scaling soil water properties and infiltration modeling1

Scaling Soil Water Properties and Infiltration Modeling1

L. R. AHUJA, J. W. NANEY, AND D. R. NiELSEN2

ABSTRACTWe examined variability and interrelation of the scaling factors

for Green-Ampt infiltration parameters of saturated hydraulic con-ductivity (K,) and wetting-front pressure head (A/) in a 9.6-ha grass-land watershed, containing three hydrologically similar silt loamMollisols. Scaling factors for different depths at a given site ap-peared to be related. The scaled mean K, decreased with depth, whilethe scale mean hf showed no trend. The K, scaling factors had greatervariability between sites than the hf factors, and the relationshipbetween the two sets of factors deviated appreciably from a 1:1 line.However, there was a linear relationship between them, which wasutilized in modeling infiltration. The composite infiltration for thewatershed calculated by using different, but related, scaling factorsfor K, and kf was very close to that obtained by using the samescaling factors, derived from either K, or hf, for both. However, thedistribution and range of the predicted infiltrations were greatly dif-ferent.

Additional Index Words: spatial variability, scaling factors, Green-Ampt model.

Ahuja, L. R., J. W. Naney, and D. R. Nielsen. 1984. Scaling soilwater properties and infiltration modeling. Soil Sci. Soc. Am. J.48:970-973.

SCALING of soil water properties is a promising sim-plified method of describing spatial variability of

these properties within a field (Warrick et al., 1977;Simmons et al., 1979; Russo and Bresler, 1981; Youngsand Price, 1981). For use in modeling infiltration orother unsaturated flow processes on a watershed scale,scaling can be combined with the frequency distri-bution analysis of scaling factors (Peck et al., 1977;Sharma and Luxmoore, 1979; Luxmoore and Sharma,1980; Warrick et al., 1977; Warrick and Amoozegar-Fard, 1979). Field technology for obtaining usefulscaling factors remains largely undeveloped. In thestudies cited above, the concept of scaling has gener-ally been tested on data pooled for all soil depths. Fora layered soil, this procedure results in assigning dif-ferent scaling factors for different soil depth intervalsat a given site. However, for modeling work a givensite has been assumed to have one scaling factor ap-plicable for all depths (Luxmoore and Sharma, 1980).This means that the scaling should be done separatelyfor different soil depth intervals, so as to obtain dif-ferent scaled mean properties for these intervals. Thescaling factors for different depths for a given site maythen be averaged to obtain one factor per site as asimplification. Alternatively, if the scaling is done ondata pooled for all depths, the average trend of changein scaling factor with depth at a given site should beincorporated in modeling. In this paper, we examinethe variability and scaling of Green-Ampt infiltrationparameters of saturated hydraulic conductivity and

1 Contribution of the Water Quality and Watershed ResearchLaboratory, USDA-ARS, Durant, OK 74702, and Dep. of Land,Air, and Water Resources, Univ. of California, Davis, CA 95616.Received 17 Feb. 1984. Approved 30 Apr. 1984.2 Soil Scientist (Physics), Geologist, and Professor of Soil and WaterSciences, respectively.

wetting-front pressure head in a small watershed, tak-ing the above points into consideration.

THEORYAccording to the extended scaling relations (Warrick et

al., 1977; Simmons et al., 1979) saturated hydraulic con-ductivity at a certain soil depth at a given site i, Ksi, may berelated to the scaled mean saturated hydraulic conductivityfor that depth in a watershed, Ksm, by:

of [1]where a, is a scaling factor for site /. Setting the average ofall a, values equal to 1.0, Ksm can be obtained from a knownset of N Ksi as:

2

[2]

The scaling factors «/ for all different sites can tnen oe oo-tained from Eq. [1].

