J. agric. Engng Res. (2001) 80 (2), 223}231doi:10.1006/jaer.2000.0673, available online at http://www.idealibrary.com onSW*Soil and Water

Two-component Transfer Function Modelling of Flow through Macroporous Soil

J. Y. Diiwu1; R. P. Rudra1; W. T. Dickinson1; G. J. Wall2

1 School of Engineering, University of Guelph, Guelph, ON, Canada N1G 2W1; e-mail of corresponding author: jdiiwu@uoguelph.ca2Land Resource Research Centre, Agriculture and Agri-Food Canada, Guelph, ON, Canada N1H 6N1; e-mail: gwall@uoguelph.ca

(Received 7 June 1999; accepted in revised form 11 October 2000; published online 21 August 2001)

The macropore and micropore domains of a soil system were considered to be hydraulically distinct. Drainageand solute transport through the soil were then characterized by mixed probability distributions. Thecomponents of the mixed probability distributions were used to represent the distinct processes of drainage andsolute transport in the macropore and micropore domains. A goodness-of-"t test showed that the lognormaldistribution was the best theoretical distribution for solute transport in each of the two domains, while thetwo-parameter gamma distribution was best for drainage in each of the domains. A validation test showed thatthe mixed probability distribution representation of the transfer function adequately incorporated dual porosityin the transfer function model for drainage and solute transport through macroporous soil.

( 2001 Silsoe Research Institute

1. Introduction

The need to protect and better manage water resourcesand environmental systems has been a driving force inresearch for modelling water #ow and solute transportthrough soil. Several approaches reported in the litera-ture for modelling solute transport through soil may becategorized as deterministic and stochastic methods(Jury, 1983; Jury & FluK hler, 1992). The deterministicmethods are based on the convection}dispersion equa-tion, which was developed from laboratory studies, andhave been modi"ed over the years to include variousdegrees of complexity (Jury, 1983; Hillel, 1980; Dyson& White, 1987; Jury et al., 1991). However, the resultingmodels have not been able to adequately represent the"eld situation because of inherent variability at "eld scale.

Stochastic modelling approaches were introduced be-cause they take into account the "eld-scale variabilityand other inherent uncertainties in the data. The transferfunction modelling approach, which is a stochastic ap-proach, was introduced in an attempt to side step theenormous data requirements of the other stochastic ap-proaches (Jury, 1982). However, the performance of thevarious forms of the transfer function model have beenmixed. For instance, the transfer function model wasapplied to simulate change in average concentration of

0021-8634/01/100223#09 $35.00/0 223

bromide with depth in a 1)44 ha "eld (Jury et al., 1982).The comparison of simulated and measured bromideconcentrations indicated that predictions were in goodagreement with measurements. Experiments were con-ducted by Dyson and White (1987) to assess the perfor-mance of the transfer function model in predicting thetransport of chloride through undisturbed structuredclay core samples, and the results compared well withthose obtained using the convection}dispersion equa-tion. In their study, Dyson and White (1987) assumedsteady-state #ow near saturation, and the input of chlor-ide at the top of the soil core as a step function. Theresults showed that the transfer function model, based onlognormal distribution of travel time, predicted break-through curves very similar to the convection}dispersionequation with optimized parameters. However, the pre-dicted resident concentrations of chloride using the twomodels were not so good. Hence, the implied assumptionof homogeneity in a soil core does a!ect the results of theconvection}dispersion and transfer function models.Also, the positively skewed bimodal travel time distribu-tion shown in some of the samples used by Dyson andWhite (1987) seems to indicate the e!ect of macroporeson solute transport, but this was not considered in thestudy. The transfer function model was applied to studysolute transport through undisturbed soil columns with

( 2001 Silsoe Research Institute

J. Y. DIIWU E¹ A¸ .224

Notation

Ama

area under the macropore component ofbreakthrough curve or subsurface hydro-graph

Ami

area under the micropore component ofbreakthrough curve or subsurface hydro-graph

