Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Hillslope Runoff Generation - Comparing Different Modeling Approaches

Hillslope Runoff Generation - Comparing Different Modeling Approaches J. Hydrol. Hydromech., 60, 2012, 2, 73­86 DOI: 10.2478/v10098-012-0007-2 HELENA PAVELKOVÁ, MICHAL DOHNAL, TOMÁS VOGEL*) Czech Technical University in Prague, Faculty of Civil Engineering, Thákurova 7, 166 29 Prague 6, Czech Republic; *) Corresponding author, Tel.: +420 22435 4341; fax: +420 224 354 793; Mailto: vogel@fsv.cvut.cz This study focuses on modeling hydrological responses of shallow hillslope soil in a headwater catchment. The research is conducted using data from the experimental site Uhlíská in Jizera Mountains, Czech Republic. To compare different approaches of runoff generation modeling, three models were used: (1) onedimensional variably saturated flow model S1D, based on the dual-continuum formulation of Richards' equation; (2) zero-dimensional nonlinear morphological element model GEOTRANSF; and (3) semidistributed model utilizing the topographic index similarity assumption - TOPMODEL. Hillslope runoff hydrographs and soil water storage variations predicted by the simplified catchment scale models (GEOTRANSF and TOPMODEL) were compared with the respective responses generated by the more physically based local scale model S1D. Both models, GEOTRANSF and TOPMODEL, were found to predict general trends of hydrographs quite satisfactorily; however their ability to correctly predict soil water storages and inter-compartment fluxes was limited. KEY WORDS: Rainfall-Runoff Modeling, Hillslope Discharge, Soil Water Storage, Transmisivity, Topographic Index, Richards' Equation, Preferential Flow. Helena Pavelková, Michal Dohnal, Tomás Vogel: TVORBA ODTOKU ZE SVAHU ­ POROVNÁNÍ ROZDÍLNÝCH MODELOVÝCH PÍSTUP. J. Hydrol. Hydromech., 60, 2012, 2; 29 lit., 8 obr., 2 tab. Studie je zamena na modelování hydrologické reakce mlké svahové pdy v pramenné cásti povodí Nisy, k výzkumu byla pouzita data z experimentálního povodí Uhlíská. Porovnání rzných konceptuálních pedstav modelování odtoku bylo uskutecnno pro: (1) jednorozmrný model promnliv nasyceného proudní S1D; (2) model zalozený na bezrozmrném nelineárním morfologickém prvku - GEOTRANSF a (3) semi-distribuovaný model vyuzívající principu podobnosti na základ topografického indexu - TOPMODEL. Hydrogramy odtoku ze svahu a zmny zásob vody v pd vypoctené zjednodusenými modely GEOTRANSF a TOPMODEL byly porovnány s odpovídajícími odezvami fyzikáln zalozeného modelu S1D. Oba modely, GEOTRANSF i TOPMODEL, byly pomrn úspsné v pedpovdi základních trend hydrogram odtoku, jejich schopnost správn pedpovídat zásoby vody v pd a toky mezi nimi vsak byla omezená. KLÍCOVÁ SLOVA: srázko-odtokové modelování, odtok ze svahu, zásoba vody v pd, transmisivita, topografický index, Richardsova rovnice, preferencní proudní. Introduction In the last few decades, processes related to runoff formation in headwater catchments, especially during extreme hydrological events, have received growing attention. A fast reaction of discharge to rainstorms is typical for small mountainous catchments. Shallow subsurface runoff is considered to be a significant component of the rainfall-runoff relationship in such catchments. It is generally assumed that this type of runoff (further on referred to as hypodermic flow) takes place after a saturated zone has formed above an impermeable or less permeable subsurface layer (hard bedrock or a less permeable soil layer). Soil water then flows laterally in the direction determined by the local slope of the soil-bedrock interface. Of the currently available models used to describe actual processes taking place at the hillslope scale, probably the most detailed one is the model based on three-dimensional Richards' equation. Such a model and other similar physically based 73 models are strongly non-linear and require iterative numerical solution. A vast number of linearized equations have to be solved when this type of model is applied to a hydrological problem, even if modest spatial dimensions are considered (such as dimensions of a single hillslope). To estimate parameters for these models, detailed knowledge of soil hydraulic characteristics is necessary. This information is commonly unavailable, especially at the entire catchment scale (Lichner et al., 2011). In order to obtain hydrological predictions at this scale, it is necessary to use simplified conceptual models. Over the years, many conceptual approaches to subsurface runoff modeling were discussed and compared in the literature. The kinematic wave equation (e.g. Fan and Bras, 1998; Troch et al., 2002) and the Boussinesq equation (e.g. Troch et al., 2003; Hilberts et al., 2004) were applied to model hypodermic flow. Richards' equation was used to describe hillslope soil water movement under variably saturated conditions e.g. by Paniconi et al. (2003), Hilberts et al. (2007) and Cordano and Rigon (2008). A combination of Richards' equation and diffusion wave equation was used by Vogel at al. (2005a, 2005b) to model hypodermic flow in the Uhlíská catchment. TOPMODEL (e.g. Beven, 2001) has been widely discussed and tested (Ambroise et al., 1996; Blazková and Beven, 1997). Applications of TOPMODEL in the Uhlíská catchment have been reported by Blazková and Beven (1997) and Blazková et al. (2002a and 2002b). Recently, Majone et al. (2010) presented a simpler model (further on referred to as GEOTRANSF) developed for modeling subsurface runoff genera- tion using non-linear relationship between runoff and soil water storage. In this paper, three models were used to simulate runoff from a hillslope: (1) one-dimensional variably saturated flow model S1D, based on dualcontinuum formulation of the Richards equation; (2) zero-dimensional non-linear morphological element model GEOTRANSF; and (3) model utilizing the topographic index similarity assumption - TOPMODEL. The main objective of this comparative study was to analyze the performance of the two simplified models (GEOTRANSF and TOPMODEL) by comparing them with the more physically based model (S1D), and to assess the ability of the models to predict soil water storages and internal fluxes in addition to discharge hydrographs. Material and methods S1D model This physically based local scale model was designed to simulate flow of water in soils with preferential pathways. It is based on the dualcontinuum approach. The approach assumes that soil water moves in two separate flow domains, one representing the soil matrix and the other representing the network of preferential pathways (denoted as SM domain and PF domain). Under saturated or nearly saturated conditions, soil water contained in the preferential flow domain flows considerably faster than that in the soil matrix domain. Soil water flow is in each of the two domains described by Richards' equation (in a similar way as in Gerke and van Genuchten, 1993). The resulting set of the two coupled governing equations can be written as: wf C f h f t h f w f K f + 1 - w f S f - w (h f - hm ) z z (1) wmCm h hm = wm K m m + 1 - wm S m + w (h f - hm ) t z z (2) where f refers to PF-domain and m to SM-domain, wm and wf ­ volume fractions of the respective domains (wm + wf = 1), C ­ the differential soil water capacity [m-1], h ­ the pressure head [m], K ­ the unsaturated hydraulic conductivity [m s-1], S ­ the local intensity of root extraction [s-1] and w ­ the 74 transfer coefficient controlling the dynamic exchange of water between the two flow domains [s-1 m-1]. The governing equations are coupled through a first order transfer term w [s-1] defined according to Gerke and van Genuchten (1993) as w = w (h f - hm ) . Vertical fluxes predicted at the lower boundary of the dual-continuum system (by S1D model) determine the recharge rate for the hypodermic flow. In general, the vertical recharge rate R [m s-1] can be calculated from a simple continuity equation formulated at the soil-bedrock interface: The dual set of governing equations for soil water flow is solved numerically by the S1D code. The most recent implementation of S1D is described in Vogel et al. (2010a) R = w f (q f 1 - q f 2 ) + wm (q m1 - qm 2 ) (3) P m f where the first term represents the recharge contributions from the PF domain and the second one from the SM domain, respectively, q1 and q2 are soil water fluxes [m s-1] above and below the soil/bedrock interface (Fig. 1). The q2 fluxes represent seepage to deeper horizons. In the present application of the model, we assume that qm2 = qm1 and qf2 = 0, i.e. the deep percolation is associated with the SM-domain flux while the PF-domain flux contributes to hypodermic flow. This leads to R = = wf qf1. S1D can handle hysteresis of soil hydraulic properties. However, this option was not used in the present study, i.e. hysteresis was assumed to be insignificant. This was partly supported by the results of our previous study (Dohnal et al., 2006). In addition, we assume that the modeled hillslope segment at Uhlíska (being short, shallow and well permeable) does not cause any substantial runoff transformation in lateral direction and the hillslope discharge can be computed from a simple quasisteady-state formula: qm1 qf1 qm2 qf2 Fig. 1. Schematic of S1D domains and fluxes. The soil is decomposed into two flow domains: the soil matrix domain (m) and preferential flow domain (f). qm1 and qf1 are the lower boundary soil water fluxes, qm2 and qf2 are the fluxes representing leakage to deeper horizons, and w is the inter-domain soil water transfer. GEOTRANSF model A simple conceptual model of subsurface runoff generation