# runoff from heterogeneous small bare catchments during soil surface sealing

TRANSCRIPT

Runoff from heterogeneous small bare catchments

during soil surface sealing

S. Assouline1 and Y. Mualem2

Received 20 September 2005; revised 8 June 2006; accepted 7 July 2006; published 14 December 2006.

[1] The combined effects of areal heterogeneity of the soil hydraulic properties and thesurface seal formation dominate the hydrological response of arid and semiarid watercatchments. Here these two phenomena were simulated to study their mutual role in runoffgeneration in small bare catchments. Seal formation during rainfall was simulatedapplying the dynamic model of Assouline and Mualem (1997). Areal heterogeneity of thesoil was represented by a lognormal distribution of the saturated hydraulic conductivity ofthe initially undisturbed soil, Ks, and by related distributions of the other soil parameters.The runoff hydrograph, at the outlet of a hypothetical bare catchment of 0.5 km2, wascalculated using the cell model of Diskin et al. (1984). Two water catchments types,homogeneous and heterogeneous, with two different soil surface states, unsealed(mulched) and ongoing dynamic sealing, were considered. Four spatial organizations ofthe 10 cells in the heterogeneous catchment, all representing the same discrete distributionof Ks, were studied. Rainfall events of two rainfall intensities, 20 and 40 mm h�1, ofdifferent durations ranging from 10 to 120 min, were applied uniformly over thecatchments to study the effect of rainfall intensity and duration on runoff characteristicsunder the different soils and catchments conditions. The state of the soil surface seal wasfound to be a dominant factor with regard to runoff generation. Relative to the runoffproduced in the homogeneous unsealed catchment under a (40 mm h�1, 45 min) rainfall,the runoff was augmented by a factor of 10 during soil surface sealing and still more, by afactor of 20, when the soil surface was already sealed. On a relative basis the impactof soil sealing on runoff is much more important than that of soil heterogeneity. The effectof areal heterogeneity on runoff seems to depend on both the soil surface condition and therainfall intensity and duration. When the soil is unsealed, the total runoff and the dischargepeak are higher for the heterogeneous catchment. When the soil surface undergoes asealing process, the total runoff and the discharge peak are higher for the heterogeneouscatchment for the lower rainfall intensity of 20 mm h�1. For the 40 mm h�1 rainfall thecatchment relative response was found to depend on the rainfall duration. A higher peakwas obtained in the homogeneous catchment for rainfall durations above 60 min, andmore runoff was produced for rainfall durations above 90 min. The spatial pattern of thecell organization in the heterogeneous catchment is an additional factor affecting thehydrological response. The hydrographs corresponding to each of four patternsrepresenting the same areal heterogeneity displayed differences regarding concentrationtime (which varied between 7 and 28 min), timing of the peak runoff (varying between 77and 146 min), and the peak discharge (varying between 0.41 and 0.57 m3 s�1).

Citation: Assouline, S., and Y. Mualem (2006), Runoff from heterogeneous small bare catchments during soil surface sealing, Water

Resour. Res., 42, W12405, doi:10.1029/2005WR004592.

1. Introduction

[2] Water availability in arid and semiarid regions can beimproved on a local or regional scale by efficient runoffharvesting. However, the efficacy of such water harvesting

systems depends on the accuracy of the predicted hydro-logical response of the water catchments to the localrainfall. Spatial and temporal variability of soil and rainfallproperties influence the infiltration process, and conse-quently, the generation of surface flow [Singh, 1997;Saghafian et al., 1995]. These variables dominate alsothe process of the soil surface sealing [Assouline, 2004].In fact, these two processes, i.e., infiltration and surfacesealing, interact strongly. However, because of their re-spective complexity, each process was generally investi-gated separately, and the quantitative aspects of theirinteractions rarely addressed explicitly. A concise review

1Department of Environmental Physics, Institute of Soil, Water andEnvironmental Sciences, Agricultural Research Organization, VolcaniCenter, Bet Dagan, Israel.

2Seagram Center for Soil and Water Sciences, Faculty of Agriculture,Hebrew University of Jerusalem, Rehovot, Israel.

Copyright 2006 by the American Geophysical Union.0043-1397/06/2005WR004592

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WATER RESOURCES RESEARCH, VOL. 42, W12405, doi:10.1029/2005WR004592, 2006

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of results from studies dealing with the effect of spatialvariability or soil sealing on infiltration and runoff ispresented hereafter.[3] Soil hydraulic properties, and particularly, the satu-

rated hydraulic conductivity, Ks, often present a signifi-cant areal heterogeneity at the field scale [Nielsen et al.,1973; Warrick and Nielsen, 1980; Greminger et al., 1985;Logsdon and Jaynes, 1996]. Several studies have shownthat when average soil properties are used instead ofspatially variable properties, large differences are observedin terms of infiltration rates [Russo and Bresler, 1982;Sivapalan and Wood, 1986; Sullivan et al., 1996; Shin etal., 1998]. Generally, accounting for areal heterogeneityleads to shorter ponding times [Smith and Hebbert,1979; Assouline and Mualem, 2002]. Milly and Eagleson[1988] showed that spatial variability of soil propertiesresulted in decreased cumulative infiltration and increasedsurface runoff. Binley et al. [1989] reported that increasingthe variance of the hydraulic conductivity generallyincreases the peak discharge and the runoff volume.Freeze [1980] found that the representation of a hetero-geneous hillslope by an ‘‘equivalent’’ homogeneous onemay lead to large errors in the characteristics of the runoffhydrograph. Saghafian et al. [1995] have shown thatin watersheds, with spatially distributed soil saturatedhydraulic conductivity, lumped values of Ks underestimatethe peak discharges and the runoff volumes. Seguis et al.[2002] found that hydrographs from heterogeneous catch-ments are very close to those obtained assuming uniformones when rainfall is long and intense. For most otherrainfall events, runoff increases with increased variabilityKs.[4] Other studies have indicated also the importance of