A*

generalized symbol representing eitherA

maor A

mibi

ordinates of breakthrough curve or sub-surface hydrograph

Ci(t) input concentration of solute, lg cm~3

Co(t) output concentration of solute, lg cm~3

f (t) travel time probability density function/soil system response function to soluteinput

f (t)dt proportion of solute molecules exitingfrom soil system

fma

(t) probability density function for solutetransport in macropore domain

fmi

(t) probability density function for solutetransport in micropore domain

g (t) soil system response function toin"ltration

gma

(t) probability density function for drainagein macropore domain

gmi

(t) probability density function for drainagein micropore domain

I(t) in"ltration rate, cmmin~1

IP

performance index of modelQ (t) drainage #ux, cmmin~1

t travel time, min(t, t#dt) time interval

*t time stepX

obsobserved #ow or solute concentration

Xpred

predicted #ow or solute concentrationDX

obs!

Xpred

D absolute di!erence between Xobs

andX

predama

relative weight for solute transport inmacropore domain

ami

relative weight for solute transport inmicropore domain

b scale parameterdma

relative weight for drainage in macro-pore domain

dmi

relative weight for drainage in micro-pore domain

g shape parameterk mean of logarithm of travel time

kma"rst moment of f

ma(t)

kmi"rst moment of f

mi(t)

p standard deveation of logarithm oftravel time

p2ma

second central moment of fma

(t)p2

misecond central moment of f

mi(t)

q integration variable

a step input of bromide, chloride and bacteria. It wasfound that the assumption of a lognormal probabilitydistribution might not be valid when the two-domainapproach is used in modelling (White et al., 1986).

The various versions of the transfer function modelreported in the literature have not performed as well asexpected in the presence of two-domain #ow (Dyson& White, 1987; White et al., 1986; Jury et al., 1990).Hence, for the transfer function model to adequatelyincorporate "eld-scale heterogeneity there is the need totake account of layering and macroporosity in the devel-opment of the model. In this paper, the two-componenttransfer function model, based on the concept of dualporosity, is being proposed. In the two-component trans-fer function modelling approach the travel time densityfunction is represented by a mixed probability densityfunction. One of the components of the mixed probabil-ity density function represents the net e!ect of the trans-port processes in the macropore domain, while the othercomponent represents the net e!ect of processes in themicropore domain.

2. Materials and Methods

2.1. Data collection

The data used in the development of the model wereobtained during rainfall simulation on 1 m by 1 m plotsunder no tillage treatment. The plots were constructed byinstalling large steel plates approximately of the samedimensions as the plots, at a depth of about 55 cm in thesoil pro"le to serve as catchment pans to collect subsur-face #ow during rainfall event. The steel plates wereinstalled by gently driving them horizontally into place inthe soil pro"le using hydraulic jacks. Plastic bottles wereinstalled in a pit for collecting the subsurface #ow forsubsequent sampling. At the soil surface, aprons wereplaced around the plots to direct surface #ow into aV-shaped trough from which samples were collected at1 min intervals to determine volume of runo! generatedand the concentration of tracer present in the runo!.

Soil in the study site was classi"ed as silt loam, accord-ing to the USDA soil classi"cation system (Day, 1965).

225TRANSFER FUNCTION MODELLING OF FLOW

However, the spatial variability of particle size distribu-tion over the study site was fairly high, with the coe$c-ient of variation for sand ranging from 7 to 33%, that forsilt ranging from 2 to 10%, and that for clay between4 and 43%. Organic matter content also varied highly,with coe$cient of variation between 4 and 21% (Diiwu,1997).

Rainfall intensity of 15)6 cmh~1 was simulated byusing the Guelph Rainfall Simulator II with a 12)7 mmfull jet nozzle that was maintained at a height of 1)5 mabove the soil surface and was operated at a pressurebetween 48)27 and 55)16 kPa (Tossel et al., 1987). Waterfor rainfall simulation was supplied by a pump at a rateof 2)4 m3 h~1. The simulator was turned on and left torun for about 15 min, during which time a ponding depthwas established and maintained. Surface and subsurface#ows were sampled for #ow volume and bromide con-centration at an interval of 1 min, over a 45 min periodfrom the beginning of simulation. Four rainfall simula-tions were carried out in 1993, and also in 1994. Thesubsurface hydrographs and breakthrough curves ob-tained from the 1993 rainfall simulations were par-titioned into macropore and micropore componentsusing a hydrograph separation technique (Diiwu, 1997;Diiwu et al., 2000). The partitioned 1993 data was usedfor calibration of the model, and the 1994 data was usedfor validation.