The wetting-front pressure head, hp in the Green-Amptmodel of infiltration into soil with initial moisture contentat field capacity and below may be approximated as (Meinand Larson, 1973):

[ K ( h ) / K s ] d h [3]

where K(h) is the unsaturated hydraulic conductivity as afunction of soil-water pressure head h. According to the ex-tended scaling theory, soil-water pressure head at any givendegree of saturation, S, at a fixed depth of site /, h(S)h maybe related to the watershed scaled mean pressure head valuefor the given depth and S, h(S)m, as

•t = h(S)m/ai [4]On this basis, hf for a fixed depth and site /, hf,, is related towatershed mean hf for that depth, hfm, as:

hfi = hfjcti. [5]The hrm can be obtained from N hfi values by the methodused for obtaining Ksm.

METHODS AND MATERIALSIn a 9.6-ha watershed near Chickasha, OK, containing

mainly three hydrologically similar Udertic Paleustolls, siltloam soil types (Sharma et al., 1980), four undisturbed soilcores were taken in 150-mm depth increments (down to the600-mm depth) at each of 15 sites spread over the wa-tershed. A specially designed double-wall sampler operatedhydraulically was used for this purpose. The diameter of thecores was 75 mm. The middle 75-mm length of each corewas used to measure saturated hydraulic conductivity andsoil water content-pressure head relationship in the labora-tory. The hydraulic conductivity was measured with a con-stant-head permeameter (Klute, 1965). Water retention atpressure heads of 0.0, -1.5, -3.0, -6.0, -15.0, -33.0, and—100 kPa was measured using a Tempe cell (Reginato andvan Bavel, 1962). These water content-pressure relation-ships, $(h), were used to determine the relative unsaturatedconductivity function K(h)/Ks by using the method of Camp-bell (1974). In almost all cases, a straight line could be fittedto the log-log plot of the 6(h) data points for pressures lessthan —6.0 kPa. The air-entry pressure was determined byextrapolation of this fitted line to saturated soil water con-

970

AHUJA ET AL.: SCALING SOIL WATER PROPERTIES AND INFILTRATION MODELING 971

Table 1—Variation in scaling factors of Ks and hffor different soil depth intervals and sites.ajK, ajhf

150- 300- 450-Siteno. 0-1 50 mm 300mm 450mm 600mm

1 0.756 0.561 0.287 0.2392 0.780 0.483 0.091 0.2393 1.106 0.383 0.497 0.3384 0.232 0.152 0.064 »T5 1.212 1.141 1.698 0.2396 0.497 0.615 0.786 2.8297 0.653 0.753 1.2348 0.9029 0.678 0.725 0.563 0.535

10 1.978 1.757 0.314 0.33811 0.670 0.383 0.064 0.67612 1.314 0.165 0.06413 1.501 2.543 3.316 1.85214 1.620 1.775 1.674 1.09615 1.101 2.564 3.347 2.619

Mean 1.000 1.000 1.000 1.000Std.Dev. 0.467 0.829 1.136 0.982Ksm(mmihr) 8.92 4.76 2.43 0.17A^lmmHjO)

t Missing values.

tent. The wetting-front pressure head, hf, was determinedusing Eq. [3]. It should be noted that the determination ofK(h)/Ks by Campbell's method, and hence of hfby Eq. [3],depends only on 6(h), and is independent of Ks. The scalingsof Ks and hr were done using Eq. [1] and [2], and [5], re-spectively, for each soil depth increment, separately. Forsome comparisons, average scaling factors were also ob-tained for h(S) functions for the upper 35% of the saturationrange (S between 1.0 and 0.65) by the least-squares proce-dure of Russo and Bresler (1981).

RESULTS AND DISCUSSIONThe scaling factors for different sites and depths ob-

4oiYiA/1 Trr\m Jf mr\n h /^oto V%T7 t»*«ao+i«rt ar\f*V\ o*-\i1 ^^vx+n

Mean

0.4610.3980.5810.1491.0731.1820.8800.9020.6251.0970.4480.5142.3031.5412.408

150-0-1 50 mm 300mm

1.225 0.6080.853 0.7890.923 1.1321.163 0.7051.447 0.9460.809 0.8240.680 0.6810.7420.833 0.7320.979 1.0380.974 0.8590.974 1.3380.818 1.4801.623 1.4810.958 1.3871.000 1.0000.262 0.311

-704.6 -608.0

"0 2.4 +

6-CO '*'

