Water Infiltration into Soil in Response to Ponded-Water Head

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    Water Infiltration into Soil in Response to Ponded-Water HeadD. Swartzendruber* and W. L. Hogarth

    ABSTRACTThe pressure head of water ponded on the soil surface can increase

    the infiltration of water into soil, but the effect has often been com-plicated to describe mathematically. This study was conducted todevise a simpler mathematical description without undue sacrificeof accuracy. A new three-parameter infiltration equation was ex-amined for its capability in describing the effect of soil-surface-pond-ed water head, h, on the cumulative quantity of water infiltrated intothe soil with time. The infiltration equation, as reduced to two-pa-rameter dimensionless form, was fitted by nonlinear least squaresto dimensionless data generated from mathematical descriptions ofinfiltration that included the effect of h in somewhat complicatedparametric form. The fitted two-parameter equation gave an excel-lent description of all the generated data, in terms of both goodnessof fit and in recovery of the dimensionless ponded head p used asan input into the generated data. From an overall perspective, re-covery of h was achieved within a relative error of 1.4% acrossthe complete range of the .generated data, thus validating the newand relatively simple equation in its description of the general effectof ponded head on the cumulative infiltration process.

    THE EFFECT OF A NONZERO HEAD of ponded wateron the infiltration of water into soil has not onlybeen included in the first theoretically derived infil-tration equation (Green and Ampt, 1911), but has alsobeen recurrently handled in more general treatmentssince then. Philip (I958a) provided a rigorous and gen-eral analysis within the framework of his solution bypower series involving time exponents in multiples ofone-half. The effects, however, were not simple; thatis, for increased ponded-water head, the first two coef-ficients in the series were both increased, but the thirdcoefficient was decreased.D. Swartzendruber, Dep. of Agronomy, Univ. of Nebraska, Lincoln,NE 68583; and W.L. Hogarth, Division of Australian Environmen-tal Studies, Griffith Univ., Nathan, Queensland, 4111, Australia.Contribution from Division of Australian Environmental Studies,Griffith Univ. Received 15 Oct. 1990. 'Corresponding author.Published in Soil Sci. Soc. Am. J. 55:1511-1515 (1991).

    In the recent work of Hogarth et al. (1989), the effectof ponded head was assessed by using the parametricequations of Parlange et al. (1985). By their very na-ture, however, these parametric equations are morecomplicated to use than a single equation. Our objec-tive in this study was to provide a simplified mathe-matical form that would retain acceptable generalityand accuracy. To this end, we examined and evaluateda recently derived infiltration equation.

    THEORETICAL CONSIDERATIONSThe new infiltration equation, obtained as an exact inte-

    gration of the quasi solution (Swartzendruber, 1987b) ofRichards' (1931) equation, is expressible most simply by thethree-parameter form

    = (S/A0)[\ - [1]where / is the cumulative quantity (volume of water per unitcross-sectional bulk area of soil) of water infiltrated into thesoil in time t after the initial and instantaneous ponding ofwater on the soil surface. The constants S, KQ, and A0 are thesorptivity, the sated (near-saturated) hydraulic conductivity,and a time-decay constant arising from the quasi solution,respectively. Furthermore, on the basis of the quasi solutiongiving rise to Eq. [1], S and A0 in principle do include theeffect of the constant ponded head, h, of water covering thesoil surface, whereas KQ is generally construed to be indepen-dent not only of h but of all other components of hydraulichead as well. Note also that the small-time form (gravity ef-fects negligible) of Eq. [1], obtained by allowing t > 0, be-comes

    / = St1'2, [2]as would be expected.

    For investigating the relationship of 5 to h, we proceedfrom the work of Hogarth et al. (1989), beginning with

    S2 = SI + 2K0h(60 - 6n), [3]where we explicitly recognize 5 as the total sorptivity andSs as the sorptivity of the soil alone (with h = 0 for aninfinitesimally thin ponding of water), 00 is the sated (near-saturated) constant volumetric water content of the soil due



    to surface ponding, and 0n is the constant initial soil watercontent prior to surface ponding. For economy of symbols,we define 5W as the component of sorptivity due to h, or

    5W = [2K0h(00 - 0n)]'/2so that Eq. [3] becomes

    S2 = 5? + SI


    [5]A dimensionless measure of the ponded head, 7, was em-ployed by Hogarth et al. (1989), namely

    7 = S2JS\ [6]but we utilize here the more direct alternative

    p = S*/Sl [7]through which, in conjunction with Eq. [5], S is expressibleby

    s = [8]Combining Eq. [6] and [7] enables the conversion of 7 to pby

    7 = p/(l + p). [9]The particular dimensionless^formulations by Hogarth et

    al. (1989) of their cumulative quantity of infiltration 7* andtime f* caused 7 to be embedded in both I* and f. To avoidthis, we define the dimensionless cumulative quantity of in-filtration Y and time T by

    Y =



    and recast the two parametric equations used previously(Parlange et al., 1985; Hogarth et al., 1989). The latter au-thors' /*, p, and 5, respectively, but with initial hydraulic conductivityK(6n) = K{ = 0 retained, to obtain finallyY =

