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

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<ul><li><p>ain the transfer function model for drainage and solute transport through macroporous soil.( 2001 Silsoe Research Institute1. Introduction</p><p>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.</p><p>Stochastic modelling approaches were introduced be-</p><p>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</p><p>Two-component Transfer Function Mode</p><p>J. Y. Diiwu1; R. P. Rudra1; W</p><p>1 School of Engineering, University of Guelph, Guelph, ON, Canad2Land Resource Research Centre, Agriculture and Agri-Food Can</p><p>(Received 7 June 1999; accepted in revised form 11</p><p>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</p><p>0021-8634/01/100223#09 $35.00/0 223.idealibrary.com on</p><p>lling of Flow through Macroporous Soil</p><p>. T. Dickinson1; G. J. Wall2</p><p>a N1G 2W1; e-mail of corresponding author: jdiiwu@uoguelph.caada, Guelph, ON, Canada N1H 6N1; e-mail: gwall@uoguelph.ca</p><p>October 2000; published online 21 August 2001)</p><p>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</p><p>( 2001 Silsoe Research Institute</p></li><li><p>from soil systemma</p><p>pore domain</p><p>Ufma</p><p>(t) probability density function for solutetransport in macropore domain</p><p>fmi</p><p>(t) probability density function for solutetransport in micropore domain</p><p>g (t) soil system response function toin"ltration</p><p>gma</p><p>(t) probability density function for drainagein macropore domain</p><p>gmi</p><p>(t) probability density function for drainagein micropore domain</p><p>I(t) in"ltration rate, cmmin~1</p><p>dmi</p><p>relative weight for drainage in micro-pore domain</p><p>g shape parameterk mean of logarithm of travel time</p><p>kma</p><p>"rst moment of fma</p><p>(t)kmi</p><p>"rst moment of fmi</p><p>(t)p standard deveation of logarithm of</p><p>travel timep2ma</p><p>second central moment of fma</p><p>(t)p2</p><p>misecond central moment of f</p><p>mi(t)</p><p>q integration variable</p><p>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).</p><p>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-</p><p>2. Materials and Methods</p><p>2.1. Data collection</p><p>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</p><p>Ama</p><p>area under the macropore component ofbreakthrough curve or subsurface hydro-graph</p><p>Ami</p><p>area under the micropore component ofbreakthrough curve or subsurface hydro-graph</p><p>A*</p><p>generalized symbol representing eitherA</p><p>maor A</p><p>mibi</p><p>ordinates of breakthrough curve or sub-surface hydrograph</p><p>Ci(t) input concentration of solute, lg cm~3</p><p>Co(t) output concentration of solute, lg cm~3</p><p>f (t) travel time probability density function/soil system response function to soluteinput</p><p>f (t)dt proportion of solute molecules exiting</p><p>J. Y. DIIW224port processes in the macropore domain, while the othercomponent represents the net e!ect of processes in themicropore domain.tion</p><p>IP</p><p>performance index of modelQ (t) drainage #ux, cmmin~1</p><p>t travel time, min(t, t#dt) time interval</p><p>*t time stepX</p><p>obsobserved #ow or solute concentration</p><p>Xpred</p><p>predicted #ow or solute concentrationDX</p><p>obs!</p><p>Xpred</p><p>D absolute di!erence between Xobs</p><p>andX</p><p>predama</p><p>relative weight for solute transport inmacropore domain</p><p>ami</p><p>relative weight for solute transport inmicropore domain</p><p>b scale parameterd relative weight for drainage in macro-</p><p>E A .and the concentration of tracer present in the runo!.Soil in the study site was classi"ed as silt loam, accord-</p><p>ing to the USDA soil classi"cation system (Day, 1965).</p></li><li><p>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).</p><p>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.</p><p>2.2. Modelling solute transport through soil by transferfunction technique</p><p>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</p><p>i(t) and C</p><p>o(t), respectively.</p><p>Then Ci(t) and C</p><p>o(t) are related by the expression</p><p>Co(t)"PCi(t!q) f (q) dq (1)</p><p>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;</p><p>TRANSFER FUNCTIONSposito & Jury, 1988).The solute travel time probability density function is</p><p>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.</p><p>2.3. Modelling drainage through soil by transferfunction technique</p><p>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:</p><p>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.</p><p>2.4. Formulation of transfer function model usinga mixed probability density function</p><p>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-</p><p>225ODELLING OF FLOWsional #ow is assumed.The soil system is also assumed to be linear and its</p><p>subsystems, that is macropore and micropore domains,</p></li><li><p>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.</p><p>2.4.1. Accounting for dual-porosity in modelWater #ow in the macropore domain has been recog-</p><p>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).</p><p>Following Moran (1959), the expression a1f1(t)#</p><p>a2f2(t)#2#a</p><p>nfn(t) is called a mixture of probability</p><p>density functions f1(t), f</p><p>2(t),2, fn(t) if a1, a2,2, an are</p><p>non-negative and a1#a</p><p>2#2#a</p><p>n"1. Such a prob-</p><p>ability density function is called a mixed or compoundprobability den...</p></li></ul>

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