a simplified model for estimating field-scale surface runoff hydrographs

HYDROLOGICAL PROCESSESHydrol. Process. 21, 1772–1779 (2007)Published online 7 December 2006 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/hyp.6345

A simplified model for estimating field-scale surface runoffhydrographs

Renato Morbidelli,1 Corrado Corradini1* and Rao S. Govindaraju2

1 Department of Civil and Environmental Engineering, University of Perugia, via G. Duranti 93, 06125 Perugia, Italy2 School of Civil Engineering, Purdue University, West Lafayette, IN 47907

Abstract:

The problem of obtaining field-scale surface response to rainfall events is complicated by the spatial variability of infiltrationcharacteristics of the soil and rainfall. In this paper, we develop and test a simplified model for generating surface runoffover fields with spatial variation in both rainfall rate and saturated hydraulic conductivities. The model is able to representthe effects of local variation in infiltration, as well as the run-on effect that controls infiltration of excess water from saturatedupstream areas. The effective rainfall excess is routed to the slope outlet using a simplified solution of the kinematic waveapproximation. Model results are compared to averaged hydrographs from numerically-intensive Monte–Carlo simulations forobserved and design rainfall events and soil patterns that are typical of Central Italy. The simplified model is found to yieldsatisfactory results at a relatively small computational expense. A proposal to include a simple channel routing scheme is alsopresented as a prelude to extend this conceptualization to watershed scales. Copyright 2006 John Wiley & Sons, Ltd.

KEY WORDS hydrology; surface runoff modelling; infiltration modelling; watershed modelling

Received 27 September 2005; Accepted 2 May 2006

INTRODUCTION

The prediction of hydrological response to rainfall eventsis one of the fundamental problems in hydrological prac-tice both in small watersheds for design of commonhydraulic structures as well as in large basins for on-line flood forecasting. Lumped and conceptual modelstend to be grossly simplified in the representation of boththe basin elements and the physical processes involved,and often do not perform well because of their inabil-ity to represent the spatial heterogeneity exhibited bythe hydrogeomorphologic properties of a watershed andothers physical quantities. Therefore, use of distributedmodels is generally recommended as being more appro-priate in watershed hydrology.

Physically based distributed models cannot be emp-loyed in many practical applications (Beven, 1989;Binley et al., 1989a,b) not only because of the prohibitivecomputational effort but also because of the lack ofappropriate experimental data describing the space–timevariation of soil hydraulic properties and rainfall. Thelatter does not allow for the full potential of physics-based models to be realized in most applications. Thus,other kinds of distributed models, involving some sim-plifications in the representation of both the geometricfeatures of the watershed and the physical processes, arecommonly used.

Before such simplified models can become useful,there are several issues that need to be resolved for an

* Correspondence to: Corrado Corradini, Department of Civil and Envi-ronmental Engineering, University of Perugia, via G. Duranti 93, 06125Perugia, Italy. E-mail: [email protected]

effective prediction of watershed discharge. For exam-ple, it is not clear what level of spatial aggregationshould be employed-probably different levels of disag-gregation could be used within different regions of agiven basin. Many distributed models use abstractionsof the basin geometry based on networks of planes andchannels (Hager, 1984; Woolhiser et al., 1990; Singh,1996; Melone et al., 1998) chosen subjectively. Further-more, following the analysis by Woolhiser and Liggett(1967), the non-linear kinematic wave theory is usuallyapplied for the description of effective rainfall-overlandflow transformation, but its numerical solution for realwatersheds is typically too expensive in terms of compu-tational effort.

For approximating the hydrological response of awatershed as a combination of responses from planes andchannels, an objective correspondence of each elementwithin specific watershed regions has to be defined.In addition, an appropriate modelling framework thatprovides effective rainfall over the planes, and describesthe transformation of effective rainfall to direct runoffthrough overland flow and channel flow mechanisms isrequired. Such a model should eliminate shortcomingsof current models that disregard the effects of spatialvariability of local infiltration over a plane, and adopt anon-linear kinematic wave solution that is not well-suitedfor real applications at watershed scale. In this light,the complex interaction between infiltration and overlandflow precludes analytical solutions of this system inthe presence of random spatial variability of infiltrationproperties. This problem is further complicated by therun-on process that incorporates infiltration of excess

Copyright 2006 John Wiley & Sons, Ltd.

