Lal R., Shukla M.K. Principles of Soil Physics

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FIGURE 14.7 Infiltration through (a)

a natural homogeneous soil (b) Green–

Ampt approximation.

where K

is the saturated hydraulic conductivity (LT

−1

) of wetted region 0<x<L, ∆H is the

hydraulic head difference between wetting front (W

) and soil surface (W

). Figure 14.7

explains the natural soil water profile wetting and the corresponding Green–Ampt profile.

The infiltration will increase the moisture content of the soil profile. Assuming a

uniformly wetted zone up to the wetting front with cumulative infiltration (I) equal to the

product of wetting front depth (L) and increase in moisture content (∆θ). Similarly,

infiltration rate equals the rate of change in water storage per unit time:

(14.3)

where I is the total or cumulative infiltration within a selected time interval, θ

and θ

are

the saturated moisture content and initial moisture content, respectively (∆θ=θ

−θ

Equating the right-hand side of Eqs. (14.2) and (14.3):

(14.4)

After rearranging the equation so that factors dependent on L and time are on one side,

Eq. (14.4) can be integrated to the limits of 0 to L and 0 to t as follows

(14.5)

Eq. (14.6) is obtained after integrating Eq. (14.5) and from Eq. (13.24)

(14.6)

Principles of soil physics 384

where D(θ)

is soil-water diffusivity of the wet soil region

(14.7)

The cumulative infiltration (I=L∆θ) can be obtained from Eq. (4.6) after multiplying the

second term with ∆θ/Dθ and rearranging first and second terms.

I=∆θ(2D(θ)

0.5

(14.8)

and the infiltration rate

(14.9)

The infiltration rate of soil is proportional to t

−0.5

. Once the gravity component of total

hydraulic head is taken into consideration, or in other words, for vertical infiltration, Eqs.

(14.2) and (14.3) can be written as

(14.10)

(14.11)

equating the right-hand side of Eqs. (14.10) and (14.11) and after rearrangement of terms

and applying integration limits

(14.12)

(14.13)

As t increases, the second term in Eq. (14.13) increases much more slowly as compared

to the first term. So at a very large time from the start of the infiltration experiment, the

Eq. (14.13) can be approximated by

(14.14)

(14.15)

and in terms of infiltration rate

i≈K

(14.16)

Water infiltration in soil 385

where b is a constant. The Green and Ampt model uses an approximate description of

actual flow regimes. It also requires the value of effective wetting from suction (Φ

According to Green and Ampt (1911) and Hillel and Gardner (1970), the value of Φ

for

initially dry soil may be of the order of 50–100 cm. However, in actual field conditions,

where initial moisture content of soil profile is not uniform, Φ

may be difficult to

define. One of the alternatives is the indirect evaluation of Φ

(Chong et al., 1982).

According to Philip (1966), if diffusivity is assumed to be concentrated at the wet end of

the moisture range, the Green–Ampt equation corresponds to the nonlinear diffusion

description of infiltration. However, real soils do not manifest a delta function D(θ). A

common form of Green-Ampt equation with i=dI/dt and

idt is given below:

(14.17)

Philips Model

The Philip algebraic infiltration equation uses mathematical approximation for the

infiltration process in a soil matrix. There are two forms of the Philip model: horizontal

and vertical infiltration.

For horizontal infiltration, Philip (1957) used the Richards equation with K(θ) (second

term on right-hand side) removed and showed that infiltration rate can be expressed as

(14.18)

where S is soil-water sorptivity and S=f(θ

, θ

). θ

and θ

are boundary and initial moisture

content. Since S is constant over time, the cumulative infiltration

I(t)=S t

0.5

(14.19)

The sorptivity, which will be more important during the initial period, can be defined as

(14.20)

For vertical infiltration,

the Philip equation consists of two separate parts: one for a short

duration and the other part for long times after infiltration commenced. The solution

describes the time dependence of cumulative infiltration by means of an infinite power

series.

(14.21)

where coefficient j

(θ) are calculated from K(θ) and D(θ), the coefficients are termed as

sorptivity (S). Transferring S into Eq. (14.21) and expanding

I(t)=S t

0.5

+(A

+K(θ)

)+A

3/2

+…+A

(14.22)

Principles of soil physics 386

Differentiating equation (14.22) with respect to time gives the solution for infiltration

rate.

