estimating water application efficiency in irrigation checks

Irrig Sci (1983) 4:137-146 Irrigation

clence © Springer-Verlag 1983

Estimating Water Application Efficiency in Irrigation Checks

S. K. Gupta, R. N. Pandey, and R. K. Batta*

Central Soil Salinity Research Institute, Karnal 132001, Haryana, India

Received October 4, 1979

Summary. Equations to predict the percolation loss and water application efficiency in irrigation checks have been derived assuming the advance of the water front to be of the form /= a t b. Simple equations based on water distribution geometry in irrigation checks have been suggested for situations where the advance function is not known. The observed field data and the comparison with exact solution indicate that elliptic and parabolic geometry of intake profiles give a better prediction accuracy compared to the straight line intake profile previously used.

In spite of its limitations and the fact that other more-efficient irrigation methods are now available, irrigation in checks or basins is still very common. This system of irrigation consists in dividing the area to be irrigated into square or rectangular plots with the minimum possible slope along the plot length. The irrigation water is applied from one end of the plot. The plot size is governed by the water supply and infiltration characteristics of the soil. The main problem in the design is the determination of check dimensions for the desired water application efficiency under a given set of conditions. Considering a constant infiltration rate Israelson (1958) proposed a simple method for check system design. Bishop (1961) and Murty and Agarwal (1970) proposed improved methods by considering a variable infiltration function. Bishop (1961) assumed the advance of the water front in checks as a linear function of time whereas Murty and Agarwal (1970) considered an instantaneous advance and derived expressions for predicting percolation losses. Under field conditions the advance of the water front is neither linear nor instantaneous but is approximated by an empirical relation l = a t b. Therefore, percolation losses obtained from Bishop (1961) and Murty and Agarwal (1970) will not be realistic.

* Drainage Engineer, Senior Drainage Engineer and Scientist S-l, respectively

138 S.K. Gupta et al.

Considering variable infiltration and advance functions, simplified equations for predicting percolation losses have been derived in this paper. The results obtained from the proposed equations are compared with those obtained from Bishop's (1961) and Murty and Agarwal's (1970) equations. Based on the water distribution geometry two more equations are derived to be used under conditions where advance functions are not known. Water application efficiencies have been measured in checks and these values were compared with the predicted values.

Theory

It has been observed (Criddle et al. 1956) that infiltration characteristics of a soil can be represented by an empirical equation of the form:

:=X:. (i)

Where

I = infiltration rate, cm/min K = constant, representing infiltration rate at unit time, era/rain n = exponent, cm/min ~ t = time, rain

integrating (1) from 0 to to, yields

K y = t~ +~ (2)

n + l

y = cumulative depth of infiltrated water at time, t (cm)

If the water front advance in a check is assumed to follow a empirical relation of the form (Criddle et al. 1956),

l = a t b (3)

in which

I = length of advance in time t, m a = length of advance in unit time, m b = exponent, m -~

The average ponding time, ta, at each point on the check surface during the period of advance could be obtained as

1 L 1/b

Where

(4)

(5)

L = total length of the check (m),

Estimating Water Application Efficiency in Irrigation Checks 139

Water surface profile Water surface profile\ /'during advance during recession \ HEAD END ~ !

-]rT . . . . . .

t2

t I + t 2

Fig. 1. Definition sketch indicating opportunity time

TAIL BUND

L A - Percolation losses

and

tx = total time of advance for check length, rain.

To apply the required depth of irrigation at the tail end of the check, water is allowed to stand for a period of q. Therefore the ponding time, 7; at any point in the check is

T = t, + q. (6)

Since no runoff is allowed from the tail end of the check, and water maintains the same level throughout, it is assumed that the water level recession at each point along the check length will occur simultaneously (Fig. 1). This assumption appears to be somewhat inconsistant as some grade in the check is essential for (3) to hold true. However, the validity of this assumption will be tested by comparing the equation developed herein with the equation developed by integrating the depth of the water infiltrated. The comparison of the equation with the observed data will provide a more rigorous test of the assumption.

