**1. Introduction**

The process of infiltration of water into the soil is highly dependent on soil hydraulic properties that are generally variable in space, both in the vertical and horizontal directions. In natural conditions, the net rainfall reaching the soil is also a ffected by the vegetation cover that produces rainfall interception, sheltering the soil surface from the impact of falling drops. Vegetation also provides root systems that generate preferential subsurface flow paths.

Historically, solutions to infiltration problems have been represented through analytical, numerical, conceptual and empirical mathematical formulations. Analytical solutions provide estimates of infiltration rate or cumulative infiltration as functions of time, usually by simplifications on the soil water content profile during the study period. Powerful computers use numerical simulations of unsaturated soil domains in a single vertical direction or in multiple spatial dimensions, allowing for the use of complex initial and boundary conditions. Conceptual models try to balance the reduction of process complexity with a satisfactory representation of physical reality, obtaining simplified problem formulations. Finally, empirical infiltration models involve parameters fitted to the measured infiltration, but they have limited power as predictive tools because the same model cannot be used in di fferent catchments.

The infiltration process has been deeply analyzed since early parts of last century at the point (or local) scale and lately also at a field scale, even though most models assumed a horizontal soil

surface. At the local scale, when single storms are considered, classical equations (e.g., References [1–6]) or more recent formulations (e.g., Reference [7]) are generally adopted, while in the presence of events with consecutive soil water infiltration-redistribution cycles, conceptual/semi-analytical models, such as the one described by Reference [8], become necessary. Upscaling of these local models to obtain real (or field) models has been performed considering both vertically homogeneous [9–11] and layered [12] soils.

All the aforementioned models, as well as many others in the scientific literature, consider a soil surface that is oriented horizontally, while in most practical conditions the infiltration process occurs on surfaces with significant gradients [13]. The results obtained in the latter condition are not conclusive [14] and a physically-based approach with the ability to justify the experimental results needs to be developed. Table 1 shows a summary of theoretical (both analytical and conceptual) and experimental (carried out in both laboratory and field) studies dealing with the role of slope in infiltration. A comparison of these analyses, especially when carried out in natural fields, is confounded by several factors such as soil type and microtopography, rainfall intensity and duration, and presence of vegetation, to name a few. However, there exist laboratory experiments (e.g., Reference [15]) designed to exclude the abovementioned e ffects, showing a significant reduction of infiltration with slope, beyond the value expected when a steady state condition of soil saturation is assumed. This result is more pronounced for bare and clay soils rather than in vegetated and coarse-textured soils. Instead, most studies in Table 1 showing that infiltration increases with increasing slope were characterized by the formation of rills and/or a sealing layer. Nevertheless, some studies reported in Table 1 recommended the adoption of empirical corrections for the saturated hydraulic conductivity, *Ks*, but these corrections are not based on a theoretical approach and cannot be extended for general use.


**Table 1.** Studies dealing with the relation between slope and infiltration.

Further theoretical developments are needed to understand the complex processes of infiltration over sloping surfaces and to obtain a model that is su fficiently representative of the available experimental results. A clear physical interpretation is required to support the prevailing hypothesis of decreasing infiltration with increasing slope and to indicate specific laboratory experiments needed to assess the proposed model.

The main objective of this paper is to present a conceptual model able to justify the reduction of infiltration (with increasing gradients) obtained in the absence of secondary disturbance e ffects. The basic idea of the proposed model would serve as a potential starting point to stimulate the development of new experiments to identify a model closer to physical reality.

## **2. Basic Equations**

A study of the slope-infiltration interaction should be carried out by considering the coupling of the fundamental processes occurring at and immediately below the soil surface (Figure 1), i.e., infiltration and surface runoff.

**Figure 1.** Schematic representation of the infiltration process over a sloping surface in a Cartesian coordinate system.

Under conditions of a homogeneous and isotropic soil and considering a smooth soil surface, the best theoretical representation of the involved processes can be obtained through the Richards equation for the sub-surface flow and the Saint-Venant equations for the overland flow. More specifically, the uni-dimensional Richards equation for a flat surface can be rewritten, in Cartesian coordinates (*x*\*, *z*\*) shown in Figure 1, in the form (Reference [23]):

$$\frac{\partial \partial}{\partial t} = \frac{\partial}{\partial z^\star} \left[ D \frac{\partial \theta}{\partial z^\star} \right] - \frac{dK}{d\theta} \frac{\partial \theta}{\partial z^\star} \cos \gamma\_\prime \tag{1}$$

where *t* is the time, θ the volumetric soil water content, *K* the hydraulic conductivity, *D* the soil water diffusivity equal to *K*(θ)(*d*ψ(θ)/*d*θ) with ψ soil suction head, and γ the slope angle. Equation (1), under appropriate initial and boundary conditions, is identical to the classical Richards equation but with *K* in the new coordinate system substituted by *K* cosγ.

The Saint-Venant equations, with a lateral inflow *ql* per unit length that does not contribute any additional momentum to the flow, may be written as:

$$\frac{\partial \underline{Q}}{\partial \mathbf{x}^\*} + \frac{\partial A}{\partial t} = q\_{l\nu} \tag{2}$$

$$S\_f = S\_0 - \frac{\partial h}{\partial \mathbf{x}^\*} - \frac{\upsilon}{\mathcal{g}} \frac{\partial \upsilon}{\partial \mathbf{x}^\*} - \frac{1}{\mathcal{g}} \frac{\partial \upsilon}{\partial t},\tag{3}$$

where *Q* is the discharge, *A* the area of a cross-section, *v* the velocity, *Sf* the friction slope, *S0* the surface slope, *h* the flow depth with reference to the bottom, and *g* the gravity acceleration.