2.2. Modelling solute transport through soil by transferfunction technique

Consider a soil system into which a solute enters onlythrough the inlet surface at an initial time of 0, and thatthe only means by which the solute leaves the system,after some time t later, is through an exit boundary.Suppose that the input and output concentrations inlg cm~3 of the solute are C

i(t) and C

o(t), respectively.

Then Ci(t) and C

o(t) are related by the expression

Co(t)"PCi

(t!q) f (q) dq (1)

Here t is called the travel time, q is a generalized integra-tion variable applicable in other domains as well as thetemporal domain, and f (t) is the travel time probabilitydensity function of the solute. Also, f (t) is called theimpulse response function (or system response function)corresponding to a narrow pulse input of solute. Equa-tion (1) is the transfer function model for solute transportthrough the soil system (Jury, 1982; Jury & Sposito, 1985;Sposito & Jury, 1988).

The solute travel time probability density function isa relationship between the processes involved in trans-

mitting the solute from the inlet surface through the soilsystem to the exit surface and the probability laws gov-erning such processes. The product f (t)dt gives the prob-ability that a solute molecule entering the soil system atan initial time of 0 exits from it at some time q during thetime interval (t, t#dt) (Jury, 1982; Jury & Sposito, 1985;Sposito & Jury, 1988). In other words, this product is theproportion of the total population of solute moleculesthat will exit from the soil system during that time inter-val. Hence, the travel time probability density function isa mathematical tool of practical signi"cance: knowledgeof it does not only help to characterize the soil systemthrough which the solute is transmitted, but also servesas a tool for water resource and environmental systemsmanagement.

2.3. Modelling drainage through soil by transferfunction technique

It has been suggested by Besbes and Marsily (1984)that, based on the assumption of a linear process anexpression similar to Eqn (1) can be applied to relatedrainage at the observation depth to in"ltration throughthe soil surface:

Q(t)"PI (t!q) g(q) dq (2)

where: Q(t) is the drainage #ux as a function of time, I(t)is the in"ltration rate and g (t) is the impulse responsefunction of the soil system to in"ltration. Here g (t) maybe considered to be the travel time probability densityfunction of water as it is transmitted from the soil surfaceto the observation depth at which the catchment panswere installed. The movement of a water molecule fromthe soil surface to the observation depth is characterizedby g (t), as the latter represents the processes in#uencingsuch movement.

2.4. Formulation of transfer function model usinga mixed probability density function

For the purpose of model development the soil pro"leis assumed to be a closed system to which input is onlythrough the inlet surface which, in this case is the soilsurface. Output from the system is only through the exitsurface at the catchment pan. Once a molecule is intro-duced into such a system it may leave the system onlythrough the exit surface, but not through any otherboundaries nor by any other means. Hence, one-dimen-sional #ow is assumed.

The soil system is also assumed to be linear and itssubsystems, that is macropore and micropore domains,

J. Y. DIIWU E¹ A¸ .226

are assumed to be time invariant. It is also assumed thatthe transmission of solute and water through the soilsystem are linear processes. Because the path a particularsolute or water molecule has to travel from the inletsurface to the exit surface may be tortuous, and itstransport is in#uenced by several random processes with-in the soil system, the travel time t is considered asa stochastic variable subject to the probability distribu-tion represented by f (t) for a solute molecule or g (t) fora water molecule.

