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VII. Application of Infiltration Models to the Design of Trickle Irrigation Systems

VII. Application of Infiltration Models to the Design of Trickle Irrigation Systems

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TRICKLE-DRIP IRRIGATION



377



A. TRANSIENT STATE INFILTRATION



In the previous sections, analytical and numerical models of the more natural

transient infiltration from trickle sources were discussed (Warrick and Lomen,

1974; Brandt ef ul., 1971). From these models wetting fronts and the pattern of

two-dimensional distribution of a specific soil-water variable [S(O), p ( B ) , or B ]

can be obtained for soils with various hydraulic properties and for different

trickle discharges. The results, such as those presented in Figs. 7 and 8, can be

applied in order to estimate the actual spacing between emitters for any given set

of crop, soil, irrigation water quantity, and trickle discharge. Thus, for a particular crop and given field conditions, the optimal combination of discharge rate,

amount of irrigation water, and emitter spacing can be obtained. For these

purposes comprehensive numerical models (Brandt et ul., 1971; Bresler et al.,

1971; Bresler, 1975) are preferred. However, in some specific problems when

analytical solutions exist, they may also be applied to a complex field problem.

An additional useful application of linearized analytical solution to transient

flow problems is that they can serve as a check for more comprehensive

numerical models (Warrick and Lomen, 1974).



B. STEADY INFILTRATION



The conditions of steady infiltration are not generally met in the field but may

be approximated when irrigation is continuous for a relatively long time, or in

intermittent irrigation when irrigation is very frequent. Water content pulses

resulting from intermittent infdtration may be damped out a few centimeters

from the source for many soils (Rawlins, 1973). This makes it possible to

consider flow beyond this region to be essentially steady for continuous irrigation when infiltration takes place during a relatively long time, and for intermittent irrigation when irrigation frequency is of the order of a day or less. Of

course, a truly steady flow can never be achieved during trickle infiltration but it

represents an asymptotic case which can be used for practical purposes of

designing trickle irrigation.

The case of steady infiltration from a shallow circular ponded area on the

horizontal surface of a semi-infinite soil, as treated by Wooding (1968), is

considered to be a useful designing tool appropriate for application purposes.

T h s is so because the actual trickle source does not behave as an idealized point

source as assumed by other steady models, but the water is spaced over a finite

circular water-saturated area on the soil surface. This radial saturated water entry

zone is initially vely small but its radius p ( t ) becomes larger as time increases and

approaches a constant value p u at some finite infiltration time (Fig. 6). The



378



ESHEL BRESLER



ultimate size of this zone depends primarily on the saturated hydraulic conductivity of the soil and the discharge rate of the dripper. An estimate of the

relationships between the ultimate radius (p,) of the ponded-saturated water

entry zone, the soil hydraulic properties K , and a, and trickle discharge Q, may

be obtained from Eq. (64) of Wooding (1968) as



Q = T P ~ K+, 4K,pU/a



(27)



Equation (27) is equivalent to the boundary conditions in Eq. (17f) for the time

in which the ultimate values of vertical water fluxes and hydraulic gradients are

approached throughout the ponded area (Fig. 5). For this ultimate time we have

from Eq. ( 1 7 0 (neglecting E),

Q = n p i K , -+ nK,pi



Icl



(28)



where Ic 1 is the absolute value of the average vertical component of the pressure

head gradient in the ponded zone defined by



In Eqs. (27) and (28) the first term on the right-hand side (RHS) of both

equations represents the flow as a result of gravitation only, while the second

term on the RHS represents the flow as a result of the pressure head gradients in

the saturated water entry zone. Solving the quadratic Eqs. (27) and (28) for the

positive root of p u , the ultimate size of the saturated zone is obtained as



It is clear from Eq. (30) that p u becomes larger as Q increases and K , decreases

and that: (a) when Q + 0 then p u + 0; (b) when Q is very large, p u is large and

the effects of a are negligibly small; and (c) when vertical pressure gradients

are negligibly small, pu reaches its largest value of (Q/zKs)"'. Also p u decreases

as a increases.



C. ESTIMATION OF SPACING BETWEEN EMITTERS



To estimate the actual spacing between emitters for a given set of field

conditions, the hydraulic properties of the soil [K(8),p ( 8 ) or a, and K,] , and

the desired value of critical water pressure p = p c midway between emitters must

be known or estimated for any particular case.

