A quasi-steady state integral model for closed-end border irrigation

Agricultural Water Management, 11 (1986) 39--57

39

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

A QUASI-STEADY STATE INTEGRAL MODEL FOR CLOSED-END B O R D E R IRRIGATION

RAMA S. RAM l, VIJAY P. SINGH 2 and SHYAM N. PRASAD 3

'P.O. Box 47612, San Antonio, T X 78265 (U.S.A.) 2 Department o f Civil Engineering, Louisiana State University, Baton Rouge, L A 70803 (U.S.A.) 3Department o f Civil Engineering, University o f Mississippi, University, MS 386 77 (U.S.A.) (Accepted 23 October 1985)

ABSTRACT Ram, R.S., Singh, V.P. and Prasad, S.N., 1986. A quasi-steady state integral model for closed-end border irrigation. Agric. Water Manage., 11: 39--57. A quasi-steady state integral (QSSI) model was developed for irrigation on closed-end (CE) borders. A semi-analytical method was used for solving the governing equations. The model results compared favorably with experimental data from 18 experimental CE borders. The absolute average percent deviation (APD) between calculated and observed advance times varied between 5.3 and 28.5. The APD for recession times varied between 1.0 and 37.3. The calculated advance times were found to be consistently smaller than observed values for these borders. For constant infiltration parameters, the border bed roughness was found to be the single most important parameter affecting model results.

INTRODUCTION

The dike at the downstream end of a closed-end iCE) border alters the hydraulics of flow. A survey of literature (Jensen, 1982; Ram et al., 1983) indicates that most of the research on hydraulics of border irrigation pertains to freely draining (FD) borders and that hydraulics of CE border irrigation and its design have received considerably less attention. For example, the kinematic wave models (Chert, 1966, 1970; Smith, 1972; Cunge and Woolhiser, 1975; Sherman and Singh, 1978, 1982; B.J. Chen et al., 1981; Singh and Ram, 1983) all have been developed for FD borders. Likewise, the zeroinertia (ZI) models by Strelkoff and Katopodes {1977) are largely confined to FD borders or channels. Strelkoff (1977) dealt with the recession phase on a CE border using mass conservation. Clemmens (1979) extended the ZI model of Strelkoff and Katopodes (1977) to CE border irrigation. It is, however, not clear what

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© 1986 Elsevier Science Publishers B.V.

40 modifications were applied. Many procedures of designing such borders neglect the recession phase (Murty, 1969, 1970; Chauhan and Singh, 1974; U.S. Soil Conservation Service, 1974; Ram et al., 1976; Tyagi, 1976; Murty and Agarwal, 1979). A mathematical model for CE borders simulating the entire irrigation cycle does not appear to have been developed. Furthermore, most models, even for FD border irrigation, have been based on the St. Venant equations of continuity and momentum. The objective of this study is to develop a mathematical model for CE border irrigation employing a quasi-steady state formulation of the St. Venant equations. This formulation allows reduction of the integral equations to ordinary differential equations which are then solved partly analytically and partly numerically. This method of solution is accurate, efficient and easily amenable to computer programming, and may be superior to the one presented by Strelkoff and Katopodes (1977) for FD borders. The proposed quasi-steady state integral (QSSI) model is verified on 18 experimental borders. A sensitivity analysis is performed to find out what parameters are important. FLOW IN A CE BORDER Consider a wide rectangular border having a small slope So and length L, with x = 0 at the head. The bed has some initial moisture content and a uniform roughness C. At time t = 0, an inflow q(O,t), which lasts for a specified time, is introduced at x = O. The water enters the border with velocity v(0,t) and depth of flow h(O,t). Part of q(O,t) flows in the border as surface flow and a part infiltrates into the soil. The surface flow undergoes various phases which are advance, storage, vertical recession and horizontal recession. The solution domain for these phases, as shown in Fig. 1, can be partitioned into four separate domains: the advance phase DI, the storage t=T3 --

"--/t,r(x).~..~

/

I= Tr

~

I

I-T 2 --

D2

t,O ~ [ • =0

0,

I- . . . .

TIME I OF OPPORTUNITY I

DI

J

I -~---

1

HORIZONTAL

I RECESSION I HASE

I I

D3

l= TI

~

J [ I

ERTICAL RECESSION F HASE

--1--~---

STORAGE PHASE

~

I i=----- L, I=L-Lr

ADVANCE PHASE

x=L

Fig. 1. Solution domain for f l o w in a closed-end border.