(14.23)

In practice the Eqs. (14.22) and (14.23) can be approximated by two parameter models as

follows

I(t)=S t

0.5

+A t

(14.24)

(14.25)

where A is a constant known as soil-water transmissivity and it approaches K

of the soil

profile as t→ ∞. It has the dimensions of LT

−1

. According to Philip (1969), as t

approaches infinity, the infiltration rate decreases to a finally asymptotic value and the

coefficient A can be replaced with saturated hydraulic conductivity of upper layer (K

)

I(t)=St

0.5

(14.26)

(14.27)

For very large time

i(t)=K

(14.28)

The magnitude of A in equation (14.25) is (A

(θ)+ε), where K

(θ) is the hydraulic

conductivity and ε is the truncation error (Kutilek and Nielson, 1994). The value of S

estimated from Eq. (14.27) is quite reliable, however, truncation errors influence the

estimated value of A. To overcome this problem Kutilek and Crejka (1987) proposed to

use first three terms of the Philip’s series solution [Eq. (14.23)] as follows

(14.29)

(14.30)

where C

is the estimate of (A

) and C

the value of (A

+ε

) where ε

is truncation

error for having used three terms. Another model proposed by Swartzendruber (1987)

uses the adjusted Philip time series solution to derive the following infiltration models

(14.31)

(14.32)

Water infiltration in soil 387

where A

is a constant. Equations (14.26), (14.27), (14.29), (14.30), (14.31), and (14.32)

also provide an initial infinite infiltration at t=0 and a constant equilibrium infiltration

rate at large t. Substituting 4K

/3S for A

into Eqs. (14.31) and (14.32) results in two-

parameter Stroosnijder (1976) infiltration model. Using the horizontal solution of Philip

(1957), Brutsaert (1977) added a correction for force of gravity to arrive at following

infiltration equations

(14.33)

(14.34)

Brutsaert (1977) proposed B values to be 1/3, 2/3, or 1, but for most practical purposes

recommended B=1 (Kutilek and Nielsen, 1991). All these models are time dependent and

provide an infinite initial infiltration at t=Ø and a finite steady state infiltration at large t.

14.3.2 Empirical Models

There are three principal empirical models.

Kostiakov Model

The Kostiakov (1932) equation for cumulative infiltration (I) and infiltration rate (i) can

be expressed as follows

I=Bt

−n

(14.35)

i=B′t

−n−1

(14.36)

where parameters B, B′, and n are constants. These parameters do not have a physical

meaning and can be obtained by fitting the equation to the experimental data. It can be

inferred from Eq. (14.35) that at t=0, I approaches ∞. However, as t increases further, i

tend to become zero. Therefore, Eqs. (14.35) and (14.36) explain the horizontal

infiltration. However, for vertical infiltration, the Kostiakov (1932) equation is

inadequate. To overcome the problem, Kostiakov proposed a maximum time range of

application with t

max

=(B/K

)

1/n

and Mezencev (1948) included another coefficient, i

which essentially shifts the axis for infiltration rate equations and for large times

infiltration approaches a finite steady state infiltration rate.

I=i

t+Bt

−n

(14.37)

i=i

+B′t

−n−1

(14.38)

Principles of soil physics 388

Horton Model

Horton (1940) equations for cumulative infiltration (I) and infiltration rate (i) are:

(14.39)

i=i

t+(i

−i

−kt

(14.40)

where i

, is initial infiltration rate at t=0, and i

is final constant infiltration rate after a

long time from the start of infiltration. The constant “k” determines the rate at which i

approaches i

. Unlike other equations, the Horton equation has a finite infiltration rate, i

at t=0. The equation is somewhat cumbersome as it has three constants, which need be

evaluated experimentally. According to Horton (1940), the decrease in infiltration rate

with time can be described by a number of factors, such as, the closure of soil pores by a

swelling soil or erosional deposit, compaction due to raindrop impact, etc.