the spatial organization in heterogeneous catchments onrunoff production [Smith and Hebbert, 1979; Hawkinsand Cundy, 1987; Merz and Plate, 1997]. Woolhiser et al.[1996] found that runoff hydrographs are strongly affectedby the specific distribution of the hydraulic conductivity forsmall runoff events, the major impact being obtained forcases when Ks decrease downslope. Seguis et al. [2002]reported that comparing the effect of stochastic and deter-ministic spatial distribution of Ks on runoff generated byhillslopes, outflow was higher when the less infiltrativesurfaces were located downhill, whatever the rainfall andvariation coefficient were. These results are in agreementwith those resulting from the reanalysis suggested byWoolhiser et al. [1996] to the data of Smith and Hebbert[1979].[5] Surface sealing of bare soils exposed to rainfall

strongly affects runoff generation in bare semiarid andarid catchments by reducing infiltration rates [Morin andBenyamini, 1977; Mualem and Assouline, 1991; Agassi etal., 1996]. The progress in modeling infiltration during soilsealing can be found in different reviews [Ahuja andSwartzendruber, 1973; Mualem and Assouline, 1996;Assouline, 2004]. The problem of calculating the infiltrationrate in crusted soils was first addressed by Hillel andGardner [1969, 1970], presuming that a sealed, or crusted,soil can be modeled as a uniform soil profile capped with asaturated thin layer of low permeability and prescribedconstant properties. This approach was applied to simulateinfiltration during the dynamic stage of seal formation, and

time-dependent seal hydraulic conductivity functions wereincorporated in the models [Farrell and Larson, 1972;Whisler et al., 1979; Moore, 1981; Ahuja, 1983; Brakensiekand Rawls, 1983; Vandevaere et al., 1998]. However, inthese studies, the seal layer properties are still arbitrarilychosen.[6] A different approach was developed by Baumhardt et

al. [1990] and Assouline and Mualem [1997]. They addressthe problem of infiltration during seal formation, whileadjoining to the seal layer complete hydraulic propertiesrelated to the characteristics of the specific soil-rainfallsystem. These studies solve numerically the one-dimensionalwater flow equation, also known as Richard’s equation.However, the model of Baumhardt et al. [1990] considersthe seal as a homogeneous layer and still maintains theassumption that its thickness is 0.5 cm, as earlier studiesdid. The model of Assouline and Mualem [1997] considersthe seal layer as a disturbed zone where all of the hydraulicparameters differ from those of the undisturbed zone below.Within this zone, the bulk density is assumed to decreaseexponentially with depth, and the changes in the hydraulicproperties are related to the change in the bulk density[Mualem and Assouline, 1989]. The dynamic model ofAssouline and Mualem [1997] was calibrated and validatedagainst data from laboratory as well as field experiments fordifferent soil and rainfall conditions. It was found wellapplicable for simulation and prediction of infiltration bothunder saturated and unsaturated flow conditions. It was alsofound to account formost of themain factors affecting rainfallinduced structural seal formation [Assouline and Mualem,2000].[7] Assouline and Mualem [2002] have applied the dy-

namic model to evaluate the combined effect of surfacesealing and soil heterogeneity on infiltration. When infiltra-tion in a field with an unsealed soil surface was comparedwith that in a field where surface sealing was occurring, thedifference between the ponding times vanished when fieldheterogeneity was accounted for. When soil surface sealingis considered, field heterogeneity led to a shorter pondingtime and a higher final steady state infiltration rate com-paratively to the case of a homogeneous field. The relativeeffect of field heterogeneity on the infiltration curveincreases with rainfall intensity. These results indicate thatthe combined effects of areal heterogeneity and seal forma-tion can have a significant impact on the hydrologicalresponse of water catchments.[8] The following study is carried out to achieve, for the

first time, a quantitative analysis of the combined effect ofsoil areal heterogeneity and surface seal formation on thehydrological response of small bare watersheds. Since wedecided to focus on small catchments, spatial and temporalvariability of rainfall will not be considered. The objectivesof this study are (1) to simulate the runoff hydrograph fromsmall homogeneous and heterogeneous bare catchmentsexposed to uniform rainfall at constant intensity, both underunsealed and surface sealing conditions, (2) to quantify thecombined effect of areal heterogeneity and surface sealingon the rainfall-runoff relationship for different rainfall con-ditions, while applying the concept that the seal hydraulicproperties are directly evolving from those of the initial soil,and (3) to demonstrate the effect of the spatial organization

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in heterogeneous catchments on the runoff hydrographwhen soil sealing is accounted for.

2. Rainfall-Runoff Modeling

2.1. Conceptual Representation of the Catchment

[9] The catchment is partitioned into subareas of uniformproperties, which are interconnected to reflect the topologyof the main drainage pattern of the catchment. A uniformrainfall of constant intensity and prescribed kinetic energy isapplied to the whole catchment. In the case of a heteroge-neous catchment, different soil properties are assigned to thesubareas randomly or according to any predeterminedspatial organization. Therefore, when soil surface sealingis accounted for, a different seal layer develops at a specificrate in each subarea, depending on its initial soil properties(see physical background and detailed formulations ofMualem and Assouline [1989] and Assouline and Mualem[1997]). Accordingly, the infiltration curve, describing thechanges in the infiltration rates during rainfall, can becomputed for the specific soil properties of every subareain the catchment and the specific characteristics of therainfall. In the case of a homogeneous catchment, allsubareas are characterized by the same soil properties,representing the mean values of the distributions appliedfor the heterogeneous catchment. Consequently, in the lattercase, a uniform seal layer develops at the same rate in allsubareas, and the same infiltration curve applies for thewhole catchment. For both the homogeneous and theheterogeneous catchments, the direct surface runoff in eachsubarea is determined, and routed through the channels ofthe drainage network to the outlet. Runon within thesubareas is not accounted for.