41

phase D2, the vertical recession phase D3, and the horizontal recession phase D4. The domain D I is for 0 < t < T, where T denotes the advance time. Let w (x) denotes the time history of the advance front; its inverse x = s (t) specifies the location of the advance front. The domain D~ is b o u n d e d by T < t < T~, where T~ is the duration of q(0,t). The domain D3 is for T~ < t < T2, where T2 marks the end of the vertical recession and the beginning of the horizontal recession. The domain D4 is b o u n d e d by T2 < t < r(x), where r(x) gives the time history of the drying front, as it traverses from x = 0 to x = L, representing the interface between the part of the border with h (x,t) = 0 and the part of the border with h(x,t) >10. Its inverse x = y(t) specifies the location of the drying front. From Fig. 1, it is clear that the horizontal recession is different from that of an FD border. Due to the obstruction at the downstream end, the recession curve undergoes an abrupt change of direction at time t = Tr which marks the beginning of the recession of impounded water. The recession curve t = r(x) continues to progress until the impounded water recedes at x = L. Many workers (Ram and Singh, 1982a, b) prefer to partition the horizontal recession phase into: (a) horizontal recession for T2 < t ~< Tr; and (b) recession of impounded water for Tr < t < T3. We define the time To = t - w ( x ) to denote the infiltration opportunity time at a point x in the border, that is, the time that water has covered the point x. The infiltration rate is assumed to depend only on the difference between the total elapsed time and the advance time. A QUASI-STEADY STATE INTEGRAL (QSSI) MODEL

To derive the governing equations in integral form, we consider a fluid element between two cross-sections at xl and x2 as shown in Fig. 2. Let v(x,t) be the mean velocity at any point during the time increment dr, and d(x,t) the depth of water infiltrated into the soil. Following Ram et al. (1983), the continuity equation can be written as: d

x2

-Z :

(h(x,t)+d(x,t))

dx+h(x2,t)

v(x2,t)-h(xl,t)

v(xt,t) = 0

(1)

vx 1

The m o m e n t u m equation, assuming quasi-steady state conditions, can be written as: 2

X2

C v~(x,t) dx = f Xl

gSo h(x,t) d x - ((h(x2,t) v2(x2,t) X I

-

h(xl,t) v2(x~,t)) - ~I g(h2(x2,t) - h2(x~,t)))

(2)

where C is border bed roughness, g acceleration due to gravity, and So bed slope.

42

h(x,t)

z=C

8order ned

x

SO So~=Sino * a

....--

°<',." L

I

I"

x=x °z

Fig. 2. Definition sketch for a representative element of spatially varied unsteady flow on a small border. We have y e t to formulate an equation for the free boundary x = s ( t ) or t = w(x), where w ( x ) denotes the time history of the advance front; its inverse x = s (t) specifies the location of the advance front. When v (s (t), t) is the velocity at the tip o f the advance front, then: ds(t) dt

(3)

• = v(s(t),t)

The initial and b o u n d a r y conditions to be satisfied can be specified as: w(0) = 0;

O<.x=s(t)<~L,

h ( 0 , t ) = h0;

0 < t < 7":

= 0;

h(s(t),t)

h ( O , t ) = 0; s(T) h(L,t)

(5) (6)

O <. s ( t ) < L

t ~> T2

(7) (8)

= L = hi(t),

r(0) = T2, h(L,t)

O < t <. T,

(4)

0<~ t<~ T

= 0,

(9)

T < t < T3 O <. x l = s ( t ) < L ,

T: < t < T3

xz = L,

t >>- T:

(10) (11)

where h~ is the depth of water at the downstream end of the border, T the time when water reaches the downstream end, T2 the time when vertical recession ends, and T3 the time when horizontal recession ends. Equations (1) to (3), and the b o u n d a r y conditions in equations (4) to (11), constitute the QSSI model for CE border irrigation.