2.4.1. Accounting for dual-porosity in modelWater #ow in the macropore domain has been recog-

nized by various investigators to be non-Darcian (Jury& FluK hler, 1992; Beven & Germann, 1981; Germann& Beven, 1981; Chen & Wagenet, 1992). Macropores areknown to conduct water rapidly through unsaturatedsoil ahead of the wetting front in the soil matrix (Ger-mann, 1988; Kung, 1990). Even in some instances, #ow inmacropores has been found to be quasi or even fullyturbulent in either the saturated or unsaturated zone(Chen & Wagenet, 1992). These observations are in sharpcontrast with the nature of #ow in the micropore domain,that is Darcian and laminar. The distinction betweenthe processes in the two domains justi"es the use ofthe dual-porosity concept on which the development ofthe proposed mixed probability distribution is based.The fact that the processes in#uencing water and solutetransmission in the macropore domain are distinct fromthose in the micropore domain suggests that the formerprocesses belong to a statistical population distinctfrom that of which the processes in the microporedomain are a sample. Hence, the probability densityfunctions representing such processes must also bedistinct from each other. The two probability densityfunctions are then super-imposed to represent the overallprocess in the soil system (Singh, 1968; Cooke &Mostaghimi, 1994).

Following Moran (1959), the expression a1f1(t)#

a2f2(t)#2#a

nfn(t) is called a mixture of probability

density functions f1(t), f

2(t),2, f

n(t) if a

1, a

2,2, a

nare

non-negative and a1#a

2#2#a

n"1. Such a prob-

ability density function is called a mixed or compoundprobability density function, and a

1, a

2,2, a

nare called

the relative weights of the compound probability densityfunctions. In this paper, the component probability den-sity functions for solute transport in the macropore andmicropore domains are denoted by f

ma(t) and f

mi(t), re-

spectively, while the corresponding relative weights aredenoted by a

maand a

mi. Hence, the proposed mixed

probability density function for solute transport in thecomposite soil system is of the form

f (t)"ama

fma

(t)#ami

fmi

(t) (3)

A similar expression for mixed probability density func-tion for drainage is

g (t)"dma

gma

(t)#dmi

gmi

(t) (4)

where gma

(t) and gmi

(t) are the drainage probability den-sity functions for the macropore and micropore domains,respectively, with corresponding relative weights d

maand d

mi.

2.4.2. Component probability density functions of modelThe probability density functions f (t) and g (t) need to

be represented by parametric frequency distributions. Inthis paper, the normal, lognormal and two-parametergamma distributions were tested for goodness-of-"t tothe experimental data obtained as described in Section2.1. These three theoretical distributions were selectedsince they are the most commonly used in drainage andsolute transport studies. The choice is supported byStedinger et al. (1993).

As a "rst step, normal and lognormal probability plotswere done using solute travel time data for the macro-pore and micropore domains. The lognormal probabilityplot was the best for the two domains. The Kol-mogorov}Smirnov one-sample test for normality wasthen done on the raw data as well as the log-transformeddata. At the 90 and 95% con"dence levels the hypothesisof normality of the raw data was rejected in favour of thelog-transformed data. The lognormal probability densityfunction was, therefore, selected as the theoretical prob-ability density function of solute travel time in both themacropore and micropore domains. Similar analysis wasdone using drainage data for the macropore and micro-pore domains. The normal, lognormal and gamma prob-ability plots did not yield a best-"t distribution fordrainage in the two domains. Moreover, the Kol-mogorov}Smirnov one-sample test rejected the hypothe-sis of normality of the raw data and log-transformed data(Haan, 1977). The gamma probability density functionwas, therefore, selected as the representative theoreticaldistribution for drainage in both the macropore andmicropore domains (Besbes & Marsily, 1984).

2.5. Calibration of model

The calibration of the proposed mixed travel timedistributions involved determination of the moments ofthe macropore and micropore components of the mixeddistribution as well as the corresponding relative weights.The data used for this were those obtained by rainfallsimulation in 1993 (see Section 2.1) and separated intomacropore and micropore components by a hydrographseparation technique (Diiwu, 1997; Diiwu et al., 2000).

