MATHEMATICAL EQUATIONS OF THE SPREAD OF POLLUTION IN SOILS

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1 MATHEMATICAL EQUATIONS OF THE SPREAD OF POLLUTION IN SOILS F. Stagnitti Deakin University, Shool Of Life & Environmental Siene, Warrnambool, Australia J.-Y. Parlange, T. S. Steenhuis Cornell University, College Of Engineering, Ithaa, NY, USA D. A. Barry Éole Polytehnique Fédérale de Lausanne, Switzerland L.Li and D. A. Lokington University of Queensland, Australia, G. C. Sander Loughborough University, Leiestershire, UK Keywords: Solute transport, Soil pollution, Mathematial models, Convetive-Diffusive equation, Snow plow, Preursor, Adsorption, Vadose zone. Pélet number, Maropores, Infiltration, Erosion, Preferential flow Contents 1. Introdution 2. Convetive-Diffusive Equation 3. Effets of Boundary Conditions 4. Chemial Reations 5. Nonlinear Adsorption 6. Two Speies Competition 7. Interation of Surfae Water and Chemial Transport in Soils 8. Column Flow 9. Transient Unsaturated Water and Solute Transport 1. Sale Dependent Solutions 11. Transient Solution Profiles 12. Soure Solutions 13. Conlusion Glossary Bibliography Biographial Skethes Summary Models desribing the spread of pollution through soils are disussed. The spread of pollution is soils is ontrolled by the flow of water and, in most ases, is desribed by the onvetive-dispersive equation. First we onsider ases when the water veloity is assumed onstant. Effets of boundary onditions, hemial reations, adsorption and speies

2 ompetition are desribed for this ase. Then two other ases are disussed; (1) when hydrology ontrols solute transport and (2) when the onvetive-dispersive equation is less important. In the former ase, erosion due to raindrop impat and the transport of pollutants adsorbed on fine partiles is disussed. In the latter ase, preferential flows, whih an be linked to either strutural voids in the soil (e.g. maropores, raks, et) or to flow instability are onsidered. Mathematial expressions desribing these ases are presented. In the final setion of this hapter we present a disussion of ases when Rihards equation ontrols water movement. When Rihards equation is used, it is diffiult to analyze solute transport due to the strongly nonlinear nature of the equation. However, a few exat analytial solutions have been obtained reently and are presented here. 1. Introdution The study of hemial transport in soils is important for a number of reasons. Some hemials are important as they are required for soil and plant health (e.g. mironutrients). Other hemials may be highly toxi partiularly if they are present in high onentrations. A hemial beomes a pollutant if its onentration exeeds some presribed water quality standard, or if a benefiial water use has been impaired, and if the ause is indued by human ativity. The study of the fate of hemials and hemial pollution in soil is vital for sustaining agriultural produtivity and land utility. The geologial media between the land surfae and the regional water table below is alled the unsaturated zone or vadose zone (Stephens, 1996). The word vadose is derived from the Latin word vadosus meaning shallow (Looney and Falta, 2a). In aord with its definition and meaning, the vadose zone inludes the rop root layer, the intermediate zone between the root layer and the apillary fringe above the saturated water table. This zone therefore plays an integral role in the global hydrologial yle ontrolling surfae water infiltration, runoff and evaporation and hene the availability of soil water and nutrients to plants. Initial investigations of this zone were foused on water availability to rops and optimal management of the root zone. However, in reent years muh more attention has foused on hemial transport in and through this zone as a result of inreased use of agrohemials suh as fertilizers and pestiides and inreased demands to store and dispose of industrial and muniipal wastes suh as sewage. This zone is typially the first subsurfae environment to enounter surfae applied agrohemials