43 MATHEMATICAL

SOLUTIONS

Various solutions of differential equations for border irrigation problems have been attempted in the past by means of numerical techniques. However, none of them appear to be based on the various conservation integrals which are valid for shallow water flow as applied in border irrigation. The usefulness of the conservation integrals has been widely recognized in studying the phenomenon of nonlinear water waves (Whitham, 1974). Equations (1) and (2) are the first two integrals applicable to flows over porous beds. They have the same measures as the moment integrals obtained when utilizing the method of weighted residuals (Ames, 1965) in solving partial differential equations numerically. In fact, our solution stems from combining these two concepts. Therefore the following series solutions are proposed for all domains:

h(x,t) = ~ Ab(t) Hb(x) b=O

(12)

ao

v(x,t) = ~ Bb(t) b=O

Vb(x)

(13)

where A b and B b are unknown functions of time which may be determined from a system of ordinary differential equations by the method of weighted residuals (Ames, 1965), and Hb and Vb constitute complete sets. The choice of Hb(X) and Vb(X) depends upon regularities of the expected solutions and the depth-velocity relationship given by:

qo = hovo

(14)

where qo = q(O,t) and Vo = v(O,t). In the past, many investigators have studied the criteria for the selection of the sets of functions to be used in the method of weighted residuals. In most cases, a set of eigenfunctions are obtained which are simple, explicit and consist of polynomials, Fourier or Bessel series. For nonlinear problems, there are no simple guidelines for the selection of the functions Hb(X) and Vb(X). The situation, however, is not as critical as it appears, since many investigators have studied in the past several variants of the same problem analytically. For example, the Lewis-Milne model (Lewis and Milne, 1938) assumes the water depth h to be constant, whereas in the dam break problem friction, infiltration and the bed slope are assumed to be zero. These two problems provide the two extremes of our model and, therefore, our choice of the functions Hb and Vb are guided by the analytical features which these models possess. In fact, conservation integrals similar to equations (1) and (2) were utilized by Singh and Prasad (1983) to obtain the mean depth in the Lewis-Milne equation.

44

The solution procedure can be summarized as follows. Since there are four distinct solution domains Dk with transition times so small that they give rise to finite discontinuities. The free surface and the velocity distributions are quite distinct but their individual variations are rather smooth in all the domains. Depending on the degrees of freedom and due to changes in the boundary conditions during transition, unknown time dependent parameters are introduced which either adjust the depth of flow or the flow velocity at the ends. Due to complex nonlinearity in the governing equations, only three terms in each case for h and v can be utilized. The continuity equation and the quasi-steady state approximation of the momentum equation, given by equations (1) and {2), are integrated using equations (12) and (13) respectively for the depth and velocity distributions. The two equations obtained after integration are solved numerically using appropriate initial and boundary conditions expressed by equations (4) and (11). In this manner solutions in the various domains are obtained.

Domain D, For convenience a constant inflow rate q (0, t) = q0 at x = 0 was assumed. The depth and velocity distributions between x = 0 and x =s(t) were selected following the procedure outlined above:

h(x,t) = ho cos~'gx~ \2s ]

(15)

v(x,~) = ~ l - c o s

~-~ + v0 l - - s

(16)

in which s represents s(t), the free boundary. Equations (15) and (16) satisfy the initial and boundary conditions given by equations (3) to (6) and (14), and represent a solution of the linearized version of the St. Venant system of shallow water wave equations. It is noted that this selection of h (x, t) and v(x,t) is not unique but was found to be appropriate. Substituting equations (15) and (16) in equation (1) and changing the limits of integration from (x, to x2) to (0 to s): d

dt

o

h0cos(~X~ dx ~2s ]

qo

(d(x, To)) dx

(17)

"~o

Here To is time of opportunity. Integrating equation (17) further within the limits 0 to t: 2sho

t t r qo dt = - ~

d

{d(x, To)) dx dT0

(18)

45 Rearranging equation (18) gives:

ho = ~s

qo t -

~

(d(x, To)) dx dT0

(19)

The term under the integral sign is the total infiltrated volume of water over the advance distance s. Substituting equations (15) and (16) in equation (2) and using the limits of integration from 0 to s:

)

cos( ))at+

0

$

= gSo f ho cos(

\2s 1

0

d x + ~ I gq___~ v~2o + qoVo

(20)

Integrating equation (20) and rearranging yields:

i gd -

-

+

-

~

) +

qoVo

2 V:o

(21)

Since ho and v0 are uniquely related by equation (14), equations (19) and (21) can be solved for ho and s as follows: (1) We calculate the critical depth of flow h ( 0 , t ) for the inflow qo and then the critical velocity v (0, t) as: h(O,t) = " g "

(22a)

v(0,t) = (gho) ' n

(22b)

(2) We assume a grid spacing Ax in the x-direction, and let s ( t l ) = A x for some small time t,. We further assume that the velocity v(0,t) is critical for the period t, and is expressed as v(O, tl). We then calculate the time t, as: t~ =

s(tO - Ax v(0,t0

(23)

