Hydrological Theory
Calculating Effective Rainfall
The Green and Ampt Method 

From the MIDUSS Version 2
Reference Manual  Chapter 7
(c) Copyright Alan A. Smith Inc. 
The basic assumption
behind the Green and Ampt equation is that
water infiltrates into (relatively) dry soil as a sharp wetting
front. Figure 77 below illustrates the variation in moisture
content q
with depth z
below the surface, at a point in time when the front has progressed a
distance L.
Figure 77  Wetting
front of the Green & Ampt model
The passage of this
front causes the moisture content to increase from an initial value
qi
to a saturated value
qs.
This difference is defined as the moisture or water deficit
M,
ie
Typically for dry
soils
M
has a value in the range 0.2 <
M
< 0.5 depending on the soil voids ratio, with lower values for
prewetted soil.
If the hydraulic
conductivity of the soil is
K
(mm/hour or inches/hr) then by Darcy's law,
where represents
the hydraulic gradient.
The head causing
infiltration is given by equation [729].
where
h0
= depth of surface ponding
(usually neglected)
L
= depth of water already infiltrated
S
= suction
head at the wetting front.
The suction head
S
(millimetres or inches) is due to
capillary attraction in the soil voids and is large for fine grained
soils such as clays and small for sandy soils.
The total infiltrated
volume between the surface of the soil and the wetting front is
defined by equation [730].
The infiltration rate
f =
dF/dt
is then given by [731].
In order to calculate the effective
rainfall, this equation must be solved for each time step in the storm
hyetograph. As illustrated in the Figure 78, three cases must be
considered in which the infiltration rates at times and
are
denoted by and
respectively,
and the rainfall intensity is
assumed to be constant during the time step. Each case is considered
separately.
Figure 78  Three cases
of the Green & Ampt model.
Case
(1)
i.e.
the rainfall intensity exceeds the infiltration capacity of the soil
throughout the whole time step so that ponding
must occur for the entire time.
Case
(2)
i.e.
at the beginning of the time step the
infiltration capacity exceeds
the rainfall intensity but this changes before the time step is
completed. Ponding will start during the
time .
Case (3)
i.e.
the rainfall infiltrates for the entire time step and no
ponding occurs.
The solution algorithm
used can be summarized as follows.
(i)
If
ij>fj
then case (1) holds and
The effective rainfall
is then given by [7.33].
If ij
£
fj then either case (2) or
case (3) applies. If we assume that case (3) applies  i.e. all the
rainfall infiltrates during time Dt
 then we can estimate:
and
(iii) Test if
i_{j}
£
f_{j+1}
also. If so, then case (3) is true and:
If
i_{j}
> f_{j+1} as
computed in step (ii) then case (2) holds. The volume required to
cause surface ponding to occur is
calculated as:
The time to the start
of ponding
dt
can then be found from equation [738].
Then:
and the effective
rainfall can be estimated as:
The application of
this algorithm to each time step in the storm hyetograph produces an
effective rainfall hyetograph for either the impervious or pervious
surface. If the surface depression storage is finite this is
subtracted from the initial elements of the hyetograph. The remaining
effective rainfall produces the direct runoff hydrograph.
In the Green and
Ampt method you are prompted to supply
values for a total of five parameters. These are:
1.
Manning's 'n' roughness coefficient
2.
Water (or Moisture) deficit
M
(say 0.0 to 0.6)
3.
Suction head
S
(mm or inches)
4.
Soil conductivity
K
(mm/hour or inches/hour)
5.
Surface depression storage depth (mm or inches)
Parameters for the Green & Ampt equation
Figure 79 – Schematic
representation of the Green & Ampt
parameters
The schematic of
Figure 79 shows the fractions of solid material, moisture and air or
vapour in the soil. The voids ratio of
the soil
h
is typically between 0.4 to 0.5.
Within the voids of a dried sample there is a certain volume of
residual moisture
qr.
The remaining 'fillable' voids comprise
the effective porosity
qe
=
h

qr
and typically varies from 0.31 to 0.48. Now if the initial moisture
is denoted by
qi
the soil moisture deficit
M
=
qe

qi
.
The effective
saturation is denoted by
Se = (q

qr)/
qe
and can be used to estimate the
suction head at the wetting front as described by Brooks and Corey,
1964.  (see references)
Some typical values
suggested by Rawls, Brakensiek and Miller
(1983) (see references) are shown below.
Soil type 
Porosity 
Effective
porosity 
Suction
head 
Hydraulic
conductivity 



mm 
mm/h 
Sand 
0.437 
0.417 
49.5 
117.8 
loamy sand 
0.437 
0.401 
61.3 
29.9 
sandy loam 
0.453 
0.412 
110.1 
10.9 
loam 
0.463 
0.434 
88.9 
3.4 
silt loam 
0.501 
0.486 
166.8 
6.5 
sandy clay loam 
0.398 
0.330 
218.5 
1.5 
clay loam 
0.464 
0.309 
208.8 
1.0 
silty
clay loam 
0.471 
0.432 
273.0 
1.0 
sandy clay 
0.430 
0.321 
239.0 
0.6 
silty
clay 
0.479 
0.423 
292.2 
0.5 
clay 
0.475 
0.385 
316.3 
0.3 
