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

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column using tensiometers and gammaray scanning or any other nondestructive method

of moisture content measurement.

FIGURE 13.7 Schematic of a system

for the measurement of unsaturated

hydraulic conductivity under steady

state condition.

The pressure outflow method determines the relationship between Φm and θ by

subjecting a saturated soil core to successive increments of Φm applied or pneumatic

pressure P. The increments in pressure heads are kept small so that D can be assumed

constant over the change in θ due to change in Φ

. Gardner (1956) solved the diffusion

form of the Richards equation and gave the following solution for θ vs D relationship and

unsaturated hydraulic conductivity determination.

(13.30)

(13.31)

where V

is the total outflow volume of water, V

is the total volume of soil in the core, V

the outflow volume at time t, L is the length of sample, and ∆P is air pressure difference.

The D(θ) obtained from Eq. (13.24) is related to ∆θ for applied ∆P (or Φ

equal to

pneumatic pressure, P). The measurements for a tension plate device must be made for

both sorption and desorption and perhaps intermediate scanning, also (Hillel, 1980). The

pressure outflow method has a major drawback in that D must be constant across ∆P.

Doering (1965) proposed the one step outflow method, which eliminated this limitation.

The method employs

Gardner’s (1962) solution and considers relatively larger ∆P, therefore relatively more

accurate.

Principles of soil physics 364

(13.32)

where θ

is the average value of θ across soil column and θ

is the final moisture content.

Transforming the Eq. (13.32) in terms of measured set of effluent volumes will result in

the following relationship

(13.33)

where V is the cumulative outflow, which can be calculated by

(13.34)

Another equation which relates D(θ) to soil water sorptivity or S(θ) was proposed by

Dirksen (1975).

where ∆θ=0

−θ

and θ

is soil’s initial moisture content. The d(log S

)/dθ in the above

equation can be evaluated by fitting a polynomial to the measured sets of S and θ values.

Measurement of unsaturated hydraulic function (K(θ) and D(θ)) is easier in the lab;

however, the lab measurements are made on a discrete and small sample, away from the

natural continuum. Since the variability of soil and, therefore, that of soil’s hydraulic

functions is well established and well known, field methods for the measurement of soil’s

hydraulic functions appear as useful techniques if a preassessment of variability is made.

13.7.3 Unsteady State Boltzmann Transform Method

The Boltzmann transform method assumes a semi-infinite uniform soil. The flow through

the soil is one-dimensional and horizontal or verti-cal with the gravity component

assumed insignificant. The initial and final moisture contents are constant. If initial

moisture content (θ0) is higher than final (θ

), outflow or drainage occurs, and if θ

>θ

infiltration happens. Substituting the Boltzmann variable k=xt

−1/2

in Eq. (13.24), and

upon integration, following equation for soil-water diffusivity is obtained.

(13.36)

Water flow in unsaturated soils 365

where D(θ) and dλ/dθ are evaluated at the moisture content θ′. Methods proposed by

Bruce and Klute (1956) or Whisler et al. (1968) can be used to estimate the diffusivity

function.

13.7.4 In Situ Methods

The in situ methods address several of the disadvantages of the laboratory methods.

However, the spatial and temporal variations make it difficult to determine exact

boundary conditions. For example in case of a steady infiltration through soil profile, the

essential and limited condition is that flow should be steady both at the upper and lower

end of soil profile, which is extremely difficult to achieve in a heterogeneous field.

However, keeping the fluxes large, so that errors due to tortuosity of flow paths are small,

can minimize the problems due to heterogeneity.

Sprinkler Method

The sprinkler method, or sprinkler infiltrometer method, makes a uniform application of

water on the soil surface (Peterson and Bubenzer, 1986). Water is applied at a rate

slightly lower than the effective hydraulic conductivity (K

) of the soil. Both θ and Φ

increase gradually and suction gradients become zero. The flow through soil profile is

only due to gravity and hydraulic gradient of flow becomes unity. Under this situation

flux through soil profile is essentially equal to the hydraulic conductivity of soil. This

provides one set of value of Φ

and θ. The experiment can be repeated for different

steady rates of water application and correspond-ing values of Φ

and θ can be obtained.

