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Reassessment of a semi-analytical field-scale infiltration model through experiments under natural rainfall events
Journal of Hydrology 565 (2018) 835–845
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Reassessment of a semi-analytical field-scale infiltration model through experiments under natural rainfall events
T
⁎
Alessia Flamminia, , Renato Morbidellia, Carla Saltalippia, Tommaso Picciafuocoa, Corrado Corradinia, Rao S. Govindarajub a b
Dept. of Civil and Environmental Engineering, University of Perugia, via G. Duranti 93, 06125 Perugia, Italy Lyles School of Civil Engineering, Purdue University, West Lafayette, IN 47907, United States
A R T I C LE I N FO
A B S T R A C T
This manuscript was handled by Dr Marco Borga, Editor-in-Chief
Operative distributed/semi-distributed rainfall-runoff formulations need a mathematically tractable component for estimating the expected areal-average infiltration rate, 〈I 〉, while point infiltration models are usually adopted for this purpose. A model for 〈I 〉, suitable for hydrological applications, was proposed by Morbidelli et al. (2006, Hydrological Processes, 20: 1465–1481). The main model tenet is the use of cumulative infiltration as an independent variable and replacement of time by its expected value. While this model - as with many other models for field-scale infiltration - was tested through Monte Carlo methods, it remains untested with experimental observations. In this study, a reassessment of the model’s applicability is conducted through experimental evidence from several rainfall-runoff events observed under natural conditions at the plot scale over a slight slope characterized by a silty loam soil. During each experiment rainfall rate, surface runoff at the plot outlet and soil water content vertical profiles were monitored at short time intervals. An overall analysis of our results suggests that the model reliably simulates 〈I 〉 and surface runoff as functions of time and is also suitable for applications under time-dependent rainfall events. The process of redistribution under rainfall hiatus is not addressed by this model.
Keywords: Areal infiltration Infiltration modelling Rainfall-runoff experiments
1. Introduction Simulation of many natural hydrological processes at field and watershed scales relies upon proper representation of areal average infiltration (Govindaraju et al., 2001). Areal-average infiltration at the field-scale has a crucial role in surface runoff prediction by distributed models that represent the hydrological response of a watershed through a superposition of field-scale responses. Field-scale infiltration also governs the evolution of soil water content (Corradini, 2014), the transport of pollutants in the vadose zone (Fiori and de Barros, 2015) and the recharge of aquifers (Smerdon and Drews, 2017). Areal averaging is a popular method to incorporate the natural spatial variability of hydraulic soil properties (Warrick and Nielsen, 1980; Sharma et al., 1987; Loague and Gander, 1990), primarily that of the soil saturated hydraulic conductivity, Ks (Russo and Bresler, 1981, 1982), of the initial soil moisture content, θin , and rainfall rate, r (Corradini and Singh, 1985). At the field scale, the effects of the spatial heterogeneity of θin (Morbidelli et al., 2012) and r (Morbidelli et al., 2006) on the areal infiltration are significantly lower than those due to the variability of Ks , mainly because of the much larger coefficient of variation of Ks . ⁎
The spatial variability of Ks is generally represented by a random field characterized by a log-normal probability density function, pdf, with a level of spatial correlation that can have an important effect on the variance of the field-scale infiltration (Govindaraju et al., 2001). Consequently, the mathematical problem concerning areal infiltration cannot be properly solved analytically and requires Monte-Carlo simulations of Ks along with a local rainfall infiltration model (Philip, 1957, 1969; Mein and Larson, 1973; Smith and Parlange, 1978; Melone et al., 2006). This procedure was adopted, for instance, by Smith and Hebbert (1979), Maller and Sharma (1984), Milly and Eagleson (1982) and Saghafian et al. (1995). Thus, an appropriate estimate of the ensemble-averaged infiltration at the field scale is obtained, but this method is too complex for large hydrological applications. Some alternative modeling approaches have therefore been proposed to obtain approximate solutions in the absence of the run-on process that represents the infiltration of overland flow running downslope into pervious downstream areas. Along these lines significant and widely recognized contributions were presented by Sivapalan and Wood (1986), Smith and Goodrich (2000) and Govindaraju et al. (2001). Sivapalan and Wood (1986) developed approximate relations for
Corresponding author. E-mail address: alessia.fl[email protected] (A. Flammini).
https://doi.org/10.1016/j.jhydrol.2018.08.073 Received 22 May 2018; Received in revised form 27 August 2018; Accepted 28 August 2018 Available online 01 September 2018 0022-1694/ © 2018 Elsevier B.V. All rights reserved.
Journal of Hydrology 565 (2018) 835–845
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Nomenclature
〈r 〉 I 〈I 〉 〈In〉 K 〈Ks〉 f F F0
Subscripts in s r p i sim ob c
initial condition saturated condition residual condition ponding condition i-th spatial cell simulated value observed value critical value
θ Ψ ε N rmin
Functions and variables fK
probability density function of saturated hydraulic conductivity g[.] function representing the run-on process H[.] Heaviside step function MKs[Ka,ω]moments of saturated hydraulic conductivity with arguments Ka and ω t time t0 time of beginning of a rainfall event t* time scaled to the time to ponding 〈t 〉 expected time Δt time step r rainfall rate
v0 Q
expected areal-average rainfall rate areal-average infiltration rate expected areal-average infiltration rate ensemble-averaged areal infiltration rate hydraulic conductivity expected areal-average saturated hydraulic conductivity infiltration rate cumulative infiltration depth cumulative infiltration depth at the beginning of a rainfall event volumetric soil water content water suction head relative error discrete number of spatial cells and rmin+R extreme values of the uniform probability density function of rainfall rate run-on in terms of discharge per unit surface cumulative surface runoff
Parameters Ψb λ c and d a, b, cG cv
air entry suction Brooks-Corey pore size distribution parameter empirical coefficients parameters of a Gamma function coefficient of variation of a random variable
local infiltration model and an equation describing the surface runoff routing through the study area is required. Alternatively, a simplified solution can be obtained by a mathematical model set up by Morbidelli et al. (2006) who combined the above-mentioned semi-analytical formulation of Govindaraju et al. (2006) with a semi-empirical component for the run-on process. The model is fairly simple and requires the solution of a set of algebraic equations for given values of the first two moments of the probability density functions of both Ks and r . Most of the models available for determining the areal-average infiltration have been developed and/or tested through Monte-Carlo techniques but, to our knowledge, in depth studies of their accuracy in simulating observed hydrological processes have not been carried out. The main objective of this paper is to test the model formulated by Govindaraju et al. (2006) and then extended by Morbidelli et al. (2006) using natural rainfall-runoff events observed by an experimental system specifically designed at the plot scale. Starting from a natural soil, an artificial plot with bounded sides, uniform soil grain size distribution and significant horizontal spatial variability of Ks has been used. Natural events consistent with the basic assumptions earlier adopted to develop the model, namely absence of a surface sealing layer as well as of long periods of soil water redistribution, have been selected. The overall model accuracy and the effectiveness of the crucial model components are investigated in this paper.