and ontaminants and hene all surfae and subsurfae hemial onentrations and subsequent environmental impats are inextriably linked to the physial, biologial and hemial dynamis inluding sorption-desorption, volatilization, photolysis and degradation (Looney and Falta, 2a). Our urrent understanding of physial and hemial proesses in the vadose zone results largely from more than 7 years of mathematial modeling of variably saturated flow using Rihards equation (Rihards, 1931) oupled with the Fikian-based onvetion-dispersion equation for solute transport. Analytial and numerial solutions of these lassial equations are widely used to study and predit water flow and solute transport for speifi laboratory and field experiments and to extrapolate these results for other experiments in different soils, rops and limati onditions. However, many reent studies have demonstrated that the assumptions impliitly adopted in the Rihards and onvetivedispersion equations are limiting the sope and appliation of solutions to these equations for many agriultural and forestry management strategies. The spread of solute and

3 pollutants in soils is ompliated by non-random spatial and temporal variations of physial, hemial and biologial omponents of soils (Hill & Parlange, 1972; Stagnitti, et al. 1995; Parlange, et al.1996; Stagnitti, et al. 1995). One manifestation of spatial and temporal heterogeneity in soils is the phenomenon of preferential flow, a general term used to desribe a variety of physial and hemial non-equilibrium flow proesses. This hapter introdues a number of useful equations to study the spread of hemials and pollution in soils. It begins with the lassial desriptions of onvetion and dispersion and then introdues reent extensions, new solutions and new models inorporating important mehanisms suh as preferential flow. 2. Convetive-Diffusive Equation Chemials arried by water in and on the soil an follow a variety of paths. In the simplest ase, a one-dimensional onvetion takes plae whih an be aompanied by dispersion, adsorption and hemial reation. Desription of ontaminant transport in terms of an average water veloity, v, by the onvetive-dispersive equation remains the first and fundamental transport equation to be onsidered. In its simplest form, the onentration of a hemial moved in one-dimension in the x diretion, an be desribed by the following equation, + v = D f a t x x x t where v is the average water veloity, D is the dispersion, a the onentration of adsorbed hemial and f is the irreversible reation deay rate. In general, v, D, f, and a ould be omplex funtions of, x, and t. Consequently no exat simple analytial solution exists for all ases. When f is a linear funtion of, D and v are onstant and a = ( R-1), where R is the retardation fator (assumed onstant), then Eq. (1) is linear with onstant oeffiients. In this ase many exat solutions an be obtained, e.g. using Laplae transforms. van Genuhten and Alvers (1982) have presented numerous exat solutions for Eq. (1) with onstant oeffiients for a wide variety of boundary and initial onditions. 3. Effets of Boundary Conditions In a typial laboratory experiment involving a olumn with a finite length L, boundary onditions at x = and x = L must be speified. At x =, depending on the method of injetion, will be imposed, for instane if water with a onstant onentration is pushed at a onstant rate, then the flux of material entering the olumn is onstant, or, (1) f ( D/ v) = at x= x (2) where f is alled the flux onentration (Kreft and Zuber, 1978, 1986; van Genuhten and Parker, 1984) and is then alled the resident onentration. Interestingly when Eq. (1)