was suggested by Majone et al. (2005, 2010). In this paper, we refer to this model as GEOTRANSF - according to Majone et al. (2005). In GEOTRANSF, locally generated vertical flux at the base of the soil profile is estimated from: Q(t ) = AR(t ) (4) where A is the hillslope contributing area [m2]. The local saturated zone storage SSZ [m], representing the amount of water which takes part in hypodermic flow, is determined by the depth of hypodermic stream hSZ [m] and the effective porosity [­], i.e.: SSZ (t ) = hSZ (t ) (5) S(t) - S0 q0 exp qh (t) = m q 0 for S S0 for S < S0 (7) If Q is calculated from (4), we can estimate hSZ from the kinematic wave approximation of Darcy's law (i.e., vertically integrated Darcy equation stripped of the pressure gradient): Q(t ) z = K eff hSZ (t ) B x (6) where m is an empirical parameter [m], S ­ the soil water storage [m], q0 ­ the specific discharge [m s-1], associated with a minimum threshold value of soil water storage S0. The vertical soil water flux is divided into two components (Fig. 2): (i) a fast flow through macropores recharging hypodermic flow: where Keff is the effective saturated hydraulic conductivity [m s-1], B ­ the local width of the hillslope [m] and z/x is equal to the cosine of the local hillslope angle. qs (t) = (1- c p )qh (t) (8) and (ii) a deep percolation component representing seepage to underlying horizons: 75 qdp (t) = c p qh (t) (9) where cp is the partition coefficient [­]. Water storage in the soil profile is calculated from the mass balance equation: S = P - E - qh t (10) where P is the infiltration intensity [m s-1] due to rainfall and E is the actual evapotranspiration intensity [m s-1]. Majone et al. (2010) used instantaneous unit hydrograph method to transform the recharge signal qs(t) into streamflow hydrograph. In our study, GEOTRANSF is first used to calculate the hypodermic flow recharge R(t) = qs(t). The hillslope discharge hydrograph is then simply estimated by applying the quasi-steady-state assumption (4). ilar values of I are considered to belong to the same topographic index class. TOPMODEL (e.g. Beven, 2001) is based on three basic assumptions: (i) quasi-steady-state relationship between the saturated zone recharge and the hillslope discharge, (ii) approximation of the effective hydraulic gradient by the local surface topographic slope and (iii) a simplified relationship between the saturated zone transmissivity and the local storage deficit. The TOPMODEL version used in this study assumes exponential relationship between transmissivity and storage deficit. According to the quasi-steady-state assumption, the hillslope discharge caused by saturated subsurface runoff can be estimated from: q j (t) = a j r(t) (12) where q is the subsurface discharge per unit contour length [m2 s-1] and r ­ the saturated zone recharge rate [m s-1], assumed to be spatially uniform. If exponential relationship between the transmissivity and storage deficit is assumed, local saturated zone storage deficit D [m] can be calculated from the corresponding value of soil-topographic index and the average storage deficit of the catchment: Non-linear reservoir D j - D = -m(I j - I ) (13) qh qs qdp Fig. 2. Schematic of GEOTRANSF fluxes. TOPMODEL TOPMODEL utilizes a concept of topographic index originally developed by Kirkby (1975) and extended to the soil-topographic index by Beven (1986): where m [m] is a parameter controlling the rate of decline of transmissivity with increasing storage deficit and the overbars refer to the catchment scale averages. The version of TOPMODEL applied in this paper (see Beven, 2001) uses three individually balanced water storages (Fig. 3): (i) spatially uniform variably saturated root zone storage, (ii) spatially distributed gravity-drainable unsaturated zone storage and (iii) catchment scale saturated zone storage represented by the average storage deficit. The root zone water balance, is calculated according to: S RZ S = P - E - RZ t t SRZ = max(SRZ ­ SRZ max, 0) (14) (15) I j = ln aj T0 tan j (11) where I is the topographic index, aj is the contributing area of the hillslope per unit contour length [m] drained through point j, T0 ­ the lateral transmissivity at full soil saturation [m2 s-1], ­ the local angle of the hillslope. The locations exhibiting sim76 where SRZ is the instantaneous water storage in the root zone [m], assumed to be uniform over the catchment, and SRZ max is the maximum value of SRZ. The amount of water in excess of SRZ max is redirected to the gravity-drainable unsaturated zone storage. The local water balance in the gravity-drainable compartment of unsaturated zone is evaluated for each topographic index class i by solving: In analogy to Eq. (5), this can be also expressed in terms of hypodermic flow depth: SUZ t S RZ - qv i t Di = (hSZ max - hSZi ) (23) (16) where qv is the vertical saturated zone recharge by gravity drainage from unsaturated zone [m s-1], which is estimated as: qvi = SUZi td Di (17) where SUZ is the water storage in the gravitydrainable compartment of the unsaturated zone [m], and td is a parameter [s m-1] interpreted as the mean residence time in SUZ per unit of saturated zone deficit. The catchment scale saturated zone recharge Qv [m3 s-1] is equal to the sum of local recharges, which can be expressed as the sum of all topographic index class contributions: Qv = Ai qvi (18) where Ai is the area associated with topographic index class i [m2]. Finally, the catchment scale water balance for the saturated zone is: D - Dt-t A t = (Qb - Qv )t-t . t (19) The saturated zone discharge at the catchment scale, Qb [m3 s-1], can be calculated from the average storage deficit D as: Qb = Q0 e- D/m . where hSZ max is equal to the maximum depth of the saturated zone, i.e. in our case the depth of the soil profile (about 75 cm). An important feature of TOPMODEL is the algorithm used for the determination of variable source areas contributing to rapid surface runoff (cf. Figs. 3 and 4). According to TOPMODEL, these areas are formed wherever the local saturated zone storage deficit D, calculated from (13), becomes zero (or negative). The saturation excess runoff generated over the variable source areas (Fig. 4) is then added to the combined (subsurface plus surface) discharge Q(t). Alternatively, surface runoff may be generated via the infiltration excess mechanism. However, this runoff mechanism is rarely if ever observed at Uhlíská and was not considered in the present application of TOPMODEL. Note the discrepancy between the postulated spatial uniformity of the saturated zone recharge, expressed in (12), and the distributed character of recharge rates calculated from (17). According to the author of TOPMODEL: "The water table is assumed to take up a shape AS IF the recharge rate was uniform across the catchment (even if the recharge is actually heterogeneous). Homogeneity is not the only problem, in fact ­ a kinematic analysis shows that if there is strong variation in recharge in space or time then it might actually take quite a long time to achieve such a steady state, while differences in soil transmissivity will also have an effect" (Beven, personal communication, 2012). (20) P E saturation excess The discharge corresponding to zero average deficit, Q0 [m3 s-1], is determined as: Q0 = Ae- I (21) RZ UZ where I is the mean value of the soil-topographic index over the catchment area A. The local saturated zone deficit can be interpreted as the difference between maximum possible saturated storage SSZ max and actual storage SSZ, i.e.: qv SZ Di = S SZ max - S SZi (22) Fig. 3. Schematic of TOPMODEL storages and fluxes (RZ = root zone storage, UZ = unsaturated zone storage, SZ = saturated zone storage). Experimental site and measurements The above described models were tested using data from the experimental catchment Uhlíská, located in the Jizera Mountains in northeast Czech Republic. The Uhlíská catchment lies in the upper part of the Cerná Nisa catchment where basic hydrological and climatological conditions have been monitored since 1982 by the Czech Hydrometeorological Institute. The Uhlíská catchment has an area of 1.78 km2 and an average altitude of 822 m above sea level. Average annual temperature is 4.7°C, annual precipitation exceeding 1300 mm. The catchment is generally covered by snow from November to April. The forest at Uhlíská was damaged acid deposition in 80's. The catchment is now covered by young spruce forest with grass undergrowth. Hydropedological survey has been conducted to determine infiltration and retention characteristics of the soil profile (Sanda, 1999; Zumr et al., 2007). The soil on the weathered porphyric granite bedrock is classified as Dystric Cambisol. The soil type varies from loamy sand to sandy loam and has a very coarse skeleton. The experimental site is equipped with automatic data collection systems for precipitation, temperature, soil water pressure, soil moisture, subsurface discharge, and other hydrological variables (Sanda, 1999). Shallow subsurface runoff is measured in a trench situated at the experimental hillslope Tomsovka. The average slope near the trench is 14%. The soil profile is approximately 75 cm deep followed by weathered bedrock. The trench consists of two sections, each four meters wide (denoted as section A and section B), and both equipped with a tipping bucket flow gauge for measuring subsurface runoff. It is assumed that each section of the trench collects water from an area of about 100 m2 (Císlerová et al., 1997, Sanda et al., 2006; Sanda and Císlerová, 2009). Model application Three different models - S1D, GEOTRANSF and TOPMODEL - were