A MODEL FOR ESTIMATING FIELD-SCALE SURFACE RUNOFF HYDROGRAPHS 1773

water from upstream saturated areas (Morbidelli et al.,2006). Thus, effects of these heterogeneities have tobe represented through numerical solutions embeddedin a Monte–Carlo framework. However, the extensivecomputer effort required to consider many syntheticrealizations even at the scale of a single slope (Julienand Moglen, 1990; Ogden et al., 1995; Saghafian et al.,1995) makes this procedure practically inapplicable atwatershed scales. Therefore, a modelling technique thatdoes not require Monte–Carlo simulations, and does notinvolve the coupled solution of infiltration and overlandflow equations is desirable. To be more useful, thistechnique must further account for spatial variability ofrainfall and infiltration properties at the field-scale todetermine the rainfall-excess that can be routed over thesoil surface to eventually yield field-scale surface runoff.This would be an important first step for upscaling thisprocedure to obtain watershed response. To the best ofour knowledge, this problem has not been satisfactorilyaddressed at the field-scale.

The main objective of this paper is to develop andtest such a model that describes the field-scale surfacerunoff from a plane. In order to develop a basic build-ing block that can then used for straightforward applica-tion to watersheds, it is necessary to describe a simpleway of channel routing. Thus, a complete model wouldcombine two components: a first component that pro-vides the expected areal-average infiltration rate overthe slope (Morbidelli et al., 2006), and a second com-ponent describing the transformation of effective rain-fall to overland flow, to determine lateral inflow intochannels, and the subsequent flow routing through thechannel (Govindaraju et al., 1999). Such an approachwould meet the aforementioned requirements. Notably,the first model component replaces the actual spatial dis-tribution of effective rainfall by an effective spatiallyuniform value which is then used as input to the secondmodel component. In this paper, the focus is on determin-ing the field-scale surface response of a plane as a firststep. This model was tested by comparing results withthose obtained from an extensive set of Monte–Carlosimulations involving the routing of local effective rain-fall to the slope outlet through numerical solutions. TheMonte–Carlo simulations are based on the description ofspatial variation of rainfall excess and are treated as abenchmark for comparison purposes. In order to focuson field-scale surface response, we do not explicitly con-sider the routing of water through channel elements asthat could confound our interpretation of the effects thatare prevalent at the field-scale. However, we suggest amethod for accomplishing this towards the end of thepaper for completion.

THE PROPOSED MODEL

The model relies upon a first component to yield theexpected areal-average infiltration rate, and a secondcomponent which routes surface runoff through the

plane. The first component is formulated by a semi-analytical contribution to describe field-scale infiltration(Govindaraju et al., 2006), and is augmented by an empir-ical term that synthesizes the infiltration of surface waterfrom saturated areas running downslope over a pervioussoil identified as the run-on process (Morbidelli et al.,2006). It allows for representation of local heterogeneityeffects including run-on, and leads to computation of anareal-average effective rainfall required as input to thesecond model component. The second component usesapproximate solutions of the kinematic wave equationsfor planes and channels based on the similarity profiles(Govindaraju et al., 1999). Because the application ofthe model is based on field-scale averages of the effec-tive rainfall field, the overland flow hydrograph at theoutlet of the plane is likely to experience a distortion.Thus, Monte–Carlo simulations were performed to testthe model.

It is well-known that the soil-saturated hydraulic con-ductivity, Ks, exhibits the maximum spatial variabilityamong all the infiltration parameters (Russo and Bresler,1981) and, in addition, that rainfall also exhibits spatialheterogeneity (Goodrich et al., 1995; Krajewski et al.,2003). For a given plane, we consider the soil to bevertically homogeneous, while Ks over the soil surfaceis represented by a stochastically homogeneous randomfield with a log-normal probability density function (pdf)characterized by the mean value hKsi and the coefficientof variation cv�Ks�.

Given a rainfall rate, r, randomly varying in spaceand characterized by a uniform pdf with mean value hriand coefficient of variation cv(r), the local generation ofsurface runoff is governed by infiltration of both rainfalland surface water (run-on).

