An analytical closed border irrigation model I. Theory

Agricultural Water Management, 15 (1989) 223-241

223

Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

An Analytical Closed Border Irrigation Model I. Theory VIJAY P. SINGH and FANG X. YU

Department of Civil Engineering, Louisiana State University, Baton Rouge, LA 70803 (U.S.A.) (Accepted 23 February 1988)

ABSTRACT Singh, V.P. and Yu, F.X., 1989. An analytical closed border irrigation model. I. Theory. Agric. Water Manage., 15: 223-241. This paper, Part I in a series of two, describes an analytical model for the entire irrigation cycle of closed borders, using a volume balance approach. The irrigation cycle is simulated in five phases: advance, storage, vertical recession, horizontal recession, and impounding recession. The model will be verified with experimental data in Part II.

INTRODUCTION

Knowledge of movement and distribution of water over and below the soil surface is required for a proper design of closed-border irrigation. The bulk of literature on border irrigation is devoted to only one or two phases of the irrigation cycle, especially the advance phase (Davis, 1961; Fok and Bishop, 1965; Wu, 1972; Bassett and McCool, 1973; Clemmens and Strelkoff, 1979; Bishop et al., 1981; Strelkoff and Clemmens, 1981; Sherman and Singh, 1978, 1982). Comparatively little has been reported on the whole cycle of the closed-border irrigation. Furthermore, most mathematical models are numerical (Bassett, 1972; Fangmeier and Strelkoff, 1979; Elliott et al., 1982; Walker and Humpherys, 1983; Singh and Ram, 1983) and often too expensive to be used for large-scale design purposes. Some of the complicated models even do not produce accurate results, due to the high accuracy they demand in estimation of infiltration, roughness and geometric characteristics, as well as initial and boundary conditions. Because of the inaccuracies incorporated in these parameters, it is reasoned that sufficiently accurate closed-form solutions can be derived for the entire irrigation cycle by using the volume balance approach. This approach has been successfully employed by Fok and Bishop (1965) and Wu ( 1972 ) to simulate the advance and recession phases independently. Strelkoff (1977) developed an analytical border irrigation model to simulate 0378-3774/89/$03.50

© 1989 Elsevier Science Publishers B.V.

224 the vertical (depletion) and horizontal recession phases. Several impractical assumptions limited the use of this model. The assumption that water surface would become horizontal at the end of storage phases is not practical. The reason is that for long borders (say more than 200 m), water depth at the downstream end will exceed the height of the downstream dyke. For short borders, the opportunity time function will become crucial in computing recession time. Therefore, the assumption that the advance time function t is a constant and equals one-third of the total advance time is not reasonable. For practical border lengths, say 100-200 m, the assumption that the water surface at the end of vertical recession will become horizontal rarely holds. Therefore, the model does not give satisfactory results for most practical closed-border irrigations. Many investigators utilized the volume balance approach to derive storage shape factors which are dependent on soil type, crop and topographic variables (Chen, 1965; Fok and Bishop, 1965; Singh and Chauhan, 1972; Singh and Ram, 1983). Derivation of the storage shape factors requires the specifications of: (a) the water surface profile, (b) the advance function, and (c) the intake rate function. As discussed by Singh and Ram (1983), different forms of shape factors result from different specifications of the above three functions. It may, however, be preferable to define shape curves of the surface and subsurface flows because the advance and infiltration rate may change a great deal from one border to another; the surface and subsurface water depths are shallow and always constrained by an upper bound (normal inflow depth and maximum infiltration depth at the upstream end in the advance phase) and a lower bound (zero depths at the advance front). Therefore, simple shape curves may be proposed for surface and subsurface water profiles which may reasonably simulate border irrigation. The objective of this paper is to develop a simple and relatively accurate model to simulate the entire closed-border irrigation process by using shape curves, which can be used for the design of farm irrigation systems. This paper, Part I in a series of two, develops a set of closed form equations for all five phases of the closed-border irrigation cycle. DERIVATIONOF MATHEMATICALMODEL The following assumptions are made for deriving the model: (1) The inflow rate is constant. (2) The border is homogeneous, i.e., its slope, width, composite roughness and intake rate do not change with location. (3) The border is wide enough, and there is no cross slope, so that side effects can be neglected. (4) The bed slope is small such that sin ~ ~ tan ~ ~ So.