applied to rainfall-runoff data observed at the Uhlíská catchment in vegetation seasons 2000, 2001 and 2002. The models were used to simulate runoff from a micro-catchment, representing the Tomsovka hillslope site. The size of the micro-catchment was 78 set equal to the estimated contributing area of one of the two sections of the experimental trench. Both hillslope segments (draining to the respective section of the trench) were considered to be 75 cm deep, 25 m long and 4 m wide, with a constant slope of tan = 0.14. The initial soil water storage conditions applied at the beginning of vegetation seasons corresponded to the soil water pressures measured by tensiometers. The soil-atmosphere interactions involved natural rainfall and potential evapotranspiration estimated by Penman-Monteith equation. Hourly averaged rainfall and daily averaged evapotranspiration intensities were utilized as input data for all three models. TOPMODEL as well as GEOTRANSF were run with one-hour time steps. S1D uses adaptive time step associated with the numerical solution. The S1D model parameters (Tab. 1) were taken from our previous study (Vogel et al., 2010b) conducted for the same experimental site. The model parameters of GEOTRANSF and TOPMODEL were calibrated to match the runoff hydrographs observed in the season 2000. The parameters were optimized by means of the Levenberg-Marquardt algorithm (Marquardt, 1963). Alternatively, the calibration of GEOTRANSF and TOPMODEL was performed for the subsurface runoff hydrographs generated by the S1D model (for the season 2000), instead of using the observed hydrographs. In this case, the S1D model served as a substitute for the real hillslope. The main advantage of this approach was a significant reduction of complexity and the associated uncertainty in the substitute system, which allowed a more straightforward comparison of the simulated internal fluxes and soil water storages, otherwise unavailable for the real system (such as comparison of SRZ and SUZ storages in TOPMODEL with SM-domain and PFdomain storages in S1D). In GEOTRANSF, a single parameter was calibrated - the parameter m. The calibration yielded a value of m = 0.00803 m (for the GEOTRANSF vs. S1D calibration). The minimum specific discharge q0 and minimum soil water storage S0 were both set equal to zero. In TOPMODEL, the calibration of the parameters m and td resulted in m = 0.005 m and td = 244 h m-1 (for the TOPMODEL vs. S1D calibration). Note that the parameter m plays in TOPMODEL similar role as in GEOTRANSF (see Eqs. (20) and (7)). The root zone storage SRZ max was set equal to 0.165 m. The lateral transmissivity at full saturation was estimated as T0 = 0.781 m2 h-1. The maximum root zone storage SRZ max was assumed to be equal to the soil water field capacity, i.e. the amount of water stored in initially fully saturated soil after 48 hours of free draining. The value of SRZ max was determined by integrating the soil water content profile (in SM-domain) at the end of 48hour drainage period simulated by S1D model. The lateral transmissivity at full saturation T0 was assumed to be equal to the product of the lateral saturated hydraulic conductivity of the network of effective preferential pathways, estimated as Keff = 2500 cm d-1 and the total depth of the soil profile (75 cm). Since the soil-bedrock interface at Tomsovka is semipervious, part of the infiltrating water seeps to deeper horizons and does not contribute to the shallow subsurface discharge. To respect that, vertical flow qv calculated in TOPMODEL was divided into T a b l e 1. Soil hydraulic parameters. Depth [cm] Matrix 1 2 3 4 Preferential ­ 0­8 8 ­ 20 20 ­ 70 70 ­ 75 0 ­ 75 r [­] 0.20 0.20 0.20 0.20 0.01 s [­ ] 0.55 0.54 0.49 0.41 0.60 the deep percolation part qdp and the saturated zone recharge part qs using the same partitioning procedure as in GEOTRANSF (Eq. (8)). The catchment scale saturated zone recharge was therefore calculated as: Qv = Ai qsi , qsi = (1- c p )qvi . (24) The value of partition coefficient cp = 0.36, determined by S1D model, was also applied in GEOTRANSF and TOPMODEL. To calculate the local soil-topographic index and the local saturated zone deficit in TOPMODEL, the contributing hillslope area at Tomsovka was divided into five segments (Fig. 4) along the slope length (each 5 m long). To compare the models, three performance criteria were used in this study: (i) coefficient of determination R2, (ii) root mean square error RMSE, and (iii) the Nash-Sutcliffe efficiency NSE. Domain Layer [cm ] 0.050 0.050 0.020 0.020 0.050 -1 n [­ ] 2.00 1.50 1.20 1.20 3.00 Ks [cm d ] 567 67 17 1.3 5000 -1 hs [cm] 0.00 ­0.69 ­1.48 ­1.88 0.00 r and s are the residual and saturated water contents, hs ­ the air-entry value, and and n ­ empirical parameters (Vogel et al., 2001). Results and discussion Specific subsurface runoff simulated by the S1D model is compared to the observed hillslope discharge in Fig. 5. It is obvious that major hillslope responses occur after abundant rainfall distributed over a longer period of time. The simulated and measured hydrographs are of a similar shape; their rising limbs are very steep. The agreement between the observed and simulated discharges is not perfect, however the model is capable of reproducing the basic character of hillslope responses. Following model performance indices were determined for the three vegetation seasons (2000, 2001, 2002): R2 = (0.87, 0.74, 0.90), NSE = (0.86, 0.72, 0.64) and RMSE = (0.14, 0.42, 1.7) cm d-1. Fig. 5 also compares the simulated changes in soil water storage, represented by the amount of water contained in the SM domain, with the observed soil water storage variations. The observed storages were converted from the measured soil water pressures using laboratory determined retention curves. Soil water pressures were measured at five locations each instrumented with three tensiometers. The determination of soil water storage by this approach was rather problematic due to episodic failures of tensiometers and uncertainties related to the schematization of the soil profile and to the representativeness of the available retention curves. The model performance criteria for GEOTRANSF and TOPMODEL are shown in Tab. 2. Both models succeeded in simulating the hillslope discharges relatively well. In most cases there was a slightly better agreement between the hydrographs predicted by GEOTRANSF and TOPMODEL and those generated by S1D (GEOTRANSF and TOP79 MODEL vs. S1D) than between the simulated and observed hydrographs (GEOTRANSF and TOPMODEL vs. data). The simulated hydrographs are compared in Fig. 6. The corresponding soil water storages are shown in Fig. 7. Fig. 7 shows relatively good agreement between the SM-domain storage simulated by S1D and the total soil water storage predicted by GEOTRANSF. This seems to indicate that the soil water storage responses at Tomsovka can be successfully modeled by a zero-dimensional non-linear compartment approach. The TOPMODEL soil water storage is divided into two components: the RZ storage and the UZ storage. This division seems to resemble the S1D dual storage - consisting of SM-domain storage and PF-domain storage. However, as it can be seen in Figs. 7 and 8, the actual functioning of the corresponding storages is quite different. The most notable difference is reflected in the cutoff value of the TOPMODEL RZ storage, caused by the introduction of the field capacity as an upper limit for the RZ storage (SRZ max). The comparison of UZ storage in TOPMODEL and PF-domain storage in S1D (Fig. 8a) reveals completely different character of storage responses. This is because the PFdomain storage contains not only gravity-drainable water but also water retained by capillary forces. The TOPMODEL UZ storage is represented by a fictitious reservoir (Eqs. (16) and (17)), which delays the RZ-storage overspills, however in a much less continuous way compared to the vertically distributed S1D PF-domain storage. In Fig. 8b, saturated zone deficits generated by TOPMODEL (Eq. (13)) are compared with those obtained by applying the S1D model together with Eqs. (4), (6) and (23). Both deficits are evaluated at the hillslope base. The figure shows substantially larger variation range for the TOPMODEL deficit than for the S1D deficit. The scale of the former is controlled by the empirical parameter m, but even more importantly, by the selected exponential relationship between transmissivity and storage deficit. More comparable results could probably be ob- tained with a linear transmissivity-deficit relationship, which would also be in a better agreement with our implicit assumption that the effective lateral saturated conductivity is independent of z (e.g. in (6)). Both S1D and TOPMODEL are capable of predicting the beginning of surface runoff and the amount of locally generated saturation excess water in case that the soil profile becomes fully saturated. Although such an event may have occurred during the simulated period (e.g. in July 2000, September 2001 or August 2002), the available data are insufficient to make a conclusive quantitative comparison between the observed and simulated variables. While S1D predicted no surface runoff over the simulation period, TOPMODEL did predict overland flow due to saturation excess on several occasions. An example of the development of variable saturation area, as predicted by TOPMODEL, is shown in Fig. 4. Another such event occurred on July 17, 2001, as indicated by the sharp decrease of saturated zone deficit to zero in Fig. 8b. Time 08:00 09:00 10:00 11:00 12:00 Fig. 4. Variable source area simulated by TOPMODEL during