# Rainfall Infiltration Modeling: A Review

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Basic Physical Models for Infiltration

_{0}, is given by Equation (1) applied at the soil surface.

_{1}= dθ/dψ and C

_{2}= dK/dψ with a typical assumption that θ and K are unique functions of ψ thus neglecting hysteresis in these functions. The initial condition at time t = 0 for z ≥ 0 is ψ = ψ

_{i}, and the upper boundary conditions at the soil surface, where z = 0, are:

_{0}= r, 0 < t ≤ t

_{p}; θ

_{0}= θ

_{s}, t

_{p}< t ≤ t

_{r}; q

_{0}= 0, t

_{r}< t,

_{p}is the time to ponding, and t

_{r}is the duration of rainfall. Hereafter the subscripts i and s denote initial and saturation quantities, respectively, while 0 stands for quantities at the soil surface. The lower boundary conditions at a depth z

_{b}which is not reached by the wetting front is ψ(z

_{b}) = ψ

_{i}for t > 0. The soil water hydraulic properties can be represented by the following parameterized forms [22]:

_{b}is the air entry head, θ

_{r}is the residual volumetric water content, c, λ, and d are empirical coefficients, and b = 3 and a = 2 according to Burdine’s method [101]. For particular values of the parameters, Equations (5a) and (5b) reduce to the well-known equations proposed by References [101,102]. For two-layered soils, two additional conditions are required at the interface between the two layers:

_{c}is the interface depth.

## 3. Point Infiltration Modeling for Homogeneous Soils

#### 3.1. Horton Empirical Equation

_{c}, exponentially decreases as follows (see also Figure 2):

_{0}and f

_{f}represent the initial and final values of f

_{c}, respectively, and α is the decay constant. When t→∞, f

_{f}can be considered equal to the saturated hydraulic conductivity of the soil. If K and D are independent of θ, References [107,108] demonstrated that Equation (8) can be obtained from Equation (7).

#### 3.2. Philip Equation

_{s}to 0.66 K

_{s}. For t→∞, Equation (9) is replaced by f

_{c}= K

_{s}. The integration of Equation (9) yields the cumulative infiltration:

_{s}, surface saturation occurs at a time t

_{p}> 0 and, following [45], infiltration can be described through an equivalent time origin, t

_{0}, for potential infiltration after ponding as:

_{p}, 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]:

_{av}is the soil water matric capillary head at the wetting front.

#### 3.3. Green–Ampt Model

_{c}= dF/dt. The resulting equation [45] is:

_{s}, that begins at the time t = 0, surface saturation is reached at a time t

_{p}> 0. For t ≤ t

_{p}

_{,}the infiltration rate q

_{0}is equal to r and later to the infiltration capacity. Mein and Larson [104] formulated this process through Equation (15) as:

_{p}as:

_{p}

_{,}Equation (16) becomes:

_{c}.

#### 3.4. Parlange–Lisle–Braddock–Smith Model

_{i}t is the cumulative dynamic infiltration rate, α is a parameter linked to the behavior of hydraulic conductivity and soil water diffusivity as functions of θ, and G is the integral capillary drive defined by:

_{p}and f

_{c}for any rainfall pattern, and for t > t

_{p}can be rewritten under the condition of surface saturation in a time dependent form as:

_{d}= K

_{s}− K

_{i}and F′

_{p}= F′(t

_{p}). The quantities F′

_{p}and t

_{p}are the values of F′ and t at ponding, respectively, at which Equation (20) with f

_{c}= r(t

_{p}) is first satisfied. The value of α usually ranges from 0.8 to 0.85 [3].

#### 3.5. Corradini–Melone–Smith Semi-Analytical/Conceptual Model

_{s}, separated by periods with r = 0 (see Figure 4). We denote by t

_{1}the duration of the first pulse, t

_{2}and t

_{3}, the beginning and end of the second pulse, respectively, and t

_{4}the beginning of the third pulse. The model was derived considering a soil with a constant value of θ

_{i}and combining the depth-integrated forms of Darcy’s law and the continuity equation. In addition, as the event progresses in time, a dynamic wetting profile, of lowest depth Z and represented by a distorted rectangle through a shape factor β(θ

_{0}) ≤ 1, was assumed. The resulting ordinary differential equation is:

_{i},θ

_{0}) is expressed by Equation (21) modified by the substitution of θ

_{s}with θ

_{0}and K

_{s}with K

_{0}. Equation (23) can be applied for 0 < t < t

_{2}, and the profile shape of θ(z) is approximated [23] by:

_{0}= r, with F′ = (r-K

_{i})t, it gives θ

_{0}(t) until time to ponding, t′

_{p}, corresponding to θ

_{0}= θ

_{s}and dθ

_{0}/dt = 0. Then, for t′

_{p}< t ≤ t

_{1}, with θ

_{0}= θ

_{s}and dθ

_{0}/dt = 0, it provides the infiltration capacity (q

_{0}= f

_{c}) and for t

_{1}< t < t

_{2}, with q

_{0}= 0, it gives dθ

_{0}/dt < 0 thus describing the redistribution process.

