for these flow conditions, theimmobile water content, him, was considered as a time-dependentflow-controlled attribute, given by him(x, t) = Min[hFC(x),h(x, t)]. Inaddition, in order to avoid negative mobile water content,hm(x, t) = h(x, t) � him(x, t), hm(x, t) was selected as hm(x, t)=Min[h(x, t) � him(x, t),hcr], where hcr = 0.01.

Fig. 1. Contours of the distribution of the mobile-region solute concentration (inmolc/m3) in a vertical x1x2-plane of the two-region flow system located at x3 = 4 m,for the steady-state flow scenarios, for 3 different values of the pressure headimposed on the soil surface, w0, �0.1 m (a), �1.2 m (b) and �3.2 m (c). Results aredepicted for the FTES formation. Centroids of the solute plumes are located atx = 2 m. Concentration exaggeration 102�.

4. Results and discussions

In the following, the attention is focused on larger-scale, inte-grated quantities of the solute transport, which provide measuresof the solute plume’s mass, location and spread; they can be ex-pressed in terms of the moments of the spatial distribution ofthe total resident solute concentration, c(x, t) = [hm(x, t)cm(x, t)+him(x)cim(x, t)]/h(x,t) given [10] as:

MðtÞ ¼Z

hðx; tÞcðx; tÞdx ð6aÞ

RðtÞ ¼ 1M

Zhðx; tÞcðx; tÞxdx ð6bÞ

S0ijðtÞ ¼1M

Zhðx; tÞcðx; tÞ½xi � RiðtÞ�½xj � RjðtÞ�dx ð6cÞ

where h(x, t) = hm(x, t)+him(x) is the total volumetric water content,M is the total mass of the solute, R(t) = (R1,R2, R3) is the coordinatevector of the centroid of the solute plume, and S0ijðtÞ (i,j = 1,2,3) aresecond spatial moments.

Another entity of interest, that is addressed in the presentstudy, relevant to the problem of groundwater contamination bysolutes moving through the vadose zone, is the solute mass dis-charge, S(L, t), monitored at a given horizontal control plane (CP)located at an arbitrary vertical distance, L, from the soil surface, ob-tained by averaging the simulated solute mass flux, s1(x, t), over theentire CP, i.e.,

SðL; tÞ ¼Z L22

L21

Z L32

L31

s1ðL; x2; x3; tÞdx2dx3 ð7aÞ

where L22 � L21 and L32 � L31 are the horizontal extents of the CPand s(x, t) is given as:

1

D. Russo / Advances in Water Resources 47 (2012) 31–42 35

s1ðx; tÞ ¼ hmðx; tÞX3

j¼1

D1jðx; tÞ@cmðx; tÞ@x1

þ u1ðx; tÞhmðx; tÞcmðx; tÞ ð7bÞ

4.1. Steady-state flows

Following Russo [37], a series of simulations, each associatedwith different predetermined pressure head, w0, were performed.In each simulation, after a steady-state flow regime wasestablished, a pulse of a passive solute (with concentrationc0 = 10molc/m3) was injected into the flow domain through a4.1 m � 4.1 m planar inlet zone located at the soil surface(x1 = 0), whose origin is located at x02 = 1.95 m and x03 = 1.95 min the x2 and the x3 directions, respectively.

Flow-attributes (i.e., pressure head, w(x) and hydraulic conduc-tivity, K = K[w(x)]) of the bimodal, heterogeneous, steady-stateflow system associated with the FTES- and the CTES-formationspertinent to the present study, were analyzed by Russo [37]. Inthe following, the emphasis is on the effect of the mass exchange

Fig. 2. Contours of the distribution of the mobile-region solute concentration (inmolc/m3) in a vertical x1x2-plane of the two-region flow system located at x3 = 4 m,for the steady-state flow scenarios, for 3 different values of the pressure headimposed on the soil surface, w0, �0.1 m (a), �1.2 m (b) and �3.2 m (c). Results aredepicted for the CTES formation. Centroids of the solute plumes are located atx1 = 2 m.Concentration exaggeration 102�.

between the mobile and immobile regions of the flow domain onsolute transport in variably saturated, two-region, composite, bi-modal, heterogeneous formations. Figs. 1 and 2 display snapshotsof the solute resident concentration in the mobile region, cm(x, t),of the FTES- and the CTES-formations, respectively, for P⁄ = 0.10,R1(t) = 2 m and for different values of w0.

Note that with respect to solute transport, in the one-region(mobile) flow domain analyzed by Russo [37], the disparity be-tween the responses of the flow systems associated with the twoformations is large when the formations are relatively wet or rela-tively dry. At intermediate saturations, however, the disparity be-tween the two formations is substantially reduced, in agreementwith the crossover behavior [37] attributed to the concave natureof the K–w relationships in these formations [34,36,37]. In contrast,in the two-region, mobile-immobile flow domain analyzed here,the crossover behavior is not evident, and the disparity betweenthe flow systems’ responses associated with the two formations(Figs. 1 and 2) decreases as the formations become drier.

The longitudinal component, S⁄11(t), of the spatial covariancetensor, S⁄ij(t) = S’ij(t) � S’ij(0) (i,j = 1,2,3), (Fig. 3) and the scaledmean solute breakthrough curves (BTCs) (Figs. 4), further quantifythe solute transport behavior. Fig. 3 suggests that as expected, theinteraction between the mobile and the immobile regions in thetwo-region flow domain considerably increases the plume spreadin the longitudinal direction, as compared with the one-region flowdomain (the insets in Fig. 3). For both flow domains, however, inthe FTES-formation (Fig. 3a), the rate of increase of S11 withincreasing R1 is a non-monotonous, concave function of w0, reach-ing a minimum at w0 = w’c, while in the CTES- formation (Fig. 3b),the rate of increase of S11 with increasing R1 increases withdecreasing w0. Note that in the two-region flow domain, w’c is

0 0.5 1. 0 1.5 2.00

0.4

0.8

1.2

(b)

# −ψ0(m) 0.1 0.4 0.8 1.2 1.6 2.4 3.2

Dis

plac

emen

t Cov

aria

nce

(m2 )

Travel Distance (m)

0.4

0.8

1.2

(a)

0 1 20

0.4

0.8

0 1 20

0.4

0.8

Fig. 3. Displacement variance of the solute plume in the vertical direction as afunction of the travel distance for the FTES (a) and the CTES (b) formations and fordifferent values of the pressure head imposed on the soil surface, w0. Results aredepicted for the two-region and the one-region (the insets) flow domains.