Since water remains standing at any point for a total average period, 1; the depth (DT) of water infiltrated during this period may be obtained using (2):

K K [ t l b ] "+I DT = _ _ Tn+ 1 = t2 + 4 - - - (7)

n + l n + l b + l

Similarly the required depth of irrigation, D,., in time t 2 is K

D,. = - - (t2) ~ + ~. ( 8 ) n ÷ l

Therefore, the fraction of the applied water lost through percolation, P, is

n + l t ~ + t ~ - t l b + l J - n + ~ - ~ (q)n+z p = (9) K [ _ _ _ t lb] "+'

n + l t 2 + 4 b + l


Representing t2/t 1 by R, (9) may be expressed as

( 1 ) n + l - N n + l R + - - l + b

P = ( R + 1 - - ~ 1 )n+ , ( 0 < b = < l ) . (10)

Thus the water application efficiency E (= 1 - p) is

Rn+l E = ( 1 ]n+l ( 0 < b = < 1). (11)

R + TT~- /

Equations (10) and (11) clearly indicate that the percolation loss and water application efficiency in a check will depend both on soil infiltration and water front advance characteristics.

For given infiltration and advance characteristics and a desired application efficiency, (11) yields

El~n+ 1 R = (1 + b)(1 - E l/n+l) " (12)

Once the value of R is known for the desired efficiency, the length of the check for a given depth of irrigation may be calculated using (2) and (3).

Geometric Configuration of Water Distribution and Percolation Loss Evaluation

From Fig. 1, it is seen that the profile of water distribution beyond the desired depth of irrigation may be approximated by a parabola or an ellipse. The percolation loss, P, then may be obtained by

Area A P = (13)

Area A + Area B

and the resulting expressions for the two cases are:

Case 1. Water profile is represented by an ellipse: Using (2) and (13) the percolation loss is

(R + 1) "+I -R n+l P = (R + 1) n+l + 0.273 R n+l (14)

and the application efficiency is 1.273 R n+ l

E = (R+ 1) n+~ +0.273 R n+~ (15)

Case 2. Water distribution profile is represented by a parabola: For this case the percolation loss is:

( R + I ) n + l - R n+l P = (R + 1) n+l + 0.5 R n+l (16)

and the application efficiency is 1,5 R n+ 1

E = ( R + I ) n+ l+0 .5R n+l " (17)


Simpl(fication of Percolation Loss Equations

Expanding binomial ly and retaining only the first two terms of the expansion (10),

(14) and (16) may be further simplified as

n + l p = ( i s )

R(1 + b ) + n + 1

n + l P (19)

1.273 R + n + 1

n + l P - (20)

1 . 5 R + n + l

Materials and Methods

To test the validity of equations developed in this paper an experiment on check irrigation was performed in an area of 0,15 ha. The three sizes of the checks used under this study were I0 x 10, 10 x 15 and 10 x 20 m -° each replicated three times. Average point infiltration in the area was measured using double ring infiltrometers at the centre of each plot except in case of 10 x 20 m s plots. In these plots infiltration was measured at two points 5 m offset on both sides of the centre along the length of check. The average infiltration rates of all the replications for each treatment were plotted against time on log-log scale and values of K and n were obtained from the eyefitted straightline relationships.

The uncropped checks were used to determine the water application efficiency. The stream size used in this study was 5 1 per s. Advance in each of the checks was measured at 2 m interval and at the end of each check. The values of a and b were obtained by plotting the advance data on log-log scale. The soil moisture data to a depth of 120 cm at 15 cm interval were collected before and after the irrigation at five predetermined points in each check. The change in soil moisture was calculated from the change in the moisture content of the profile. Water application efficiency in each plot was calculated to obtain the average value for each treatment by comparing the average amount of water retained in 0-30 cm profile with that of the average total water applied at the head end. The soil moisture deficit needed to attain field capacity was 15% by volume and the necessary depth of water application was 4.5 cm.

Results and Discussion

The exact equations derived to calculate the percolation losses and application efficiency during irrigation in checks are given by (10) and (11). It may be seen that the equations derived are independent of the parameters a and K and are dependent only on exponents b and n of the advance and infil tration relations. As the physical conditions of the soil surface and initial moisture contents are likely to have major effect on K and a, these parameters are not likely to affect the prediction behaviour of the equations derived herein. It is therefore to be expected that the use of these equations would provide a good estimate of the percolation losses under many field situations.

