Integral equation solutions to surface irrigation

J. agric. Engng Res. (1989) 42,251-265

Integral Equation

Solutions to Surface Irrigation J. MOHAN REDDY*

The existing equations for computation of flow in border irrigation were modified for application to furrow irrigation, and common equations for computation of flow in surface irrigation (borders and furrows) with slope greater than 0.1 percent were derived. The solutions were discussed in terms of integral equations, and the performance of the model in estimating advance and recession was compared with field data and found to be acceptable.

1. Introduction

Mathematical modelling is a useful tool in the design and operation of surface irrigation systems. Several methods-hydrodynamic, zero-inertia, kinematic-wave and volumebalance-are available to compute flow in surface irrigation systems; however, the volume-balance method is the most economical of all and is reasonably accurate when used under appropriate conditions; hence, it can be used in the modelling of flow in surface irrigation systems. During the last three decades several volume-balance equations have been proposed to compute advance’-’ and recessiona” in border irrigation, and advance in furrow irrigation.“-‘4 The solution techniques ranged from simple recursive estimations to series expansions. The solutions presented by Strelkoff’ are comprehensive, and cover the advance and recession (freely draining and ponded cases) phases of border irrigation. The objective of this paper is to modify these equations and present a single set of equations for computation of advance and recession in both borders and furrows, and discuss the solutions in terms of integral equations. An integral equation is one in which the unknown function appears under the integral sign, as shown below: $(t) = Y(f) + E ’k(t, z)@(z) dz I0

(1)

in which $ is the unknown function, Y is a known function, E is a complex or real parameter, k(t, z) is called the kernel function which depends on the current variable and the variable of integration, and t is the variable upper limit of integration. Eqn (1) is a linear integral equation. The region of integration in this case coincides with the domain of dependence and therefore, does not have a fixed upper limit. This is analogous to the case of linear initial value hyperbolic partial differential equation (propagation type) in which the domain of dependence is part of the solution. The solution at any time t depends on the values of x(z) on 0 c z 6 t and not on values where t > r. A nonlinear integral equation is given by Q(t) = Y(t) +

E

’G[t,

I0

q(t)]

dt

(2)

G is a nonlinear function. For a general classification of integral equations refer to Golberg.15

in which

* Departmentof Agricultural Engineering, University of Wyoming, Laramie, WY 82071, USA Received 5 January 1988; accepted in revised form 18 December

1988

251 C021-8634/89/040251+ 15 $03.00/O

0 1989 The British Society for Research in Agricultural Engineering

252

INTEGRAL

EQUATION

SOLUTIONS

TO

IRRIGATION

Notation A a

area of surface flow, m2 exponent of infiltration function b, a’ limits of integration C a constant c basic or steady state infiltration rate, m3 m-l SC’ G a nonlinear function I ;F;ag;i:filtration rate,

Jd k

infiltration rate, m3 s-l m-l approximate value of the integral function Infiltration coefficient, 3

-1 S-a

L Enzh of field, m 1 inundated portion of field

length, m a constant m an exponent n Manning’s roughness parameter 4 inflow rate per unit width or individual furrows, m2 s-l or m3 s-’ qR rate of runoff from the downstream end, m* s-l or m3 s-’ R hydraulic radius, m SJ slope of the field SY rate of change of depth with distance term representing the effect of subsurface flow on advance rate s variable of integration t time, s L3 time of inflow cutoff, s fL time of advance to end of field, s tR time of recession at upstream end of field, s volume of water applied to the field, m3 volume of runoff, m3 M

K

volume of runoff before start of recession, m3 WCO) volume of surface storage at time of cutoff, m3 volume of surface storage at !&RI beginning of recession, m3 volume of subsurface stov,ctR) rage, m3 wi weighting factor j x advance distance, m Y depth of flow, m Yd depth of flow at downstream end of field, m YU depth of flow at upstream end of field, m3/m depth of infiltration, m3/m depth of infiltration at the upstream end of field, m 6 variable of integration in the Gaussian Quadrature technique an exponent exponent and coefficient, respectively, in defining the relationship between depth and area rl surface shape factor degree of parabola gamma function complex or real parameter E in Volterra integral unknown function in Volterra integral v known function in Volterra integral variable of integration a constant in Philip’s solution for advance subsurface shape factor constant and exponent, respectively, in defining the relationship between area and hydraulic radius Quadrature point j an exponent

P,::

ry

I

J.

