Mein_Larson_infiltration.pdf

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Hydrological Sciences - Journal - des Sciences Hydrologiques,3 5 ,2, 4/1990

Derivat ion of an expl ici t equat ion forinfil tration on t h e b a s i s o f t h e M e i n - L a r s o n

m o d e l

BERMHARD H. SCHMEDInstitut fur Hydraulik, Gemsserkunde und Wasserwirtschaft,Technische Universitt Wien, A-1040 Vienna, Austria

Ab strac t Proceeding from the well-known infiltration model byMein & Larson (1973) and several extensions, explicit equationsfor time-dependent infiltration rate and cumulative infiltrationare proposed in this paper. Although the time-range of validity islimited due to the chosen mode of derivation, the formulae seemwell suited for application in the course of storm runoff modellingfor small catchments. Major advantages of the proposed equationsare constituted by the comparatively simple structure and by ahigh degree of versatility allowing a considerable number of effectsto be explicitly accounted for. The parameters involved can beidentified from measurements. Rules of thumb regarding therange of applicability are given.

Etablissement d 'une quation explicite dcrivant rinfiltration surla base du modle de Mein-Larson

Rsum Cet article propose des quations explicites destines l'valuation du taux d'infiltration et de l'infiltration cumulative enfonction du temps. Cette mthode repose sur le modle deMein-Larson et ses diffrentes extensions. Maigre la validitlimite dans le temps de ces quations, due la mthode choisie,les formules sont bien appropries pour les modles d'coulementd'averse dans les petits bassins hydrologiques. Les principauxavantages des quations proposes rsident dans la simplicit de

leur structure et leur haut degr de versatilit qui permet de tenircompte directement d'un nombre maximum de donnes influantsur le modle. Les paramtres utiliss peuven t tre identifisd'aprs les donnes mesures. On trouv era enfin un e rglepratique concernant leur champ d'application.

INTRODUCTION

Mathematical simulation of overland flow has received a great deal of interestfor about two decades. Since a hydrologically realistic view of this process

cannot content itself with the calculation of flows over impermeable planes,the infiltration compo nent gains in importance. Althoug h the Richardsequation describing soil water movement has been known since 1931, thecomplexity of overland flow calculation justifies the development and

Open for discussion until 1 October 1990 197


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BernhardH. Schmid 198

subsequent use of approximate infiltration models, especially if kinematicshock waves are to be included in the analysis. One of the best knownphysically-based infiltration models accounting for both pre-ponding andpost-ponding stages was presented by Mein & Larson (1973). Comparisonof the results obtained by this algorithm with solutions of the Richardsequation was favourable and extensions regarding time-varying rainfallintensity (James & Larson, 1976; Schmid & Gutknecht, 1988) as well as amore general expression for ponding time (Smith & Parlange, 1978) havesince been proposed. Con tributions by Chu (1978) and Kutflek (1980)are noteworthy.

In the context of semi-analytic overland flow models involving shockdynamics an explicit expression for rainfall excess and therefo re infiltrationrate as well as cumulative infiltration would be preferable to an iterationalgorithm. For example, a necessary criterion of kinematic shock formationdue to the intersection of a characteristic originating from the f-axis atsome time t = tBQ (corresponding to a water depth hBQ) and theso-called limiting characteristic starting fromx = 0 at ponding time t !treWdt (D

P

where re denotes the time-dependent rate of rainfall excess (rainfall intensity

minus infiltration rate). In this instance it is obvious that an explicitexpression for cumulative infiltration will greatly facilitate the practicalevaluation and subsequent discussion of equation (1 ). Several other examplescould be given in the course of a full treatment of kinematic shocks, which is,however, beyond the scope of this paper. The reade r is referred to Schmid(1990) for further information.

Since the Mein-Larson model together with the extensions mentionedhas proved a versatile and sufficiently accurate method, the development ofan explicit equation on the basis of this method seems justified and may, asindicated above, lead to a useful tool for further overland flow studies.

FUNDAMENTALS

The basic equations corresponding to those derived by Mein & Larson (1973)can be given as:

infiltration capacity:

/ c = ^ - [ 1 + (5ov + ^ 3 (2)

infiltration rate:

fr = min(r, fc) (3)


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199 Derivation of an explicit equation forinfiltration

continuity:

c ^ ^ - r p / o . - e . ) (4)

with r = rainfall intensity;/ = infiltration capacity;/ = infiltration rate ;S^ = average suction at the wetting front;h = water depth at the ground surface;z. = depth of the wetting front (zero datum is ground level);

saturated vertical conductivity of the soil;celerity of the wetting front;antecedent rainfall intensity;volumetric water content of the soil at saturation; andinitial volumetric water content.

Soil water hysteresis need not be considered, since only the imbibition case istreated here. Thus, the suction 4> < 0 can be described as a single-valuedfunction of the relative hydraulic conductivitykr = KiByK^ with K dependingon the volumetric water content 9.S can now be defined by:

s

Csri

* o v =r, i

fl.0

r, i

*(fcr) kr (5)

with kfi = r./K . Following the recommendation by Mein & Larson (1973),the minimum lower boundary of the integral in equation (5) will be taken as0.01 in order to avoid difficulties during numerical integration.