225

Advance phase Let the advance function be denoted by x = x (t). At time t, the advance front reaches the position x = S, as shown in Fig. 1. Then at any point x = ~ (0 < ~ < S ), the surface water depth is h(~, t), and the accumulated intake (infiltrated) depth is H(~, t). The continuity equation then is: qot=

fo

h(~,t) d~+

f:

H(~,t) d~

(1)

where qo is the unit width inflow rate*. The accumulated intake depth H(~, t) at a point ~ and time t can be expressed by Kostiakov's (1932) equation as: H(~, t) --g[t-~(~)]A=K[T(~, t ) ] A

(2)

where K and A are infiltration parameter and time exponent, respectively, and are usually assumed constant; ~(~) is the time history of advance front which is the inverse function x (t); and ~(~, t) is the opportunity time function, that is, the time interval that water has covered the point ~ at time t. It is assumed that the intake depth profile (subsurface shape curve) is of the quadratic type: H(~, t)=Co+Cl~+C2~ 2,

O<_~<_S

(3)

in which Co, C1 and C2 are constants that can be determined by using the following conditions: H(0, t)=Kt A

(4a)

H(S, t)=0

(4b)

0H(C, t) I

--0

l =o

(4c)

Fig. 1. Surface and subsurface profiles during advance phase. *All the quantities in this paper having dimensions of time and length are measured in minutes and meters, respectively.

226

Though the condition in equation (4c) is an approximation, it would deviate little from reality because the inflow rate is much larger than the infiltration rate, and because of the property of Kostiakov's equation. When (4a), (4b) and (4c) are applied to (3), the constants are obtained as Co = Kt A, C1 = 0 and C2 = - Co/S 2. Therefore, equation (3) becomes:

g(~, t ) = K t A 1 -

(5)

Integrating (5):

f:H(~, t)d~=~KtAS

(6)

The surface storage can be defined as:

o~h(~, t) d~=/~S

(7)

where/~ is the average water depth on the surface at time t. On substituting (6) and (7) into (1), the advance function can be expressed as:

S-

qot -

2

h+-3Kt

(8) A

From a geometric point of view, (8) can be written as:

S

qot h+H

(9)

where/~ is the average infiltrated water depth below the surface at time t. Several empirical expressions have been reported for/~ a n d / t which can be considered as special cases of (8) and (9). Christiansen et al. (1966) gave for /~: fl-

Kt A I+A

(10)

which is the same as (8) if A--0.5. Fok and Bishop (1965) improved (10) by adding a correction factor F as:

/--I--

FKt ~ I+A

(11)

where

F=b(I+A)

b+l + ~

""]

(12)

227

in which b is a constant stemming from the empirical advance equation: (13)

x=ct b

Based on experimental data, Fok and Bishop ( 1965 ) gave the following expression for b: (14)

b=e -°'sA

For computing/~, Fok and Bishop (1965) used the equation:

_ho

(15)

-l+b

where ho is the normal inflow depth. Equations (10), (11) and (15) are all empirical in nature, and the parameters in these equations do not have precise physical meaning. Field observations show that the normal flow depth first appears at the upstream end, and then extends downward. Therefore,/~will gradually increase. Based on this phenomenon, two assumptions are made for computing/~ in this paper: (1)/~varies linearly with S, as shown in Fig. 2, and subject to the condition: /~1s=o = o~ho

(16)

where ~ is a constant (0.5 < ~ < 1 ). (2) After the advance front has traveled long enough to say L, the average water depth, reaches 0.95 of the normal flow depth ho, which can be expressed by Manning's equation:

~.

Nqo

.,~3/5

ho = \60 (So)O.~]

(17)

where N is Manning's roughness coefficient;/{ remains constant thereafter. The above two assumptions lead to the following expressions: /{(S)=~ho-~ (0"95-~)h°-s, L

for S<_L

(18a)

I~(S)=0.95ho,

for S > L

(lSb)

-~-~o.95.o

0.95h o

s(t) Le

Fig. 2. Relationship between average water depth on soil surface and advance distance.