2.2. Areal Heterogeneity of the Soil HydraulicProperties

[10] The soil hydraulic properties are expressed in termsof the models of Brooks and Corey [1964] for the retentioncurve and Mualem [1976] for the hydraulic conductivityfunction. Thus the soil hydraulic properties are character-ized by five constants: the saturated hydraulic conductivity,Ks, the saturated water content, qs, the residual watercontent, qr, the air entry value, ya and the pore sizedistribution parameter, l. In the unsealed heterogeneouscatchment, the hydraulic properties vary in space but areuniform along the vertical profile. The areal heterogeneityof the catchment is represented as in the work of Assoulineand Mualem [2002]. Lognormal probability distributionfunction, f(Ks), is applied for the saturated hydraulic con-ductivity, Ks:

f Ksð Þ ¼ 2p0:5Ks

� ��1exp� ln Ksð Þ � ln K*s

� �h i2=2

� �ð1Þ

with Ks* denoting the mean of f (Ks).[11] The other soil parameters, qs, qr, ya, and l, are

related to Ks, according to the following relationships:

ya ¼ y*a Ks=K*s

� �0:23ð2Þ

l ¼ l* Ks=K*s

� ��0:17ð3Þ

qs ¼ 0:36� 0:0165 log Ksð Þ; qs K*s

� �¼ q*s ð4Þ

qr ¼ �0:004� 0:010 log Ksð Þ; qr K*s

� �¼ q*r ð5Þ

where qs*, qr*, ya* and l*, the mean values of the respectivedistributions, are equal to the values of the correspondingparameters representing the undisturbed soil of thehomogeneous catchment. The constants in equations (4)and (5) correspond to Ks units in cm s�1. The distributionsf(qs), f(qr), f(ya), and f(l) are derived by adjoining thecalculated probability f(Ks) from equation (1) for each Ks

value to the corresponding soil parameter values calculatedfor that Ks by equations (2)–(5).[12] During soil surface sealing, the soil parameters vary

both in time and in depth. Hence the seal formation leads toone more dynamic dimension of heterogeneity. In such acase, when areal heterogeneity is accounted for the initialstate of the water catchment, the soil parameters vary withtime to be spatially distributed, a fact that significantlyaugments the complexity of the solution.

2.3. Seal Layer

[13] The dynamic model of Assouline and Mualem[1997] relates surface sealing to the initial soil mechanicaland hydraulic properties, as well as the physical character-istics of the rainfall. At the soil surface, the soil disturbanceresulting from the raindrop impacts is expressed in terms ofthe bulk density increase, Dr0, taken as a function ofrainfall intensity, I, and time of exposure to rainfall, t:

Dr0 I ; tð Þ ¼ Dr*0 1� e�x Ið Þt� �

ð6Þ

where Dr0* is the maximal increase reached after a longexposure to rainfall, and x, a soil-rainfall characteristic. Theexpression of x in equation (6) is given by

x ¼ 6wkI

dmaxDr*t ri;y ið Þ

Z dmax

0

d2f d; Ið Þdd ð7Þ

where f(d,I) is the raindrop size distribution, dmax, themaximal drop diameter, k, a rainfall parameter interrelatingraindrop velocity and drop diameter d, w, a fittingparameter, and t(ri, y i), the initial soil shear strength.Below the soil surface, the bulk density, r(I,h,t), decreasesexponentially with depth, h, within the disturbed zone[Mualem and Assouline, 1989]:

r I ; h; tð Þ ¼ ri þDr0 I ; tð Þe�g Ið Þh; g Ið Þ ¼ �ln 10�3� �

=dc Ið Þð8Þ

where ri is the initial soil bulk density, and g(I) is directlyrelated to the seal thickness, dc(I). When the seal layer iscompletely formed, Dr0(I,t) = Dr0* for t � 0 and for everyI. The hydraulic properties of the seal layer, namely, theretention curve qc(y , r), and the conductivity function,Kc(y , r), are related to the bulk density, r(I,h,t), of theseal layer according to the relationships presented inAppendix A. Once the bivariant functions qc(y , r) andKc(y , r) are determined, a well-defined flow system is

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attained which allow the application of the flow equationunder sealing or sealed conditions.

2.4. Infiltration Curve

[14] The solution of the flow problem relies on thenumerical solution of Richard’s equation governing tran-sient, one-dimensional vertical water flow in saturated-unsaturated soil for specific initial and boundary conditions:

C yð Þ @y@t

¼ @

@ZK yð Þ @y

@Zþ 1

�� �; C yð Þ ¼ @q yð Þ

@yð9Þ

[15] The finite difference scheme applied for the numer-ical solution of equation (9) was found to be stable andaccurate to simulate wetting processes in sealed and un-sealed soils [Mualem et al., 1993; Assouline and Mualem,2001].[16] To simplify the numerical solution when soil sealing

is considered, a uniform equivalent layer is applied toreplace the nonuniform seal, thereby generating a twouniform layers flow system. For the undisturbed soil (Z ��dc), the flow equation (equation (9)) is solved with thehydraulic properties of the undisturbed soil. For the uniformseal layer (0 > Z > �dc), it is solved with the equivalent sealfunctions, qc(y) and Kc(y) (Appendix A). The effect of thissimplification of the flow system on the simulated infiltra-tion curve has been discussed by Assouline and Mualem[2001]. The conditions for the solution of equation (9) arepresented in Appendix B.[17] When the catchment is homogeneous, the initial soil

