SW—Soil and Water: Two-component Transfer Function Modelling of Flow through Macroporous Soil

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  • ain the transfer function model for drainage and solute transport through macroporous soil.( 2001 Silsoe Research Institute1. 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-

    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 twoJ. agric. Engng Res. (2001) 80 (2), 223}231doi:10.1006/jaer.2000.0673, available online at http://wwwSW*Soil and Water

    Two-component Transfer Function Mode

    J. Y. Diiwu1; R. P. Rudra1; W

    1 School of Engineering, University of Guelph, Guelph, ON, Canad2Land Resource Research Centre, Agriculture and Agri-Food Can

    (Received 7 June 1999; accepted in revised form 11

    The macropore and micropore domains of a soil systemand solute transport through the soil were then chcomponents of the mixed probability distributions weresolute transport in the macropore and micropore domdistribution was the best theoretical distribution for sotwo-parameter gamma distribution was best for drainagthe mixed probability distribution representation of the tcause 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.idealibrary.com on

    lling of Flow through Macroporous Soil

    . T. Dickinson1; G. J. Wall2

    a N1G 2W1; e-mail of corresponding author: jdiiwu@uoguelph.caada, Guelph, ON, Canada N1H 6N1; e-mail: gwall@uoguelph.ca

    October 2000; published online 21 August 2001)

    were considered to be hydraulically distinct. Drainagearacterized by mixed probability distributions. Theused to represent the distinct processes of drainage andins. A goodness-of-"t test showed that the lognormallute transport in each of the two domains, while thee in each of the domains. A validation test showed thatransfer function adequately incorporated dual porositymodels 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

  • from soil systemma

    pore domain

    Ufma

    (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

    dmi

    relative weight for drainage in micro-pore domain

    g shape parameterk mean of logarithm of travel time

    kma

    "rst moment of fma

    (t)kmi

    "rst moment of fmi

    (t)p standard deveation of logarithm of

    travel timep2ma

    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-

    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! generatedNota

    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 exiting

    J. Y. DIIW224port processes in the macropore domain, while the othercomponent represents the net e!ect of processes in themicropore domain.tion

    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 parameterd relative weight for drainage in macro-

    E A .and 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).

  • MHowever, 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;

    TRANSFER FUNCTIONSposito & Jury, 1988).The solute travel time probability density function is

    a 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-

    225ODELLING OF FLOWsional #ow is assumed.The soil system is also assumed to be linear and its

    subsystems, that is macropore and micropore domains,

  • 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, fn(t) if a1, a2,2, an are

    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, an are 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 the

    J. Y. DIIW226composite 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 rainfall

    U E A .simulation in 1993 (see Section 2.1) and separated intomacropore and micropore components by a hydrographseparation technique (Diiwu, 1997; Diiwu et al., 2000).

  • M2.5.1. 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 either

    Ama

    or Ami

    and *t is the time step. The symbols bi,

    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.

  • 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

    n

    J. Y. DIIWU E A .228*k, mean of logarithm of travel time. s p, standard deviatio

    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 plotTablFitted 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.

    1 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 valuesof logarithm of travel time.

    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 domaine 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

    contributing 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

  • 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.

    plot 3 was the most likely to have contributed the highestproportion of mass of solute observed at the catchment

    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

    229TRANSFER FUNCTION MODELLING OF FLOWpan 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 thoseFig. 1. Comparison of observed and predicted subsurface hy-drographs: , observed yow; , predicted yow(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 twoFig. 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

    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 #ow

    J. Y. DIIW230through 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 soluteconcentrations 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.

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    231TRANSFER FUNCTION MODELLING OF FLOW

    Notation1. Introduction2. Materials and Methods3. Results and discussionTable 1Table 2Table 3Figure 1Figure 2Table 4

    4. ConclusionsAcknowledgementsReferences

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