M.

253

REDDY

In the derivation of general equation for computation of advance and recession in surface irrigation (horders and furrows), the following relationships were assumed between depth of flow, area, and hydraulic radius: Y=ICAO

(3)

AzR”3 = fl h

(44

or

A = (q@‘AI(S,/41’A (4b) in which y is normal depth of flow, m, q is inflow rate, m3 s-l, R is hydraulic radius, m, A is normal area of surface flow, m2, assumed to be constant during the time of inflow into the field, S, is field slope, n is Manning’s roughness parameter, and K, /3, p and A are empirical parameters whose values depend upon the flow cross-section. The values of K, /3, and p are equal to one whereas A is equal to 3.33 in the case of border irrigation. 2. Volume-balance

equations

In surface irrigation, from the time water is introduced at one end in a dry field, until it spreads and reaches the downstream end of the field is called the advance phase. As the inflow continues, the applied water alters the storage of surface and subsurface water, this is described as the storage phase. Once the inflow is cut off, the ponding depth slowly decreases. This is called the depletion phase. After some time, the ponding depth at the upstream end or downstream end drops to zero, marking that the water is beginning to recede from the surface through runoff and/or infiltration. The process of water movement from the soil surface is called the recession phase. These phases of surface irrigation are depicted in Fig. 1. This section presents the mathematical expression for the different phases.

-

-

_

Recession

phase

Depletion

phase

storage phase lnflltrotlon opportunity time Advance phase

Distance

down field

Fig. 1. Definition sketch showing different phases of surface irrigation

254

INTEGRAL

2.1. Advance The volume-balance given as

EQUATION

SOLUTIONS

TO

IRRIGATION

phase

equation of surface irrigation during the advance phase (Fig. 2) is x qt =

I0

x A(t, s) ds +

I0

z(t, s) ds

(5)

in which t is the time of inflow, s, z is the infiltrated volume of water, m3 m-‘, and x is the advance distance, m. Eqn (5) can also be written as: A(t, s) ds +

z(t, r)Frdr

in which t is the advance time to distance s, s, and ds/dr is the advance rate, m s-l, which is the solution sought. Before any solution can be found, the function A(t, s) and z(t, t) must be given. The assumption made in the application of volume-balance methods to sufficiently sloping borders is to have normal depth (area) of flow at the upstream end of the field, and Eqns (4) is used to calculate the flow area. Similarly, the infiltration at any particular point is given as z(t, r) =

z(t-r) ()

for or

t>t t
The zone of integration is presented in Fig. 3. Given these functions, Eqn (6) is written, with an appropriate surface shape factor n, which is defined later, as f qt=rjAx+ The second term on the right-hand-side

i0

z(t - t) dx/dt

dt

(7)

of Eqn (7) can be integrated

qt = VAX + z(t - +x(r)

’+

I0

‘dz

-x(t) I Oh

by parts to yield

dt

(8)

YA

----i) 4

x

Fig. 2. Volume-balance during advance phase showing surface and subsurface profiles

255

.I. M. REDDY

T

Fig. 3. Zone of integration (after Leszyczynski5)

in which the second term on the right-hand-side drops out when the limits of integration are substituted: the resulting equation is re-arranged to obtain (9) in which the unknown function n(t) integral equation.

is under the integral sign; hence, Eqn (9) is an

2.2. Runoff and recession phase Runoff from the downstream end occurs at the end of advance phase in the case of freely draining fields. Since the inflow rate and the infiltration rate are known at the end of advance, the runoff rate can be approximated using the following equation:9 qR = q - [i(t) + i(t - f,)]L/2

