modeling infiltration with varying hydraulic conductwity under simulated rainfall conditions

JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATIONVOL 34, NO.2 AMERICAN WATER RESOURCES ASSOCIATION A1RIL 1998

MODELING INFILTRATION WITH VARYING HYDRAULICCONDUCTWITY UNDER SIMULATED RAINFALL CONDITIONS1

Ram Gupta, Ramesh Rudra, and Trevor Dkkinson2

ABSTRACT Physically-based models are extensively used to simu-late the infiltration process under varied field conditions. Mostmodels are based on the deterministic nature of input parametersrelated to the flow process (such as hydraulic conductivity). Thesemodels yield poor predictions of infiltration rates because they donot include the field-scale variations of flow parameters. The paperpresents an approach for integrating the field-scale variability ofhydraulic conductivity with an infiltration model to simulate infil-tration under the rainfall conditions. A model describing the spatialstructure of hydraulic conductivity has been developed usingstochastic techniques. The stochastic structure of hydraulic conduc-tivity was then incorporated in the Green-Ampt and Mein-Larsoninfiltration model. The model outputs on the instantaneous infiltra-tion rates and cumulative infiltration were evaluated using thefield infiltration data measured under simulated rainfall condi-tions. The results show that the combined model is capable of rep.resenting the instantaneous infiltration rates and cumulativeinfiltration of the study soils. The model may, therefore, be used tosimulate the rainfall infiltration process for spatially-variable soilsunder the field conditions.(KEY TERMS: rainfall simulator; infiltration; spatial variation;hydraulic conductivity; Green.Ampt and Mein-Larson model.)

INTRODUCTION

The infiltration process is an integral part of hydro-logic components of nonpoint source pollution (NPS)models. Many empirical and analytical equations areused to simulate infiltration process in NPS models.The Green-Ampt equation (1911) is a physically basedequation that is widely used in these models. The soilhydraulic parameters used by the Green-Ampt equa-tion have been found to exhibit a large degree ofvariability under field conditions (Biggar and Nielsen,1976; Jensen and Butts, 1986), which substantiallyaffect the soil-water movement in surface and subsur-face soils (Smith and Hebbert, 1979; Bresler and

Dagan, 1983). Many studies have shown that the pre-diction accuracy of NPS model may be improved byincorporating the spatially variable characteristics ofinput parameters (Biggar and Nielsen, 1976; Rudraet al., 1986). Stochastic techniques offer a mathemati-cal framework for dealing with such parameter vari-ability by considering them as random variables(Sharma et al., 1983; Webster, 1977). The inclusion ofhydraulic conductivity, described using such stochas-tic techniques, in the infiltration models offers a pos-sibility for better predictions of infiltration processunder spatially variable conditions.

Various empirical and physically based models areused to describe infiltration process on field scale.Richards' infiltration equation is the governing equa-tion for the infiltration of water in the soil profile(Mein and Larson, 1973; Smith and Parlange, 1978).The Green and Ampt (1911) equation is a solution ofthis equation, and is one of the physically-basedapproximations of the infiltration process. The infil-tration equation uses Darcy's law to estimate infiltra-tion rate into a deep homogeneous soil having asharply defined wetting front under a ponded condi-tion. This approach was extended by Mein and Larson(1973) to account for surface ponding with steadyrainfall intensity. Their approach included infiltrationcases both prior to and after the occurrence of pond-ing on soil surface. The resultant equation is knownas Green-Ampt and Mein-Larson (GAML) infiltrationequation, and is used in the present study for infiltra-tion simulations.

The objectives of this study were: (1) to describethe spatial structure of saturated hydraulic conduc-tivity using stochastic techniques; (2) to develop ahybrid infiltration model by combining the variable

'Paper No. 96143 of the Journal of the American Water Resources Association. Discussions are open until December 1, 1998.2Environmental Engineer, Chesapeake Bay Local Assistance Department, 805 East Broad St., Suite 701, Richmond, Virginia 23219-1924;

and Professors, School of Engineering, University of Guelph, Guelph, Ontario, Canada N1G 2W1.

JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION 279 JAWRA

Gupta, Rudra, and Dickinson

characteristics of hydraulic conductivity with GAMLinfiltration model; and (3) to evaluate the models bycomparing model results with field-measured infiltra-tion data on instantaneous and cumulative rainfallrates.

EXPERIMENTAL MATERIALS AND METHODS

Description of Study Area

Field experiments were conducted on a fine sandy-loam soil in a conventionally tilled area located at theGreenbelt Farm of Agriculture Canada, Ottawa(Canada). The field has a subsurface drainage systeminstalled in 1974, and has been monitored for surfaceand ground-water investigations using four observa-tion wells located at nine different sites within thearea. The area has been cropped with continuoussilage corn (Zea mays L.) since 1971. The crop was 40days old at the time of the experiments. Selectedphysical properties of soil in the experimental areaare given in Table 1.

TABLE 1. Physical and Chemical Propertiesof Soil at the Experimental Site.

Depth(cm)

Sand(%)

Silt(%)

Clay(%) pH

OM(%)

0-10 63.85 26.80 9.40 4.9 3.4

10-20 64.46 27.00 8.60 5.2 3.1

20-30 66.31 29.00 4.70 5.4 —

30-40 60.55 30.60 8.80 5.4 —

40-50 44.30 36.30 19.40 6.0 —

Rainfall In filtration Experiments

For the study, two transects (AB and CD) in field 2and 4 were selected (Figure 1). The rainfall simula-tion experiments were conducted on the ten locationsalong these transects at a 75-rn spacing in the X-direction. Each location had a battery of five test plotsof 1- x 1-rn size. Infiltration tests were carried outsequentially in summer of 1990, and were completedwithin a two-day period. The test plots were preparedby inserting prefabricated metal frames into theground to a depth of 5.0 to 7.5 cm along the boundaryof the test plot. The frames were fastened at cornersto prevent flow of water away from the test-plots. A

V-shaped trough attached with a vacuum pump wasinstalled along the lower edge of each plot to collectrunoff water. Soil disturbance was minimized insideand outside the test plots. Liquid sealant was used toseal gaps between the metal frames and the soil sur-face. Preparation of a test plot and conducting one setof infiltration tests took about 125 minutes.

The Rainfall Simulator (RS) designed and devel-oped at the University of Guelph, Guelph (Canada)for micro-scale laboratory and field studies, was usedto simulate rainfall events. The RS is a portable typeof simulator and involves a continuous-spray, wide-angle, and full-jet nozzle. The hydraulic characteris-tics of the nozzle are given in Table 2. Physicalconditions of soil surface at the time of experimentsare given in Table 3. The simulated rainfall wasapplied at a constant intensity of 16.02 cm/h for 15-20minutes. Fifty sets of data, one from each test-plot, oninfiltration measurements were collected. The data onvolume of infiltrating water were computed by sub-tracting the runoff volume from the inflow volume foreach time interval. The volume of water, thusobtained, was converted to instantaneous infiltrationrate by dividing the cumulative volume by the dura-tion of the interval. By using data, the instantaneousinfiltration rate curves were obtained for each testplot. The curves were then averaged to obtain a singlecurve for the entire study area. Another curve usingthe cumulative infiltration data was obtained byusing a similar procedure. The two curves, thusobtained, were considered to represent the instanta-neous infiltration rates and the cumulative infiltra-tion behavior of the entire study area. The data oninstantaneous infiltration rates in conjunction withthe Philip's two-term infiltration equation (Philip,1957) were employed to obtain saturated hydraulicconductivity, K, values for each plot. The K values,thus obtained, are termed here as experimentally-derived K, and were used to determine their spatiallyvariable structures.