This method, although simple, requires large number of closely spaced sprinklers

supplying water at a constant rate. Since measurement of Φ

and θ need to be made for a

wide range, the method poses a problem for sprinkler application rate of less than 1

mm/h. The exposed soil surface tends to disperse and seal the pores due to the impact of

rain droplets, thus reducing the infiltrability of soil. The steady flux is very difficult to

achieve for a layered soil.

Crust-Topped Method

The crust-topped method employs a less permeable crust of topsoil, which reduces the

flux density, soil wetness, and corresponding K(θ) and D(θ) values of the infiltrating

profile due to the steep hydraulic gradient across the less permeable crust of topsoil. The

impeding layer induces the suction in the subsoil, which increases with the increasing

hydraulic resistance of the crust. Once steady infiltration is established flux and

conductivity of subsoil becomes equal. The Φ

and θ can also be measured

simultaneously using tensiometer and nondestructive moisture content measurement

device. This method can work for a wide range of Φ

and θ measurements, thus

eliminating the range limitation of the sprinkler method described above. For very high

suctions, the measurements may take long time and accurate measurement of flux may

become difficult to achieve. Evaporation may also become significant if proper care is

not taken. The crust-topped method can be applied while using double ring infiltrometers,

tension infiltrometers, or disc permeameters (Perroux and White, 1988).

Principles of soil physics 366

Internal Drainage Method

The internal drainage method for the measurement of K(θ) and D(θ) where water

movement is purely by drainage and not by evaporation was developed by Watson (1966)

and known as the instantaneous profile method. In this method, plots 5 m×5 m or 10

m×10 m in cross section are selected and instrumented with a tensiometer and

nondestructive moisture content measurement devices at various depths. Water is applied

to the field till the tensiometer readings at each depth become constant. The top of the

plot is covered with plastic or any other material so that loss of water due to evaporation

is not significant. The readings of Φ

and θ are recorded at various depths simultaneously

at specified time intervals. It has been generally found that during internal drainage in a

deep, wetted soil, the hydraulic gradient deep in the soil layer is unity. Therefore, K(θ) is

equal to the time rate of change of moisture content as given below:

13.37

where, W is the total moisture content of profile to depth z

(13.38)

and the total moisture content change per unit time can be obtained by integrating

between successive soil layers up to the depth z.

The D(θ) can be determined by the time rate of change of matric suction, and hydraulic

gradient (Gardner, 1970)

If the hydraulic gradient is near unity, then the inverse term in Eq. (13.40) becomes one

and D(θ) can be calculated by the time rate of change of suction only.

13.7.5 Functional Relationships

Some common functional relationships are described below.

Soil Water Retention Models

One of the most simple and popular functions for describing θ(Φ

) has been the equation

of Brooks and Corey (1964), herein referred to as the BC equation

Water flow in unsaturated soils 367

(13.41)

where s is the effective degree of saturation, also called the reduced moisture content

(0≤s≤1), θ

and θ

are the residual and saturated water contents, respectively; Φ

is the

air entry or bubbling pressure, a is an empirical parameter (L−

) whose inverse is often

referred to as the air entry value or (Φma), and λ is a pore-size distribution parameter

affecting the slope of the retention function. Remember for notational convenience, a and

are taken as positive for unsaturated soils.

Several continuously differentiate (smooth) equations have been proposed to improve

the description of soil-moisture retention near saturation. A related smooth function with

attractive properties is the equation of van Genuchten (1980), herein referred to as the

VG equation:

s=[1+(αΦ

)

]

−m

(13.42)

where α, n, and m are empirical constants affecting the shape of the SMCC. The limiting

curve follows from Eq. (13.42) by removing the factor 1 from the denominator. This

shows that the VG- and BC-functions become equivalent at low s when λ=mn.

