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Infiltration
1
Purpose of infiltration models




Irrigation planning
Drainage calculations
Hydrological models
Soil erosion models
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Measurement
 double ring method - gives values which are
too high for soil erosion studies, so best use
rainfall simulators for this purpose - but
method is fine for surface irrigation design
 rainfall simulator (best for sprinkler irrigation
design and hydrological modelling)
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Double ring method
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Modeling
Some workers have used texture, bulk density
(e.g.) to predict infiltration but reality is
complicated by e.g. roots, fauna, tillage system,
etc. so be cautious with equations
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Philip’s equation (1957)
Used Darcy-Richard equation and continuity
equation to show that accumulated infiltration:
1
2
I  St  At  k 3 t  k 4 t  k 5 t 2  ...
3
2
2
5
I is the accumulated infiltration
S is called the sorptivity (related to rate at
which soil absorbs water) measured by horizontal
infiltration
A is related to saturated hydraulic conductivity
(rate at which water passes through soil).
At very long times, A approaches the hydraulic
conductivity.
11
The infiltration rate, dI/dt at some time, t is
given by:
dI
dt
 St
1
2
 12
A
To solve equation for accumulated I, tabulate t
and t1/2 for each I, then treat as independent
variables, x1 and x2 so
I = Sx1 + Ax2
This can be solved in Minitab (not Excel) under
“Stats - regression” (untick intercept box)
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Kostiakov (1932)
I=cta
dI/dt = i = c a t a-1
Infiltration rate declines with time so a - l is
always negative.
Thus, a is always less than 1.
This means that for large times, the rate is 0,
which is not true either experimentally or
physically.
For example, if a = 0.5, then a-1 = - 0.5
so infiltration rate = constant x 1/ t 0.5.
When t = 100, 1/t0.5 = 0.1.
When t=10000, 1/t 0.5 = 0.01, and so on
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To find a and c, tabulate logarithms of I and t in
Excel.
If y = log I and x = log t, and k=log c then
y = k + ax so plot y against x and plot the curves
and “Add Trendline” using the “include equation”
option.
The slope of the line is a and the intercept is k
from which you can calculate c
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Parameters for Kostiakov's and Philip's equations
for a variety of soils
[I in inches, t in minutes]
Kostiakov parameters
Soil type
Honeoye gravely silt
loam
Aiken clay loam
Palouse silt loam
Crown sandy loam
Cecil clay loam
Parsons fine sandy
loam
Houston clay
Philip's parameters
a
c
S
A
0.82
0.66
0.91
0.08
0.52
0.25
0.24
1.05
0.073
0.64
0.55
0.70
0.08
0.17
0.65
0.285
0.332
0.433
0.039
0.035
0.0047
0.029
-0.024
-0.0013
0.88
0.055
0.343
-0.0193
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Distribution of Kostiakov’s parameters for some soil textural
groups (I in mm, t in minutes)
0.8
0.7
SL
CL
ZL
0.6
ZL
a
0.5
0.4
0.3
0.2
fSL
CL
0.1
Clay
0
0
5
10
15
20
25
30
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c
Green-Ampt (1911)
Based on analysis of the physics but not as thorough
as Philip’s :i = ic + b/I
ic is the asymptotic infiltration rate at t =  and I is
very large.
However, when t=0, and therefore I=0, the
equation predicts infiltration rate is infinite which is
also not correct.
The equation can also be expressed as:
I = ic - B/t
where B = gradient of infiltration curve against
1/time.
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To solve, plot I against 1/t and find the slope
and the intercept
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Horton (1940)
i = ic + (i0 - ic)e -kt
i = i0 at t = 0;
ic is the “final” infiltration rate
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k is a measure of the rate at which
infiltration approaches final value;
mathematically consistent but cumbersome
to use in practice because of 3 variables
and the exponential curve
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There could be more elegent solutions but the
following is a workable method.
Calculate i over each time increment. The time
that applies will be midway between the
observed times for accumulated infiltration.
