SOIL SCIENCE SOCIETY OF AMERICAJOURNAL

VOL. 49 MARCH-APRIL 1985 No. 2

DIVISION S-l-SOIL PHYSICS

Flux and Water Content Relation at the Soil Surface1

J.-Y. PARLANCE, W. L. HOGARTH, J. F. BOULIER, J. TOUMA, R. HAVERKAMP, AND G. VACHAUD2

ABSTRACTA general analytical equation is presented which relates the sur-

face water content and surface flux. The result is obtained for gen-eral soil properties and arbitrary dependence of the flux on time.The equation is approximate but optimal at least in the short timelimit. Its accuracy is assessed by comparison with numerical andexperimental results for Grenoble sand and when the surface fluxis constant. The analytical and numerical results differ by a smallamount which would however be significant if the analytical resultwas used to predict the time at which a certain water content isreached at the surface, for instance the value of time at ponding. Itis shown that the numerical solution has an intrinsic error in theshort time limit. This suggests that the analytical result may be moreaccurate than the numerical solution. This is reinforced by the goodagreement between the numerical results and an earlier approximateanalytical solution which is known to be often inaccurate, for in-stance if it is used to estimate the value of the sorptivity. Finally,careful experimental observation shows closer agreement with theanalytical result than with the numerical solution.

Additional Index Words: Grenoble sand, soil physics, incipientponding, surface runoff, surface erosion.

Parlange, J.Y., W.L. Hogarth, J.F. Boulier, J. Touma, R. Haver-kamp, and G. Vachaud. 1985. Flux and water content relation atthe soil surface. Soil Sci. Soc. Am. J. 49:285-288.

THE PREDICTION of the increase of the water con-tent, 05, at a soil surface for a surface flux mea-

sured as a function of time, represents an importantproblem of soil physics. Of particular importance isthe determination of the time, tp, when incipientponding first occurs, which determines the onset ofsurface runoff and possible surface erosion. In an ear-

1 Contribution from the School of Australian EnvironmentalStudies, Griffith Univ., Brisbane, Qld 4111, Australia; and Institutde Mecanique, B.P. 68, 38402 Saint Martin D'Heres, Cedex, Gren-oble, France. Received 2 Nov. 1983. Approved 14 Aug. 1984.2 Professor and Lecturer, The School of Australian Environmen-tal Studies, Griffith Univ.; and Postgraduate Student, AssistantTeacher, Research Scientist and Research Scientist, Institut de Me-canique, Grenoble, respectively.

Her study (Smith and Parlange, 1977), gravity wasconsidered for two limiting cases, i.e. when the con-ductivity, K(0), varies slowly with water content 0, orwhen dK/dQ varies like the diffusivity D(0). It appearsthat many soils have a behaviour intermediary be-tween those two limits (Smith and Parlange, 1977,1978). In addition, when gravity is not negligible,Smith and Parlange (1977) limited themselves to thecase when the surface flux varied like a power of time.

In the present study, we do not limit ourselves toany particular soil behaviour, and the time depen-dence of the surface flux is arbitrary. We develop ananalytical expression which relates water content andsurface flux. Previous numerical and analytical modelsare used for comparison, and some recent experimen-tal results (Boulier et al., 1984) are also discussed.

ANALYSISWe consider Richards' equation governing infiltration in

one dimension under the form (e.g., Haverkamp et al., 1977),

dQ/dt = d[D (d&/dz)]/dz - (dK/d®)(d<d/dz) [1]We assume that initially 0 = 6,, a constant. Following

Parlange (1980), but for a water content at the surface beinga function of time rather than constant, Eq. [1] is integratedbetween z and z —» oo to obtain

f°'U, 0,) dz']/dt = - D (d@/dz)

+ (K ~ K,), [2]where Kt = A"(0,). Note that 0 and A: in Eq. [1] have beenreplaced by (0 — 0,) and (K — K,) respectively, which ispossible since 0, is assumed to be constant. Equation [2] isnow integrated between z = 0 and z —> co, yielding

, foo poo r@sd { I dz [ \ (0 - 0,) dz'}}/dt =

»/u »/z */6iDdQ

K-K,)dz. [3]The left-hand side of Eq. [3] is integrated by parts, and wefinally obtain

285

286 SOIL SCI. SOC. AM. J., VOL. 49, 1985

r°° r®sd { JQ (0 - 0,) zdz\/dt - Je Dd® =

- JJV - K,)(dz/d®)d® . [4]We now have to estimate the left and right-hand sides ofEq. [4]. Let us replace the left-hand side by the approxi-mation,

\d [i2/(®s - e,)]/dt - 0 - 20,y(6S - 0,)] Dd&}/2 ,

where the cumulative infiltration, i, is given by

Cmi = \ (0 - 0,)Jo v " dz.

