Solutions of Flow Equations and Measurements in the Alpine Toce Valley


AND RENATO SANTANGELO1 1 Dipartimento di Ingegneria dei Materiali e dell'Ambiente - Osservatorio Geofisico, Universita di Modena e Reggio Emilia, Via Vignolese 905, 41100 Modena, Italy, 2ISMAR, Grandi Masse, CNR, S. Polo 1364, 30125 Venezia, Italy


Hydrological models are based on the efficient and robust description of the different aspects of the hydrological cycle achieved by the parameterisation of the major pathways in this cycle: precipitation and evaporation (Brutsaert 1991). While rainfall data are, almost everywhere, easily available, evaporation measurements are still rare. However, on a global basis, evaporation is a component of the hydrological cycle almost as important as precipitation. In fact, continental precipitation is of the order of 0.80 m/y, and evaporation amounts to, roughly, 0.50 m/y (Brutsaert 1991). The surface soil moisture and the exchange of heat and moisture between the land surface and the atmosphere are of great importance in different fields like hydrology, meteorology and agriculture. The knowledge of the state of saturation of a soil and its spatial and temporal trends is a key factor to improve hydrological models (flood forecast) and meteorological numerical weather prediction (NWP) models. In fact, the atmosphere and the underlying land surfaces represent a heavily coupled system (Eagleson 1978; Brubaker and Entekhabi 1995; Brubaker and Entekhabi 1996). The evaporation process consists of two main consecutive stages. In the first stage, when the soil is wet and conductive enough to supply water at a rate commensurate with the evaporative demand, the evaporation rate is limited by external meteorological conditions (atmosphere limited stage). During the second stage of evaporation, the evaporation rate is limited by the rate at which the soil can deliver moisture towards the evaporation zone (soil limited stage; Hillel 1980b). Furthermore, the soil water content and other soil properties determine the runoff production in response to atmospheric precipitation. Nowadays, the capacity of meteorological models to provide accurate quantitative rainfall forecasts at the scale of flood-prone basins remains rather limited, especially for small mountain catchments in the Mediterranean regions. The lead times necessary to save property, and sometimes lives, imply that forecasters cannot rely only on observed rainfall. Obled and Djerboua (2000) suggest that to foresee much beyond the response time of the catchment (few hours), accurate models, knowledge of soil characteristics and hydrological observations are requested.

These aspects were the main scientific objectives concerning the hydrological tasks of the MAP programme. The MAP-hydrology research activities also

Climate and Hydrology in Mountain Areas. Edited by C. de Jong, D. Collins and R. Ranzi © 2005 John Wiley & Sons, Ltd aimed to improve the understanding of orographically influenced precipitation events and related flooding episodes and to improve the numerical prediction of moist processes in regions with complex topography, including interactions with land-surface processes (Binder 1996). The Lago Maggiore and, in particular, the Ticino-Toce watershed (CH-I) was one of the test sites of the MAP Special Observing Period (SOP; Binder and Schar 1996). The climatology of the southern slope of the Alps clearly shows distinct local precipitation maxima (Bougeault et al. 1998), and one of these, both for precipitation amounts and for frequency of heavy precipitation, occurs in the Lago Maggiore area (Canton Ticino - Northern part of the Piedmont region). The MAP-SOP (7 September-15 November 1999) was a very large experimental effort over the Alps mountain range, during which several Italian teams (Hydrology Working Group and Planetary Boundary Layer Working Group) operated jointly in the Lago Maggiore target area.

In the first part of this chapter, a user-friendly algorithm is presented to evaluate the water mass balance at the soil surface. The mass balance is obtained by means of soil moisture measurements at different depths. The soil moisture is measured by means of time domain reflectometry (TDR), which is a relatively new technique based on measuring the apparent dielectric constant of the soil. The apparent dielectric constant is related to the propagation velocity of an electromagnetic pulse travelling in the soil. The relationship between the dielectric constant and the soil volumetric water content is described by Topp etal. (1980) and Ledieu etal. (1986), among others, in an empirical fashion using both polynomial and linear forms. The algorithm to estimate the water mass balance at the surface was applied to the soil moisture data collected at a hydrological station installed in a wide meadow located between the mountain slope and the Toce River. A short drought period (in July) and the major precipitation event (IOP-02, in September) may be seen in the soil moisture data set measured in this Alpine Valley.

