*3.2. Philip Equation*

A widely adopted analytical solution of the Richards equation was proposed by References [9–11,109] under the conditions of vertically homogeneous soil, constant initial moisture content, and saturated soil surface with immediate ponding. For early to intermediate times the semi-analytical solution for the infiltration capacity, based on a series expansion and truncation after the first two terms, is expressed as:

*fc* = 12*St*−1/2 + *A* , (9)

where *S* is the sorptivity, depending on soil properties and initial moisture content, and *A* is a quantity ranging from 0.38 *Ks* to 0.66 *Ks*. For *t*→∞, Equation (9) is replaced by *fc* = *Ks*. The integration of Equation (9) yields the cumulative infiltration:

$$F = St^{1/2} + At.\tag{10}$$

Philip's model was extended for applications to less restrictive conditions. For a constant rainfall rate *r* > *Ks*, surface saturation occurs at a time *tp* > 0 and, following [45], infiltration can be described through an equivalent time origin, *t*0, for potential infiltration after ponding as:

$$t\_p = \frac{S^2\left(r - \frac{A}{2}\right)}{2r\left(r - A\right)^2},\tag{11}$$

$$t\_0 = t\_p - \frac{1}{4A^2} \left[ \left( S^2 + 4AF\_p \right)^{1/2} - S \right]^2,\tag{12}$$

$$f\_{\mathcal{E}} = \frac{1}{2} \mathcal{S} (t - t\_0)^{-1/2} + A\_{\prime} \ \ t \ > \ t\_{\mathcal{P}} \ . \tag{13}$$

For unsteady rainfall under the condition of a continuously saturated surface for *t* > *tp*, the infiltration process can be represented adopting a similar procedure. Generally, *S* and *A* are derived from the calibration of hydrological models; however, *S* can also be approximated as [110]:

$$S = \left[ 2(\phi - \theta\_i) K\_s |\psi\_{av}| \right]^{1/2} \,, \tag{14}$$

where *φ* is the soil porosity and *ψav* is the soil water matric capillary head at the wetting front.