areal average and variance of infiltration rate considering a random variability of Ks and steady rainfall invariant in space. Monte-Carlo simulations were used to validate these relations but a single realization was used for the averaging procedure and the involved errors were not clearly expressed. Smith and Goodrich (2000) developed a parametric formulation for areal-average infiltration, obtained through the Latin Hypercube sampling method and only one realization of Ks for each combination of mean and coefficient of variation. Govindaraju et al. (2001) presented a semi-analytical/conceptual model for the expected areal-average infiltration obtained with Ks as a random variable and uniform rainfall. Three different model versions were proposed: the first one based on the assumption of cumulative infiltration as the independent variable associated with an expected infiltration time, the second and third versions were obtained through a representation of point cumulative infiltration by a series expansion and a parameterized expression, respectively. Some approaches for estimating the areal-average infiltration in the case of a joint random horizontal variability of Ks and r have been also proposed. Wood et al. (1986) found expressions for areal mean and variance of infiltration rate but with the same limitations aforementioned for Sivapalan and Wood (1986). Castelli (1996) proposed an analytical formulation for areal-average infiltration but its application is limited by the simplified representation of infiltration at the local scale. Govindaraju et al. (2006) developed a semi-analytical model under the condition of a joint spatial heterogeneity of Ks and r both assumed as random variables with pdf of lognormal type for Ks and uniform type between two extreme values for r . The mathematical formulation used cumulative infiltration as the independent variable along with a formulation for expected time. The aforementioned modeling approaches assume negligible runon, thus limiting their application. This limitation is tested by the results from investigations performed using Monte-Carlo simulations (Saghafian et al., 1995; Woolhiser et al., 1996; Corradini et al., 1998; Nahar et al., 2004). However, when including the run-on process, the estimate of the ensemble-averaged infiltration by Monte-Carlo technique becomes very complex considering that a coupled solution of a
2. A short account of a semi-analytical areal-average infiltration modeling and its reassessment for application to real rainfall patterns Building on the conceptualization of Govindaraju et al. (2006), Morbidelli et al. (2006) proposed an approach for representing direct rainfall infiltration with the run-on process. For the purpose of this study all the variables are considered to be independent. For negligible effects of the spatial heterogeneity of θin , considering a total of N cells and a given realization of r and Ks the areal-average infiltration rate, I , at a given cumulative infiltration, F , can be written through a sum of the contributions of unsaturated and saturated cells 836
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as:
< t (F ) >= N
I (F ) =
1⎧ ∑ (ri + v0 )[1−H (F −Fpi)] + N ⎨ i=1 ⎩
⎭
i=1
(
1 2RFc2
−
2R
(
R 2
) ) {M
1 Fc
(
−
1 Fc
( ) {M
+
1 M [r F , Fc K s min c
rmin + R R 1 RFc
〈I (t ) 〉 ≃ 〈In (t ) 〉 + g [t ∗; cv (Ks ), cv (r )]
{1−MK s [(rmin + R) Fc , 0]}+
+
K s [(rmin
K s [(rmin
1]
∫0
Ka
K ωfK (K ) dK
Fc , j]}
(5)
where 〈I 〉 is the expected areal-average infiltration rate, cv (Ks ) and cv (r ) denote the coefficients of variation of Ks and r , respectively, and t ∗ is a time scaled to the time to ponding. The explicit formulation of g [.] is given in the Appendix. For spatially uniform rainfall and negligible run-on, in the absence of the empirical term in Eq. (5) and considering that:
+ R) Fc , 1]−MK s [rmin Fc , 1]}+
+ R) Fc , 2]−MK s [rmin Fc , 2]}+ (2)
MK s [(rmin + R) Fc , ω]−MK s [rmin Fc , ω] = 0
where rmin and rmin + R are the extreme values of a uniform pdf of r ; F Fc = [Ψ (θ − θ ) + F ] , with θs soil water content at saturation and Ψ soil s in suction head, linked with the critical values of saturated hydraulic conductivity, K c = Fc r , which determine surface saturation for a given F ; and MK s [.,.] is expressed by:
MK s [K a, ω] =
)
where 〈r 〉 and 〈t (F )〉 represent the expected field-scale rainfall rate and the expected time associated to a given F value, respectively, estimated through the averaging procedure over the ensemble of realizations of Ks and r . Because of the spatial variability of both Ks and r , the time t associated with a given F takes on different values in each cell. Therefore the expected time was assumed to approximate the actual relation 〈In (t )〉 for the ensemble-averaged areal infiltration rate as a function of time by that obtained from combining 〈In (F )〉 and 〈t (F )〉. We also note that the series in the last term of Eq. (4) converges rapidly and can be approximated by the first five terms for most applications. The inclusion of the run-on process makes the problem analytically intractable, therefore Morbidelli et al. (2006) proposed to represent its effects through an empirical term, g [.], additional to 〈In (t ) 〉:
{MK s [(rmin + R) Fc , 0]−MK s [rmin Fc , 0]}+
+ rmin +
θ
(4)
{MK s [(rmin + R) Fc , 2]−MK s [rmin Fc , 2]}+
2 rmin
Ψ θ
1 {MK s [ < r > (j + 1) < r > j + 1
(1)
where subscript i is adopted for variables in the i-th spatial cell and p at the ponding time. H [.] is the Heaviside step function, fi is the local infiltration rate and v0 is the discharge per unit surface representing run-on. As N → ∞, Eq. (1) becomes an integral equation which involves the pdfs of Ks and r and provides the expected areal-average infiltration for a given value of F. Starting from this integral equation, under the conditions of negligible run-on and r invariant with time, Govindaraju et al. (2006) using the extended Green-Ampt model (Mein and Larson, 1973) and choosing F as the independent variable derived the following equation for ensemble-averaged areal infiltration rate, 〈In (F ) 〉, from infinite realizations at the field-scale:
< In (F ) >=
Fc , 0]}+
( − ) + ⎡F + Ψ (θs−θin ) ln Ψ (θ −s θ )in+ F ⎤· s in ⎣ ⎦ ∞ · {MK s [ < r > Fc , −1]} + ΨΔθ ∑ j = 1
N
∑ fi H (F −Fpi) ⎫⎬
F {1−MK s [ < r >
for
ω = 1, 2
(6)
Eq. (2) can be rewritten as:
< I (F ) > = r {1−MK s [rFc , 0]} +
Ψ (θs−θin ) + F MK s [rFc , 1] F
(7)
and Eq. (4) keeps the same form with 〈r 〉 substituted by r . For local rainfall rate variable with time and represented by a stepwise function, it is first required to reformulate the extended GreenAmpt model for steady rainfall. Considering the possible existence of a cumulative infiltration F0 at the beginning of a rainfall event, t0 , the cumulative infiltration in the ith cell during the unsaturated stage is expressed as:
(3)
where K a and ω represent the first and second argument, respectively, of the MK s function in Eq. (2) and fK (K ) is the pdf of Ks . Furthermore, a relation between F and the corresponding expected time 〈t (F )〉 was proposed as:
Fi (t ) = Fi0 + ri (t −t0),
t ≤ tpi, Fi < Fpi
Fig. 1. Grain size distribution of the soil used for the experiments. 837
(8)
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For spatially variable rainfall this leads to rewriting Eq. (4) as:
< t (F ) > =
F t0− < r0>
+
F {1−MK s [< r
+ ⎡F + Ψ (θs−θin ) ln ⎣ ∞ + Ψ (θs−θin ) ∑ j = 1
(
Table 1 Main characteristics of the study soil: θr and θs are the residual and saturated soil water contents, respectively, Ψb the air entry head, λ the Brooks–Corey pore size distribution index, c and d empirical coefficients (Smith et al., 1993).
> Fc , 0]}+
Ψ (θs − θin ) Ψ (θs − θin) + F
)⎤⎦·{M
1 {MK s [< r (j + 1) < r > j + 1
K s [
> Fc , −1]}+
> Fc , j]} (9)
while Eqs. (2) and (5) remain unchanged. For spatially variable and unsteady rainfall we have generally available a stepwise function 〈r (t )〉 represented by successive pulses, thus Eqs. (2), (5) and (9) can be successively applied within each time step. Specifically, at the end of the first time step with < t (F ) = t0 + Δt , F (t0 + Δt ) is obtained as solution of Eq. (9) and then from Eqs. (2) and (5), 〈In (F )〉 and 〈I (t0 + Δt )〉, respectively, can be computed. The values of F (t0 + Δt ) and t0 + Δt are then used as new F0 and t0 for next time step with a new expected value of 〈r 〉. The same procedure can be used for larger times. Finally, we note that the model application is fairly simple because it requires the solution of ordinary algebraic equations (Eqs. (2), (5) and (9) for unsteady rainfall) with very limited computational effort. The first two moments of the pdfs of Ks and r have to be assessed on the basis of direct measurements or values determined in similar areas or using pedotransfer functions.
Soil components
Clay
Weight (%) Hydraulic properties
28 θr 0.070
θs 0.36
Silt
Sand
57 Ψb (mm) −500
15 λ 0.2
c 5
d (mm) 50
wind and temperature fluctuations. The measurements of rainfall and surface runoff used in this study were made in 2013/2014. This experimental plot (using natural rainfall as the water source) was later modified by the inclusion of an artificial rainfall system (Morbidelli et al., 2017). In this study a tipping bucket system (Fig. 2) was used to record the overland flow discharging from the plot while the infiltrated water flow discharging from the drainage layer was not examined. Later, Morbidelli et al. (2017) used additional tipping buckets to analyze intermediate and deep flow. In this study infiltration has been computed as difference between the experimental values of rainfall and overland flow. According to the USDA soil classification (Linsley et al., 1992), this soil, whose components are shown in Table 1, was a silty loam. Its characteristics incorporated in the functional forms expressing the hydraulic properties (Smith et al., 1993) are shown in Table 1, with values taken out from Morbidelli et al. (2011, 2014) and Flammini et al. (2018). As to Ks , the expected areal-average value, 〈Ks〉, was determined in the absence of entrapped air by Morbidelli et al. (2017) who performed deep flow measurements for steady conditions under artificial rainfalls of long duration. In the same work Morbidelli et al. (2017) determined the spatial variability of Ks from measurements carried out using classical devices (double ring infiltrometer, CSIRO permeameter and Guelph permeameter). Four vertical profiles of soil moisture were monitored in the locations shown in Fig. 2 through the Time Reflectometry Technique (TDR). Each profile was determined by four buriable three-rod waveguides of length 20 cm inserted horizontally at 5, 15, 25 and 35 cm below the soil surface. The measurements of soil water content, obtained from the TDR signal through the universal calibration curve of Topp et al. (1980), were recorded at intervals of 30 min. The whole system was
3. Experimental system A closed plot with internal dimensions of 8.1 m × 8.7 m was constructed with impermeable base and sides. A 15 cm gravel drainage layer (grain size of few centimeters) was placed on the impermeable base. A geotextile with a 3 mm depth was placed on the upper surface of the drainage layer to prevent the passage of fine particles. A natural soil was first split up into several diameter classes and then carefully meshed to create a soil without roots and stones. Therefore the soil was characterized by a spatially uniform grain size distribution (Fig. 1) even though differences in soil packing could not be completely eliminated. This soil was placed on the geotextile to an evenly packed thickness of 70 cm. The bare upper surface of the experimental plot had a slope of 4%. The impermeable sides and base were designed to prevent overland flow or groundwater entry from outside the plot area. The upper bare surface of the plot received natural rainfall and was exposed to natural
Fig. 2. Schematic representation of the experimental plot and monitoring system: locations of soil moisture vertical profile measurements (v1-v4), and tipping bucket systems for the measurement of rainfall and runoff. 838
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5. Experimental results
used to investigate the movement of the wetting front and to estimate cumulative infiltration during a rainfall event. Measurements of rainfall rate and surface runoff at the soil surface outlet were performed with time resolution of 5 min by tipping bucket sensors (Fig. 2). Air temperature, relative humidity and wind speed were also observed at time intervals of 5 min. Measurements of pan evaporation were also performed.