During Monte–Carlo simulations, for each realizationof Ks and r, local infiltration rate and surface runoffwere estimated by coupling the classical Green–Amptequation for infiltration capacity and the numerical solu-tion of the non-linear kinematic wave; the latter wasthen used to develop the surface runoff hydrograph atthe slope base as input into the channel. With the inclu-sion of run-on, a serious computational burden is incurredwhen solving each realization of the local rainfall excesscoupled with kinematic approximation of surface routing(Morbidelli et al., 2006). By averaging the hydrographsover the ensemble of realizations, we obtained the field-scale hydrograph used for testing the proposed model.

A short account of the field-scale infiltration component

Morbidelli et al. (2006) expressed the expected areal-average infiltration rate, hIi, as:

hI�t�i ¾D hIn�t�i C hriatŁb exp��ctŁ� �1�

where hIn�t�i represents the areal-average infiltration rateobtained by neglecting the run-on process; a, b and c aremodel parameters; and tŁ is the ratio between the timeand the time to ponding obtained using hKsi.

Copyright 2006 John Wiley & Sons, Ltd. Hydrol. Process. 21, 1772–1779 (2007)DOI: 10.1002/hyp

1774 R. MORBIDELLI, C. CORRADINI AND R. S. GOVINDARAJU

The quantity hIn�t�i is computed by the Govindarajuet al. (2006) model which really provides an approxima-tion of hIn�t�i through a relation for hIn�F�i coupled witha relation between time and cumulative infiltration, F. ForhIn�F�i we have:

hIn�F�i D 1

2RF2c

fGKs [�rmin C R�Fc, 2] �GKs [rminFc, 2]g

� r2min

2RfGKs [�rmin C R�Fc, 0] �GKs [rminFc, 0]g

C(rmin C R

2

)f1 �GKs [�rmin C R�Fc, 0]g

C 1

Fc

(rmin C R

R

)fGKs [�rmin C R�Fc, 1]

�GKs [�rminFc�, 1]g

� 1

Fc

(1

RFc

)fGKs [�rmin C R�Fc, 2] �GKs [�rminFc�, 2]g

C 1

FcGKs [rminFc, 1] �2�

with rmin and rmin C R being extreme values of the pdfof r, and the GKs and Fc functions are defined as:

GKs �K1, �� D∫ K1

0K� fK�K� dK �3�

where K1 and � represent the first and the second argu-ment of the GKs function in Equation (2), respectively;fK�K� is the pdf of Ks, and:

Fc D F

� C F�4�

where � is the difference between saturated, �s, andinitial, �i, water contents.

For the expected values of t(F) we have:

ht�F�i D F

hri f1 �GKs [hriFc, 0]g

C[FC � ln

( �

� C F

)]fGKs [hriFc],�1g

C �1∑jD1

1

�jC 1�hrijC1 fGKs [hriFc, j]g �5�

The quantities a, b and c of Equation (1) for �i − �swere expressed by:

a D 2Ð8 [cv(r) C cv�Ks�]0Ð36 �6�

b D 5Ð35 � 6Ð32 [cv(r) cv�Ks�] �7�

c D 2Ð7 C 0Ð3[ hri/hKsi

cv(r) cv�Ks�

]0Ð3�8�

with a tending to zero for �i ! �s, for which using somecalibration events over a fine textured soil we proposehere the following explicit relation:

a D 2Ð8 [cv(r) C cv�Ks�]0Ð36

[1 �

(�i � �r�s � �r

)3Ð78]

�9�

A short description of the flow routing componentthrough the plane

The non-linear kinematic wave model describing themovement of water over plane sections with flow resis-tance expressed by the Manning law can be written as(Govindaraju et al., 1990; Singh, 1996):

∂y

∂tC S

120

n

∂ym

∂xD i�x, t� �10�

where y is the flow depth, i is the net lateral inflow, S0

is the plane slope, x is the horizontal spatial coordinatein the downslope direction, n is the Manning coefficientand m is 5/3.