36 D. Russo / Advances in Water Resources 47 (2012) 31–42

larger and the rate of increase of S11 with decreasing w0 (whenw0 < w’c) is larger than in the one-region flow domain. Conse-quently, in contrast to the one-region flow domain, in the two-region flow domain the disparity between the two formationsdecreases with decreasing water saturation (i.e., smaller imposedw0), in agreement with the plumes depicted in Figs. 1 and 2.

The solute BTCs depicted in Fig. 4, are expressed in terms of thescaled solute mass discharge, S(t,L)(L/M0V1), as a function of thescaled travel time, tV1/L, where L is the distance to the CP, M0 isthe total solute mass injected into the flow system and V1 is thelongitudinal (vertical) component of the effective solute velocityvector, V, calculated as V = [dR1/dt,dR2/dt,dR3/dt]T. Fig. 4 suggeststhat as expected, the interaction between the two regions of theflow domain considerably increases the skewing of the scaled BTCsas compared with the one-region flow domain (the insets in Fig. 4).The increased skewing is expressed by an earlier breakthrough, anenhanced peak arrival and a longer tailing. For both flow domains,in the FTES-formation (Fig. 4a), the skewing of the scaled BTCs is aconcave function of w0, reaching a minimum at w0 = w00c. Contraryto the one-region flow domain, however, in the two-region flowdomain, the skewing of the scaled BTC decreases slightly withdecreasing w0 (when w0 > w00c) and increases substantially withdecreasing w0 (when w0 < w00c). In the CTES-formation (Fig. 4b),similar to the one-region flow domain, the skewing of the scaledBTC increases monotonously with decreasing w0, at a rate fasterin the two-region flow domain than in the one-region flow domain.Consequently, in contrast to the one-region flow domain in whichthe disparity between the BTCs associated with the two formationsis larger when the formations are relatively wet or relatively dry,(with a minimum at intermittent water contents in the vicinity

0 1 2 3 40

0.4

0.8

1.2

1.6 (b)

# −ψ0(m) 0.1 0.4 0.8 1.2 1.6 2.4 3.2

Scal

ed S

olut

e D

isch

arge

Scaled Travel Time

0

0.4

0.8

1.2

1.6 (a)

0 1 2 3 40

0.5

1.0

1.5

0 1 2 3 40

0.5

1.0

1.5

Fig. 4. Scaled solute BTC at an horizontal CP located at L = 1.25 m for the FTES (a)and the CTES (b) formations and for selected values of the pressure head imposedon the soil surface, w0. Results are depicted for the two-region and the one-region(the insets) flow domains.

of w0 = w00c), in the two-region flow domain, the difference be-tween the BTCs associated with the two formations decreases asthe formations become drier.

4.2. Transient flows

Following Russo [37], crop, agricultural practice and climaticconditions, typical of the coastal region of Israel, (a citrus orchardirrigated with sub-canopy micro-sprinklers irrigation system)were taken into account in this set of simulations. Meteorologicaldata from Bet Dagan were used to estimate potential evapotrans-piration rates, esp(t), (with annual amount of ETp =

Resp(t)dt =

830 mm), and to provide rain rates and amounts with annualrainfall amount of 550 mm. In order to obtain different soil waterregimes, three different annual amounts of irrigation water, QI,corresponding to QI/ETp = 1.1, 1.3 and 1.5 (each associated withirrigation’s schedule of irrigation every 3 days and irrigationintensity of 5 mm/hr), were considered here.

Assuming a complete cover of the wetted soil surface by thetrees’ canopy, the assumption of negligibly small soil evaporation(i.e., ETp = Tp, where Tp =

Rsp(t)dt) is the potential transpiration)

was adopted here. The root effectiveness function in (2) was esti-mated from the root distribution data of Mantell and Goell [22]and water uptake by plant roots was implemented by an approachsimilar to the maximization iterative approach suggested by Neu-man et al. [23]; for more details see Russo et al. [44]. Using thesame initially-solute-free flow domain employed in the steady-state simulations (with a uniform initial pressure head, wi = �2 m),flow and transport were simulated starting on April 1st. After fewirrigation events, a pulse of a passive solute (with concentrationc0 = 10molc/m3) was injected into the flow domain through thesame planar inlet zone as in the steady-state simulations.

Flow-attributes of the heterogeneous, transient flow systemassociated with the FTES- and the CTES-formations, pertinent tothe present study, were analyzed by Russo [37]. In the following,the focus is on the effect of the interaction between the mobileand immobile regions of the flow domain on solute transport.Fig. 5 displays snapshots of the solute resident concentrations inthe mobile region for formations associated with the FTES- andthe CTES-formations, respectively, for QI/ETp = 1.5 and R1(t)=2 m.Fig. 5 suggests that the agreement between the concentration’sdistributions associated with the two formations is relatively good,in agreement with the results for the one-region flow domain ana-lyzed by Russo [37]. Similar results were obtained for QI/ETp = 1.1and QI/ETp = 1.3.