3 0.4* '*/

3 o/ .0 0.4 0.8

300-450mm

1.1560.6230.8410.4410.9501.0400.772-

1.1522.3220.6930.5950.7471.2601.408

1.0000.471

-652.8

/'.*> ' 4 4 •

1.2 1.6 2.0

450-600 mm Mean

0.659 0.9120.488 0.6881.105 1.000

0.7701.168 1.1281.325 0.999

0.7110.742

1.271 0.9970.451 1.1980.876 0.851

0.9690.901 0.9870.971 1.3341.785 1.3851.0000.392

-815.6

/ • 0- 1 50 mm/ o 150-300

+ 300-4504 450-60O&

0

2.4 2.8 3.2z/increment separately are presented in Table 1. Thescaled mean properties Ksm and hfm are also given. TheKs scaling factors show higher variability between sitesthan the hf factors. In both cases, the variability be-tween sites is greater in the lower 300 mm of soil pro-file than in the upper layers. There is also an appre-ciable scatter in scaling factors for different depths ofa given site. However, the analysis of variance of dataindicated that the mean sum of squares for sites wasnearly 5.6 times that of the residual in case of Ks scal-ing factors (significant at 99% level of probability), and1.7 times in the case of hf factors (significant at 90%level). This showed that the variability between sitesis a major component of the total variability, more soin Ks factors. This finding may be a crude justificationfor averaging scaling factors for different depths of agiven site as a simplification, as has been done in someeariler studies. The scaled mean Ksm decreases withdepth, while the hfm shows no trend.

The scaling factors for /z/at different depths are plot-ted against the corresponding factors for Ks in Fig. 1.The relationship between the two sets of factors hasa large scatter, but deviates appreciably from a.1:1line. The data for different depth intervals in Fig. 1show no tendency of being separated; they fall withinthe variability of each other. The relationship of av-erage /z/and Ks scaling factors, averaged for the foursoil depths at a given site, is presented in Fig. 2. The

SCALING FACTORS FOR Ks, o»(Ks)Fig. 1—Scaling factors for the wetting front pressure head, hf, plotted

against the scaling factors for hydraulic conductivity, Kn of thewatershed.

0.4 OS 1.2 1.6 2D 2.4 2B 3.2DEPTH-AVERAGED SCALING

FACTORS FOR Ks, 3r(Ks)Fig. 2—A plot of depth-averaged scaling factors for A/ and K, of the

watershed.

plot indicates a linear relationship between hf and Ksfactors, as shown by a least-squares line fitted to thedata. A similar plot of average h(S) scaling factorsagainst Ks factors yielded a relationship that was veryclose to that between hf and Ks factors (shown as adashed line in Fig. 2). Superposition of Fig. 2 on Fig.1 indicated that the straight lines in Fig. 2 also fairlywell represented the unaveraged data in Fig. 1. This

972 SOIL SCI. SOC. AM. J., VOL. 48, 1984

10.0

V)ac.g

^P"3J"Kuj oC!) U-<cc.% «

- 3 - 2 - 1 0 1 2 3PROBABILITY VARIABLE. (Y-Y)/ay

Fig. 3—Fractile diagram of the logarithms of scaling factors for K,.

was satisfying in that we were not off-base in aver-aging.

The data in Fig. 1 and 2 indicate that the scalingfactors for Ks and hf are not equal, although it wouldbe convenient if these factors were equal, as has beenshown or assumed in some previous studies (Sharmaand Luxmoore, 1979; Luxmoore and Sharma, 1980;Warrick and Amoozegar-Fard, 1979). For applicationof the scaling theory to dissimilar field soils, there neednot be a requirement that the factors for Ks and hfmust be equal. The fact that the two sets of factorsappear to be linearly related may, nevertheless, offeran advantage for modeling infiltration. In order to de-termine the extent of error involved in the predictedinfiltration when a certain simplifying assumption ismade in relating Ks and hf factors, the following caseswere studied.

