Derive total infiltration from Horton's equation

Horton's equation

The infiltration capacity (rate) at time $t$ is

$$f_p = f_c + (f_0 - f_c)\,e^{-Kt}$$

where $f_0$ is the initial capacity, $f_c$ the final (constant) capacity and $K$ the decay constant (h⁻¹).

Total (cumulative) infiltration

Total infiltration in time $t$ is the integral of the rate (assuming rain is always above capacity):

$$F_p(t) = \int_0^t f_p\,dt = \int_0^t \left[f_c + (f_0 - f_c)e^{-Kt}\right]dt$$ $$F_p(t) = f_c\,t + (f_0 - f_c)\left[\frac{e^{-Kt}}{-K}\right]_0^t$$ $$\boxed{F_p(t) = f_c\,t + \frac{f_0 - f_c}{K}\left(1 - e^{-Kt}\right)}$$

As $t \to \infty$, $F_p \to f_c t + (f_0 - f_c)/K$, so the cumulative curve becomes a straight line of slope $f_c$.

Graph

Infiltration capacity $f_p$ falls from $f_0$ to $f_c$ (sketch drawn for $f_0 = 4$, $f_c = 1$, $K = 1.5$ h⁻¹):

 f (cm/h)
  |*
  |
  | *
  |  *
  |   *
  |    *
  |     *
  |      **
  |        *****
  |             ****************************  fc
  +----------------------------------------- t

Total infiltration $F_p$ rises quickly at first (slope $f_0$) and then increases at the constant slope $f_c$:

 F (cm)
  |                                      ***
  |                                  ****
  |                             *****         final slope = fc
  |                         ****
  |                     ****
  |                *****
  |             ***
  |         ****
  |      ***
  |    **
  |  **           initial slope = f0
  | *
  |*
  +----------------------------------------- t

The area under the $f_p$-$t$ curve up to time $t$ equals $F_p(t)$.