227TRANSFER FUNCTION MODELLING OF FLOW

2.5.1. Moments of mixed probability density functionThe mean and variance of the mixed probability distri-

bution are of the form (Singh & Sinclair, 1972)

k"ama

kma

#ami

kmi

(5)

p2"ama

p2ma#a

mip2mi#a

maami

(kma!k

mi)2 (6)

where kma

and kmi

denote the "rst moments of the com-ponents of the mixed distribution for solute transport inthe macropore and micropore domains, while p2

maand

p2mi

denote the corresponding second central moments.The expressions for the moments of the mixed distribu-tion for drainage are similar to Eqns (5) and (6) withama

and ami

replaced by dma

and dmi

, respectively.

2.5.2. Relative weights of components of mixedprobability density function

The relative weights of the components of the mixedprobability distribution are needed for the drainage andsolute transport models which are based on the conceptof dual porosity. The relative weights are measures of therelative contributions of the macropore and microporedomains to drainage and solute transport through thesoil system. For solute transport these relative weightsare de"ned as

ama"

Ama

Ama#A

mi

(7)

ami"

Ami

Ama#A

mi

(8)

where Ama

and Ami

denote, respectively, the areas underthe macropore and micropore components of the break-through curve for the particular event. The expressionsfor d

maand d

mifor drainage are similar to Eqns (7) and (8)

with Ama

and Ami

denoting the areas under the macro-pore and micropore components of the subsurface hy-drograph for the particular event. The areas A

maand

Ami

were each obtained by means of the trapezoidal rule(Abramowitz & Stegun, 1970):

A*"DtC

bo#b

m2

#b1#b

2#2#b

m~1D (9)

where A*

is a generalized symbol representing eitherA

maor A

miand *t is the time step. The symbols b

i,

i"0, 1, 2,2, m, denote the ordinates of the break-through curve or subsurface hydrograph, depending onwhether a

maand a

mior d

maand d

miare being computed.

2.6. <alidation of model

The performance criteria used to evaluate the validityof the proposed model includes comparison of the

predicted and observed subsurface hydrographs andbreakthrough curves for selected rainfall simulationevents, and evaluation of a performance index I

Pde"ned

as

Ip"1!

+DXobs

!Xpred

D+X

obs

(10)

where Xobs

is the observed #ow or solute concentration,X

predis the predicted #ow or solute concentration, and

DXobs

!Xpred

D is the absolute di!erence between Xobs

andX

pred. This expression de"nes an index which indicates

the deviation of the predicted data from the observeddata. The value I

P"1 indicates perfect performance of

the model, a situation hardly attained in the "eld due toextreme variability in physical and hydraulic propertiesof "eld soil (Diiwu et al., 1998).

3. Results and discussion

3.1. Parameters of model

The parameters of the lognormal probability densityfunction were "tted for solute transport while those forthe two-parameter gamma probability density functionwere "tted for drainage using the method of moments.The "rst moment of travel time is the time that the centreof mass of the drainage or solute concentration distribu-tion arrives at the observation depth, in this case thecatchment pan. The second central moment of traveltime gives the time required to transport all the solutemolecules through the soil pro"le or completely drain thepro"le of the water applied. For every probability distri-bution, the parameters are related to the "rst and secondcentral moments (Haan, 1977). In the case of the lognor-mal distribution, the parameters are the mean and stan-dard deviation of the log-transformed solute travel times,while for the gamma distribution they are the shape andscale parameters. The "tted parameters of the lognormaland gamma probability density functions are presentedin Tables 1 and 2, respectively.

For each simulation event the "tted parameters of thelognormal probability density function of solute traveltime in the micropore domain are higher than in themacropore domain (see Table 1). These trends are ex-pected since solute peak concentrations would be ob-tained much earlier in the macropore domain than in themicropore domain, and also solute transport in the for-mer domain is faster than in the latter domain. In the caseof drainage in the macropore and micropore domainsduring each simulation event, the "tted parameters of thegamma travel time probability density function present-ed in Table 2 indicate that the shape parameter is greater

Table 1Fitted parameters of lognormal probability density function

Macropore domain Micropore domain

Plot no. Date k* ps k* ps

1 26 May !1)431 0)044 !1)978 0)1882 26 May !0)744 0)051 !0)947 0)2541 1 June !1)618 0)009 !1)922 0)1322 1 June !1)654 0)267 !2)774 0)3333 1 June !0)001 0)0001 !1)193 0)0931 8 June !1)414 0)028 !1)974 0)0292 8 June !1)67 0)061 !2)377 0)0753 8 June !1)109 0)009 !1)913 0)3031 7 July !0)708 0)015 !1)792 0)0022 7 July !1)758 0)149 !2)695 0)206

*k, mean of logarithm of travel time. s p, standard deviation of logarithm of travel time.