Hydraulic Conductivity Models

The model of Mualem (1976) for predicting the relative hydraulic conductivity, K, is

(13.43)

with

(13.44)

where K

is the saturated hydraulic conductivity, s is degree of saturation, and l is a pore-

connectivity parameter equal to 0.5 as an average for many soils estimated by Mualem

(1976). Substituting the inverse of the VG equation into Eq. (13.43) then integrating and

then substituting the K=0 leads to the restriction m=11/n, and the Eq. (13.43) reduces to

the following expression for K:

K(s)=K

[1−(1−s

1/m

)

]

(13.45)

In terms of pressure head

(13.46)

Principles of soil physics 368

and for soil-water diffusivity, following equation can be derived from Eqs. (13.23) and

(13.42)

(13.47)

When the BC retention function is substituted into (13.44) the following hydraulic

conductivity function with respect to moisture content, pressure head and soil-water

diffusivity equations are obtained

(13.48)

(13.49)

(13.50)

The predictive equations for K(θ) used thus far assume that K

is a well-defined and

easily measured soil hydraulic parameter. This assumption is probably correct for many

repacked, coarse-textured, and other soils characterized by relatively narrow pore-size

distributions. However, direct field measurement of K

is generally very difficult for

undisturbed and especially structured field soils. Also K(θ) near saturation is determined

primarily by soil’s structural properties, which are subject to considerable spatial

variability in the field (van Genuchten et al., 1991). However, soil’s textural properties

are less variable and have a more dominant effect on K(θ) in the dry range. The rapid

decrease of the predicted K(θ) near saturation when n is relatively small is intuitively

realistic. It suggests that K(θ) near saturation is determined by only a very few large

macropores or cracks which may have little relation to the overall pore-size distribution

that determines the general shape of the predicted conductivity curve at intermediate

moisture contents. Thus, it seems more accurate to match the predicted and observed

unsaturated hydraulic conductivity functions at moisture content somewhat less than

saturation. The same holds for the θ

, which is best regarded as an empirical parameter to

be used in the context of a specific water retention model, and hence must be fitted to

observed unsaturated soil water retention data points.

The model of Burdine (1953) can be written in a general form as follows

(13.51)

where

(13.52)

as in Eq. (13.50) the pore-connectivity parameter l accounts for the presence of a

torturous flow path. A variety of values have been suggested for l; Burdine (1953)

Water flow in unsaturated soils 369

assumed a value of 2. Results analogous to those for Mualem’s model can also be derived

for Burdine’s model and can be referred in the user’s manual for RETC code (van

Genuchten et al., 1991).

Empirical Approaches to Estimating Unsaturated Hydraulic

Conductivity

The hydraulic properties can also be expressed in terms of relatively simple mathematical

expressions (Mualem, 1989). A list of empirical models for unsaturated hydraulic

conductivity is given in Table 13.2.

TABLE 13.2 Empirical Expressions for

Unsaturated Hydraulic Conductivity

Model Boundary condition Reference

K(θ)

n=3.5 Averjanov (1950)

s=(θ−θ

)/(θ

−θ

)

K(θ)=α|Φ

|−

Wind (1955)

K(θ)

=exp(αΦm)

Gardner (1958)

K(θ)=K

For Φ

≥Φ

Brooks and Corey

(1964)

K(θ)

=(Φ

/Φ

mCT

)−n For Φ

<Φ

King (1964)

K(θ)=K

For Φ

≥Φ

mCT

K(θ)

=exp[α(Φ

−Φ

mCT

)] For Φ

≤Φ

≤

mCT

Rijtema (1965)

K(θ)=K(θ)

(Φ

/Φ

)

−n

For Φ

<Φ

Note: sinh(x), cosh(x), tanh(x), coth(x), sech(x), and csch(x) are hyperbolic trigonometric functions

and are defined in terms of the natural exponential function e

(e.g., hyperbolic sine of x or

sinh(x)=[e

−e

−x

]/2 and cosh(x)=[e

−x

]/2).

Example 13.1

Calculate the unsaturated hydraulic conductivity parameters from Φ

−θ relationship for a

sandy loam soil given in the following table. The saturated hydraulic conductivity of the

soil was 4×10−

cm sec

−1

Solutions

Principles of soil physics 370

For water content=0.44

The pore class increment (i) and denominator index (j) both are equal to 1. Also θ

=θ

therefore, the hydraulic conductivity is K= K

=4×10−

cm sec−

For water content=0.43

The pore class i=2, therefore in the numerator 2j+1−2i=1, 3, 5, 7…25 [Eq. (13.29]

For water content=0.40

The pore class i=3, therefore in the numerator 2j+ 1−2i=1, 3, 5, 7… 23

These calculations can progress till we get to the last moisture content value of 0.02.

