0t 0z 0 V V0t 1 ep t z 0t 0838

where p, which is suitable to write as 4 ■ a2 ■ D, is a positive constant with the dimension of one over time.

With the boundary conditions described in (8.38), (8.37) reduces to (8.35) and yields:

erfc

erfc

Solution (8.39) holds for fi < (V2/(4 • D)). For fi > (V2/(4 • D)), the solution (here not reported) can be expressed by the error function of complex variable. For fi = (V2/(4 • D)), Equation (8.39) reduces to:

which permits to compute the time t0 after which the inflection point appears. From the second derivative of (8.40), t0 = 2.25 • (D/V2) is obtained. Figure 8.7, obtained using V = 4 • 10-7 m • s-1 and D = 10-8 m2 • s-1, shows the behaviour of three soil moisture profiles obtained by Equation (8.39) for three different values of the time (1, 3, 5 days) and for two values of the parameter fi. The dashed profiles are obtained for fi = 4 • 10-6 s-1 (i.e. fi = (V2/(4 • D))), which means t0 « 1.6 days. The curve for t = 1 day (t < t0) does

Figure 8.7 Behaviour of the theoretical soil moisture profile obtained by Equation (8.39) for three different values of the time (1, 3, 5 days) and for two values of the parameter fi. The dashed curves are for fi = 4 • 10-6 s-1 and the solid lines are for fi = 2 • 10-6 s-1

Figure 8.7 Behaviour of the theoretical soil moisture profile obtained by Equation (8.39) for three different values of the time (1, 3, 5 days) and for two values of the parameter fi. The dashed curves are for fi = 4 • 10-6 s-1 and the solid lines are for fi = 2 • 10-6 s-1

not have the inflection point, which is present in the other two curves, corresponding to t = 3 and t = 5 days, that is, t > to. The solid line profiles are obtained for

days) is located closer to the surface with respect to the previous case.

Experimental data

During the period of measurements at the Pallanzeno station, two were the main hydrological events. The first happened during the summer period (dry event) before the MAP-SOP and the second during the second Intensive Observing Period (IOP-02, wet event).

Dry event

The trend of the experimental data collected during the dry event (21-25 July 1999) suggests a time dependence of the soil moisture at the surface. This experimental case may be modelled in a way similar to the example of Section 8.5.2. Using the following boundary conditions:

û = l,t = 0,z > 0; û = û0(t) = e z = 0,t > 0

the theoretical soil moisture simply results:

erfc

In fact, treating a dry case, the theoretical soil moisture changes from 1 to 0 and the solution (8.42) is one minus solution (8.39).

Figure 8.8 shows the evolution of the soil moisture in the upper 40 cm of soil. The experimental data (symbols) are the mean daily profiles observed at Pallanzeno from July 21 to July 25. The solid lines are the corresponding theoretical trends obtained from the analytical solution (8.42) and from (8.12) assuming 9m = 5%,9M = 30%,D = 5 • 10-9m2 s-1, V = 4 • 10-7ms-1 and fi = 1.6 • 10-6 s-1. The value of fi was chosen on the basis of the experimental soil water content values measured at 5 cm below the surface. The condition fi < (V2/(4 • D)) is verified by the values of parameters V and D. The mean hydraulic diffusivity introduced in the linearized flow equation was computed by the relationship suggested by Crank (1956) for the drying processes. The values of 0m ,9m and V were derived from the experimental data. The matching of linear solution and experimental data is mainly affected by the non-uniform initial condition (21 July 1999) and by the inhomogeneity of the soil layer (see also the section below). Furthermore, a not-short period (five days) is modelled.

Wet event

During I0P-02 (19-21 September 1999), the heaviest precipitation event of the MAP-SOP, a cumulative precipitation of 225.4 mm was recorded at Pallanzeno. The observations describing the behaviour of the soil moisture in the 0-40 cm layer, during the initial part of the event (from September 19 at 16:00 h to September 20 at 08:00 h), are compared with a solution derived from the previously described methodology (Section 8.5.2).