Spatial Structure of Hydraulic Conductivity

The data series containing the experimentally-derived K5 values was assumed to consist of the deter-ministic and the stochastic components. Thesecomponents may be described by different models.The deterministic component was described using aFourier series. The series has been used to describeperiodicity of soil physical properties in spatialdomain (Webster, 1977) and of soil temperature inthe temporal domain (Persaud and Chang, 1983). Theseries containing two harmonic terms has been foundadequate for describing the deterministic component

JAWRA 280 JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION

310 m

V

L x

Modeling Infiltration With Varying Hydraulic Conductivity Under Simulated Rainfall Conditions

450 m

+ Battery of Five Rainfall Simulation Runs.

Figure 1. Layout of Rainfall Simulation Experiments at Greenbelt Farm of Agriculture Canada, Ottawa (Canada).

of hydraulic conductivity data (Gupta et al., 1994).The series of following form was used in the presentstudy:

X(x) = A1 + [C1 Cos (2it x / N) + D1 Sin (2ir x / N)

+ C2 Cos (4ic x /N) + D2 Sin (4ic x / N)] (1)

where x is the distance coordinate from initial point(1 � x � N); X(x) is deterministic component of K8 val-ues at point x (1 � x < N); N is number of infiltrationtest-sites along a transect; A1 is mean of hydraulicconductivity data of a transect; C1 and C2 are modelcoefficients of cosine terms; and D1 and D2 are themodel coefficients of sine terms. The X(x) componentwas computed for each measuring point, and wasremoved from the experimentally-derived K8 values.

TABLE 2. Hydraulic Characteristics of RainfallSimulator (Tossell et al., 1987).

Characteristic Value

Nozzle Diameter 12.7 mmNozzle Height Above Plot 1500 mmSimulated Rainfall Intensity 160.2 mm/hTest-Plot Size 1-rn x 1-rn

Kinetic Energy Flux Density 0.7570 w/m2

TABLE 3. Physical Properties of Soil at Timeof Rainfall Simulation Experiments.

Item Value

Bulk Density 1390-1460 kg/rn3Antecedent Soil Moisture Content 18.40-2 1.60 % (vol.)

Soil Surface Conditions Dry and undisturbedDuration of One Rainfall Run 60-80 minutes

The stochastic component is represented by asecond-order auto-regressive model having the follow-ing form:

S(x) = X (Sx - Sxi) + X2 (Sx - S2) + a (2)

where X1 and X2 are the first and second order modelparameters, respectively; S , S,1, and Sx2 are thestochastic components of hydraulic conductivity atpoints x, x-1, and x-2, respectively; and ax is the resid-uals or independent subcomponent at point x. Thecoefficients X1 and X2 of Equation (2) were derivedfrom the (ACR)1 and (ACR)2 values (i.e., ACR at lag 1and lag 2, respectively) given as follows (Box andJenkins, 1976):

= (ACR)1 [1 - (ACR)2] / [1 - (ACR)12]

= [(ACR)2 - (ACR)12] / [1 - (ACR)12]

(3a)

(3b)


I— fl4+ + + * 0•••

3

2+ + * + +

A B

I


The formulated models of deterministic and stochasticcomponents were used to obtain hydraulic conductivi-ty values at each measuring point. The models werestatistically evaluated by comparing model generatedvalues with the experimentally derived data ofhydraulic conductivity. Two tests were performed forthe evaluation. The first was an independence ofresidual series. The model data were subtracted fromthe experimentally-derived values of hydraulic con-ductivity. The ACRs of the series, thus obtained, werecomputed and tested for their significance at the 0.05level. The second test was descriptive statistics (i.e.,geometric mean, coefficient of variation, and ACRs).Descriptive statistics of both data sets were tested atthe 0.05 level of significance. Once the tests were sat-isfied and the model adequacy was confirmed, thecomputed values of hydraulic conductivity were incor-porated in the infiltration model for simulating theinstantaneous infiltration rates and cumulative infil-tration at various time steps.

