A Distributed Converging Overland Flow Model 2. Effectof ... - OAKTrust

VOL.

12, NO. 5

WATER

RESOURCES

RESEARCH

OCTOBER

1976

A Distributed ConvergingOverland Flow Model 2.

Effect of Infiltration

BERNARD SHERMAN AND VIJAY P. SINGH New Mexico Institute of Mining and Technology,Socorro,New Mexico 87801

The overlandflow on an infiltrating convergingsurfaceis studied.Mathematical solutionsare developed to studythe effectof infiltrationon nonlinearoverlandflow dynamics.To developmathematical solutions,infiltrationand rainfall are representedby simpletime and spaceinvariant functions.For complexrainfall and infiltration functions,explicit solutionsare not feasible.

INTRODUCTION

Overland flow and infiltration have beenextensivelystudied as separatecomponentsof the hydrologiccycle[Woolhiserand Liggett, 1967; Woolhiser, 1969; Kibler and Woolhiser, 1970; Singh,1974;Lane, 1975;Philip, 1957;HanksandBowers,1962; Whislerand Klute, 1965; Rubin, 1966]. A combined study of these phasesis required for modeling overland flow. With a few exceptions,notably the work by Smith [1970] and Smith and Woolhiser[1971], the conventionalapproach [Wooding, 1965;Eagleson,1972;Singh, 1975] to combiningthesephases has been through the familiar notion of so-calledrainfall excess.In this approach,infiltration is independentlydetermined and subtracted from rainfall; the residual is termed rainfall excess,which forms input to the overlandflow model. It seems to us that this conceptof rainfall excessis more an artificethan a reality. The processesof rainfall, infiltration, and runoff occur concurrently in nature and therefore warrant a combined study. The purposeof this paper, part 2 of a series,is to considerinfiltration in the convergingoverland flow model and then develop mathematical solutions for overland flow. The mathematical treatment developedhere is usefulin studying the effectof infiltration on nonlinearwatersheddynamics.

where u is the local mean velocity and a and n are kinematic wave parameters.As before,q(x, t) = 0 when t > T and n > 1. The boundary conditions are

h(x, O) = 0

(3)

h(0, t) = 0

0 < t < T

It is plausibleon physicalgroundsthat therewill be a curvet = tø(x) in{t > T, 0 T. We distinguishthree casesA, B•, and B2 (Figure 2) which dependon the relativedispositionof the threecurvest = tø(x), t = T, and t = t(x, 0); t = t(x, x*) is the prolongation of t = t(x, O) to the right of x = x*. In caseA, tø(x) > T > t(x, 0), 0 < x < L(1 - r). In caseB•, tø(x) > T, and tø(x) > t(x, 0), but t = T and t / t(x, 0) intersectat x / x*; i.e., T = t(x*, 0), and 0 < x* < L(1 - r). In caseB:, tø(x) > T, but t = Tand t = t(x, O) intersectat x = x*, and t = tø(x) and t - t(x, x*) intersectat x = ./; i.e., tø(./) = t(œ,x*), and 0 < œ < L(1 - r). Since tø(x) and t(x, 0) are not known until we have solved the problem, it appearsthat we cannot distinguishthesecases beforehand.But in the specialcasewhich we considerin this three casesbeforehand. The domains D•, D:, and Do in caseA and the domains D•, D•:, D:, and Do in casesB• and B• are

indicated in Figure 2. In caseA the solutionsin D: and Do when q and f are constantare obtained from the discussionin part 1. Let

d

L--

dx

x

or(x)

< 0

(15)

the curvest = t(x, Xo) do not intersectin Ds (Appendix B of part 1). The free boundary t = tø(x) is now determinedby

(1 - p)L:-

(L - Xo*)•' + p(L - x) • = 0

(16)

and (14). EliminatingXo* between(16) and (14), we get (here w - n-1(2/f)("-1)/n.)

tø(x) = T+ w

),/,, L--r/

ß (L- r/)'" -- (L -- x)2

dr/ (17)

where

X(x) = L-

[(1 - jo)L9'-J-jo(L- x)2]1/2

As in part 1, for fixedx, h(x, t) is an increasingfunctionof t in Do, independentof t in D2, and a decreasingfunctionof t in D• (Figure 3). The criterion for distinguishingbetweencaseA and casesB• and B2is, as in part 1, obtained from

T, = 1

Then in D: the solutionis given by (13) and (14) of part 1, • and 7 being replacedby •* and T*'

(14)

The curvest = t(x, to)do not intersectin D:, the curvest = t(x, x0*) do not intersectin D•, and on the assumption

paper,q(x, t) andf(x, t) bothconstant,wecandistinguish the

•, =

ß

t

Fig. 1. Rainfall and infiltration, constantin spaceand time.

q, = q_ ]

•

T='r*

[

ce(r/),/,, 1 'L'•--(L- r/) 2

dr/ (18)

If (18) doesnot have a root between0 and L(1 - r), then we are in case A; if there is such a root x*, then we are in case Bx

h(x = [ .(x)(L--( --x) 3 ' t,,)

(9)

or caseB:. If F(x) is the right sideof (18), then caseA occursif and only if F[L(1 - r)] < T, and case Bx or caseB: occursif and only if F[L(1 - r)] > T. To distinguishbetweencasesBx and B:, we note, referringto (13), that