(10)

in which qR is the runoff rate, m3 s-‘, L is the length of run, m, and i is the infiltration rate, m3 s-’ m-l. The area of flow at the upstream end starts to decrease once the inflow into the field is cutoff. The time from supply cutoff till the area of flow at the upstream end drops to zero is called the recession-lag time (f,,). This is the time taken to drain the volume of water enclosed by area ABCDE (Fig. 4). A linear profile was assumed in the calculation of this volume of surface storage which is a function of flow area and the length of run. Using Eqn (3) the expression for surface storage is given as

v,0co)=

g$ KYuWS- (ydIK)“fqI(yu

- yd)

(11)

in which Vv(tm) is the volume of surface storage (ABCDE) at the time of cutoff, m3, and y, and yd are normal flow depths at the upstream and downstream ends of the field, m, respectively. Recession at the upstream end begins at the end of recession-lag time, and is given as

(12)

256

INTEGRAL

EQUATION

SOLUTIONS

TO IRRIGATION

Fig. 4. Schematic surface profile: depletion and recession phase

in which tR is the time of recession (zero depth) at the upstream end of the field, s, and V,,(t,,)/q represents the recession-lag time, s. At the downstream-end, runoff continues until the flow area gradually drops to zero by the end of irrigation. The surface storage volume in the furrow (or border) at the beginning of recession (DCEF), assuming that the rate of change, S,,, of depth with distance is uniform over the field length (Fig. 4) and Eqns (3) and (11) in effect, is given by (13) where s

Y

=Y&R) L

in which V,(t,) is the volume of surface storage at the beginning of recession, m3, and yd(fR) is the normal depth of flow for the runoff rate at time tR, m. The rate of recession is proportional to the rate of change of surface storage9 which is given by the following differential equation:

in which I is the length of the inundated portion of the field, m, and i is the average infiltration rate in the field and is a function of time, m3 s-’ m-l. The second term on the right-hand-side of Eqn (15) is the runoff rate at the downstream end. The assumption here is that the depth of flow at the down-stream end, yd(t) = S,f(t), now a function of time, is always at normal depth. Division of Eqn (15) by f*‘@and simplification results in the following nonlinear ordinary differential equation:

in which M = (S,/K)“@

(17)

257

J. M. REDDY

and

Integration of Eqn (16) yields

which is a nonlinear homogeneous yolterra integral equation of second kind. Eqn (19) is similar to Eqn (2). By assuming Z is constant, which in many cases does approach constancy with large infiltration times, Eqn (16) can be transformed into a variables separable form: dl IW-WB + c@-WW

dt

=

(20)

or (21) in which m = (/3 - 1)/b and cx= (A - 2)/2/X Runoff rate during the recession phase is assumed to be proportional depth of flow at the downstream-end, and is given as

to the normal

The rate is dependent upon Z(t). The volume of runoff, V,, m3, can be obtained by integrating Eqn (22) with an appropriate l(t) function. The other approach of length of the inundated portion of the field, 1, as shown below: I”

1+ cl”-”

(23)

in which w = [A/(2/3) - m]. Integrating Eqn (23) results in

=+ I

L

V,,

V,

MC

1”

[ (1+ cl=-*)

dz

in which V,r is the volume of runoff, m3, up to time tR, and is computed using v, = V(L) - V,(G - v,O,)

(25)

in which V(t,) is the volume of water applied to the field, m3, V,(t,) is the volume of surface storage at time tR, m3, and V,(t,) is the infiltrated volume up to time tR, m3. 3. Solution techniques This section deals with the techniques for solving the advance and recession phases of surface irrigation. Equation (9) is similar to Eqn (1). The most general technique for solving Volterra equations of second kind is the method of successive substitutions3P5 in which the initial guess is improved by a series of substitutions and integrations. The

