1

1 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Hydrology

[7-2]Infiltration Models

Mohammad N. Almasri

2 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Horton Infiltration Model

Horton (1933) suggested that infiltration capacity rapidly declines during the early part of a storm (rainfall event) and then tends towards an approximately constant value after a couple of hours for the remainder of the event

Horton Model is an empirical formula that says that infiltration capacity starts at a constant rate (f0) and is decreasing exponentially with time (t)

After some time when the soil saturation level reaches a certain value, the rate of infiltration will level off to the rate fc which is the minimum asymptotic value of infiltration

2

3 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Horton Infiltration Model

4 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Horton Infiltration Model

Horton equation for determining the infiltration capacity is:

wherefp = the infiltration capacity (potential) at some time

(L/T)k = a constant representing the rate of decrease in the

infiltration capacity (decay coefficient) and depends on soil characteristics (1/T)

fc = a final or equilibrium infiltration capacity (L/T)f0 = the initial infiltration capacity (L/T)t = the time (T)

ktc0cp e)ff(ff −−+=

3

5 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Horton Infiltration Model – Cumulative Infiltration

Cumulative infiltration F(t): the total amount of water entering the soil

F(t) can be found using the following equation:

A relationship exists between cumulative infiltration and infiltration capacity:

dt)t(f)t(Ft

0∫=

kf

)ffln(kf

kf)ffln(

kf)t(F p

cpc0

c0c −−−⎥⎦

⎤⎢⎣⎡ +−=

6 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Horton Infiltration Model – Ponding Time

Time of ponding according to Horton’s model is given by the following equation:

Time of ponding is the elapsed time between the time rainfall begins and the time water begins to pond on the soil surface

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−

=c0

cp ff

filnk1t

4

7 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Horton Infiltration Model – Example [1]

A catchment soil has the following Horton infiltration parameters: f0 = 100 mm/h, fc = 20 mm/h, and k = 2 min-1

Plot the infiltration capacity curve with time for this catchment

Plot the potential cumulative infiltration for this catchment

8 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Horton Infiltration Model – Example [1]

5

9 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Horton Infiltration Model – Example [2]

Find out the sensitivity of the infiltration capacity curve to different decay coefficients (k) assuming that f0= 2.9 in/h and fc= 0.5 in/h

Assume k values = 0.15, 0.30, and 0.45 hour-1

10 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Horton Infiltration Model – Example [3]

rainfall

rainfall

rainfall

6

11 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Horton Infiltration Model – Example [3]

rainfall

Infiltration

rainfall

Infiltration

rainfall

Infiltration

Actual infiltration = rainfall intensity

Actual infiltration = infiltration capacity

12 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Horton Infiltration Model – Example [4]

Rainfall

7

13 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Green-Ampt Model

Green-Ampt model is based on a simple conceptualization of an infiltrating front in a dry soil with an approximated sharp interface

This sharp interface is the wetting front that divides the soil of moisture content Өibelow from saturated soil with moisture content of Өs (equals porosity)

The wetting front has penetrated a depth of L in time tsince infiltration beganWater is ponded to a depth of H

Wet Zone

14 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Green-Ampt Model

Apparently, total cumulative infiltration after time t(since infiltration began) equals:

F(t) = Lt × (θs - θi) = Lt × ∆ӨIn addition, we can implement Darcy’s Law where the infiltration rate depends on the hydraulic conductivity of the soil along with the head difference, or:

where ψ is the wetting front capillary pressure head

8

15 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Өi: Initial water content (-)

Өs: Saturation water content (-)

Ψ: Wetting front capillary pressure head (L)

K: Hydraulic conductivity (L/T)

Green-Ampt Model

Cumulative infiltration F(t) in Green-Ampt model is given by the following equation:

While infiltration rate f(t) is given by the following equation:

These two equations are used under the assumption that water is ponded to a small but negligibledepth on the soil surface

16 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Green-Ampt Model

When water is applied at a rate higher than K (i>K) of the soil, ponding occurs

To find the ponding time and the cumulative infiltration at ponding (using Green-Ampt model) use the following equations:

Ponding time

Cumulative infiltration at the ponding time

9

17 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Green-Ampt Model – Example [1]

Compute the infiltration rate and cumulative infiltration after one hour of infiltration into a soil that initially had a water content of 0.1 and a saturated water content of 0.44. The average wetting front capillary pressure head is 16.7 cm and the hydraulic conductivity is 0.65 cm/hr

18 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Green-Ampt Model – Example [1]

To find the cumulative infiltration, we have to use the following equation:

If we substitute in the above equation, we arrive at:

Solving the above equation iteratively gives the cumulative infiltration (F) which equals 3.167 cmNow, compute the infiltration rate using the following equation