As expected it may be seen that (10) for b = 1 and b = 0 reduces to the equations proposed by Bishop (1961) and Murty and Agarwal (1970) respectively. However, the serious disadvantage of these equations is that they give the percolation loss at two extreme conditions of advance i.e. l inear (b = 1) and instantaneous (b = 0) which seldom occur under field conditions. The percolation losses obtained


Table 1. Percolation losses as calculated by different equations for various n and R = 1

Infiltration Percolation losses (%) constant (n) Eq. (10) Pb* Eq. (10) Eq. (14) Eq. (16)

b = l b = 0

- 0.1 30.5 30.2 46.4 40.3 36.4 - 0.3 24.7 23.8 38.3 32.8 29.3 - 0.5 18.2 17.2 29.2 24.5 21.6 - 0.7 11.4 10.4 19.4 15.3 13.3 - 0.9 3.9 3.4 6.6 5.2 4.5

* Pb = Percolation losses are calculated by Bishop's (1961) approach

Table 2, Calculated percolation losses from exact and simplified equations for various values ofR (n = - 0.46 and b = 0,73)

R Percolat ion Loss(%)

Eq.* Eq. Eq.

(10) (18) (14) (19) (16) (20)

1.31 17.9 19.2 22.0 24.4 19.1 21.6 1.95 13.1 13.8 16.5 17.9 14.5 15.6 3.39 8.2 8.4 10.5 11.1 9.2 9.6

* Equations (18), (19), and (20) are simplified forms of Eqs. (10), (14), and (16) respectively

from (10) for b = 1 and b = 0, Bishop's equation and (14) and (16) are shown in Table 1, for different values of n, and R = 1. F rom the table it may be seen that percolation values obtained from (14) and (16) are in between those obta ined from (10) for b = 0 and b = 1. The values for b = l are nearly equal to the values obtained from Bishop's equat ion indicating the inherent assumption of l inear advance in it. The percolat ion losses for b = 0 will not be different from those obtained from Murty and Agarwal 's equation as the two expressions are essentially the same. This indicates that (14) and (16) could be used to obta in a better approximat ion of percolat ion losses in case the value of b is not known. The use of these equations should be preferred rather than assigning the extreme values to the advance exponent in (10).

To test the s implif ied equations (18), (19) and (20) the predic ted percola t ion losses from these equations are compared in Table 2 with those obtained from (10),:,(! 4) and (16). It may be observed from this table that the s impl i f ied equations give comparable results for field values of b, n and R. Though some error may be introduced at higher values of n or lower values of b for all pract ical purposes this is unimpor tant as the extreme values of b and n are rarely en- countered.

To see the effect of b on percolat ion losses (10) was evaluated for various values of b and n at R = 1. The results are presented in Fig. 2. It may be observed from


~ 6 0

5 o _.J

c 4 0 o i

~ 30

~- 20

I0

The Number On Curves Are Values Of n

-O-f ~ " " " - - - ~ ~ - 0"3 ~ - - - e 5

0,7

~j I I i, I I I I 0 . 1 I 0 0£ 0"3 04 0"5 (>6 (>7 0"8 I'0

0"9

Value of b

Fig. 2. Effect o f n & b on percolation losses

Fig. 2, that the percolation loss decreases as the value of b increases. The losses on the other hand increase as the value of n decreases. The average infiltration and length of advance rate equations were evaluated and are reported in Fig. 3. The fit of the data around these averages is quite good indicating the validity of the use of these equations for evaluating the water application efficiency. It may be added that when individual values from each replication are plotted more scattering of the data is observed (the range of variation of individual values from the average is shown in Fig. 3) but the position of the average line is not changed. The value of R in the three check sizes varied from 1.31 to 3.39. The calculated application efficiencies for each case along with the observed values are presented in Table 3. The observed values are quite close to the predicted values though on the lower side. This is probably due to imperfect levelling which resulted in accumulation of water at the tail end, remaining on the soil surface after the rest of the check had dried. This also indicates that as the value of R decreases the water application efficiency decreases which suggests that to achieve higher efficiency we should try to reduce advance time for the same depth of irrigation to be applied either by changing the length of the check or by increasing the stream size.