[5]

[6]To justify the approximation of Eq. [5], note first that the

estimate of the left-hand side of Eq. [4] is exact for all timesif 0 varies discontinuously from 0, to @s, i.e. when D is adelta function at 0 = Qs, since z = i/(Qs — 0,) in that case.In addition, when gravity is negligible (Smith and Parlange,1977), the estimate is optimal, i.e. it does not depend sig-nificantly on the shape of the profile when D differs slightlyfrom a delta function. Hence, by using this estimate in Eq.[4], we shall automatically obtain a relation between / and®s which is optimal for short times. The estimate of theright-hand side of Eq. [4] was given in Parlange et al. (1982)and will not be repeated here. Eqs. [5], [9], and [12] of Par-lange et al. (1982) show that

d® ^ [(®s + 0 - 20,)/

s - e,-)] Dd® {l - a-\di/dt)(Ks - K,rl ln{\Ks+ a(Ks - K,)

where (Parlange et al., 1982)[7]

- K)/(KS - K,)] d® [8]and Ks = K(QS).

Combining the left-and right-hand sides of Eq. [4] givenby Eq. [5] and [7] yields

a 5

/{I - i(d®s/dt)/[2(®s - ®i)- K,)} ln{l + a(Ks - ^/[(di/dt) - Ks + K,]} , [9]where S2 stands for (Parlange et al., 1982)

65(05 + 0 - 2®i)Dd®. [10]S2

At ponding, if 05 is the saturated water content, then 5 be-comes the sorptivity and Eq. [10] is known to be extremelyaccurate in general (Brutsaert, 1976; Elrick and Robin, 1981)and can be used for the Grenoble sand considered later(Touma et al., 1984). Equation [9] provides the relation weseek. If 0s(;) is known, then Eq. [9] is an ordinary differentialequation for i(t) to be solved using Runge-Kutta formulaewith /'(O) = 0. If, on the other hand, i(t) is given, then Eq.[9] yields 0^0 with ©^(O) = 0, using an iterative technique.Note that if /(<), for instance, is given, it does not have toobey any special law.

COMPARISON WITH NUMERICAL RESULTSAND DISCUSSION

Here we consider the case of constant flux, i.e. dijdt is constant in Eq. [9]. Numerically, explicit condi-tions must be considered. We shall take a "Grenoblesand" which is described in detail by Touma et al.

( 1 984). In particular, it was found that soil-water prop-erties could be described accurately by analyticalexpressions following the prescription of Van Genu-chten ( 1 980) for capillary pressure (h) and Brooks andCorey (1964) for conductivity, or

and- 0,)/(0Sat -

K = E®F,

[11]

[12]with 0Sat = 0.312, 0r = 0.0265, A = 0.0437 cm-', B= 2.2223, C = 0.55 (= 1 - B-'), E = 18130 cm/hr, F = 6.07. To complete the formulation, we take0, = 0.07 and di/dt + Kt = 8.25 cm/hr, which isabout half the saturated conductivity. The soil-waterdiffusivity is denned by

D = K dh/d® . [13]Note that the rapid variation of D with 0 necessaryto apply Eq. [9] follows from the three equations above.The applicability of Eq. [9] to the Grenoble sand wasalso verified by Touma et al. (1984) for the simplercase when ®s is constant.

Under those conditions, Eq. [9] is easily integrated,and the resulting ®$(t) is indicated in Table 1. Thisintegration is achieved by noting that, during the in-itial stages of infiltration, gravity is negligible. Then,if D at 0 = 0, is nonzero, the diffusion equation hasa solution, t~l/2(® — 0,), which is a function of z/\ft,valid for t very small. Thus, calling Q the flux, i.e. Q= di/dt + KJ, Eq. [9] can be simplified in the limit t—» 0. The right-hand side of that equation becomesS2/(2 di/dt), so that S2 and thus 0| vary like t, and theleft-hand side then becomes / (1 — 1/4). In the limitt —> 0, Eq. [9] can be written:

(Q ~ Ki)2 t = 25*73 . [14]