⎟⎟⎠

⎞⎜⎜⎝

⎛−×

+−×+×=)1.044.0(7.16

F1ln)1.044.0(7.16165.0F

f = 1.815 cm/h

10

19 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Green-Ampt Model – Example [1]

You can use the method of successive substitution to find the value of F

You start by assuming a value of F (for the LHS), compute the RHS, and assume that the new value of F is the RHS value and so on

20 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Green-Ampt Model – Example [1]

11

21 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Green-Ampt Model – Example [2]

Calculate the cumulative infiltration and the infiltration rate after one hour of rainfall of intensity 5 cm/hr. The soil had a water content of 0.1 and a saturated water content of 0.44. The average wetting front capillary pressure head is 16.7 cm and the hydraulic conductivity is 0.65 cm/hr

22 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Green-Ampt Model – Example [2]

Here i=5 cm/hr > K; therefore tp is first computed and then Fp

Now, use the following equation to find the cumulative infiltration after 1 hour of rainfall

This gives a value of F = 3.02 cm

12

23 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Green-Ampt Model – Example [2]

The use of the following equation provides a values of f = 1.87 cm/hr

24 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Green-Ampt Model – Example [3]

For the following soil properties, do the following: Determine the amount of water when ponding occursDetermine the time to pondingPlot the cumulative infiltration function

K = 1.97 cm/hr, θi = 0.318, θs = 0.518, i = 7.88 cm/hr, Ψ = 9.37 cm

13

25 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Green-Ampt Model – Example [3]

We know that:

Rearrange this equation to get:

K)t(f)(K)t(F is

−θ−θψ

=

Substituting the parameter values gives a cumulative value of infiltration of 0.625 cm

To compute the time of ponding (tp), we use:

itF pp =tp = 0.625 / 7.88 = 0.079 hours

26 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Green-Ampt Model – Example [3]

This means that until 0.625 cm has infiltrated, the rate of infiltration equals the rainfall rate

After that, the actual infiltration rate declines with time

Use the following two equations to find out the values of F(t) and f(t)

14

27 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Green-Ampt Model – Example [3]

28 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Phi Index (Φ)

Φ Index is the average rainfall intensity above which the volume of rainfall equals the volume of runoff

The hashed area above the dashed line represents measured runoff over the catchment area

The unhashed area below the line is the measured rainfall that did not appear as runoff but represents all the losses including interception, evaporation and infiltration

Time

Time

Inte

nsity

Φ indexLosses

Run

off

(Rai

nfal

l exc

ess)

Inte

nsity Hyetograph

15

29 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Phi Index (Φ)

To determine the Φ Index for a given storm, the amount of observed runoff is determined and the difference between this quantity and the total gauged rainfall is then calculated

The volume of loss is then distributed uniformlyacross the storm pattern

It should be kept in mind that Φ Index varies as the storm intensity varies with time and thus Φ Index is of limited value and that many determinations should be made and averaged before the index is used

30 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Phi Index (Φ) – Example [1]

The rainfall intensities during each 30 min of a 150-min storm over a 500-acre basin are 5.5, 3, 1, 3.5, and 2 in/hr, respectively

The direct runoff from the basin is 105 acre-ft

Determine Φ Index for the basin

16

31 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Phi Index (Φ) – Example [1]

Find the total rainfall as follows:30/60 × (5.5 + 3 + 1 + 3.5 + 2) = 7.5 in or 0.625 ft

Rainfall volume = 500 × 0.625 = 312.5 acre-ft

Runoff volume = 105 acre-ft

Volume under Φ Index = 312.5 – 105 = 207.5 acre-ft

Infiltration depth (losses depth) = 207.5/500 = 0.415 ft or 5 in

Φ Index = 5 × (1/150) × (60) = 1.98 in/hr

32 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Phi Index (Φ) – Example [2]

You have two storm events of 75 mm of a total duration of 6hours as shown in the figures

Both produced a total runoff equivalent to 33 mm

Find out the Φ Index for the two storm events

17

33 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Phi Index (Φ) – Example [2]

Φ index

Φ = 8 mm/h Φ = 9 mm/h

34 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Phi Index (Φ) – Example [3]

Compute the depth of runoff and the infiltration considering the rainfall event summarized in the table

Assume a Φ Index value of 0.6

18

35 Hydrology – Fall 2007 – [7-2] Infiltration Models Mohammad N. Almasri, PhD An-Najah National University

Phi Index (Φ) – Example [3]

Compute the intensity for each durationIf the Φ Index is higher than the rainfall intensity then the infiltration equals the rainfallIf the Φ Index is less than the rainfall intensity then the infiltration equals the Φ Index Net rainfall intensity is the rainfall intensity - ΦIndex