The boundary condition is given by:

y0�t� D y�0, t� D

nq�0, t�

S120

1/�mC1�

�11�

where q(0,t) is the flow discharge into the plane at theupstream end, and the initial condition is expressed in thegeneral form:

y�x, 0� D yiniz�x� �12�

We prescribe the solution for the flow depth to be ofthe form:

y�x, t� D y0�t�C y1�t� sin(�x

2L

)�13�

where L is the length of the plane and y1 is function oftime only. The spatial distribution of the depth at anygiven time is described by the sine term. Equation (13)is essentially employing the principle of superposition.

Substituting Equation (13) into Equation (10) and inte-grating over the spatial domain results in the followingordinary differential equation for y1�t�:

dy1

dtC �

2L

S120

n[�y0 C y1�

mC1 � ymC10 ]

� �

2i�t�C �

2

dy0

dtD 0 �14�

which can be integrated numerically with very littlecomputational effort. In Equation (14), i�t� represent thespatially averaged net lateral inflow per unit area reachingthe overland flow element.

The initial condition for Equation (14) is:

y1�0� D(�

2

)[y � y0�0�] �15�

where y is the spatial averaged value of the initial flowdepth. Once y1�t� is obtained from Equation (14), thetotal flow discharge from the overland section may becomputed as:

qoutflow D S120

n[y0�t�C y1�t�]

mC1. �16�



MODEL TESTING

Using the Monte–Carlo simulations the model was testedover a soil representative of a clay loam (henceforth des-ignated as Soil B) typical of soils in Central Italy the mainhydraulic features of which, required for the estimate ofinfiltration rate, are given in Table I. As described ear-lier, simulations are needed for comparison because of thedistortion of effective rainfall over the plane. Therefore,we compare the overland flow hydrographs obtained bythe proposed model with those obtained from averagesof Monte–Carlo simulations at the slope outlet.

A variety of numerical experiments corresponding todifferent combinations of the pdf of r and Ks for manyrainfall patterns were performed. A 50-m long and 20-mwide plane divided into cells to represent the spatialvariability of r and K were considered. To satisfy stabilityrequirements of the coupled numerical solution of theinfiltration and kinematic wave equations, we selecteda 5-m long and 2-m wide grid. A Manning roughnesscoefficient n D 0Ð15 (metric units) was chosen for all theexperiments.

Table I. Hydraulic data used for the study soil

Property Soil B

Saturated water content, �s 0Ð3325Residual water content, �r 0Ð1225Initial water content, �I 0Ð1659Suction head, (mm) 1116Saturated hydraulic conductivity: mean

value, hKsi (mm h�1)0Ð75

Saturated hydraulic conductivity:coefficient of variation, cv (Ks)

0Ð3–1Ð0

In order to examine the accuracy of the proposed modelin terms of the surface runoff hydrograph, we showresults for five experiments, which involve moderatevalues of the coefficients of variation of r and Ks andare representative of a large range of conditions. Theexperiments include time varying rainfall patterns of bothreal and design rainfall events from Central Italy.

Figure 1 shows the surface runoff hydrograph under anobserved moderate rainfall event of long duration. As itcan be seen the proposed model adequately estimates thepeak flow while the volume of overland flow is overes-timated by 27%. Since the model component describingthe effective rainfall-surface runoff transformation con-serves the water volume, this error may be attributed tothe approximation of the expected areal-average infiltra-tion by the first model component. We found that thefield-scale infiltration was underestimated by the modelin the period between 9 and 11 h. As indicated in theinset of Figure 1, this is a period of heavy rainfall forthis event.

Figures 2 and 3 compare the surface hydrographs at theslope outlet generated by two heavy observed rainfalls ofshort duration. The hydrographs given by the proposedmodel exhibit appropriate shape with errors in the totalvolume of surface water within 14%. This result isexpected since most of the error in infiltration volumes,and consecutively surface runoff volumes, is incurredby representing the spatially varying excess rainfall bya uniform effective value. When rainfall intensities arehigh compared to the mean of Ks, the effect of thiserror is diminished and the approximation of Equation (1)becomes more accurate.

