Chapter 5 Infiltration

5.1 Infiltration Processes

5.1.1 Definition

Infiltration is the flow of water through the soil surface into a porous medium under gravity action and pressure effects.

5.1.2 Infiltration characteristics

The infiltration capacity is the maximum rate at which water can be absorbed by a given soil per unit area under given conditions.

Figure 5.1 The infiltration process depending on soil type and flow [Musy,2001]

Infiltration regime i(t) depends on the supply regime (irrigation, rain), but also on soil properties. The cumulative infiltration I(t), is the total amount of water infiltrated during a given period.

(5.1)

where:
	I(t)	the cumulative infiltration during the t period (mm)
	i(t)	the infiltration regime during the t period (mm/h)

Hydraulic conductivity at saturation k_s, is an essential parameter of infiltration. It represents the limiting value of infiltration if the soil is saturated and homogenous. Percolation is the vertical water flow in soils (porous unsaturated environment) on the groundwater layer under the influence of gravity. This process follows infiltration and has a major influence on the underground layer water supply.

Net rain is the amount of rain that falls to the ground surface during a shower. The clear rain is deduced from the total rain, diminished by the intercepted fraction of vegetation and that which is stored in ground depressions. The difference between the infiltrated rain and the drained rain on the ground surface is called production function.

5.1.3 Factors which influence infiltration

The main factors that influence the infiltration are:

the soil type (texture, structure, hydrodynamic characteristics). The soil characteristics influence capillary forces and adsorption;
the soil coverage. Vegetation has positive influence on infiltration by increasing the time of water penetration in soil;
the topography and morphology of slopes;
the flow supply (rain intensity, irrigation flow);
the initial condition of soil humidity. Soil humidity is an important factor of infiltration regime. The infiltration regime evolves differently in time for dry or wet soils;
soil compaction due to rain drop impact and other effects. The use of hard agricultural equipment can have consequences on the surface layer of soil.

Figure 5.2 The infiltration regime depending on time for different types of soil [Musy,2001]

5.2 Models Used to Estimate Infiltration Rates

Infiltration processes can be estimated by means of different models:

models based on empirical relations involving 2, 3 or 4 parameters;
physically based models

5.2.1 Models based on empirical relations

Empirical relations show a decrease of infiltration depending on initial time (either exponential or quadratic function of time), which tends to a limit value, generally k_s, but near 0. An empirical relation is the Horton formula, where the infiltration capacity can be expressed as following [Drobot, 1999]:

(5.2)

(5.3)

where:
	i(t)	infiltration capacity at time t (mm/h)
	i₀	initial infiltration capacity depending on soil type (mm/h)
	t	time from the beginning of the shower (h)
	I(t)	total quantity of infiltrated water from initial time until the moment t (mm water column)
	γ	empirical constant depending on soil type (min^-1)

This formula is not linear and it presents certain practical difficulties. Through linearization, we obtain:

(5.4)

As logarith, we get:

(5.5)

The formula of the Institute of Soil and Water Management of the EPFL is:

(5.6)

where:
	i(t)	infiltration capacity at time t (mm/h)
	i_f	final infiltration capacity (mm/h)
	a, b	correction coefficients

The relation is a little different from that of Horton. There are just two parameters. This relation has the advantage of allowing the search of functional relations between the limit/final capacity of infiltration and soil texture, and also between the parameter a and the amount of soil humidity. Other formulas can be used to determine the infiltration regime of water from soil.

5.2.2 Physically based models

These models describe in a simplified manner the water movement in soils, especially at the humidity front level, depending on certain physical parameters.

Table 5.1 Main functions used at infiltration [Musy,2001]

Author

Function

Legend

Horton

i(t)	-	infiltration capacity during time [cm/s]
i₀	-	initial infiltration capacity [cm/s]
i_f	-	final infiltration capacity [cm/s]
γ	-	constant depending on the soil type

Kostiakov

parameter depending on soil conditions

Dvorak-Mezencev

i₁	-	infiltration capacity at time t=1min [cm/s]
t	-	time [s]
b	-	constant

Holtan

c	-	factor variable from 0.25 to 0.8
w	-	Holtan equation flow factor
n	-	experimental constant approximately = 1.4

Philip

s	-	sorptivity [cms^-0.5]
A	-	gravity component depending on hydraulic conductivity at saturation [cm/s]