(3) The cumulative infiltration is computed by the Kostyakov equation: (24)

d = KT~

where K and a axe infiltration constants and To is replaced by t,. (4) We recalculate the time t~ as: tl =

Sh ho(t~) + Sz d(0,t~)

qo

s(tl)

(25)

46

where Sh and Sz are profile shape factors and d(O,t~) is the cumulative infiltration at time t~. We then use smaller of the t w o values of t~. (5) We use a value of Sh = 0.7 in equation {25) for constant s ( t l ) = A x , and calculate Sz (Katopodes and Strelkoff, 1977) as: 1

(26)

Sz = ~

l+a

in which a is the same as in equation (24). (6) We use a border bed roughness C as specified by: g C = --

(27a)

with V 1/6 (27b)

Ch = nm

where G is the normal depth of flow at the upstream end, nm Manning's roughness coefficient, Ch Chezy's roughness coefficient and g acceleration due to gravity. (7) We calculate h0 from equation (19) and use it to calculate Vo from equation {14). (8) Using a 4-th order Runge-Kutta method, we calculate the time t from equation {21) for a specified As (= Ax). (9) We iterate steps (7) and (8) until the advancing front reaches the downstream end of the border. D o m a i n D2

When water reaches the downstream end of the border, the velocity of the advance front goes to zero and the depth of water starts to rise at the downstream end. The initial and b o u n d a r y conditions at time T when water reaches the end of the border can be specified as: h(0,T) = ho(T) h(L,T) = hi(T)

(28) = 0

(29)

To satisfy these conditions, the depth and velocity distributions between x = 0 and x = L are selected following the general procedure: h = ho cos

(x)

v -- v0 1 - ~

+

L

(30)

(31)

47

where vo is the velocity at the upstream end of the border at any time t and h l the depth of flow at the downstream end of the border. Substituting equations (30)--(31) in equation (1) and using the limits of integration from 0toL: d dt

i[ 0

+ --~-j dx - qo = -

h0 cos

-~o (d(x, To)) dx

(32)

Integrating equation (32) further between the limits T and t and substituting the conditions given by equations (28)--(29): h, = 4h0(T) + --

(d(x, To)) dx dT0

0(t-T)-

(33)

Substituting equations (30)--(31) in equation (2) and using the limits of integration as 0 to L: c

vo

-

ax

= gSo

- g

2

o cosi'X

\2LI

(h~ - h2o)

+

+

ax

hov~o

(34)

Integration of equation (34), substitution of conditions given by equation (28) and (29) and rearranging give: h : + [SoL( 4 ho(T___~}+

L /-

~- h°+ - - + g

~

~

+ 7r

+ [ ( 4 h 0 ( T-~) _ + 2VSt2L ! --32(~)]

L

/J

= 0

ho

(35)

where Vs is volume of water in surface storage expressed as: t

Vs = q ° ( t - T ) - ;

L

d

fo d--To (d(x, To)) dxo dT0

(36)

Equations (33), (35) and (36) are solved to obtain h0 and h,. This specifies the depth distribution in domain Dz. For known h0, the velocity Vo is calculated using equation (14). The velocitydistribution is obtained from equation (31). The steps involvedin solvingequations (33), (35) and (36) are: (1) We choose a small time increment At and compute ITs using equation (36). (2) We compute h0 from equation (35) by using the standard NewtonRaphson scheme (Burden et al., 1978).

48 (3) Using the value of h0 as obtained in step (2) we c o m p u t e h~ from equation (33). (4) The computations are continued for the duration of irrigation t = T~.

Domain D3 The initial and b o u n d a r y conditions for the domain Da can be written as:

h(O,T,) = ho(T~)

(37)

h(L, T1) = h~(T~)

(38)

The depth of water in this domain is also given b y equation {30). The velocity distribution can be expressed as: v = v0 1 -

;

x>0

(39)

The depth at the section x = x~ at time t~ ~< t ~< T: can be expressed as:

h(xl,t) = ho cos

(-~Z)

hlxl

+ --

(40)

L

The depth at xl corresponding to T~ can also be determined from:

h(x,,t) = x,So

(41)

In the following it is assumed that the section x, is represented by x* when t = T,. The total volume of water Vw to be infiltrated during the vertical recession phase can be expressed as:

- ~xl*

h0cos

+

L

(42)

Simplifying equation (42):