Using this information one can calculate wetted soil volume (Fig. 8) or the

distribution of p , 0 , or S, i.e., p(r,z), 0(r,z) (Fig. 7), or S(g,a), from which the

desired spacing can then be determined. For convenience and general designing



TRICKLE-DRIP IRRIGATION



379



purposes, it is sometimes sufficient to have the distribution of p , 8, or S at the

soil surface, as given by 8(x,O) in Fig. 7 or by S(l,O) in Fig. 11. Figure 11 was

reconstructed from Wooding (1968) and gives the calculated distribution of

S(t,O)/S, for seven values of a = apu/2. Because a unique relationship between

S and 8 and between 0 and p (nonhysteritic case) exists, the data in Fig. 11

represent the soil surface distribution of water content as well as the water



DIMENSIONLESS RADIUS E



FIG. 11. Soil surface values of S/S, as a function of 5 = r / p , for seven different values of

labeling the lines).



a = aU/2(the numbers



380



ESHEL BRESLER



pressure head. To obtain p(r,O) and e(r,O) values from Fig. 11, when a K,, and

pu are known, one has to use Fig. 11, Eq. ( 2 2 ) and the relationships



The values of 0 can then be taken from the O ( p ) relationship.

It should be noted that due to the linear steady state form of Eq. (23),

solutions from a single source as given in Fig. 11 can be added together to solve

for a more realistic situation of multiple sources field. Furthermore, the general

solution to the linearized steady state form of Eq. (23) or the flow equation (1 5)

can be used for many particular and practical field problems, although it

includes some errors owing to the linearization procedure and to the steady flow

assumption. The assumptions of no surface water flux away from the water

entry zone and of a constant (Y value may also be inadequate for some field

problems (Bresler, 1977). In addition, it cannot be claimed that Eq. (22) is

universally exact, but it does model the observed rapid nonlinear decrease of K

with p in unsaturated soil (Philip, 1968). Values of a are generally smaller in

fine-textured material and larger in coarse-textured material (Bresler, 1977),

where gravity may be relatively more important than capillarity for water

movement. However, in spite of these limitations the linearized steady state

solution may be a useful alternative to the nonsteady numerical solution in

determining the spacing between emitters in the irrigated field according to the

rate of discharge, the hydraulic properties of the soil, and the desired soil water

pressure between emitters.

The general procedure of calculating the spacing between emitters as a function of discharge and midway critical pressuree pc involves the following steps

(Bresler, 1977):

(1) Estimate K ( 0 ) and p ( 0 ) or K , and a for any given soil (e.g., Figs. 2 and 3

of Bresler e l al., 1971 or Table I of Bresler, 1977).

( 2 ) Estimate p u from Eq. (30) as a function of Q for any soil using the predetermined value of K , and a,or K ( 0 ) and p(0), and calculate a = p u a / 2 .

(3) Select a critical value of 0, to be used and estimate p c from the p ( 0 , )

characteristic curve (e.g., Bresler er aZ., 1971, Fig. 2) or select directly a value of

pc to be used.

(4) Calculate K(p,) from Eq. (22), S, from Eq. (32), and S(p,)/S, from Eq.

(3 1) for any soil.

(5) Obtain ,$(pc)from Fig. 11 using S(P,)/S, and a = apu/2 or from a figure

similar to Fig. 7 or 8. Note that S@,)/S, = [K(p,) - K O ] / ( K , - K O ) X K(pc)/K,

= exp (olpc).

(6) Calculate the distance (half spacing) between emitters as a function of the



38 1



TRICKLE-DRIP IRRIGATION



I0



DISCHARGE 0 f l i t G / h o i r r )



."



FIG. 12. Estimated spacing between emitters r = rc(Pc,Q) = d / 2 for sandy soil (K,= 0.13

cm minute-', cx = 0.09), as a function of trickle discharge, Q, and selected values of

pore-water pressure head (p = p,) midway between emitters (the numbers labeling the lines).



selected pressure head and trickle discharge [d = 2r,(p,,Q)] from rc = t C p u , or

obtain it directly from a figure similar to Fig. 7 or 8.

(7) Graph the results (e.g., Fig. 12).

A figure such as Fig. 12 may be used as a practical nomogram to calculate the

distance between emitters if their rate of discharge is known, to select the

desired discharge when spacing is given, or to find the best spacing-discharge

combination when a certain economic criterion has to be achieved. In applying

the results as given in Fig. 12 only one iso-p, line must be used, since any d-Q

combination depends on the chosen value of p c . This value must therefore be

determined before any d-Q selection is made. The selection of p c is somewhat

arbitrary and depends on the safety factor that one would like to choose. This is

so because of the uncertainty involved in the response of plants to p, and to the

degree of partial wetting of their root zone. It should also be remembered that in

calculating the data of Fig. 12 from Wooding's method a single emitter (no

interaction) was assumed. With this assumption a safety factor has already been



382



ESHEL BRESLER



taken into account regardless of the value of p c chosen. As a general guideline it

is recommended that the value of p c to be selected, be the one corresponding to

the transition between the lowest and highest values of a p / & (i.e., when lgrad 0 I

starts to increase steeply with r). This is, of course, an approximate average p c

value which may be lower (more negative) for less sensitive crops and must be

higher when a specific crop sensitive to soil-water stress is grown.