Case 1—Cumulative frequency distribution of log-arithms of depth-averaged Ks scaling factors was cal-culated, and approximated by a straight line (Fig. 3).This distribution was divided into 10 segments of equalprobability density, and a median a(Ks) was calculatedfor each segment (Sharma and Luxmoore, 1979). Amedian a(hj) for each segment was calculated from themedian a(Ks) using the relationship of Fig. 2. Proba-bility densities associated with median a(hj) were thesame as for corresponding a(Ks). Using these data andscaled mean Ks and /?/ values for the 0 to 150-mm soildepth increment, as an example, a cumulative infil-tration curve was calculated for each segment, fromwhich a probability-density-averaged composite curvewas obtained. The integrated Green-Ampt equation(Corey, 1977, p. 194-195) with ponding depth set equalto zero was used for infiltration calculations.

Case 2—The calculations made in Case 1 were re-peated except the median a(hj) values were set equalto the median a(Ks) values.

Case 3—The calculation^ made in Case 1 were re-peated except the median a(Ks) values were set equalto the median a(hj) values calculated for Case 1.

The composite cumulative infiltration curves upto3500 s for the three cases are shown in Fig. 4. Therange of infiltration values at 3500 s are also given foreach case. The results indicate that setting a(hj) valuesequal to a(Ks) values (Case 2) reduces the compositecumulative infiltration only to a small extent, as com-

2K23sjj"3ZOO

PSto:85 'o%c1?0

9

Range of qat 3500 sec

• Case 1 17.4-201.0- + Case 2 33.5-153.0

0 CaseS 68.0- 90.8 **

*

**

**

?, . . . . i , . . . i ,

' '.

_

•

.

.

10 I02 I03

TIME, SEC10"

Fig. 4—Composite cumulative infiltration for the watershed and therange of the spatially-variable cumulative infiltration at 3500 secfor the three scaling-factor combinations described in the text.

pared with using different values for a(hj) and a(Ks)(Case 1). The range of q values was decreased in Case2. On the other hand, compared with Case 1, settinga(K,) equal to a(/z/) values (Case 3) increased the com-posite cumulative infiltration, but decreased the rangeof q values even more than in Case 2. However, thedifference in composite infiltration between the lattertwo cases was only 2.6% at 3500 s.

CONCLUSIONSThe results of this study indicate that the scaling

factors for different depths at a given site may have adegree of relationship. The scaling factors for the sat-urated hydraulic conductivity (JQ in a watershed mayvary much more than the factors for the wetting-frontpressure head (/z/) or the water characteristic curve forthe watershed. This results in a departure from a 1:1relationship in the scaling factors for the two varia-bles. However, there may be a relationship betweenthe two sets of scaling factors, which can be utilizedin modeling infiltration. The use of the Ks scaling fac-tors for both Ks and hfi or vice versa, in conjunctionwith the respective scaled mean Ks and hfi may resultin a negligible error in the composite cumulative in-filtration. However, the distribution and range of thepredicted infiltrations will be greatly different. UsingKs factors for both Ks and /z/will be better than usinghf factors for both.

MCCONNAUGHEY & BOULDIN: THE PREDICTION OF PARTIAL ANAEROBIOSIS IN SATURATED SOIL 973

9. Sharma, M.L., and R.J. Luxmoore. 1979. Soil spatial variabilityand its consequences on simulated water balance. Water Re-sour. Res. 15:1567-1573.

10. Sharma, M.L., G.A. Gander, and C.G. Hunt. 1980. Spatial var-iability of infiltration in a watershed. J. Hydrol. 45:101-122.

11. Simmons, C.S., D.R. Nielsen, and J.W. Biggar. 1979. Scalingof field-measured soil water properties. Hilgardia 47:77-154.

12. Warrick, A.W., and A. Amoozegar-Fard. 1979. Infiltration and

drainage calculations using spatially scaled hydraulic proper-ties. Water Resour. Res. 15:1116-1120.

13. Warrick, A.W., G.J. Mullen, and D.R. Nielsen. 1977. Scalingfield-measured soil hydraulic properties using a similar mediaconcept. Water Resour. Res. 13:355-362.

14. Youngs, E.G., and R.I. Price. 1981. Scaling of infiltration be-havior in dissimilar porous materials. Water Resour. Res.17:1065-1070.