properties for all subareas are characterized by the constantmean values Ks*, qs*, qr*, yo* and l*. The computedinfiltration for the whole catchment, in this case, is obtaineddirectly from the solution of equation (9). When thecatchment is heterogeneous, the procedure follows the stepspresented by Assouline and Mualem [2002].[18] 1. Transform the continuous f(Ks) (equation (1)) into

a discrete distribution ofKsi values by truncating f(Ks) at Ks =10 Ks* and dividing it into N different intervals (i = 1, 2,.., N),delimited by (N + 1) points (n = 1, 2,. . .., N + 1). TheKsi value characterizing every interval (or each uniformsubarea in the catchment) is the arithmetic mean of theKs values at the respective limits of each interval. The corre-sponding relative weight, mi, for each subarea i is given by

mi ¼ Fi=XN

i¼1Fi with Fi ¼

ZKsnþ1

Ksn

f Ksð ÞdKs andXN

i¼1mi ¼ 1

ð10Þ

[19] 2. Generate the corresponding discrete distributionsof qs, qr, ya, and l, according to Equations (2)–(5) andusing the discrete distribution of Ks.[20] 3. Solve equation (9) for each uniform subarea in the

catchment.

2.5. Runoff Hydrograph

[21] The cell model version of Diskin et al. [1984], basedon the model of Diskin and Simpson [1978], is applied tocompute the runoff hydrograph at the outlet of the catch-ment. In the model, each cell represents a uniform subarea.All the cells are interconnected to form a branching struc-

ture reproducing the main routing network of the catchment.Two types of cells are recognized: (1) exterior cells at theboundary of the catchment, which have only rainfall excessas input, and (2) interior cells having both rainfall excessand channel inflow as inputs. Each cell simulates theprocesses producing the outflow at the outlet of the naturalsubarea considered. This outflow forms the channel inflowinput for the next downstream cell. For an interior cell withmore than one upstream cell, the channel inflow is the sumof the outflows of the upstream cells.[22] The rainfall excess input to each cell is transformed

into the corresponding output by routing it through a pair oflinear reservoirs in series, which represent overland andchannel flow in the subarea. For a given cell, the constant ofone reservoir, k, is assumed to be proportional to the squareroot of the ratio between the cell area and the mean area ofall the cells in the catchment. Furthermore, the constant ofthe second reservoir is assumed to be (k/10). Therefore thecorresponding impulse response function for transformingthe excess rainfall input to each cell, GR(t), is given by

GR tð Þ ¼ e� t=kð Þ � e� 10 t=kð Þh i

=0:9k ð11Þ

where t is the time from start to input.[23] The total channel inflow input to each interior cell is

routed in the cell by a lag and route procedure based on amodel consisting of a linear channel and a linear reservoir inseries. The channel input is first delayed by a time delay, d,and then routed through the reservoir with a constant t.Therefore the corresponding impulse response function fortransforming the channel input of each cell,GC(t), is given by

GC tð Þ ¼ t�1e� t�dð Þ=t½ ; t > d

GC tð Þ ¼ 0 ; t � dð12Þ

[24] The total output of each cell is equal to the sum ofthe two outputs, which are derived from the rainfall excessfunction and from the channel inflow function by convolu-tion with their impulse response functions, equations (11)and (12), respectively.[25] The excess rainfall in a given cell is determined by

the difference between the applied rainfall and abstractions.In the cell model, two types of abstractions are considered:(1) the initial abstraction, aP, accounting for infiltratedwater until ponding and surface storage, expressed in [L],and (2) a constant rate abstraction, aS, accounting for thesteady infiltration rate, expressed in [L T�1]. Accordingly,the infiltration curve is approximated by an equivalent stepfunction that conserves the cumulative infiltration duringthe whole rainfall event. In this study, surface storage isneglected. Consequently, aP represents the cumulative in-filtration until ponding time and aS, the constant infiltrationrate that conserves the cumulative infiltration during theremaining part of the rainfall event. However, unlike theversion of Diskin et al. [1984], where aP and aS areconstants for all the cells, the applied cell model is modifiedto account for the catchment’s heterogeneity. Specific aP

and aS are determined for each cell, according to thecorresponding infiltration curves that result from the areal

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distribution of the soil and the seal hydraulic properties inthe catchment.

3. Model Application

[26] The cell model is applied for a hypothetical smallbare catchment of a total area, A, of 0.5 km2, which canrepresent small water harvesting catchments in arid andsemiarid regions. The catchment was divided into N =10 cells, with a mean cell area, AM, of 0.05 km2. Thebranching structure of the catchment is shown in Figure 1(pattern 1). The specific area, Ai, of each cell i, is set equalto (miA) to infer to the catchment the spatial variabilityresulting from the discrete distribution of Ks (equation (10)and Table 1). The routing parameters of each cell aredetermined according to its area, Ai, based on the relation-ships suggested by Diskin et al. [1984]:

ki ¼ kM Ai=AMð Þ0:5 ð13Þ

di ¼ dM Ai=AMð Þ0:5 ð14Þ

ti ¼ tM Ai=AMð Þ0:5 ð15Þ

[27] The catchment parameters kM, dM and tM were takenequal to 0.80 h, 0.1 h, and 0.5 h, based on an attempt toapply the cell model to a small loess bare catchment of0.8 km2 prone to soil surface sealing [Dody, 1986].[28] The initial soil properties of the catchment were

taken similar to those of the Atwood silty clay loam soil,for which the dynamic seal model was previously calibrated[Assouline and Mualem, 1997]. The soil parameters neededto define the retention curve and the hydraulic conductivityfunction of the initial undisturbed Atwood soil and thecorresponding seal layer, are presented in Table 2. Rainfallwas applied uniformly over all the catchment at two rainfallintensities, 20 and 40 mm h�1. Different rainfall durations,ranging from 10 to 120 min. were simulated. Two catch-ment types, homogeneous and heterogeneous, with twodifferent soil surface conditions, unsealed (mulched) anddynamic soil sealing, were considered. In the heterogeneouscatchment case, three additional patterns of the branchingstructure of the catchment (Figure 1), representing differentdeterministic spatial organizations of the same discretedistribution of Ks, were also considered.