As above discussed the model solution requires the assessment of 〈Ks〉 under conditions associated to natural rainfall-runoff events. Its value was determined by the following calibration procedure. Eight rainfall-runoff events were selected to estimate 〈Ks〉. The value of 〈Ks〉 which gave the best agreement between model simulations and experimental results obtained for both surface runoff and areal-average infiltration was adopted. This optimal value was found to be 1 mmh−1. The quantities εQ and εF represent the relative errors in the cumulative surface runoff Q , and cumulative infiltration F , respectively, and are defined as follows for any event:
4. Selected rainfall-runoff events and assessment of model input data The model solution in terms of 〈I 〉 as a function of expected time 〈t 〉 requires knowledge of (1) the first moments of the pdfs of Ks and r , and (2) the second moments henceforth expressed through the associated coefficients of variation cv (Ks ) and cv (r ) . In this investigation, 〈r 〉 has been assumed equal to the value observed by the available raingauge and cv (r ) has been assessed as 0.05 on the basis of earlier unpublished measurements performed in a similar area. This estimate is representative for frontal rainfalls over the experimental plot that is weakly influenced by local orography (Corradini, 1985). In addition, in the experiments of this work 〈Ks〉 was expected to have a value less than that earlier obtained in the same plot by Morbidelli et al. (2017) because they worked under different conditions. Specifically they used steady conditions with saturation of the whole soil layer, produced through artificial rainfalls of long duration, and in the absence of entrapped air. Therefore, because of the possible presence of entrapped air under natural rainfall events, the model has to be applied using for 〈Ks〉 the expected saturated hydraulic conductivity at natural saturation. It has been deduced by adjustment through calibration events of the experimental value obtained by Morbidelli et al. (2017). This calibration procedure, to a certain extent, could also correct different structural deficiencies of the model. However, the physical meaning of 〈Ks〉 should not be lost because, as later shown, the 〈Ks〉 calibrated value was found to be in a typical range of fine-textured soils. In any case, we note that the value of 〈Ks〉 cannot be derived through an upscaling of a classical functional form (f.i. Eq. (5), Corradini et al., 1997) for point hydraulic conductivity. This is due to the fact that such a functional form involves spatially variable soil hydraulic properties through a nonlinear dependence. The quantity cv (Ks ) was taken equal to 1.0, in agreement with measurements available in the literature for similar soil types (Smettem and Clother, 1989; Mohanty et al., 1994). This choice was also suggested by the experimental results, obtained through classical devices by Morbidelli et al. (2017), from which similar levels of heterogeneity of Ks at the plot scale were observed. Further quantities required by the model are θin and Ψ . The initial soil water content was selected as the spatially averaged value obtained from the TDR measurements. The suction head was deduced as a numerical solution of the net capillary drive (Melone et al., 2006) referred to the range θin−θs and expressed through the functional forms of the soil hydraulic properties. Sixteen natural rainfall-runoff events were selected for 〈Ks〉 calibration of and testing. We have selected all the events occurred in the analysis period (autumn-winter) for which there was a significant production of surface runoff (at least 0.5 mm in depth). Furthermore we note that in this period heavy rainfalls were not observed. Table 2 summarizes these events that have been grouped so as to have two sets with comparable characteristics in terms of total rainfall and observed surface runoff as well as of the parameters θin and Ψ estimated in advance. Most events started from high values of θin with respect to the saturated value θs = 0.36. All the events involved moderate rainfall rates occurred during the period Autumn-Spring when the soil surface was not affected by cracks and formation of sealing layers. The absence of a crust was deduced from the observed shapes of the vertical profiles (v1-v4) of θ which were found to be typical of vertically homogeneous soils.
εQ = (Qsim−Qob)/Qob
(10)
εF = (Fsim−Fob)/Fob
(11)
where subscripts “sim” and “ob” stand for “simulated” and “observed”, respectively. As can be seen in Table 3, on the average, fairly limited relative errors in the cumulative surface runoff (|εQ| = 0.24) and very low relative errors in the cumulative infiltration (|εF | = 0.06) were obtained from calibration events. Furthermore, Fig. 3 shows that the temporal evolution of runoff and field-scale infiltration depth (obtained as difference of rainfall and runoff being evaporation negligible) are wellrepresented by the model, particularly for the infiltration depth. A lower accuracy in the simulation of surface runoff was found for the event of February 26, 2014 with εQ = 0.65 even if the shape of the hydrograph (Fig. 4a) appears to be reasonable considering that at least the periods with and without runoff are well identified. For the same event, the model behavior was found to be satisfactory in terms of both cumulative infiltration depth (εF = 0.08) and evolution of infiltration depth (Fig. 4b). A set of eight rainfall-runoff events with characteristics similar, in terms of range of total rainfall and surface runoff, to those of the Table 2 Characteristics of the selected rainfall-runoff events, divided into calibration and validation groups; θin and Ψ are the initial soil water content and suction head at the wetting front, respectively, in the experimental plot.
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Table 3 Comparison of results derived from experiments and simulated by the model for the rainfall-runoff events used in the calibration and validation stages. The quantities εQ and εF represent the relative errors in the cumulative runoff and cumulative infiltration, respectively.
equation and then laboratory experiments (Melone et al., 2006), was applied. Fig. 7a displays an appropriate simulation of the vertical profile v1 carried out adopting the value Ks = 4 mmh−1 obtained by a calibration procedure and using the soil characteristics given in Table 1. Furthermore, with the same value of Ks the point model simulated adequately the vertical profile after a redistribution period of 12 h (Fig. 7b). The same approach used for the vertical profile v2 provided Ks = 0.4 mmh−1. In Figs. 6 and 7 θin is represented by an interpolated constant value which fitted properly its observed value at t = 0 . This is required by the vertical profile shape adopted in the selected point infiltration model. Furthermore, the existence of a large spatial heterogeneity of Ks was supported by the very different cumulative infiltration values which were found equal to 7.4, 0.3, 3.7 and 5.4 mm in the verticals v1, v2, v3 and v4 of Fig. 2, respectively. For the sake of simplicity, frequently point infiltration models are used at larger spatial scales. In the case the point infiltration model developed by Corradini et al. (1997) was applied with Ks uniform through the plot and equal to 4 mmh−1 surface runoff would not be generated. The same model used with Ks spatially uniform and equal to 1 mmh−1, obtained by the calibration performed for 〈Ks〉, would produce as surface runoff of 0.5 mm, while the observed runoff was of 2.36 mm. In the same context, the point infiltration model applied with a Ks specifically calibrated through the available events (Table 2) would provide substantial distortions of the surface runoff hydrographs. This agrees with the results earlier obtained by Corradini et al. (2002) through Monte Carlo simulations. After all, it is widely recognized that the use of a single space-invariant value of Ks in substitution of a random field determines an early runoff generation in more permeable soil points and a delayed runoff production in less permeable soil points, with the result of a direct hydrograph narrower than the actual one (Corradini et al., 2002). The above results indicate that the stochastic problem of areal infiltration linked with the spatial heterogeneity of Ks cannot be replaced by a deterministic approximation based on the use of a single value of Ks considered uniform through the study field. On the other hand, the
calibration events was selected to test the adopted model. The comparison of the simulated results and data derived from these experiments, carried out under natural conditions at the plot scale, may be used to assess model reliability in the simulation of infiltration rate and surface runoff production. In addition, this comparison can enable us to evaluate the efficiency of the specific approximations utilized in developing the model framework. The results obtained by this comparison for the cumulative depths of infiltration and surface runoff are summarized in Table 3 with |εQ| = 0.27, |εF |=0.05 and, as for the calibration events, there is not a significant trend of the errors εQ and εF . Therefore, the model behavior for the calibration and validation events was found to be comparable in terms of cumulative depths of both runoff and infiltration for natural rainfall events. Four representative rainfall-runoff events were selected to illustrate the validation of the model in Fig. 5, where a comparison of observed and modelled values of surface runoff and infiltration is shown. The errors in the surface runoff are appreciable, but the shape of the computed histograms is well simulated. In addition, observed histograms of the infiltration depth are reproduced with a great accuracy. We note that the event of April 4, 2014 was characterized by the lowest model accuracy for the surface runoff but the hydrograph shape was fairly well simulated and that, in any case, the infiltration depth as a function of time was well-reproduced. 6. Discussion of results The adopted model relies upon the existence of spatial variability of Ks through the study plot. This variability can also be highlighted through an analysis of the observed vertical profiles of soil moisture content. For the sample event of October 9, 2013, earlier used in the calibration stage, Fig. 6 illustrates the significant difference between the profile shapes observed in the representative verticals v1 and v2 of Fig. 2. This difference can be substantially ascribed to a spatial variation of Ks , as it can be deduced by the simulation of the local vertical profiles by an accurate point infiltration model. Specifically, the model proposed by Corradini et al. (1997), validated using the Richards 840
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Fig. 3. Calibration stage: comparison of results derived from experiments and simulated by the model for four representative rainfall-runoff events. (a) Surface runoff depth and (b) infiltration depth, as functions of time. The rainfall patterns are shown.