The results depicted in Fig. 5 are to be expected based on thesteady-state flow simulations. Results of the latter, suggest thatthe agreement between the responses of the FTES- and the CTES-formations increases with decreasing mean water saturation(when a portion of the water-filled-pore-space is immobile; thecase analyzed here), or when the mean pressure head of the flowsystem approaches the mean pressure head at which a crossoveroccurs (when the entire water-filled-pore-space is mobile; the caseanalyzed by Russo [37]). Consequently, in realistic, transient, non-monotonous flow systems associated with intermittent water sat-urations due to the time-dependent forcing conditions imposed onthe soil surface and water extraction by plant roots, the response ofthe composite formations is expected to be essentially indepen-dent of the texture of of the embedded soil.

Profiles of the horizontally-averaged, mean values and standarddeviations of the solute concentration for the formations associ-ated with the FTES- and the CTES formations, and for the one-and the two-region flow domains, are depicted in Fig. 6 for selectedvalues of QI/ETp; the centroids of the solute plumes associated withthese profiles are located at x1 = 2 m. For both flow domains, for agiven QI/ETp ratio, the agreement between the concentration’s

Fig. 5. Contours of the distribution of the mobile-region solute concentration (inmolc/m3) in a vertical x1x2-plane of the two-region flow system located at x3 = 4 m,for the FTES (a) and the CTES (b) formations. Results are depicted for transient flowassociated with relative amount of applied water, QI/ETp = 1.5. Centroids of thesolute plumes are located at x1 = 2 m.Concentration exaggeration 102�.

5

4

3

2

1

00 0.01 0.02

(a)

Soil

Dep

th (

m)

0 0.01 0.02

Concentration (molc/m3)

(b)

# QI/ETp 1.1 1.3 1.5

5

4

3

2

1

00 0.02 0.04

(c)0 0.02 0.04

(d)

5

0 0 0.004

5

0 0 0.004

5

0 0 0.015

5

0 0 0.015

Fig. 6. Profiles of the horizontally-averaged, mean values and the standarddeviations (the insets) of the solute concentration for the FTES (a,c) and the CTES(b,d) formations and for transient flows associated with selected values of QI/ETp.Results are depicted for the one-region (c,d) and the two-region (a,b) flow domains;the centroids of the solute plumes associated with the profiles are located atx1 = 2 m.

D. Russo / Advances in Water Resources 47 (2012) 31–42 37

profiles associated with the two formations is quite good althoughpeak values associated with the FTES-formation are slightly smal-ler than their counterparts associated with the CTES-formation.

For the one-region flow domain analyzed by Russo [37](Fig. 6c,d), the mean solute concentration profiles exhibit thesymmetrical Fickian distributions typical of the solution of theone-region CDE with constant coefficients, with zero concentrationat the soil surface and concentration peaks that decrease withincreasing QI/ETp. The respective standard deviation profiles (theinsets in Fig. 6c,d) are less symmetric and more irregular with anapparent bimodality which might be explained by the dissipatingaction of the local dispersion [17].

For the two-region flow domain (Fig. 6a,b), the profiles of themean solute concentration in the mobile region are much moreskewed than their counterparts associated with the one-regionflow domain, exhibiting non-zero concentrations at the soil surfacethat decrease with increasing QI/ETp and concentration peaks thatincrease with increasing QI/ETp. The respective standard deviationprofiles (the insets in Fig. 6a,b), are more irregular and exhibit sec-ondary peak close to the soil surface, which, in turn, may be ex-plained [40] by the exchange of mass between the mobile andimmobile regions of the formation combined with transient, non-monotonous water flow originating from the periodic flux imposedon the soil surface and water extraction by plant roots.

Because of the latter, the total solute mass, hmcm + himcim, in theupper part of the soil profile may substantially increase duringredistribution periods and is only partially leached during the infil-tration periods. During the latter periods, inasmuch as the mobilewater content is smaller than the total water content, the solutein the mobile region of the two-region flow domain, may travelfaster than if the soil had contained only a single mobile region.Consequently, the profiles of the mean and the standard deviationof the solute concentration in the mobile region of the two-region

flow domain (Fig. 6a,b) are much more skewed than their counter-parts associated with the one-region flow domain (Fig. 6c,d). Theseresults are in qualitative agreement with the results of the analysesof flow and transport in a variably saturated, two-region, uni-modal, heterogeneous formation (i.e., a formation with soilaggregates and/or sand particles within which the water-filled-pore-space is essentially stagnant, that has no embedded soil withcontrasting conductivities; e.g., [43].

It should be emphasized, that when the contrast between meanconductivities of the embedded and the background soils of thecomposite formation is sufficiently large (i.e., few orders of magni-tude), the one-region bimodal flow system may exhibit featurestypical of a two-region, mobile-immobile behavior (e.g., theconcentration profiles in Fig. 6). The present analyses, however,are restricted to limited-contrast situations (i.e., contrasts of Ks2/Ks1 � 0.033 and Ks2/Ks1 � 30 for the FTES- and the CTES-formations,respectively, Table 1). Furthermore, in variably saturated, hetero-geneous bimodal formations, the value of j = |logK2 � logK1| (andthe logK variance of the composite formation) are concave func-tions of the mean pressure head, W, which exhibit minima at cer-tain critical pressure heads [34,36], and may exceed theircounterparts in saturated flow when the formation further dries,particularly in the CTES-formations. For both the FTES- and theCTES-formations, however, at intermittent water saturations, com-mon in realistic, transient flow systems associated with water up-take by plant roots, j may be quite small. Consequently, in the

0 1 2 3 4 50

0.4

0.8

1.2(b)

# QI/ETp 1.1 1.3 1.5

Scal

ed S

olut

e D

isch

arge

Scaled Travel Time

0

0.4

0.8

1.2(a)

0 1 2 3 4 50

0.5

1.0

0 1 2 3 4 50

0.5

1.0

38 D. Russo / Advances in Water Resources 47 (2012) 31–42

present study, the one-region bimodal flow system did not exhibitfeatures typical of a two-region, mobile-immobile behavior (seethe concentration profiles in the insets of Fig. 6).

