WATER
RESOURCES
RESEARCH,
VOL.
19, NO. 6, PAGES 1599-1612, DECEMBER
1983
A Kinematic Model for Surface Irrigation'
Verification by Experimental Data
VIJAY P. SlNGH
DepartmentofCivil Engineering,LouisianaState University
RAMA S. RAM
Departmentof Mathematics,Alcorn State University
A kinematic model for surfaceirrigation is verified by experimentaldata obtained for 31 borders.
These borders are of varied characteristics.Calculated values of advance times, water surface profiles
when water reachesthe end of the border, and recessiontimesare comparedwith their observations.The
predictionerror in most casesremainsbelow 20% for the advancetime and below 15% for the recession
time.The water surfaceprofilescomputedby the modelagreewith observedprofilesreasonablywell. For
the data analyzedhere the kinematicwave model is found to be sufficientlyaccuratefor modelingthe
entireirrigation cycleexceptfor the verticalrecession.
Hart et al., 1968; Kincaid, 1970; Kincaid et al., 1972], he
concluded
that for many casesit is unnecessaryto solve the
A knowledgeof advance,recessiondistribution of depth of
complete
hydrodynamic
equationsfor irrigation advanceand
water and distribution of infiltrated water is required for an
optimal design of surface irrigation. One way to determine that kinematic wave theory would yield satisfactoryresults.
thesedesignvariablesis by usingmathematical models.There Woolhiser[1970] expressedthat the applicabilityof kinematic
are many models of surfaceirrigation. Most of these models wave theory in tracingthe advancingfront is open to question
can be classifiedin order of increasingcomplexity as (1) stor- becausethe kinematic assumptionis clearly invalid in the image models,(2) kinematicmodels,(3) zero-inertiamodels,and mediate region of the front. Smith, using the data of Tinney
(4) hydrodynamicmodels.A recentstudy by Ram [1982] sur- and Basserr[1961], computedthe percentageerror in locating
veys these models critically. Basserret al. [1981] have dis- the front by the kinematic wave method as a function of kincussedthe current state of the art of hydraulics of surface ematic flow number [Woolhiser and Liggett, 1967]. He
irrigation. For the sake of completeness
a brief review of only showed that the error decreasedexponentially with the increasingnumber and that it was less than 5% beyond the
kinematic wave modelsis givenhere.
INTRODUCTION
Although kinematic wave theory [Liqhthill and Whirham,
1955] has extensivelybeen utilized in modelingbasin hydrology [Sinqh,1978; Sinqhand Aqiralioqlu,1980; Woolhiser,1982]
and predicting flood movement in rivers [Fread, 1982], its
applicationin surfaceirrigation has beensomehwatlimited. C.
L. Chen [Chen, 1966, 1970] is perhapsthe first to have usedit
to solve analytically the problem of irrigation advanceover a
wide porous bed. He concluded that the kinematic wave
method may only be valid for supercriticalflow but expressed
doubts about its reliability. Woolhiser[1970] questionedthis
conclusionand showedthat the method would also be applicable to subcritical flow but would give poor resultsif water
pondedat the downstreamboundaryand a movingbackwater
extendedover an appreciablelength of the field. Chen [1970]
assumedthe advancingfront to be a characteristicand did not
verify his solutions by experimentaldata. Woolhiser [1970]
showedthis assumptionto be incorrect.
In 1972, R. E. Smith published his numerical work on
border irrigation advanceand ephemeralflood waves based
on kinematic wave theory [Smith, 1972]. The method of
characteristics
and
the Lax-Wendroff
scheme
were
used to
determineirrigation advance.By usingthe field data of Criddle et al. [1956] and experimentaldata of Kincaid [1970] and
comparing with other methods [Wilke and $merdon, 1965;
Copyright 1983 by the AmericanGeophysicalUnion.
Paper number 3W1327.
0043-1397/83/003W- 1327505.00
value of 4.
These studies were confined only to the advance phase of
the irrigation cycle [Basserrand McCool, 1973]. Cunge and
Woolhiser [1975] were the first to have derived, under the
assumptionof constant inflow and constant infiltration, analytical solutionsin dimensionless
form for advance,recession,
infiltration opportunity time, and depth of flow. No verification was, however, done to show that the kinematic wave
approximationis suitablefor all phasesof the irrigation cycle.
Sherman and $ingh [1978, 1982] and $ingh and Sherman
[1983] made a comprehensivemathematical study of kinematicwave modelingof surfaceirrigation. Two mathematical
issueswere addressed:(1) formulation of free boundary problems using kinematic wave theory and full hydrodynamic
equations and (2) solution of the free boundary problems
using kinematic wave equations. Depending upon the variability of infiltration ratef and the kinematic friction parameter
/•, three caseswere distinguished:(1)fand/• were constant,(2)
/• was stationarybut spacedependent,andf was constant,and
(3)fwas time variant but spaceindependentand was constant.
Each
of these cases was considered
when
the duration
of
inflow was (1) infinite, (2) finite but greater than characteristic
time, and (3) finite but less than characteristictime. Explicit
solutions were obtained when f was constant, and an approach was suggestedwhenf was time dependent.These solutions were consideredfor advance,storage,depletion, and recessionphasesof the irrigation cycle but were not verified
using field data. Evidently, the solutions obtained by Cunge
and Woolhiser[1975] are specialcasesof the above solutions.
1599
1600
SINGH AND RAM: KINEMATIC MODEL FOR SURFACE IRRIGATION
Some of the solutions obtained by Sherman and Singh
[1978] have sincebeentestedin a limited way. For example,
B. J. Chen, R. C. McCann, and V. P. Singh [Chen et al., 1981]
testedthe solutionsfor the advancedphaseby usingdata from
one border of Kincaid [1970]. Numerical solutionswere obtained for variablef by the method of characteristics
and the
Lax-Wendroffmethod.Good agreementbetweenobservations
and simulationswas found and supported the findings of
Smith [1972]. In a study on evaluationof modelsof border
irrigationrecession
(verticalas well as horizontal),Ram and
Singh[1982] testedthe solutionsfor the horizontalrecession
phaseby usingfour sets of data on open bordersof Roth
[1971] and compared them with a number of recession
models. These solutions yielded better results than other
modelsand were in good agreementwith observations.To
avoid confusionin the ensuingdiscussion,we define vertical
and horizontal
t :T 2
Horizontol
Recession
Phose
t=T I
Time
D2
Storage Phase
of
Opportunity
t=T
D•
Advance
Phase
t=O
x=O
x=L
Fig. 1. Solutiondomainfor flowin a freelydrainingborder.
recession. The vertical recession starts at the
cessationof inflow and continuesuntil the depth of flow at the
upstreamend becomeszero. The horizontal recessionstarts
phases,which include advance,storage,and recession.These
with the depthof flow becomingzeroat the upstreamendand constitutethe irrigation cycle,as shownin Figure 1.