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Fig. 4. Calibration stage: comparison of results derived from the experiment and simulated by the model for the rainfall-runoff event of February 26, 2014. (a) Surface runoff rate and (b) infiltration rate, as functions of time. The rainfall pattern is also shown.
validation) one event was expected to be influenced by this process. In this context, for example, a comparison of the events of October 5, 2013 (θin = 0.238) and March 4, 2014 (θin = 0.328) reveals an appreciable model performance (Fig. 5). This suggests that the two model components, the semi-analytical and the empirical ones, interact fairly well to represent field-scale behavior.
investigated areal model as a whole represents the main features of the surface runoff hydrograph satisfactorily, and simulates the observed infiltration patterns with a high accuracy. In addition, on the basis of the characteristics of the available experiments, it is also possible to deduce the satisfactory reliability of the model components because: 1. the hyetographs shown in Figs. 3 and 5 highlight that the model has been calibrated and checked with appropriate results using timedependent natural rainfall events. Therefore, the stepwise approach adopted to extend the basic model developed for steady rainfall to the formulation for variable rainfall rate appears to be a viable approach; 2. the trend of the “observed” areal infiltration as a function of time does not experience significant distortions (Figs. 3b and 5b). This indicates that the crucial assumption of F as independent variable combined with a formulation of the expected time provides a suitable mathematical approximation of the physical process; 3. through Monte Carlo simulations, Morbidelli et al. (2006) showed that run-on has a significant effect for moderate storms and high values of cv (Ks ) and/or cv (r ) , provided θin ≪ θs . Furthermore, they found the run-on effect to be negligible for θin → θs . Therefore for most events of Table 2, characterized by values of θin (in the range ~0.30–0.33) very close to θs (0.36), the role of run-on was substantially limited. On the other hand, in each group (calibration and
7. Conclusions The natural spatial variability of hydraulic soil properties, particularly that of the saturated hydraulic conductivity, makes the implementation of distributed/semi-distributed rainfall-runoff models with point infiltration approaches largely inapplicable. In this regard, areal infiltration models that reduce the stochastic problem to a tractable deterministic problem incurring modest computational effort are desirable. This study highlights the performance of such a model for an artificial field plot characterized by a vertically homogeneous soil under numerous time-varying rainfall events. The results reveal the model, reassessed for application to unsteady rainfall patterns, to show promise in terms of simulation of areal infiltration and surface runoff. The comparable performance associated with events with negligible and significant run-on effect suggests that semi-analytical and empirical model components contribute to field-scale infiltration appropriately. 842
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Fig. 5. Validation stage: comparison of results derived from experiments and simulated by the model for four representative rainfall-runoff events. (a) Surface runoff depth and (b) infiltration depth, as functions of time. The rainfall patterns are also shown.
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Fig. 6. Event October 9, 2013: observed soil moisture values at the end of rainfall and at different depths along two (v1-v2) of the four verticals of Fig. 2.
Fig. 7. Event October 9, 2013: comparison of soil water profile simulated by the model for local infiltration by Corradini et al. (1997) and soil moisture measurements obtained at different depths along the vertical v1 of Fig. 2 (a) at the end of the event (t = 1.5 h) and (b) during the redistribution stage (t = 13.5 h). In the simulation, the soil parameters of Table 1, the calibrated value of Ks = 4 mmh−1 and the initial soil water content locally observed (θin = 0.3045) were assumed.
precipitation areas. Finally, the model was formulated for vertically homogeneous soils and events not involving redistribution of soil water content within a single rainfall-runoff event. The last process can significantly affect the model results mainly for long redistribution periods. Furthermore, in the representation of the expected areal-average infiltration for cases with formation of a thin crust or a more permeable upper layer, due to tilled soils or/and presence of vegetation, the adoption of a two-layered soil with uniform Ks in each layer (with Ks = 〈Ks〉 in the top layer) can be considered a satisfactory first approximation for 〈I (t )〉.
Furthermore, good agreement between observed and simulated infiltration and surface runoff depths as functions of time highlights the applicability of the framework of the model, which relies upon the adoption of cumulative infiltration as an independent variable linked with the expected time. In addition, results indicate the model is also suitable for applications under conditions of time-dependent rainfall events. The solution of the mathematical model is simple and requires the knowledge of the first two moments of pdfs of both Ks and r . The tested model enables us to link the representation of infiltration and Hortonian surface runoff at different spatial scales. In this context, we remark that these processes are quantified at field scale starting from a classical point infiltration approach. Then, distributed/semi-distributed formulations of the hydrological response to rainfall at the watershed scale can be developed through an ensemble of field-scale elements, each one with spatially uniform soil typology. Of course, this scheme has to be consistent with the representation of homogeneous
Acknowledgment This research was mainly financed by the Ministry of Education, University and Research, Italy (PRIN 2015).
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Appendix Morbidelli et al. (2006) deduced the empirical term in the right hand side of Eq. (5) through a variety of Monte-Carlo simulations. In explicit form this term was expressed by a Gamma function (Askey and Roy, 2010) as:
g [t ∗; cv (Ks ), cv (r )] = < r > a (t ∗)bexp (−cG t ∗)
(A.1)
with the parameters a , b and cG given by the following relations:
a = 2.8[cv (r ) + cv (Ks )]0.36
(A.2)
b = 5.35−6.32[cv (r ) cv (Ks )]
(A.3) 0.3
< r > / < Ks > ⎤ cG = 2.7 + 0.3 ⎡ ⎢ cv (r ) cv (Ks ) ⎦ ⎥ ⎣
(A.4)
where 〈Ks〉 stands for the expected field-scale value of Ks . Both 〈Ks〉 and 〈r 〉 are expressed in units of millimeters per hour. In Eq. (A.1) t ∗ = t / tp , with tp – the time to ponding - computed by the extended Green-Ampt relation (Mein and Larson, 1973) using the expected values of Ks and r as:
tp =
Ks Ψ (θs−θin ) 〈r 〉 ( 〈r 〉−〈Ks 〉 )
(A.5)
Eq. (A.2) holds for θi ≪ θs while for θi → θs we have a → 0, furthermore Eq. (A.3) is undefined for cv (r ) and/or cv (Ks ) equal to 0.
input to field-scale infiltration and surface runoff models. Water Resour. Manage. 26 (7), 1793–1807. Morbidelli, R., Saltalippi, C., Flammini, A., Rossi, E., Corradini, C., 2014. Soil water content vertical profiles under natural conditions: matching of experiments and simulations by a conceptual model. Hydrol. Process. 28, 4732–4742. Morbidelli, R., Saltalippi, C., Flammini, A., Cifrodelli, M., Picciafuoco, T., Corradini, C., Govindaraju, R.S., 2017. In situ measurements of soil saturated hydraulic conductivity: Assessment of reliability through rainfall–runoff experiments. Hydrol. Process. 31, 3084–3094. Nahar, N., Govindaraju, R.S., Corradini, C., Morbidelli, R., 2004. Role of run-on for describing field-scale infiltration and overland flow over spatially variable soils. J. Hydrol. 286, 36–51. Philip, J.R., 1957. The theory of infiltration, 1, The infiltration equation and its solution. Soil Sci. 83, 345–357. Philip, J.R., 1969. Theory of infiltration. Adv. Hydrosci. 5, 215–216. Russo, D., Bresler, E., 1981. Soil hydraulic properties as stochastic processes, 1, Analysis of field spatial variability. Soil Sci. Soc. Am. J. 45, 682–687. Russo, D., Bresler, E., 1982. A univariate versus a multivariate parameter distribution in a stochastic-conceptual analysis of unsaturated flow. Water Resour. Res. 18 (3), 483–488. Saghafian, B., Julien, P.Y., Ogden, F.L., 1995. Similarity in catchment response, 1, Stationary rainstorms. Water Resour. Res. 31 (6), 1533–1541. Sharma, M.L., Barron, R.J.W., Fernie, M.S., 1987. Areal distribution of infiltration parameters and some soil physical properties in lateritic catchments. J. Hydrol. 94, 109–127. Sivapalan, M., Wood, E.F., 1986. Spatial heterogeneity and scale in the infiltration response of catchments. Scale problems in hydrology. In: Dordrecht, Holland, Gupta, V.K. (Eds.), Water Science and Technology Library. D. Reidel Publishing Company, pp. 81–106. Smerdon, B.D., Drews, J.E., 2017. Groundwater recharge: the intersection between humanity and hydrogeology. J. Hydrol. 555, 909–911. Smettem, K.R.J., Clother, B.I.E., 1989. Measuring unsaturated sorptivity and hydraulic conductivity using multiple disc permeameters. J. Soil Sci. 40, 563–568. Smith, R.E., Parlange, J.-Y., 1978. A parameter-efficient hydrologic infiltration model. Water Resour. Res. 14 (3), 533–538. Smith, R.E., Hebbert, R.H.B., 1979. A Monte Carlo analysis of the hydrologic effects of spatial variability of infiltration. Water Resour. Res. 15 (2), 419–429. Smith, R.E., Corradini, C., Melone, F., 1993. Modeling infiltration for multistorm runoff events. Water Resour. Res. 29 (1), 133–144. Smith, R.E., Goodrich, D.C., 2000. A model to simulate rainfall excess patterns on randomly heterogeneous areas. J. Hydrol. Eng. 5 (4), 355–362. Topp, G.C., Davis, J.L., Annan, A.P., 1980. Electromagnetic determination of soil water content: measurements in coaxial transmission lines. Water Resour. Res. 16, 574–582. Warrick, A.W., Nielsen, D.R., 1980. Spatial variability of soil physical properties in the field. In: Hillel, D. (Ed.), Applications of Soil Physics. Academic Press, New York, pp. 319–344. Wood, E.F., Sivapalan, M., Beven, K., 1986. Scale effects in infiltration and runoff production. Proc. of the Symposium on Conjunctive Water Use. IAHS Publ. N. 156, Budapest. Woolhiser, D.A., Smith, R.E., Giraldez, J.-V., 1996. Effects of spatial variability of saturated hydraulic conductivity on Hortonian overland flow. Water Resour. Res. 32 (3), 671–678.