At very small times, Eq. [14] provides a first ap-proximation for the time, t, as a function of ®s, whichwe call t,. To improve this approximation and get abetter approximation, t2(®s), we can operate as fol-lows: Note that to calculate /,, the left-hand side ofEq. [9] was replaced by i (1 — 1/4). To calculate t2,we now replace the left-hand side by / {l — [d \n(®s- ®i)/d In f,]/2}, still keeping 52/(2 di/dt) as an ap-proximation for the right-hand side of Eq. [9]. Thisnew equation, replacing Eq. [14], yields t2(®s) easily.If so desired, din t^ above could be replaced by d Int2, and the process repeated to obtain t3(®s), and soon. Numerically, convergence was obtained rapidly forsmall times and t2 was close to t\, so that t3 was es-sentially the same as t2. For instance, for ®s = 0.08855,Eq. [14] yields for the present example, t\ = 0.106 sand \d In (®s - ®t)/d In t{}/2 =* 0.205. Note that thislast number is less than 1/4, i.e. the value used tocalculate /,, as expected, since it must approach zeroas t —» oo. Using these values, we obtain t2 = /, (1 —l/4)/(l - 0.205) = 0.1 s, which is close to 0.106, asexpected.

The values, ®s = 0.08855 at t = 0.1 s, can then beused as starting point for the integration of the com-plete Eq. [9] with Runge-Kutta formulae. To maintaina four-figure accuracy, it is necessary to take a timestep, A/ = 0.01 s at t =s 0.1 s, until ®s increases less

PARLANCE ET AL.: FLUX AND WATER CONTENT AT THE SOIL SURFACE 287

Table 1. Value of 9g at the surface as a function of time asobtained by the optimal and numerical methods.

0.3 6 (caf/eaf)

Time, sec <5>s(Eq.t91) Qs (Numerical)

612182430364248546072849610812015018024030036048060084010801680228028803600

0.14480.15970.16920.17620.18180.18650.19050.19400.19710.19990.20490.20910.21270.21590.21880.22490.22990.23760.24340.24800.25490.25990.26660.27090.27660.27920.28040.2810

0.09820.12370.14320.15590.16430.17090.17660.18180.18630.19040.19720.20000.20700.21100.21450.22170.22730.23580.24210.24700.25410.25920.26590.27020.27590.27860.27990.2807

rapidly, allowing a larger Af. This was confirmed bychanging the time step to A/ = 0.001 s. To check thevalidity of the above technique, the whole procedurewas repeated with a starting point much closer to theinitial values. This, of course, required a smaller stepsize, but it was found that the figures given in Table1 were accurate to four decimal places.

We may note that the influence of a in Eq. [9], asdetermined by Eq. [6] for the Grenoble sand as a func-tion of ©5(0, is weak. This is obvious for early timeswhen gravity is negligible and a does not appear inthe governing equation. We have also integrated Eq.[9] with a constant and taken its final value, for longtimes, without affecting the fourth decimal of Q^t).

Table 1 also indicates values of ®s obtained by thedirect numerical integration of Eq. [1]. The methodfollows that described by Haverkamp et al. (1977,1981), with a time step of 0.5 s and spatial step of10~2 m. Reduction of the steps produces no significantchanges in the results. The numerical solution disa-grees significantly with the results obtained from Eq.[9]. Up to times of 60s, the values of ©5 differ by oneon the second decimal place, and by one of the thirddecimal place up to times of 7 min.

We suspect that this discrepancy is most likely dueto errors in the numerical solution during the earlystages of infiltration. This is because very steep spatialand temporal gradients are hard to handle numeri-cally; for instance, as t — » 0, Eq. [9] shows that d®s/dt — » co. To analyse the direct numerical solution ofEq. [1], we shall first show that it is almost identicalto an earlier analytical approximation (Haverkamp etal., 1977),

2 =-[Ktf)-K,]} dp . [15]

The relation, z(0, 0S), of Eq. [15] was first proposed

Z(cro)Fig. 1.—Sketch of experimental results (——), numerical results (—

— —) and analytical results from Eq. [15] (- - -), giving ® as afunction depth, z, for different times.