When water is released,there is a front wall of water which
continuestill the depth of flow becomeszero at the downadvances down the channel. This front wall of water is the
streamend of a freelydrainingborder.If the borderis bunded
at the downstreamend, the water getsimpoundedagainstthe
bund and the horizontal recessionis complete as soon as the
water surfaceprofilebecomeshorizontal.
This reviewpointsout that dependingupon the variability
of infiltration and inflow, two typesof solutionsof kinematic
wave equationshave been sought:(1) analyticalsolutions
when these are constant and (2) numerical solutionswhen
these are variable.
Since infiltration
and inflow are seldom
constant,analyticalsolutionsare of limited value to surface
irrigationdesign.Numericalsolutionsare difficultto develop
becausethe entire solution domain of irrigation is not known
beforehandand is expensiveto apply. This studyattemptsto
advancefront, that is, the moving interfacebetweenthe watercoveredand -uncoveredparts of the channel.Let x = s(t) or,
inversely,t = •(x) denote the history of this front; this time
history is the advancefunction.This front is a free boundary
which has to be determined along with the depth h(x, t) and
velocity u(x, t) for a specifiedho(t) or q(0, t). The advance
phasecontinuesuntil the water reachesthe downstreamend
of the channel. As shown in the figure, this phase is representedby the domain D•, which is boundedby x = 0, t = •(x),
x - L, and t - T; L denotesthe length of the field and T the
advance time.
Let f(r) be the infiltration rate at time r; the quantity
developa methodwhichis part numericaland part analytical r = t- •(x) denotes the infiltration opportunity time at a
to solve kinematic wave equations when infiltration and point x in the field, that is, the interval of time that water has
inflow are time varying. Some advantagesof both types of coveredthe point x, where t is the total time elapsedsincethe
solutions are thus combined in this method. It is simpler and inflow began.The infiltration rate f(r) is assumedto depend
more efficient than the numerical method proposedby Sher- only on the differencer betweenthe total elapsedtime and the
manandSingh[1982]. Althoughthe mathematicsof kinematic
wave approximationin respectof surfaceirrigationis understood reasonablywell, a comprehensivetesting of this approximationin modelingthe entireirrigationcycleis lacking.
This paper attemptsto do this testingusing the kinematic
wavemodelreportedby ShermanandSingh[1978, 1982].
advancetime, that is, it is time dependentbut independentof
xforx
> 0.
If the inflow q(0, t) is continued after water has reachedthe
downstream end of the channel, then the water will continue
to accumulatein the channel.This water buildup constitutes
the storagephase.As shown in Figure 1, this phaseis representedby domain D2, which is boundedby x = 0, T < t < T•
and x = L. As soon as the inflow is cut off at t- T•, q(0,
KINEMATIC WAVE MODEL
t) = 0, t > T•, the storagephaseendsand the recessionphase
begins.The depth of flow at x = 0, t - T• goesto zero instanThe kinematic wave model is formulated and discussedby taneously.This implies that the kinematic wave assumption
Sherman and Singh [1978, 1982] and Singh and Sherman does not accommodatevertical recession.As time progresses,
[1983]. We refer to their work for the background infor- the zero depth travelsdownward.This zero depth is calledthe
mation but briefly outline below the formulation of the model. drying front. The movementof this drying front characterizes
Surfaceirrigationessentiallyinvolvesthe flow of water down a the horizontal recession and continues until all the water is
plane or channel with a small slope and porous bed. It is drained out of the channel. We let t = •(x) denote the time
assumedthat the channel under considerationis initially dry history of the drying front, that is, the moving interfacebeand is rectangular,having uniform crosssection.Let x be the tweenthe part of the channelwith h(x, t)= 0 and the part of
distancealong the channel which may extend indefinitelyto the channelwith h(x, t)> 0. This time history t = •(x), T• <
the right of its head at x = 0. At time t = 0, water is released t < T2 is a free boundarywhosedeterminationis a part of the
at the head x = 0 of the field. The water inflow at x - 0 has a
solutionfor the recessionphase.This phaseis representedby
known time-dependentdepth ho(t)or rate q(0, t). The inflow of domain D3, which is bounded by x = 0, T• < t < •(x), and
water at x -- 0 lastsfor a specifiedlengthof time T•. DependThe depth of water h(x, t) and the unknown time history
ing upon the duration of inflow and the boundaryconditions
at the end of the channelat x - L, the flow undergoesvarious •(x) are subjectto the followingkinematicwave formulation
SINGH AND RAM: KINEMATIC
MODEL FOR SURFACE IRRIGATION
over the leading column at a given position in x and movesas
the wave front. This processcontinuesuntil the water reaches
[Shermanand Sinqh,1978]:
t•h
t•
t3•'
+ •xx
(]•h")
= - fit - •(x)] h(0,
t)= ho(t
)
0<t<T•
1601
h(0, t)=0
the end of the border.
(1)
t>T•
To formulate the KWT method, we considerthat in (1),/• is
constant.For a finite grid spacingAx and At, (1) can be written as
= {fih"-'[x,
=0
(2)
Ah
Ah
A'•+ nj•hn-•
•Ax = - f(:)
wheren e (1, 3] and fi > 0 are kinematicwaveparameters.
(5)
where
NUMERICAL SOLUTIONS
Equations(1)-(2) are validin the solutionregioncomprised
Ah = hi+ 1 - hi
i _>1
(6)
of the domainsD•, D2, and D3. The solutionto theseequations can be found in each of thesedomainsby usingthe where i is a dummy index and assumeh = hi. Substituting(6)
methodof characteristics.