4 is linear with onstant oeffiients then f obeys a similar equation with the boundary ondition for f being simpler than for at x =. A standard initial ondition might be = f = at t = (3) The boundary ondition at x = L presents some diffiulties (Parlange et al. 1992). If disontinuity in is not permitted at the end of the olumn and there is no dispersion in the olletor, then = at x = L or f = at x = L x and this is true as long as the Pélet number P= / e vl D is not too small, e.g. at least 4 or more. Pélet numbers of at least 4 are ommonly found exept when maropores e.g. raks are present in the olumn and the effetive dispersion is dominated by diffusion between raks and the soil matrix surrounding them. In that ase, P > 4 e it an be shown (4) (Parlange and Starr, 1975; 1978; Parlange, et al. 1982) that taking f as the solution for a semi-infinite olumn f i.e. for L, leads to a, from the definition of f in Eq. (2), whih satisfies Eq. (1) with an error of order exp(- P e) << 1. For instane taking f = and saling time by R, for a semi-infinite olumn, x tv vx x + tv 2 f / = erf exp erf + 4Dt D 4Dt then Eq. (2) gives ( L tv) exp vx L + tv v 4 f exp erf Dt π Dt = + D 4Dt D L+ tv L+ tv erf 4Dt 4Dt 2 1 exp ( x+ tv) vx x + tv v exp erf Dt π 4Dt + + D 4Dt D x+ tv x+ tv erf 4Dt 4Dt (5) (6) derived from the general solution of Eq. (2) vx L f vx = f + exp exp dx D (7) x x D given Eq. (6) when Eq. (5) holds. It is onvenient to define as the solution for when L, then we obtain at one

5 vx ( L) = + exp [ f ( x= L) ( x= L) ] (8) D Eq. (8) will often be useful in pratie when f and are known. For instane for zero and first order kinetis (i.e. f onstant or proportional to ), f and are established in losed form (Parlange et al. 1982, 1992; Parlange and Starr, 1878; van Genuhten and Alves, 1982) and yield a simple expression for, with a slight error beause f is not exatly equal to f. In partiular at x = L, the breakthrough urve is very simply obtained by x ( = L) = ( x= L) ( x= L) (9) f f Not surprisingly the maximum error of Eq. (8) is at x = L. In Table 1 we onsider the ase when P = 4 e, the minimum aeptable value, and the three ases when f = ; fl / v =.2; and fl / v = 1, whih orresponds to no, zero and first order kinetis respetively. The maximum error is at time vt 2 / D= 4 and the absolute errors are about.2 in all three ases. vl / D = 4 f4 x ( = L) exat no kinetis zeroth order first order Table 1: Solutions of Eq. (8) for no, zero and first order kinetis The exat values presented in Table 1 are from van Genuhten and Avles (1982) where they are tabulated using somewhat omplex series solutions. If the Pélet number P e is very small and dispersion results from moleular diffusion between mobile and immobile water, Eq. (1) should not be used by itself and the region of immobile water should be taken into aount expliitly. This onept may have been first introdued by Passioura (1971) for aggregated porous media. The hemial in immobile water is a omponent of a, and expliitly written as ( θim / θm) im / t in Eq. (1), where θ im and θ m are the water ontents in the immobile and mobile regions and im is the onentration of the hemial in the immobile region. An additional equation is now required to find im. If the exhange proess between the two regions is slow ompared to transport proesses then the rate limited exhange is often modeled by the following equation θ / t = k ( ), (1) im im m im

6 where k is a rate oeffiient. Parker and Valohi (1986), van Genuhten and Dalton (1986) and Brusseau et al. (1994) have disussed various mass transfer proesses and their models. The other limiting ase when the exhange proess is rapid would lead to loal equilibrium although this limit does not seem to hold in pratie. Readers are referred to Wallah and Parlange (1998; 2) for a thorough disussion of the exhange proesses between the two regions. 