References Askey, R.A., Roy, R., Olver, F.W.J., Lozier, D.M., Boisvert, R.F., Clark, C.W., 2010. Series expansions. NIST Handbook of Mathematical Functions. Cambridge University Press. Castelli, F., 1996. A simplified stochastic model for infiltration into a heterogeneous soil forced by random precipitation. Adv. Water Resour. 19 (3), 133–144. Corradini, C., 1985. Analysis of the effects of orography on surface rainfall by a parameterized numerical model. J. Hydrol. 77 (1–4), 19–30. Corradini, C., 2014. Soil moisture in the development of hydrological processes and its determination at different spatial scales. J. Hydrol. 516, 1–5. Corradini, C., Singh, V.P., 1985. Effect of spatial variability of effective rainfall on direct runoff by a geomorphologic approach. J. Hydrol. 81 (1–2), 27–43. Corradini, C., Melone, F., Smith, R.E., 1997. A unified model for infiltration and redistribution during complex rainfall patterns. J. Hydrol. 192, 104–124. Corradini, C., Morbidelli, R., Melone, F., 1998. On the interaction between infiltration and Hortonian runoff. J. Hydrol. 204, 52–67. Corradini, C., Govindaraju, R.S., Morbidelli, R., 2002. Simplified modelling of areal average infiltration at the hillslope scale. Hydrol. Process. 16, 1757–1770. Fiori, A., de Barros, F.P.J., 2015. Groundwater flow and transport in aquifers: insights from modeling and characterization at the field scale. J. Hydrol. 531 (1), 1. Flammini, A., Corradini, C., Morbidelli, R., Saltalippi, C., Picciafuoco, T., Giráldez, J.V., 2018. Experimental analyses of the evaporation dynamics in bare soils under natural conditions. Water Resour. Manage. 32 (3), 1153–1166. Govindaraju, R.S., Morbidelli, R., Corradini, C., 2001. Areal infiltration modelling over soils with spatially correlated hydraulic conductivities. J. Hydrol. Eng. 6 (2), 150–158. Govindaraju, R.S., Corradini, C., Morbidelli, R., 2006. A semi-analytical model of expected areal-average infiltration under spatial heterogeneity of rainfall and soil saturated hydraulic conductivity. J. Hydrol. 316, 184–194. Linsley, R.K., Franzini, J.B., Freyberg, D.L., Tchobanoglous, G., 1992. Water – Resources Engineering. McGraw-Hill, Singapore. Loague, K., Gander, G.A., 1990. R-5 revisited, 1, Spatial variability of infiltration on a small rangeland catchment. Water Resour. Res. 26 (5), 957–971. Maller, R.A., Sharma, M.L., 1984. Aspects of rainfall excess from spatially varying hydrological parameters. J. Hydrol., Amsterdam 67, 115–127. Mein, R.G., Larson, C.L., 1973. Modeling infiltration during a steady rain. Water Resour. Res. 9, 384–394. Melone, F., Morbidelli, R., Corradini, C., Saltalippi, C., 2006. Laboratory experimental check of a conceptual model for infiltration under complex rainfall patterns. Hydrol. Process. 20 (3), 439–452. Milly, P.C.D., Eagleson, P.S., 1982. Infiltration and evaporation at inhomogeneous land surfaces. MIT Ralph M. Parsons Laboratory Rep. No. 278, Massachusetts Institute of Technology, Boston. Mohanty, B.P., Ankeny, M.D., Horton, R., Kanwar, R.S., 1994. Spatial analysis of hydraulic conductivity measured using disc infiltrometers. Water Resour. Res. 30, 2489–2498. Morbidelli, R., Corradini, C., Govindaraju, R.S., 2006. A field-scale infiltration model accounting for spatial heterogeneity of rainfall and soil saturated hydraulic conductivity. Hydrol. Process. 20, 1465–1481. Morbidelli, R., Corradini, C., Saltalippi, C., Flammini, A., Rossi, E., 2011. Infiltration-soil moisture redistribution under natural conditions: experimental evidence as a guideline for realizing simulation models. Hydrol. Earth Syst. Sci. 15, 1–9. Morbidelli, R., Corradini, C., Saltalippi, C., Brocca, L., 2012. Initial soil water content as
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Research papers
Reassessment of a semi-analytical field-scale infiltration model through experiments under natural rainfall events
T
⁎
Alessia Flamminia, , Renato Morbidellia, Carla Saltalippia, Tommaso Picciafuocoa, Corrado Corradinia, Rao S. Govindarajub a b
Dept. of Civil and Environmental Engineering, University of Perugia, via G. Duranti 93, 06125 Perugia, Italy Lyles School of Civil Engineering, Purdue University, West Lafayette, IN 47907, United States
A R T I C LE I N FO
A B S T R A C T
This manuscript was handled by Dr Marco Borga, Editor-in-Chief
Operative distributed/semi-distributed rainfall-runoff formulations need a mathematically tractable component for estimating the expected areal-average infiltration rate, 〈I 〉, while point infiltration models are usually adopted for this purpose. A model for 〈I 〉, suitable for hydrological applications, was proposed by Morbidelli et al. (2006, Hydrological Processes, 20: 1465–1481). The main model tenet is the use of cumulative infiltration as an independent variable and replacement of time by its expected value. While this model - as with many other models for field-scale infiltration - was tested through Monte Carlo methods, it remains untested with experimental observations. In this study, a reassessment of the model’s applicability is conducted through experimental evidence from several rainfall-runoff events observed under natural conditions at the plot scale over a slight slope characterized by a silty loam soil. During each experiment rainfall rate, surface runoff at the plot outlet and soil water content vertical profiles were monitored at short time intervals. An overall analysis of our results suggests that the model reliably simulates 〈I 〉 and surface runoff as functions of time and is also suitable for applications under time-dependent rainfall events. The process of redistribution under rainfall hiatus is not addressed by this model.
Keywords: Areal infiltration Infiltration modelling Rainfall-runoff experiments
1. Introduction Simulation of many natural hydrological processes at field and watershed scales relies upon proper representation of areal average infiltration (Govindaraju et al., 2001). Areal-average infiltration at the field-scale has a crucial role in surface runoff prediction by distributed models that represent the hydrological response of a watershed through a superposition of field-scale responses. Field-scale infiltration also governs the evolution of soil water content (Corradini, 2014), the transport of pollutants in the vadose zone (Fiori and de Barros, 2015) and the recharge of aquifers (Smerdon and Drews, 2017). Areal averaging is a popular method to incorporate the natural spatial variability of hydraulic soil properties (Warrick and Nielsen, 1980; Sharma et al., 1987; Loague and Gander, 1990), primarily that of the soil saturated hydraulic conductivity, Ks (Russo and Bresler, 1981, 1982), of the initial soil moisture content, θin , and rainfall rate, r (Corradini and Singh, 1985). At the field scale, the effects of the spatial heterogeneity of θin (Morbidelli et al., 2012) and r (Morbidelli et al., 2006) on the areal infiltration are significantly lower than those due to the variability of Ks , mainly because of the much larger coefficient of variation of Ks . ⁎
The spatial variability of Ks is generally represented by a random field characterized by a log-normal probability density function, pdf, with a level of spatial correlation that can have an important effect on the variance of the field-scale infiltration (Govindaraju et al., 2001). Consequently, the mathematical problem concerning areal infiltration cannot be properly solved analytically and requires Monte-Carlo simulations of Ks along with a local rainfall infiltration model (Philip, 1957, 1969; Mein and Larson, 1973; Smith and Parlange, 1978; Melone et al., 2006). This procedure was adopted, for instance, by Smith and Hebbert (1979), Maller and Sharma (1984), Milly and Eagleson (1982) and Saghafian et al. (1995). Thus, an appropriate estimate of the ensemble-averaged infiltration at the field scale is obtained, but this method is too complex for large hydrological applications. Some alternative modeling approaches have therefore been proposed to obtain approximate solutions in the absence of the run-on process that represents the infiltration of overland flow running downslope into pervious downstream areas. Along these lines significant and widely recognized contributions were presented by Sivapalan and Wood (1986), Smith and Goodrich (2000) and Govindaraju et al. (2001). Sivapalan and Wood (1986) developed approximate relations for
Corresponding author. E-mail address: alessia.fl[email protected] (A. Flammini).
https://doi.org/10.1016/j.jhydrol.2018.08.073 Received 22 May 2018; Received in revised form 27 August 2018; Accepted 28 August 2018 Available online 01 September 2018 0022-1694/ © 2018 Elsevier B.V. All rights reserved.