D. NUMERICAL EXAMPLE



Evaluation of the applicability of any estimation method must be obtained by

comparing its results with actual trickle infiltration field data. Since a good

agreement exists between the numerical method and experimental field and

laboratory results (Bresler et al., 1971), the steady state results of Wooding

(1968) were compared with simulated transient flow data obtained by the

numerical method proposed by Brandt et al.. (1971). This is, of course, a first

approach which should be fully examined in the field.

The parametric soil data given in Table I1 were taken from Bresler et al. (1971)

for the two soils studied. In calculating K(p,) it was assumed that the relationship (22) holds for values of p which are smaller than the air entry value of pa of

each soil. Values of S were calculated from Eq. (31), with K values substituted

from Eq. (22) and S, values from Eq. (32), or simply by S/S, = exp(.p,).

The soil parameters given in Table I1 and Figs. 1 to 3 in Bresler el al. (1971)

were used to calculate the midway distance r, corresponding to p c for the two

soils and two different trickle discharges using the data of Fig. 11 and the results

of the numerical solution (Num. Soln.) of Brandt et al. (1971).

The data presented in Table I11 show that with the described estimation

methods one is able to calculate radial distances at the soil surface which

correspond to a given soil-water pressure (or water content). It should be

emphasized again that the proposed methods are good approximations to actual

field conditions as long as the solutions simulate the actual field situations. This

was previously shown in some limited, specific cases (Bresler et al., 1971).



E. EFFECT O F SOIL HYDRAULIC PROPERTIES

O N SPACING-DISCHARGE RELATIONSHIPS



The effects of soil hydraulic properties on the spacing-discharge relationships

are demonstrated in Figs. 6 and 13. Figure 6 , which represents the radius of the

saturated-ponded water entry zone as a function of time for two discharge rates

and the two soils, demonstrates mainly the effect of the saturated K , on the

spacing-discharge relationships. The effect of (Y (or K(p) and p(O)] on the



TABLE I1

Parametric Soil Data



Gilat (loam)

Mahal-Sinai (sand)



-40

-15



0.014

0.142



0.56

3.15



0.025

0.062



0.28

0.11



-80

-5 5



-40

4 0



5.15 X

2.35 x lo-'



0.367

0.166



TABLE 111

Midway Distances (rc) between Trickle Emitters Calculated by Linearized Steady State Solution (Lin. Soln.) and Numerical Transient State

Solution (Num. Soh.) for Two Soils and Two Rates of Discharge

Q

Pu

Pu

rcVc)

'c(Pc)

(Lin. Soln.)

(Num. S o h )

a

(Uter/

(Lin. Soh.)

(Num. Soln.)

W C )

Soil

hour)

(cm)

(cm)

(apu/2)'

(Fig. 11)'

(cm)

(cm)

Cilat (loam)



Nahal-Sinai (sand)



4

20



21.1

65.2



21.0

61.1



0.32

0.98



1.49

1.22



31.4

79.5



33.1

66.1



4



5.7

18.9



5.9

17.9



0.18

0.59



3.55

2.10



20.2

39.7



20.7

30.3



20



aDimensionless.



TRICKLE-DRIP IRRIGATION

250



0



I



-



4



I

I

p c = -70 cm



0



DISCHARGE



12



Q



385



I



16



20



liter



(-) hour



FIG. 13. Distance between emitters ( d ) as a function of trickle discharge Q) for two soils

and two values of pC After Bresler (1977).



spacing-distance (d-Q) relationships for a given p c is illustrated in Fig. 13.

According to Philip (1968), a is a measure of the relative importance of gravity

and capillary for water movement in a particular soil. In soil with a high a value

gravity tends to dominate and in soil with low (Y values capillary tends to

dominate. This tendency is more pronounced at smaller rates of discharge (Q).

Figure 13 also illustrates the effects of soil-water properties and the rate of

discharge on the selection of spacing between emitters. Larger spacing is permitted in soils with lower values of K , and a (see Table I1 and Fig. 13) and also

when the crop grown is not sensitive to water stresses and/or to partial soil

wetting (higher p c values are permitted). Closer spacing is required for soils

having higher K , and 01 values when a sensitive crop is being grown. For any

given soil the emitter spacing can be increased as the discharge rate becomes

higher. However, since the rate of growth of rc or d with Q decreases as Q increases (Figs. 12 and 13), the proper choice of Q chiefly depends on some

optimization criteria in the engineering design of the field irrigation system.