The characteristic
equationsof (1) into (5) and solvingfor hi+ •,
can be written
as
dx/dt= nfihn- •
(3)
dh/dt= - fit - •(x)]
(4)
ShermanandSinqh[1982] proposeda numericalmethodto
solve(3)-(4) for the entireirrigationcycle.Chenet al. [1981]
solvedtheseequationsfor the advancephaseby a simpler
method.This studymodifiesthis latter methodby makingit
computationally
moreefficientfor the advancephaseand developsa semianalytical
methodfor the recession
phase.The
resultingmethodis thuspart numericalandpart analytical.
(Ax)hi + (At)nj•hin --(Ax At)f(:)
hi+
•=
Ax+(At)
nj•hi
n-•
(7)
Equation (2), which expressesthe inverse of the velocity of
kinematic shock,can be used to determineAt for a specified
grid spacingAx:
a•i+ 1 -- •i+ 1 -- •i-- axf(•hin- 1)
(8)
where • and •+• are times when wave front reachesxi and
x•+ •. Note that (At)i+• -(A•)•+• is dependentupon the position and depth of the front column. Equations(7)-(8) can be
solvedto yield the advancecurve t = •(x) as follows.
Domain D a
1. We assumethat the inflow q(0, t) - q0 is constant.The
Central to the solution to be obtained in this domain is the corresponding
depth of flow at the upstreamend can be calcudetermination of the time history of the advance front, lated usingManning's equation:
t = •(x). We developa numericalmethod,designatedas kinG -' (flqo)
ø'6
(9)
ematic wave train method (KWT), to solve the initial value
problemgiven by (1)-(2) that yields this time history. We whereG is normal depth of flow at the upstreamend. In (9),
hypothesizethat water moves in a seriesof columns of finite is expressedas
width, as shownin Figure 2a. The front column moveswith
[} -- rim/So
0'5
(1O)
kinematicvelocity.When it movesto a new position,all the
followingcolumnsmove forward,each occupyingthe place where rtmis Manning's roughnesscoefficient.
vacatedby its immediatepredecessor,
as shownin Figure2b.
2. For a specifiedAx and known initial conditionh = G at
The movementcontinuesuntil the heightof the front column •i = 0, the advancetime •i + • can be calculatedusing(8).
(the depthof the wavefront) reducesto a minimumspecified 3. For computingf(:) in (7) it is assumedthat it is conby hc,asshownin Figure2c.Then the precedingcolumntakes stant for sometime :c, 0 _<: _<:c and followsthe Kostyakov
equationfor: >_:c:
f (:) = aK: a- •
where K and a are infiltration
I
Water
x
Column
I
(al
2
3
i-3
i-2
?.
3
i-3
i-2
i-I
2
3
i-3
i-2
i-I
x=O
T
constants
(11)
to be determined
empirically; :c can be specifiedfor a particular soil, and
(at)•+ • - (a•),+ • = :.
4. The water column at xi was advancedto xi+ • by simultaneouslysolving(7)-(8) in conjunctionwith (14).
5. The solution of (7)-(8) was iterated using: = (At)z+•
-A•i+ • until the desireddegreeof accuracyin hi+• for a
specifiedx was obtained.In the presentanalysis,five iterations
were found sufficientto guaranteea precisionof the order of
Column
Waterx
I
x•O
h"G
I
6. The water column of height h•+• at xi+ • was permitted
to advanceto x•+2 by (7)-(8). The desireddegreeof precision
in h•+2 was obtained by step 5. This processwas continued
until the height of the front column (depth of the advancing
front) reacheda specifiedvalue hc(Figure 2c).A ratio between
the height of the front column(depth of the advancingfront)
or tip depth (hc)and the normal depth of flow at the upstream
end (G) can be used to control the tip depth in the solution.
This ratio can be expressedas
W_•ter
Column
hc
•
I
10-5 m.
i
Fig. 2. Movement of water columns.
X
1602
SINGH AND RAM: KINEMATIC MODEL FOR SURFACE IRRIGATION
f
•2:•(x2)'
h:h(x2'
TI+At)=0
For a hydrodynamicmodel, Rt rangesbetween0.10 to 0.15
[Kincaid, 1970]. However,in this analysis,Rt = 0.05 yielded
better results.
7. Once hc was reached,the water column immediately
t=T I
behind the front was permittedto take over and was allowed
j+l
to advance. the infiltration opportunity time at any point,
when the front was at the position xi. 2, was calculatedby
t: ti,j
•i+ 1 -- •1' •i+ 1 -- •2 ''' •i+ 1 -- •i- 1, •i+ 1 -- •i (Figure 3).
Thus we see that for the first column the infiltration opj-I -Advance
Curve
portunity time is always •i+ •- •i = ti+ • subjectto its minimum valuezc. Similarly,for the secondand third columnsthe
infiltration opportunity times are •i+ • - •i- • and •i+ • - •i-2.
•
I
I
I
I
= X
2
3
i-I
i
i+l
A similar pattern followsfor all precedingcolumns.The infilx=O
x:x2
x:x i
x=k
tration rate for any column at xi is an averageof the rates at
xi and xi_ 2. The depth of the first columnat xi+ • is hi+• and
Fig. 4. Characteristic
solutions
for thedomainsD•, D2,and D 3.
is superceded
by the secondcolumnwith depthhi,if hi+• < hc.
Now the secondcolumn plays the role of the first column,the
Equations(13) and (16), coupledwith t = •(x), weresimultathird column plays the role of the secondcolumn, and so on.
neously
solved to obtain h(x, t) along the characteristicsfor
This processwas continued until water reached the downtime t, 0 _<t < T. The progressof characteristiccurvesalong
stream end of the border.
the x axis is shownin Figure 4. Thesecharacteristiccurvesare
After determination of t = •(x), the solution for h(x, t) in
boundedby 0 < t _<T, x = 0, and t = •(x).
domain D• was obtained by the method of characteristics
using(3)-(4). Equation (3) can be expressed
numericallyas
Domain De
•t•-t(-x
z,••
Characteristic•
(At)i+1,J= Ax/(n•hijn2)
(13)
where(At)i+•,• is the time requiredfor the water wave at
t = t•,x = 0, to travelfromiAxto (i + 1)Ax,andhiuthedepth
of water at lax at timejar as shownin Figure 4.