The model was also tested under rainfall patternstypically used for deriving design hydrographs. Figures 4and 5 illustrate the accuracy of the model when an

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0.0018

0.002

0 4 6 8 10 12

Time (h)

Dis

char

ge

(m3 /

s)

0

2

4

6

8

10

12

1 2 3 4 5 6 7 8 9 10 11 12

2

Rai

nfa

ll R

ate

(mm

)

Time (h)

Proposed

Monte Carlo

Figure 1. Comparison of the surface runoff hydrographs at the slope outlet, computed with the proposed model and the Monte–Carlo technique,produced by the natural rainfall event is shown in the upper part of the figure. The latter was observed at the Assisi station (Central Italy) on

August 1, 2002. Soil B with coefficients of variation of Ks and r equal to 0Ð3 and 0Ð45, respectively



0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

Time (h)

Dis

char

ge

(m3 /

s)

Time (1/4 h)

1 2 3 4 5 6

Rai

nfa

ll R

ate

(mm

)

9080706050403020100

Proposed

Monte carlo

Figure 2. Comparison of the surface runoff hydrographs at the slope outlet, computed with the proposed model and the Monte–Carlo technique,produced by the natural rainfall event is shown in the upper part of the figure. The latter was observed at the P. Pietra station (Central Italy) on

June 17, 2003. Soil B with coefficients of variation of Ks and r equal to 0Ð3 and 0Ð45, respectively

Time (h)

Dis

char

ge

(m3 /

s)

0 0.5 1 1.5 2 2.5

0.006

0.005

0.004

0.003

0.002

0.001

0

1 2 3 4

Time (1/4 h)

Rai

nfa

ll R

ate

(mm

)

0

10

20

30

40

50

60

Proposed

Monte Carlo

Figure 3. Comparison of the surface runoff hydrographs at the slope outlet, computed with the proposed model and the Monte–Carlo technique,produced by the natural rainfall event is shown in the upper part of the figure. The latter was observed at the Monte Bibbico station (Central Italy) on

May 15, 1998. Soil B with coefficients of variation of Ks and r equal to 0Ð3 and 0Ð45, respectively

alternating block and a time-increasing rainfall ratedistribution are involved, respectively. The predictedhydrographs experience minor distortions compared tothose obtained by Monte–Carlo simulations, and estimatepeak flows and volumes of surface water with goodaccuracy. The surface runoff at the slope outlet can beused as net lateral inflow into the channel and thenrouted to the outlet by the similarity profile componentwhich was found (Govindaraju et al., 1999) to be very

accurate in an earlier comparative analysis with numericalsolutions of the kinematic wave equations obtained by theKINEROS model (Smith et al., 1995).

An overall analysis of the aforementioned resultsindicates that the proposed model predicts the effectiverainfall formation and its routing through the plane withconsiderable accuracy and therefore it can be consideredsuitable as basic element for developing models ofoverland flow predictions at watershed scale.



Time (h)

Dis

char

ge

(m3 /

s)

1 2 3 4 5 6 7 8 9 10

0.012

0.01

0.008

0.006

0.004

0.002

00 1 2 3 4 5 6 7 8 9 10

Proposed

Monte carlo

05

1015202530354045

Rai

nfa

ll R

ate

(mm

)

Time (h)

Figure 4. Comparison of the surface runoff hydrographs at the slope outlet, computed with the proposed model and the Monte–Carlo technique,produced by the design rainfall event is shown in the upper part of the figure. The latter was obtained from the intensity-duration curve of themeasuring station of Perugia (Central Italy) with 10 years return period, considering an alternating block time evolution. Soil B with coefficients of

variation of Ks and r equal to 0Ð3 and 0Ð45, respectively

30

Dis

char

ge

(m3 /

s)

0.012

0.01

0.008

0.006

0.004

0.002

0

Time (h)

0 2 4 6 8 10

Proposed

Monte carlo

05

10152025

354045

1 2 3 4 5 6 7 8 9 10

Time (h)

Rai

nfa

ll R

ate

(mm

)

Figure 5. Comparison of the surface runoff hydrographs at the slope outlet, computed with the proposed model and the Monte–Carlo technique,produced by the design rainfall event is shown in the upper part of the figure. The latter was obtained from the intensity-duration curve of themeasuring station of Perugia (Central Italy) with 10 years return period, considering a time-increasing evolution. Soil B with coefficients of variation

of Ks and r equal to 0Ð3 and 0Ð45, respectively

MODEL IMPLEMENTATION AT WATERSHEDSCALE

Once a network of planes and channels representativeof the actual watershed network has been determined,the model for estimating the surface runoff at theoutlet may be easily accomplished on the basis of theproposed model for the two components of infiltrationand flow routing, the latter extended to describe thechannel routing by following Govindaraju et al. (1999).As in the overland flow case, for a channel with a

regular trapezoidal cross section, the following ordinarydifferential equation may be obtained:

[bL C 2y0Lz C 4y1Lz

�

]dy0

dtC

[2Lb

�C 4y0Lz

�

C Lzy1

]dy1

dtC S

120 [b�y0 C y1�C z�y0 C y1�

2]�mC1�

n[bC 2�y0 C y1�√

1 C z2]m

� q�0, t�� Lr�t� D 0 �17�



where b is the bottom width, z is the side slope, L is thechannel length, S0 is the channel slope and r�t� is thespatially averaged net lateral inflow into the stream perunit length.

The general nature of initial and boundary conditionsfor the surface runoff and streamflow sections makesthem suitable for estimating the hydrological responseto rainfall events. Specifically, the first term on theright-hand side of Equation (13) enables us to consideran upstream inflow to planes and channels. This iscrucial for each internal channel, characterized by ageomorphological order greater than 1, and is an essentialrequirement also for long planes. In fact, Govindarajuet al. (1999) show that, in order to avoid the possibilityof misleading results, when the rainfall duration is lessthan the time of concentration of a given plane, thenthe plane has to be further divided into a cascade ofplanes. Lastly, the possibility of creating more elementsof smaller sizes is very important in order to allowspatial heterogeneity at smaller scales to be adequatelyrepresented, because the spatial effects are integrated outwithin each element.

CONCLUSIONS

An adequate model for simulating field-scale surfacerunoff hydrograph should consider that in the presenceof spatial variability of infiltration parameters and rainfallrate we can have considerable infiltration of overland flowrunning downslope into pervious saturated or unsaturatedareas (run-on process) which should be described bycoupled numerical solutions of infiltration and surfacerunoff equations. Furthermore, because of the randomnature of this spatial heterogeneity these solutions shouldbe embedded in a Monte–Carlo framework with manysynthetic realizations. Such a type of model wouldbe too complex for practical applications and evenfor research purposes. Therefore, appropriate simplifiedmodels, in any case close to physical reality, have to bedeveloped.

The goal of this paper was to develop and test asimplified model of field-scale surface runoff that incor-porates run-on and local scale variability of both satu-rated hydraulic conductivity and rainfall rate. This is notcommon in hydrologic modelling because of the prob-lems aforementioned; solved here by using an empiri-cal representation of run-on, which leads to remove therequirement of a coupled solution for infiltration andsurface runoff, and by incorporating a semi-analyticalcomponent for rainfall infiltration, which provides theexpected areal-average infiltration rate without employ-ing the Monte–Carlo procedure too expensive in termsof computational effort. By using observed and synthetictime-varying rainfall events and soil properties typicallyencountered in Central Italy, model results were com-pared to results from Monte–Carlo simulations. On thebasis of the numerical results, the following conclusionscan be drawn:

1. The simplified model is able to do a credible jobof replicating the field-scale runoff hydrograph overa plane. Important attributes of the hydrograph suchas volume, peak flow rate and time base are gen-erally well reproduced. This is particularly true forshorter high-intensity rainfall events when the field-scale approximation of Equation (1) is more accu-rate. For longer events, especially with low rain-fall intensity, the model is in any case sufficientlyaccurate but the runoff volume is not well con-served.

2. The proposal to extend this to a watershed scale modelprovides important cautionary note as to how spatialdiscretization should be achieved based on temporaldiscretization of rainfall and time of concentration forthe plane and channel elements.

The model proposed in this paper presents an importantfirst step in demonstrating that a simplified conceptualiza-tion is possible to obtain field-scale surface runoff overfields with spatially-varying infiltration properties withinclusion of a run-on component. The problem of extend-ing this to small watersheds will be further explored in afuture study.

ACKNOWLEDGEMENT

This work was funded partly by the National ResearchCouncil of Italy, Special Project GNDCI.

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a simplified model for estimating field-scale surface runoff hydrographs

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