Vw(r,) - 2Lho sin(.x*,

(43)

Similarly at any time t, Vw can be expressed as:

2Lho Yw(t~)

=

s i n \ ~ - ~ ] - ~ hoxl c o s ( ~ - )

(44)

7f

The total infiltrated VQ is expressed as: L

t

VQ = fo ~, ~

d

(d(x, To)) dTo dx

(45)

49

Inserting equations (43)--(45) in equation (1):

lrrx, \

2Lho

sinl~/ \ 2L ]

_

1

~ hox~ c o s

7tXl

2L

=

Vw(T,)

-

VQ

(46)

We assume that the total infiltrated volume at any time during the vertical recession phase can be expressed as:

VQ = q o ( t - T,)

(47)

The velocity distribution for x > 0 is expressed by equation {39). It is not valid at x = 0. From equations (40)- (41): h0 cos

(~L) + hlxi = XlSo

(48)

L

We assume that velocity v0 remains constant and can be expressed as: Vo -

q0 G

(49)

Where G is the normal depth of flow at the upstream end of the border. Substituting equations (30) and (39) in equation (2) and using the limits of integration as 0 to L:

c/o

<'-"

So _g 2

+T}

(h~ - h2o) - hov2o

(50)

g (h ~, - h~o) - hov~o

(51)

Solving equation (50):

<:,.,,< 3- = gSoLr2,_,,,o 7r + ,,s,.,} Rearranging terms:

h~ = 2S0 [ 2Lho + h l L 1 + h 2o - 2 [ hove+ CLv:ol ~- ~ 2J 7 3 J

(52)

From equations (46) and (52) we can compute ho and hi in the following steps: (1) From known initial conditions given by equations (37) and (38) we calculate x, = x* at time t = T, using equation (48) and the Newton-Raphson method. (2) We calculate Vw(T1) from equation (43). (3) We assume a small time increment A t and calculate VQ from equation (45). (4) We calculate ho from equation (46).

50 (5) We further calculate Vo using equation (49) and assume it to be constant throughout time. (6) We substitute Vo and ho in equation (52) and calculate h~. (7) Using h0 and hi as in steps (4)--(6), we calculate xl from equation (48). (8) We repeat steps (3)--(7) until ho = 0. (9) Thus, depth and velocity distributions for known values of ho, h~ and Vo are specified by equations (30) and {39). D o m a i n D4

The initial and boundary conditions for this domain can be expressed as: h(O, T2) = 0

(53)

h ( L , T2) = h , ( T 2 )

(54)

dy

v(y(t),t)

= --

(55)

h(y(t),t)

= 0

(56)

dt

where y ( t ) is the location of the free boundary t = r(x) during the horizontal recession phase and T2 the time at the end of the vertical recession. The depth and velocity distributions were selected following the general procedure as:

h

-

hi L-y -

-

(x

dy dt v

=

- -

y -L

x

+

-

y)

(57)

dy

dy dt

dt

y -L

(58)

Substituting equation (57) in equation (1) and integrating with changed limits of y to L: d ) h, d-~ y ~

(x-y)

dx = -

fyL d ~

(59)

( d ( x , To)) d x

Integrating equation (59) from T2 to t, and substituting the initial conditions given by equation (54): h, - L---~y L.

2

~

-~o

( d ( x , To)) d x

dTo

(60)

51

The term (L hi(T2))~2 represents the initial volume of water in surface storage at the end of the vertical recession at t = T2. Replacing (L h, (T2))/2 by Vs(T2) equation (60) can be rewritten as: h,-

Vs(T2)-

L-y

(d(x, To)) dx dro

(61)

2

The term under the integral sign in equation (61) is the total volume of infiltration. Now we substitute equations (57)-(58) in equation (2) and integrate from y to L:

dy

C J y L ~ - - £ x+ dt

y : L

dx

= gSo

L

h,

(x-y)

f L-----y-

dx

gh~'

2

(62)

Simplifying equation (62) and rearranging the terms:

1[

=-~ gSoh, " - ~

]

- ?gh*,

(63)

The term h , ( ( L - y ) / 2 ) in equation (63} is the volume of water in surface storage at any time t = T2. This can be written as: Vs(t) = Vs(T2)- YQ

(64)

In equation (64), VQ can be written as: t L

VQ = fT2 fy ~

d

(d(x, To)) dx aT0

(65)

Substituting equation (64) in equation (61) we get:

"~ /

y

L - y +

-3"(-'L : y) 1

C [gSo(Vs(T2) - VQ) - ~ gh2ol

(66)

52 where Vs(T2) is the total volume of water in surface storage at the end of vertical recession, t = T2, that can be expressed as: Vs(T2) -

2L ho(T1)

+

hi (T1) L 2

Vw(T~)

(67)

We use equations (61) and (66) to solve for h~ and y as follows: (1) We calculate Vs(T2) using equations (43) and (67) in which h0 and h~ are specified by equations {37)--(38). (2) We take a small grid spacing AX in the x-direction, and solve equation (66) by the 4-th order Runge-Kutta method for time t using previous values of h~ and assuming VQ = 0. (3) Using calculated time from equation (66) for specified Ax, we calculate hi using equation (61). (4) We calculate VQ from equation {65). (5) We use h~ and VQ as in steps (3)--(4) and calculate time t for another Ax from equation (66). (6) We repeat steps (3)--(5) until all the water is drained from the field. EXPERIMENTAL DATA Eighteen sets of data, designated as R-1 through R-18, were used to verify the QSSI model. These data are due to Ram and Lal (1969, 1971) and Ram {1969, 1972). Half of the data sets (R-1 to R-9) are from non-vegetated borders, and half (R-10 to R-18) from vegetated {wheat crop) borders. These borders are 100 m long and 6 m wide, and have rails on each side of the border for precise leveling. The inflow was measured by a 90 ° V-notch weir before flowing into a distribution channel installed at the upstream end of the border 1 m up the first station. This ensured uniform entry of water at the upstream end of each border. The water depth was measured by point gauges at each station at every 20 m. PARAMETER ESTIMATION The QSSI model, expressed by equations (1) to (3), contains two unknown infiltration parameters K and a of the Kostyakov equation and one unknown bed roughness parameter C. Ram (1969) estimated K and a for these borders by a volume balance method. Ram et al. (1983) estimated C from nm by equation (27) by assuming it to remain constant during the irrigation cycle. MODEL VERIFICATION The solutions in the various domains of the irrigation cycle were obtained using all 18 sets of data. These included calculated values of advance time,

53

water surface profile and recession time against distances along the length of the border. For comparison of calculated and observed values of advance and recession times, absolute percent deviation PD* and absolute average percent deviation APD** were used. PD

= I(observed quantity - computed quantity)/observed quantity l N

APD = ~

PDi/N (N, number of observed values)

i=l

The calculations were based on the observed advance times taken every 20 m. Advance

The calculated times compared well with the observed times but were consistently lower. The PD and APD between calculated and observed advance times ranged respectively between 0.0 and 45.2, and 5.3 and 27.8, although in a majority of cases the PD and APD were below 20. High PD values were in either early or the late stages of advance for almost all cases of the data sets. Calculated and observed advance times are plotted in Fig. 3 for sample data sets R-13 to R-15. MODEL: QSSI OBSERVED DATA o •

200

- -

- - - -

DATA SET R - 1 3 DATA SET R-14 DATA SET R - 15

CALCULATED CURVE ADVANCE RECESSION R-14

j#R-,5 160

/I/ / /

.C t 2 0 E

~" *~-

1 /

/

v-

/R-13

/

/

/

/

I"

/

A

/

~/,

o

/R-15

80

40

20

40 60 DISTANCE, m

80

I00

Fig. 3. Advance and recession curves for the data sets R-13, R-14 and R-15.

54

Water surface profile Observed and calculated water surface profiles are plotted in Fig. 4 for the sample data sets R-13 to R-15 corresponding to the time when the water reached the downstream end of the border. The c o m p u t e d profiles were lower than observed profiles all along the border. It may be possible to improve the c o m p u t e d profiles b y expressing the depth distribution to more closely agree with the field data. However, the model results as evident from the calculated advance times were acceptable for practical purposes. MODEL: OSSI OBSERVED DATA ~, DATA SET R" 13, TIME = SO.Omln. o DATA SET R-14, TIME = 60.Omm • DATA SET R-IS, T~ME= 96.0rain - - CACULATED PROFILES 0.05 R-13

~,

E. 0 0 3 I I-n ,,, 0.02 Q

"

~

~ •

~

= •

0 l0 I

0

20

t I 40 60 DISTANCE, m

80

I00

Fig. 4. Water surface profiles for the data sets R-13, R-14 and R-15.