Integrating (4), we get
Domain D3
f{x - •l'x(s)]}ds
•i+1
h[x(t),t] = hl'x(ti),ti] -
The characteristicsoriginating from the t axis, T < t < T•,
were determinedusing (13) and (16) in the samemanner as in
domain D2.
(14)
The solutionin D3 is comprisedof h(x, t) and t = •(x). The
depth of water at x = 0, t = T• is zero. Central to the solution
here is the determinationof t = •(x). For a specifiedx, t = •(x)
can be calculatedby
We set F(s) = •[x(s)], ti _<s _<ti+• and perform a linear interpolation for F [Chen et al., 1981],
r(s) = r*s + •
+ [(A+
(15)
To start the computationwe assumedthat f• =f•+ 2. At each
point xi, five iterations were performedto give a precisionof
where
r* = •i+l-- •i
(at) +
the orderof 10- 5 min.The recession
curvet = •(x) wastraced
fi• = •i+ • -- (ti+ •)r*
ti is th time at whichthejth wavereachesthe point iAx, •i the
time at which the advancefront reachesthe point iAx and
(At)i+•.• is asin (13).Using(14)and(11)wecanwrite
K
hi+
•.j= h,u 1- r* [(ti+
•.•- •i+•)a_ (tiu- •i)a] (16)
until x = L, as shownin Figure 4.
To obtain h(x, t) in D3, we solved for the characteristics
issuingfrom t = t(x2, r•) = t2, x = x2, as shown in Figure 4.
At x = x2 = Ax, t = t(x2, T•) obtainedfrom the characteristic
emanatingfrom x = 0, t = T• in D2. For an incrementaltime
At = •i+ •-•i
for specifiedAx = x2, h(x2, r• + At)= 0, we
allowed the depth h(x2, r•) to linearly approach h(x2, r•
+ At) = 0 in a specifiednumber of stepsp. The time •2 can be
obtainedfrom (17) as
FAx
]ø.6
•2: T,+L//f(-•-[)2/sj
'rc
+l-ti
The time for the characteristicoriginatingat t = T• to reach x
is t2. Thus we calculated
t:•(x)
ti_l:•i_ I j-I ....
Water
i-4
i-$
i-2
i-I
i
(18)
Column
i+l
Fig. 3. Kinematic wave train solutionfor the freeboundaryt = •(x).
P, = ((2 - t2)/p
(19)
P2 = h(x2, T,)/p
(20)
The characteristics
were allowed to originateat x = x2 with
the depth of flow and the correspondingtime givenby
ß
t2,j = t2 + jp•
(21)
h2,•= h(Ax,T•) - JP2
(22)
$INGH AND RAM' KINEMATIC MODEL FOR SURFACE IRRIGATION
wherej = 1, 2, 3,..., p. Thesecharacteristics
were allowedto
progressdown the slopeby simultaneously
solving(13)-(16).
These equationscan be iterated five times to calculatethe
depthof flowwith an accuracy
of the orderof 10-5 m. The
time of occurrence
of zero depthhi+•,s- 0 is the recession
time at xi + •. In this analysisa valueof p - 25 wasused.
EXPERIMENTAL
DATA
Data on two types of borderswere used in this study: (1)
borders with no bund at the downstream end, also called
freely draining borders ['Kincaid, 1970; Roth, 1971; Roth et al.,
19743; and (2) borders with a bund at the downstream end,
also calledclosedend borders['Ram,1969, 1972; Ram and Lal,
19713.It may be pointed out that the kinematicwave model is
applicable to the advance phase only in case of closed end
borders. Therefore
its verification
will be confined
to the ad-
vancephasefor theseborders.
Four sets of data are due to Roth [1971] and Roth et al.
[1974]. These,referredto as data setsRoth-8, Roth-9, Roth-10
and Roth-11, are summarized in Table 1. These data were
collected on nonvegetatedborders (soil classifiedas sandy
loam, bulk density 1.4).The bordersdefinea flow 5.89 m wide
and 91.46 m long with 1.239 m of extensionon each of the
upstream and the downstreamends of the border. At these
ends,sills were provided to insure uniform entry of water at
the upstream end and to eliminate the drawdown effect of
outflow at the downstreamend. Inflow was measuredby a
10-cm propellermeter and the outflow by a triangular critical
depth flume.
Nine sets of data used in this study are due to Kincaid
['1970].These,referredto as data setsK-l-K-9, werecollected
for irrigationson vegetated(bromegrass,bromegrassalfalfa,
grain sorghum,barley)bordersand are presentedin Table 1.
During each irrigation, the inflow to the borders was measured with a Parshall flume and runoff, if any, with broad
crestedrectangularweir set at the averageelevation of the
downstreamend of the border. The inflow had one entry point
to the entire border without the spreadingbasinat the upper
end. Thus uniform entry of water was not achieved.The depth
was measuredwith staff gageson steelbenchmarks set at the
averagecross-sectional
elevationof selectedstationsalongthe
border [Howe and Heermann,1970].
The data on closed end borders used here are due to Ram
[1969-]. He collecteddata on 18 irrigations; these data, referred to as data sets R-I-R-18, are summarized in Tables 2
and 3. The data setsR-l-R-9 (Table 2) are on nonvegetated
borders,and R-10-R-18 (Table 3) on vegetated(wheat crop)
borders.The bordersare 100 m long and 6 m wide and have
rails on each side of the borders for precise leveling. The
inflow was measuredby a 90ø V-notch weir beforeflowing
into a distributionchannelinstalledat the upstreamend of the
border 1 m up the first station.This insureduniformentry of
water at the upstreamof each border. The water depth was
measuredby point gaugesat eachstationat every20 m.
PARAMETER ESTIMATION
The kinematicwave model, expressedby (1)-(2) and (11),
containstwo unknown infiltration parametersK and a of the
Kostyakov equationand two unknown friction parametersn
and/• of the stage-discharge
relation.The valuesof theseparametersfor eachdata setare givenin Tables 1-3.