Plot i against these values for time and use Excel
to Add Trendline using a degree 5 polynomial
remembering to use the “include equation” option.
Assume that i0 is the intercept given by this
polynomial.
The value for ic must be less than the final
observed value. Set up a range of possible values
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from 0.5 x final value to 1.0 x final value.
By taking natural logarithms of each side of the
equation, it can be shown that
ln (i-ic) = ln (i0 - ic) - kt
Rearranging this,
ln (i0 - ic) - ln (i-ic) = kt
You can now tabulate the left hand side against
time. Use Excel to calculate the Correlation
Coefficient for column corresponding to each
different value of ic. The best value for ic will be
the one with the highest correlation coefficient.
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Otherwise use programmes like SAS
Holtan (1961)
i = ic + a (M - I)n
 M is the water storage capacity (total
porosity antecedent water content), above
first impeding stratum.
 No meaning if no impeding layer
 Only holds for 0 < I < M so I > M must set
i = ic
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To solve, again ic must be less than the final
observed infiltration rate so try values in the
range between from 0.5 x final rate and 1.0 x
final rate.
For each value of ic plot log (i-ic) against
log (M-I).
This should be a straight line.
Use the correlation coefficient for the columns
corresponding to each value of ic to determine
the best line.
The slope will be n and the intercept will be log
a.
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Use of infiltration in hydrological modelling
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Typical infiltration curves
After 5 hours,
 high intake-rate soils may have infiltration rate
of 60 to 100 mm h-1
 medium intake-rate soils may have infiltration
rate of 25 mm h-1
 low intake-rate soils may have infiltration rate
of 1 to 4 mm h-1
Amounts of infiltration may be:
 200 mm after 2 hrs for high intake-rate;
 200 mm after 6 hrs for moderate intake-rate;
 200 mm after 25 hrs for a low intake-rate
No universally recognised classification system. 26
Classification of infiltration curves
The following curves are suggested as the basis of a
possible system:
100
90
1
80
2
60
50
3
40
4
30
5
20
6
10
7
110
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105
100
90
85
80
75
70
65
60
55
45
40
35
30
25
20
15
10
5
50
time (mins)
95
8
0
0
Infiltration (mm)
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These are based on the following values for Philips’
A&S
Curve
8
7
6
5
4
3
2
1
Philips’ S
0.31
0.56
0.86
1.35
2.09
3.3
5.08
8
Philips’ A
0.0062
0.013
0.025
0.044
0.076
0.132
0.234
0.41
I=At+St0.5
where t in mins, I in mm
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Curve 1 or faster:
Curves 1 to 2:
Curves 2 to 3:
Curves 3 to 4:
Curves 4 to 5:
Curves 5 to 6:
Curves 6 to 7:
Curves 7 to 8:
Curve 8 or slower:
extremely rapid
very rapid
rapid
moderately rapid
moderate
moderately slow
slow
very slow
extremely slow
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Infiltrationrate
6
The slope of the preceding
accumulated infiltration
curves look like this
5
Infiltrationrate(mm/min)
4
3
2
1
1
2
5
4
3
0
1
11
21
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time(mins)
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30
61
It should be noted that the distribution of
infiltration rates is log-normal and skewed.
The results of one experiment at a single site are
shown in the following diagram.
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30
25
percentage of sample
20
15
10
5
0
< 85
85 - 120
120 - 160
160 - 220
220 - 310
310 - 560
time taken for 200 mm to infiltrate (mins)
560 - 1200
1200 - 3200
32 over 3200
SCS Curve Number concept
Rainfall (P) ends up as either :
• total runoff (Q),
• retention (G),
• initial abstraction (Ia).
Ia is the abstraction corresponding to losses from
a combination of early infiltration (before runoff),
interception (on vegetation) and surface retention
(puddles).
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The model is based on the observation that ratio of
actual infiltration (G) to the maximum potential
infiltration (S) is equal to the ratio of the actual runoff
(Q) to the maximum potential runoff (P – Ia).
