The infiltration of available water from the surface into the root zone is determined using the Green-Ampt equation. This equation is the result of applying the one-dimensional version of Darcy's equation in the vertical direction

$\displaystyle{q_{i} = -K(\psi)\left[\frac{d\psi}{dz} - 1\right],}$     (1)

where $\psi$ is the total pressure head and $K(\psi)$ is the hydraulic conductivity.

The Green-Ampt model for surface infiltration assumes a piston-like wetting front as depicted in the figure to the right. Assuming $q_{i}$ is a function of time only and integrating Equation (1) along the instantaneous wetted profile gives

$\displaystyle{\int_0^{L_f} q_{i} dz = -\int_0^{L_f} K(\psi)\frac{\partial \psi}{\partial z} dz + \int_0^{L_f} K(\psi) dz}$      (2)

Using the change of variable $\psi = \psi(z)$, so that $\frac{\partial \psi}{\partial z} dz = d\psi$, the fact that $\psi(0) = H_p$ ($H_p$ is the pressure head due to the water ponded at the surface), and $\psi(L_f) = \psi_i$ ($\psi_i$ is the initial pressure head at $L_f$, the depth of the wetting front), Equation (2) becomes

$\displaystyle{q_{i}\int_0^{L_f} dz = -\int_{H_p}^{\psi_i} K(\psi) d\psi + \int_0^{L_f} K(\psi)dz.}$      (3)

The first integral on the right-hand-side may be written as

$\displaystyle{-\int_{H_p}^{\psi_i} K(\psi) d\psi = -\int_{H_p}^0 K(\psi)d\psi - \int_0^{\psi_i}K(\psi) d\psi.}$      (4)

It follows that for $0 \le \psi \le H_p$, $K(\psi) = K_s$, the hydraulic conductivity at saturation. (The pressure head is negative if and only if the soil is unsaturated.) Also, letting $k_{rw}$ represent the relative permeability of the soil, we have

$\displaystyle{\int_0^{\psi_i} K(\psi)d\psi = K_s\int_0^{\psi_i} k_{rw} d\psi}$      (5)

Consequently,

$\displaystyle{-\int_{H_p}^{\psi_i}K(\psi)d\psi = K_s(H_p - \int_0^{\psi_i} k_{rw}d\psi)} $     (6)

Using the concept of relative permeability again, we may write

$\displaystyle{\int_0^{L_f} K(\psi) d\psi = K_s\int_0^{L_f} k_{rw} dz.}$      (7)

Substituting Equation (6) and Equation (7) into Equation (3) gives

$\displaystyle{q_{i}\int_0^{L_f} dz = K_s(H_p + \int_0^{L_f} k_{rw} dz - \int_0^{\psi_i}k_{rw} d\psi ).}$      (8)

Using the concept, once again, of an abrupt moisture change in the Green-Ampt profile, let $k_{rw} = 1$ on $0 \le z \le L_f$. Then, Equation (8) becomes

$\displaystyle{q_{i}\int_0^{L_f} dz = K_s\left(H_p + L_f - \int_0^{\psi_i} k_{rw} d \psi \right).}$      (9)

The integral

$\displaystyle{\int_0^{\psi_i} k_{rw} d\psi = H_f}$      (10)

is often referred to as the "negative pressure head at the wetting front" (see, for example Neuman). Using $H_f$ and the fact that $\int_0^{L_f} dz = L_f$, Equation (9) may be written as

$\displaystyle{q_{i} = K_s \left[\frac{H_p + L_f - H_f}{L_f}\right]}$,      (11)

which is a common form of the Green-Ampt equation for infiltration at the surface.

In the present model, $H_p$ is assumed to be negligible, and the saturated hydraulic conductivity $K_s$ is given by

$K_s = K_0e^{fz}$     (12)

to account for the variation in hydraulic conductivity in a class of non-uniform soils (Beven). Using a generalization of the relationship for a layered soil (Childs and Bybordi), the infiltration rate is given by

$\displaystyle{q_{i} = \frac{dI}{dt} = \frac{L_f + H_f}{\int_0^{L_f} \frac{dz}{K(z)}}}$,     (13)

where $I$ is the cumulative infiltration. Assuming that at the ponding time $t_p$ the cumulative infiltration is $I_p$ (Mein and Larson), the wetting front has penetrated to a depth of

$\displaystyle{L_p = \frac{I_p}{\Delta \theta}}$,     (14)

where $\Delta \theta$ is the change in soil porosity across the wetting front.

For a given constant rainfall rate $q_r = r$, the time to ponding is determined by substituting $\frac{dI}{dt} = r$ and the relationship in Equation (14) into Equation (13) and integrating to give

$\displaystyle{r = \frac{K_0f(H_f + I_p/\Delta \theta)}{1 - e^{-fI_p/\Delta \theta}}}$       (15)

Equation (15) may be solved for $I_p$ which in turn is used to determine $t_p$ through the relation

$I_p = rt_p$      (16)

Calculating the time to ponding determines if the current rainfall rate $r=q_r$ results in ponding in the present time interval. If not, then $q_{i} = r$. Otherwise, ponding begins at some time in the current interval including the possibility of ponding occurring for the entire interval.

The infiltration excess available for runoff is simply any positive difference between the rainfall rate $q_r$ and the infiltration rate $q_i$. That is,

$q_{ie} = \mbox{max}(q_r - q_i, 0)$     (17)

In summary, the fluxes $q_i$ and $q_{ie}$ from the surface are determined in the manner outline below.

For a time interval $\Delta t$ beginning at time $t$ with rainfall rate $q_r(t) = r$:

  1. If ponding has occurred in the previous time step then solve

    $\displaystyle{q_i(t) = \frac{dI(t)}{dt} = \frac{K_0f\left(H_f + \frac{I(t)}{\Delta \theta}\right)}{1 - e^{\frac{-fI(t)}{\Delta \theta}}}}$

    for $q_i(t)$ by an iterative technique such as Newton-Raphson. Then

    $q_{ie}(t) = \mbox{max}(q_r(t) - q_i(t), 0) $

  2. If ponding had not occurred in the previous time step, determine if ponding will occur in the current time step using Equations (15) and (16) to calculate $t_p$.