258

procedure

INTEGRAL

EQUATION

SOLUTIONS

TO

IRRIGATION

is given as follows: Let the initial value of x(t) be x()(t) =

5

(26)

Then, xl

=

-$ - $l’(t - t)“-l~o(z)

dz

x2 = 5

- $

I(t - ~)~-‘xr(r) dz

x, = $

- 2

/‘(t - r)o-%,-i(t)

dr

0

After n iterations,

(29)

the solution converges to the following5: (30)

in which (31) I is the gamma function, and 5 is defined as follows: 5 = kI (I+ a)l(rlA)

(32)

in which k, m3 SC m-l, and a are the coefficient and exponent, respectively, in the Kostiakov infiltration function, z = kt”. The solution presented above is more general. But when the kernel is a polynomial in (t - r), i.e. k(t, z) = k(t - t), (33) the advance problem belongs to an important class of Volterra integral equations called equations of the Faltung (convolution) type. Since Eqn (9) is linear, the Laplace transformation technique can be used for solving it: UW)]

= lrn e-Y(t)

dt = F(s)

(34)

in which F(t) is an unknown function. The inverse of Eqn (34) is given as iY’[F(s)]

= F(t)

(35) For the Kostiakov type infiltration function, the Laplace transform of Eqn (6) is given as:

L[x(t)]= 4[#(l:

Es-“)

1

(36)

The solution for x(t) is obtained by the inverse Laplace Transformation of Eqn (36). This solution will be identical to Eqn (30). These series solutions are convergent; however, the convergence is not rapid at large t. Philip and Farre14 presented a new series solution for large t. This series was found to be satisfactory16 when Eta/z(2 + a) 3 1. In Eqn (30), the series S, represents the effect of cumulative infiltration depth on advance distance. By choosing an appropriate shape factor for the infiltrated profile, the

259

J. M. REDDY

series S, can be replaced by a simple analytical expression for the infiltrated volume. By assuming a linear advance rate, the subsurface shape factor for the Kostiakov type infiltration function is given as 5;= l/(1 + a)

(37)

in which 5; is the subsurface shape factor. However, for large infiltration times, the most appropriate infiltration function is the Kostiakov-Lewis type, which is given below: z = kt” + cc

(38) in which c is the basic or steady state infiltration rate, m3 s-l m-‘. Using Eqns (37) and (38), and again assuming linear advance rate, the volume of water infiltrated into the soil over a distance of x can be calculated using V, = [
(39) in which V, is the volume of water infiltrated into the soil, m3. However, during the simulation it was found that the above assumption on advance rate consistently underestimated the advance time. Better advance predictions were obtained by setting the value of I; equal to 0.75-0.80, in the following equation: V, = Qkt” + ct)x

(40) In Eqn (40), 5‘= 0.75 for fine textured soils, and 5 = 0.80 for light textured soils. Similarly, by assuming that the surface water profile can be described by a monomial power law of y, the surface shape factor is given by” 1 /3+A-2 V=l+y=/3+i-l

(41)

in which n is the surface shape factor, and y = l/(/3 + A - 2). With Eqns (40) and (41) in effect, the advance equation takes the following form: x(t) =

4t

(42)

VA + C(kt” + ct)

The recession and runoff equations for freely draining cases are nonlinear integral equations. In the case of border irrigation a simple analytical solution exists to solve the recession and runoff integrals.‘* However, in the case of furrow irrigation, the nonlinear integral equation does not lend itself to an analytical solution. Hence, Eqns (21) and (24) are numerically integrated using the Gaussian-Quadrature technique which is given below: J = i

Wjp(6j)

(43)

j=l

for approximating

an integral of the form

in which J is the approximate

value of the integral function;

Wj is the weighting factor;

~(6~) is the value of the given function at the given value of the variable (dj); b and d are

limits of integration; rewritten as

and N is the number of quadrature

points. Therefore,

Eqn (21) is

(44)