The longitudinal component, S⁄11(t), of the spatial covariancetensor (6c) and the scaled solute BTCs associated with the FTES-and the CTES-formations are depicted in Figs. 7 and 8, respectively,for three different values of the relative amount of the irrigationwater, QI/ETp. These figures suggest that as expected, under tran-sient, non-monotonous flow conditions, the exchange of mass be-tween the two regions combined with a time-dependent, mobilewater content, hm(x, t) = h(x, t)-him(x), considerably increase boththe solute plume spread in the longitudinal direction and theskewing of the solute BTCs, as compared with the one-region flowdomain (the insets in Figs. 7 and 8). Furthermore, the results de-picted in Figs. 7 and 8 suggest that similar to the one-region flowdomains [37] and in agreement with the results depicted inFig. 5, under conditions of transient, non-monotonous flow, alsoin two-region flow domains, the disparity between the responsesof the FTES- and the CTES-formations is considerably reduced.

Note, however, that in both formations, the shape of the scaledsolute BTCs in the two-region flow domains (Fig. 8) are much lesssensitive to changes in the QI/ETp ratio than their counterparts inthe one-region flow domains (the insets in Fig. 8). This behaviormay be attributed to water extraction by plant roots, which, inturn, depends on both the soil water pressure head and the soilsolution concentration (2).

In the one-region flow domain, in which the solute is almostcompletely leached from the upper part of the soil profile(Fig. 6c,d), water extraction by the plant roots is essentially not af-fected by the solute concentration; consequently, for the range of

0 0 .5 1. 0 1 .5 2.00

0.4

0.8

1.2(b)

# QI/ETp 1.1 1.3 1.5

Dis

plac

emen

t Cov

aria

nce

(m2

)

Travel Distance (m)

0

0.4

0.8

1.2 (a)

0 1 20

0.2

0.4

0 1 20

0.2

0.4

Fig. 7. Displacement variance of the solute plume in the vertical direction as afunction of the travel distance for the FTES (a) and the CTES (b) formations and fortransient flows associated with selected values of QI/ETp. Results are depicted forthe two-region and the one-region (the insets) flow domains.

Fig. 8. Scaled solute BTC at an horizontal CP located at L = 1.25 m for the FTES (a)and the CTES (b) formations and for transient flows associated with selected valuesof QI/ETp. Results are depicted for the two-region and the one-region (the insets)flow domains.

the QI/ETp values examined, ETa = ETp, (where ETa =Resa(t)dt is

the actual annual evapotranspiration and esa(t) = sa(t) =R

Sw(x; t)dxis the actual evapotranspiration rate) independent of QI/ETp. In thetwo-region flow domain, however, water extraction by plant rootsis affected by the solute concentration residing in the upper part ofthe soil profile (Fig. 6a,b). Consequently, for the range of the QI/ETpratios examined, the actual evapotranspiration, ETa, is smaller thanETp and decreases with decreasing QI/ETp. Therefore, for a given QI/ETp ratio, the relative net applied water, QNAW/QI, where QNAW =QI � ETa is the net applied water, is larger in the two-region flowdomain than in the one-region flow domain, particularly whenthe QI/ETp ratio is relatively small.

4.3. One-region versus two-region flow domains

The results presented in the previous sections suggest that thecrossover behavior, attributed to the concave nature of the K–wrelationships in variably saturated, composite, bimodal, heteroge-neous formations, which characterizes the transport in these for-mations (when they consist of a single, mobile region [37]), isnot apparent when the composite bimodal formations consist ofmobile and immobile regions. In other words, mass exchange be-tween the two regions of the flow domain masks features of thetransport related to the unsaturated hydraulic conductivity of thecomposite formation; this is valid for both steady state and tran-sient, non-monotonous flows.

Further insight regarding this finding, is obtained by referring tothe dependence of the mobile water content and solute mass fluxon the mean pressure head of the composite flow system. Themean mobile water content, hm, depicted as a function of mean

D. Russo / Advances in Water Resources 47 (2012) 31–42 39

pressure head in Fig. 9 (which represent both steady-state andtransient flows, denoted by lines and symbols, respectively) isbased on averaging of the mobile water content over the embed-ded soil volume only (Fig. 9a) and over the entire flow domain(Fig. 9b). Note that in the transient case (associated with three dif-ferent values of QI/ETp) in which hm = hm(x, t), the mean values de-picted in Fig. 9 correspond to elapsed times at which the respectivesolute plumes’ centroids reach the vertical position x1 = 2 m.

Fig. 9 suggests, that in the case of the two-region flow domain,when the formation is relatively wet, the mobile water content,hm = h � him, associated with the embedded soil only (Fig. 9a) ismuch larger in the FTES-formation than in the CTES-formation.However, the disparity between the two formations decreases rap-idly as the mean pressure head, W, decreases and essentiallydiminishes when W < Wc (Wc � �1.5 m). This is in contrast withthe one-region-flow domain (the inset in Fig. 9a) in which thesubstantial difference between the embedded soil’s mobile watercontent, hm = h � hr, associated with the two formations, persistsfor values of W much lower than Wc. Note that although for bothformations the embedded soil volume occupies only 10% of the to-tal flow domain volume, the difference between the one- and thetwo-region flow domains is still evident when the entire compositeflow domain is considered (Fig. 9b).

Solute mass flux (Eq. (7b)) in heterogeneous formations is dom-inated by its advective component, which, in turn, depends on themobile region’s water content, hm, and solute concentration, cm,and on the Eulerian velocity vector related to the hydraulicconductivity through Darcy’s law. The mean values and standarddeviations of the solute mass flux depicted as functions of themean pressure head in Figs. 10 and 11, are based on averaging ofthe solute mass flux over the embedded soil volume only

0 1 2 3 40

0.1

0.2(b)

Mean Pressure Head (m)

Mea

n M

obile

Wat

er C

onte

nt (

m3/m

3)

0

0.1

0.2(a)

FTE CTE

0 1 2 3 40

0.2

0.4

0 1 2 3 40

0.2

0.4

Fig. 9. Mean mobile water content obtained by averaging over the embedded soilvolume only (a) and over the entire flow domain (b) as functions of the meanpressure head. Results are depicted for the two-region and the one-region (theinsets) flow domains for both steady state and transient flows (denoted by lines andthe symbols, respectively), for the FTES and the CTES formations (solid line, fulltriangle and dashed line, open triangle, respectively).