4. Chemial Reations The disussion till now only onsidered linear equations. If the dispersion term in Eq. (1) an be negleted, at least as a first approximation, then the method of harateristis may be employed to find solutions of the governing equations. Although the method is very general we shall onentrate on a partiular ase whih illustrates the method quite learly and also makes use of results obtained in the previous setion. First onsider the ase of no adsorption, D and v onstant and f an arbitrary funtion of. Barry et al. (1993) solved this ase to analyze nitrogen transport and kinetis in soils (also see MLaren (1976), Starr et al. (1974), Starr and Parlange (1975) for appliations). When experimental observations are used to estimate f it is partiularly onvenient to use steady state results. Under steady state, Eq. (1) redues to, 2 d d v = D f. (11) 2 dx dx The method of harateristis ignoring dispersion yields a simple solution g1 d x= v, (12) f () where g 1 is a first approximation to the value of at x =. The effet of dispersion an signifiantly lower the onentration at x = below. Indeed it an be shown that ignoring dispersion in the differential results is far less aurate than when ignoring it in the boundary ondition at x = (Parlange et al. 1982, 1984; Barry et al. 1986). It is also lear that ignoring d 2 / dx 2 in Eq. (11) means that we are not onsidering any boundary ondition suh that with a olumn of length L, Eq. (4) for instane is irrelevant. Indeed it is lear from Eq. (12) that x is infinite at =, sine f () =. Thus Eq. (12) is relevant for a semi-infinite olumn. As we previously exhibited in the linear ase, will be lose to so long as the solution is not in the boundary layer near x = L, whih has a thikness of order D/ v, as shown by Eq. (8). Thus when using experimental data together with Eq. (12), it is ruial to hek, a posteriori if neessary, that all data of interest are suffiiently far from the end of the olumn. At the surfae ondition, Eq. (4) yields

7 D g1 = f( g 2 1), (13) v Clearly the proedure will be valid as long as the orretion due to dispersion is small or 2 v D f >>. (14) / ( ) 1 Brams and MLaren (1974) used Eq. (12) in their analysis but did not orret g 1 and used g 1 = whih is far less aurate than Eq. (13). It is now easy to iterate using Eq. (12) to estimate 2 2 Dd / dx in Eq. (11) or, g2 d D f ( g2) x= v + ln f() v f(), (15) where g2 is the new onentration at x =, or from Eq. (4) g = 2 Df( g2) df 2 v + D d = g2. (16) Note that Eq. (15) still holds for a semi-infinite olumn only. To estimate the auray of this proedure onsider the more realisti ase of Mihaelis- Menten kinetis when f f() = K +, (17) where f and K are the limiting rate onstant and the saturation onstant respetively. This funtion an be introdued easily into Eqs. (12), (13), (15), and (16) and in partiular g d 1 [ g K ln( g/ ) ] f() = f + (18) for use in Eqs. (12) and (15). Consider a reasonable example of v / f = 1 m and K = and values of D/ v varying from.5 to 5 m. Hene v 2 / Df( ) in Eq. (14) varies between 4, a very large value, and 4, a moderate value. The values of / at, 5, and 1 m are obtained numerially (Barry et al. 1993), i.e. they are onsidered exat. Then we take the results at 5 and 1 m and onsider them as observed and we ask the question what are the errors assoiated with those values if we use them to predit v/ f and k/ when Eqs. (12) and (13) or Eqs. (15) and (16) are hosen. The errors are presented in Table 2 and rounded up to the next whole perent. It is interesting to note that

8 either Eq. (13) and Eq. (16) produe insignifiant errors in the predition of () / as either g1/ or g2/, so in pratie using the simpler Eq. (13) rather than Eq. (16) is justified and reommended. In all ases, Eq. (15) is obviously quite reliable, although not shown in Table 2, the error in K/ inreases rapidly when D/ v > 5. Therefore the limit 2 of use for Eq. (15) and (16), or (13), is v/ Df( ) 4. On the other hand the limit of usability of Eq. (12) is quite poor, even for v 2 / Df( ) as high as 4 the estimate of K/ is not aeptable. This is of ourse expeted, as at least a first order orretion due to diffusion should be inluded. More details an be found in Barry et al. (1993). D/ v ()/ (5) / (1) / Using Eqs. (12) and (13): Relative Errors in % v / f K / -2 7 Using Eqs. (15) and (16): Relative Errors in % v / f K / Table 2: Exat onentrations at x =,5,and 1 m. When v/ f = 1 m and K / = 1, for various values of D/ v. The errors in parameters estimation are at the onentrations at 5 and 1 m TO ACCESS ALL THE 31 PAGES OF THIS CHAPTER, Visit: Bibliography Ahuja, L. R. (199). Modeling soluble hemial-transfer to runoff with rainfall impat as a diffusion proess. Soil Siene Soiety of Ameria Journal 54(2), pp