Simulation of Infiltration by GAML Model

The Green-Ampt infiltration equation has beenfound suitable for a variety of field conditions, andtherefore was selected in the present study. The equa-tion may be expressed as:

f=Ks(1+Sau*M/F)

where f is infiltration rate (cm/mm); K8 is saturatedhydraulic conductivity of wetted zone (cm/mm); Say issoil-water tension at the wetting front (cm); M is ini-tial moisture deficit (percent); and F is the cumulativeinfiltration (cm). Equation (4) can also be expressed interms of cumulative infiltration by substitutingf = dF/dt and integrating over the time period withthe boundary conditions: F = 0 at t = 0 and F = fat t =oo. The solution, thus obtained, may be written:

K8 t = F - Say M * In [1 +F / (M*S0)]

where F is cumulative infiltration (cm) and t is theelapsed time (mm). Mein and Larson (1973) modifiedEquation (5) by considering surface ponding duringinfiltration process. The resultant infiltration equa-tion was expressed as:

= MSav (R/K8 - 1)' (6)

where F is cumulative infiltration at time of ponding(cm) and R is rainfall intensity (cm/mm). Equation (6)is further modified to predict infiltration rates aftersurface ponding by translating the time scale. The

resulting model has been termed as the Green-Amptand Mein-Larson (GAML) model. The model wasextended by Tan (1987) to the specific case of a lay-ered soil profile. Tan's version of the model for a sin-gle layer has been used here.

To apply the model, it is necessary to determine anumber of input parameters. The parameters on soilmoisture deficit (M) and soil-moisture tension at thewetting front (Say) were obtained from an analysis ofsoil samples collected from the study area. The soil-moisture tension at the wetting front is equal to thearea under the relative hydraulic conductivity versussoil-moisture tension curve, and may be expressed as:

Sauf Kr d'I' (7)

where 'P1 is soil moisture tension prior to experiment(cm), Kr is relative hydraulic conductivity (K('P)/K8),and 'P is the soil-moisture tension at saturation (cm).An exponential relationship of unsaturated hydraulicconductivity with soil-moisture tension was assumed,and Say was obtained by solving Equation (7).

To obtain infiltration rates for a spatially variablesoil, the values of hydraulic conductivity obtainedfrom the stochastic models were incorporated into theGAML model and infiltration rates were computed. Atotal of 40 computer simulations runs were performed

(4) and the output on instantaneous infiltration ratesand the cumulative infiltration were obtained at vari-ous time steps. The outputs, were used to determinetheir mean and the mean ± one standard deviationvalues at each time step, and were then comparedwith the field measured infiltration data of the corre-sponding time step.

RESULTS AND DISCUSSION

(5) Variation and Spatial Structure of SaturatedHydraulic Conductivity

The infiltration rate at steady-state was found tovary from 0.075 to 0.132 cm/mm, with a mean of 0.116cm/mm and a coefficient of variation of 11.34 percent.But the K8 values were found to vary over a widerrange from 0.042 to 0.261 cm/mm, with a mean of0.194 cm/mm and a coefficient of variation of 25.05percent. The auto correlation coefficients (ACRs) val-ues for the K8 data were 0.2 15 and 0.154 at lags 1 and2, respectively (1 lag = 75 m). The non-zero values ofthe ACRs indicated that the K5 values were not com-pletely random, but had a mutual dependence on the



values measured at the adjoining points. The coeffi-cient of variation and non-zero values of ACRs indi-cate that the K8 values not only exhibit variations inspatial domain, but also have a mutual dependenceon their adjoining test sites. Hence, K8 may be ana-lyzed using the stochastic techniques.

The K5 estimates constituted a database for thestochastic analysis. The K5 exhibited greater varia-tions in X-direction compared to those in Y-direction.Hence, a one-dimension model has been used todescribe deterministic and stochastic structures of theK9 values. The coefficients of model described byEquation (1) were determined separately for the ABand CD transects having a spatial origin at the lefthand corner of the fields (Figure 1). The correspond-ing coefficients were averaged, and the deterministiccomponent, thus obtained, is expressed as:

X(x) = 0.1636 + [0.0253 Cos (2ic x / N)

- 0.0194 Sin (2t x / N) - 0.0176

Cos (4iv x / N) - 0.0310 Sin (4ic x / N)] (8)

The first term in Equation (8) represents the arith-metic mean of hydraulic conductivity values; the sec-ond and third terms represent the first harmoniccontribution; and the fourth and fifth terms representthe second harmonic contribution. The analysis ofvariance showed that the deterministic componentaccounted for 15 percent of the total variance in thehydraulic conductivity values. The remaining vari-ance not accounted for by the two harmonics wasattributed to the stochastic variations in the data.