J. Y. DIIWU E¹ A¸ .228

in the macropore domain than in the micropore domain;however, there is no clear trend for the scale parameter.

The computed relative weights of the proposed mixedprobability density functions are presented in Table 3.The relative weights presented here are speci"c for thetest "eld used in the study; they, therefore, need to becomputed when the mixed distribution form of the trans-fer function model is to be applied at another site. Thenumber and spatial distribution of macropores in the soilpro"le depend on such factors as climate conditions,biological activities in the soil and land use. Hence, therelative weights, which depend on the spatial distributionof macropores, would need to be checked occasionally tomake sure that they have not changed with time.

The relative weights for drainage indicate that for plot1 the probability of subsurface #ow observed at thecatchment pan being macropore #ow was highest at 55%on 1 June and lowest at 19% on 7 July. These values

TablFitted parameters of gamma

Macropo

Plot no. Date g

1 26 May 5)0962 26 May 1)9521 1 June 4)0052 1 June !0)6113 1 June 1)2751 8 June 1)2552 8 June 0)5983 8 June 1)4041 7 July 0)8412 7 July 0)674

Note: g, shape parameter; b, scale parameter.

indicate that for plot 1 macropore #ow was most likely tobe observed on 1 June and least likely on 7 June. For plot2, macropore #ow was more likely to be observed on7 July and least likely on 1 June. In the case of plot 3,macropore #ow was more likely on 8 June than on 7 July.In the "eld, macropore #ow was most likely to be ob-served during the rainfall simulation of 1 June on plot1 than for any other rainfall simulation event.

The relative weights for solute transport indicate thatfor plot 1 the macropore domain was most likely to havecontributed highly to the mass of solute observed at thecatchment pan on 26 May and least likely on 7 July. Forplot 2 the contribution of the macropore domain wasmost likely on 26 May and least likely on 7 July. In thecase of plot 3, the probability of the macropore domaincontributing to the total mass of solute transport to thecatchment pan was 60% on 1 June as against 50%on 8 June. Of the three plots, the macropore domain on

e 2probability density function

re domain Micropore domain

b g b

48)773 !0)446 12)30226)597 0)764 29)89254)998 !0)551 12)1413)892 !0)38 21)379

11)375 1)208 45)99718)036 0)076 22)41815)983 !0)439 18)10016)809 !0)271 16)20616)586 0)411 32)80316)736 !0)553 14)414

Table 3Relative weights of components of mixed probability density function

Drainage travel time Solute travel time

Plot no. Date dma

dmi

ama

ami

1 26 May 0)402 0)598 0)573 0)4272 26 May 0)296 0)705 0)565 0)4351 1 June 0)553 0)447 0)505 0)4952 1 June 0)058 0)942 0)464 0)5363 1 June 0)388 0)612 0)601 0)3991 8 June 0)229 0)772 0)488 0)5122 8 June 0)246 0)754 0)507 0)4933 8 June 0)425 0)574 0)496 0)5041 7 July 0)186 0)814 0)356 0)6442 7 July 0)411 0)589 0)441 0)559

Note: dma

, relative weight for drainage in macropore domain; dmi

, relative weight for drainage in micropore domain; ama

,relative weight for solute transport in macropore domain; a

mi, relative weight for solute transport in micropore domain.

229TRANSFER FUNCTION MODELLING OF FLOW

plot 3 was the most likely to have contributed the highestproportion of mass of solute observed at the catchmentpan on 1 June than for any other simulation event. Thesevalues seem to indicate that high drainage rates do notnecessarily coincide with high solute transport rates.