rainfall-runoff episode which occurred on September 11, 2001. The hillslope segment is 25 m long and consists of 5 elements. Conclusions The studied models were compared in terms of their performance when applied to the simulations of hillslope discharges, observed at the experimental hillslope site. In addition, the adopted methodology allowed us to analyze and compare internal T a b l e 2. Model performance evaluated for GEOTRANSF and TOPMODEL. Model parameters were alternatively calibrated against the observed hillslope discharge hydrographs and the hydrographs generated by S1D model. In both cases, the vegetation season 2000 was used as a calibration period. RMSE is given in cm d-1. GEOTRANSF vs. data 2000 2001 2002 R 0.81 0.62 0.82 TOPMODEL vs. data R 0.82 0.70 0.86 GEOTRANSF vs. S1D R 0.87 0.78 0.96 TOPMODEL vs. S1D R2 0.95 0.77 0.71 NSE 0.92 0.76 065 RMSE 0.09 0.28 0.74 NSE 0.74 0.54 0.44 RMSE 0.18 0.54 2.11 NSE 0.75 0.60 0.62 RMSE 0.18 0.50 2.19 NSE 0.82 0.75 0.85 RMSE 0.14 0.29 0.48 Section A 18 Section B S1D 12 Fig. 5. S1D vs. data in 2001: (a) subsurface hillslope discharge, (b) soil water storage - the shaded area reflects the spatial variability of soil water storage, evaluated from multiple tensiometer measurements. model fluxes and storages. This is particularly important when internal variables are subject to physical interpretation, such as in TOPMODEL, where the saturation storage deficits are used to determine the occurrence and distribution of variable source areas during rainfalls. The results confirm that subsurface runoff dominates hydrological responses of the studied hillslope. Considering the uncertainties in the measured data and model parameters, all applied models were able to simulate the hillslope responses to precipitation relatively well. Zero-dimensional and semi-distributed catchment scale models with limited amount of parameters were quite successful in predicting hillslope discharge hydrographs but less so in predicting soil water storage variations. By choosing a model for hydrological modeling, it is necessary to consider not only the model performance but also the number of model parameters and the procedures necessary to determine their values. In that respect, the GEOTRANSF model, in spite of its simplicity, proved to be a promising tool for modeling hydrological responses of small mountainous catchments with shallow highly permeable soils. Acknowledgement. The study was supported by the Ministry of Environment of the Czech Republic, Project No. SP/2e7/229/07 and by the Czech Science Foundation, Project No. 205/08/1174. We thank Dr. Martin Sanda (Czech Technical University in Prague) for supervising the field observations at the Tomsovka site. 81 Rainfall intensity (cm d-1) 4 0 9-Jul-00 14-Jul-00 19-Jul-00 24-Jul-00 29-Jul-00 3-Aug-00 8-Aug-00 0 29-Aug-01 3-Sep-01 8-Sep-01 13-Sep-01 18-Sep-01 23-Sep-01 28-Sep-01 18 9 0 30-Jul-02 4-Aug-02 9-Aug-02 14-Aug-02 19-Aug-02 24-Aug-02 29-Aug-02 Fig. 6. Subsurface hillslope discharge (30-day details). 8 1-May-00 25-May-00 18-Jun-00 12-Jul-00 5-Aug-00 29-Aug-00 22-Sep-00 16-Oct-00 15 13 11 1-May-01 22 25-May-01 18-Jun-01 12-Jul-01 5-Aug-01 29-Aug-01 22-Sep-01 16-Oct-01 7 1-May-02 25-May-02 18-Jun-02 12-Jul-02 5-Aug-02 29-Aug-02 22-Sep-02 16-Oct-02 Fig. 7. Simulated SM storage in S1D, total soil water storage in GEOTRANSF, and root zone storage in TOPMODEL. Gravity drainable storage (cm) Topmodel S1D 1 (a) Saturated zone storage deficit (cm) (b) 1 Topmodel S1D 0 1-May-00 25-May-00 18-Jun-00 12-Jul-00 5-Aug-00 0 29-Aug-00 22-Sep-00 16-Oct-00 Fig. 8. Simulated gravity-drainable storage in TOPMODEL vs. PF storage in S1D (a) and saturated zone storage deficits determined using TOPMODEL and S1D (b). List of symbols a A B Cf Cm cp D ­ contributing hillslope area per unit contour length [m], ­ contributing hillslope area [m2], ­ hillslope width [m], ­ soil water capacity of preferential flow domain [m-1], ­ soil water capacity of soil matrix domain [m-1], ­ partition coefficient (GEOTRANSF) [­], ­ local saturated zone storage deficit (TOPMODEL) [m], ­ average storage deficit (TOPMODEL) [m], ­ actual evapotranspiration [m s-1], ­ soil water pressure head in PF domain [m], ­ soil water pressure head in SM domain [m], ­ depth of the saturated hypodermic stream [m], ­ air-entry value [m], ­ soil-topographic index (TOPMODEL) [s m-1], ­ average topographic index (TOPMODEL) [s m-1], ­ effective hydraulic conductivity for hypodermic flow [m s-1], ­ PF-domain hydraulic conductivity [m s-1], ­ SM-domain hydraulic conductivity [m s-1], D E hf hm hsz hs I I Keff Kf Km ­ saturated hydraulic conductivity [m s-1], ­ empirical parameters in GEOTRANSF and TOPMODEL [m], n ­ empirical parameter of van Genuchten retention function [­], NSE ­ Nash-Sutcliffe efficiency [­], P ­ infiltration intensity [m s-1], ­ deep percolation (GEOTRANSF) [m s-1], qdp ­ vertical soil water flux in PF domain [m s-1], qf ­ vertical soil water flux (GEOTRANSF) [m s-1], qh ­ vertical soil water flux in SM domain [m s-1], qm ­ macropore component of soil water flux (GEOqs TRANSF) [m s-1], ­ saturated zone recharge (TOPMODEL) [m s-1], qv ­ specific discharge associated with S0 (GEOTRANSF) q0 [m s-1], Q ­ hillslope discharge [m3 s-1], ­ saturated zone discharge (TOPMODEL) [m3 s-1], Qb ­ saturated zone recharge (TOPMODEL) [m3 s-1], Qv ­ discharge corresponding to zero average deficit Q0 (TOPMODEL) [m3 s-1], R ­ saturated zone recharge [m s-1], ­ coefficient of determination [­], R2 RMSE ­ Root Mean Square Error [m s-1], Ks m Saturated zone storage deficit (cm) ­ S1D Hillslope runoff generation ­ comparing different modeling approaches. S Sf Sm SSZ SRZ SUZ S0 t td T0 wf wm w s r w ­ soil water storage [m], ­ root water uptake in PF-domain [s-1], ­ root water uptake in SM-domain [s-1], ­ saturated zone storage (TOPMODEL) [m], ­ soil water storage in the root zone (TOPMODEL) [m], ­ soil water storage in unsaturated zone (TOPMODEL) [m], ­ minimum threshold value of soil water storage (GEOTRANSF) [m], ­ time [s], ­ empirical parameter (TOPMODEL) [s m-1], ­ lateral transmissivity at full soil saturation (TOPMODEL) [m2 s-1], ­ volume fraction of PF-domain [­], ­ volume fraction of SM-domain [­], ­ empirical parameter of van Genuchten retention function [m-1], ­ inter-domain soil water transfer coefficient [s-1 m-1], ­ hillslope angle [­], ­ saturated water content [m3 m-3], ­ residual water content [m3 m-3], ­ first order soil water transfer term [s-1], ­ effective porosity for hypodermic flow [­]. DOHNAL M., DUSEK J., VOGEL T., 2006: The impact of the retention curve hysteresis on prediction of soil water dynamics. J. Hydrol. Hydromech., 54, 258­268. FAN Y., BRAS R., 1998: Analytical solutions to hillslope subsurface storm flow and saturation overland flow. Water Resour. Res., 34, 921-927. GERKE H. H., VAN GENUCHTEN M. Th., 1993: A dualporosity model for simulating the preferential movement of water and solutes in structured porous media. Water Resour. Res., 29, 305­319. HILBERTS A. G. J., VAN LOON E. E., TROCH P. A., PANICONI C., 2004: The hillslope-storage Boussinesq model for non-constant bedrock slope. J. Hydrol., 291, 160-173. HILBERTS A. G. J., TROCH P. A., PANICONI C., BOL J., 2007: Low-dimensional modeling of hillslope subsurface flow: Relationship between rainfall, recharge, and unsaturated storage dynamics. Water Resour. Res., 43, W03445. KIRKBY M. J., 1975: Hydrograph modelling strategies, in Process in Physical and Human Geography, edited by R. Peel, M. Chisholm, and P. Haggett, p. 69-90, Heinemann, London. LICHNER L., ELDRIDGE D. J., SCHACHT K., ZHUKOVA N., HOLKO L., SIR M., PECHO J., 2011: Grass Cover Influences Hydrophysical Parameters and Heterogeneity of Water Flow in a Sandy Soil. Pedosphere, 21, 719-729. MAJONE B., BERTAGNOLI A., BELLIN A., RINALDO A., 2005: GEOTRANSF: a continuous non-linear hydrological model. Eos Trans. AGU, 86 (52), Fall Meet. Suppl., Abstract H23C-1441. MAJONE B., BERTAGNOLI A., BELLIN A., 2010: A nonlinear runoff generation model in a small Alpine catchment. J. Hydrol., 385, 300-312. MARQUARDT D., 1963: An Algorithm for Least-Squares Estimation of Nonlinear Parameters, SIAM J. on Applied Mathematics, 11, 431-441. PANICONI C., TROCH P. A., VAN LOON E. E., HILBERTS A. G. J., 2003: Hillslope-storage Boussinesq model for subsurface flow and variable source areas along complex hillslopes: 2. Intercomparison with a three-dimensional Richards equation model. Water Resour. Res., 39, Article No. 1317. TROCH P. A., VAN LOON E. E., HILBERTS A. G. J., 2002: Analytical solutions to a hillslope-storage kinematic wave equation for subsurface flow. Adv. Water Resour., 25, 637-649. TROCH P. A., PANICONI C., VAN LOON E. E., 2003: Hillslope-storage Boussinesq model for subsurface flow and variable source areas along complex hillslopes: 1. Formulation and characteristic response. Water Resour. Res., 39, Article No. 1316. SANDA M., 1999, Tvorba podpovrchového odtoku na svahu. [Doctoral thesis.], CTU in Prague, Prague. SANDA M., HRNCÍ M., NOVÁK L., CÍSLEROVÁ M., 2006: Impact of the soil profile on the rainfall-runoff process. [In Czech.] J. Hydrol. Hydromech., 54, 183-191. SANDA M., CÍSLEROVÁ M., 2009: Transforming hydrographs in the hillslope subsurface. J. Hydrol. Hydromech., 57, 4, 264­275. VOGEL T., VAN GENUCHTEN M.TH., CÍSLEROVÁ M., 2001: Effects of the Shape of the Soil Hydraulic Functions near Saturation on Variably-Saturated Flow Predictions. Adv. Water Resour., 24, 133-144. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Hydrology and Hydromechanics de Gruyter

Hillslope Runoff Generation - Comparing Different Modeling Approaches

Loading next page...