The dependent part of stochastic component wasanalyzed using auto-correlations of the data. The cor-relogram showing the variations in ACRs at differentlags is presented in Figure 2. The correlogram showsthat the ACRs at each point, except few, have anasymptotically decreasing trend with increase in dis-tance between the two consecutive measurements(i.e., lag). The ACR at lag 2 is 0.175, compared to0.2 15 at lag 1. These values indicate that the correla-tion between two measured values decreases withincrease in lag. The decreasing trend in ACRs withlag revealed that the dependence between two neigh-boring values is greater than the dependence betweenvalues lying further apart. Stochastic component of

Figure 2. Correlograms of Model Generated and Experimentally-Derived Values of Hydraulic Conductivity.


1

0.80.60 0.4c)

-0.6

-0.8

Lag


K8 having such trend in ACRs was expressed by thefollowing model:

S(x) = 0.4054 (S - S1) - 0.2616 (S - S2) + a (9)

The first and second terms of Equation (9) constitutethe dependent part, and the third term represents theindependent part of the stochastic component. Thedependent part accounted for 62 percent of the totalvariance explained by stochastic component (i.e., 53percent of the total variance in the raw data). Theremaining 38 percent of the variance explained by thestochastic component (i.e., 32 percent of the total vari-ance) cannot be explained by Equation (9), and wasattributed to the independent part of the model.These results reflect a high spatial variability in thehydraulic conductivity values and a relatively lowdegree of dependence on the adjoining measurementpoints.

The models, given by Equations (8) and (9), wereemployed to obtain hydraulic conductivity values ateach measurement point. The values, thus obtained,were tested for the model adequacy. First, the residu-al series, a, was tested for its statistical indepen-dence. The test of series revealed that the ACRs have

a decreasing trend with increasing lag (Figure 3). Theconfidence limits drawn on ACRs indicate that theACRs are not significantly different from zero at the0.05 level. The results reveal that the residuals areindependent and approximately equal to zero, indicat-ing the non-significant differences between the exper-imental and generated K5 values. Second, thedescriptive statistics (i.e., geometric mean, ACRs, andcoefficient of variation) of the model values werecompared with their respective values of the experi-mentally-derived hydraulic conductivity data, and arepresented in Table 4. Least Significance Difference(LSD) tests showed that the differences in the statisti-cal parameters of the two data sets were non-signifi-cant at 0.05 level, indicating a statistical similaritybetween the modeled and the experimentally-derivedK8 values (Table 4).

These tests showed that the statistical properties ofthe model data were same as those of the experimen-tal data, indicating the adequacy of the formulatedmodels. Hence, the formulated model could be used todescribe the stochastic structure of hydraulic conduc-tivity of the study soil.

Figure 3. Correlogram of Residual Series of Hydraulic Conductivity With Upper and Lower Confidence Limits.


--1

0.8I-0) 0.40-4UI-

10

-0.4

-0.6

ACR

Upper limit-9-Lower linii

Lag


TABLE 4. Statistics of Modeled and ExperimentallyDerived Values of Hydraulic Conductivity.

StatisticalParameters

ExperimentallyDerivedValues

ModeledValues

Geometric Mean 0.194 cm/mm 0.223 cm/mm

Auto Correlation Coefficients

(a) Lag 1(b) Lag2(c) Residuals at Lag 1

0.2150.1540.0034

0.256

0.165

0.0045

Coefficient of Variation 25.05% 32.13%

In filtration Simulation Incorporating SpatialVariations of Hydraulic Conductivity

Since the infiltration rate values exhibited less spa-tial variation than the K5 values, it was assumed thata single curve obtained by averaging the measureddata of 50 test plots would represent the infiltrationbehavior of the entire study area. Therefore, two suchcurves, one for instantaneous infiltration rate andanother for cumulative infiltration, were obtainedfrom the 50 test plots of data. The model outputs oninstantaneous infiltration rate and cumulative infil-tration were analyzed separately to determine themean and the mean ± one standard deviation limits.The results were compared with the average andcumulative infiltration behavior of the study area(Figures 4 and 5).

