**5. Experimental System**

The calibration process of the λ parameter requires the use of results obtained by laboratory experiments carried out through a physical model [26]. The adopted equipment is 1.52 m long, 1.22 m wide, and 0.78 m deep, with a tray angle adjustable up to 30◦. As it can be seen in Figure 5, all boundaries of the physical model are impermeable except for the soil surface. A small gravel layer (with thickness 7 cm) is placed at the bottom of the soil. Surface and percolated/deep flows are measured by two calibrated sensors based on a tipping-bucket mechanism. Different steady rainfall rates, sufficiently uniform over the slope with values up to 80 mm <sup>h</sup>−1, can be generated using special sprinklers, a pump, and a manual manometer. The various natural soils used for the experiments are characterized by the grain size distributions shown in Figure 6 with saturated hydraulic conductivity associated to horizontal surface, *Ks*, equal to 2.93 mm <sup>h</sup>−1, 3.20 mm <sup>h</sup>−1, 10.37 mm h−<sup>1</sup> and 17.00 mm h−<sup>1</sup> for soil 1, soil 2, soil 3, and soil 4, respectively [26,32–36]. Each experimental run lasted more than 24 h, and the rainfall that was applied during the first 8–10 h produced extended periods with steady conditions in the absence of direct rainfall infiltration. Before the beginning of each experiment, rainfall was applied in order to have high soil water content at any depth. Furthermore, in the time period between two successive experiments, the surface water content was kept sufficiently high in order to avoid the formation of cracks. The measurements of deep flow for different gradients obtained under steady conditions are assumed equal to *Kse*. Soil erosion did not affect the experiments as checked by an analysis of the surface water cloudiness.

**Figure 5.** A view of the physical model adopted in the laboratory experiments for slope-infiltration studies.

**Figure 6.** Grain size distribution of the soils used in the laboratory experiments.

#### **6. Analysis of Results**

Some experimental results on the laboratory system described above were deduced by Reference [26]. Three natural soils were used in different sets of laboratory experiments. The first set of experiments (performed by using soil 1 of Figure 6, labeled as "Clay Loam" soil according to USDA soil classification) consisted of 24 trials with slopes in the range 1◦–15◦ and rainfall rates in the range 10–20 mm h−1. The second set (by using soil 2 of Figure 6, labeled as "Loam" soil by USDA) consisted of 8 trials with slopes and rainfall rates in the ranges 1◦–15◦ and 10–15 mm <sup>h</sup>−1, respectively. Finally, the third set of experiments (soil 3 of Figure 6, labeled as "Sandy Loam" soil by USDA) consisted of 18 trials where slopes and rainfall rates were in the ranges 1◦–10◦ and 20–30 mm <sup>h</sup>−1, respectively. As it can be seen, the adopted surface slopes and soil types were not exhaustive because they didn't involve slopes exceeding 15◦ and didn't consider a very coarse textured soil.

Therefore, a new set of experiments has been carried out with a fourth soil type (soil 4 of Figure 6, labeled as "Sandy" soil by USDA) considering the surface slope in the range 10◦–26◦ and a very heavy rainfall rate (about 60 mm <sup>h</sup>−1). The main results obtained by these new experiments are summarized in Table 2 and confirm that for a fixed reference rainfall rate the slope gradient has a negative influence on the steady deep flow. These results are consistent with those of Reference [26]. Even if they refer to conditions dominated by gravitational effects, the theoretical formulations earlier proposed provide incorrect simulations.

**Table 2.** Steady surface and deep flows for different laboratory slope gradients under a rainfall rate of about 60 mm h−1. Soil 4 of Figure 6 is shown.


The estimate of λ requires us to subdivide the experiments selected here, excluding those carried out with almost horizontal soil surface that have been used for the determination of *Ks* (14 experiments with slope equal to 1◦, as specified in References [15] and [26]), into two groups (with comparable general features) to be used in the calibration and validation phases, respectively (Table 3).

Through an inverse procedure, for each laboratory experiment of the selected calibration set a specific λ parameter can be derived from Equation (4). Specifically, as a first approximation, *vl* is assumed to be equal to *Ks* because a better approach closer to physical reality would require a joint solution of Equations (1) and (2), which is blocked, as pointed out in Section 2, by the difficult representation of their interaction. Then, the term on the left, *P*(*v* ≤ *vl*), is expressed by the ratio between the observed steady deep flow, *df*, and *Ks*. Therefore, λ is the only unknown quantity in Equation (4).

Considering the laboratory experiments 1–20 of Table 3, the following general relation between slope and λ has been obtained:

$$
\lambda = 0.9861e^{-0.139 \times \text{slope}},
\tag{5}
$$

where the *slope* is expressed in (◦). Figure 7 shows the accuracy level of this interpolating function. The scattering of the λ values at given slope angles may be linked with other influential factors not represented in our approach such as, for example, the surface roughness that is expected to be variable from bare soil to another because of their different structure.

The model validation has been made using the laboratory experiments 21–40 of Table 3, through the relative error of the steady deep flow, <sup>ε</sup>*df*, defined as follows:

$$
\varepsilon\_{df} = 100 \frac{\left(\mathcal{K}\_{\text{sc}} - d\_f\right)}{d\_f},
\tag{6}
$$

with *Kse* given by:

$$K\_{\rm sc} = K\_{\rm s} \Big(1 - e^{-\lambda K\_{\rm s}}\Big). \tag{7}$$

As shown in Table 4, <sup>ε</sup>*df* is within the range of −28.9%/+22.6% with an algebraic mean value of 1.0% (or 9.9% when the absolute value of each <sup>ε</sup>*df* is considered). In a few cases, the relative error is significant, and therefore, we note that the steady deep flow is generally well reproduced by the proposed model (see also Figure 8).

Finally, from the experimental results, it comes out that increasing the rainfall rate becomes larger for the steady surface flow while the steady deep flow experiences minor changes. More specifically, combining for each slope the results shown in Table 4 for the steady surface and deep flow with the values of rainfall rate given in Table 3, it can be deduced as doubled values of rainfall rate and surface runoff do not determine in the average a clear trend of the deep flow. This outcome may be ascribed to the fact that increasing the water depth on the surface increases its average velocity on the slope, but the velocity of the small layer that affects infiltration does not experience significant variation.


**Table 3.** Main characteristics of the selected laboratory experiments (from Reference [26] and Table 2), subdivided into calibration and validation sets.

**Figure 7.** The λ parameter of Equation (4) obtained for the calibration experiments performed at the different slope angles. The best interpolating function is also plotted.

**Table 4.** Model validation synthesized through the relative error of the steady deep flow, <sup>ε</sup>*df* (Equations (6) and (7)).


**Figure 8.** Comparison of the observed and simulated steady deep flow for the different validation experiments.
