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VII. Application of Infiltration Models to the Design of Trickle Irrigation Systems
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
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
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
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
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
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
(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
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
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
Parametric Soil Data
2.35 x lo-'
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
(Num. S o h )
W C )
p c = -70 cm
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-
tion coefficient for dividing flow, and pipe roughness coefficient. The HazenWilliams equation accounting for dividing flow between emitters is
HL = 2.78.
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
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
N = -L
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
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,
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
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
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
= 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
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
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