 
/lp/de-gruyter/hillslope-runoff-generation-comparing-different-modeling-approaches-VfZr03W0Lo

References (32)

Publisher
de Gruyter
Copyright
Copyright © 2012 by the
ISSN
0042-790X
DOI
10.2478/v10098-012-0007-2
Publisher site
See Article on Publisher Site

Abstract

J. Hydrol. Hydromech., 60, 2012, 2, 73­86 DOI: 10.2478/v10098-012-0007-2 HELENA PAVELKOVÁ, MICHAL DOHNAL, TOMÁS VOGEL*) Czech Technical University in Prague, Faculty of Civil Engineering, Thákurova 7, 166 29 Prague 6, Czech Republic; *) Corresponding author, Tel.: +420 22435 4341; fax: +420 224 354 793; Mailto: vogel@fsv.cvut.cz This study focuses on modeling hydrological responses of shallow hillslope soil in a headwater catchment. The research is conducted using data from the experimental site Uhlíská in Jizera Mountains, Czech Republic. To compare different approaches of runoff generation modeling, three models were used: (1) onedimensional variably saturated flow model S1D, based on the dual-continuum formulation of Richards' equation; (2) zero-dimensional nonlinear morphological element model GEOTRANSF; and (3) semidistributed model utilizing the topographic index similarity assumption - TOPMODEL. Hillslope runoff hydrographs and soil water storage variations predicted by the simplified catchment scale models (GEOTRANSF and TOPMODEL) were compared with the respective responses generated by the more physically based local scale model S1D. Both models, GEOTRANSF and TOPMODEL, were found to predict general trends of hydrographs quite satisfactorily; however their ability to correctly predict soil water storages and inter-compartment fluxes was limited. KEY WORDS: Rainfall-Runoff Modeling, Hillslope Discharge, Soil Water Storage, Transmisivity, Topographic Index, Richards' Equation, Preferential Flow. Helena Pavelková, Michal Dohnal, Tomás Vogel: TVORBA ODTOKU ZE SVAHU ­ POROVNÁNÍ ROZDÍLNÝCH MODELOVÝCH PÍSTUP. J. Hydrol. Hydromech., 60, 2012, 2; 29 lit., 8 obr., 2 tab. Studie je zamena na modelování hydrologické reakce mlké svahové pdy v pramenné cásti povodí Nisy, k výzkumu byla pouzita data z experimentálního povodí Uhlíská. Porovnání rzných konceptuálních pedstav modelování odtoku bylo uskutecnno pro: (1) jednorozmrný model promnliv nasyceného proudní S1D; (2) model zalozený na bezrozmrném nelineárním morfologickém prvku - GEOTRANSF a (3) semi-distribuovaný model vyuzívající principu podobnosti na základ topografického indexu - TOPMODEL. Hydrogramy odtoku ze svahu a zmny zásob vody v pd vypoctené zjednodusenými modely GEOTRANSF a TOPMODEL byly porovnány s odpovídajícími odezvami fyzikáln zalozeného modelu S1D. Oba modely, GEOTRANSF i TOPMODEL, byly pomrn úspsné v pedpovdi základních trend hydrogram odtoku, jejich schopnost správn pedpovídat zásoby vody v pd a toky mezi nimi vsak byla omezená. KLÍCOVÁ SLOVA: srázko-odtokové modelování, odtok ze svahu, zásoba vody v pd, transmisivita, topografický index, Richardsova rovnice, preferencní proudní. Introduction In the last few decades, processes related to runoff formation in headwater catchments, especially during extreme hydrological events, have received growing attention. A fast reaction of discharge to rainstorms is typical for small mountainous catchments. Shallow subsurface runoff is considered to be a significant component of the rainfall-runoff relationship in such catchments. It is generally assumed that this type of runoff (further on referred to as hypodermic flow) takes place after a saturated zone has formed above an impermeable or less permeable subsurface layer (hard bedrock or a less permeable soil layer). Soil water then flows laterally in the direction determined by the local slope of the soil-bedrock interface. Of the currently available models used to describe actual processes taking place at the hillslope scale, probably the most detailed one is the model based on three-dimensional Richards' equation. Such a model and other similar physically based 73 models are strongly non-linear and require iterative numerical solution. A vast number of linearized equations have to be solved when this type of model is applied to a hydrological problem, even if modest spatial dimensions are considered (such as dimensions of a single hillslope). To estimate parameters for these models, detailed knowledge of soil hydraulic characteristics is necessary. This information is commonly unavailable, especially at the entire catchment scale (Lichner et al., 2011). In order to obtain hydrological predictions at this scale, it is necessary to use simplified conceptual models. Over the years, many conceptual approaches to subsurface runoff modeling were discussed and compared in the literature. The kinematic wave equation (e.g. Fan and Bras, 1998; Troch et al., 2002) and the Boussinesq equation (e.g. Troch et al., 2003; Hilberts et al., 2004) were applied to model hypodermic flow. Richards' equation was used to describe hillslope soil water movement under variably saturated conditions e.g. by Paniconi et al. (2003), Hilberts et al. (2007) and Cordano and Rigon (2008). A combination of Richards' equation and diffusion wave equation was used by Vogel at al. (2005a, 2005b) to model hypodermic flow in the Uhlíská catchment. TOPMODEL (e.g. Beven, 2001) has been widely discussed and tested (Ambroise et al., 1996; Blazková and Beven, 1997). Applications of TOPMODEL in the Uhlíská catchment have been reported by Blazková and Beven (1997) and Blazková et al. (2002a and 2002b). Recently, Majone et al. (2010) presented a simpler model (further on referred to as GEOTRANSF) developed for modeling subsurface runoff genera- tion using non-linear relationship between runoff and soil water storage. In this paper, three models were used to simulate runoff from a hillslope: (1) one-dimensional variably saturated flow model S1D, based on dualcontinuum formulation of the Richards equation; (2) zero-dimensional non-linear morphological element model GEOTRANSF; and (3) model utilizing the topographic index similarity assumption - TOPMODEL. The main objective of this comparative study was to analyze the performance of the two simplified models (GEOTRANSF and TOPMODEL) by comparing them with the more physically based model (S1D), and to assess the ability of the models to predict soil water storages and internal fluxes in addition to discharge hydrographs. Material and methods S1D model This physically based local scale model was designed to simulate flow of water in soils with preferential pathways. It is based on the dualcontinuum approach. The approach assumes that soil water moves in two separate flow domains, one representing the soil matrix and the other representing the network of preferential pathways (denoted as SM domain and PF domain). Under saturated or nearly saturated conditions, soil water contained in the preferential flow domain flows considerably faster than that in the soil matrix domain. Soil water flow is in each of the two domains described by Richards' equation (in a similar way as in Gerke and van Genuchten, 1993). The resulting set of the two coupled governing equations can be written as: wf C f h f t h f w f K f + 1 - w f S f - w (h f - hm ) z z (1) wmCm h hm = wm K m m + 1 - wm S m + w (h f - hm ) t z z (2) where f refers to PF-domain and m to SM-domain, wm and wf ­ volume fractions of the respective domains (wm + wf = 1), C ­ the differential soil water capacity [m-1], h ­ the pressure head [m], K ­ the unsaturated hydraulic conductivity [m s-1], S ­ the local intensity of root extraction [s-1] and w ­ the 74 transfer coefficient controlling the dynamic exchange of water between the two flow domains [s-1 m-1]. The governing equations are coupled through a first order transfer term w [s-1] defined according to Gerke and van Genuchten (1993) as w = w (h f - hm ) . Vertical fluxes predicted at the lower boundary of the dual-continuum system (by S1D model) determine the recharge rate for the hypodermic flow. In general, the vertical recharge rate R [m s-1] can be calculated from a simple continuity equation formulated at the soil-bedrock interface: The dual set of governing equations for soil water flow is solved numerically by the S1D code. The most recent implementation of S1D is described in Vogel et al. (2010a) R = w f (q f 1 - q f 2 ) + wm (q m1 - qm 2 ) (3) P m f where the first term represents the recharge contributions from the PF domain and the second one from the SM domain, respectively, q1 and q2 are soil water fluxes [m s-1] above and below the soil/bedrock interface (Fig. 1). The q2 fluxes represent seepage to deeper horizons. In the present application of the model, we assume that qm2 = qm1 and qf2 = 0, i.e. the deep percolation is associated with the SM-domain flux while the PF-domain flux contributes to hypodermic flow. This leads to R = = wf qf1. S1D can handle hysteresis of soil hydraulic properties. However, this option was not used in the present study, i.e. hysteresis was assumed to be insignificant. This was partly supported by the results of our previous study (Dohnal et al., 2006). In addition, we assume that the modeled hillslope segment at Uhlíska (being short, shallow and well permeable) does not cause any substantial runoff transformation in lateral direction and the hillslope discharge can be computed from a simple quasisteady-state formula: qm1 qf1 qm2 qf2 Fig. 1. Schematic of S1D domains and fluxes. The soil is decomposed into two flow domains: the soil matrix domain (m) and preferential flow domain (f). qm1 and qf1 are the lower boundary soil water fluxes, qm2 and qf2 are the fluxes representing leakage to deeper horizons, and w is the inter-domain soil water transfer. GEOTRANSF