Infiltration Parameters
For the data setsRoth-8-Roth-11, Roth [1971] estimatedK
1603
1604
SINGH AND RAM.' KINEMATIC MODEL FOR SURFACE IRRIGATION
TABLE 2. Irrigation Data for NonvegetatedBorderswith Bund at the Downstream End
Data
Parameters
Inflowrateqo,m3/m/min
Infiltration constant K, m/min a
Infiltration exponenta
Depth at the upstreamend, G, m
Manning's roughnesscoefficientnm
Chezy'sroughness
coefficient
Ch,m•/2/s
Borderbed slopeSo,m/m
Kinematicfriction//, mx/3/min
Border length L, m
Length of one reach, m
Length from the upstreamend where
impounding starts,L - l, m
Duration of irrigation, min
Number
of stations
Set
R- 1
R-2
R-3
R-4
R-5
R-6
R-7
R-8
R-9
0.1600
0.0039
0.56700
0.0255
0.0590
9.26
0.0050
72.4
100.0
10.0
80.0
0.1200
0.00450
0.57400
0.0230
0.0660
8.11
0.0050
64.5
100.0
10.0
82.0
0.0800
0.00460
0.59000
0.0150
0.0480
10.26
0.0050
87.7
100.0
10.0
88.0
0.1600
0.00479
0.60500
0.0350
0.0770
7.44
0.0030
47.7
100.0
10.0
70.0
0.1200
0.00464
0.58800
0.0328
0.0920
6.15
0.0030
35.7
100.0
10.0
70.0
0.0800
0.00361
0.61500
0.0370
0.1000
5.49
0.0030
32.9
100.0
10.0
74.5
0.1600
0.00402
0.69000
0.0500
0.0800
7.54
0.0010
23.6
100.0
10.0
35.0
0.1200
0.00330
0.69000
0.0390
0.0710
8.21
0.0010
26.8
100.0
10.0
38.0
0.0800
0.00630
0.52700
0.0310
0.0730
7.72
0.0010
26.2
100.0
10.0
40.5
22.5
11
37.0
11
59.0
11
35.5
11
50.0
11
74.0
11
50.0
11
59.0
11
95.0
11
After Ram [1969, 1972]; Ram and Lal [1971].
and a by a volume balanceanalysis.This was basedon the
assumptionthat the water surfaceand infiltration profilesin
the last 9.5 m reach, over which the water was advancing,had
a profile shapefactor of 0.7. For the data setsK-l-K-9, Kincaid [1970] determined K and a using the method discussed
by Gilley [1968]. For the remainingdata setsR-I-R-18, these
parameterswereestimatedby Ram [1969, 1972], usingagain a
volume balance method.
in space and time. The stage-discharge
relation was representedby Manning'sequation.Therefore,n = 5/3, and
]• = (1/nmXS
f)1/2
(23)
wherenmis Manning's
roughness
coefficient
and St slopeof
the energyline.
Fundamental to determinationof nmis to obtain the slope
of energyline St. For the data setsRoth-8-Roth-11,Roth
Since the Kostyakov equation gives very high infiltration [1971] computed nmin spaceand time for each irrigation by
rate in the beginning,even exceedingthe infiltration capacity assumingflow to be steadyand uniformand satisfyingKruse's
of the soil surface,a minimum time of opportunity must be criteria [Kruse, 1960]. The valuesof nmpresentedin Table 1
specifiednear the wetting front. This time was found by trial are averagedover spaceand time. Kincaid [1970], for his data
and error for each data set representinga specificlocation so setsK-l-K-9, computedSt by taking it as the slopeof a
as to obtain the least error between observed time of advance
straightline fitted by the leastsquaresregressionthroughthe
and the time of advance calculated by the kinematic wave total head data. Energy gradients were calculated for the
model. This time is 0.1 min for the data sets Roth-8-Roth-11,
entire profile and for each 30.48-m incrementwhich were then
5.0 min for K-l-K-9, and 0.5 min for R-I-R-18.
usedto estimate nm.Its averagevaluesare shown in Table 1.
For thedatasetsR-I-R-18, Ram[1969,1972]assumed
St to
RoughnessParameters
be the sameas the bed slope.The valuesof nmwere basedon
Although the bed roughnesschangesthroughout the irri- the normal depth of flow at the upstreamend when water
gation cycle [Roth, 1971],/•(x, t) was assumedto be constant reachedthe downstreamend and are givenin Tables2 and 3.
TABLE 3. Irrigation Data for Vegetated(Wheat Crop) Borderswith Bund at the DownstreamEnd
Data
Parameters
R-10
Inflowrateqo,m3/m/min
0.1600
0.00440
0.62000
0.0381
0.1140
Chezy'sroughness
coefficient
Ch,m•/2/s 5.07
0.0050
Borderbed slopeSo,m/m
37.1
Kinematicfrictionfi, mx/3/min
100.0
Border length L, m
10.0
Length of one reach,m
70.0
Length from the upstreamend where
Infiltration constantK, m/mina
Infiltration exponent a
Depth at the upstreamend, G, m
Manning'sroughnesscoefficientnm
Set
R-11
R-12
R-13
R-14
R-15
R-16
R-17
R-18
0.1200
0.00360
0.63000
0.0350
0.1320
4.32
0.0050
32.0
100.0
10.0
71.5
0.0800
0.00530
0.53300
0.0300
0.1540
3.63
0.0050
27.6
100.0
10.0
76.0
0.1600
0.00390
0.67400
0.0450
0.1170
5.10
0.0030
28.1
100.0
10.0
57.5
0.1200
0.00405
0.60000
0.0430
0.1450
4.10
0.0030
22.7
100.0
10.0
60.0
0.0800
0.00609
0.53300
0.0395
0.1887
3.10
0.0030
17.5
100.0
10.0
65.0
0.1600
0.00415
0.6400
0.0715
0.1460
4.41
0.0010
13.0
100.0
10.0
10.0
0.1200
0.00340
0.6900
0.0525
0.1160
5.26
0.0010
16.3
100.0
10.0
15.5
0.0800
0.00460
0.58500
0.0440
0.1300
4.57
0.0010
14.6
100.0
10.0
35.0
impoundingstarts,L - l, m
Duration of irrigation,min
41.0
51.0
75.0
50.0
60.0
96.0
60.0
77.0
Number
11
11
11
11
11
11
11
11
of stations
After Ram [1969, 1972]; Ram andLal [1971].