Recession Calculated and observed recession times are plotted in Fig. 3 for the sample data sets R-13 to R-15. The PD and APD b e t w e e n calculated and observed recession times ranged respectively between 0.0 and 79.5 (except PD = 136 for the data set R-1 at x = 80 m) and 1.0 and 37.3. In the beginning of the border calculated recession times agreed quite closely with observed recession times. Toward the downstream end the agreement became poor. SENSITIVITY A N A L Y S I S

The sensitivity of advance and recession times c o m p u t e d by the QSSI model was evaluated with respect to the parameters K, a and C, and grid spacing Ax. The data set R-13 was used. C was varied as 0.01, 0.1, 0.3, 0.6, 1.1 and 1.5 for different values of Ax as 0.5, 1.0, 2.0, 5.0 and 10.0 m. K was chosen as 0.001, 0.004 and 0.007 whereas a as 0.3, 0.7 and 0.8.

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Effect of bed roughness C By taking K and a as constants, Ax = 2.0 m, and varying C, advance and recession times were c o m p u t e d by the QSSI model. The advance time decreased as C increased. The recession time also decreased with increasing C but only up to a certain value b e y o n d which it became almost independent of C. For C = 1.1 and Ax = 2.0 m calculated and observed advance times were in close agreement. However, calculated recession times exceeded observations whereas the reverse was true in case of advance. This indicated that roughness during recession was less than that during advance. This agreed with the results of kinematic-wave solutions (Ram et al., 1983).

Effect of grid spacing Ax By varying the grid spacing and keeping other parameters constant, advance and recession times were c o m p u t e d by the QSSI model. The advance time was almost independent of the grid spacing for the range of values used. There was a small change in recession with a significant change in Ax. It increased as Ax decreased from 1.0 m to 2.0 m but increased when Ax increased from 2.0 m to 5.0 m. The effect of varying C and Ax on APD between c o m p u t e d and observed advance as well as recession times was evaluated. The APD was minimum (9.48) for advance of the data set R-13 if C was 1.50 and Ax was 0.5 m. It was minimum (11.96) for recession if Ax = 1.0 and C = 1.5.

Effect of infiltration parameters K and a The effect of changes in K and a on advance and recession times were also evaluated. An increase of K from 0.001 to 0.007 increased advance time from 24.9 to 74.5 min. The actual value of K was 0.0039. It was f o u n d that for calculated advance and recession times to match with observations, K should be more than 0.001 during advance and less than 0.004 during recession. This was consistent because infiltration rate in the recession phase was less than that in the advance phase. Similar results were obtained by varying a between 0.3 and 0.8. The actual value of a was 0.674. Calculated advance times were closer to observations for a ~ 0.8 but recession times were for a < 0.7. This implies lower infiltration rate during the recession phase. These results indicate that advance times are very sensitive to changes in K and a whereas recession times are but to a considerably less extent. CONCLUSIONS

The following conclusions can be drawn from this study. (1) The QSSI model predicted the advance time satisfactorily for the data analyzed here. In most cases the prediction error remained below 20% for non-vegetated borders and below 30% for vegetated borders.

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(2) The model predicted vertical as well as horizontal recession reasonably well. The prediction error was small in the beginning stages of recession in almost all cases but grew with the progress of recession in some cases. (3) The agreement between observed and computed water surface profiles was satisfactory. Since the depth of flow was normally very small, a small error in prediction appeared large. (4) For the data analyzed here the model was adequate for modeling the entire irrigation cycle. ACKNOWLEDGEMENT