4. Results and Discussion

4.1. Effect of Seal Formation on Runoff

[29] The effect of soil surface conditions on the hydro-logical response of a small homogeneous bare catchmentis expressed in terms of the cumulative runoff generatedfor the case where 30 mm of rainfall with the intensityI = 40 mm h�1 are applied to a catchment characterized by apattern 1 network (Figure 1). Rainfall duration of 45 min isconsidered to represent the mean lifetime of convectivestorms cells [Harrold, 1973].The results are depicted inFigure 2. The unsealed (mulched) catchment produces verylittle runoff (0.8 mm when expressed as runoff discharge perunit area of the catchment). When a seal layer develops at thesoil surface during rainfall, runoff production is increasedmore then tenfold (9.1 mm). By comparison, for the casewhere the soil surface has been already sealed (by previousrain events), the runoff is still higher (17.8 mm), almosttwice the amount produced during the seal formation with a

Figure 1. Four patterns representing different spatialorganizations of the cells in the heterogeneous catchment.

Table 1. Initial Saturated Hydraulic Conductivity and the Relative

Area of Each Cell in the Watershed

Cell Number i Ksi, mm h�1 Ai/A

1 1.0 0.122 3.0 0.183 5.0 0.154 7.0 0.125 9.5 0.116 13.0 0.107 17.5 0.088 25.0 0.079 40.0 0.0510 60.0 0.02

Table 2. Parameters of the Hydraulic Properties of the Atwood

Silty Clay Loam and of the Final Seal Based on the Data of

Baumhardt [1985] and on the Calibration of the Dynamic Model

for I = 40 mm h�1a

Parameter Value

rs, Mg m�3 2.65rb, Mg m�3 1.40Ks*, mm h�1 7.0qs*, m

3 m�3 0.42qr*, m

3 m�3 0.02ya*, m �0.30l* 0.158Dro*, Mg m�3 0.40dc, m 0.071Ksc, mm h�1 0.42qsc, m

3 m�3 0.397qrc, m

3 m�3 0.021yac, m �0.35lc 0.142

aFor final seal, see equations (1) to (8) and Appendix A. For calibrationof the dynamic model, see Assouline and Mualem [1997].

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runoff coefficient (the ratio between runoff and rainfall) of0.59. Generally speaking, it is well known that soil surfacesealing or the presence of a seal layer over a bare watershedaffect infiltration. These results provide a quantitative esti-mate of the relative effect of soil surface condition on theexpected runoff production, and additional evidence why itcannot be neglected when it comes to runoff prediction orwater harvesting estimation and design.[30] The combined effect of seal formation and catchment

heterogeneity on runoff generation is expressed in terms ofthe runoff hydrograph resulting from the application of30 mm of rainfall at I = 40 mm h�1 to a small bare catch-ment characterized by a pattern 1 network (Figure 1). Theresults are shown in Figure 3 for the homogeneous (lumpedsoil hydraulic properties represented by their mean values inall the cells) and for the heterogeneous (distributed soilhydraulic properties) catchments. For both cases, two soilsurface conditions were considered: unsealed (mulched) andwhen soil sealing is taking place. The results indicate thatcatchment heterogeneity has a significant effect on thecharacteristics of the runoff hydrograph. For the unsealedsoil case, the peak discharge in the heterogeneous catchmentis 0.20 m3/s, while it is only 0.04 m3/s for the homogeneouscatchment case. Also, runoff appears earlier at the outletwhen the distributed approach is applied. In terms ofcumulative runoff, 4.2 mm of runoff are produced in theheterogeneous catchment compared to the 0.8 mm amountgenerated in the homogeneous one. Therefore, assuming ahomogeneous catchment may lead to a large error on runoffestimates, in agreement with the results of Freeze [1980]and Saghafian et al. [1995]. When soil sealing is accountedfor, the heterogeneity effect is reduced. Runoff appears atthe outlet practically at the same time, for both the homo-geneous and the heterogeneous catchment. More runoff isstill generated in the heterogeneous catchment (10.8 mm),but the difference with the result of the lumped approach(9.1 mm) is small and less dramatic than in the case of theunsealed soil surface. An interesting point, revealed by theresults in Figure 3, is the fact that in heterogeneous smallbare catchments prone to soil sealing (which is likely to bethe case under natural conditions) neglecting soil heteroge-neity may not affect too much the estimated hydrologicalresponse of the catchment while neglecting soil sealing will

introduce a significant error in runoff hydrograph and totalrunoff predictions.[31] To quantify the combined effect of soil surface

sealing and catchment heterogeneity on the runoff hydro-graph, two indexes, the variability effect index (IV) and thesoil sealing effect index (IS) are defined:

IV tð Þ ¼ HD tð Þ � HL tð Þ½ =HLp ; IS tð Þ ¼ HS tð Þ � HM tð Þ½ =HMp

ð16Þ

where HD and HL are the values of the hydrograph for thedistributed and the lumped approaches respectively; HLp

represents the peak discharge resulting from the lumpedapproach; HS and HM are the values of the hydrograph forthe sealing and the mulched (unsealed) soil surfaceconditions respectively; and HMp represent the peakdischarge for the mulched case. The variations of the IV (t)and IS(t) curves corresponding to the runoff events ofFigure 3 are shown in Figures 4a and 4b. Considering thevariability effect (Figure 4a), IV (t) for the unsealed soilsurface is positive for the whole runoff event, and presents abell shape with a peak of 3.8. When soil sealing isaccounted for, IV (t) oscillates around zero, with a negativepart at the rising limb of the hydrograph, and a positive partfor the peak discharge, and becomes rather negligible for thefalling limb of the hydrograph. Considering the soil sealingeffect (Figure 4b), IS (t) is always positive, and is alsoalways higher when the lumped approach rather than thedistributed one is applied. The respective peak values are9.8 and 1.6, and the maximum effect of soil sealing in ahomogeneous catchment is sixfold the effect of a heter-ogeneous one. These results indicate that under soil surfacesealing conditions, the impact of catchment heterogeneityon runoff is reduced. However, they also indicate that,independently of the degree of heterogeneity of a catch-ment, the impact of soil surface sealing, when and if thisoccurs, on runoff is a significantly more important factor.

4.2. Effect of Rainfall Intensity and Duration

[32] The effect of rainfall intensity on the hydrologicalresponse of homogeneous and heterogeneous catchments

Figure 2. Cumulative runoff versus time for threesurface conditions in the case where 30 mm of rainfallat I = 40 mm h�1 are applied to the homogeneous (lumped)pattern 1 catchment.

Figure 3. Runoff hydrographs resulting from the applica-tion of 30 mm of rainfall at I = 40 mm h�1 to a pattern 1catchment for the homogeneous (lumped) and the hetero-geneous (distributed) cases and the unsealed and the surfacesealing conditions.

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(pattern 2 in Figure 1) during seal formation are presented inFigure 5a for the case where 30 mm of rainfall are applied atI = 20 mm h�1 and I = 40 mm h�1. For both catchments,higher rainfall intensity produces more runoff, that occurssooner and reaches sooner a higher peak discharge. For bothrainfall intensities, the lumped approach leads to less runoffthan the distributed one. However, the difference betweenthe hydrographs resulting from the two approaches dimin-ishes for the higher rainfall intensity.[33] The effect of rainfall intensity on the impact of

catchment heterogeneity in terms of the variability effectindex IV (equation (16)) corresponding to the runoff eventsof Figure 5a is shown in Figure 5b. Except for the earlystages, the IV (t) values for I = 20 mm h�1 are higher thanthose for I = 40 mm h�1, due to the fact that runoff occurredsooner for the higher intensity. Also, the fluctuating shapeof IV (t) remains similar for the two rainfall intensitiesalthough the amplitude of the fluctuations is much lesspronounced for the higher one. The IV (t) corresponding to adifferent spatial organization of the cells in the catchment(pattern 1) and corresponding to the runoff events inFigure 3 is also depicted in Figure 5b. The effect of thespecific structure of the catchment’s heterogeneity is clearlyapparent, as IV (t) corresponding to the pattern 1 networkpresents opposite trends to IV (t) related to the pattern 2network. This indicates that, from an engineering point ofview, applying lumped hydraulic properties to a heteroge-

neous catchment can lead to either overestimation or under-estimation of the timing of peak discharge depending on thespecific spatial organization of the catchment.[34] Rainfall duration is a key variable when it comes to

evaluate rainfall-runoff relationships. The effect of rainfallduration was investigated by applying the two rainfall in-tensities for various durations, ranging from 10 to 120 min.The results in terms of the runoff coefficient (the ratiobetween runoff and rainfall) are depicted in Figure 6a, forthe soil surface sealing case. The general trend is that runoffcoefficients increase with rainfall intensity and duration,although the rate of increase diminishes with the rainfallduration. Basically, the distributed approach leads to higherrunoff coefficients than the lumped one, with the differencebeing maximal for the short rainfall durations. This reflectsthe effect of heterogeneity on the ponding time, at whichrunoff is initiated, when soil surface sealing is accounted for[Assouline and Mualem, 2002]. For the low rainfall inten-sity, the difference between the two approaches diminishesgradually with the duration of rainfall, but the runoffcoefficient for the heterogeneous catchment remains higherthan that for the homogeneous one for the whole range ofduration considered. The higher rainfall intensity acceleratesthe reduction of the difference between the two approaches

Figure 4. (a) Variability effect index (IV) versus time forthe unsealed and the surface sealing conditions based on therunoff hydrographs depicted in Figure 3. (b) Soil surfacesealing effect index (IS) versus time for the homogeneous(lumped) and the heterogeneous (distributed) catchmentsbased on the runoff hydrographs depicted in Figure 3.

Figure 5. (a) Runoff hydrographs resulting from theapplication of 30 mm of rainfall at I = 20 mm h�1 andI = 40 mm h�1 to a pattern 2 catchment for the homoge-neous (lumped) and the heterogeneous (distributed) casesduring seal formation. (b) Variability effect index (IV) versustime for two rainfall intensities and two patterns ofcatchment networks based on the runoff hydrographsdepicted in Figure 3 and Figure 5a.