228

Substituting (18a) and (18b) into (8) and provided that the border width is W and inflow rate is Q, the general advance function can be expressed as:

\21o5

L[4Q(O'95-a)h°t-LW ~-(2KtA+aho) j _aho_~Kt A S-

S<_L

2(0.95-a)ho L

Qt

S-

W(0.95ho +

~Kt a )'

L
(19a)

(19b)

or inversely, the advance time t can be computed by iteration as: 2

LW{[ (1.9-2a)hOSL tK+ 1 --

2 A baho+~KtK]

-(~KtA+~hof}

4Q(0.95-a)ho to =0,

tK+l--

Q

,

to=t[S=L, L
S
(19c)

(19d)

where Le is the border length. Equation (19b) shows theoretically that if the border length is long enough, and t is sufficiently long, say 0.95ho/(2KtA/3) < 1/20 or t> (28.5ho/K)l/A,then the advance function can be approximated by:

S = ~3O t 1-A

(20)

Equation (20) is the same as the frequently used empirical equation (13) if c= 3Q/(2 WK) and b -- 1 - A. Furthermore, (20) suggests that when t is sufficiently large, the advance distance is mainly determined by the size of the inflow rate and the infiltration characteristics. Of course, these conditions rarely occur.

Equation (14) may be approximated by retaining the first three terms of the series expansion as: b=e

-°'6A ~

1 --0.6A + 0.18A 2

(21)

From (15), one can show that the advance rate predicted by the model of Fok and Bishop (1965) is larger than that of the proposed model. Calibration of the proposed model using ten data sets from Roth (1971) and Atchison (1973), given in Table 1 (four data sets are for nonvegetated borders; six data sets are for vegetated borders), produced values of L and a (by try-

229

and-error method to obtain the least squares sum of errors between observed and calculated values) as: L = 7 5 m for nonvegetated border, L = 2 2 5 m for vegetated border, and a = 0.62 for both vegetated and nonvegetated borders. A comparison between observed and calibrated values of S showed that the average relative e r r o r (ARE) was less than 6% and the average absolute difference ( A D ) was less than 2.5 m for the ten experimental borders ( 110 calibrated and observed advance front positions were analyzed). Further analysis showed that the model accuracy was relatively insensitive to changes in the factors and L. For instance, if L changed from 100 m to 1000 m on vegetated borders, ARE for the experimental data changed from 5.4% to 9.7%, AD changed from 2.14 to 5.02 m. It will be shown in Part II that the error in prediction of advance (using 15 closed experimental borders) was less than 8%. TABLE 1

Irrigation datafor vegetated and nonvegetated borders (after Roth, 1971;Atchison, 1973) Parameter

Data sets Roth-8 (1)

Roth-9 (2)

R o t h - l l At-17 (3) (4)

At-1 (5)

At-2 (6)

At-3 (7)

At-4 (8)

At-5 (9)

At-6 (10)

0.6201

0.8529

0.858

0.8478

0.8461

0.8427

0.6813

0.8478

0.5097

0.9803

0.0010

0.0010

0.0010

0.0011

0.0011

0.0011

0.0011

0.0011

0.0011

0.0011

100

100

100

100

100

100

100

100

100

100

0.035

0.029

0.060

0.211

0.107

0.098

0.119

0.092

0.134

Inflow rate, Q (m3/min)

Border slope, So ( m / m )

Border length, Le (m) Average roughness, Manning's N Infiltration constant, K, ( m / m i n a)

0.0170

0.01152 0.01426 0.01643

0.00466 0.00238 0.00223 0.00192 0.00180 0.00183 0.00165

0.394

0.358

0.266

0.385

0.599

0.540

0.517

0.495

0.444

0.644

44.7

41.2

39.5

28.9

47.3

33.7

34.9

31.5

37.3

29.5

181.4

179.7

179.3

190

1.6

0.8

1.7

Infiltration exponent, A

Total advance time (min) Duration of inflow (rain) Vertical recession time (min)

Border widths for all data sets are 19.33 ft.

6.0

147

28.0

130

23.0

140

32.0

140

35.0

140

24.0

245

21.0

230

0

~

~'~

~"~'4¢7"i'DIl~'tT/'4tT'l"¢~tl"t'~(~tl'4"¢l'4\\l

Fig. 3. Water surface profile during storage phase.