9 Ahuja, L. R. and Lehman, O. R. (1983). The extent and nature of rainfall-soil interation in the release of soluble hemials to runoff. Journal of Environmental Quality 12(1), pp Ahuja, L. R., Ross, J. D. and Lehman, O. R. (1981). A theoretial-analysis of interflow of water through surfae soil horizons with impliations for movement of hemials in field runoff. Water Resoures Researh 17(1), pp Baker, R. S. and Hillel, D. (199). Laboratory tests of a theory of fingering during infiltration into layered soils. Soil Siene Soiety Of Ameria Journal 54(1), pp Barry, D. A., Parlange, J. Y., Haverkamp, R. and Ross, P. J. (1995). Infiltration under ponded onditions: 4. An expliit preditive infiltration formula. Soil Siene 16(1), pp Barry, D. A., Parlange, J. Y., Li, L., Prommer, H., Cunningham, C. J. and Stagnitti, E. (2). Analytial approximations for real values of the Lambert W-funtion. Mathematis and Computers in Simulation 53(1-2), pp Barry, D. A., Parlange, J. Y., Sander, G. C. and Sivaplan, M. (1993). A lass of exat-solutions for rihards equation. Journal of Hydrology 142(1-4), pp Barry, D. A., Parlange, J. Y. and Starr, J. L. (1986). Interpolation method for solving the transport-equation in soil olumns with irreversible kinetis. Soil Siene 142(5), pp Barry, D. A., Parlange, J. Y. and Starr, J. L. (1987). Numerial-analysis of the preursor effet. Soil Siene 143(5), pp Barry, D. A. and Sander, G. C. (1991). Exat-solutions for water infiltration with an arbitrary surfae flux or nonlinear solute adsorption. Water Resoures Researh 27(1), pp Barry, D. A. and Sposito, G. (1989). Analytial solution of a onvetion-dispersion model with timedependent transport-oeffiients. Water Resoures Researh 25(12), pp Barry, D. A., Starr, J. L., Parlange, J. Y. and Braddok, R. D. (1983). Numerial-analysis of the snow-plow effet. Soil Siene Soiety of Ameria Journal 47(5), pp Bauters, T. W. J., DiCarlo, D. A., Steenhuis, T. S. and Parlange, J. Y. (1998). Preferential flow in waterrepellent sands. Soil Siene Soiety of Ameria Journal 62(5), pp Bauters, T. W. J., DiCarlo, D. A., Steenhuis, T. S. and Parlange, J. Y. (2). Soil water ontent dependent wetting front harateristis in sands. Journal of Hydrology 231, pp Bear, J. (1972). Dynamis of Fluids in Porous Media. New York, Elsevier, 764 pp. [A lassial authoritative text with many solutions to porous media flow problems.] Bond, R. D. (1964). The influene of the miroflora on physial properties of soils. II. Field studies on water repellent soils. Australian Journal of Soil Researh 2, pp Bond, W. J. (1986). Veloity-dependent hydrodynami dispersion during unsteady, unsaturated soil water flow: Experiments. Water Resoures Researh 22, pp Bond, W. J. and Smiles, D. E. (1983). Influene of veloity on hydrodynami dispersion during unsteady soilwater flow. Soil Siene Soiety Of Ameria Journal 47(3), pp Brams, W. H. and MLaren, A. D. (1974). Phosphatase reations in olumns of soil. Soil Biology & Biohemistry 6(3), pp Brue, R. R. and Klute, A. (1956). The measurement of soil moisture diffusivity. Soil Siene Soiety of Ameria Journal 2( ). [A seminal paper on soil-water diffusivity] Brusseau, M. L., Gerstl, Z., Augustijn, D. and Rao, P. S. C. (1994). Simulating solute transport in an aggregated soil with the dual-porosity model - measured and optimized parameter values. Journal of Hydrology 163(1-2), pp Brutsaert, W. (1968). The adaptability of an exat solution to horizontal infiltration. Water Resoures Researh 4, pp [One of several foundation papers on infiltration theory] Brutsaert, W. (1976). Conise formulation of diffusive sorption of water in a dry soil. Water Resoures