260

INTEGRAL

EQUATION

SOLUTIONS

TO IRRIGATION

Table 1 Four-point Gauss-Legendre

Quadrature

Index of the point

Quadrature point, 0,

Weighting factor, w,

1

+0.33995 -0.33995 +0.861146 -0+61146

0.652145 0.652145 0.347854 0.347854

2 3 4

where 6._

I-

8,(L-f)+L+f 2

L

1

in which ej is the quadrature point. Eqn (44) is solved for different values of IS L, to compute recession time as a function of length. Similarly, the solution of the runoff integral, Eqn (24), after applying the Quadrature technique, is given as follows: v;,(f) = V, + MC

w-4

2

sy cN wp,J .[1 + Cd_p 1

j=l

Once again, Eqn (46) is solved for different values of 1 with Sj given by Eqn (45). In Eqn (46), V, is calculated using Eqn (25). A four-point Gaussian Quadrature technique was used to solve Eqns (43) and (45). The Quadrature points, 6,, and the weighting factors, wj, are given in Table 1. 4. Example calculations This section presents the application of the techniques discussed so far for computing advance and recession in a furrow irrigation system. The data for the example problem (Benson 221) are presented in Table 2.

Table 2 Irrigation data used in simulations

Furrow irrigation t Variable

q, m3 s-l m-l s, n L, m

t,, min k, m3 m-l min-” a c, m3 m-l mine’

Border irrigation * oxlO o-001

0.024 92.0 38.0 oaKi

0.2716 0.0 0.0

;

1-o 1-o

:

3.33 * From Bassett” t From Elliott*’

Matchett-235

Printz-323

Benson -221

0@0092 om95 0.02 425.0 1364.0 oGO33 040 omOO3 2.18 0.79 1.35 3.00

0.0035 0@025 0.02 350.0

0*00114 oa44 o-02 625.0 613.0 0,018 0*02 oaoO1 l-05 0.69 0.58 2.91

110-o O-0125 oa24 oaO5 1.07 o-70 O-62 292

J. M. REDDY

261

Steps: 1. f=0*75 2. n = O-61538 [Eqn (41)] 3. A = 00005019 m* [Eqn (4b)] 4. y, = 0.0272, m [Eqn (3)] (Advance computation) 5a. z(t = 5 min) = 0.019089 m3me1 [Eqn (38)] b. x(t = 5 min) = 19.65 m [Eqn (42)] c. steps 5a to 5b were repeated until x(t) 3 625 m (Vertical recession) 6. j(t,,) = [I(&) + I(&, - t,)]/2 = OOOOOO171418 m3 s-l m-l 7. qR = 00000 686 m3 s-’ [Eqn (lo)] 8. Ad = 00007276 m* [Eqn (4b)] 9. yd = 000717 m [Eqn (3)] 10. b(fco) = l-4877 m3 [Eqn (ll)] 11. tR = 634.75 min [Eqn (12)] (Horizontal recession) 12. 1 f /3 = 1.69; l//3 = 1449 13. j(t(tR) = (I(tR) + z(tR - tL))/2 = 0-oooo01705 m3 s-l me1 14. q(tR) = 000007436 m3 s-’ [Eqn (lo)] 15. Ad(tR) = 00007689 m* [Eqn (4b)] 16. y&R) = 0007455 m [Eqn (3)] 17. S, = 0000011928 [Eqn (14)] 18. %(tR) = 0.1962 m3 [Eqn (13)] 19. vz(tR) = L[z(tR) + z(tR - tL)]/2 = 42 m3 20. V, = -0.26144 m3 [Eqn (25)] 21. M = OOOtBMO682 [Eqn (17)] 22. C = 000005546 [Eqn (18)] 23. m = -04493; (Y= 0.6594 (Application of Guassian Quadrature) 24a. Whenx=62.5m,l=L-x=562*5m b. b1 = 604.37 a2 = 583.13 6, = 620.66 LEqn (45)] a4 = 566.84 I C. pj(6j) = (l/(67 + CSJ~) PI(&) = 10.86 ~~(8~) = 21.56 ~~(8~) = 27.41 p4( 6,) = 3306 d. t(l = 562.5) = 63544 min [Eqn (44)] e. To complete the recession phase, the above procedure was repeated ing the value of x until I = 0.

by increment-

262

INTEGRAL 5.