F. DESIGNING THE LATERAL SYSTEM



The data as given in Fig. 12 or 13 can be combined with principles of

hydraulics in order to obtain diameter (0)

and length ( L ) of the lateral system

for design purposes (Bresler, 1977). For a given d-Q combination and uniformity criteria, the L-D design relationship and the necessary pressure head at the

lateral inlet depend upon emitter discharge function, elevation changes, reduc-



386



ESHEL BRESLER



tion coefficient for dividing flow, and pipe roughness coefficient. The HazenWilliams equation accounting for dividing flow between emitters is



HL = 2.78.



F*L*D-4*8'(6N/C)1.85



(33)



where HL = friction head loss in laterals (meters)

= average rate of emitter discharge (meters3 hours-')

N = number of emitters per lateral

F = reduction coefficient for dividing flow between emitters along the

lateral

L = the lateral length (meters)

D = inside diameter of the lateral pipe (meters)

C = the Hazen-Williams roughness coefficient

Equation (33) is an empirical equation. Care should be taken in using the units

specified to each of the above-dimensioned variables. The empirical values of F

and C have been tabulated (compare, e.g., Howell and Hiler, 1974). As the

spacing between emitters along the lateral is given by d = 2r, (Figs. 12 and 13),

it is possible to express N as



=4



N = -L

(34)

2r, d

where d is the distance between emitters along the lateral. Substituting L / d for N

in Eq. (33) and rearranging

L = 88.88 D'.708 (HL/F)0.351(Cd/a)"649



(35)



Knowing the Q-d relationship (Fig. 12 or 13) and assuming F to be constant

over a given range of L and d, Eq. ( 3 5 ) gives the L-D relationships for any

preselected value of head loss HL .

To select an appropriate value of HL for Eq. ( 3 9 , a proper criterion may be

based on the differences between the emitter discharge at the lateral inlet and

the downstream discharge, relative to the average discharge,

Qi-Qd



GE



Q

Here Qi is the inlet discharge, Qd is the downstream discharge, and E is a

preselected error fraction, say 0.05 or similar.

The relationship between emitter discharge rate and the hydraulic head at the

emitter may be given by the empirical expression

Q=b@



(37)



where b and p are constant characteristic of the flow regime in the emitter and H

is the hydraulic head at the emitter. Data of b and are available from the



TRICKLE-DRIP IRRIGATION



387



manufacturer but also can easily be determined experimentally in the laboratory.

Using the maximum value of e it follows from Eq. (36), using Eq. (37), that



H f = -€0+Pd

b



Since HL



(38)



= Hi - H d , then



Knowing both the pressure head at the lateral inlet and the emitter constants b

and 0,permits the calculation of HL from Eq. (39) for a given and Q. This

value of HL is then substituted into Eq. (35) to obtain the D-L relationship

needed for the lateral design. When the lateral length is given by the size of the

plot and Hi is known, the diameter D is calculated from Eq. (35). Otherwise, the

optimum economic 0-L-Hi combination has to be calculated for each field and

soil condition.

In summary, it is suggested (Bresler, 1977) that steady and nonsteady infdtration models, which are well suited for the analysis of unsaturated flow through

porous media, can be applied to design a trickle irrigation system. The two

modeling approaches make it possible to calculate the spacing between emitters

as a function of their rate of discharge, soil hydraulic properties, and crop

sensitivity to water stress. However, it is important to remember that each of the

proposed methods has certain limitations. For example, some of them involve

errors that arise from the linearization procedure and from the estimation of K ,

and a. The assumption concerning the steady state flow is a very restricting one.

It should also be emphasized that problems are involved in selecting the correct

hydraulic parameters of the soil: K(O) and p(O). In addition, seepage of unused

water below the rooting zone is not considered when S/S, (Fig. 1I), O(r,o), or

p(r,o) is taken at the soil surface. It is also emphasized that problems are

involved in seiecting the correct p c value. Additional research is needed to ascertain the validity of the views expressed in this chapter, to develop field methods

for determining the necessary soil-water parameters, and to select the best midspace pressure head for a given set of soil, climate, and crop growing conditions

(Bresler, 1977).



VIII. Water Management in Marginal Soils



Hardpans of various types, different sands and sand dunes, desert pavement of

many kinds, and saline and alkali soils are very common marginal soils in arid



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VII. Application of Infiltration Models to the Design of Trickle Irrigation Systems

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