The conductivity for this class will be

These calculations can be easily made on a spreadsheet or by writing a simple

computer program. Remember the denominator for all the calculations remains the same.

The calculated values of conductivity are listed in the table below. At least twenty sets of

moisture content vs. suction readings at a regular interval are required. However, if more

sets are available, the better.

Water content

(cm

−3

)

Suction

head (cm)

Pore class

increment (nr.)

Denominator

index 2j−1

Hydraulic

conductivity (cm

sec

−1

)

0.44 10 1 1 0.004

0.43 15 2 3 0.002

0.40 25 3 5 0.001

0.34 35 4 7 0.0006

0.30 40 5 9 0.0003

0.26 50 6 11 0.0002

0.22 60 7 13 7.05E–5

0.18 70 8 15 2.64E–6

0.14 80 9 17 6.92E–6

0.08 150 10 19 7.47E–7

Water flow in unsaturated soils 371

0.05 600 11 21 5.24E-8

0.04 1000 12 23 9.73E–9

0.03 3000 13 25 1.1E–9

0.02 5000 14 27 1.27E–10

Example 13.2

Plot the soil water characteristic curve from the Φmθ data given in Example 13.1. If

volumetric moisture content is 0.44 at saturation, calculate (a) bulk density of soil, (b)

moisture content at 1/3 bar, (c) moisture content at 15 bar, (d) available water, (e) water

release from 0.5 m soil profile, and (f) volumetric water capacity.

Solution

The soil water characteristic plotted on a semilog paper is given in figure (a) below.

(a) Assuming moisture content at saturation corresponds to porosity of soil. Bulk density

can be estimated from volumetric water content.

θ=1−ρ

/ρs;−ρ

=−ρ

(1−θ)=2.65*(1–0.44)=1.48 g cm

−3

(b) Water content at 1/3 bar or 333 cm (read from SWC)=7%

(d) Available water=b−c=6%

(e) Water release from 0.5 m soil profile=0.06*0.5=0.03 m=30 mm

(f) Volumetric water capacity (dθ/dh) curve is plotted and shown in following Fig. (b).

Example 13.3

Calculate the unsaturated hydraulic conductivity and diffusivity function for the SWC

data given in Example 13.2. Use the VG equation for the above calculation. Assume θ

for the above calculations.

Principles of soil physics 372

Solution

Step 1: Optimize the van Genuchten parameters a, n, m, and θ

for the given soil, using

RETC code (van Genuchten et al., 1991)

(a) Assume initial values of a, n, and θ

and assume m=l-l/n.

(b) Use the above data and Eqs. (13.41) and (13.42).

for each θ.

(d) Calculate the sum of squares between predicted and given h values in Example 13.2.

(e) Minimize the sum of squares by using any nonlinear optimization program, (solver

subroutine in excel, RETC program of van Genuchten et al., 1991, etc.)

(f) Obtain the final estimates of α, n, and θ

Step 2: Calculate K function using Eq. (13.47) and D function using Eq. (13.48). The

typical K and D functions obtained for the above data are presented in the following

figure.

PROBLEMS

1. Use the Φ

−θ

data in example 2 and plot Φ

−θ

curve. Read the data from curve for a

small interval so that at least 40 sets of h−θ readings are obtained. If the saturated

hydraulic conductivity of the soil is 5.4×10

−4

cm sec−

, calculate the K functions and plot

it with respect to pressure head.

2. Use the same data and optimize parameters of van Genuchten equation [(13.42)-

(13.43)]. Assume θ

as 0.46. Obtain and Plot the K and D functions.

3. Using Eqs. (13.31) and (13.32), calculate the dimensions of K and D.

4. Consider a 50 m long horizontal sandy loam column. Atmospheric pressure is

maintained at one end of this column, while −150 cm is maintained at the other. Calculate

the direction and magnitude of flux density. If K(Φ

) function is given as (a) K=2, (b)

K=2+0.01 (Φ

), and (c) K=3 exp(0.05 Φ

)cm h

−1

. Plot a graph between K(Φ

) and Φ

(x).

Water flow in unsaturated soils 373