In the second part of the chapter, the Richards equation is taken into account. Numerical solutions of partial differential equations can be obtained using different finite-difference and finite-element approximations. Nevertheless, analytical solutions are of great interest because they allow insight into the physics of the processes. Here, two different approaches to obtain exact solutions of the Richards equation are presented. One is used to solve the non-linear equation in which the gravity term is neglected (diffusion equation); the other allows to derive solutions to the linearized Richards equation. The solution of the non-linear one-dimensional equation is based on the method suggested by Philip (1960). This method allows obtaining the soil water content evolution if the hydraulic diffusivity is known; vice versa it will give the diffusivity if the soil water content is known. This procedure can be used to create a table of hydraulic diffusivity functions on the basis of the experimental data features. During drying periods, the soil moisture vertical profile may present an inflection point; however, this kind of solution cannot be obtained if the hydraulic diffusivity is a monotonic increasing function of the soil volumetric water content. Another characteristic of all the solutions obtained using this method is that the cumulative evaporation is always proportional to the square root of the time. The second method is based on the linearised Richards equation. With this approximation, arbitrary initial and boundary conditions can be assumed obtaining valid solutions that represent the experimental data both during infiltration and evaporation periods. The solutions of the linearised Richards equation may be derived also using input fluxes at the surface (Warrick 1975; Basha 1999; Chen et al. 2001).


The research field site was located in a wide meadow in front of the hydropower plant of Pallanzeno (Long. 8.260°E, Lat. 46.047°N), in the Toce River valley, which is a classical glacial basin located in the North Piedmont (Italy), see Table 8.1 (Ranzi et al. 2003) and Figure 8.1. This test site is at 250 m (a.s.l.) and is located between the

Table 8.1 Basin characteristics


Toce at Candoglia

Name of the area

Val d'Ossola

Mountain range

Northern Italian Alps

Elevation range of the basin


(m a.s.l.)

Elevation range of


experimental sites (m a.s.l.)


45° 54'-46° 28' N


7° 52'-8° 29' E

Area (km2)




Glaciers and permanent snow



Dominant vegetation type

Deciduous and coniferous


Forests (%)


Mean runoff at catchment


outlet (mm)

Mean precipitation (mm)


Figure 8.1 General map of Italy and detailed map of the Po basin. The study area (Toce River Basin) is located in the small rectangular frame on the left

mountain slope and the Toce River (about 300 m away). The grass of the meadow was regularly cut so the grass height varied from 10 to 30 cm.

An automatic station was set up installing 15 buriable probes connected to a TDR system (Soilmoisture Equipment Corporation 2000) by a multiplexer. Twelve probes were installed horizontally at the following depths [cm]: 5, 7.5, 10, 12.5, 15, 20, 25, 30, 35, 40, 47 and 50. Three probes were installed vertically to measure the mean soil moisture of three successive soil layers: 0-20, 25-45 and 50-70 cm. The measurements were collected automatically (at a time step of 4 h) starting at the end of March and ending on November 15, 1999 (end of the MAP-SOP). Air temperature and precipitation data were available from the station at the hydropower plant.

According to the USDA soil texture classification, the first 40 cm of soil of the study site (Pallanzeno) is a silty loam poor in organic matter. Three soil samples were collected at three different depths. The upper depth (0 -10 cm) had a higher percentage of clay compared to the two lower depths while the soil organic matter content decreases as usual from the surface

Table 8.2 USDA textural classification

Depth [cm]


Silt [%]

Clay [%]

Organic matter

















to the deep layers (see Table 8.2). In Table 8.3, two different values (in situ and in laboratory) of the hydraulic conductivity at the saturation (Ks) are reported; the superficial layer (0 -14 cm) was measured in laboratory while the value for the deeper layer was obtained by means of an in situ infiltration test. The real value of Ks is expected to be between the two measured values reported in Table 8.3. Because of the difficulties related to the in situ test, the value of the saturated hydraulic conductivity (Ks) is, most likely, closer to the laboratory value (Falappi etal. 2000). The soil water content at saturation (0s) has been assumed to be equal to 92.5% of porosity (Van Genuchten etal.

Table 8.3 Soil hydraulic characteristics



Ks (laboratory)

2.26 10-7 [ms-1]

Ks (in situ)

1.89 10-4 [ms-1]

O (porosity)


9s = 0.925 O


9max (measured)

56 [%]

1991); this value equals the maximum soil water content measured at Pallanzeno during the IOP-02. The porosity and the soil water content at saturation values are also reported in Table 8.3. The soil analyses and the water retention curves have been carried out by the research unit of the University of Brescia and Istituto Agrario di San Michele all'Adige (Falappi et al. 2000; Eccel et al. 2000).

The soil moisture characteristic curve and the unsaturated hydraulic conductivity can be described respectively by the following power functions (Campbell 1985).

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