A straightforward analysis of Equations (1)–(3) along with initial and boundary conditions highlights that they cannot interact adequately. Specifically, the solution of the Richards equation may produce effects on the Saint-Venant equations (e.g., when the soil surface is saturated, the rainfall

excess determined through Equation (1) and its boundary conditions could be identified with *ql*, that moves along the slope according to Equations (2) and (3)). On the other hand, there is no possibility for the Saint-Venant equations to produce, from a mathematical point of view, any effect on the upper boundary conditions of the Richards equation, therefore precluding a possible interaction between the overland flow velocity and infiltration. Currently, we do not have a good way of defining the interface conditions between surface and subsurface flows at the soil surface.

#### **3. The Conceptual Model**

The basic element of the proposed model is inspired by the following observation. When a golf player is completing a hole, the last shot over the green has to be realized with a threshold velocity. If the ball is characterized by a velocity greater than this threshold value, it does not enter the hole (Figure 2) even though the direction is correct. A similar effect may be applicable to water moving on a porous surface.

Let us consider steady conditions with a small layer of overland flow generated by a rainfall rate, *r*. In line with the aforementioned abstraction, water "particles" can be roughly considered like balls running over a soil surface and drawn in the soil pores by gravity but in the presence of an interaction among the liquid water "particles". Furthermore, it is logical to consider that the arrival of new "particles" close to the pores depends on the persistence of overland flow under a given rainfall rate. The last condition assures the existence of "particles" that potentially may fall in the pores.

For a horizontal soil surface, considering that (1) all the "particles" may potentially enter the pores, because their velocity over the surface is practically equal to zero, and (2) the rainfall excess cannot enter the pores, the proposed model becomes unnecessary.

The interest in this conceptual model arises in case the soil surface possesses a slope, as in many natural conditions. With rainfall excess, water particles move downstream. Depending on friction slope, fluid viscosity, and other local conditions, each infinitesimal "particle" of water is characterized by a specific velocity (Figure 3). Consequently, it can be assumed that the "particles" move over the slope with different velocities that in the average increase with the distance from the soil surface.

The quantity "particle velocity" in the small layer that produces subsurface flow may be assumed as a stochastic variable characterized by a specific cumulative probability that for simplicity can be expressed through an exponential term. The last choice, somewhat arbitrary, could be changed without significant alterations of the proposed conceptual model. We have:

$$P(\upsilon \le \upsilon\_l) = 1 - e^{-\lambda \upsilon\_l} \tag{4}$$

where λ (>0) is the parameter of the probability distribution, *v* the independent variable, and *vl* a specific value of *v* representing the maximum velocity that allows infiltration of "particles" in a given pore.

Therefore "particles" with velocity less than *vl* may enter a specific pore, while "particles" with a velocity greater than *vl* continue their run over the surface. Only the fraction *P*(*v* ≤ *vl*) may fall into the pore producing subsurface flow.

As for a golf player, the threshold velocity depends on the hole diameter. In the proposed conceptual model, *vl* depends on the pore dimensions that are linked with soil texture and particle layout in the soil matrix. This means that *vl* may be linked with *Ks*. Furthermore, *P*(*v* ≤ *vl*) changes with increasing slope because of the increasing velocity of "particles". Consequently, for steeper slopes, the probability to have values of *v* lower than *vl* decreases and Equation (4) suggests that λ becomes smaller (Figure 4). The probability *P*(*v* ≤ *vl*) is also influenced, particularly in the presence of vegetation, by the surface roughness, but bare soils are considered here.

**Figure 2.** Movement of balls in a golf game: Balls can fall into the hole or not depending on their velocity (*<sup>v</sup>*2>>*v*1).

Hence, a given surface slope determines a specific *P*(*v* ≤ *vl*) through a specific λ value, while the soil structure, characterized by a well-known *Ks*, affects the threshold velocity *vl*. As a final result, under steady conditions, infiltration of water into a slope, *Kse*, can be obtained as *Kse* = *Ks* × *P*(*v* < *vl*), with *Ks* that represents the saturated hydraulic conductivity for a horizontal soil surface. The quantity *Kse* may be considered as an effective saturated hydraulic conductivity depending on the soil gradient.

**Figure 3.** Soil surface with significant slope and "particles" characterized by various velocities.

**Figure 4.** Example of three different cumulative probability functions of the "particle" velocities associated to different slopes (slope 1 < slope 2 < slope 3) with the corresponding λ values in Equation (4) equal to 4, 2, and 1 s/cm, respectively.

## **4. Model Parameters**

The key parameters of the model are the threshold velocity, *vl*, and the decay parameter of the cumulative probability, λ. A simple approach to obtain their values can be to fix one of them and to determine the other using a calibration procedure based on the use of experimental data. As discussed above, the threshold velocity is physically linked with the pore diameters and therefore with soil texture and particle layout, which in turn influence *Ks*. We assume *vl* equal to *Ks* even though this choice will affect the estimate of λ, that should depend only on the slope while its value obtained through calibration will also adjust for this non-optimal hypothesis. Really, this assumption does not reflect the physical reality because *vl* depends on the pore diameters that in a given soil is a random variable and therefore should be represented by a stochastic approach. Therefore, our rough simplification with a sole value of *vl* equal to *Ks* is equivalent to considering all soil pores characterized by a representative diameter.