EQUATION

SOLUTIONS

TO IRRIGATION

Results and discussion

Four example problems, one border and three furrow irrigation considered to test the accuracy of the volume-balance model presented in results from the model were compared with field data, and other available Walker and Humpherys”) mathematical models. The data used in presented in Table 2.

systems, were this paper. The (Bassett,” and the study are

5.1. Border irrigation Example 1. This example is provided by WSU test B2 (Bassett”). The results from the model were compared with the observed values, and are presented in Fig. 5. As can be seen, the deviations in prediction of advance are negligible. However, the difference between the predicted and observed values of recession, particularly at the downstream end of the field, was about 10%. Similarly, the runoff volume computed by the writer was 3 m3, and that computed by Bassettlg was 2.5 m3, a difference of 6.3% of the total volume of water applied. 5.2. Furrow irrigation The data for all the furrow irrigation simulations were obtained from Elliott,*’ and the results were compared with the kinematic-wave model developed by Walker and Humpherys. *’ Examnle 2. Benson 221. The calculated advance time to the downstream end of the field WA 535min, where as the observed advance time was 513 min (Fig. 6), which

Dlstonce,

Fig. 5. Advance

m

and recession field test: freely draining border (WSU-B2). equation model; (- -) Bassett’s model

??

observed;

(-)

integral

263

J. M. REDDY

I40

__---__c_r-5------e_--3

Distance,

Fig. 6. Advance

m

and recession field test: furrow irrigation (Benson 221). equation model; (- -) Kinematic-wave model

??

observed;

(-)

integral

(-)

integral

200 . 100

??

I

0 I 200 Distance,

Fig. 7. Advance

_-I

I 300

I

I 400

m

and recession field test: furrow irrigation (Print2 323). equation model; (- -) Kinematic-wave model

??

observed;

264

INTEGRAL 800

I

I

700 -

600

I

I

I

I

EQUATION

I

I

SOLUTIONS

I

I

I

TO IRRIGATION

I

__________-___------___

L I_

500 ; i-

400-

c 300 -

200 -

100

‘00

300 Distance,

400

500

600

m

Fig. 8. Advance and recession field test: furrow irrigation (Matchett 235). equation model; (- -) Kinematic-wave model

??

observed; (-)

integral

resulted in a difference of 4.3%. The difference in the prediction of recession times was insi nificant. The observed volume of runoff was 2 m3 where as the predicted volume was 0 m!?. This resulted in an error of 4.8% of the total volume of water applied to the furrow (42 m”). Example 3, Printz 323. As shown in Fig. 7, the difference between the observed and model predicted advance and recession was negligible. The runoff volume predicted by the writer was O-76 m3, and the observed runoff was l-20 m3, a difference of 1.9% of the total volume of water applied to the furrow (19 m’). Example 4, Matchett 235. The computed advance time to the end of the field was 220 min where as the observed advance time was 250 min, a deviation of 12%. However, there was no difference between the model predicted and observed recession times (Fig. 8). The measured runoff volume was 29.9 m3, and the runoff volume predicted by the model was 33.5 m3, an error of 4.8% of the total water volume applied to the furrow (75.3 m’). From the above results it is clear that the performance of the model in predicting advance and recession phases of surface irrigation was reasonably accurate. However, the model predicted recession times were consistently faster than the observed recession. This may be due to the overestimation of infiltrated volume of water into the rootzone by neglecting the effect of decreasing wetted perimeter (during the recession phase) on infiltration rate per unit furrow length. Similar argument holds true for the underestimation of runoff by the model in two cases of furrow irrigation simulations. The accuracy of the predicted runoff volume could be slightly increased by introducing a constant into the recession and runoff integrals to account for the decreasing wetted perimeter. However, since the error in estimating the runoff volume was not large and the