(Figs. 10a and 11a) and over the entire flow domain (Figs. 10band 11b). Note that the mean values and the standard deviationsdepicted in Figs. 10 and 11, represent both steady-state and tran-sient flows (denoted by lines and symbols, respectively); they cor-respond to the elapsed times at which the respective soluteplumes’ centroids reach the vertical position, x1 = 2 m.

Figs. 10 and 11 follow the same pattern and clearly demonstratethat the crossover behavior, characterizing the bimodal heteroge-neous formations associated with one-region mobile flow domain,disappears when a two-region, mobile-immobile flow domain ex-ists. In the latter situation, since the fraction of the total water con-tent which is stagnant, i.e., him/(hm + him), increases as theformation becomes dryer, the exchange of mass between the mo-bile and the immobile regions of the formation decreases boththe solute concentration and the solute mass flux in the mobile re-gion. This is particularly so when the embedded soil is fine-tex-tured associated with considerably larger immobile watercontent and mass transfer coefficient as compared with their coun-terparts associated with the coarse-textured embedded soil(Table 1).

The absence of crossover behavior in the two-region flow do-mains, therefore, is attributed to the fact, that as water saturationdecreases, the hydraulic conductivity of the fine-textured embed-ded soil is not small enough (as compared with the hydraulic con-ductivity of the background soil) to act as a capture zone for thesolute particles. Hence, as water saturation decreases, mass trans-fer from the mobile region to the immobile region of the fine-textured embedded soil dominates the capture zone for the soluteparticles. The latter process may persist for a considerable range ofdecreasing water saturation.

0 1 2 3 4

10-7

10-6

10-5

10-4

10-3

10-2

(b)

Mean Pressure Head (m)

Mea

n So

lute

Mas

s Fl

ux (

mol

c/d

)

10-7

10-6

10-5

10-4

10-3

10-2

(a)

FTE CTE

0 1 2 3 410-9

10-5

10-1

0 1 2 3 410-9

10-5

10-1

Fig. 10. Mean solute mass flux obtained by averaging over the embedded soilvolume only (a) and over the entire flow domain (b) as functions of the meanpressure head. Results are depicted for the two-region and the one-region (theinsets) flow domains for both steady state and transient flows (denoted by lines andthe symbols, respectively), for the FTES and the CTES formations (solid line, fulltriangle and dashed line, open triangle, respectively).

0 1 2 3 410-14

10-10

10-6

10-2

(b)

Mean Pressure Head (m)

Solu

te M

ass

Flux

Var

ianc

e (m

olc/d

)2

10-14

10-10

10-6

10-2

(a)

FTE CTE

0 1 2 3 410-15

10-11

10-7

10-3

0 1 2 3 410-15

10-11

10-7

10-3

Fig. 11. Solute mass flux standard deviation obtained by averaging over theembedded soil volume only (a) and over the entire flow domain (b) as functions ofthe mean pressure head. Results are depicted for the two-region and the one-region(the insets) flow domains for both steady state and transient flows (denoted bylines and the symbols, respectively), for the FTES and the CTES formations (solidline, full triangle and dashed line, open triangle, respectively).

0 1 2 3 4 50

0.5

1.0

1.5

ra=0.062mm, ψFC=-6.67m

ra=0.125mm, ψFC=-3.33m

ra=0.25mm, ψFC=-1.67m

FTE CTE

QI/ETp=1.5

Scal

ed S

olut

e D

isch

arge

Scaled Travel Time

Fig. 12. Scaled solute BTC at horizontal CP located at L = 1.25 m for the FTES and theCTES formations (solid and dashed lines, respectively) for selected values of averageradius of the soil particles, ra and the pressure head at ‘‘field capacity’’, wFC. Resultsare depicted for the transient flow scenario associated with QI/ETp = 1.5.

40 D. Russo / Advances in Water Resources 47 (2012) 31–42

4.4. The dependence of the solute BTC on c(x) and him(x)

Transport in two-region, mobile immobile flow domains isaffected by the mass transfer coefficient, c = c(x), (5) and theimmobile water content, him = him(x). The former depends on boththe average radius of the soil aggregates and/or sand particles, ra,and the immobile him = him(x); in the present study, the latter de-pends on a selected value of the pressure head, wFC. The analysespresented in the previous sections are based on single values ofboth ra (0.125 mm) and wFC (�3.33 m).

It is important, therefore, to analyze the sensitivity of theresponse of the flow system to the values of ra and wFC. For simplic-ity, the average size of the soil aggregate and/or sand particle istreated here as an operationally defined, condition-dependent

Table 3Mean values and standard deviation (SD) of the immobile water content, him, and the massselected values of ra and wFC

Soil parameter Background soil Fine-textured e

Mean SD Mean

ra = 0.0625 mm, wFC = �6.67 mhim 0.162 0.037 0.386c (d�1) 139.21 191.54 2659ra = 0.125 mm, wFC = �3.33 ma

him 0.203 0.048 0.427c (d�1) 89.96 98.53 964.5ra = 0.250 mm, wFC = �1.67 mhim 0.258 0.053 0.454c (d�1) 54.28 41.03 306.3

a Default values.

mathematical artifice; in line with the Laplace equation for capil-lary pressure [3], the values of ra and |wFC| were considered asinversely proportional. Using three different values of ra

(ra = 0.0625 mm, 0.125 mm (default) and 0.25 mm) and the corre-sponding values of wFC (wFC = �6.67 m, �3.33 m (default), and�1.67 m, respectively), and considering the FTES- and the CTES-formations, the transient flow scenario associated with QI/ETp = 1.5 was employed for this purpose. Statistics of him and cassociated with the selected values of ra and wFC, employed inthe sensitivity analysis, are given in Table 3.