10 Researh 12(6), pp [An elegant proedure to alulate sorptivity.] Chuoke, R. L., Vanmeurs, P. and Vanderpoel, C. (1959). The instability of slow, immisible, visous liquidliquid displaements in permeable media. Transations of the Amerian Institute Of Mining And Metallurgial Engineers 216, pp de Rooij, G. H. (2). Modeling fingered flow of water in soils owing to wetting front instability: A review. Journal of Hydrology 231, pp [A omprehensive and ritial review.] Deinert, M., Parlange, J.-Y., Steenhuis, T. S., Selker, J. and Cady, K. B. (22). Real-time measurement of water profiles in a sand using neutron radiography. Hydrology Days, J.A. Ramirez, Ed. AGU Pub. 22, pp Deinert, M. R., Parlange, J. Y., Steenhuis, T., Throop, J., Unlu, K. and Cady, K. B. (24). Measurement of fluid ontents and wetting front profiles by real-time neutron radiography. Journal of Hydrology 29(3-4), pp DiCarlo, D. A., Bauters, T. W. J., Darnault, C. J. G., Steenhuis, T. S. and Parlange, J. Y. (1999). Lateral expansion of preferential flow paths in sands. Water Resoures Researh 35(2), pp DiCarlo, D. A., Bauters, T. W. J., Darnault, C. J. G., Wong, E., Bierk, B. R., Steenhuis, T. S. and Parlange, J. Y. (2). Surfatant-indued hanges in gravity fingering of water through a light oil. Journal of Contaminant Hydrology 41(3-4), pp DiCarlo, D. A., Bauters, T. W. J., Steenhuis, T. S., Parlange, J. Y. and Bierk, B. R. (1997). High-speed measurements of three-phase flow using synhrotron X rays. Water Resoures Researh 33(4), pp Fritsh, F. N., Shafer, R. E. and Crowley, W. P. (1973). Algorithm 443, solution of the transendental equation wew = x. Communiations of the ACM 16, pp Fujita, H. (1952a). The exat pattern of a onentration-dependent diffusion in a semi-infinite medium.1. Textile Researh Journal 22(11), pp [This work and that of Fujita (1952b, 1954) presented innovative solutions to the diffusion equation, whih are now known as the Fujita solutions.] Fujita, H. (1952b). The exat pattern of a onentration-dependent diffusion in a semi-infinite medium.2. Textile Researh Journal 22(12), pp Fujita, H. (1954). The exat pattern of onentration dependent diffusion in a semi-infinite medium. Textile Researh Journal 24, pp Gao, B., Walter, M. T., Steenhuis, T. S., Hogarth, W. L. and Parlange, J. Y. (24). Rainfall indued hemial transport from soil to runoff: theory and experiments. Journal of Hydrology 295(1-4), pp Gao, B., Walter, M. T., Steenhuis, T. S., Parlange, J. Y., Nakano, K., Rose, C. W. and Hogarth, W. L. (23). Investigating ponding depth and soil detahability for a mehanisti erosion model. Journal of Hydrology 277(1-2), pp Gardner, W. R. (1958). Some steady state solutions of the unsaturated moisture flow equation with appliation to evaporation from a water-table. Soil Siene 85, pp [The publiation of this work marked an important event at the time in soil physis researh] Gelhar, L. W., Welty, C. and Rehfeldt, K. L. (1992). A ritial review of data on field-sale dispersion in aquifers. Water Resoures Researh 28, pp Glass, R. J., Parlange, J.-Y. and Steenhuis, T. S. (1989a). Wetting front instability, 1. Theoretial disussion and dimensional analysis. Water Resoures Researh 25, pp [This and a series of papers that follow by Glass et al. and papers also by Selker et al. (1992ab; 1996) extended and refined the foundational work of Hill and Parlange, (1972).] Glass, R. J., Steenhuis, T. S. and Parlange, J.-Y. (1989b). Mehanism for finger persistene in homogeneous, unsaturated, porous media: Theory and verifiation. Soil Siene 148, pp Glass, R. J., Steenhuis, T. S. and Parlange, J.-Y. (1989). Wetting front instability, 2. Experimental determination of relationships between systems parameters and two-dimensional unstable flow field behaviour in initially dry porous media. Water Resoure Researh 25, pp