105.0
11
SINGH AND RAM: KINEMATIC MODEL FOR SURFACEIRRIGATION
1605
tedagainst
distances
alongthelengthof theborder.In addition,for comparison
of calculated
andobserved
valuesof ad-
MODEL VERIFICATION
The solutionsin the variousdomainsof the irrigation cycle
vanceand recession
times,absolutepercentdeviationPD and
absolute
average
percentdeviationAPD wereused.Thesecalverifythesesolutions,
calculated
(Cal) and observed
(Obs) culations were based on the observed advance times taken
values of advance times, water surfaceprofiles when water every15.2m for thedatasetsK-l-K-7 andevery.30.5m for
wereobtainedusing31 setsof data,asgivenin Tables1-3. To
reachedthe end of the border,and recessiontimeswere plot-
the data sets K-8-K-9.
When observed advance times were
TABLE 4. Absolute
PercentDeviations
(PD) Between
Calculated
andObserved
Advance
andRecession Timesfor the Data Setson FreelyDrainingBorders
Distance
from the
Roth-8
AdvanceTime,
RecessionTime,
min
min
Upstream
End, m
Cal
Obs
PD*
Obs
Cal
PD
0.0
9.1
18.3
27.4
36.6
45.7
54.9
64.0
73.2
82.3
0.0
1.6
4.7
8.6
12.3
15.6
19.2
23.8
27.2
31.1
0.0
2.0
4.2
7.3
10.8
14.4
18.7
23.7
30.2
36.2
0.0
19.7
11.5
17.6
14.2
8.6
2.6
0.3
9.9
14.0
183.0
188.0
191.0
193.0
195.0
197.0
!98.0
198.0
199.0
199.0
181.4
187.8
191.1
193.8
196.2
198.3
200.2
202.1
203.8
205.4
0.9
0.1
0.1
0.4
0.6
0.7
1.4
2.0
2.6
3.2
91.5
35.4
44.1
19.8
200.0
207.0
3.5
Absolute average
percent
deviation
Roth-9
.........
10.73
0.0
9.1
18.3
0.0
2.1
5.9
27.4
36.6
45.7
54.9
64.0
73.2
82.3
91.5
......
1.36
0.0
2.2
5.2
0.0
1.9
12.8
180.5
189.0
193.0
179.9
191.1
196.9
0.3
1.1
2.1
10.1
8.5
18.7
196.0
201.7
2.9
14.3
19.5
24.4
28.2
32.3
37.8
42.4
12.2
16.3
20.3
24.3
29.9
35.0
41.0
17.4
19.4
20.1
13.1
8.0
7.9
3.3
199.0
202.0
205.0
208.0
210.0
212.0
214.0
205.8
209.5
212.9
216.,1
219.1
221.9
224.7
3,4
3.7
3.8
3.9
4.3
4.7
5.0
0.0
1.4
3.3
6.9
8.9
13.1
15.1
0.0
2.8
5.6
8.4
11.2
14.1
16.9
Absolute average
percent
deviation
Roth-10
.........
0.0
9.1
18.3
27.4
36.6
45.7
54.9
11.15
0.0
48.9
41.3
18.0
20.4
7.4
10.5
64.0
18.3
19.8
7.8
73.2
82.3
91.5
20.5
23.3
25.7
22.9
25.6
28.8
10.7
9.1
10.9
......
3.21
182.0
188.0
194.0
199.0
204.0
208.0
211.0
179.0
197.9
207.7
215.6
222.5
228.7
234.5
1.6
5.3
7.1
8.3
9.1
10.0
11.0
.214.0
23•.8
12.1
217.0
220.0
223.0
244.9
249.7
254.3
12.9
13.5
14.1
Absolute average
percent
deviation
Roth-ll
.........
0.0
9.1
16.81
0.0
1.9
0.0
2.3
18.3
517
5.6
27.4
36.6
9.4
12.9
9.0
12.6
0.0
15.1
0.9
4.7
2.6
......
181.0
189.0
9.54
179.3
191.9
0.9
1.6
194.0
198.5
2.3
198.0
202.0
.203.7
208.4
2.0
3.1
45.7
16.5
16.5
0..3
205.0
212.5
3.7
54.9
20.4
20.2
0.8
208.0
216.4
4.0
64.0
24.3
24.3
0.1
210.0
219.9
4.7
73.2
82.3
28.5
32.8
28.9
33.7
1.5
2.5
212.0
214.0
223.3
226.6
5.3
5.9
91.5
37.4
39.4
5.0
216.0
229.7
6.3
,
Absolute average
percent
deviation
.........
3.04
......
3.71
AfterRoth[1971]'
Rothetal.[1974]'Zc= 0.1min,Rt = 0.05,
Ax= 1.524
m'model
iskinematic.
*Absolute
percent
deviation
= [(observed
quantity
-computed
quantity)/observed
quantity].
1606
SINGHAND RAM.'KINEMATICMODELFORSURFACE
IRRIGATION
ß
$INGH AND RAM: KINEMATIC MODEL FOR SURFACE IRRIGATION
<
1607
1608
SINGH AND RAM' KINEMATIC MODEL FOR SURFACEIRRIGATION
Observed
240
Model:
Data
a,
Data Set Roth-9
0
Data Set Roth- 8
•
Calculated
Curve
Advance
K•nemat•c
Observed
KWT
-------Recession
•- r- •' % •'o o
•-
Data
Data Set R-
ß
Data
Set
60 /"•'
Recession
R-3
R-3•-
j.
Roth-9
Curves
Advance (KWT)
R-
0
o
Calculated
Data
Set
'• •
*
ß
*
'• •' -- '"'
•.
•R-2
160
••
•
•
o
•
o
o
o
o
E 120
80
_
0
0
I •,•
20
I
I
I
40
60
80
DISTANCE,
20
40
g•
øth-8
I
60
DISTANCE,
80
I
m
Fi•. 7. Advance
andrecession
curves
for thedatasetsE-I, E-•, and
E-3.
I0 0
rn
Fig. 5. Advanceand recession
curvesfor the data setsRoth-8and
Roth-9.