    P/(2(dY/dT - 1)]+ (1/25) ln[l + d/(dY/dT - 1)] [12]

    andT = [1/25(1 - 8)] ln[l + 5/(dY/dT- 1)]+ p/[2(dY/dT - 1)] - {[1 + (1 - S)p]/2(l - 8)}

    l/(dY/dT - 1)]. [13]The soil-characterizing parameter 8 ranges from a maximumof one to a minimum of zero, with o > 0 reducing Eq. [12]and [13] to the Green and Ampt (1911) equation

    7 - [(p + 1)12} ln[l + 2Yf(p + 1)] = T. [14]; As shown in the Appendix, the I(f) relationship from Eq.[12] and [13] at very small times (gravity effects negligible)

    is the same for all 5 (0 < 8 < 1), and thus includes the Greenand Ampt Eq. [14] (6 0). This therefore generalizes to all5 the customary Green and Ampt (1911) small-time I(t),which is

    7 = [2K0(h [15]where P is the constant suction head associated with thewetting front, not only as originally conceived (Green andAmpt, 1911) but also as reiterated since then (Philip, 1954,1958b). Comparing Eq. [15] with Eq. [A4] (Appendix) gives

    Ss(l + p)1'2 = [2K0(h + P)(60 - 0n)]1/2- [16]Appropriate combination of Eq. [4], [7], and [16] yields

    P = h/PSI - 0n)

    and from Eq. [8] and [16], we note that5 = [2K0(h + P)(B0 -


    [19]which is the desired relationship between S and h for a giveno, P, and (00 - J.

    Although P has been construed as fictitious by two author s(Philip, 1958b, p. 333; Panikar and Nanjappa, 1977, p. 13),a recent rigorous derivation (Swartzendruber, 1987a) showssuch a contention to be superfluous. That is, the expression


    = A51 J.K(r)dr, [20]

    where T = r(6) is the suction-head function, K(r) is the hy-draulic conductivity as a function of T, and rn = r(0J. Thisis quite general and not limited to the Green and Amptassumption, and does confer on P the meaning of a weightedmean value of r across the interval (0, rn) with weightingfactor K(T)/KQ. Philip (1958b) found, for what he designatedas Yolo light clay, that the S calculated from Eq. [19] wasalways within the range 0.5 to 1.2% of his exact valuesof S, for h ranging from 0.10 to 2.00 m. Such remarkableaccuracy is supported by our present finding that Eq. [19] isimplied by the small-time I(f) of Eq. [12] and [13] indepen-dently of the soil as characterized by 5. Finally, to give someidea of the value of P, we note from Philip (1958b) that 5S= 125.38 nm/s1'2, f0 = 123.0 nm/s, and 00 - 0n = 0.2574,with Eq. [18] then giving P =- 248.3 mm for Yolo light clay.

    The use of Eq. [8], [10], and [11] in Eq. [1] yieldsY = -'[! - exp(-r/2)] + T, [21]

    wheren = /(! + p)1/2


    As the dimensionless counterpart of Eq. [1], Eq. [21] is atwo-parameter form with n and a as the dimensionless pa-rameters. Because Ss and K0 are constants for a given soil,a is a dimensionless form of A0 by Eq. [23]. Also, p is adimensionless form of h by Eq. [17], S/SS = (1 + p?n is adimensionless form of S by Eq. [8], and n'1 = (1 + p)l/2/afrom Eq. [22] is a dimensionless form of the quantity S/A0appearing in Eq. [1].

    If fitted to a dimensionless data set (Y,T) generated insome fashion for given p from such equations as Eq. [12],[13], or [14], Eq. [21] could then be assessed for its validityand applicability. In addition to the goodness of fit as ex-pressed by the residual sum of squares, we could also ex-amine how well the given p is recovered. Recognizing thatthe p embedded in the fitted Eq. [21] by Eq. [22] would notrecover the given p exactly, we designate the p in Eq. [22]as pc and solve for it, obtaining

    pc = (a/n)2 - 1. [24]Using the fitted values of a and n in Eq. [24] thus yields pcfor comparison with the given p used to generate the dataset to which Eq. [21] was fitted.

    METHODSTo generate (Y, T) data for fitting by Eq. [21 ], we employed

    the two parametric equations of Hogarth et al. (1989) recasthere as Eq. [12] and [13]. For given 5 and/?, 51 pairs of(Y,T)were determined from Eq. [12] and [13] by computerizedNewton-Raphson and bisection methods, with common T


    values varying from 10~5 to 105. The 51 (Y,T) pairs wereobtained at each of 13 values of p ranging from 0 to 100,and carried out for 5 = 0.850 and 0.425. For 5 = 0, the 51(Y,T) pairs were obtained at each of the 13/7 values by usingEq. [14] since, even for 5 as small as 0.425, the calculationsbecame very time consuming. The value of 5 = 0.850 wassuggested by Parlange et al. (1982). By taking 5 = 0 and0.850, we are considering the extremes of soil types. Thevalue of 5 = 0.425 gives an intermediate value between theextremes.