The results shown in Figure 4 reveal that themodel overpredicted the infiltration rate for the firstthree time steps and underpredicted for the remain-ing time steps of the infiltration run. The overpredic-tion of infiltration during the initial time steps maybe attributed to the inability of the model to ade-quately account the crop characteristics (e.g., cropstage, leaf area, root density and distribution) and soilsurface detention. The interception of rain water bycrop canopy and soil surface affect the infiltration pro-cess, resulting lower measured infiltration rates dur-ing initial stages. However, Figure 4 shows that theobserved values fall within the range of one standarddeviation around the mean of model values. A varia-tion of this magnitude is acceptable under the fieldconditions. Further, the differences between the mod-eled and experimental values were found statisticallynon-significant at 0.05 level. A comparison based onthe cumulative infiltration values yielded similarresults (Figure 5). Therefore, it may be inferred that astochastic model, describing the spatial structure ofsaturated hydraulic conductivity, combined with theGAML infiltration model, may be used to represent

the infiltration process at the study area. The studyresults are applicable on the scale of measurementselected in the study. To extrapolate the results on awatershed scale, infiltration studies must be conduct-ed to obtain spatial structures of K8 which can repre-sent the soils of the watershed. The resultantstochastic models with the GAML model may then beused to describe the infiltration process on the water-shed scale.

SUMMARY AND CONCLUSIONS

The paper presents the results of a simulationstudy conducted by integrating the spatial variabilityof saturated hydraulic conductivity, K8, with aninfiltration model. Infiltration experiments were con-ducted at 50 test sites at the Greenbelt Farm of Agri-culture Canada, Ottawa (Canada) using a portablerainfall simulator. The spatially variable characteris-tics of K8 were described using stochastic techniques.Infiltration process was simulated using a Green-Ampt and Mein-Larson (GAML) infiltration model.The simulation results on instantaneous infiltrationrates and the cumulative infiltration were comparedwith the field measured data. Study leads to the fol-lowing conclusions:

(1) The values of saturated hydraulic conductivityon the test sites varied with a coefficient of variationof 25 percent, and were mutually dependent on theirvalues measured at the adjoining sites.

(2) The spatial variability in hydraulic conductivi-ty may be represented by a deterministic component,described by a two-harmonic Fourier series, and astochastic component, described by a dependent andan independent part.

(3) The integration of deterministic stochastichydraulic conductivity with the GAML infiltrationmodel represents the instantaneous infiltration ratesand cumulative infiltration adequately for all timesteps, except a few, at the study site. The applicationof a deterministic-stochastic hydraulic conductivity inGAML infiltration model provides acceptable resultson infiltration process under simulated rainfall condi-tions. Therefore, the GAML model incorporating thestochasticity of hydraulic conductivity may be consid-ered suitable for describing the infiltration processunder the field conditions. However, to extend theresults on watershed scale, the spatial structure of Kssuitable for representing the watershed soils must befirst obtained. The resultant stochastic models withthe GAML model may then be used to describe theinfiltration process on the watershed scale.

JOURNAL OF THE AMERICAN WATER RESOURCES AssociATioN 285 JAWRA

JE°0.8

0 2 4 6 8 10 12 14 16Time (miii)

Figure 5. Comparison of Modeled and Observed Values of Cumulative Infiltration at Experimental Site.



0.5

It20.3ci

C'I-ci.

00.1

0

4.0

p

Model

Mean + Sc

Mean - sdp

Exp.

0 2 4 6 8 10 12 14 16Time (miii)

Figure 4. Comparison of Modeled and Observed Values of Infiltration Rates at Experimental Site.

p

Model-t

Mean + Si

Mean - sdp

Exp.