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Nomenclature
〈r 〉 I 〈I 〉 〈In〉 K 〈Ks〉 f F F0
Subscripts in s r p i sim ob c
initial condition saturated condition residual condition ponding condition i-th spatial cell simulated value observed value critical value
θ Ψ ε N rmin
Functions and variables fK
probability density function of saturated hydraulic conductivity g[.] function representing the run-on process H[.] Heaviside step function MKs[Ka,ω]moments of saturated hydraulic conductivity with arguments Ka and ω t time t0 time of beginning of a rainfall event t* time scaled to the time to ponding 〈t 〉 expected time Δt time step r rainfall rate
v0 Q
expected areal-average rainfall rate areal-average infiltration rate expected areal-average infiltration rate ensemble-averaged areal infiltration rate hydraulic conductivity expected areal-average saturated hydraulic conductivity infiltration rate cumulative infiltration depth cumulative infiltration depth at the beginning of a rainfall event volumetric soil water content water suction head relative error discrete number of spatial cells and rmin+R extreme values of the uniform probability density function of rainfall rate run-on in terms of discharge per unit surface cumulative surface runoff
Parameters Ψb λ c and d a, b, cG cv
air entry suction Brooks-Corey pore size distribution parameter empirical coefficients parameters of a Gamma function coefficient of variation of a random variable
local infiltration model and an equation describing the surface runoff routing through the study area is required. Alternatively, a simplified solution can be obtained by a mathematical model set up by Morbidelli et al. (2006) who combined the above-mentioned semi-analytical formulation of Govindaraju et al. (2006) with a semi-empirical component for the run-on process. The model is fairly simple and requires the solution of a set of algebraic equations for given values of the first two moments of the probability density functions of both Ks and r . Most of the models available for determining the areal-average infiltration have been developed and/or tested through Monte-Carlo techniques but, to our knowledge, in depth studies of their accuracy in simulating observed hydrological processes have not been carried out. The main objective of this paper is to test the model formulated by Govindaraju et al. (2006) and then extended by Morbidelli et al. (2006) using natural rainfall-runoff events observed by an experimental system specifically designed at the plot scale. Starting from a natural soil, an artificial plot with bounded sides, uniform soil grain size distribution and significant horizontal spatial variability of Ks has been used. Natural events consistent with the basic assumptions earlier adopted to develop the model, namely absence of a surface sealing layer as well as of long periods of soil water redistribution, have been selected. The overall model accuracy and the effectiveness of the crucial model components are investigated in this paper.
areal average and variance of infiltration rate considering a random variability of Ks and steady rainfall invariant in space. Monte-Carlo simulations were used to validate these relations but a single realization was used for the averaging procedure and the involved errors were not clearly expressed. Smith and Goodrich (2000) developed a parametric formulation for areal-average infiltration, obtained through the Latin Hypercube sampling method and only one realization of Ks for each combination of mean and coefficient of variation. Govindaraju et al. (2001) presented a semi-analytical/conceptual model for the expected areal-average infiltration obtained with Ks as a random variable and uniform rainfall. Three different model versions were proposed: the first one based on the assumption of cumulative infiltration as the independent variable associated with an expected infiltration time, the second and third versions were obtained through a representation of point cumulative infiltration by a series expansion and a parameterized expression, respectively. Some approaches for estimating the areal-average infiltration in the case of a joint random horizontal variability of Ks and r have been also proposed. Wood et al. (1986) found expressions for areal mean and variance of infiltration rate but with the same limitations aforementioned for Sivapalan and Wood (1986). Castelli (1996) proposed an analytical formulation for areal-average infiltration but its application is limited by the simplified representation of infiltration at the local scale. Govindaraju et al. (2006) developed a semi-analytical model under the condition of a joint spatial heterogeneity of Ks and r both assumed as random variables with pdf of lognormal type for Ks and uniform type between two extreme values for r . The mathematical formulation used cumulative infiltration as the independent variable along with a formulation for expected time. The aforementioned modeling approaches assume negligible runon, thus limiting their application. This limitation is tested by the results from investigations performed using Monte-Carlo simulations (Saghafian et al., 1995; Woolhiser et al., 1996; Corradini et al., 1998; Nahar et al., 2004). However, when including the run-on process, the estimate of the ensemble-averaged infiltration by Monte-Carlo technique becomes very complex considering that a coupled solution of a
2. A short account of a semi-analytical areal-average infiltration modeling and its reassessment for application to real rainfall patterns Building on the conceptualization of Govindaraju et al. (2006), Morbidelli et al. (2006) proposed an approach for representing direct rainfall infiltration with the run-on process. For the purpose of this study all the variables are considered to be independent. For negligible effects of the spatial heterogeneity of θin , considering a total of N cells and a given realization of r and Ks the areal-average infiltration rate, I , at a given cumulative infiltration, F , can be written through a sum of the contributions of unsaturated and saturated cells 836
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A. Flammini et al.
as:
< t (F ) >= N
I (F ) =
1⎧ ∑ (ri + v0 )[1−H (F −Fpi)] + N ⎨ i=1 ⎩
⎭
i=1
(
1 2RFc2
−
2R
(
R 2
) ) {M
1 Fc
(
−
1 Fc
( ) {M
+
1 M [r F , Fc K s min c
rmin + R R 1 RFc
〈I (t ) 〉 ≃ 〈In (t ) 〉 + g [t ∗; cv (Ks ), cv (r )]
{1−MK s [(rmin + R) Fc , 0]}+
+
K s [(rmin
K s [(rmin
1]
∫0
Ka
K ωfK (K ) dK
Fc , j]}
(5)
where 〈I 〉 is the expected areal-average infiltration rate, cv (Ks ) and cv (r ) denote the coefficients of variation of Ks and r , respectively, and t ∗ is a time scaled to the time to ponding. The explicit formulation of g [.] is given in the Appendix. For spatially uniform rainfall and negligible run-on, in the absence of the empirical term in Eq. (5) and considering that:
+ R) Fc , 1]−MK s [rmin Fc , 1]}+
+ R) Fc , 2]−MK s [rmin Fc , 2]}+ (2)
MK s [(rmin + R) Fc , ω]−MK s [rmin Fc , ω] = 0
where rmin and rmin + R are the extreme values of a uniform pdf of r ; F Fc = [Ψ (θ − θ ) + F ] , with θs soil water content at saturation and Ψ soil s in suction head, linked with the critical values of saturated hydraulic conductivity, K c = Fc r , which determine surface saturation for a given F ; and MK s [.,.] is expressed by:
MK s [K a, ω] =
)
where 〈r 〉 and 〈t (F )〉 represent the expected field-scale rainfall rate and the expected time associated to a given F value, respectively, estimated through the averaging procedure over the ensemble of realizations of Ks and r . Because of the spatial variability of both Ks and r , the time t associated with a given F takes on different values in each cell. Therefore the expected time was assumed to approximate the actual relation 〈In (t )〉 for the ensemble-averaged areal infiltration rate as a function of time by that obtained from combining 〈In (F )〉 and 〈t (F )〉. We also note that the series in the last term of Eq. (4) converges rapidly and can be approximated by the first five terms for most applications. The inclusion of the run-on process makes the problem analytically intractable, therefore Morbidelli et al. (2006) proposed to represent its effects through an empirical term, g [.], additional to 〈In (t ) 〉:
{MK s [(rmin + R) Fc , 0]−MK s [rmin Fc , 0]}+
+ rmin +
θ
(4)
{MK s [(rmin + R) Fc , 2]−MK s [rmin Fc , 2]}+
2 rmin
Ψ θ
1 {MK s [ < r > (j + 1) < r > j + 1
(1)
where subscript i is adopted for variables in the i-th spatial cell and p at the ponding time. H [.] is the Heaviside step function, fi is the local infiltration rate and v0 is the discharge per unit surface representing run-on. As N → ∞, Eq. (1) becomes an integral equation which involves the pdfs of Ks and r and provides the expected areal-average infiltration for a given value of F. Starting from this integral equation, under the conditions of negligible run-on and r invariant with time, Govindaraju et al. (2006) using the extended Green-Ampt model (Mein and Larson, 1973) and choosing F as the independent variable derived the following equation for ensemble-averaged areal infiltration rate, 〈In (F ) 〉, from infinite realizations at the field-scale:
< In (F ) >=
Fc , 0]}+
( − ) + ⎡F + Ψ (θs−θin ) ln Ψ (θ −s θ )in+ F ⎤· s in ⎣ ⎦ ∞ · {MK s [ < r > Fc , −1]} + ΨΔθ ∑ j = 1
N
∑ fi H (F −Fpi) ⎫⎬
F {1−MK s [ < r >
for
ω = 1, 2
(6)
Eq. (2) can be rewritten as:
< I (F ) > = r {1−MK s [rFc , 0]} +
Ψ (θs−θin ) + F MK s [rFc , 1] F
(7)
and Eq. (4) keeps the same form with 〈r 〉 substituted by r . For local rainfall rate variable with time and represented by a stepwise function, it is first required to reformulate the extended GreenAmpt model for steady rainfall. Considering the possible existence of a cumulative infiltration F0 at the beginning of a rainfall event, t0 , the cumulative infiltration in the ith cell during the unsaturated stage is expressed as:
(3)
where K a and ω represent the first and second argument, respectively, of the MK s function in Eq. (2) and fK (K ) is the pdf of Ks . Furthermore, a relation between F and the corresponding expected time 〈t (F )〉 was proposed as:
Fi (t ) = Fi0 + ri (t −t0),
t ≤ tpi, Fi < Fpi
Fig. 1. Grain size distribution of the soil used for the experiments. 837
(8)
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A. Flammini et al.
For spatially variable rainfall this leads to rewriting Eq. (4) as:
< t (F ) > =
F t0− < r0>
+
F {1−MK s [< r
+ ⎡F + Ψ (θs−θin ) ln ⎣ ∞ + Ψ (θs−θin ) ∑ j = 1
(
Table 1 Main characteristics of the study soil: θr and θs are the residual and saturated soil water contents, respectively, Ψb the air entry head, λ the Brooks–Corey pore size distribution index, c and d empirical coefficients (Smith et al., 1993).
