# Initial Soil Water Content as Input to Field-Scale Infiltration and Surface Runoff Models

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<ul><li><p>Initial Soil Water Content as Input to Field-ScaleInfiltration and Surface Runoff Models</p><p>Renato Morbidelli & Corrado Corradini &Carla Saltalippi & Luca Brocca</p><p>Received: 9 March 2011 /Accepted: 13 January 2012 /Published online: 1 March 2012# Springer Science+Business Media B.V. 2012</p><p>Abstract Evidence is given of the role of initial soil moisture content, i, in determining thesurface runoff hydrograph at field scale, that is a crucial element when distributed models forthe estimate of basin response to rainfall have to be formulated. This analysis relies uponsimulations performed by a model that, because of the necessity of representing theinfiltration of surface water running downslope into pervious saturated or unsaturated areas,uses a coupled solution of a semi-analytical/conceptual approach for local infiltration and anonlinear kinematic wave equation for overland flow. The model was applied to actualspatial distributions of i, earlier observed over different fields, as well as to a uniform valueof i assumed equal to the average value or to the value observed in a site characterized bytemporal stability. Our results indicate that the surface runoff hydrograph at a slope outlet ischaracterized by a low sensitivity to the horizontal heterogeneity of i, at least in the cases ofpractical hydrological interest. In fact, in these cases the correct hydrograph can be simulatedwith considerable accuracy replacing the actual distribution of i by the correspondingaverage value. Moreover, the surface hydrograph is sufficiently well reproduced eventhough a single value of i, observed at a site anyhow selected in the field of interest, isused. In particular, this extreme simplification leads to errors in magnitude on peak runoffand total volume of surface water with values typically within 10% and 15%, respectively.</p><p>Keywords Hydrology . Surface runoff modeling . Soil water content</p><p>1 Introduction</p><p>The estimate of rainfall infiltration at field scale is a fundamental topic in hydrologicalapplications based on the use of rainfall-runoff transformations. In fact, the latter is frequently</p><p>Water Resour Manage (2012) 26:17931807DOI 10.1007/s11269-012-9986-3</p><p>R. Morbidelli (*) : C. Corradini : C. SaltalippiDepartment of Civil and Environmental Engineering, University of Perugia,via G. Duranti 93, 06125 Perugia, Italye-mail: renato@unipg.it</p><p>L. BroccaResearch Institute for Geo-Hydrological Protection, National Research Council,Via Madonna Alta 126, 06128 Perugia, Italy</p></li><li><p>represented through abstractions of the basin geometry into networks of elements with planesand channels (Hager 1984; Woolhiser et al. 1990; Singh 1996; Melone et al. 1998). Further, aconceptual approximation of the hydrological response from this basin structure requires toidentify: (1) an objective correspondence of each element with a specific watershed region, (2)an appropriate formulation of effective rainfall over the planes, and (3) the transformation ofeffective rainfall to direct runoff, through the mechanisms of overland flow and channel flow.</p><p>In this context, infiltration is generally the main loss that limits the development ofeffective rainfall at field scale. Therefore, the estimate of the areal-average infiltration rate, I,is required. That becomes complicated by the spatial variability of soil hydraulic properties(Zhao et al. 2010), rainfall rate and initial soil water content. The spatial variability ofsaturated hydraulic conductivity, Ks, is generally assumed as the most important in compar-ison with those of the other soil hydraulic properties that influence the infiltration process(Russo and Bresler 1981; Dagan and Bresler 1983). At the same scale the spatial variabilityof rainfall rate, r, is considered less significant, also in the case of convective rainfalls(Goodrich et al. 1995; Morbidelli et al. 2006). Many studies have shown that the spatialvariability of Ks, assumed as a stochastic variable with a lognormal probability densityfunction (Nielsen et al. 1973; Warrick and Nielsen 1980; Sharma et al. 1987; Loague andGander 1990), can significantly affect the response of a slope to a uniform rainfall (Binley etal. 1989a,b; Saghafian et al. 1995; Corradini et al. 1998; Merz et al. 2002). Under theseconditions, several semi-analytical (Sivapalan and Wood 1986; Govindaraju et al. 2001) andsemi-empirical models (Smith and Goodrich 2000; Corradini et al. 2002) were proposed forthe estimate of the areal-average infiltration rate. Some formulations concerning the effectsof the joint spatial variability of Ks and r were also developed. Wood et al. (1986) used thetwo-term Philip equation to find approximate expressions for areal mean and variance ofinfiltration rates, however, their averaging operation occurred in space over a single reali-zation. Their relationship for areal average infiltration was validated by comparison withMonte-Carlo simulations, but the magnitude and the trend of the resulting errors were notspecified. Furthermore, Castelli (1996) developed a theoretical approach but under verysimplified conditions, particularly for the formulation of local infiltration rate. Govindarajuet al. (2006) and Morbidelli et al. (2006) proposed a more complete modelling to obtain theareal average infiltration, with a semi-analytical formulation and an additional componentthat describes empirically the run-on process consisting in infiltration of surface waterrunning from saturated areas downslope over a pervious soil. The possibility to replacethe spatially varying infiltration rate by the average infiltration rate was investigated byTayfur and Singh (2004) but specifically in the context of models for sediment transport.However, all these models dont consider the possible effects of spatial variability of initialsoil water content, i, that has to be appropriately clarified.</p><p>In a simulation study, using the Coweeta catchment topography, North Carolina, Graysonet al. (1995) considered two patterns of soil moisture with the same properties of mean,variance and correlation length. They showed that, assuming a pattern spatially random andthe other organized by a wetness index, very different responses to given rainfall inputs wereproduced. Similar results were also obtained by Merz and Plate (1997), Merz and Brdossy(1998) and Bronstert and Brdossy (1999). However, rather different results were derived inother investigations (Goodrich et al. 1994; Aubert et al. 2003; Longobardi et al. 2003;Corradini et al. 2008; Brocca et al. 2009b). In particular, for two sub-basins of the WalnutGulch experimental watershed, Goodrich et al. (1994) found that the knowledge of theremotely sensed average initial soil moisture was sufficient as input to rainfall-runoff modelsin semi-arid regions, provided the rainfall was known with a great spatial resolution.Grayson and Western (1998) suggested that a network with a limited number of moisture</p><p>1794 R. Morbidelli et al.</p></li><li><p>sensors could provide soil moisture time series potentially usable as antecedent conditiondata. Therefore, these indicative wetness conditions derived by satellite, and/or a few localmeasurements at surface (see also Tombul 2007), might be sufficient to establish antecedentconditions for a rainfall-runoff event in alternative to detailed spatial measurements (Korenet al. 2008). Recently, Brocca et al. (2009a, 2010) from spot measurements carried out indifferent experimental plots located in the Upper Tiber River (Central Italy) investigated thespatial pattern of the volumetric soil moisture at the surface. The results stated that, at fieldscale, i can be usually assumed as a random variable characterized by a probability densityfunction of normal type, set equal to zero for negative values, and coefficient of variationequal to0.1. Moreover, they found that a limited number of sampling points representativeof all the values observed in each experimental field was sufficient to determine averageconditions at basin scale (see also Vachaud et al. 1985), but they did not quantify theinfluence of this approximation on overland flow generation.</p><p>Considering the results aforementioned someway contrasting, the main objective of thispaper is to improve the knowledge of the role of spatial variability of initial soil moisture onthe overland flow generated at the field scale. The proposed study takes into account theinfiltration of rainfall and, in addition, of overland flow running downslope into pervioussaturated or unsaturated areas (run-on process) that none of the above-mentioned inves-tigations considered earlier. Theoretical results have been obtained starting from the Broccaet al. (2009a, 2010) experimental data and combining a local infiltration model earlierproposed for complex rainfall patterns (Corradini et al. 1997) with a nonlinear kinematicwave approximation for overland flow. The second objective is to address the problemconcerning the number of soil moisture measurements to be sampled for an appropriateestimate of the field scale surface runoff hydrograph.</p><p>2 Statement of the Problem</p><p>As a geometrical simplification of a natural hillslope we represent it by a single plane ofslope S0. The soil is assumed to be vertically homogeneous, but the initial soil water contentis considered to vary through the soil surface according to experimental observations byBrocca et al. (2009a, 2010). Given a time varying rainfall rate, r, the generation of surfacerunoff will be governed by the combined effects of the processes of rainfall infiltration andrun-on. Figure 1 provides a simple scheme of the interacting processes along a strip of the</p><p>rainfall</p><p>saturated cell</p><p>unsaturated cell</p><p>runon</p><p>infiltration</p><p>overland flow</p><p>slope outlet</p><p>t1</p><p>t2</p><p>t3</p><p>rainfall</p><p>saturated cell</p><p>unsaturated cell</p><p>runon</p><p>infiltration</p><p>overland flow</p><p>slope outlet</p><p>t1</p><p>t2</p><p>t3</p><p>Fig. 1 Schematic representationof the mechanisms contributingto the generation of the surfacerunoff hydrograph at the fieldscale. A single strip at threedifferent times t1< t2</p></li><li><p>slope at three different times t1< t2< t3. As it can be seen, in this example at the time t1 thereare only unsaturated cells subjected to rainfall infiltration, while at the time t2 there aresaturated cells with rainfall infiltration and unsaturated cells with infiltration due to bothrainfall and run-on; lastly, at the time t3 all the strips contribute to the overland flow at theslope outlet.