Storage phase The storage phase begins when the advance front reaches the downstream end x=Le, and time t= Ta. This phase continues until the inflow is cut off at time t= Ts. At the end of the advance phase, the volume of water stored on surface may be computed by: 2

A

Va =]~Le=qoTa-~gTa Le

(22)

As explained before, the normal inflow depth will expand continuously down the border as the advance time elapses. In fact, the depth of advance front will rapidly go up to more than 80% of the normal depth within less than 1 or 2 min. Furthermore, when the advance front is blocked by a dyke, the flow depth will rise more quickly. Therefore, for simplicity of simulation, we may idealize the surface profile to be a straight line with a characteristic flow depth, ha, at the downstream end (though it is usually assumed to be zero at the moment

t=Ta): ha=2l~-ho -2q°Ta Le

~KT~-ho

(23)

where ]~ is obtained from (22). The surface water profile, as shown in Fig. 3, at time t = Ta, may be simulated as:

h(x, Ta)=ho-

h0 - h a Le

x

(24)

Equation (24) should be reasonably accurate except near the downstream end. At time t = ts ( Ta < ts < Ts), the continuity equation ( 1 ) becomes:

qot~ -

(ho +ho)L 2

~-Vi~

(25)

where he is the water depth at the downstream end at time t= t~, Vi~is the total infiltrated water in the border during time t~. The following conditions determine the assumed parabolic subsurface water profile as expressed in (3):

231

H(O, t~)=Kt A H(Le, t~) =K(t~ - Ta)A

(26b)

OH(~, 0~ t~) ~=o = 0

(26c)

(26a)

Then we have:

H(',t~)=KtA-K[t~

A _

-~

-

Ta)

A

lq ,

O~,~Le

(27)

Integrating (27) with respect to ~ from 0 to L e yields: 1

a

Via =-~KLe[2t~

+ (t~-T~) A ]

(28)

Substituting (28) into (25) and solving for he, we get the water depth at the downstream end at time ts (Fig. 3) as: 2 A he= 2qo -~ t~ -~K[Zt~ + (t~-Ta) A] -ho

(29)

Since we assumed that the water depth at the upstream end is ho and the water depth at the downstream end can be computed by (29), the surface profile may be simulated as:

h(x, t )=ho-

2ho--

+gK[2t

+ (ts--TDAI ,

O
(3O)

If the inflow is turned offas soon as the advance front reaches the downstream end at x = Le (t~ = T~), (29) and (30) are the same as (23) and (24), respectively.

Vertical recessionphase As soon as the inflow is cut off, the vertical recession phase begins. This phase can be described by depleting the triangular volume of water V1, as shown in Fig. 4, and making the water depth at the upstream end zero: 1 Lv V1 =~ho

(31)

where Lv is the position that the horizontal line, passing through the origin 0, intersects the water surface profile represented by (30), in which t~ equals T~: Lv -

ho 2ho 2qo T~ 2K A So-~ L~L~ + ~ e [2T~ +(T~-TA)a]

(32)

232

Ih

Fig. 4. Depleting volumes of water and water surface profile at the end of vertical recession phase.

where the depth at x = L, is: (33)

h, =L&

Similarly, the subsurface profile is assumed to be of the parabolic type. At time T, when vertical recession is finished, the profile can be expressed, similar to (27), as: H(x, T,) =KTtL -K Integrating

Tv-CC?-Ta) LE

1x2,

Olx
(34)

(27) and (34) from x = 0 to x = L: (35)

(36) The volume x=L,is:

during vertical

of water infiltrated

recession

between

x = 0 and

(37)

=KL”(T$

-T:)

-$$T$ e

-T;-(Tv-Ta)A+(Ts-T,)A]

Let h, remain unchanged during the vertical recession umes to be depleted to the downstream part, x > L,, is:

=;hoLv-KL,(T:-Tf)+,

phase. Then the vol..