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and the two curves intersect at a given duration fromwhich the homogeneous catchment is producing morerunoff than the heterogeneous one. For the conditions ofthis simulation, the duration at which this intersection occursis 90 min. The peak discharges versus rainfall duration areshown in Figure 6b. For the investigated range of rainfalldurations, peak discharges increase practically linearly withrainfall duration, the slope being steeper as rainfall intensityis higher. Here too, the maximal difference between the twoapproaches occurs at the short durations. For the low rainfallintensity, higher peak discharges are obtained when thedistributed approach is applied for all rainfall durations.However, for the higher rainfall intensity, an intersection isobtained at the 60 min duration, from which higher peakdischarges resulted from the homogeneous catchment.[35] The impact of the soil surface condition on the effect

of rainfall duration is illustrated in Figure 7 for the higherrainfall intensity. In terms of runoff coefficients (Figure 7a),it appears that the intersection between the runoff coefficientcurves of the lumped and the distributed approaches doesnot occur for the unsealed soil case. In this case, thecalculated runoff coefficients are significantly higher forthe distributed approach, and consequently more runoff isgenerated by this approach for the whole duration rangeinvestigated. A similar result is obtained when peak dis-charges are considered (Figure 7b).[36] These results can explain some of the nuances in

reported observations on the effect of catchment heteroge-neity on runoff. The more common conclusion is that peakdischarge and runoff volume generally increased withvarying hydraulic conductivity [Milly and Eagleson, 1988;Binley et al., 1989; Saghafian et al., 1995]. On the other

hand, Seguis et al. [2002] linked the effect of heterogeneityon runoff to rainfall duration, stressing that hydrographsfrom heterogeneous catchments are very close to thoseobtained assuming uniform ones when rainfall is long andintense. These conclusions seem to be conflicting, at leastfor the long and intense rainfall conditions. However, theresults in Figures 6 and 7 show that soil surface conditionscan affect the impact of heterogeneity on runoff character-istics, which may reconcile the contradictory conclusions.The results for the lower rainfall intensity are in agreementwith the previously stated common conclusion, indepen-dently of the rainfall duration and the soil surface condi-tions. For the higher rainfall intensity, which can beconsidered as simulating an intense rainfall event, thiscommon conclusion seems to be valid only for the unsealedsoil case. The conclusion of Seguis et al. [2002] wouldcorrespond more to the case where soil surface sealingoccurred. Considering that their study was related to thesemiarid zone of the Sahel, where soils are highly sensitiveto seal formation [Hoogmoed and Stroosnijder, 1984; Perezet al., 1999], it is very likely that their observations reflectthe combined effect of soil sealing and catchment hetero-geneity on runoff production.

4.3. Effect of the Spatial Organization of the Cells inthe Catchment

[37] The four arbitrarily chosen patterns of drainage net-works (Figure 1) illustrate a few of the possibilities fordifferent spatial organizations that can still be characterizedby the same statistical distribution of the soil properties overthe catchment. The runoff hydrographs corresponding toeach of these patterns, resulting from the application of

Figure 6. (a) Runoff coefficient and (b) peak dischargeversus rainfall duration for the homogeneous (lumped) andthe heterogeneous (distributed) pattern 2 catchments duringseal formation for the two rainfall intensities.

Figure 7. (a) Runoff coefficient and (b) peak dischargeversus rainfall duration for the homogeneous (lumped) andthe heterogeneous (distributed) pattern 2 catchments andfor the unsealed and the surface sealing conditions, for I =40 mm h�1.

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30 mm of rainfall at I = 40 mm h�1 where seal formation istaking place, are depicted in Figure 8. Large differences canbe observed in all hydrograph parameters, i.e., time ofrunoff occurrence at the outlet, time to peak discharge,and peak discharge. For these four patterns, the time ofoccurrence of the runoff varies between 7 to 28 min, thetime to peak discharge varies between 77 to 146 min, andthe peak discharge varies between 0.41 and 0.57 m3/s. Theeffect of the representation of the variability in the hetero-geneous catchment on the parameters of the runoff hydro-graph is not limited only to the surface sealing case and it isobserved also for the unsealed soil case (results not shown).[38] The hydrograph corresponding to the homogeneous

catchment with soil sealing conditions in Figure 3, whichresults from applying the infiltration curve obtained for theaverage hydraulic properties to all the cells in the catch-ment, fit within the range defined by the four hydrographsin Figure 8. Although this range represents only 4 realiza-tions of a larger number of possible spatial organizations,the fact that the homogeneous hydrograph fit within therange may lead to the conclusion that in catchments wheresurface sealing occurs, the lumped approach may serve as afirst approximation of the hydrological response of hetero-geneous catchments. This would not be valid for the casewhere the unsealed soil surface condition prevails.

5. Summary and Conclusions

[39] The present study of the combined effect of soilsurface conditions and areal heterogeneity of soil propertieson runoff from small bare catchments yields some conclu-sions that can contribute to improve the conception ofrunoff generation, and the design of water harvestingsystems. Soil surface condition, and explicitly here sealformation or the presence of a crust layer, cannot beneglected when it comes to estimate runoff production.When catchment heterogeneity is also relevant, the relativeimpact of areal heterogeneity on runoff is much morepronounced when the soil surface is not subject to sealformation. It appears that, at least for the conditions appliedin this study, the impact of soil sealing on runoff is muchmore important than that of soil heterogeneity. Therefore,

while catchments heterogeneity may be neglected in somecases, neglecting soil surface sealing can introduce signif-icant errors in all predicted parameters of the hydrologicalresponse of the catchment and, consequently, probably alsoin the design of the related water harvesting system. For theunsealed soil case, more runoff and higher peak dischargesresulted from the distributed approach, for the two rainfallintensities and the whole range of durations considered.Under soil sealing conditions, the distributed approach alsoled to more runoff production and to higher peak dischargefor the whole range of durations only for the lower rainfallintensity case. For the higher rainfall intensity case, morerunoff was produced by the lumped approach for rainfallduration above 90 min, and higher peak discharges, fordurations above 60 min. Consequently, and unlike for theunsealed soil surface case, assuming a homogeneous catch-ment could serve as a first approximation of the runoffhydrograph of bare heterogeneous watersheds prone to sealformation and exposed to high rainfall intensities.[40] Another factor found to affect the hydrological

response of the catchments is the deterministic spatialorganizations of the catchment heterogeneity. Differentspatial organizations characterized by the same statisticaldistribution of the soil properties within the catchmentpresented substantial differences in terms of time of occur-rence of runoff, time to peak flow, and peak flow rate,which might be important in some cases from the engineer-ing point of view.