model A simple conceptual model of subsurface runoff generation was suggested by Majone et al. (2005, 2010). In this paper, we refer to this model as GEOTRANSF - according to Majone et al. (2005). In GEOTRANSF, locally generated vertical flux at the base of the soil profile is estimated from: Q(t ) = AR(t ) (4) where A is the hillslope contributing area [m2]. The local saturated zone storage SSZ [m], representing the amount of water which takes part in hypodermic flow, is determined by the depth of hypodermic stream hSZ [m] and the effective porosity [­], i.e.: SSZ (t ) = hSZ (t ) (5) S(t) - S0 q0 exp qh (t) = m q 0 for S S0 for S < S0 (7) If Q is calculated from (4), we can estimate hSZ from the kinematic wave approximation of Darcy's law (i.e., vertically integrated Darcy equation stripped of the pressure gradient): Q(t ) z = K eff hSZ (t ) B x (6) where m is an empirical parameter [m], S ­ the soil water storage [m], q0 ­ the specific discharge [m s-1], associated with a minimum threshold value of soil water storage S0. The vertical soil water flux is divided into two components (Fig. 2): (i) a fast flow through macropores recharging hypodermic flow: where Keff is the effective saturated hydraulic conductivity [m s-1], B ­ the local width of the hillslope [m] and z/x is equal to the cosine of the local hillslope angle. qs (t) = (1- c p )qh (t) (8) and (ii) a deep percolation component representing seepage to underlying horizons: 75 qdp (t) = c p qh (t) (9) where cp is the partition coefficient [­]. Water storage in the soil profile is calculated from the mass balance equation: S = P - E - qh t (10) where P is the infiltration intensity [m s-1] due to rainfall and E is the actual evapotranspiration intensity [m s-1]. Majone et al. (2010) used instantaneous unit hydrograph method to transform the recharge signal qs(t) into streamflow hydrograph. In our study, GEOTRANSF is first used to calculate the hypodermic flow recharge R(t) = qs(t). The hillslope discharge hydrograph is then simply estimated by applying the quasi-steady-state assumption (4). ilar values of I are considered to belong to the same topographic index class. TOPMODEL (e.g. Beven, 2001) is based on three basic assumptions: (i) quasi-steady-state relationship between the saturated zone recharge and the hillslope discharge, (ii) approximation of the effective hydraulic gradient by the local surface topographic slope and (iii) a simplified relationship between the saturated zone transmissivity and the local storage deficit. The TOPMODEL version used in this study assumes exponential relationship between transmissivity and storage deficit. According to the quasi-steady-state assumption, the hillslope discharge caused by saturated subsurface runoff can be estimated from: q j (t) = a j r(t) (12) where q is the subsurface discharge per unit contour length [m2 s-1] and r ­ the saturated zone recharge rate [m s-1], assumed to be spatially uniform. If exponential relationship between the transmissivity and storage deficit is assumed, local saturated zone storage deficit D [m] can be calculated from the corresponding value of soil-topographic index and the average storage deficit of the catchment: Non-linear reservoir D j - D = -m(I j - I ) (13) qh qs qdp Fig. 2. Schematic of GEOTRANSF fluxes. TOPMODEL TOPMODEL utilizes a concept of topographic index originally developed by Kirkby (1975) and extended to the soil-topographic index by Beven (1986): where m [m] is a parameter controlling the rate of decline of transmissivity with increasing storage deficit and the overbars refer to the catchment scale averages. The version of TOPMODEL applied in this paper (see Beven, 2001) uses three individually balanced water storages (Fig. 3): (i) spatially uniform variably saturated root zone storage, (ii) spatially distributed gravity-drainable unsaturated zone storage and (iii) catchment scale saturated zone storage represented by the average storage deficit. The root zone water balance, is calculated according to: S RZ S = P - E - RZ t t SRZ = max(SRZ ­ SRZ max, 0) (14) (15) I j = ln aj T0 tan j (11) where I is the topographic index, aj is the contributing area of the hillslope per unit contour length [m] drained through point j, T0 ­ the lateral transmissivity at full soil saturation [m2 s-1], ­ the local angle of the hillslope. The locations exhibiting sim76 where SRZ is the instantaneous water storage in the root zone [m], assumed to be uniform over the catchment, and SRZ max is the maximum value of SRZ. The amount of water in excess of SRZ max is redirected to the gravity-drainable unsaturated zone storage. The local water balance in the gravity-drainable compartment of unsaturated zone is evaluated for each topographic index class i by solving: In analogy to Eq. (5), this can be also expressed in terms of hypodermic flow depth: SUZ t S RZ - qv i t Di = (hSZ max - hSZi ) (23) (16) where qv is the vertical saturated zone recharge by gravity drainage from unsaturated zone [m s-1], which is estimated as: qvi = SUZi td Di (17) where SUZ is the water storage in the gravitydrainable compartment of the unsaturated zone [m], and td is a parameter [s m-1] interpreted as the mean residence time in SUZ per unit of saturated zone deficit. The catchment scale saturated zone recharge Qv [m3 s-1] is equal to the sum of local recharges, which can be expressed as the sum of all topographic index class contributions: Qv = Ai qvi (18) where Ai is the area associated with topographic index class i [m2]. Finally, the catchment scale water balance for the saturated zone is: D - Dt-t A t = (Qb - Qv )t-t . t (19) The saturated zone discharge at the catchment scale, Qb [m3 s-1], can be calculated from the average storage deficit D as: Qb = Q0 e- D/m . where hSZ max is equal to the maximum depth of the saturated zone, i.e. in our case the depth of the soil profile (about 75 cm). An important feature of TOPMODEL is the algorithm used for the determination of variable source areas contributing to rapid surface runoff (cf. Figs. 3 and 4). According to TOPMODEL, these areas are formed wherever the local saturated zone storage deficit D, calculated from (13), becomes zero (or negative). The saturation excess runoff generated over the variable source areas (Fig. 4) is then added to the combined (subsurface plus surface) discharge Q(t). Alternatively, surface runoff may be generated via the infiltration excess mechanism. However, this runoff mechanism is rarely if ever observed at Uhlíská and was not considered in the present application of TOPMODEL. Note the discrepancy between the postulated spatial uniformity of the saturated zone recharge, expressed in (12), and the distributed character of recharge rates calculated from (17). According to the author of TOPMODEL: "The water table is assumed to take up a shape AS IF the recharge rate was uniform across the catchment (even if the recharge is actually heterogeneous). Homogeneity is not the only problem, in fact ­ a kinematic analysis shows that if there is strong variation in recharge in space or time then it might actually take quite a long time to achieve such a steady state, while differences in soil transmissivity will also have an effect" (Beven, personal communication, 2012). (20) P E saturation excess The discharge corresponding to zero average deficit, Q0 [m3 s-1], is determined as: Q0 = Ae- I (21) RZ UZ where I is the mean value of the soil-topographic index over the catchment area A. The local saturated zone deficit can be interpreted as the difference between maximum possible saturated storage SSZ max and actual storage SSZ, i.e.: qv SZ Di = S SZ max - S SZi (22) Fig. 3. Schematic of TOPMODEL storages and fluxes (RZ = root zone storage, UZ = unsaturated zone storage, SZ = saturated zone storage). Experimental site and measurements The above described models were tested using data from the experimental catchment Uhlíská, located in the Jizera Mountains in northeast Czech Republic. The Uhlíská catchment lies in the upper part of the Cerná Nisa catchment where basic hydrological and climatological conditions have been monitored since 1982 by the Czech Hydrometeorological Institute. The Uhlíská catchment has an area of 1.78 km2 and an average altitude of 822 m above sea level. Average annual temperature is 4.7°C, annual precipitation exceeding 1300 mm. The catchment is generally covered by snow from November to April. The forest at Uhlíská was damaged acid deposition in 80's. The catchment is now covered by young spruce forest with grass undergrowth. Hydropedological survey has been conducted to determine infiltration and retention characteristics of the soil profile (Sanda, 1999; Zumr et al., 2007). The soil on the weathered porphyric granite bedrock is classified as Dystric Cambisol. The soil type varies from loamy sand to sandy loam and has a very coarse skeleton. The experimental site is equipped with automatic data collection systems for precipitation, temperature, soil water pressure, soil moisture, subsurface discharge, and other hydrological variables (Sanda, 1999). Shallow subsurface runoff is measured in a trench situated at the experimental hillslope Tomsovka. The average slope near the trench is 14%. The soil profile is approximately 75 cm deep followed by weathered bedrock. The trench consists of two sections, each four meters wide (denoted as section A and section B), and both equipped with a tipping bucket flow gauge for measuring subsurface runoff. It is assumed that each section of the trench collects water from an area of about 100 m2 (Císlerová et al., 1997, Sanda et al., 2006; Sanda and Císlerová, 2009). Model