The results of the analysis (Fig. 12) clearly demonstrate the ef-fect of the selected values of ra and wFC on the solute breakthrough;i.e., increasing ra and wFC (associated with larger him and smaller c,see Table 3) considerably increases the skewing of the solute BTC.Furthermore, the results depicted in Fig. 12 suggest that the soluteBTC in variably saturated, mobile–immobile, bimodal, heteroge-neous formations associated with intermediate water saturations(attributed to the time-dependent flux imposed on the soil surfaceand water extraction by plant roots), is essentially independent ofthe embedded soil texture for a substantial range of ra and wFC val-ues. The results depicted in Fig. 12 and the results of additionalanalyses (not shown here), based on a single value of ra and differ-ent values of wFC (i.e., in this case the average size of the soil aggre-gate and/or sand particle is being treated as an actual physicalentity), strength the generality of the main finding of the presentstudy.

transfer coefficient, c, for the background soil and the two different embedded soils for

mbedded soil Coarse-textured embedded soil

SD Mean SD

0.035 0.0817 0.0145983.2 6.697 5.980

0.026 0.0847 0.0148210.6 2.052 2.681

0.030 0.0936 0.020343.40 1.050 2.632

D. Russo / Advances in Water Resources 47 (2012) 31–42 41

5. Summary and concluding remarks

Considering variably saturated, composite, bimodal, heteroge-neous formations, the main purpose of the present study was toextend the previous analyses [37] to the case in which part ofthe water-filled pore space of each of the sub-soils of the compos-ite formation is stagnant, and, to investigate the effects of theinteraction between the mobile and the immobile regions of thewater-filled pore space on solute transport in these formations.Following Russo [37], the present analyses consider formationswith relatively small volume fraction of the fine- and coarse-tex-tured embedded soils; furthermore, the analyses are restricted tolimited-contrasts situations. The main results of the present studysummarized as follows:

(i) Similar to unimodal heterogeneous formations associatedwith two-region flow domains (e.g., [43]), in bimodal, heteroge-neous formations associated with two-region flow domains, boththe spread of the solute plume in the longitudinal direction andthe skewing of the solute BTCs are considerably larger as comparedwith their counterparts in bimodal, one-region flow domains, par-ticularly under transient, non-monotonous flow conditionsassociated with a time-dependent mobile water region.

(ii) Mass exchange between the mobile and the immobile re-gions in the two-region, bimodal flow domains masks features ofthe transport which exist in one-region, bimodal flow domains,which, in turn, are related to characteristics of the unsaturatedhydraulic conductivity in variably saturated, composite, bimodalheterogeneous formations. In particular, the crossover behaviorthat characterizes one-region, bimodal flow domains, disappearsin two-region, bimodal flow domains.

The first result attributed to the mass exchange between themobile and the immobile regions of the two-region-flow-domain,which, in turn, may spread solute ahead and behind the soluteplume centroid asymmetrically, and, consequently, enhances boththe solute spread and the skewing of the solute BTC. Regarding thesecond result, in one-region-flow-domains, the embedded soil(which may act as a capture zone for the solute particles) enhancesboth the solute spread and the skewing of its breakthrough whenthe formations are relatively wet or relatively dry (for the FTES-and the CTES-formations, respectively) and has a small effect onthe transport at intermediate water saturations.

In two-region, mobile-immobile flow domains, however, aswater saturation decreases, the hydraulic conductivity of thefine-textured embedded soil ceases to be small enough (in compar-ison with the hydraulic conductivity of the background soil) to actas a capture zone for the solute particles. Consequently, masstransfer from the mobile region to the immobile region of thefine-textured embedded soil takes over as the dominant mecha-nism for the capture zone, and, in turn, may persist for a consider-able range of decreasing water saturation. Therefore, the differencebetween the responses of the FTES- and the CTES-formationsdiminishes with decreasing water saturation. The latter finding isvalid for both steady state and transient, non-monotonous flowsand for a substantial range of the immobile water contents andthe mass transfer coefficients.

It should be emphasized that the present study, as well as theprevious study of flow and transport in variably saturated bimodalheterogeneous formations [37], are restricted to limited contrastssituations, i.e., to composite formations whose bi-modality mani-fest as a mixture of different soil materials (e.g., [29]); these situa-tions, however, are of definite interest in applications. The ‘singlerealization’ approach, adopted here, which, in turn, requires thefulfillment of ergodic conditions, may further restrict the general-ity of the results of the present study. The intensive computationaldemand associated with the relatively large number of 3-D simu-lations needed to establish relationships between quantities of

interest and the mean pressure head (in the steady-state simula-tions) or relative amounts of irrigation water (in the transient sim-ulations) for the two different bimodal formations, restricts thesize of the 3-D flow domain. Consequently, the solute source atthe soil surface extended only over two correlation length-scalesof the composite formation in each of the lateral transverse direc-tions. Therefore, the cases examined and the derived solutions maybe not ergodic and may restrict the generality of the results of thepresent study.

This drawback is compensated in part by the fact that the rela-tively small correlation length-scale of the composite formation inthe vertical direction (Iyv � 0.4 m,L1/Iyv > 10) may allow the solutedisplaced downwards to sample the formation heterogeneity effec-tively. Results of the analyses of Russo et al. [46] suggest that whenL1/Iyv is relatively large, the response of the spatially heterogeneousflow system may be quite robust to the selected realization. Fur-thermore, I would like to emphasize that the aforementioned lim-itations may be compensated in part by the fact that the numericalsimulations performed in this study are fully 3-D (i.e., the fluid par-ticles can circumvent zones of low conductivity laterally in twodirections); they may address real world situations, as they areable to simulate simultaneously complex, spatially heterogeneous(with randomness extended to all parameters, with cross-correla-tion between them) 3-D transient flows in variably saturated, bi-modal formations subject to temporally-variable irrigation and/orrainfall, evaporation and water uptake by roots.