> Fc , 0]}+
Ψ (θs − θin ) Ψ (θs − θin) + F
)⎤⎦·{M
1 {MK s [< r (j + 1) < r > j + 1
K s [
> Fc , −1]}+
> Fc , j]} (9)
while Eqs. (2) and (5) remain unchanged. For spatially variable and unsteady rainfall we have generally available a stepwise function 〈r (t )〉 represented by successive pulses, thus Eqs. (2), (5) and (9) can be successively applied within each time step. Specifically, at the end of the first time step with < t (F ) = t0 + Δt , F (t0 + Δt ) is obtained as solution of Eq. (9) and then from Eqs. (2) and (5), 〈In (F )〉 and 〈I (t0 + Δt )〉, respectively, can be computed. The values of F (t0 + Δt ) and t0 + Δt are then used as new F0 and t0 for next time step with a new expected value of 〈r 〉. The same procedure can be used for larger times. Finally, we note that the model application is fairly simple because it requires the solution of ordinary algebraic equations (Eqs. (2), (5) and (9) for unsteady rainfall) with very limited computational effort. The first two moments of the pdfs of Ks and r have to be assessed on the basis of direct measurements or values determined in similar areas or using pedotransfer functions.
Soil components
Clay
Weight (%) Hydraulic properties
28 θr 0.070
θs 0.36
Silt
Sand
57 Ψb (mm) −500
15 λ 0.2
c 5
d (mm) 50
wind and temperature fluctuations. The measurements of rainfall and surface runoff used in this study were made in 2013/2014. This experimental plot (using natural rainfall as the water source) was later modified by the inclusion of an artificial rainfall system (Morbidelli et al., 2017). In this study a tipping bucket system (Fig. 2) was used to record the overland flow discharging from the plot while the infiltrated water flow discharging from the drainage layer was not examined. Later, Morbidelli et al. (2017) used additional tipping buckets to analyze intermediate and deep flow. In this study infiltration has been computed as difference between the experimental values of rainfall and overland flow. According to the USDA soil classification (Linsley et al., 1992), this soil, whose components are shown in Table 1, was a silty loam. Its characteristics incorporated in the functional forms expressing the hydraulic properties (Smith et al., 1993) are shown in Table 1, with values taken out from Morbidelli et al. (2011, 2014) and Flammini et al. (2018). As to Ks , the expected areal-average value, 〈Ks〉, was determined in the absence of entrapped air by Morbidelli et al. (2017) who performed deep flow measurements for steady conditions under artificial rainfalls of long duration. In the same work Morbidelli et al. (2017) determined the spatial variability of Ks from measurements carried out using classical devices (double ring infiltrometer, CSIRO permeameter and Guelph permeameter). Four vertical profiles of soil moisture were monitored in the locations shown in Fig. 2 through the Time Reflectometry Technique (TDR). Each profile was determined by four buriable three-rod waveguides of length 20 cm inserted horizontally at 5, 15, 25 and 35 cm below the soil surface. The measurements of soil water content, obtained from the TDR signal through the universal calibration curve of Topp et al. (1980), were recorded at intervals of 30 min. The whole system was
3. Experimental system A closed plot with internal dimensions of 8.1 m × 8.7 m was constructed with impermeable base and sides. A 15 cm gravel drainage layer (grain size of few centimeters) was placed on the impermeable base. A geotextile with a 3 mm depth was placed on the upper surface of the drainage layer to prevent the passage of fine particles. A natural soil was first split up into several diameter classes and then carefully meshed to create a soil without roots and stones. Therefore the soil was characterized by a spatially uniform grain size distribution (Fig. 1) even though differences in soil packing could not be completely eliminated. This soil was placed on the geotextile to an evenly packed thickness of 70 cm. The bare upper surface of the experimental plot had a slope of 4%. The impermeable sides and base were designed to prevent overland flow or groundwater entry from outside the plot area. The upper bare surface of the plot received natural rainfall and was exposed to natural
Fig. 2. Schematic representation of the experimental plot and monitoring system: locations of soil moisture vertical profile measurements (v1-v4), and tipping bucket systems for the measurement of rainfall and runoff. 838
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5. Experimental results
used to investigate the movement of the wetting front and to estimate cumulative infiltration during a rainfall event. Measurements of rainfall rate and surface runoff at the soil surface outlet were performed with time resolution of 5 min by tipping bucket sensors (Fig. 2). Air temperature, relative humidity and wind speed were also observed at time intervals of 5 min. Measurements of pan evaporation were also performed.
As above discussed the model solution requires the assessment of 〈Ks〉 under conditions associated to natural rainfall-runoff events. Its value was determined by the following calibration procedure. Eight rainfall-runoff events were selected to estimate 〈Ks〉. The value of 〈Ks〉 which gave the best agreement between model simulations and experimental results obtained for both surface runoff and areal-average infiltration was adopted. This optimal value was found to be 1 mmh−1. The quantities εQ and εF represent the relative errors in the cumulative surface runoff Q , and cumulative infiltration F , respectively, and are defined as follows for any event:
4. Selected rainfall-runoff events and assessment of model input data The model solution in terms of 〈I 〉 as a function of expected time 〈t 〉 requires knowledge of (1) the first moments of the pdfs of Ks and r , and (2) the second moments henceforth expressed through the associated coefficients of variation cv (Ks ) and cv (r ) . In this investigation, 〈r 〉 has been assumed equal to the value observed by the available raingauge and cv (r ) has been assessed as 0.05 on the basis of earlier unpublished measurements performed in a similar area. This estimate is representative for frontal rainfalls over the experimental plot that is weakly influenced by local orography (Corradini, 1985). In addition, in the experiments of this work 〈Ks〉 was expected to have a value less than that earlier obtained in the same plot by Morbidelli et al. (2017) because they worked under different conditions. Specifically they used steady conditions with saturation of the whole soil layer, produced through artificial rainfalls of long duration, and in the absence of entrapped air. Therefore, because of the possible presence of entrapped air under natural rainfall events, the model has to be applied using for 〈Ks〉 the expected saturated hydraulic conductivity at natural saturation. It has been deduced by adjustment through calibration events of the experimental value obtained by Morbidelli et al. (2017). This calibration procedure, to a certain extent, could also correct different structural deficiencies of the model. However, the physical meaning of 〈Ks〉 should not be lost because, as later shown, the 〈Ks〉 calibrated value was found to be in a typical range of fine-textured soils. In any case, we note that the value of 〈Ks〉 cannot be derived through an upscaling of a classical functional form (f.i. Eq. (5), Corradini et al., 1997) for point hydraulic conductivity. This is due to the fact that such a functional form involves spatially variable soil hydraulic properties through a nonlinear dependence. The quantity cv (Ks ) was taken equal to 1.0, in agreement with measurements available in the literature for similar soil types (Smettem and Clother, 1989; Mohanty et al., 1994). This choice was also suggested by the experimental results, obtained through classical devices by Morbidelli et al. (2017), from which similar levels of heterogeneity of Ks at the plot scale were observed. Further quantities required by the model are θin and Ψ . The initial soil water content was selected as the spatially averaged value obtained from the TDR measurements. The suction head was deduced as a numerical solution of the net capillary drive (Melone et al., 2006) referred to the range θin−θs and expressed through the functional forms of the soil hydraulic properties. Sixteen natural rainfall-runoff events were selected for 〈Ks〉 calibration of and testing. We have selected all the events occurred in the analysis period (autumn-winter) for which there was a significant production of surface runoff (at least 0.5 mm in depth). Furthermore we note that in this period heavy rainfalls were not observed. Table 2 summarizes these events that have been grouped so as to have two sets with comparable characteristics in terms of total rainfall and observed surface runoff as well as of the parameters θin and Ψ estimated in advance. Most events started from high values of θin with respect to the saturated value θs = 0.36. All the events involved moderate rainfall rates occurred during the period Autumn-Spring when the soil surface was not affected by cracks and formation of sealing layers. The absence of a crust was deduced from the observed shapes of the vertical profiles (v1-v4) of θ which were found to be typical of vertically homogeneous soils.
εQ = (Qsim−Qob)/Qob
(10)
εF = (Fsim−Fob)/Fob
(11)
where subscripts “sim” and “ob” stand for “simulated” and “observed”, respectively. As can be seen in Table 3, on the average, fairly limited relative errors in the cumulative surface runoff (|εQ| = 0.24) and very low relative errors in the cumulative infiltration (|εF | = 0.06) were obtained from calibration events. Furthermore, Fig. 3 shows that the temporal evolution of runoff and field-scale infiltration depth (obtained as difference of rainfall and runoff being evaporation negligible) are wellrepresented by the model, particularly for the infiltration depth. A lower accuracy in the simulation of surface runoff was found for the event of February 26, 2014 with εQ = 0.65 even if the shape of the hydrograph (Fig. 4a) appears to be reasonable considering that at least the periods with and without runoff are well identified. For the same event, the model behavior was found to be satisfactory in terms of both cumulative infiltration depth (εF = 0.08) and evolution of infiltration depth (Fig. 4b). A set of eight rainfall-runoff events with characteristics similar, in terms of range of total rainfall and surface runoff, to those of the Table 2 Characteristics of the selected rainfall-runoff events, divided into calibration and validation groups; θin and Ψ are the initial soil water content and suction head at the wetting front, respectively, in the experimental plot.