KL’[T:-T$-(Tv-Ta)A+(Ts-Ta)A] e (33)

On the other hand, volume V, can be calculated

by using Manning’s

equation:

233

60~¢

V2-

~ 0.5~, 5/3

~-30 ]

~v

ZN

(Tv -T~)

(39)

where fl is a parameter of Manning's roughness coefficient, which modifies Manning's equation for the case of transitional flow as assumed in the recession phases. Manning's roughness factor, N, corresponds to the turbulent flow. Cowan (1950) and Palmer (1946) have stated that this factor represents the net effect of all the factors causing retardation of flow. However,flowin border irrigation can be turbulent (Michael and Pandya, 1971; Roth, 1971), laminar and transitional (Bowman, 1960; Myers, 1959), accordingto the criteria described by Chow (1959). Since most models of border irrigation are based on turbulent flow (Michael and Pandya, 1971; Ram and Lal, 1971; Strelkoff, 1977; Sherman and Singh, 1978, 1982; Strelkoff and Clemmens, 1981), these models require modificationwhen the flowis laminar or transitional. To properly account for changing flowbehavior, we define:

N1 =fiN

(40)

where fl is a dimensionless factor. The value of the parameter fl was derived from the ten freely draining experimental data sets given in Table 1. For nonvegetated borders, fl was found to be 2.12, and for vegetated borders, fl--3.05. Average relative error between calculated and observed horizontal recession times for the ten data sets (110 computed and observed values were compared) was 12.9%. Average absolute difference was 5.59 min. Equating (38) and (39) to solve for Tv yields:

Tv=Ts-+ 60(So)O.5h~/3fiN

{1-~hoLv -

KL,,( T¢ - T~ )

.4_KL~[TA _ 3L2e ~__. - T ~A - (T~--Ta)A+ (Ts-T~) A]

(41)

Equation (41) is an implicit equation of Tv. It is more convenient to compute the net vertical recession time Tsv (where TN,,= Tv- T~):

ZN 1 L TN~ ----60 (So) °'ShSv/3 ~-~ho v-KLv[TNv+

Ts) A_ T~]

KL~ } +-~e[(TNv+Ts)A--T~--(TN,,+T~--T,)A+(T~--T,)A] If

T~= Ta, (42) becomes:

(42)

234

[1

TNv-- 60(So)O.ShSv/3 ~hoLv-KLv[ (TNv + T.) A- TA ]

KL3v (TNv+T,)A--T~v--Ta]}

(43)

If TNv is small, say less than 5 min, as is usually the case for nonvegetated, steep borders, TNv in (41) can be approximately computed by setting Ts= T., T~ - T2 ~ A T 2 - 1 TNv and neglecting the term TNv with exponent, hence:

0.5flNhoLv 60 (So)°5h~/3 TNv

-

-

flNKALv 14 60(So)O.Sh~/3T~ -A

fiNICAL 3v 180(So)O.ShSv/aL2T1-A

(44)

Comparatively, (41) or (42) are more accurate than Shockley et al.'s (1964) model:

ho

(45)

TNv -- 120So qo or the SCS (U.S. Soil Conservation Service, 1974) model:

qO.2N 1.2 TNv - - ~

(46)

These two models assume that the sum of the outflow rate at x=L, and the infiltration rate from x--0 to x - L v during vertical recession time is equal to the value qo. Computations show that these two models are usable only when the bed slope is steep, or when the vertical recession time is shorter (say less than 5 min). When vertical recession time is longer, say longer than 5 min, the proposed model is more accurate. It will be shown in Part II that the average prediction error by (42) or (43) was less than 2% and the average absolute error was only 1.2 min.

Impounding position The impounding position Lp is marked by the tail point at which the surface water profile becomes horizontal, as shown in Fig. 5. To make the infiltration function integrable without losing accuracy and physical meaning, let us suppose the opportunity time function ~(x) at time Tv to be linear as shown in Fig. 6. Thus: ~(x,

Ta Tv)--Tv---x Le

(47)

235 h(x,t)

ft

kv

Xh-

-.- T v

~

,

,

(

~

Fig. 5. Schematic surface profiles at the end of vertical recession after impounding. ke

To Tv t

Fig. 6. Opportunity time at the end of vertical recession phase.