Holtan Model

The Holtan equation (1961) for infiltration rate in soil matrix is again a two-form

mathematical equation as given below

(14.41)

where ic is the final constant rate of infiltration and

is the available porosity as

depleted by infiltration volumes, which can be expressed as

(14.42)

where M is the moisture storage capacity of the soil above the first impeding stratum or

control layer. It can also be expressed as total porosity—antecedent moisture content of

soil, in depth units. I is the cumulative infiltration at that time.

i=i

for I>M

(14.43)

As long as 0≤I≤M, the Eq. (14.40) is consistent, however, for I>M, the (M−1)

becomes

positive or negative depending upon the value of exponent n. Also in absence of an

impeding layer, Holtan (1961) did not discuss the meaning of M. Huggings and Monk

(1967) reported that effective depth is a function of land use and soil management.

According to Holtan and Creitz (1967), the control depth could be the depth down to B-

horizon and the parameter n could be assumed constant equal to 1.4 for all soils. The

parameters of empirical infiltration equations are time dependent. The Horton (1940) and

Holtan (1961) provide a finite infiltration rate both at t=0 and t=∞. A comparison of all

the infiltration models discussed above is given in Table 14.2 and can also be referred in

Davidson and Selim (1986); Haverkamp et al. (1988) and Shukla et al. (2003b) among

others.

Water infiltration in soil 389

14.4 INFILTRATION INTO LAYERED PROFILE

The infiltration process occurs in natural soils, which are neither uniform in texture nor

homogeneous. They comprise several layers or horizons of different texture, bulk density,

and moisture content extending up to variable depths. This nonhomogeneity of soil

profile has a pronounced effect on the water infiltration process. Infiltration primarily

depends on the relationship between hydraulic conductivity [K(θ)] and gradients of

suction (Φ

) in each layer. According to Miller and Gardner (1962), metric suction (Φ

)

and hydraulic head (H) remain continuous throughout the soil profile regardless of the

sequence of layers. However, soil’s moisture content (θ) and hydraulic conductivities

[K(θ)] at the soil layer interface exhibit sharp discontinuity (Fig. 14.8). For a two-layer

soil system, Takagi (1960) demonstrated that if an upper soil layer is less pervious than a

lower, a continuous negative pressure develops in the lower profile, which extends deep

inside the soil matrix (Fig. 14.8a).

The layering, as shown in Fig. 14.8, can also cause the instability of wetting front. At

the interface, pressure head is too small in the upper layer to force the entry of water into

coarser pores of the impeding layer. As θ increases in the upper layer, the pressure head

also increases and water enters the impeding soil via finer pores. Since the supply of

water from

TABLE 14.2 Comparison of Various Infiltration

Equations

Parameter Green–Ampt

(1911)

Philip (1957) Kostiakov

(1932)

Horton (1940) Holtan (1961)

Number 2 2 2 3 4

Theory Physically based Physically based Empirical Empirical Empirical

Surface

ponding

Required Required Not

required

Not required Not required

Initial

infiltration

Infinite Infinite Infinite Finite Finite

At large

time

Steady

infiltration =

hydraulic

conductivity

)

Steady

infiltration =

hydraulic

conductivity

)

Zero Steady

infiltration =

hydraulic

conductivity

)

Steady

infiltration =

hydraulic

conductivity

)

Principles of soil physics 390

FIGURE 14.8 (a) The discontinuity of

water content across a layer when

conductivity of top layer was smaller

than the lower layer (based on Takagi,

1960; Kutilek and Nielsen, 1994); (b)

zone of saturation when conductivity

of top is higher than the impeding

layer.

topsoil is limited by the K

for that soil, water conduction in the impeding layer is mainly

through preferential domains. This type of flow is also termed as “finger flow” and points

out towards the hydrodynamic instabilities or inabilities to describe infiltration through

layered soil profile (also refer Fig. 14.6d),

If we reverse the layers in Fig. 14.8, topsoil is more pervious than the impeding soil

(Fig. 14.8b). A zone of Φ

>0, is formed on top of the impeding low conductivity layer.

After a large time since infiltration began, a zone of saturation as shown in Fig. 14.6b is

formed on either side of the layer. This temporary water table development is also termed

the perched water table. In dealing with nonhomogeneous soil profiles it is usually

convenient to divide the profile into homogeneous layers. The simplest example is a

crust-topped profile, which we will discuss now.