by Parlange (1972) for constant flux infiltration. Theimprovement brought by Haverkamp et al. (1977) wasthe determination of Q^t) by mass conservation, whichdiffers from &^(t) calculated by Parlange (1972) by asmall but significant amount. The form given in Eq.[15], and some variations of it, have also been usedby other authors, e.g. White (1979) and Perroux et al.(1981). Figure 1 shows some profiles calculated usingthe White procedure (1979) from Eq. [15] with asso-ciated 05 value, and from numerical integration of Eq.[1]. Both are in essential agreement, except near thewetting front, i.e. for 0 =a 0;, where the numericalsolution predicts values of z slightly larger than thosepredicted by Eq. [15]. This is quite significant, as Eq.[15] predicts a wetting front already ahead of the ac-tual wetting front. This is because, to obtain Eq. [15]from Eq. [1], it is necessary to replace d®/dt by [(Q —Ki)/(&s ~ <=>,)] d®/dz, whereas dQ/dt is obviously largernear the wetting front. Thus, the denominator in Eq.[15] is too small and z too large near the wetting front.The error involved will be quantified further in a par-ticular example below. Since Eq. [15] is taken to con-serve mass, the incorrect position of the wetting frontmust be compensated by a lower value of Q&). Thisqualitative difference is also observed in the compar-ison between the integration of Eq. [9] and the nu-merical integration of Eq. [1] as indicated in Table 1.Further careful experiments following the techniquedescribed in Boulier et al. (1984) verify this result.Figure 1 shows the experimental profiles correspond-ing to the short time results of Boulier et al. (1984),

288 SOIL SCI. SOC. AM. J., VOL. 49, 1985

which all have a wetting front behind those predictednumerically or by Eq. [15]. Note, however, that as tgets larger, the relative error becomes less. Again, bothTable 1 and Fig. 1 show this property. This decreaseof the relative error is most likely associated to thefact that, as / becomes larger, gradients are less steep.Also, the dependence of z on 0, given by Eq. [15],becomes more accurate for longer times (Parlange,1972). It must be emphasized that the experimentaldata are subject to experimental errors and, conse-quently, do not yield a proof in absolute sense. How-ever, the systematic tendency of the discrepancy be-tween numerical and experimental profiles seems tobe sufficiently marked to exclude experimental erroras the only influence.

However, it is possible to quantify exactly the errorof Eq. [15], and by extension of the numerical resultfor a particular case. Consider then the standard casewhen water constant is fixed at the soil surface. Then,by definition of the sorptivity,

Q- Kt = Sr'/2/2, [16]i.e. Q is now variable, but @s is constant. In the earlystages, Q is very large, and the gravity term in Eq. [ 15]is negligible. This equation can then be written as:

&r'/2/2 = (&s -•es

£>(/3)/G3- 0,W/3.[17]

Mass conservation gives at once, by integration of Eq.[17], remembering that S = $zt~l/2 dQ, and integratingthe right-hand side of Eq. [17] by parts,

^/2 = (05 - 9,) f°SDd@ , [18]•/ wj

which yields values of S that are too large (Elrick andRobin, 1981). Thus, for 0S fixed, and to conserve mass,the front will be too far. Specifically, if D obeys therelation,

D = D0 exp n (0 — [19]with n of order eight (Reichardt et al, 1972), then Eq.[ 18] yields estimates of S2, differing by about 6%, asmall but significant error, to the values of S2 to beinserted in Eq. [9]. By comparison, Eq. [10] yields anessentially exact result.

In conclusion, it appears that Eq. [9], which appliesfor arbitrary surface flux and soil properties, is in bet-ter agreement with experimental observations than thebest available numerical scheme. An earlier analyticalresult given in Eq. [15] can be inaccurate. However,it is simple to apply, and should be used as long asone is aware of its limitations. It gives a similar errorto the direct numerical integration of Eq. [1]. The er-ror in the direct numerical integration of Eq. [ 1 ] islinked to the difficulty in handling high gradients, andthe error will tend to be larger when surface fluxes are

larger. The coherence of results obtained by the nu-merical scheme used here for the direct integration ofEq. [1], and by the earlier analytical solution, couldeasily, but mistakenly, have been construed as a guar-antee of their accuracy. This shows that great care mustbe taken in the evaluation oof approximate results. Ofcourse, a better numerical scheme, which is not af-fected by the rapid changes in early times, will be de-vised in the future, but, eventually, experimentationmust have the final word. Eq. [9] would then seem tobe the most reliable method when an accurate valueof 0,${?) is needed; for instance, to predict ponding time,especially when ponding occurs in a few minutes un-der high-intensity rainfall.