Absolutepercentdeviationsbetweencalculatedand observedadvancetimes were compbtedfor all the data setsas
not availableat regularintervals,they wereinterpolatedfrom
shown in Tables 4-6. The values of PD are between 0.0 and
48.9 for the data sets Roth-8-Roth-11.
For these data the
figures[Kincaid,1970].No data wereavailableon recession absoluteaveragepercentdeviation(APD) rangesfrom 3.04 to
16.81(Table4). The PD and APD are between0.0 and 191.1
time and water surfaceprofilefor the data setsK-l-K-7.
and 9.2 and 75.5 for the data sets K-l-K-9
Advance
and between 0.00-
53.3 and 3.60-18.39 for the data setsR-I-R-18, respectively
Thecalculated
andobserved
advance
timesaregivenin (Tables5 and 6). High PD valuesin almostall casesarein the
Table4 for the data setsRoth-8-Roth-11,in Table 5 for the
initialstages.Sincethe advancetimeis smallin thebeginning,
evena smallabsoluteerror (AER) will causehigh PD. APD is
Theseare plottedagainstdistances
for the sampledata sets below25.69exceptfor the data setsK-l, K-2, K-3, and K-4,
Roth-8-Roth-9 in Figure 5, for the data sets K-4-K-6 in where the APD values are 75.52, 31.62, 36.36, and 43.9, respecFigure6, and for the data setsR-l-R-3 and R-10-R-12 in tively (Table 5).
Figures7 and 8. For the data setsRoth-8-Roth-ll and R-IR-18, calculated,advancetimes are in good agreementwith
data setsK-l-K-9, and in Table 6 for the data setsR-I-R-18.
observed times. For the data sets K-l-K-9
the observed and
Model
computedtimesdo not compareveryclosely.
Observed
ß
DataSetR-12
I
•0 Data
Data Set
Set KK- 4
5
/
IOO
a
Calculated
Data
Data Set R-IO
Data Set R-II
120
IOO
:- Model
K•nemahc
/
Observed
Data
/
K•nemat•c
•
o
Curves
•
Advance (KWT)
----
Recession
r- R_12....•"'
ß
ß
ß
...
80,r- -Advance
ß Calculated
DotoSetK-6
I
Curve
(KWT)A
40
O
o
A
O
OO
ß
K-6
20
J
0
0
40
80
120
160
200
•
240
DISTANCE,
m
Fig.6. Advance
curves
forthedatasetsK-4,K-5,andK-6.
100
DISTANCE,
rn
Fig.8. Advance
andrecession
curves
forthedatasets
R-10,R-11,
andR-12.
SINGH
Model
ß•
0
.04
-
•
•
.o2
•
.01 -
KINEMATIC
MODEL
FOR SURFACE
o
Data Set R-I,
Dato Set R-3, Time = 59.0 rmn
Calculated Profile (KWT)
o
F F
•
0
Roth-9
.0•
0
;
0
0
20
40
60
DISTANCE,
FiB. •.
Time= 22. Stain
.04
I 0 Data
Set
R-2,
T•rne
=37.0rain
ß
• •
•
1609
Model
K• nemat•c
Observed Data
ß•
•
•
IRRIGATION
K•nemat•c
Observed Data
Data Set R-9, T•me -- 41 0 rnln
Data Set R-IO,Time =28.Stain
Calculated Profile (KWT)
o
•
AND RAM'
80
20
40
I00
60
DISTANCE,
80
I00
m
m
Watc• surfacep•o•]csfo• the data setsEoth-9 and Eoth-]O.
Fig. 11. Water surfaceprofilesfor the data setsR-l, R-2, and R-3.
Water SurfaceProfile
thehydrodynamic
(HDK)model
ofKincaid
etai.[1972],
the
Observedand calculatedwater surfaceprofiles are plotted
in Figures9 and 10 for the data setsRoth-8-Roth-ll and in
Figures 11 and 12 for the data setsR-l-R-3 and R-10-R-12.
Theseprofilescorrespondto the time when water reachesthe
end of the border. These figures consistentlyshow that in
domain D•, the calculateddepths are lessthan the observed
depths on the border but the agreementbetweenthe two is
satisfactory.For a quantitativecomparison,no statisticalmeasureswere computed,for the depth of flow affectsadvancefor
which comparisonsare discussed
already.
zero inertia models of Strelkoff and Katopodes[1977] and of
Ram et al. [1983] designatedrespectivelyas ZIS and ZIR, and
the kinematic wave (KWS) model of Smith[1972]. Two setsof
data to which applicationsof thesemodelshave beenreported
in irrigation literature were selected:Roth-9 and K-6. Figure
13 comparesfor the data set Roth-9 values of observedadvancewith thosecomputedby the KW, ZIS, and ZIR models.
The ZIS and ZIR models are comparable.The values of advancecomputedby thesemodelsare in closeagreementwith
observations. The KW
model is the least accurate of the three
for this set of data. As seen from Table 4, the PD of the KW
Recession
model increasesin the beginningup to about 20 min and then
decreases
significantly.
The observedand computed recessiontimes are given in
Figure
14
showsa comparisonof the observedadvancefor
Tables4 and 7 for differentdata sets.Theseare plotted for the
sampledata setsRoth-8-Roth-9 in Figure 5 and for the data the data set K-6 with the advance computed by the KW,
setsR-l-R-3 and R-10-R-12 in Figures7 and 8. For the data KWS, ZIR, and HDK models. The KW and KWS models
setsR-l-R-18 the observedrecessiontimes are available (and yield comparable results. The ZIR and HDK models give
thereforeplotted) only to the point wherethe impoundingof comparableresultswhich agree closelywith observationsup
to about 25 min but begin to diverge thereafter. The KW
water starts. The observed and calculated recession times for
model
giveslarge errors in the beginningup to about 25 min,
all the data sets except Roth-10, R-6, and R-12 are in close
as
seen
from Table 5, but yields sufficientlyaccurateresults
agreement.
Absolute percent deviations between calculated and observedrecessiontimes were computed for all the data setsas
given in Tables 4 and 7. The PD and APD ranged between
0.0-14.1
and 1.36-9.54
for the data sets Roth-8-Roth-ll
and
between 0.0-21.1 and 4.72-13.49 for the data sets R-I-R-18,
respectively.This showsa good agreementbetweencalculated
and observed recession times.
thereafter.