_{p}, but reinfiltration occurs according to two alternative approaches determined by a comparison of r and the downward redistribution rate, D

_{F}(t = t

_{2}), expressed by:

_{F}, the reinfiltrated water is distributed to the whole dynamic profile and θ

_{0}(t) can be still computed by Equation (23), whereas for r > D

_{F}, the profile of θ(z, t = t

_{2}) is assumed temporarily invariant and starts a superimposed secondary wetting profile which advances alongside the pre-existing profile according to Equation (23) modified by substituting θ

_{i}with θ

_{0}(t

_{2}) and F′ with F′

_{2t}accumulated for t ≥ t

_{2}. If the secondary profile reaches the depth of the first steady one, the compound profile reduces to a single profile and then Equation (23) can be again applied (see Figure 3 for a schematic representation of θ(z,t)). On the other hand, if at t = t

_{3}the secondary profile has not caught up with the first one, redistribution is first applied to the secondary profile and then re-established to the single profile in the successive rainfall hiatus. Finally, in the case at t = t

_{4}, the θ(z) profile is still compound and r is larger than D

_{F}(t

_{4}), and a procedure of consolidation that merges the composite profile is applied early to avoid the formation of a further additional profile.

## 4. Point Infiltration Modeling for Vertically Non-Uniform Soils

#### 4.1. Green–Ampt-Based Model for a Layered Soil

_{2}is the depth of the wetting front below the interface. Equation (27) can be solved at each time for L

_{2}by successive substitutions; L

_{2}is then used in Equations (28) and (29) to determine F and f

_{c}, respectively.

#### 4.2. Corradini–Melone–Smith Semi-Analytical/Conceptual Model for a Two-Layered Soil

_{2}= ψ

_{i}. The initial condition is ψ

_{1}= ψ

_{2}= ψ

_{i}constant at t = 0, and at the interface q(Z

_{c}) is approximated through the downward flux in the upper layer as:

_{c},ψ

_{10}) is expressed by Equation (21) modified by the substitutions of D(θ)dθ with K(ψ)dψ, θ

_{s}with ψ

_{10}, and θ

_{i}with ψ

_{c}. As long as water does not infiltrate in the lower layer, the model presented in [26] is used, then starting from the time t

_{c}when the wetting front enters the lower layer, the following system of two ordinary differential equations may be applied:

_{2}defined as:

_{1}(ψ

_{10}) = dθ

_{10}/dψ

_{10}, C

_{1}(ψ

_{c}) = dθ

_{1c}/dψ

_{1c}. The quantity γ represents a conceptual proportion of the upper layer where θ is increasing due to rainfall and is assumed equal to 0.85 [42], β

_{2}and p

_{2}are determined by Equations (25a–c) but applied using θ

_{20}, θ

_{2s}, θ

_{2i}, and θ

_{2r}and substituting r with q(Z

_{c}). On the basis of the same stepwise rainfall pattern earlier adopted to explain the model presented in [26], Equations (31) and (32) may be used for t

_{c}< t < t

_{2}. Then, the two-layer model has to be applied in each layer by analogy with the procedure described for homogeneous soils; in particular, compound and consolidated profiles develop in each layer. In the underlying soil, the generation of additional profiles occurs through q(Z

_{c}) in substitution of r. On the basis of the described steps, model application to arbitrary rainfall patterns is straightforward. The solution of the above system, Equations (31) and (32), may be obtained by a library routine for the Runge–Kutta–Verner fifth-order method with a variable time step. Calibration and testing of the model were performed through a comparison with numerical solutions of the Richards equation. Three soils (clay loam, silty loam, and sandy loam) with a variety of thicknesses were combined to realize two-layered soils, where either layer was more permeable, that were selected as test cases. In all instances, the simulations involved the cycle of infiltration–redistribution–reinfiltration. The infiltration rate as well as the water content at the surface and interface were found to be very accurately estimated by the semi-analytical/conceptual model.