An illustration of equation (5) is given in Fig. 1.

kr0 kr /l 1.0

Fig. 1 Illustration of the integral term in equation (5).


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BernhardH. Schmid 200

DERIVATION OF THE EXPLICIT EQUATION

Preparatory to the subsequent analysis, a relation for the time of ponding (t)must be obtained. If the rainfall intensity at ponding time is denoted by r.the basic equation may be written as:

'p'ffiJ-U'p) (6)Therefore, from equation (2):

After rearrangement of terms, the depth of the wetting front at ponding time(z ) can be computed by:

s,p - x (S ^

+ / )t


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201 Derivation of anexplicit equation for infiltration

with FI,P= I'" O - r J * and , - r(tp). n

For the special case ofr = constant, the time of ponding can be givenas:

p sv

V*h)-(*,- e.)

(r - KJ-ir - r.)(10)

Smith & Parlange (1978) pointed out that an equation like equation(9)can be used successfully to determine the time of ponding for arbitraryrainfall patterns, r = r(t). After t has been calculated, the only restrictiontobe imposed on the arbitrariness of time-varying rainfall intensity is the

condition r( t > t ) fc(t > t ), which will be satisfied in most cases due to therapid decay of infiltration capacity after ponding.For t Z t , f c can be replaced by f r in equation (2), which, in turn, is

equal to the time-derivative of the cumulative infiltration. DenotingF1 + F tas F (total shaded area in Fig.2) and employing a relationship for z ' similarto equation (8) for z . equation (2) may be rewrittenin the following form:

dF1l +


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Bernhardt!. Schmid 202

which, except for the inclusion of r., agrees with previously derived results (e.g.Chu, 1978).

For clarity, replacement of the term [K -A/(K - r ()] by B seems

indicated. Thus, equation (13) becomes:

F t - IMn( l + FXIB) - Flp + I M n ( l +F lp/B) = K^ -A (f - tp)/B

(14)

Recalling that F t = F t + F j . cancelling and rearranging terms leads to:

Fx - B-\n[l+Fxl(B * F ^ K ^ A ClB (15)

with t P .At this point, there are several possibilities of expanding the logarithminto a Taylor series. After investigation of various alternatives, the mostsatisfactory way was considered to be:

In 1 +B + F.hp .

lB + F.hP 2

1B + F.hp .

+ 0B + F.hp J

(16)

With regard to equation (16) three aspects should be pointed out:(a) the range of convergence of the infinite series is limited byFxl

(B + F1 ) 1. Thus, in spite of secon d-orde r accuracy, the correctasymptotic behaviour offf -* K^ for t -* will no t be repro duced ;

(b) consideration of the above limit leads to the postulate tha t the secondterm of the argument of the logarithm should be as low as possible.Tha t is why ejuatio n (16). is superior to any form containing theargument (1 + FJB), since FJB > FXI{B + Fx \ Derivation of such anexpression is, of course, possible, but leads to poor results; and

(c) a Taylor series witho ut an upp er limit to the range of convergence hasalso been studied. Although this method was able to reproduce thecorrect asymptotic behaviour of the infiltration rate, it was rejected forthe following reason: the series consisted of rational functions so thatany approximation beyond the first order led to an equation of adegree larger than or equal to three. While the first-order approximation, as expected, turned out as too rough, analytical solutions tothird-order equations are too complex to be of practical interest, sothat, finally, a decision in favour of equation (16) was made.

Substitution of the truncated series of equation (16) into equation (15) leads

to :

BB + F.hP

B+

2 B + F.hp JKw- A - tIB (17)


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from which the required explicit relations for the total cumulative infiltration,F, and the infiltration rate, respectively, are finally derived:

F~ \Pr(t)dt + r,t +K-


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Bernhardt H. Schmid 204

g * - >

acrc

I H

3-

2-

0-

0

- | r

8 12 16 20 24Time (min)

Fig. 3 Columbia sandy loam: infiltration rates under steady rain.r = 3.336 mm min'1; 9 . = 0.125 (case A) and 9 . = 0.200 (case B),respectively; full lines= explicit equation (19); dashed lines =iteration algorithm; anddash-dotted lines = Richards equation.

co

C D

>

J3

3o

Time (min)Fig. 4 Colum bia sandy loam: cumulative infiltration under steadyrain, r - 3.336 mm min'1; 9 . = 0.125 (case A) and Q{ = 0.200 (caseB), respectively; full lines= explicit equation (18); dashed lines=iteration algorithm; anddash-dotted lines = Richards equation.

In general, it may be inferred from Table 1 that errors related to thecumulative infiltration are smaller than those related to the correspondinginfiltration rate. Accordingly, the range of applicability of equation (18) willbe wider than that of equation (19).