SOURCES OF ERROR
There can be a multitude of reasonsfor discrepanciesbetween
observations
and
the results
obtained
from
the KW
model. Some of these reasons are discussed here. One of the
assumptionsin the KW model is that the depth of surfaceflow
A COMPARISON WITH OTHER MODELS
A limited comparisonof irrigation advanceyielded by the
kinematic wave (KW) model was made with that yielded by
.O6
I
'•
Model.
K•nemahc
Observed
Data
Data Set R- I0, T•rne: 41 0 rmn
0
DataSetR-II ,Time= 510m,n
ß
Doto Set R-12,T•me = 75.Omen
-- Colculoted
Profile
(KWT)
Model'
Observed
0
ß•
•
.03
[
E .02
r- RothI0
.04
K•nemat•c
E
Data
_
Data Set R-8, TI me = 44 I rn•n
Data Set R-II, T•me=39 4rain
Calculated Profile (KWT)
,,
'•
'•
_Roth-II
n
01
Ixl'
o
0
I
I
I
20
40
60
C3 .02
0
0
DISTANCE,
I
80
I00
m
Fig. 10. Water surfaceprofilesfor the data setsRoth-8 and Roth-I l.
0
20
40
60
DISTANCE,
80
I00
m
Fig. 12. Water surfaceprofiles for the data sets R-10, R-11, and
R-12.
1610
SINGH AND RAM: KINEMATIC MODEL FOR SURFACE IRRIGATION
8O
ß•
Observed
Colculoted
•
Doto Set Roth-9
Advonce
Curve
6O
.E
E
DEL
u• 40
2O
0
0
20
i
i
I
i
40
60
80
I00
DISTANCE,
rn
Fig. 13. Advancecurvesfor the data setRoth-9.
increasesfasterthan it doesin caseof hydrodynamicand zero
inertia models.The consequence
is sloweradvance,especially
in the initial stagesof irrigation, as seenfrom Tables4-6. This
effect becomesmore pronouncedas the bed slope decreases
and precludesuseof the KW model for zero slopebordersof
small lengths(or level basins).
The KW model assumesuniform entry of water into the
border at its upstream.This assumptionis nearly satisfiedby
the Roth and R data sets but not by the K data sets.This
partly explainsthe larger differencesobservedat early times
for the K data sets.Furthermore, values of re, Rt, and Ax
changefrom one border to another. In the KW model these
were assumed to be constant for each of the three sets of data.
Irrigation advanceand recessionare found to be very sensitive
to/• and K. For example,/• is different for advancefrom that
for recession. Their
accurate
determination
is one of the most
important considerationsaffectingaccuracyof the KW model.
For each border, averageestimatesof theseparameterswere
used.An investigationinto parameter sensitivity,error analysis, and comparisonof models is currently being carried out,
the resultsof whichwill be reportedseparately.
CONCLUSIONS
The followingconclusionscan be drawn from this study.
1. The kinematic wave model predicts the advance time
sufficiently accurately for the data analyzed here. In most
8O
ß•
0
ß
•
Observed
Dote Set K- 6
KWS MODEL
ZlR
MODEL
Colculoted
Advonce Curve
6O
KW
MO•DEL
u• 40
ß
20
I
40
80
i
I
I
I
120
160
200
24O
DISTANCE,
rn
Fig. 14. Advancecurvesfor the data set K-6.
o o. o.o. o.o o o o o o. o
'•
SINGH AND RAM: KINEMATIC MODEL FOR SURFACE IRRIGATION
1611
casesthe prediction error remains below 20%. A large error
normally occursin the beginningof irrigation.
2. The model predicts the horizontal recessiontime reasonablywell. In most casesthe predictionerror remainsbelow
15%.
3. The model is not capableof accommodatingthe vertical
recession.
4. The agreementbetweenobservedand computed water
surfaceprofilesis satisfactory.Since the depth of flow is normally very small,a small error in predictionmay appear large.
5. For the data analyzed here, the model is sufficiently
accuratefor modelingthe entire irrigation cycleexceptfor the
vertical recession.
Acknowledgment.This study was supportedin part by funds provided by the National ScienceFoundation under the project Free
Boundary Problems in Water ResourceEngineering,NSF-ENG-7923345.
REFERENCES
Bassett, D. L., and D. K. McCool, A mathematical model of water
advance and flow in small earth channels, project completion
report, Dep. of Agric. Eng., Wash. State Univ., Pullman, 1973.
Bassett,D. L., D. D. Fangmeier,and T. Strelkoff, Hydraulics of sur-
face irrigation, in Designand Operationof Farm Irrigation Systems,
edited by M. E. Jension,pp. 447-498, AmericanSocietyof Agricultural Engineers,St. Joseph,Michigan, 1981.
Chen, B. J., R. C. McCann, and V. P. Singh, Numerical solutionsto
the kinematic model of surfaceirrigation, Tech. Rep. MSSU-EIRSCE-81-1, Eng. and Ind. Res. St., Miss. State Univ., Mississippi
State, 1981.
Chen, C. L., Mathematical hydraulicsof surfaceirrigation. Tech. Rep.
PR-WRll-2, 98 pp., Utah Water Res. Lab., Utah State Univ.,
Logan, 1966.
Chen, C. L., Surfaceirrigation usingkinematic-wavemethod. Journal
of the Irrigation and Drainage Division, J. lrrig. Drain. Div. Am.
Soc.Civ. Eng.,96(IR 1), 39-48, 1970.
Criddle, W. D., S. Davis, C. H. Pair, and D. G. Shockley,Methods for
evaluatingirrigation systems,Agric. Handb.82, Soil Conserv.Serv.,
U.S. Dep. of Agric.,Washington,D.C., 1956.
Cunge, J. A., and D. A. Woolhiser,Irrigation systems,in Unsteady
Flow in Open Channels,edited by K. Mahmood and V. Yevfevieh,
pp. 522-537, Water ResourcesPublications,Fort Collins, Colo.,
1975.
Fread, D. L., Flood routing: A synopsisof past, present and future
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(ReceivedOctober 29, 1982;
revisedJuly 21, 1983;
acceptedAugust4, 1983.)