Using Kostiakov's equation, the volumes of water infiltrated by the end of the vertical recession phase can be obtained by substituting (47) into (2) and then integrating from x = 0 to x-- Le, giving: Yi (Le, T v ) -

KLe

(I+A)Ta

[T~v +A

-

-

( T v - Ta) l + a ]

(48)

The volumes remaining on the surface at the end of the vertical recession are:

VR =qoTs - Yi (Le) =qoTs

KLe (I+A)T~

(49)

[TI+A-(Tv-Ta)I+A ]

By application of the continuity equation and using the linear profile assumption (see Fig. 4), we have: VR --

(h'e +hv) (Le - L v ) hvLv ~ 2 2

(50)

where h'e is the water depth at the downstream end at time t--Tv. Equating (49) to (50) and after rearranging terms, we get the depth at the downstream end as:

236

2qoT~ hl -

2KLe [TI+ ~ _ (Tv__Ta)I+A]_hvLv

(I+A)Ta

/%

Lo-Lv

(51)

The water surface profile may be approximately simulated as:

h(x, Tv)=xSo,

x<_Lv

(52a)

h(x, Tv)=hv+h'~-h~ (x-Lv), Lv<_x<_L~ L~-Lv

(52b)

Let the impounding time be tp. The opportunity time function at time tp (see Fig. 7) may be represented as:

Ta

v(~, t . ) = T v - - ~ + - ~ p p ( t p - T v ) ,

T.

T(~, tp) ----tp --~-~e~,

~
(53)

Lp >~_
(54)

The volume infiltrated between x=O and x=Lp and during the time period from t=Tv to t=tp can be obtained by substituting (47) and (53) into (2), then subtracting and integrating from ~= 0 to ~= L.: A

__

K

(l+A,(,.57.

A

L~)r(L Va

~I+A

"1

(I+A)Ta The volume infiltrated between Lp and Le and from t = Tv to t=tp can be ob0

~

Lp

Le

To Tv

t

'p

Fig. 7. Opportunity time at the time of impounding.

x

P

237 tained by substituting an assumed linear opportunity time function, similar to (47), into (2), differentiating it. and then integrating it as follows: ~tp~

V4(Lp' tp)= JTvJLp[,

KA

~'--A

t.--

-- ( I + A ) T a

dx dt

L.

- ( t p - T a ) l+a

--I+A

It is assumed that when depleting volumes 1/5 (dashed parts in Fig. 5), the water depth h. at x=I_v remains constant and can be computed by (52b):

Vs

hg

(58/

- 2So

The average outflow rate when depleting V5 is:

qeS--

60 (So) 1/2,h5/3 vp

(591

The average outflow rate when depleting V6 is: So~

60(So) 1/2

qe6 =-~JL.-(h.lSo) fin 45(So)lZ2h~/3 2ZN

[(/-V ~)So] 5/~ d~ -

(60)

where fl is a parameter of Manning's roughness factor. It should have a different value from the one in vertical recession phase, because the downstream water influences the outflow at point Lp and because the water movement further decreases. Let the infiltrated water during depletion of 1/5 be Vis, and during depletion of 1/6 be V~6,then the following two equations should be satisfied by Vi5 and Vi6: Vi5 "JPVi6 = V3(L.. tp)

y -yi

t ----tp

(61) -T.

(62)

238 Solving (61) and (62) for Vis and Vi6: Vi5 - VSL6 + V6(le5 -C1~5 V3(Lp, tp) - (tp - Tv)q~sCie6

(63)

(~e6 -- qe5 Vie __

V 6 q e 5 "~-

Vs(~e6--(~e6V3(Lp, t p ) - ( t p - T~)c/~5c/~6 (~e5 -- (~e6

(64)

Thus, the impounding position Lp and time tp should satisfy the following two continuity equations: 1 V R - V3(np, tp) - V4(np, tp)--~(Le -Lp)2So

(65a)

or Lp_Le_

{2 [ YI{-

\

Y3(Lp, t p ) - Y4(Lp, tp)] ~o.5 So

]

(65b)

and

vs-Yi+

t p _ (~e5

y+-yi~

+- (~e6

+-Tv

(66)