0.0


LiTERATURE CITED NOTATIONS

Biggar, R. E. and D. R. Nielsen, 1976. The Spatial Variability of the A1 = mean of the hydraulic conductivity dataLeaching Characteristics of a Field Soil. Water ResourcesResearch 12:78-84. a = residuals or independent sub-component

Box, J. E. P. and G. M. Jenkins, 1976. Time Series Analysis — Fore-casting and Control. Holden Days, Inc., San Francisco, Califor- C1 and C2 = model coefficients for cosine termsnia.

Bresler, E. and G. Dagan, 1983. Unsaturated Flow in Spatially D1 and D2 = model coefficients for sine termsVariable Fields. 2. Application of Water Flow Models to VariousFields. Water Resources Research 19:421-428. f = infiltration rate (cm/mm)

Green, W. H. and G. Ampt, 1911. Studies of Soil Physics. Part I.The Flow of Air and Water Through Soils. Journal of Agricultur- F = cumulative infiltration (cm)al Sciences 4:1-24.

Gupta, R. K., R. P. Rudra, W. T. Dickinson, and D. E. Elrick, 1994. Kr = relative hydraulic conductivity (K('P)/K8)Spatial Patterns of Three Infiltration Parameters, CanadianJournal of Agricultural Engineering 36(1):9-14. K9 = saturated hydraulic conductivity (cm/mm)

Jensen, K. H. and M. B. Butts, 1986. Modelling of UnsaturatedFlow in Heterogeneous Soil. Nordic Hydrology 281-294. M = initial moisture deficit (percent)

Mein, R. G. and C. L. Larson, 1973. Modelling Infiltration DuringSteady Rain. Water Resources Research 9(2):384-394. N = spatial sample size of hydraulic conductivity

Persaud, N. and A. C. Chang, 1983. Estimating Soil Temperature measurementsby Linear Filtering of Measured Air Temperature. Soil Sci. Soc.Am. J. Proc. 47:841-846. = soil moisture tension at wetting front (cm)

Philip, J. R., 1957. The Theory of Infiltration. I. The InfiltrationEquation and Its Solution. Soil Sci. 83:345-357. S(x) = stochastic component of hydraulic conductivity

Rudra, R. P., W. T. Dickinson, D. J. Clark, and G. J. Wall, 1986. at point xGAMES — A Screening Model of Soil Erosion and Fluvial Sedi-mentation on Agricultural Watersheds. Canadian Water t = time from beginning of the rainfall event (mm)Resources Journal 11:58-71.

Sharma, M. L., R. J. W. Ban-on, and E. S. De Boer, 1983. Spatial x = distance coordinate from initial point (1 � x N)Structure and Variability of Infiltration Parameters. In:Advances in Infiltration. Proc. National Conference on Advances X(x) = deterministiccomponent of hydraulicin Infiltration, Am. Soc. of Agricultural. Eng., St. Joseph, Michi- conductivity at point xgan, pp. 113-121.

Smith, R. E. and R. H. B. Hebbert, 1979. A Monte Carlo Analysis of ? and ? = first and second order stochastic modelthe Hydrologic Effects of Spatial Variability of Infiltration, parametersWater Resources Research 15:419-429.

Smith, R. E. and J. Y. Parlange, 1978. A Parameter Efficient 'P = soil moisture tension (cm)Hydrologic Infiltration Model. Water Resources Research14:533-538. 'P1 = soil moisture tension prior to experiment (cm)

Tan, Kim-Ann, 1987. Modelling of Rainfall Infiltration UnderThawing Soil Conditions. M.S. Thesis, University of Guelph,Guelph, Ontario, Canada, 164 pp.

Tossell, R., W. T. Dickinson, R. P. Rudra, and G. J. Wall, 1987. APortable Rainfall Simulator. Canadian Agricultural Engineering29:155-162.

Webster, R., 1977. Spectral Analysis of Gilgai Soil. Aust. Journal ofSoil Research 15:19 1-204.


modeling infiltration with varying hydraulic conductwity under simulated rainfall conditions

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