265

J. M. REDDY

duration of recession phase incorporated into the model.

in furrow

irrigation

is not very

long,

this

aspect

was not

6. Conclusions Considering the irregularities in slope, furrow geometry, and the difficulty in locating the trailing edge during the recession phase, the performance of the model can be considered acceptable for computation of flow in surface irrigation (borders and furrows), provided the field is sufficiently steep (slope greater than O-1 percent) and rough for the depth at the upper end of the field to rise quickly to normal depth. The derived equations are simple, and the solutions can be obtained on a hand calculator. Furthermore, because of the acceptable accuracy of the model, it can be used in the design of surface irrigation systems for non-cracking soils. References Hall, W. A. Estimating

irrigation border flow. Agricultural Engineering 1956, 37(4): 263-265 Ostromeeki, J. Method of computing the border flow irrigation system. International Commission on Irrigation and Drainage Bulletin 1960,9: 73-77 Bomky-&es&h, K. Hydraulic examination of border flow irrigation. Hydrologiai Kozlony 1960,40: 16-27 (Hungarian) Philip, J. R, Farrell, D. A. General solution of the infiltration-advance problem in irrigation hydraulics. Journal of Geophysical Research 1964, 69(4): 621-237 B. Solving the differential equation of gradually varied unsteady flow in Leszczyn&i, border-strip irrigation. Archiwum Hydrotechniki 1967, 14(3): 393-426 (Polish) Hart, W. E.; Bassett, D. L.; SheIkoff, T. Surface irrigation hydraulics-kinematics. Journal of Irrigation and Drainage Engineering 1968, w(4): 419-440 KatyaI, A. K.; Kijne, J. W. Prediction of the advancing wetting front in border-strip irrigation. Irrigation Science 1980, 1: 177-184 Wu, I. P. Recession flow in surface irrigation. Journal of Irrigation and Drainage Engineering 1972,%(l): 77-90 Strelkolf, T. Algebraic computation of flow in border irrigation. Journal of Irrigation and Drainage Engineering 1977,103(3): 357-377 Reddy, J. M. Computation of recession and runoff integrals. Journal of Irrigation and Drainage Engineering 1980,106(4): 367-370 Davis, J. R. Estimating rate of advance for irrigation furrows. Transactions of the American Society of Agricultural Engineers 1961, 1: 52-57 Krivovjaz, S. M. Calculation of furrow irrigation. Gidrotekhnika i Melioratsiia 1961, U(1): 12-23 (Russian) VIadimireseu I. Rational formulae for the hydraulic computation of long irrigation furrows. Fifth Coniess of International Commission on Irrigation and Drainage 1963, R16: 259-275 (Romanian) ECShafei, Y. Z. General solution for the evaluation of advance of water in furrow irrigation: theory and practice. Kulturtechnik Flurbereinigung 1980, 21: 8-19. Golberg, M. A. Solution Methods for Integral Equations. Plenum Press 1979, New York Michael, A. M. Water front advance in irrigation borders. Journal of Agricultural Engineering Research 1971, 16(l): 62-71 Elliott, R. L. Zero-inertia furrow irrigation modeling applied to the derivation of infiltration parameters. PhD dissertation 1981, Colorado State University, Fort Collins, Colorado Gii, M. A. Solution of recession and runoff integrals. Journal of Irrigation and Drainage Engineering 1983, 109(3): 335-337 Bassett, D. L. A dynamic model of overland flow in border irrigation. PhD dissertation 1973, University of Idaho, Moscow, Idaho WaIker, W. R.; Humpherys, A. S. Kinematic-wave furrow irrigation model. Journal of Irrigation and Drainage Engineering 1983,109(4): 377-392 Elliott, R. L. Furrow irrigation field evaluation data. Department of Agricultural and Chemical Engineering 1980, Colorado State University, Fort Collins, Colorado.