As an initial value of Lp and tp to iterate (65b) and (66) for getting the desired values of Lp and tp, we may, for practical purposes, use Lp = L e / 2 and tp = 1.5Ta. In order to expedite computations, we may further initialize tp as: (67)

tp - V5 + V6 - V3 ( Lp , tp )

where (/e is the approximation of average outflow rate at x = Lp: (68)

Vs + V+ 4+- V~ ÷ V~

Horizontal recession phase

This is the period from the time the vertical recession phase ends to the time when impounding begins. Because the water depth on the surface is usually thin and is difficult to simulate accurately, we might use a shortcut to compute the horizontal recession phase without significant loss of accuracy. We simulate the process as follows (see Fig. 5): tH = Tv -~ ( tp - Tv ) X H

Lp

,

XH --
(69)

239 o

,p

ke

Fig. 8. Water surface profiles during impounding recession phase.

where tn is the time when tail water recedes at x = XH.

Impounding recessionphase This phase is completely caused by infiltration and marked by the moving tail point toward the downstream end, as shown in Fig. 8. First, we compute the time, te, at which there is no water on the surface anymore. From the continuity equation, we have: VR : V3 (Le, re)

(70)

Substituting (49) and (55) into (70):

t~=Ta+{TI+A_ ~ (I+A)(t~-Tv-Ta)KL~ x

KL~

VR~ (I+A)Ta

[T l+a - (Tv - T a ) I+A ]

(71)

because the net recession time is usually greater than, or equal to, total advance time, te=2Ta may be used as an initial value for computing te in (71). After getting the value of te, the impounding recession phase may simply be simulated by:

tr=tp-~ (te-tp)(xr-Lp) L~ - Lp

Lp<_xr<_Le, tp<_t~<_te

(72)

Thus, the entire closed border irrigation cycle can be simulated analytically. CONCLUDING REMARKS

Analytical simulation of the entire closed border irrigation process has rarely been reported in literature. The ensuing paper, Part II in this series, will demonstrate the accuracy of the proposed model by 15 experimental closed borders, with ease of application and short computer execution time. The following conclusions may be drawn from this study: The advance function, equation (19), can be adopted to any border slope from gentle to steep. The vertical recession part is valid only if the impounding point Lp is larger

240

t h a n Lv. T h i s m e a n s t h a t t h i s p a r t does n o t fit level b o r d e r s or b o r d e r s w i t h a v e r y s m a l l slope. C o m p u t a t i o n s s h o w t h a t if Lv, c o m p u t e d b y (32), is less t h a n h a l f of t h e b o r d e r l e n g t h Le, w h i c h is u s u a l l y t h e case, t h e a v e r a g e relative e r r o r (ARE) w o u l d be less t h a n 10% ( n o t i c e t h a t v e r t i c a l r e c e s s i o n t i m e is u s u a l l y very short). T h e a s s u m p t i o n s m a d e for (18) i m p l y t h a t t h e p r o p o s e d m o d e l will n o t fit for s u c h field soils t h a t h a v e large i n f i l t r a t i o n rates, for e x a m p l e sand, so t h a t n o r m a l flow d e p t h c a n n o t be f o r m e d .