It is assumed that both ratios are zero at time equal
zero and approach one for time equal infinity for an
infinite rain event.
Actual infiltration is given by rainfall minus the initial
abstraction minus the runoff, i.e.
G = (P - Ia) -Q
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Thus empirically :
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It was also found that in many situations Ia was
approximately equal to 0.2S.
This has been used in the preparation of graphs
and tables in most text books.
(If other relationships were used in the following
equations, the graphs and the tables in the handouts would
need to be recalculated.)
If Ia = 0.2 S, the previous equation becomes :-
Q
( P 0.2S) 2
P  0.8S
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Values of S were worked out for different
catchment conditions.
S is usually expressed as a “curve number”, N
such that
1000
N
S  10
When S is 0, N is 100 -> 100% runoff
Lowest N in practice is about 6.
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see handout for tables to calculate N
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Some problems with infiltration and soil water
flow models (based on Youngs, 1995)
Effect of soil air
 trapped, air-filled pores
 compression of air in front of wetting front
when there is no escape
 viscosity of air is not negligible so may be
effect even when there is an escape route
Soil heterogeneity
 spatial and vertical heterogeneity
 random but governed by laws of probability
 fingering/instability when less permeable >
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more permeable
Soil aggregation
macropores and micropores,
domain theory
bypass flow
natural processes / cultivation effects
aggregates may become isolated with little
moisture transfer
 entrapped air in aggregates may become
compressed





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Soil instability
Structural breakdown and shrinking/swelling cause
time dependent soil physical properties
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Non-Darcian flow
 Soil water may not behave as classical fluid,
especially where there are electrically
charged soil colloids;
 Reynolds's number is used to predict type of flow
(turbulent, laminar). R is a function of dimensions
of flow channel/pipe, viscosity, velocity.
Darcy's law fails at R>1, which it is during
infiltration;
 Swelling / shrinking means that the frame of
reference is moving! Geometry (shape) of structure
is not constant swelling is not uniform in all
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directions
 Clay particles in suspension also move
 In clay soils, part of the overburden is
transmitted to the soil water- has to be allowed
for in equations, proportion depends on the
moisture content
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Non-isothermal flow
 Theory assumes isothermal conditions but there
are thermal gradients near the surface and the
heat of wetting generated at the wetting front.
Hysteresis
•
Hysteresis can be incorporated but diffusivity
becomes discontinuous which makes analysis more
difficult.
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Anisotropy
 Solutions assume isotropic media - not true in
real soils
 Conductivity in horizontal direction may be
different (usually slower) from the vertical
direction
 Anisotropic because of cracks, worm holes, etc.
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Effects of structure
Structure has a great effect on infiltration.
For loam, range can be from 1.5 mm h-1 for
massive structure to 150 mm h-1.for well
structured.
Improved structure e.g. by roots and
infiltration will increase, runoff will decrease,
erosion will decrease.
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Effect of vegetation
Vegetation causes the soil to be more "open" and
increases infiltration.
One relationship is :-
K veg  Ksat (1  a )
1.4
Kveg is the conductivity with vegetation,
Ksat is the conductivity without vegetation
a is the fraction of ground covered with vegetation
at the base.
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Alteration
For water harvesting, sometimes infiltration needs
to be reduced (covered in 556 - Water Harvesting
and Use of Chemicals)
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Aqueel system
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Crusts and capping
Caps caused by intense rain on bare soil, especially
if low OM and/or high silt content.
Crustability decreases with increasing contents of
clay and OM.
Crusts can reduce infiltration by a factor of 10.
Tall vegetation with no ground cover not only
increases detachment because of drop sizes but also
surface sealing.
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Algae
mats of intertwined filamentous blue-green algae
can also reduce infiltration and hence contribute
to soil erosion
algae samples taken from near Dilling in South
Kordofan were predominantly Lyngbya spp. and
Microcoleus spp
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