Assuming the boundary condition (8.38) as fi approaches infinity, the solution of Equation (8.31) is:

The mean hydraulic diffusivity introduced in the linearised flow equation was computed by the relationship suggested by Crank (1956) for the infiltration process. The experimental data suggest the presence of a wetting front moving downward. Similarly, Equation (8.43) has an inflection point that moves downwards with a velocity related to the constant V. The velocity of the inflection point approaches V as the time increases. Identifying the inflection point with the experimental wetting front, an estimation of V is obtained. The evolution of the soil moisture in the upper 40 cm of soil is presented in Figure 8.9. The data begin on September 19 at 16:00h

Pallanzeno - July 21-25, 1999

Pallanzeno - July 21-25, 1999

25 30 35

Soil moisture (%)

Figure 8.8 Dry event. Evolution of the daily mean soil moisture at Pallanzeno in the 0-40cm soil layer from July 21 to July 25. The experimental profiles (symbols) are compared with the theoretical trends (solid lines) obtained from Equation (8.42)

25 30 35

Soil moisture (%)

Figure 8.8 Dry event. Evolution of the daily mean soil moisture at Pallanzeno in the 0-40cm soil layer from July 21 to July 25. The experimental profiles (symbols) are compared with the theoretical trends (solid lines) obtained from Equation (8.42)

Pallanzeno - September 19-20, 1999 (IOP-02)

Pallanzeno - September 19-20, 1999 (IOP-02)

40 45

Soil moisture (%)

Figure 8.9 Wet event. Evolution of the soil moisture at Pallanzeno in the 0-40 cm soil layer from September 19 at 16:00h to September 20 at 08:00 h. The experimental profiles (symbols) are compared with the theoretical trends (solid lines) obtained from Equation (8.43)

Sep. 20 h 08

Sep. 20 h 04

Sep. 19 h 24

Sep. 19 h 20

Sep. 19 h 16

40 45

Soil moisture (%)

Figure 8.9 Wet event. Evolution of the soil moisture at Pallanzeno in the 0-40 cm soil layer from September 19 at 16:00h to September 20 at 08:00 h. The experimental profiles (symbols) are compared with the theoretical trends (solid lines) obtained from Equation (8.43)

and end on September 20 at 08:00h, with a time step of four hours. The experimental profiles (symbols) are compared with the theoretical trends (solid lines) obtained from the analytical solution (8.43) and from (8.12) assuming 9m = 33%,9m = 49%,D = 5 • 10-8 m2 s-1, V = 8 • 10-6ms-1. The modelled behaviour matches the experimental data mainly at the first two time steps, that is, after four and eight hours from the beginning of precipitation, when essentially the upper part of the soil is involved in the infiltration process. Falappi et al. (2000) reported different values of the saturated hydraulic conductivity (obtained both in laboratory and in situ) depending on the depth of the soil involved in the measurement. These data suggest that the hydraulic conductivity at saturation increases with the depth of the soil layer. The main differences between the experimental data and the theoretical solution, obtained for a homogeneous soil, should therefore be related to the not perfect homogeneity of the considered soil. For example, the shape of the soil moisture profile on September 19 at 24:00h (circles) can be better fitted by the linear solution using a slightly greater value of the hydraulic diffusivity. A similar effect is obtained modifying the parameter V .

8.6 CONCLUSIONS

The experimental soil moisture measurements carried out during an international fieldwork have been presented. The solutions of the flow equation for the unsaturated zone (both for drying cases and infiltration cases) highlight different soil moisture profile according to the soil characteristics, which can be recognised in the experimental data.

The observations describing the behaviour of the moisture in the 0 -40 cm soil layer during a drought period and during the heaviest precipitation event, I0P-02 of the MAP-SOP, are compared with two different solutions derived from the methodology described in Section 8.5.2.

A water balance algorithm to estimate the cumulative evaporation from the soil water content experimental measurements was presented. From the computed cumulative evaporation, the first and the second stages of evaporation can also be distinguished in this rainy Alpine Valley.

Meteorological and hydrological working groups of the MAP experiment collaborate strongly because NWP models and runoff and/or flood events prediction models need meteorological data as well as soil hydraulic characteristics and soil water content data. Meteorological stations are widely spread over the territory; this is not the case of experimental hydrological stations. The data collected at the Pallanzeno station are still being studied, and the results must be compared with the ones obtained by the other hydrological and meteorological research units.

An interesting evolution of this work will be related to the solution of the linearised flow equation using the precipitation as boundary condition. If the intensity of precipitation is lower than the infiltration capacity of the soil, it can be assumed as the prescribed vertical flux at the air-soil interface. The main advantages of this evolution are that precipitation measurements are much more common than soil moisture measurements and that precipitation events may be approximated by means of functions, which permit to obtain closed-form solutions.

8.7 ACKNOWLEDGEMENTS

We wish to thank the Italian Electrical Energy Company ENEL for the permission to install our instrumentation on their properties and for the help given during the whole period of measure. We also thank them for their rain gauge data.

The Italian Council of Research (CNR) and the Italian Ministry of the University and of the Scientific and Technological Research (MURST), in part, funded this work.

Finally, we wish to thank the two reviewers of this work for their helpful suggestions and their help in making this paper clearer and more complete.

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