Although the findings of the present study are not the ‘‘groundtruth’’, I believe that they may indicate appropriate trends for theconsequences of characteristics of the soil, the external forces im-posed on the soil surface and the plant for the transport of solutesin near surface, variably saturated, heterogeneous formations forquite realistic conditions. Strictly controlled 3-D simulations ofwater flow and solute transport through heterogeneous, variablysaturated, bimodal formations considering macro-pores, connec-tivity, long-range-correlation and connected channels are requiredin order to further generalize the main findings of this investiga-tion. Furthermore, an important scientific issue that should inves-tigated, is under what circumstances a simplified rate-limiting,diffusion-based transport model, like the two-region, mobile-immobile model employed in the present study, can be substitutedfor a transport model in which the flow through the less conduc-tive soil is explicitly accounted for.

Acknowledgments

This is contribution 014/10 from the Institute of Soils, Waterand Environmental Sciences, the Agricultural Research Organiza-tion, Bet Dagan, Israel. Research was supported in part by a grantfrom the United States-Israel Bi-National Agricultural Researchand Development Fund (BARD). The author is grateful to Mr. AsherLaufer for his technical assistance.

References

[1] Ababou, R. Three-dimensional flow in random porous media. Ph. D. Thesis.Mass Inst of Tech. Cambridge, MA; 1988.

[2] Ball WP, Buehler C, Harmin TC, MacKay DM, Roberts PV. Characterization of asandy aquifer material at the grain scale. J Contam Hydrol 1990;5:253–95.

[3] Bear J. Dynamics of fluids in porous media. New York, N.Y.: Elsevier; 1972.[4] Beckett PHT, Webster R. Soil variability: a review. Soils Fert 1971;34:1–15.[5] Bresler E. Application of conceptual model to irrigation requirements and salt

tolerance of crops. Soil Sci Soc Am J 1987;51:788–93.[6] Byers E, Stephens DB. Statistical and stochastic analyses of hydraulic

conductivity and particle-size in a fluvial sand. Soil Sci Soc Am J1983;47:1072–81.

[7] Coats KH, Smith BD. Dead end pore volume and dispersion in porous media.Soc Pet Eng J 1964;4:73–84.

[8] Collis-George N. Drainage and soil structure – a review. Aust J Soil Res1991;29:923–33.

42 D. Russo / Advances in Water Resources 47 (2012) 31–42

[9] Dagan G. Statistical theory of groundwater flow and transport: pore tolaboratory, laboratory to formation, and formation to regional scale. WaterResour Res 1986;22:120S–34S.

[10] Dagan G. Flow and transport in porous formations. Berlin: Springer-Verlag;1989.

[11] Desbarats AJ. Numerical estimation of effective permeability in sand-shaleformations. Water Resour Res 1987;23:273–86.

[12] Desbarats AJ. Macrodispersion in sand-shale sequences. Water Resour Res1990;26:153–63.

[13] Gomez-Hernandez, Wen XH. To be or not to be multi-Gaussian? A reflectionon stochastic hydrogeology. Adv Water Res 1998;21:47–61.

[14] Guber A, Pachepsky Y, Shein E, Rawls WJ. Soil aggregates and water retention.In: Pachepsky Ya, Rawls WJ, editors. Development of pedotransfer functions insoil hydrology. Amsterdam-London-New York-Tokyo: Elsevier; 2004. p.143–51.

[15] Jones AJ, Wagenet RJ. In-situ estimation of hydraulic conductivity usingsimplified methods. Water Resour Res 1984;20:1620–6.

[16] Jury WA, Gardner WR, Gardner WH. Soil physics. New York, NY: John Wiley &Sons, Inc.; 1991.

[17] Kapoor V, Gelhar LW. Transport in three-dimensionally heterogeneousaquifers, 2. Prediction and observations of concentration fluctuations. WaterResour Res 1994;30:1789–801.

[18] Kay BD, Angers DA. Soil structure. In: Warrick AW, editor. Soil physicscompanion. Boca Raton, FL: CRC Press; 2002. p. 249–95.

[19] Khaleel R, Relyea J. Variability of Gardner’s a for coarse-textured sediments.Water Resour Res 2001;37:1567–75.

[20] Liu GC, Zheng R, Geoffrey GR, Tick JJ, Butler Jr, Gorelick SM. Relativeimportance of dispersion and rate-limited transfer in highly heterogeneousporous media: analysis of a new tracer test at the macrodispersion experiment(MADE) site. Water Resour Res 2010;46:W03524. http://dx.doi.org/10.1029/2009WR00643.

[21] Lu Z, Zhang D. On stochastic modeling of flow in multimodal heterogeneousformations. Water Resour Res 2002;38:1190. http://dx.doi.org/10.1029/2001WR00102.

[22] Mantell A, Goell A. The response of Shamouti orange trees to irrigationfrequency and depth of wetting. Prelim Rep 709, 16pp., Inst of Agr Res. VolcaniCenter, Bet Dagan, Israel, 1972.

[23] Neuman SP, Feddes RA, Bresler E. Finite element analysis of two-dimensionalflow in soils considering water uptake by roots, 1. Theory. Soil Sci Soc Am Proc1975;39:224–30.

[24] Nielsen DR, Biggar JW, Erh KT. Spatial variability of field-measured soil–waterproperties. Hilgardia 1973;42:215–60.

[25] Nimah MN, Hanks RJ. Model for estimating soil water, plant and atmosphericrelationships. 1. Description and sensitivity. Soil Sci Soc Am Proc1973;37:522–7.

[26] Perkins TK, Johnston OC. A review of diffusion and dispersion in porous media.Soc Petrol Eng J 1963;3:70–84.