application Three different models - S1D, GEOTRANSF and TOPMODEL - were applied to rainfall-runoff data observed at the Uhlíská catchment in vegetation seasons 2000, 2001 and 2002. The models were used to simulate runoff from a micro-catchment, representing the Tomsovka hillslope site. The size of the micro-catchment was 78 set equal to the estimated contributing area of one of the two sections of the experimental trench. Both hillslope segments (draining to the respective section of the trench) were considered to be 75 cm deep, 25 m long and 4 m wide, with a constant slope of tan = 0.14. The initial soil water storage conditions applied at the beginning of vegetation seasons corresponded to the soil water pressures measured by tensiometers. The soil-atmosphere interactions involved natural rainfall and potential evapotranspiration estimated by Penman-Monteith equation. Hourly averaged rainfall and daily averaged evapotranspiration intensities were utilized as input data for all three models. TOPMODEL as well as GEOTRANSF were run with one-hour time steps. S1D uses adaptive time step associated with the numerical solution. The S1D model parameters (Tab. 1) were taken from our previous study (Vogel et al., 2010b) conducted for the same experimental site. The model parameters of GEOTRANSF and TOPMODEL were calibrated to match the runoff hydrographs observed in the season 2000. The parameters were optimized by means of the Levenberg-Marquardt algorithm (Marquardt, 1963). Alternatively, the calibration of GEOTRANSF and TOPMODEL was performed for the subsurface runoff hydrographs generated by the S1D model (for the season 2000), instead of using the observed hydrographs. In this case, the S1D model served as a substitute for the real hillslope. The main advantage of this approach was a significant reduction of complexity and the associated uncertainty in the substitute system, which allowed a more straightforward comparison of the simulated internal fluxes and soil water storages, otherwise unavailable for the real system (such as comparison of SRZ and SUZ storages in TOPMODEL with SM-domain and PFdomain storages in S1D). In GEOTRANSF, a single parameter was calibrated - the parameter m. The calibration yielded a value of m = 0.00803 m (for the GEOTRANSF vs. S1D calibration). The minimum specific discharge q0 and minimum soil water storage S0 were both set equal to zero. In TOPMODEL, the calibration of the parameters m and td resulted in m = 0.005 m and td = 244 h m-1 (for the TOPMODEL vs. S1D calibration). Note that the parameter m plays in TOPMODEL similar role as in GEOTRANSF (see Eqs. (20) and (7)). The root zone storage SRZ max was set equal to 0.165 m. The lateral transmissivity at full saturation was estimated as T0 = 0.781 m2 h-1. The maximum root zone storage SRZ max was assumed to be equal to the soil water field capacity, i.e. the amount of water stored in initially fully saturated soil after 48 hours of free draining. The value of SRZ max was determined by integrating the soil water content profile (in SM-domain) at the end of 48hour drainage period simulated by S1D model. The lateral transmissivity at full saturation T0 was assumed to be equal to the product of the lateral saturated hydraulic conductivity of the network of effective preferential pathways, estimated as Keff = 2500 cm d-1 and the total depth of the soil profile (75 cm). Since the soil-bedrock interface at Tomsovka is semipervious, part of the infiltrating water seeps to deeper horizons and does not contribute to the shallow subsurface discharge. To respect that, vertical flow qv calculated in TOPMODEL was divided into T a b l e 1. Soil hydraulic parameters. Depth [cm] Matrix 1 2 3 4 Preferential ­ 0­8 8 ­ 20 20 ­ 70 70 ­ 75 0 ­ 75 r [­] 0.20 0.20 0.20 0.20 0.01 s [­ ] 0.55 0.54 0.49 0.41 0.60 the deep percolation part qdp and the saturated zone recharge part qs using the same partitioning procedure as in GEOTRANSF (Eq. (8)). The catchment scale saturated zone recharge was therefore calculated as: Qv = Ai qsi , qsi = (1- c p )qvi . (24) The value of partition coefficient cp = 0.36, determined by S1D model, was also applied in GEOTRANSF and TOPMODEL. To calculate the local soil-topographic index and the local saturated zone deficit in TOPMODEL, the contributing hillslope area at Tomsovka was divided into five segments (Fig. 4) along the slope length (each 5 m long). To compare the models, three performance criteria were used in this study: (i) coefficient of determination R2, (ii) root mean square error RMSE, and (iii) the Nash-Sutcliffe efficiency NSE. Domain Layer [cm ] 0.050 0.050 0.020 0.020 0.050 -1 n [­ ] 2.00 1.50 1.20 1.20 3.00 Ks [cm d ] 567 67 17 1.3 5000 -1 hs [cm] 0.00 ­0.69 ­1.48 ­1.88 0.00 r and s are the residual and saturated water contents, hs ­ the air-entry value, and and n ­ empirical parameters (Vogel et al., 2001). Results and discussion Specific subsurface runoff simulated by the S1D model is compared to the observed hillslope discharge in Fig. 5. It is obvious that major hillslope responses occur after abundant rainfall distributed over a longer period of time. The simulated and measured hydrographs are of a similar shape; their rising limbs are very steep. The agreement between the observed and simulated discharges is not perfect, however the model is capable of reproducing the basic character of hillslope responses. Following model performance indices were determined for the three vegetation seasons (2000, 2001, 2002): R2 = (0.87, 0.74, 0.90), NSE = (0.86, 0.72, 0.64) and RMSE = (0.14, 0.42, 1.7) cm d-1. Fig. 5 also compares the simulated changes in soil water storage, represented by the amount of water contained in the SM domain, with the observed soil water storage variations. The observed storages were converted from the measured soil water pressures using laboratory determined retention curves. Soil water pressures were measured at five locations each instrumented with three tensiometers. The determination of soil water storage by this approach was rather problematic due to episodic failures of tensiometers and uncertainties related to the schematization of the soil profile and to the representativeness of the available retention curves. The model performance criteria for GEOTRANSF and TOPMODEL are shown in Tab. 2. Both models succeeded in simulating the hillslope discharges relatively well. In most cases there was a slightly better agreement between the hydrographs predicted by GEOTRANSF and TOPMODEL and those generated by S1D (GEOTRANSF and TOP79 MODEL vs. S1D) than between the simulated and observed hydrographs (GEOTRANSF and TOPMODEL vs. data). The simulated hydrographs are compared in Fig. 6. The corresponding soil water storages are shown in Fig. 7. Fig. 7 shows relatively good agreement between the SM-domain storage simulated by S1D and the total soil water storage predicted by GEOTRANSF. This seems to indicate that the soil water storage responses at Tomsovka can be successfully modeled by a zero-dimensional non-linear compartment approach. The TOPMODEL soil water storage is divided into two components: the RZ storage and the UZ storage. This division seems to resemble the S1D dual storage - consisting of SM-domain storage and PF-domain storage. However, as it can be seen in Figs. 7 and 8, the actual functioning of the corresponding storages is quite different. The most notable difference is reflected in the cutoff value of the TOPMODEL RZ storage, caused by the introduction of the field capacity as an upper limit for the RZ storage (SRZ max). The comparison of UZ storage in TOPMODEL and PF-domain storage in S1D (Fig. 8a) reveals completely different character of storage responses. This is because the PFdomain storage contains not only gravity-drainable water but also water retained by capillary forces. The TOPMODEL UZ storage is represented by a fictitious reservoir (Eqs. (16) and (17)), which delays the RZ-storage overspills, however in a much less continuous way compared to the vertically distributed S1D PF-domain storage. In Fig. 8b, saturated zone deficits generated by TOPMODEL (Eq. (13)) are compared with those obtained by applying the S1D model together with Eqs. (4), (6) and (23). Both deficits are evaluated at the hillslope base. The figure shows substantially larger variation range for the TOPMODEL deficit than for the S1D deficit. The scale of the former is controlled by the empirical parameter m, but even more importantly, by the selected exponential relationship between transmissivity and storage deficit. More comparable results could probably be ob- tained with a linear transmissivity-deficit relationship, which would also be in a better agreement with our implicit assumption that the effective lateral saturated conductivity is independent of z (e.g. in (6)). Both S1D and TOPMODEL are capable of predicting the beginning of surface runoff and the amount of locally generated saturation excess water in case that the soil profile becomes fully saturated. Although such an event may have occurred during the simulated period (e.g. in July 2000, September 2001 or August 2002), the available data are insufficient to make a conclusive quantitative comparison between the observed and simulated variables. While S1D predicted no surface runoff over the simulation period, TOPMODEL did predict overland flow due to saturation excess on several occasions. An example of the development of variable saturation area, as predicted by TOPMODEL, is shown in Fig. 4. Another such event occurred on July 17, 2001, as indicated by the sharp decrease of saturated zone deficit to zero in Fig. 8b. Time 08:00 09:00 10:00 11:00 