Appendix A

[41] The hydraulic functions of the seal layer, qc(y , r)and Kc(q, r), are expressed using Brooks and Corey’s [1964]relationship and Mualem’s [1976] model, respectively:

qc y ; rð Þ ¼ qsc rð Þ � qrc rð Þ½ y=yac rð Þ½ �lc rð Þ þ qrc rð Þ ðA1Þ

Kc q; rð Þ ¼ Ksc rð Þ q� qrc rð Þð½ = qsc rð Þ � qrc rð Þ½ f g 2þ2:5lc rð Þ½ =lc rð Þ

ðA2Þ

where qsc is the saturated water content, qrc, the residualwater content, lc, the pore size distribution parameter, yac,the air entry value, and Ksc, the saturated hydraulicconductivity of the seal. These variables are estimated usingthe following relationships [Mualem and Assouline, 1989]:

qsc rð Þ ¼ qs �Dr I ; h; tð Þ=rs ðA3Þ

qrc rð Þ ¼ qr 1þDr I ; h; tð Þ=ri½ ðA4Þ

yac rð Þ ¼ aya 1þDr I ; h; tð Þ=ri½ b a ¼ 0:99; b ¼ 3:72 ðA5Þ

lc rð Þ ¼ l� CDr I ; h; tð Þ ðA6Þ

Ksc rð Þ ¼ Ks

qsc rð Þ � qcr rð Þqs � qr

�2:5 ya

yac rð Þ

�2 lc rð Þ 1þ lð Þl 1þ lc rð Þf g

�2ðA7Þ

Figure 8. Runoff hydrographs resulting from the applica-tion of 30 mm of rainfall at I = 40 mm h�1 to each of thefour patterns (Figure 1) of the heterogeneous catchmentduring seal formation.

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where qs is the saturated water content, qr, the residual watercontent, l, the Brooks and Corey parameter, ya, the air entryvalue, Ks, the saturated hydraulic conductivity, ri, the bulkdensity and rs, the solids density of the undisturbed soil.Dr(I,h,t) is the distributed increase in the soil bulk densitywith depth [r(I,h,t) � ri].[42] Equation (A3) assumes that the saturated water

content is equal to porosity. Equation (A4) assumes thatthe residual water content on a weight basis remainsunchanged during compaction. Equation (A5) is an empir-ical relationship based on experimental data related to theeffect of compaction on ya. The linear relationship betweenlc and Dr (equation (A6)), with the fitting parameter C, isthe simplest approximation that can be assumed since dataon which a more complex relationship can be based are notavailable. Equation (A7) follows the principles suggested byMualem [1986] relating hydraulic conductivity to poregeometry and soil structure.[43] When the equivalent uniform seal approximation is

applied, mean hydraulic functions of the equivalent seallayer, qc(y) and Kc(q), are expressed, and the mean sealhydraulic parameters for the disturbed layer of thickness dcare computed according to equations (A3)–(A6) whereDr(I,h,t) is replaced by an arithmetic average:

Dr I ; tð Þ ¼ dc Ið Þ�1

Z dc Ið Þ

0

Dr I ; h; tð Þdh ðA8Þ

and Ksc(r) is replaced by a harmonic average value:

Ksc ¼ dc Ið ÞZ dc Ið Þ

0

dh=Ksc rð Þ" #�1

ðA9Þ

Appendix B

[44] The flow equation is solved using the followingconditions. The lower boundary condition of the flowsystem, at depth L = 1.5 m, was characterized by a constanthydraulic head, yL, low enough so that the hydraulicconductivity, K(yL), and the downward flux, q(L), can beneglected:

Z ¼ �L; t � 0 ; yL ¼ const ðB1Þ

[45] The upper boundary condition switched from a Neu-man condition to a Dirichlet condition during the wettingprocess. Before ponding, while yZ=0 < 0, the upper bound-ary condition was:

Z ¼ 0 ; 0 < t < tp ; qz¼0 ¼ �I ðB2Þ

[46] At ponding time, t = tp, the boundary conditionswitched to

Z ¼ 0 ; t � tp ; yZ¼0 ¼ 0 ðB3Þ

with tp being the ponding time. In all the simulated cases,the initial condition was assumed to be identical:

Z � 0 ; t ¼ 0 ; y Zð Þt¼0¼ �50 kPa ðB4Þ

[47] The grid for the finite difference approximation ofequation (9) is irregular. In the upper 8 cm layer, the nodalspacing, DZ, is 0.2 cm, while it is 2.0 cm for all theremaining soil profile. To reduce numerical errors, the finegrid domain stretches deeper than the interface between theequivalent seal layer and the underlying undisturbed soil, tocover well the zone that includes the discontinuity in thehydraulic properties at Z = �dc. The mass balance errorsincrease with the rainfall intensity with an upper limit ofapproximately 2.0%.

[48] Acknowledgments. The authors express their sincere thanks toM. H. Diskin for the opportunity to combine the cell model in this study.This research was partly supported by a grant from the Ministry of Science,Culture and Sport of Israel and the Bundesministerium fuer Bildung andForschung (BMBF) of Germany. Their support is gratefully acknowledged.Contribution 606/05 of the Agricultural Research Organization, Institute ofSoil, Water and Environmental Sciences, Bet Dagan, Israel.

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����������������������������S. Assouline, Department of Environmental Physics, Institute of Soil,

Water and Environmental Sciences, ARO, Volcani Center, P.O.B. 6, BetDagan 50250, Israel. ([email protected])

Y. Mualem, Seagram Center for Soil and Water Sciences, Faculty ofAgriculture, Hebrew University of Jerusalem, 76100 Rehovot, Israel.

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