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Table 3 Comparison of results derived from experiments and simulated by the model for the rainfall-runoff events used in the calibration and validation stages. The quantities εQ and εF represent the relative errors in the cumulative runoff and cumulative infiltration, respectively.
equation and then laboratory experiments (Melone et al., 2006), was applied. Fig. 7a displays an appropriate simulation of the vertical profile v1 carried out adopting the value Ks = 4 mmh−1 obtained by a calibration procedure and using the soil characteristics given in Table 1. Furthermore, with the same value of Ks the point model simulated adequately the vertical profile after a redistribution period of 12 h (Fig. 7b). The same approach used for the vertical profile v2 provided Ks = 0.4 mmh−1. In Figs. 6 and 7 θin is represented by an interpolated constant value which fitted properly its observed value at t = 0 . This is required by the vertical profile shape adopted in the selected point infiltration model. Furthermore, the existence of a large spatial heterogeneity of Ks was supported by the very different cumulative infiltration values which were found equal to 7.4, 0.3, 3.7 and 5.4 mm in the verticals v1, v2, v3 and v4 of Fig. 2, respectively. For the sake of simplicity, frequently point infiltration models are used at larger spatial scales. In the case the point infiltration model developed by Corradini et al. (1997) was applied with Ks uniform through the plot and equal to 4 mmh−1 surface runoff would not be generated. The same model used with Ks spatially uniform and equal to 1 mmh−1, obtained by the calibration performed for 〈Ks〉, would produce as surface runoff of 0.5 mm, while the observed runoff was of 2.36 mm. In the same context, the point infiltration model applied with a Ks specifically calibrated through the available events (Table 2) would provide substantial distortions of the surface runoff hydrographs. This agrees with the results earlier obtained by Corradini et al. (2002) through Monte Carlo simulations. After all, it is widely recognized that the use of a single space-invariant value of Ks in substitution of a random field determines an early runoff generation in more permeable soil points and a delayed runoff production in less permeable soil points, with the result of a direct hydrograph narrower than the actual one (Corradini et al., 2002). The above results indicate that the stochastic problem of areal infiltration linked with the spatial heterogeneity of Ks cannot be replaced by a deterministic approximation based on the use of a single value of Ks considered uniform through the study field. On the other hand, the
calibration events was selected to test the adopted model. The comparison of the simulated results and data derived from these experiments, carried out under natural conditions at the plot scale, may be used to assess model reliability in the simulation of infiltration rate and surface runoff production. In addition, this comparison can enable us to evaluate the efficiency of the specific approximations utilized in developing the model framework. The results obtained by this comparison for the cumulative depths of infiltration and surface runoff are summarized in Table 3 with |εQ| = 0.27, |εF |=0.05 and, as for the calibration events, there is not a significant trend of the errors εQ and εF . Therefore, the model behavior for the calibration and validation events was found to be comparable in terms of cumulative depths of both runoff and infiltration for natural rainfall events. Four representative rainfall-runoff events were selected to illustrate the validation of the model in Fig. 5, where a comparison of observed and modelled values of surface runoff and infiltration is shown. The errors in the surface runoff are appreciable, but the shape of the computed histograms is well simulated. In addition, observed histograms of the infiltration depth are reproduced with a great accuracy. We note that the event of April 4, 2014 was characterized by the lowest model accuracy for the surface runoff but the hydrograph shape was fairly well simulated and that, in any case, the infiltration depth as a function of time was well-reproduced. 6. Discussion of results The adopted model relies upon the existence of spatial variability of Ks through the study plot. This variability can also be highlighted through an analysis of the observed vertical profiles of soil moisture content. For the sample event of October 9, 2013, earlier used in the calibration stage, Fig. 6 illustrates the significant difference between the profile shapes observed in the representative verticals v1 and v2 of Fig. 2. This difference can be substantially ascribed to a spatial variation of Ks , as it can be deduced by the simulation of the local vertical profiles by an accurate point infiltration model. Specifically, the model proposed by Corradini et al. (1997), validated using the Richards 840
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Fig. 3. Calibration stage: comparison of results derived from experiments and simulated by the model for four representative rainfall-runoff events. (a) Surface runoff depth and (b) infiltration depth, as functions of time. The rainfall patterns are shown.
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Fig. 4. Calibration stage: comparison of results derived from the experiment and simulated by the model for the rainfall-runoff event of February 26, 2014. (a) Surface runoff rate and (b) infiltration rate, as functions of time. The rainfall pattern is also shown.
validation) one event was expected to be influenced by this process. In this context, for example, a comparison of the events of October 5, 2013 (θin = 0.238) and March 4, 2014 (θin = 0.328) reveals an appreciable model performance (Fig. 5). This suggests that the two model components, the semi-analytical and the empirical ones, interact fairly well to represent field-scale behavior.
investigated areal model as a whole represents the main features of the surface runoff hydrograph satisfactorily, and simulates the observed infiltration patterns with a high accuracy. In addition, on the basis of the characteristics of the available experiments, it is also possible to deduce the satisfactory reliability of the model components because: 1. the hyetographs shown in Figs. 3 and 5 highlight that the model has been calibrated and checked with appropriate results using timedependent natural rainfall events. Therefore, the stepwise approach adopted to extend the basic model developed for steady rainfall to the formulation for variable rainfall rate appears to be a viable approach; 2. the trend of the “observed” areal infiltration as a function of time does not experience significant distortions (Figs. 3b and 5b). This indicates that the crucial assumption of F as independent variable combined with a formulation of the expected time provides a suitable mathematical approximation of the physical process; 3. through Monte Carlo simulations, Morbidelli et al. (2006) showed that run-on has a significant effect for moderate storms and high values of cv (Ks ) and/or cv (r ) , provided θin ≪ θs . Furthermore, they found the run-on effect to be negligible for θin → θs . Therefore for most events of Table 2, characterized by values of θin (in the range ~0.30–0.33) very close to θs (0.36), the role of run-on was substantially limited. On the other hand, in each group (calibration and
7. Conclusions The natural spatial variability of hydraulic soil properties, particularly that of the saturated hydraulic conductivity, makes the implementation of distributed/semi-distributed rainfall-runoff models with point infiltration approaches largely inapplicable. In this regard, areal infiltration models that reduce the stochastic problem to a tractable deterministic problem incurring modest computational effort are desirable. This study highlights the performance of such a model for an artificial field plot characterized by a vertically homogeneous soil under numerous time-varying rainfall events. The results reveal the model, reassessed for application to unsteady rainfall patterns, to show promise in terms of simulation of areal infiltration and surface runoff. The comparable performance associated with events with negligible and significant run-on effect suggests that semi-analytical and empirical model components contribute to field-scale infiltration appropriately. 842
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Fig. 5. Validation stage: comparison of results derived from experiments and simulated by the model for four representative rainfall-runoff events. (a) Surface runoff depth and (b) infiltration depth, as functions of time. The rainfall patterns are also shown.
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Fig. 6. Event October 9, 2013: observed soil moisture values at the end of rainfall and at different depths along two (v1-v2) of the four verticals of Fig. 2.
Fig. 7. Event October 9, 2013: comparison of soil water profile simulated by the model for local infiltration by Corradini et al. (1997) and soil moisture measurements obtained at different depths along the vertical v1 of Fig. 2 (a) at the end of the event (t = 1.5 h) and (b) during the redistribution stage (t = 13.5 h). In the simulation, the soil parameters of Table 1, the calibrated value of Ks = 4 mmh−1 and the initial soil water content locally observed (θin = 0.3045) were assumed.
precipitation areas. Finally, the model was formulated for vertically homogeneous soils and events not involving redistribution of soil water content within a single rainfall-runoff event. The last process can significantly affect the model results mainly for long redistribution periods. Furthermore, in the representation of the expected areal-average infiltration for cases with formation of a thin crust or a more permeable upper layer, due to tilled soils or/and presence of vegetation, the adoption of a two-layered soil with uniform Ks in each layer (with Ks = 〈Ks〉 in the top layer) can be considered a satisfactory first approximation for 〈I (t )〉.
Furthermore, good agreement between observed and simulated infiltration and surface runoff depths as functions of time highlights the applicability of the framework of the model, which relies upon the adoption of cumulative infiltration as an independent variable linked with the expected time. In addition, results indicate the model is also suitable for applications under conditions of time-dependent rainfall events. The solution of the mathematical model is simple and requires the knowledge of the first two moments of pdfs of both Ks and r . The tested model enables us to link the representation of infiltration and Hortonian surface runoff at different spatial scales. In this context, we remark that these processes are quantified at field scale starting from a classical point infiltration approach. Then, distributed/semi-distributed formulations of the hydrological response to rainfall at the watershed scale can be developed through an ensemble of field-scale elements, each one with spatially uniform soil typology. Of course, this scheme has to be consistent with the representation of homogeneous
Acknowledgment This research was mainly financed by the Ministry of Education, University and Research, Italy (PRIN 2015).
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Appendix Morbidelli et al. (2006) deduced the empirical term in the right hand side of Eq. (5) through a variety of Monte-Carlo simulations. In explicit form this term was expressed by a Gamma function (Askey and Roy, 2010) as:
g [t ∗; cv (Ks ), cv (r )] = < r > a (t ∗)bexp (−cG t ∗)
(A.1)
with the parameters a , b and cG given by the following relations:
a = 2.8[cv (r ) + cv (Ks )]0.36
(A.2)
b = 5.35−6.32[cv (r ) cv (Ks )]
(A.3) 0.3
< r > / < Ks > ⎤ cG = 2.7 + 0.3 ⎡ ⎢ cv (r ) cv (Ks ) ⎦ ⎥ ⎣
(A.4)
where 〈Ks〉 stands for the expected field-scale value of Ks . Both 〈Ks〉 and 〈r 〉 are expressed in units of millimeters per hour. In Eq. (A.1) t ∗ = t / tp , with tp – the time to ponding - computed by the extended Green-Ampt relation (Mein and Larson, 1973) using the expected values of Ks and r as:
tp =
Ks Ψ (θs−θin ) 〈r 〉 ( 〈r 〉−〈Ks 〉 )
(A.5)
Eq. (A.2) holds for θi ≪ θs while for θi → θs we have a → 0, furthermore Eq. (A.3) is undefined for cv (r ) and/or cv (Ks ) equal to 0.
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