## 5. Areal Infiltration Models over Soil with Variable Hydraulic Properties

#### 5.1. Smith and Goodrich Approach

_{s}has been proposed by Reference [60]. The authors assumed a lognormal probability density function (PDF) of K

_{s}with a mean value <K

_{s}> and a coefficient of variation CV(K

_{s}), and considered one realization of the random variable. Then, adopting the model presented in [106] (see also [22]) and the Latin Hypercube sampling method, and through a large number of simulations performed for many values of CV(K

_{s}) and rainfall rates, they developed the following effective areal relation for the scaled areal-average infiltration rate, ${I}_{e}^{*}$, linked with the corresponding scaled cumulative depth, ${F}_{e}^{*}$:

_{e}denotes the areal effective value of K

_{s}given by:

_{s}. Finally, Equation (35) may be also applied for r variable with time.

#### 5.2. Govindaraju–Corradini–Morbidelli Semi-Analytical/Conceptual Model

_{n}(F)> under the condition of negligible effects of the run-on process. The model incorporates heterogeneity of both K

_{s}and r assumed as random variables with a lognormal and a uniform PDF, respectively. The quantity <I

_{n}(F)> is estimated through the averaging procedure over the ensemble of two-dimensional realizations of K

_{s}and r. For the sake of simplicity, we first examine the model under a steady-rainfall condition, and then provide the guidelines for applications to a time-varying rainfall rate.

_{n}(F)> at a given F can be written as:

_{r}(r) and ${f}_{{K}_{s}}\left(K\right)$ are the PDFs of r and K

_{s}, respectively, with K

_{c}which denotes the maximum value of K

_{s}leading to surface saturation in the i-th cell, determined by:

_{min}and r

_{min}+ R extreme values of the PDF of r and ${M}_{{K}_{s}}$ given by:

_{a}and ω stand for the first and the second argument, respectively, of the ${M}_{{K}_{s}}$ function. To relate time to F, an implicit relation between the expected value of t, <t>, and F is provided as:

_{pa}is the time to ponding (see Equation (18)) associated with <r> and <K

_{s}>. The parameters a, b, and c are expressed by:

_{i}≪ θ

_{s}and for θ

_{i}→θ

_{s}we have a→0. Furthermore, Equation (46) is undefined for $CV\left(r\right)$ and/or $CV\left({K}_{s}\right)$ equal to 0. In Equations (43)–(46), length units are in mm and time is expressed in hours.

_{s}is fairly simple and requires limited computational effort.

_{s}(Equations (40), (42), and (43)) was validated by comparison with the results derived starting from MC sampling and using a combination of the extended Green–Ampt formulation at the local scale with the kinematic wave approximation [125] that is required to represent run-on. Through a wide variety of simulations it was shown that: (1) the model produced very accurate estimates of <I> over a clay loam soil and a sandy loam soil; (2) the spatial heterogeneity of both r and K

_{s}can be neglected only when <r> ≫ <K> or for storm durations much greater than t

_{pa}; (3) the effects on <I> produced by significant values of CV(K

_{s}) and CV(r) are similar; (4) run-on plays a significant role for moderate storms and high values of CV(K

_{s}) and CV(r); and (5) the model can be simplified using Equations (47) and (48) for CV(r) substantially less than CV(K

_{s}) and steady rainfalls.

## 6. Conclusions and Open Problems

_{s}>, CV(K

_{s}), <r>, and CV(r) together with the corresponding quantities for soil moisture content [126].

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Infiltration capacity, IC(t), and infiltration rate, f(t), of a silty loam soil under a natural rainfall event (observed in Central Italy) characterized by a variable intensity, r(t).

**Figure 4.**(

**a**) Rainfall pattern selected to describe the Corradini–Melone–Smith semi-analytical model for point infiltration. (

**b**,

**c**) Profiles of soil water content at various times indicated in (

**a**) and associated with different infiltration-redistribution stages. For symbols, see the text.

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**MDPI and ACS Style**

Morbidelli, R.; Corradini, C.; Saltalippi, C.; Flammini, A.; Dari, J.; Govindaraju, R.S.
Rainfall Infiltration Modeling: A Review. *Water* **2018**, *10*, 1873.
https://doi.org/10.3390/w10121873

**AMA Style**

Morbidelli R, Corradini C, Saltalippi C, Flammini A, Dari J, Govindaraju RS.
Rainfall Infiltration Modeling: A Review. *Water*. 2018; 10(12):1873.
https://doi.org/10.3390/w10121873

**Chicago/Turabian Style**

Morbidelli, Renato, Corrado Corradini, Carla Saltalippi, Alessia Flammini, Jacopo Dari, and Rao S. Govindaraju.
2018. "Rainfall Infiltration Modeling: A Review" *Water* 10, no. 12: 1873.
https://doi.org/10.3390/w10121873