(10)

(1 - p)L • - (L - x*) • + p(L - x): = 0

t(x, t,,) = t,, + l

ß or(r/) •/" L'•-- (L -- r/)2dr/

In Ds the solutionis givenby (18) and (19) of part 1, t5and 3/ being replacedby/5* and 7*'

.(x)(L --(L x)--x)2] h(x, Xo) =fl,[(L-Xo)2--

(19)

doesnot have a root between0 and L(1 - r) in caseB• and does have such a root ./in case B:. Such a root exists if and

only if

(1 )

L2r: < p-X(L - x*): - [(l/p)-

1]L2

or

1 - p(1 - r•') < [(L - x*)/L]: do

(12)

(20)

Thus if (20) is true, we are in caseB•.,and otherwisewe are in case B•. In case B• the intersectionof the curvest - t(x, x*)

SHERMAN ANDSINGH: FLOWMODELING, 2

899

and t = tø(x) occurs at

• = L - {(1/o)(L - x*) •-

[(l/o)-

llL:} •/:

(21)

?= T+co a(r/) '/" (L-- r/)•'-- (L-- •)• d• (22) We discussnow the solution in casesB• and B:. In both

casesthe solutionin D,, is givenby (13) and(14), in D: by (9)

and(10), andin Da by (11) and(12). It remains to determine the solutionin D•:. As in part 1, we defineXo*by T = t(Xo*, Xo);herex* • Xo*• L(1 - r). Thusfrom (12),

D• t: t(• D•

T=7*

••o * 1

ß o •(•)'/'•

L(I-r)

• -- (L- •)•••,,__,•/,, d• (23) •(L- Xo)

Fig. 2a. Solutiondomainfor caseA.

Then from (16) and (17) of part 1, h(x; Xo*, Xo)

= • (1--p)(L-xo) --(L--Xo*+ p(L-x)2 •/'• a(x)( L -- x)

(/•gl t= tø • t:t(x,x•/•

(24)

t(x;Xo*, Xo)= r -Q'y , •-•r/-•i?;;

L(L--xo* --r/ )•'+ p(L_r/)2 ßI(1--p)(L-Xo) 2-I(n-1)/n dr/

D•

': D•

(25)

It isprovedin AppendixA thatthecurvesdefinedby(23) and (25) do not, on condition(15), intersectin D•2. In caseB2,part of the boundaryof D•2is t = tø(x).Thisis obtainedby eliminating x0andx0*between(23) and(25), and

L( I-r)

X•

-x

Fig. 2b. Solutiondomainfor caseB•.

from (24),

(I

--

p)(L - Xo)-•-- •,• ,r - Xo,)2 + p(L - x) 2 = vn

From (26) we get Xo* = X(x, Xo),where

X(x, x0) = L-

[(1 -p)(L-x0)

2+p(L-x)2]

u2 (27) t= t(x,x •()

Thus t = tø(x) is definedby

T=•*

,t=tø(x ') /.•

o)a(r/)1'/" •x X(x ,x

•,'

ßo

I

L--r/

!(n-I)In dr/

(28)

ß (L- Xo) 2-- (L- r/)2

i

D•

D•

t(x, Xo) = T q- co ,/,, ß (.... > a(r/) i ß

(L- r/)•' -- (L- x)•'

dr/

(29)

In (28) and(29),)• < x < L(1 - r); when0 < x < )•, tø(x)is

x*

•'

L_(I-r)

Fig. 2c. Solutiondomainfor caseB•..

defined by (16).

The behaviorof h(x, t) asa functionof t for fixedx, 0 < x < x*, is the samein casesB• and B2asit is in caseA (Figure3). In casesB• and B2,ht(x, t) > 0 when(x, t) G Da, and ht(x, t) < 0

CONCLUDING REMARKS

The mathematicalsolutionsdevelopedabove demonstrate

when(x, t) • D•; thearguments arethesameastheyarein that simultaneousconsiderationof the rainfall and infiltration of thehydrologic cyclealtersthecharacter of overland caseA. The maximumof h(x, t) occursthereforewhen(x, t) G phases To stimulatethe watershed response realisD•2(Figure3), but it canoccuron the boundaryof D•2asin ttow dynamics. studyisessential, although wedorecogpart 1(0 = 0). Thecaseor(x)= otisdiscussed in greaterdetail tically,theircombined nize that this enhancesthe mathematicalcomplexity. in AppendixB; Figure4 illustrates the possibilities.

900

SHERMAN ANDSINGH: FLOW MODELING, 2

.-. % ,.

,-- •

o '--4O

v v •

ß

•

0•'

I

•V

• ¾1

.,_

SHERMAN AND SINGH: FLOW MODELING, 2

901

A simple calculation shows thatthesignofg'(z)andthereAs wasnotedin part 1, theanalysis in thispapercanbe by carried outontheassumption thatq(x,t) = q(x)andf(x, t) = forealsothesignof ht(x,t) is determined

f(x) withonlya slight increase in mathematical complexity. l/n Butthemainfeatures arealreadycontained in thecaseq(x,t) k(z)= n(1-- p- z2)-- [z(1--z2)n--l] = q andf(x, t) = f discussed above andin theappendices.

.•

APPENDIX A

d• (B5)

It follows fromAppendix Bofpart1thatthecurves t = t(x, For a fixedx, T _