</p><p>Local infiltration is described by assuming one-dimensional flow into independentvertically homogeneous columns, following the scheme adopted by Corradini et al.(1998). The model proposed earlier by Corradini et al. (1994) and then reformulated byCorradini et al. (1997) is selected to determine the infiltration rate in each column, consid-ering that it was found to be very accurate by using the Richards equation as a benchmark.The flow problem is simplified by assuming an initial soil water content, i, invariant withdepth, z, and approximating the dynamic wetting profile, (z), by a distorted rectanglerepresented in functional form through a shape factor (1) which depends on the surfacewater content, 0. Combining the continuity equation and a depth-integrated form of theDarcy law, the following relation is obtained:</p><p>d0dt</p><p> 0 i b 0 I 0 0 i db 0 d0 b 0 h i q0 K0 0 i G i; 0 b 0 p K0I 0</p><p> 1</p><p>where t is the time; q0 is the downward water flux at the surface; K0 is the hydraulicconductivity at the surface; I is the cumulative dynamic infiltration depth; p is a quantitydepending on the profile shape and linked with ; G(i, 0) is the net capillary drivedepending on both the suction head, , and hydraulic conductivity, K, as:</p><p>G i; 0 1K0</p><p>Zy 0 </p><p>y i </p><p>K dy 2</p><p>The model requires the knowledge of the functional forms of hydraulic soil properties,which are expressed according to the parameterization adopted by Smith et al. (1993) as:</p><p>K Ks rs r</p><p> 32 l=3</p><p>y yb rs r</p><p> c l=" #1 c= d 4</p><p>where s is the volumetric water content at natural saturation; r is the residual volumetricwater content; 1 is the pore size distribution index; yb is the air entry head, which is given ina tabular form for soil texture classes (Rawls et al. 1983); and c and d are empiricalcoefficients. The downward water flux at the soil surface, q0 , may be formally expressed as:</p><p>q0 r v0 unsaturated surface 5</p><p>q0 f saturated surface 6where v0 represents the run-on in terms of discharge per unit surface, estimated as specifiedin the following. The quantity f denotes the infiltration capacity derived from Eq. 1 with</p><p>1796 R. Morbidelli et al.</p></li><li><p>d0/dt00. All the parameters in Eqs. 14 are considered to be constant through the slope,except the initial soil water content.</p><p>Surface runoff is routed over the plane by the kinematic wave equation with flowresistance expressed by the Manning law (Woolhiser and Goodrich 1988; Singh 1996;Venkata et al. 2008). We have:</p><p>@h</p><p>@t @ah</p><p>m</p><p>@x r q0 7</p><p>where h is the depth of flow at time t and position x; a 1=n S1=20 , with n Manningroughness coefficient; m05/3 and !hm at the slope outlet represents the discharge per unitwidth. The upper boundary condition and the initial condition are:</p><p>h 0; t 0 and h x; 0 0 8By Eq. 5 we observe that over an unsaturated elementary area r-q0 of Eq. 7 is equal to -v0,</p><p>while on a saturated surface r-q0 represents the effective rainfall rate. The quantity v0 isobtained through the solution of Eq. 7 in the adjacent upstream area. With the support ofprevious numerical simulations (Schmid 1989), we neglect the influence of overland flowdepth on the infiltration rate.</p><p>The discharge at the slope outlet is given by:</p><p>Qt a hm x L; t B 9with L and B length and width of the slope, respectively. In principle Q(t) is dependent on thespatial distribution of i through Eq. 1. We note that, in principle, for natural surfaces withirregular microtopography, flow dynamics should be represented in two dimensions by thediffusion wave approximations (Govindaraju et al. 1992; Tayfur et al. 1993; Tayfur andSingh 2004), while the use of the bi-dimensional kinematic wave approximation (Tayfur2001) would be inappropriate. However, for the case of a plain with smooth surface it iswidely recognized that the one-dimensional kinematic wave approximation provides anappropriate representation of the surface runoff hydrograph (see for example Giakoumakisand Tsakiris 2001). Our mathematical formulation is supported by the fact that our primaryobjective is not to maximize the accuracy in simulating the surface runoff hydrograph but toinvestigate its sensitivity to different spatial representations of soil moisture content. On theother hand, it is expected that the choice of a slope with erratic microtopography determinesan irregular bi-dimensional overland flow and therefore mitigates the role of the spatialvariability of i.</p><p>3 Numerical Approach, Study Soils and Selected Rainfall Events</p><p>Numerical solutions of the kinematic wave equation coupled with the infiltration equationwere carried out over two different planes: the first-one, 60 m long and 50 m wide, th...</p></li></ul>

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