14.5 INFILTRATION INTO CRUSTED SOILS

A surface crust is created on the soil surface mainly by the impact of raindrops. This can

also take place due to the spontaneous slaking or breakdown of soil aggregates during

rapid wetting. The impact of raindrops may be partly reduced if the soil surface has

mulch or other vegetative cover on it. These processes are discussed in Chapter 6. A

surface crust is normally characterized by a relatively less porosity and higher bulk

density. Consequently, crust has a lower hydraulic conductivity than the soil strata

beneath. The crust can also be formed if the infiltrating solution has a high sodium

Water infiltration in soil 391

adsorption ratio (SAR) by inducing dispersion and swelling of clay particles. The

dispersion of aggregates separates the finer particles from the macroaggregate, which

then occupy the interaggregate space in the soil domain, thus reducing the hydraulic

conductivity of the crust layer. At steady infiltration rate, flux through the crust equals

flux through the impeding soil. These two fluxes can be equated using Darcy’s law:

(14.44)

where K

and K

are the conductivity of crust and impeding layer respectively, and

, L

is the thickness and R

the resistance of crust. The second term on either side

defines the gradient for both layers, respectively, with H as total hydraulic head. As

steady infiltration is approached, the gradient in the impeding layer or subcrust zone

tends to become unity and the gravitational gradient is the only effective driving gradient.

So flux through subcrust soil can be written as

q=K(θ)

(Φm

)

(14.45)

where K(θ) is the unsaturated hydraulic conductivity of subcrust zone, which is also a

function of matric potential head in the subcrust zone. If the crust remains saturated the

flux through the crust can be expressed as

(14.46)

where H

is the depth of ponding on crust surface, Φ

is suction head, and z

is the depth

or thickness of crust. If depth of ponding and crust thickness are very small and can be

neglected then

(14.47)

In terms of hydraulic resistance, the Eq. (14.46) can be written as follows provided the

hydraulic gradient of the impeding layer is unity.

(14.48)

14.6 PRECIPITATION INFILTRATION

The infiltration models assume that soil surface is held at a fixed potential (Φ

) rather

than at a fixed rate of water flux. When rainfall is the source of water, the fixed potential

condition occurs if rainfall intensity exceeds the soil infiltration rate. As a result, ponding

or positive potentials occurs on the soil surface. Ponding does not occur immediately,

because the initial infiltration into the soil is very rapid. However, if rainfall continues at

Principles of soil physics 392

a constant rate, the infiltration rate decreases and the rainfall rate exceeds infiltration rate,

thus creating ponding on soil surface.

Many researchers have studied the infiltration process under rainfall or sprinkler

irrigation [Youngs (1964), Rubin and Steinhardt (1964), Rubin (1966, 1968), Parlange

and Smith (1976), Boulier et al. (1987)]. Rubin (1968) carried out a series of rainfall-

infiltration tests under different surface boundary conditions. Under ponding conditions

the wetted profile consists of two parts, a water-saturated part and the water-unsaturated

part. The depth of the saturated zone and the steepness of infiltration curve increase

continuously downward. However, after a long time the steepness gradually decreases

and tends to be asymptotic to the time axis (Fig. 14.9). The infiltration curves when

surface ponding does not occur remain parallel to time axis (Fig. 14.9) till the rainfall or

sprinkler rate exceeds the infiltration rate. The infiltration curve becomes steep and takes

the form of a ponded infiltration curve. From Fig. 14.9, two inferences can be made: (i)

the

FIGURE 14.9 Relation between

surface flux and time during

infiltration into Rehovot sand due to

rainfall (solid line) and flooding

(dashed line). The numbers labeling

the curve indicate the ratio of the

rainfall rate to the saturated hydraulic

conductivity. (Redrawn from Rubin,

1968.)

constant rate of infiltration limited by rainfall or infiltration occurs as long as the ratio of

rainfall rate to hydraulic conductivity is less than one; (ii) the shapes of ponded

infiltration curve and the rain limited curves are similar and attain same limiting

infiltration rate, however, they do not coincide.

Water infiltration in soil 393