Dooge

a	-	constant
F_max	-	maximal retention capacity
F_t	-	water quantity retained on soil at time t

Green&Ampt

k_s	-	hydraulic conductivity at saturation [mm/h]
h₀	-	surface pressure load [mm]
h_f	-	pressure load at the humidity front [mm]
z_f	-	humidity front depths [mm]

From the models presented in Table 5.1 the following two models have been used most often:

The Philip model

Philip proposed a method of resolving the vertical infiltration for certain initial and boundary conditions. This model has introduced the notion of "sorption" that represents the soil capacity to absorb water when the flow is produced only under gradient pressure [Musy, 1998]. The infiltration can be simplified as follows:

(5.7)

where:
	i(t)	infiltration rate (cm s^-1)
	s	sorption (cm s^-0.5)
	t	time (s)
	A	the gravity component, depending on hydraulic conductivity on saturation (cm s^-1)

For horizontal infiltration the gravity gradient is not involved. Infiltration will have the following expression:

(5.8)

(5.9)

The Green and Ampt model

This model is based on hypotheses that involve a schematisation of infiltration processes (Figure 5.3):

Figure 5.3 Infiltration process schematisation according to Green and Ampt [Musy,2001]

The method's main hypotheses are:

a humidity front is perfectly defined;
a transmission zone, where in time and space water storage and hydraulic conductivity are constant;
the suction forces of the humidity front are constant;

Based on the Darcy law the model includes the hydrodynamic parameters of soil:

(5.10)

where:
	i(t)	infiltration rate (mm)
	t	time (h)
	k_s	hydraulic conductivity at saturation (mm/h)
	H₀	hydraulic total head at surface (mm)
	H_f(t)	hydraulic total head at the humidity front level (mm)
	z_f	maximum depth of the humidity front
	h₀	pressure head at surface (mm)
	h_f	pressure head of the humidity front (mm)

One of Green's and Ampt's model hypotheses stipulates that water storage from the transition zone is uniform. The cumulative infiltration I(t) results from the product of water storage and the depth of the humidity front.

(5.11)

where:
	I(t)	cumulate infiltration
	θ₀	quantity of water imposed on the surface
	θ	initial quantity of water in soil
	z_f	maximum depth of humidity front

The last two relations result in:

(5.12)

For horizontal flow infiltration has the relation:

(5.13)

For vertical flow infiltration becomes:

(5.14)

This model is satisfactory when applied to a soil with coarse texture, but it is an empirical method in which it is necessary to determine the pressure head of humidification front.

5.2.3 Variation of infiltration rates during a rainfall

The spatial and temporal variability of water quantity existing in soil is described by infiltration curves or hydric profile (Figure 5.4). These represent the vertical water distribution in soil at different periods t. In a homogeneous (uniform) soil when the soil surface is flooded, the hydric profile has three zones: a saturation zone, a transition zone and a humidity zone.

Figure 5.4 Characteristics of the hydric profile during infiltration [Musy,2001]

During a rainfall the infiltration capacity of soil decreases to a limiting value, which represents the infiltration potential at saturation. If we compare the rain intensity and the infiltration capacity of the soil, there are two possibilities:

when the rain intensity is inferior to infiltration capacity, water infiltrates faster due to the supply regime. The necessary time to equalize the infiltration capacity is variable and depends on existing soil humidity conditions or on the shower. The time taken is longer when the soil is dry and the water supply regime is similar to the hydraulic conductivity at saturation k_s;
when the rain intensity is superior to the infiltration capacity of the soil the water surplus is stocked on the surface or in ground depressions. The infiltration regime and the infiltration capacity for net storm rain are presented in the next Figure (Figure 5.5) [Musy,2001].

Figure 5.5 Infiltration regime and net storm rain [Musy,2001]

Bibliography

Drobot, R., and P. Serban. 1999. Aplicatii de hidrologie si gospodarirea apelor (Application of Hydrology and Water Resources Management). Ed. HGA, Bucuresti, Romania.

Musy, A. 1998. Hydrologie appliquée, Cours polycopié d'hydrologie générale, Lausanne, Suisse.

Musy, A. 2001. e-drologie. Ecole Polytechnique Fédérale, Lausanne, Suisse.