This study was supported in part by funds provided by the National Science Foundation under the project, Free Boundary Problems in Water Resource Engineering, NSF-ENG-79-23345. REFERENCES Ames, W.F., 1965. Nonlinear Partial Differential Equations in Engineering, Vol. I. Academic Press, New York, NY. Burden, R.L., Faiers, J.D. and Reynolds, A.C., 1978. Numerical Analysis. Prindle, Weber and Schmidt, Boston, MA. Chauhan, H.S. and Singh, P., 1974. Handbook of Agricultural Engineering: Surface Irrigation -- I, Surface Irrigation Design. Department of Agricultural Engineering, College of Technology, G.B. Pant University of Agriculture and Technology, Pantnagar, India. Chen, B.J., McCann, R.C. and Singh, V.P., 1981. Numerical solutions to the kinematic model of surface irrigation. Tech. Rep. MSSU-EIRS-CE-81-1, Engineering and Industrial Research Station, Mississippi State University, Mississippi State, MS. Chen, C.L., 1966. Mathematical hydraulics of surface irrigation. Tech. Rep. PR-WR11-2, Utah Water Research Laboratory, Utah State University, Logan, UT, 98 pp. Chen, C.L., 1970. Surface irrigation using kinematic-wave method. J. Irrig. Drain. Div. Proc. ASCE, 96(IR1): 39--48. Clemmens, A.J., 1979. Verification of zero-inertia model for border irrigation. Trans. ASAE, 22 (6): 1306--1309. Cunge, J.A. and Woolhiser, D.A., 1975. Irrigation systems. Chapter 13 in: K. Mahmood and V. Yevjevich (Editors), Unsteady Flow in Open Channels. Water Resources Publications, Fort Collins, CO, pp. 552-537. Jensen, M.E. (Editor), 1982. Design and Operation of Irrigation Systems. American Society of Agricultural Engineers, St. Joseph, MI, 829 pp. Lewis, M.R. and Milne, W.E., 1938. Analysis of border irrigation. Agric. Eng., 19: 267-272. Murty, V.V.N., 1969. Hydraulic design for check method of irrigation. J. Agric. Eng. Res., 14: 319--332. Murty, V.V.N., 1970. Percolation loss in check system of irrigation. J. Agric. Eng. Res., 15: 375--378. Murty, V.V.N. and Agarwal, M.C., 1979. A rational approach to the design of check systems of irrigation. J. Agric. Eng. Res., 15: 163--170. Ram, R.S., 1969. Hydraulics of recession flow in border irrigation system. Unpublished M. Tech. Thesis, Indian Institute of Technology, Kharagpur, India. Ram, R.S., 1972. Comparison of infiltration measurement techniques. J. Agric. Eng. India, 9: 67--75.

57 Ram, R.S. and Lal, R., 1968. Recession of impounded water in an irrigated border with closed downstream end. Harvester, 2: 206--210. Ram, R.S. and Lal, R., 1971. Recession flow in border irrigation. J. Agric. Eng. Indian Soc. Agric. Eng., 8: 62--70. Ram, R.S. and Singh, V.P., 1982a. A design procedure for closed and irrigation borders. Agric. Water Manage., 5: 1--14. Ram, R.S. and Singh, V.P., 1982b. Evaluation of models of border irrigation recession. J. Agric. Eng. Res., 27: 235--252. Ram, R.S., Jain, T.C. and Jain, K.C., 1976. Evaluation of check basin method of irrigation for wheat crop in light soils. Indian J. Agric. Sci., 46: 67--75. Ram, R.S., Singh, V.P. and Prasad, S.N., 1983. Mathematical modeling of border irrigation. Water Resour. Rep. 5, Department of Civil Engineering, Louisiana State University, Baton Rouge, LA, 302 pp. Sherman, B. and Singh, V.P., 1978. A kinematic model for surface irrigation. Water Resour. Res., 14: 357--364. Sherman, B. and Singh, V.P., 1982. A kinematic model for surface irrigation: an extension. Water Resour. Res., 18: 65~ ~,67. Singh, V.P. and Prasad, S.N., 1983. Derivation of mean depth in the Lewis-Milne equation for border irrigation. Proc. ASCE Specialty Conf. Advances in Irrigation, 20--22 July 1983, Jackson Hole, WY. American Society of Civil Engineers, New York, NY, pp. 242--249. Singh, V.P. and Ram, R.S., 1983. A kinematic model for surface irrigation: verification by experimental data. Water Resour. Res., 19: 1599--1642. Smith, R.E., 1972. Border irrigation advance and ephemeral flood waves. J. Irrig. Drain. Div. Proc. ASCE, 98(IR2): 289--307. Strelkoff, T., 1977. End depth under zero-inertia conditions. J. Hydraul. Div. Proc. ASCE, 103(HY7): 699--711. Strelkoff, T. and Katopodes, N.D., 1977. Border irrigation hydraulics with zero-inertia. J. Irrig. Drain. Div. Proc. ASCE, 103(IR3): 325--342. Tyagi, N.K., 1976. An empirical approach to design of check irrigation system. Irrig. Power J., January: 85--89. U.S. Soil Conservation Service, 1974. Border irrigation. Chapter 4 in: SCS National Engineering Handbook, Section 15. U.S. Department of Agriculture, Washington, DC. Whitham, G.B., 1974. Linear and Nonlinear Waves. Wiley, New York, NY, 636 pp.