Describe hortons equation with parameters

Question

Assignment:

Problem:

(a). Show that the ponding time under rainfall of constant intensity i for a soil described by Horton's equation with parameters f_o  (infiltration rate at time 0), f_c (initial infiltration rate) and K (infiltration decay coefficient) is given by

t_p = 1/iK [ƒ_o - i + ƒ_c 1n(ƒ_o-ƒ_c / i- ƒ_c)]

If a soil has sorptivity f_o = 4.0 cm/hr, f_c = 0.8 cm/hr and K = 1.1 cm/hr, compute the ponding time and cumulative infiltration at ponding time under a rainfall of 2.5 cm/hr. Plot your solution. Indicate the range of i for which this equation is valid.

(b) Below is a table containing data from a double-ring infiltrometer experiment conducted on a bare sandy soil in an arid region.

(i) Using the data below, construct a plot of f(t) versus time (plot as accurate as you can by hand). Estimate values of f_o, f_c, and k for the Horton's infiltration formulation (be sure to closely evaluate which k values to use in your average or aggregated k value).

(ii). Now calculate the total infiltrated depth, F(t) at 20, 60 and 90 minutes using the parameters you have obtained using Horton's equation for cumulative infiltration (F(t)).

(iii). How does these values compare to the measured field values at 20, 60 and 90 minutes?

Time (min)     Water height (cm)

0                   20
0.5                19.5
1                   18.9
2                  18.0
3                  17.2
4                  16.4
5                  15.9
10                14.2
20                11.2
30                10.1
60                 9.6
90                 9.0
120               8.9

(c). (i) Derive Green-Ampt infiltration equation and its ponding time tp.