3.2. Performance of model

The observed solute concentrations and drainage#uxes used for evaluating the proposed model were those

Fig. 1. Comparison of observed and predicted subsurface hy-drographs: , observed yow; , predicted yow

obtained by rainfall simulation in 1994 (see Section 2.1).The corresponding predicted values were obtained byapplying the discrete convolution forms of Eqns (1) and(2) in Eqns (3) and (4). The predicted and observedsubsurface hydrographs and breakthrough curves didnot deviate markedly, as shown in Figs 1 and 2. Thevalues of the performance index I

Pare presented in Table 4.

These range between 0)900 and 0)999, indicatingvery close agreement between the predicted and ob-served hydrographs and breakthrough curves. The two

Fig. 2. Comparison of observed and predicted breakthroughcurves: , observed solute concentration; , predicted

solute concentration

Table 4Performance index Ip of proposed model

Plot no. Performance index

Drainage Solute transport

1 0)996 0)9992 0)900 0)9233 0)966 0)936

J. Y. DIIWU E¹ A¸ .230

performance criteria, therefore, showed that the pro-posed model performs quite well, and so adequatelyincorporates dual porosity in the transfer function model.Between rainfall simulation plots the performance indexIP

values showed that in predicting drainage the modelperforms better on plots 1 and 3 than on plot 2. Forpredicting solute transport, the model performs best onplot 1 and worst on plot 2.

Using the observed and predicted subsurface #ows thechange in storage during each time step was estimated bymeans of water balance. For each time step the estimatedchange in storage using the observed subsurface #ow wascompared with the estimated change in storage using thepredicted subsurface #ow. It was found that the change instorage using the predicted subsurface #ows fell short by5}10% compared to the change in storage using theobserved subsurface #ows. Mass balance check was alsocarried out by computing the di!erence between mass ofsolute in the predicted solute transport and mass ofsolute in the measured surface runo! during each timestep. It was found that the shortfall in this di!erence inmass of solute compared to the mass in the measuredsolute transport varied from 3 to 8%.

4. Conclusions

A mixed probability density function has been pro-posed for formulating the transfer function model of #owthrough macroporous soil, based on the concept of dualporosity. Frequency analysis revealed that the lognormaldistribution was the best theoretical distribution for sol-ute transport in the macropore and micropore domains.In the case of drainage, a goodness-of-"t test did notreveal an appropriate theoretical distribution. The two-parameter gamma distribution was, therefore, selectedfor the probability distribution of drainage.

Comparison of observed and predicted subsurface hy-drographs and breakthrough curves and computed per-formance index values showed that even though themodel is hardly perfect, it performs fairly well. The pro-posed approach, therefore, adequately incorporates mac-ropore and micropore #ows and their associated solute

concentrations as distinct statistical populations in thetransfer function model. However, because the modelparameters depend on the occurrence and distribution ofmacropores in the "eld, the parameters are spatially andtemporally variable in any "eld. This imposes the re-quirement that the model must be recalibrated for every"eld once in a while.

Acknowledgements

The "nancial support of the Natural Science andEngineering Research Council (NSERC) of Canada isgreatly appreciated. Thanks also to Agriculture andAgri-Food Canada for making the necessary facilitiesavailable for the "eld experiments.

References

Abramowitz M; Stegun I A (1970). Handbook of MathematicalFunctions, 1046pp. Dover Publications Inc., New York

Besbes M; de Marsily G (1984). From in"ltration to recharge:use of a parametric transfer function. Journal of Hydrology,74, 271}293

Beven K; Germann P (1981). Water #ow in soil macropores II.A combined #ow model. Journal of Soil Science, 32, 15}30

Chen C; Wagenet R J (1992). Simulation of water and chemicalsin macropore soils, Part 1. Representation of the equivalentmacropore in#uence and its e!ect on soil water #ow. Journalof Hydrology, 130, 105}126

Cooke R A; Mostaghimi S (1994). MIXFIT. A microcomputer-based routine for "tting heterogeneous probability distribu-tion functions to data. Transactions of the ASAE, 37(5),1463}1472