[27] Polmann DJ, Mclaughlin D, Luis S, Gelhar LW, Ababou R. Stochastic modeling oflarge-scale flow in heterogeneous unsaturated soils. Water Resour Res1991;27:1447–58.

[28] Rao PSC, Jessup RE, Addiscott TM. Experimental and theoretical aspects ofsolute diffusion in spherical and nonspherical aggregates. Soil Sci1982;133:342–9.

[29] Ritzi RW, Dai Z, Dominic DF, Rubin Y. Spatial correlation of permeabilty incross-stratified sediment with hierarchical architecture. Water Resour Res2004;40. http://dx.doi.org/10.1029/2003WR00242.

[30] Roth K. Steady-state flow in an unsaturated two-dimensional, macroscopicallyhomogeneous, Miller-similar medium. Water Resour Res 1995;31:2127–40.

[31] Rubin Y. Flow and transport in bimodal heterogeneous formations. WaterResour Res 1995;31:2461–8.

[32] Russo D. Stochastic analysis of vadose-zone solute transport in a vertical crosssection of heterogeneous soil during nonsteady water flow. Water Resour Res1991;27:267–83.

[33] Russo D. Stochastic analysis of macrodispersion in gravity-dominated flowthrough bimodal heterogeneous unsaturated formations. Water Resour Res2002;38:1114. http://dx.doi.org/10.1029/2001WR00085.

[34] Russo D. Stochastic analysis of macrodispersion in heterogeneous bimodalvariably saturated formations: the effect of water saturation. Water Resour Res2004;40:W09201. http://dx.doi.org/10.1029/2004WR00304.

[35] Russo D. Stochastic analysis of solute mass discharge in gravity-dominatedflow through bimodal heterogeneous unsaturated formations. Vadose Zone J2005;4:939–53.

[36] Russo D. On probability distribution of hydraulic conductivity in variablysaturated bimodal heterogeneous formations. Vadose Zone J 2009;8:1–12.http://dx.doi.org/10.2136/vzj2008.011.

[37] Russo D. First-order and numerical analyses of flow and transport inheterogeneous, bimodal variably saturated formations. Water Resour Res2010;46:W06509. http://dx.doi.org/10.1029/2009WR00830.

[38] Russo D, Bouton M. Statistical analysis of spatial variability in unsaturatedflow parameters. Water Resour Res 1992;28:1911–25.

[39] Russo D, Bresler E. Soil hydraulic properties as stochastic processes: I. Ananalysis of field spatial variability. Soil Sci Soc Am J 1981;45:682–7.

[40] Russo D, Jury WA, Butters G. Numerical analysis of solute transport duringtransient irrigation, II. The effect of immobile water. Water Resour Res1989;25:2119–27.

[41] Russo D, Fiori A. Equivalent vadose zone steady-state flow, an assessment of itscapability to predict transport in a realistic combined vadose zone-groundwater flow system. Water Resour Res 2008;44:W09436. http://dx.doi.org/10.1029/2007WR00617.

[42] Russo D, Russo I, Laufer A. On the spatial variability of parameters of theunsaturated hydraulic conductivity. Water Resour Res 1997;33:946–56.

[43] Russo D, Tauber-Yasur I, Laufer A, Yaron B. Numerical analysis of field-scaletransport of bromacil. Adv Water Resour 1998;21:637–48.

[44] Russo D, Zaidel J, Laufer A. Numerical analysis of flow and transport in a three-dimensional partially saturated heterogeneous soil. Water Resour Res1998;34:1451–68.

[45] Russo D, Zaidel J, Laufer A. Numerical analysis of flow and transport in avariably saturated bimodal heterogeneous porous formation. Water ResourRes 2001;37:2127–41.

[46] Russo D, Zaidel J, Fiori A, Laufer A. Numerical analysis of flow and transportfrom a multiple – source system in a partially saturated heterogeneous soil.Water Resour Res 2006;42:W06415. http://dx.doi.org/10.1029/2006WR00492.

[47] Sudicky EA. A natural gradient experiment on solute transport in a sandaquifer: Spatial variability of hydraulic conductivity and its role in thedispersion process. Water Resour Res 1986;22:2069–82.

[48] Sudicky EA. The Laplace transform Galerkin technique for efficient time-continuous solution of solute transport in double-porosity media. Geoderma1990;46:209–32.

[49] Tseng PH, Jury WA. Comparison of transfer function and deterministicmodeling of area-average solute transport in heterogeneous field. WaterResour Res 1994;30:2051–64.

[50] van Genuchten MTh. A closed-form equation for predicting the hydraulicconductivity of unsaturated soils. Soil Sci Soc Am J 1980;44:892–8.

[51] van Genuchten MTh, Dalton FN. Models for simulating salt movement inaggregated field soils. Geoderma 1986;38:165–83.

[52] van Genuchten MTh, Wierenga PJ. Mass transfer studies in sorbing porousmedia,1. Analytical solutions. Soil Sci Soc Am J 1976;40:473–80.

[53] Winter CL, Tartakovsky DM. Mean flow in composite porous media. GeophysRes Lett 2000;27:1759–62.

[54] Winter CL, Tartakovsky DM. Groundwater flow in heterogeneous compositeaquifers. Water Resour Res 2002;38. http://dx.doi.org/10.1029/2001WR00045.

[55] Winter CL, Tartakovsky DM, Guadagnini A. Numerical solutions of momentequations for flow in heterogeneous composite aquifers. Water Resour Res2002;38. http://dx.doi.org/10.1029/2001WR00022.

[56] Wu L, Vomocil JA, Childs SW. Pore size, particle size, aggregate size and waterretention. Soil Sci Soc Am J 1990;54:952–6.

[57] Zheng C, Wang P. MT3DMS: a modular three-dimensional multi-speciestransport model for simulation of advection, dispersion, and chemicalreactions of contaminants in groundwater systems: documentation andusers guide. Contract Report SERDP-99-1. Army Engineer Research andDevelopment Center, Vicksburg, MS; 1999.