11 Hairsine, P. B. and Rose, C. W. (1991). Rainfall detahment and deposition: Sediment transport in the absene of flow-driven proesses. Soil Siene Soiety of Ameria Journal 55(2), pp Haverkamp, R., Ross, P. J. and Parlange, J.-Y. (22). Simple soil water hysteresis predition model based on theory and geometri saling. Hydrology Days, J.A. Ramirez, Ed. AGU Pub. 22, pp Heilig, A., DeBruyn, D., Walter, M. T., Rose, C. W., Parlange, J. Y., Steenhuis, T. S., Sander, G. C., Hairsine, P. B., Hogarth, W. L. and Walker, L. P. (21). Testing a mehanisti soil erosion model with a simple experiment. Journal of Hydrology 244(1-2), pp Hill, D. E. and Parlange, J.-Y. (1972). Wetting front instability in layered soils. Soil Siene Soiety of Ameria Journal 36, pp [A seminal work on establishing the onditions under whih soil-water fingering ours] Hillel, D. (1987). Unstable flow in layered soils - A review. Hydrologial Proesses 1(2), pp [A useful review of layered flow] Hillel, D. and Baker, R. S. (1988). A desriptive-theory of fingering during infiltration into layered soils. Soil Siene 146(1), pp Huang, K., van Genuhten, M. T. and Zhang, R. (1996). Exat solution for one dimensional transport with asymptoti sale dependent dispersion. Applied Mathematial Modeling 2, pp Kim, Y. J., C. J. G. Darnault, N. O. Bailey, J. Y. Parlange, and T. S. Steenhuis (25), Equation for desribing solute transport in field soils with preferential flow paths, Soil Siene Soiety of Ameria Journal, 69, pp King, J. R. (1989). Exat solutions to some nonlinear diffusion equations. Quarterly Journal of Mehanis and Applied Mathematis 42, pp Klafter, J., Blumen, A. and Shlesinger, M. F. (1987). Stohasti pathway to anomalous diffusion. Physial Review A 35, pp Kreft, A. and Zuber, A. (1978). On the physial meaning of the dispersion equation and its solution for different initial and boundary onditions. Chemial Engineering Soiety 3, pp Kreft, A. and Zuber, A. (1986). Flux-averaged and volume-averaged onentrations in ontinuum approahes to solute transport. Comment. Water Resoures Researh 22(7), pp Liu, Y., Bierk, B. R., Selker, J. S., Steenhuis T. S. and Parlange, J.-Y. (1993). High intensity X-Ray and tensiometer measurements in rapidly hanging preferential flow fields. Soil Siene Soiety of Ameria Journal 57, pp Liu, Y., Parlange, J. Y., Steenhuis, T. S. and Haverkamp, R. (1995). A soil water hysteresis model for fingered flow data. Water Resoures Researh 31(9), pp Looney, B. B. and Falta, R. W., Eds. (2). Vadose Zone: Siene and Tehnology Solutions. Colobus, Battelle Press, 589 pp. [A bakground text] MLaren, A. D. (1976). Comments on nitrate redution in unsaturated soil. Soil Siene Soiety Of Ameria Journal 4(5), pp MLaren, A. D. (1976). Rate onstants for nitrifiation and denitrifiation in soils. Radiation and Environmental Biophysis. 13(1), pp Pang, L. and Hunt, B. (21). Solutions and verifiation of a sale dependent dispersion model. Journal of Contaminant Hydrology 53, pp Parker, J. C. and Valohi, A. J. (1986). Constraints on the validity of equilibrium and 1st-order kineti transport models in strutured soils. Water Resoures Researh 22(3), pp Parlange, J.-Y. and Hill, D. E. (1976). Theoretial analysis of wetting front instability in soils. Soil Siene 122, pp [A lassi and early paper that laid the foundation for understanding unstable flow onditions on soil water transport] Parlange, J.-Y., Starr, J. L., van Genuhten, M. T., Barry, D. A. and Parker, J. C. (1992). Exit ondition for misible displaement experiments. Soil Siene 153(3), pp

12 Parlange, J. Y., Barry, D. A. and Haverkamp, R. (22). Expliit infiltration equations and the Lambert W- funtion. Advanes in Water Resoures 25(8-12), pp Parlange, J. Y., Brutsaert, W., Fink, J. P. and ElKadi, A. I. (1996). A multi-sale theory of swelling porous media.1. Appliation to one-dimensional onsolidation. Comment. Transport in Porous Media 23(3), pp Parlange, J. Y. and Fleming, J. F. (1984). 1st integrals of the infiltration equation.1. Theory. Soil Siene 137(6), pp [First in a series of papers that use integral solutions for infiltration] Parlange, J. Y., Hogarth, W. L., Barry, D. A., Parlange, M. B., Haverkamp, R., Ross, P. J., Steenhuis, T. S., DiCarlo, D. A. and Katul, G. (1999). Analytial approximation to the solutions of Rihards' equation with appliations to infiltration, ponding, and time ompression approximation. Advanes in Water Resoures 23(2), pp [Reent advanes and new solutions to Rihards equation] Parlange, J. Y. and Starr, J. L. (1975). Linear dispersion in finite olumns. Soil Siene Soiety of