REFERENCES Atchison, K.T., 1973. Retardance coefficients and other data for a vegetated irrigation border. M.S. thesis, University of Arizona, Tuscon, AZ, 58 pp. Bassett, D.L., 1972. A mathematical model of water advance in border irrigation. Trans. ASAE, 15: 992-995. Bassett, D.L. and McCool, D.K., 1973. A mathematical model of water advance and flow in small earth channels. Project completion report, Department of Agricultural Engineering, Washington State University, Pullman, WA. Bishop, A.A., Walker, W.R., Allen, N.L. and Poole, G.J., 1981. Furrow advance rates under surge flow systems. J. Irrig. Drain. Div. ASCE, 107(3): 257-264. Bowman, C.C., 1960. Manning's equation for shallow flow. In: ARS-SCS Workshop Hydraulics of Surface Irrigation, ARS 41-43, U.S. Department of Agriculture, Washington, DC, pp. 2123. Chen, C.L., 1965. Techniques of border irrigation by a hydraulic method of routing. Rep. P-WR111, Utah Water Research Laboratory, Utah State University, Logan, UT, pp. 15-16. Chow, V.T., 1959. Open-Chennel Hydraulics, McGraw Hill, New York, pp. 7-16. Christiansen, J.E., Bishop, A.A., Kiefer, F.W., Jr. and Fok, Y., 1966. Evaluation of intake rate constants as related to advance of water in surface irrigation. Trans. ASAE, 9: 671-674. Clemmens, A.J. and Strelkoff, T., 1979. Dimensionless advance for level basin irrigation. J. Irrig. Drain. Div. ASCE, 105: 259-273. Cowan, W.L., 1950. Estimating hydraulic roughness coefficient. Trans. Am. Geophys. Union, 31: 603-610. Davis, J.R., 1961. Estimating rate of advance for irrigation furrows. Trans. ASAE, 4: 52-54. Elliott, R.L., Walker, W.R. and Skogerboe, G.V., 1982. Zero-inertia modeling of furrow irrigation advance. J. Irrig. Drain. Div. ASCE, 108(3): 179-195. Fangmeier, D.D. and Strelkoff, T., 1979. Mathematical models and border irrigation design. Trans. ASAE, 22: 93-99. Fok, Y.S. and Bishop, A.A., 1965. Analysis of water advance in surface irrigation. J. Irrig. Drain. Div. ASCE, 91: 99-116. Kostiakov, A.N., 1932. On the dynamics of the coefficient of water percolation in soils and on the necessity of studying it from a dynamic point of view for purposes of amelioration. In: Trans. 6th Committee International Soc. Soil Sci., Part A, pp. 17-21 (in Russian). Michael, A.M. and Pandya, A.C., 1971. Hydraulic resistance relationship in irrigation borders. J. Agric. Eng. Res., 16: 72-80. Myers, L.E., 1959. Flow regimes in surface irrigation. Agric. Eng., 40: 11,676-677,682-683. Palmer, V.J., 1946. Retardance coefficients for low flow in channels lined with vegetation. Trans. Am. Geophys. Union, 27: 187-197. Ram, R.S., 1969. Hydraulics of recession flow in border irrigation system. M.S. thesis, Indian Institute of Technology, Kharagpur, India.

241 Ram, R.S., 1972. Comparison of infiltration measurement techniques. Indian Soc. Agric. Eng. J. Agric. Eng. 9(2): 67-75. Ram, R.S. and Lal, R., 1971. Recession flow in border irrigation. Indian Soc. Agric. Eng. J. Agric. Eng., 8(3): 62-70. Roth, R.L., 1971. Roughness during border irrigation. M.S. thesis, University of Arizona, Tucson, AZ, 78 pp. Sherman, B. and Singh, V.P., 1978. A kinematic model for surface irrigation. Water Resour. Res., 14: 357-363. Sherman, B. and Singh, V.P., 1982. A kinematic model for surface irrigation: an extension. Water Resour. Res., 18: 659-667. Shockley, D.G., Woodward, H.J. and Phelan, J.T., 1964. Quasi-rational method of border irrigation design. Trans. ASAE, 7: 420-423. Singh, P. and Chauhan, H.S., 1972. Shape factors in irrigation water advance equation. J. Irrig. Drain. Div. ASCE, 98 (3):443-458. Singh, V.P. and Ram, R.S., 1983. Some aspects of the hydraulics of border irrigation. Water Resour. Rep. 2, Department of Civil Engineering, Louisiana State University, Baton Rouge, LA, 81 pp. Strelkoff, T., 1977. Algebraic computation of flow in border irrigation. J. Irrig. Drain. Div. ASCE, 109 (IR3): 357-377. Strelkoff, T. and Clemmens, A.J., 1981. Dimensionless stream advance in sloping borders. J. Irrig. Drain. Div. ASCE, 107(IR4): 361-382. U.S. Soil Conservation Service, 1974. Border irrigation. In: National Engineering Handbook, Chapter 4, Section 15, USDA, SCS. Walker, W.R. and Humpherys, A.S., 1983. Kinematic-wave furrow irrigation model. J. Irrig. Drain. Div. ASCE, 109 (4): 377-392. Wu, I.P., 1972. Recession flow in surface irrigation. J. Irrig. Drain. Div. ASCE, 98(IR1 ): 77-90.