Day R P (1965). Particle fractionation and particle size analysis.In: Methods of Soil Analysis, Part 1 (Black C A, ed), pp545}567, Agronomy Series, No. 9. American Society of Ag-ronomy, Madison, WI

Diiwu J Y (1997). Transfer function modelling of drainage andsolute transport through layered macroporous soil. PhDThesis, School of Engineering, University of Guelph, Guelph,Ontario, Canada

Diiwu J Y; Rudra R P; Dickinson W T; Wall G J (1998).Dependence of variability of hydraulic properties on physicalproperties of "eld soil. Journal of Environmental Systems,26(2), 171}189

Diiwu J Y; Rudra R P; Dickinson W T; Wall G J (2000).Two-component analysis of #ow through macroporous soil.Journal of Agricultural Engineering Research, doi:10.1006/jaer.2000.0617

Dyson J S; White R E (1987). A comparison of the convec-tion}dispersion equation and transfer function model forpredicting chloride leaching through undisturbed, structuredclay soil. Journal of Soil Science, 40, 525}542

Germann P F (1988). Approaches to rapid and far-reachinghydrologic processes in the vadose zone. Journal of Con-taminant Hydrology, 3, 115}127

Germann P; Beven K (1981). Water #ow in soil macropores. I.An experimental approach. Journal of Soil Science, 32, 1}13

231TRANSFER FUNCTION MODELLING OF FLOW

Haan C T (1977). Statistical Methods in Hydrology, 1st Edn,378pp. The Iowa State University Press, Ames

Hillel D I (1980). Fundamentals of Soil Physics, 413pp. Aca-demic Press Inc., San Diego, CA

Jury W A (1982). Simulation of solute transport using a transferfunction model. Water Resources Research, 18, 363}368

Jury W A (1983). Chemical transport modelling: current ap-proaches and unresolved problems. In: Chemical Mobilityand Reactivity in Soil Systems (Nelson D W; Elrick D E;Tanji K K, eds), pp 49}64. American Society of Agronomy,Madison, WI

Jury W A; Dyson J S; Butters G L (1990). A transfer functionmodel of "eld scale solute transport under transient water#ow. Soil Science Society of America Journal, 54, 327}332

Jury W A; FluK hler H (1992). Transport of chemicals throughsoil: mechanisms, models, and "eld applications. Advances inAgronomy, 47, 141}201

Jury W A; Gardner W R; Gardner W H (1991). Soil Physics, 5thEdn., 328pp. John Wiley and Sons, Inc., New York

Jury W A; Sposito G (1985). Field calibration and validation ofsolute transport through soil. 1. Fundamental concepts.Water Resources Research, 22, 243}247

Jury W A; Stolzy L H; Shouse P (1982). A "eld test of thetransfer function model for predicting solute transport.Water Resources Research, 18, 369}375

Kung K-J S (1990). Preferential #ow in a sandy vadose zone. 1.Field observation. Geoderma, 46, 59}71

Moran P A (1959). The Theory of Storage, 1046pp. John Wileyand Sons, Inc., New York

Singh K P (1968). Hydrologic distributions resultingfrom mixed populations and their computer simulation.International Association of Scienti"c Hydrology, 81,671}681

Singh K P; Sinclair R A (1972). Two-distribution method for#ood-frequency analysis. Proceedings of American Society ofCivil Engineers, Hydraulics Division, 98, 29}43

Sposito G; Jury W A (1988). The lifetime probability densityfunction for solute movement in the subsurface zone. Journalof Hydrology, 102, 503}518

Stedinger J R; Vogel R M; Foufoula-Georgio E (1993). Fre-quency analysis of extreme events. In: Handbook of Hydrol-ogy (Maidment D R, ed), pp 18.1}18.66, McGraw-Hill, Inc.New York

Tossel R W; Dickinson W T; Rudra R P; Wall G J (1987).A portable rainfall simulator. Canadian Agricultural Engin-eering, 29, 155}162

White R E; Dyson J S; Haigh R A; Jury W A; Sposito G (1986).A transfer function model of solute transport through soil. 2.Illustrative applications. Water Resources Research, 22,248}254