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205 Derivation of anexplicit equation for infiltration

S8

,3*3Is

.55

ill

1

CN fr> CN > 00 ^i - < N 'd d od ^

a3a333$35

$ *-, *-( ^ --(

.ssssali

On O ** *>"i

s

SSSSSSSSSS

It J


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Bernhardt!. Schmid 206

Unfortunately, the theory of error propagation which might be used fortracing errors from the Taylor series expansion down to the final results does notyield any easily handled criterion of applicability as would be required in this

context. It may, however, be assumed that errors in the final result will depend onthe initial erro r committed in the course of the truncation of equation (16). Thus,the second term of the logarithmic argument, i.e.F t/(B + F t )= 8, (normalizedcumulative infiltration) may be taken as a measure of the expected final error.The relative initial erro r as a percentage defined by:

ln (l + S) - S + 0.5 82

ln(l + 8)100 (20)

is a single-valued function of 8, the range of which can be seen from Table2, with 8 = 1.0 as the limit of convergence of equation (16).

Table 2 Relative error depending on normalized cumulativeinfiltration

B + FIP

e(%)

0.10.20.30.40.50.60.70.80; 91.0

0.31.32.84.97.5

10.614.318.322.927.9

If F1 is approximated by the third term of the right hand side of equation(18), the maximum permitted value of t* (time counted from ponding) can beexpressed as a function of 8, which finally yields the criterion required:

1

(r P-?&-rf

8 2 . r 2 + 8 . ( 8 - 2 ) - 4 , + 2 -8 .( 1 - s j - ^ - r + 2 - 8 - ^ - r . -

2 - 8 - V i

(21)

For r. close to zero, the above equation can be simplified:


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W = 2 ' K -{r-Kf (22)

8 2 . / ^ + 8 - ( S - 2 ) - ^ + 2 - 5 . ( 1 - 6 ) . ^ - ^

If the sam e erro r margin, e.g. 10%, is applied to the infiltration ra te and thecumulative infiltration, respectively,t* mWL will be larger with regard to thelatter thus reflecting the wider range of applicability of equation (18).

Application of equation (21) to the set of data given in Table 1 showedthat the following recommendations regarding the order of magnitude oftolerable initial relative errors can be made:(a) the choice of e = 3% corresponding to 8 = 0.31 mostly leads to errors

in infiltration rate around 10% att* = t* mwi . This figure applies verywell to Guelph loam but it can be increased for coarser soils (Columbiasandy loam), e.g. e = 4% to 6% (5 = 0.36 to 0.44) and should beslightly lower for finer soils (Ida silt loam, e = 2%, 5 = 0.25). In thelatter case, errors above 10% may sometimes result, but, since theamount of rainfall excess is considerably larger than that of infiltrationin such situations, these errors cannot exert any great influence on thecalculation of surface runoff and may, therefore, still be considered astolerable; and

(b) the e value recomm ended in the context of cumulative infiltrationcalculation is 8% (6 = 0.52) reflecting the wider range of applicability ofequation (18). This figure was in good agreement with the resultsobtained for Guelph loam and Columbia sandy loam (here, again, itmay be increased by 1% to 2%). For finer soils, e should be chosen asabout 6% (8 = 0.44).

CONCLUSIONS

In the context of overland flow modelling, an explicit equation describing the

process of infiltration is desirable, especially if aspects of kinematic shockrouting are to be considered. Such an equation can be derived from thealgorithm presented by Mein & Larson (1973). It has been shown to besimple in structure and can therefore be handled easily. Results are sufficientlyaccurate within a certain range of time after ponding, which can be estimatedby means of a given rule of thum b. The equation proposed is particularlywell suited for the computation of cumulative infiltration as required in thecourse of the integration of kinematic overland flow characteristics.

Acknowledgements Thanks are due to Professor Dieter K. Gu tknech t of theInstitut fur Hydraulik, Gewasserkunde und Wasserwirtschaft, TechnischeUniversitt Wien, for kindly reviewing this paper.


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Bernhardt H. Schmid 208

REFERENCES

Chu, S. T. (1978) Infiltration during an unsteady rain.Wat Resour. Res. 14 (3 ), 461-466.Jam es, L. G. & Larson, C. L (1976) Modeling infiltration and redistribution of soil water

during intermittent application. Trans. ASAE19 (3), 482-488.Kutflek, M. (1980) Constant-rainfall infiltration. /.Hydrol. 45, 289-303.Mein, R. G. & Larson, C. L. (1973) Modeling infiltration during a steady rain.Wat. Resour. Res.

9 (2), 384-394.Schmid, B. (1990) On kinematic casca des: derivation of a generalized shock formation

criterion. /. Hydraul. Res. (in pre ss).Schmid, B. H. & Gutknecht,_D. K. (1988) Bin Ingenieurverfahren zur Infiltrationsberechnu ng

mit Taschenrechner. sterr. Wasserwirtsch. 40 (7/8), 175-183 (in German).Smith, R. E. & Parlange , J. -Y. (1978) A parameter-efficient hydrologie infiltration model.Wat.

Resour. Res. 14 (3 ), 533-538.

Received 23 September 1988; accepted 29 July 1989

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