Efficiency values calculated according to Bishop's (1961) approach were always higher than those obtained using the approach presented in this paper. The agree- ment is closest with elliptic shape in two of the checks here as it is closest with (I 1) in the third one (10 x 15 m). The assumption of parabolic shape of the profile is always the second best. However, the error among these three and above observed values is always less than 5%. Differences in predicted values of (11) and observed ones could be due to the assumption that average depth of water is applied in average ponding time over the check and also to the assumption that t~ is constant over the check. To evaluate the effect of first assumption we compare the average depth of water infiltrated during advance only from this approach to that of Christiansen et al. (1961). From appendix-A it could be seen that equation of Christiansen et al. (1961) is same as derived in this paper if only one term of Binomial expansion is considered in the evaluation. The error appears only when higher order terms are considered. In fact the predicted values of application efficiencies obtained from Christiansen's approach will lie somewhere in between those given by Bishop's (1961) and (11). This shows that in actual sense observed


r .

./• -120

• • I0

,,0~ ~ . .e., , 0 X 2 0

"8 =" ~ Average value and standard '~ • .6 daviofion of data at t=lOrrds

¢ :

and 1;= 8m

i ¢.t

.2

i -I

IO I00 TIME (Mrs)

Fig. 3. Determination of infiltration rate and length of advance equation constants

Table 3. Observed and calculated application efficiency (E%) in check irrigation

Check Application efficiency dimensions (m x m) R Observed Calculated

Eq. (11) Eb* Eq. (15) Eq. (17)

10 x 10 3.39 87.6 91.8 93.0 89.5 90.8 10 x 15 1.95 86.5 86.9 88.8 83.5 85.5 10 x 20 1.31 78.7 82.1 84.9 78.0 80.9

* Application efficiency as calculated from Bishop's (1961) approach


values are always lower than predicted values in each case under study (Table 3). This may be due to the assumption of constancy of t 2 in deriving these equations and the variability of infiltration and advance functions around the average line assumed in evaluating parameters used in prediction equation.

Conclusions

Simple equations to predict percolation losses or application efficiencies have been derived for use under check system of irrigation. The calculated values are within _+ 5% of the observed values for the three equations proposed herein.

Appendix A

Average opportunity time during advance for any length, L, of a check, border or furrow is given by (5) as:

b ) tl t a = q - t l ~ - l + b (i)

Therefore the average depth of water applied in this time is

D, - - 1 - - - . (ii) n + l l + b

Where D, = average depth of water applied during advance. Expanding (ii) bino- mially and using only first two terms, we have

Kbt~+' [1 n + l (n+ l)(nb) l D a - n + l b l + b + 2 ! ( l + b ) ' (iii)

The comparable equation for such a case using depth average approach could be given as (Christiansen et al. 1966)

Kbtf+l [1 n + l ( n + l ) n l D , - n + ] - b l + b ~- 2 . ~ - + ~ " (iv)

It could be seen that (iii and iv) differ from each other only in the last term in the bracket. But (iii) will always result in higher values to a maximum of + 5%. However, simplifying (ii) and again using binomial expansion will also result in

K t ~ + l l l ] D a - n T 1 l + b ( n + l ) " (v)

The (v) gives comparable results to (iv). This shows that the assumption of average depth of water applied in average apportunity time results in inaccuracies upto + 5% over the depth average approach. However, this also can be taken care of by applying the binomial expansion to a particular form of the equation and using only the first term of the expansion (v). This analysis suggests that the average opportunity time approach could give reasonably accurate predictions of water


infiltrated during advance only and is independent of the length of the i r r igat ion system for such a case.

References

Bishop AA (1961) Relation of intake rates to length of run in surface irrigation. J Irrig Drainage Div 83 (IRI): 23

Christiansen IE, Bishop AA, Kiefer FW, Fok Yu-Si (1966) Evaluation of intake rate constants as related to advance of water in surface irrigation. Trans Am Soc Agric Eng 9:671

Criddle WD, Davis S, Pair CH, Shockley DG (1956) Method for evaluating irrigation systems. Agricultural Handbook. Vol 82. Soil Conservation Series. United States Depart- ment of Agriculture

Israelson OW (1958) Irrigation Principles and Practices. 2nd Edn. John Wiley & Sons, New York

Murty VVN, Agarwal MC (1970) A rational approach to design of check system of irrigation. J Agric Eng Res 15:163

estimating water application efficiency in irrigation checks

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