12:00 Fig. 4. Variable source area simulated by TOPMODEL during rainfall-runoff episode which occurred on September 11, 2001. The hillslope segment is 25 m long and consists of 5 elements. Conclusions The studied models were compared in terms of their performance when applied to the simulations of hillslope discharges, observed at the experimental hillslope site. In addition, the adopted methodology allowed us to analyze and compare internal T a b l e 2. Model performance evaluated for GEOTRANSF and TOPMODEL. Model parameters were alternatively calibrated against the observed hillslope discharge hydrographs and the hydrographs generated by S1D model. In both cases, the vegetation season 2000 was used as a calibration period. RMSE is given in cm d-1. GEOTRANSF vs. data 2000 2001 2002 R 0.81 0.62 0.82 TOPMODEL vs. data R 0.82 0.70 0.86 GEOTRANSF vs. S1D R 0.87 0.78 0.96 TOPMODEL vs. S1D R2 0.95 0.77 0.71 NSE 0.92 0.76 065 RMSE 0.09 0.28 0.74 NSE 0.74 0.54 0.44 RMSE 0.18 0.54 2.11 NSE 0.75 0.60 0.62 RMSE 0.18 0.50 2.19 NSE 0.82 0.75 0.85 RMSE 0.14 0.29 0.48 Section A 18 Section B S1D 12 Fig. 5. S1D vs. data in 2001: (a) subsurface hillslope discharge, (b) soil water storage - the shaded area reflects the spatial variability of soil water storage, evaluated from multiple tensiometer measurements. model fluxes and storages. This is particularly important when internal variables are subject to physical interpretation, such as in TOPMODEL, where the saturation storage deficits are used to determine the occurrence and distribution of variable source areas during rainfalls. The results confirm that subsurface runoff dominates hydrological responses of the studied hillslope. Considering the uncertainties in the measured data and model parameters, all applied models were able to simulate the hillslope responses to precipitation relatively well. Zero-dimensional and semi-distributed catchment scale models with limited amount of parameters were quite successful in predicting hillslope discharge hydrographs but less so in predicting soil water storage variations. By choosing a model for hydrological modeling, it is necessary to consider not only the model performance but also the number of model parameters and the procedures necessary to determine their values. In that respect, the GEOTRANSF model, in spite of its simplicity, proved to be a promising tool for modeling hydrological responses of small mountainous catchments with shallow highly permeable soils. Acknowledgement. The study was supported by the Ministry of Environment of the Czech Republic, Project No. SP/2e7/229/07 and by the Czech Science Foundation, Project No. 205/08/1174. We thank Dr. Martin Sanda (Czech Technical University in Prague) for supervising the field observations at the Tomsovka site. 81 Rainfall intensity (cm d-1) 4 0 9-Jul-00 14-Jul-00 19-Jul-00 24-Jul-00 29-Jul-00 3-Aug-00 8-Aug-00 0 29-Aug-01 3-Sep-01 8-Sep-01 13-Sep-01 18-Sep-01 23-Sep-01 28-Sep-01 18 9 0 30-Jul-02 4-Aug-02 9-Aug-02 14-Aug-02 19-Aug-02 24-Aug-02 29-Aug-02 Fig. 6. Subsurface hillslope discharge (30-day details). 8 1-May-00 25-May-00 18-Jun-00 12-Jul-00 5-Aug-00 29-Aug-00 22-Sep-00 16-Oct-00 15 13 11 1-May-01 22 25-May-01 18-Jun-01 12-Jul-01 5-Aug-01 29-Aug-01 22-Sep-01 16-Oct-01 7 1-May-02 25-May-02 18-Jun-02 12-Jul-02 5-Aug-02 29-Aug-02 22-Sep-02 16-Oct-02 Fig. 7. Simulated SM storage in S1D, total soil water storage in GEOTRANSF, and root zone storage in TOPMODEL. Gravity drainable storage (cm) Topmodel S1D 1 (a) Saturated zone storage deficit (cm) (b) 1 Topmodel S1D 0 1-May-00 25-May-00 18-Jun-00 12-Jul-00 5-Aug-00 0 29-Aug-00 22-Sep-00 16-Oct-00 Fig. 8. Simulated gravity-drainable storage in TOPMODEL vs. PF storage in S1D (a) and saturated zone storage deficits determined using TOPMODEL and S1D (b). List of symbols a A B Cf Cm cp D ­ contributing hillslope area per unit contour length [m], ­ contributing hillslope area [m2], ­ hillslope width [m], ­ soil water capacity of preferential flow domain [m-1], ­ soil water capacity of soil matrix domain [m-1], ­ partition coefficient (GEOTRANSF) [­], ­ local saturated zone storage deficit (TOPMODEL) [m], ­ average storage deficit (TOPMODEL) [m], ­ actual evapotranspiration [m s-1], ­ soil water pressure head in PF domain [m], ­ soil water pressure head in SM domain [m], ­ depth of the saturated hypodermic stream [m], ­ air-entry value [m], ­ soil-topographic index (TOPMODEL) [s m-1], ­ average topographic index (TOPMODEL) [s m-1], ­ effective hydraulic conductivity for hypodermic flow [m s-1], ­ PF-domain hydraulic conductivity [m s-1], ­ SM-domain hydraulic conductivity [m s-1], D E hf hm hsz hs I I Keff Kf Km ­ saturated hydraulic conductivity [m s-1], ­ empirical parameters in GEOTRANSF and TOPMODEL [m], n ­ empirical parameter of van Genuchten retention function [­], NSE ­ Nash-Sutcliffe efficiency [­], P ­ infiltration intensity [m s-1], ­ deep percolation (GEOTRANSF) [m s-1], qdp ­ vertical soil water flux in PF domain [m s-1], qf ­ vertical soil water flux (GEOTRANSF) [m s-1], qh ­ vertical soil water flux in SM domain [m s-1], qm ­ macropore component of soil water flux (GEOqs TRANSF) [m s-1], ­ saturated zone recharge (TOPMODEL) [m s-1], qv ­ specific discharge associated with S0 (GEOTRANSF) q0 [m s-1], Q ­ hillslope discharge [m3 s-1], ­ saturated zone discharge (TOPMODEL) [m3 s-1], Qb ­ saturated zone recharge (TOPMODEL) [m3 s-1], Qv ­ discharge corresponding to zero average deficit Q0 (TOPMODEL) [m3 s-1], R ­ saturated zone recharge [m s-1], ­ coefficient of determination [­], R2 RMSE ­ Root Mean Square Error [m s-1], Ks m Saturated zone storage deficit (cm) ­ S1D Hillslope runoff generation ­ comparing different modeling approaches. S Sf Sm SSZ SRZ SUZ S0 t td T0 wf wm w s r w ­ soil water storage [m], ­ root water uptake in PF-domain [s-1], ­ root water uptake in SM-domain [s-1], ­ saturated zone storage (TOPMODEL) [m], ­ soil water storage in the root zone (TOPMODEL) [m], ­ soil water storage in unsaturated zone (TOPMODEL) [m], ­ minimum threshold value of soil water storage (GEOTRANSF) [m], ­ time [s], ­ empirical parameter (TOPMODEL) [s m-1], ­ lateral transmissivity at full soil saturation (TOPMODEL) [m2 s-1], ­ volume fraction of PF-domain [­], ­ volume fraction of SM-domain [­], ­ empirical parameter of van Genuchten retention function [m-1], ­ inter-domain soil water transfer coefficient [s-1 m-1], ­ hillslope angle [­], ­ saturated water content [m3 m-3], ­ residual water content [m3 m-3], ­ first order soil water transfer term [s-1], ­ effective porosity for hypodermic flow [­]. DOHNAL M., DUSEK J., VOGEL T., 2006: The impact of the retention curve hysteresis on prediction of soil water dynamics. J. Hydrol. Hydromech., 54, 258­268. FAN Y., BRAS R., 1998: Analytical solutions to hillslope subsurface storm flow and saturation overland flow. Water Resour. Res., 34, 921-927. GERKE H. H., VAN GENUCHTEN M. Th., 1993: A dualporosity model for simulating the preferential movement of water and solutes in structured porous media. Water Resour. Res., 29, 305­319. HILBERTS A. G. J., VAN LOON E. E., TROCH P. A., PANICONI C., 2004: The hillslope-storage Boussinesq model for non-constant bedrock slope. J. Hydrol., 291, 160-173. HILBERTS A. G. J., TROCH P. A., PANICONI C., BOL J., 2007: Low-dimensional modeling of hillslope subsurface flow: Relationship between rainfall, recharge, and unsaturated storage dynamics. Water Resour. Res., 43, W03445. KIRKBY M. J., 1975: Hydrograph modelling strategies, in Process in Physical and Human Geography, edited by R. Peel, M. Chisholm, and P. Haggett, p. 69-90, Heinemann, London. LICHNER L., ELDRIDGE D. J., SCHACHT K., ZHUKOVA N., HOLKO L., SIR M., PECHO J., 2011: Grass Cover Influences Hydrophysical Parameters and Heterogeneity of Water Flow in a Sandy Soil. Pedosphere, 21, 719-729. MAJONE B., BERTAGNOLI A., BELLIN A., RINALDO A., 2005: GEOTRANSF: a continuous non-linear hydrological model. Eos Trans. AGU, 86 (52), Fall Meet. Suppl., Abstract H23C-1441. MAJONE B., BERTAGNOLI A., BELLIN A., 2010: A nonlinear runoff generation model in a small Alpine catchment. J. Hydrol., 385, 300-312. MARQUARDT D., 1963: An Algorithm for Least-Squares Estimation of Nonlinear Parameters, SIAM J. on Applied Mathematics, 11, 431-441. PANICONI C., TROCH P. A., VAN LOON E. E., HILBERTS A. G. J., 2003: Hillslope-storage Boussinesq model for subsurface flow and variable source areas along complex hillslopes: 2. Intercomparison with a three-dimensional Richards equation model. Water Resour. Res., 39, Article No. 1317. TROCH P. A., VAN LOON E. E., HILBERTS A. G. J., 2002: Analytical solutions to a hillslope-storage kinematic wave equation for subsurface flow. Adv. Water Resour., 25, 637-649. TROCH P. A., PANICONI C., VAN LOON E. E., 2003: Hillslope-storage Boussinesq model for subsurface flow and variable source areas along complex hillslopes: 1. Formulation and characteristic response. Water Resour. Res., 39, Article No. 1316. SANDA M., 1999, Tvorba podpovrchového odtoku na svahu. [Doctoral thesis.], CTU in Prague, Prague. SANDA M., HRNCÍ M., NOVÁK L., CÍSLEROVÁ M., 2006: Impact of the soil profile on the rainfall-runoff process. [In Czech.] J. Hydrol. Hydromech., 54, 183-191. SANDA M., CÍSLEROVÁ M., 2009: Transforming hydrographs in the hillslope subsurface. J. Hydrol. Hydromech., 57, 4, 264­275. VOGEL T., VAN GENUCHTEN M.TH., CÍSLEROVÁ M., 2001: Effects of the Shape of the Soil Hydraulic Functions near Saturation on Variably-Saturated Flow Predictions. Adv. Water Resour., 24, 133-144.

Journal

Journal of Hydrology and Hydromechanicsde Gruyter

Published: Jun 1, 2012

There are no references for this article.