    The fitting of Eq. [21] to a given set of 51 (Y,T) pairs wasactually carried out in terms of natural logarithms, namelythe 51 data pairs (In Y, In T). Logarithms were employed togive equal weighing to small and large values of Y.and T.Transforming Eq. [21] logarithmically by v = In Y and u =In T yieldsv = ln{l + n exp u exp[a exp(/2)]} In .[25]Equation [25] was then fitted to each (In Y, In T) data setby nonlinear least squares, using the NLIN package program(Marquardt searching option) of the Statistical Analysis Sys-tem (SAS Institute, 1990), with derivatives dv/dn and dv/dafrom Eq. [25] as inputs. Beginning at T = 10~5, the 51 com-mon T values were selected to progress by increments of InT very nearly evenly spaced. Specifically, for any given dec-ade range of T, the values were 1.0, 1.6, 2.5, 4.0, and 6.4,each of these being multiplied by 10% where N took on thesuccessive values 5, 4, . . . , 4, and 5 for each decaderange. This is the same scheme used by Swartzendruber andClague (1989, p. 621, second column), but wherein it is er-roneously stated that the number of T values was 30 insteadof the correct number 51.

    As a check on the correctness of the overall computationalprocess, Eq. [25] was modified by setting n = a and thenfitting the resulting one-parameter equation to two data sets,for 8 = 0.850 and p = 0 in Eq. [12] and [13], and for p =0 in Eq. [14] (5 = 0). The resulting fitted a and the residualsum of squares (RSS) agreed exactly with the correspondingfindings of Swartzendruber and Clague (1989) for their datagenerated from the equations of Parlange et al. (1982) andGreen and Ampt (1911), respectively.

    To compare the pc of Eq. [24] with the input p of Eq. [12]and [13], or [14], the customary percentage error of pc>termed Ep, is

    Ep = 100(pc p)/p, [26]but Eq. [26] does fail for p = 0. Of as much or even moreinterest is an examination of how well the term (h + P) ofTable 1. Results of fitting Eq. [25] (parameters n and a) to data

    generated from Eq. [12] and [13] with soil-characterizing param-eter 5 = 0.850, using different input p (dimensionless pondedhead); pc of Eq. [24] is the recovered p, while E, and Eh+, are thepercentage errors in pc and (Ac + P), respectively.

    Eq. [19] is recovered by (/zc + P), as assessed by the (hc +P) percentage error = Eh+P = 100[(/zc + P) - (h + P)]/(h+ P), or

    Eh+P = 100(/zc - h)/(h + P). [27]If h = hc when p = pc in Eq. [17], then the resulting /zc =pj> and the h = pP from Eq. [17] itself allow Eq. [27] tobecome

    Eh+P = 100(/>c - p)l(p + 1), [28]with Eq. [28] being useable at p = 0.

    RESULTS AND DISCUSSIONFor the data generated from Eq. [12] and [13] with

    5 = 0.850, the results are presented in Table 1. Asregards goodness of fit, the largest RSS is 2.985 X 10'3at p = 0. This RSS is less than that of the correspond-ing case of the one-parameter equation of Swartzen-druber and Clague (1989) fitted to the data setgenerated from the equation of Parlange et al. (1982)with 6 = 0.850, which gave an RSS of 3.65 X id'3and a maximum relative error of 2.35%. The 18%reduction in RSS is attributable to Eq. [25] and [21]having two fitted parameters, n and a, rather than hav-ing n = a as in the single-parameter fit (Swartzen-druber and Clague, 1989). Hence, the goodness of fitof Eq. [25], and thus of Eq. [21], is deemed excellentfor all values of p in Table 1.

    For input p > 0.2 (Table 1), recovery of the inputp by the pc of Eq. [24] is within an error on the orderof 2%, but the error increases distinctly as p decreasesbelow 0.2, apparently because of the small magnitudeof p. For, examining the error Eh+P with which (hc +P) recovers (h + P), we again find it maximal at p =0 but its value is now the very acceptably small1.356%. Hence, with the error Eh+P within 1.4% forall values ofp in Table 1, the recovery of (h + P) andthus of h, by Eq. [25] and thus of Eq. [21], is thereforedeemed excellent for 6 = 0.850.

    Results are presented in Table 2 for the data gen-erated from Eq. [12] and [13] with 5 = 0.425. Withthe largest RSS of 1.576 X 10~3 being even less thanthe largest RSS of Table 1, the goodness of fit of Eq.[25] is even better than it was for the case 8 0.850.Table 2. Results of fitting Eq. [25] (parameters n and a) to data

    generated from Eq. [12] and [13] with soil-characterizing param-eter S = 0.425, using different input p (dimensionless pondedhead); pc of Eq. [24] is the recovered p, while Ep and Ettf are thepercentage errors in pc and (Ac + P), respectively.

    Input p,Eq. [12]and [13]

    Fitted parametersn a


    150.9524 151.9726121.8770102.626677.752262.320851.846044.296525.36499.18455.58664.01212.81940.5681


    Sum ofsquaresin 10~3



    PcEq. [24] Eq. [26]

    En,Eq. [28]

    Input p,Eq. [12]and [13]






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