Ameria Journal 39(5), pp Parlange, J. Y. and Starr, J. L. (1978). Dispersion in soil olumns - effet of boundary-onditions and irreversible reations. Soil Siene Soiety of Ameria Journal 42(1), pp Parlange, J. Y., Starr, J. L., Barry, D. A. and Braddok, R. D. (1982). A theoretial-study of the inlusion of dispersion in boundary-onditions and transport-equations for zero-order kinetis. Soil Siene Soiety of Ameria Journal 46(4), pp Passiour, Jb (1971). Hydrodynami dispersion in aggregated media.1. Theory. Soil Siene 111(6), pp Philip, J. R. (196). Very General Class of Exat Solutions In Conentration-Dependent Diffusion. Nature 185(478), pp [One of several foundation papers on infiltration theory] Philip, J. R. (1972). Steady infiltration from buried, surfae, and perhed point and line soures in heterogeneous soils.1. Analysis. Soil Siene Soiety Of Ameria Proeedings 36(2), pp [One of several foundation papers on infiltration theory] Pikens, J. F. and Grisak, G. E. (1981). Modeling of sale-dependent dispersion in hydrogeologi systems. Water Resoures Researh 17(6), pp Raats, P. A. C. (1973). Steady upward and downward flows in a lass Of Unsaturated Soils. Soil Siene 115(6), pp [One of several foundation papers on infiltration theory] Rihards, L. A. (1931). Capillary ondution of liquids through porous mediums. Physis 1, pp [A seminal treatise that is the foundation stone of modern soil physis.] Rimmer, A., DiCarlo, D. A., Steenhuis, T. S., Bierk, B., Durnford, D. and Parlange, J.-Y. (1998). Rapid fluid ontent measurement method for finger flow in an oil-water-sand system using synhrotron X-Rays. Journal of Contaminant Hydrology 31, pp Ritsema, C. J., Steenhuis, T. S., Parlange, J.-Y. and Dekker, L. W. (1996). Predited and observed finger diameters in field soils. Geoderma 7(2-4), pp [Detailed experimental work on soil finger phenomenon] Rogers, C., Stallybrass, M. P. and Clements, D. L. (1983). On two phase infiltration under gravity and with boundary infiltration. Nonlinear Analysis Theory Methods and Appliations 7, pp Rose, C. W. (1985). Developments in soil erosion and deposition models. Advanes in Soil Siene 2, pp [Useful artile on soil erosion models] Rupp, D. E., R. E. Peahey, K. L. Warren, and J. S. Selker (26), Diuron in surfae runoff and tile drainage from two grass-seed fields, Journal of Environmental Quality, 35, pp Saffman, P. G. and Taylor, G. I. (1958). The penetration of a fluid into a porous medium or Hele-Shaw ell ontaining a more visous liquid. Proeedings, Royal Soiety, London, A245, pp Sander, G. C. and Braddok, R. D. (25). Similarity solutions for unsaturated water and solute transport. Advanes in Water Resoures 1, pp [An exellent reent paper with new approahes to

13 similarity solutions to unsaturated flow problems] Sander, G. C., Hairsine, P. B., Rose, C. W., Cassidy, D., Parlange, J.-Y., Hogarth, W. L. and Lisle, I. G. (1996). Unsteady soil erosion model, analytial solutions and omparison with experimental results. Journal of Hydrology 178, pp Sander, G. C., Parlange, J. Y., Kuhnel, V., Hogarth, W. L., Lokington, D. and Okane, J. P. J. (1988). Exat Nonlinear Solution for Constant Flux Infiltration. Journal of Hydrology 97(3-4), pp Selker, J. S., Lelerq, P., Parlange, J.-Y. and Steenhuis, T. S. (1992a). Fingered flow in two dimensions, 1. Measurement of matri potential. Water Resoures Researh 28(9), pp [This and a series of papers that follow by Selker (199b;1996) provide exellent analysis of and solutions to omplex soil olumn experiments. Also see papers by Glass et al. (1989ab).] Selker, J. S., Parlange, J.-Y. and Steenhuis, T. S. (1992b). Fingered flow in two dimensions, 2. Prediting finger moisture profile. Water Resoures Researh 28(9), pp Selker, J. S., Steenhuis, T. S. and Parlange, J.-Y. (1992). Wetting front instability in homogeneous sand soils under ontinuous infiltration. Soil Siene Soiety of Ameria Journal 56(5), pp Selker, J. S., Steenhuis, T. S. and Parlange, J.-Y. (1996). An engineering approah to fingered vadose pollutant transport. Geoderma 7, pp Smiles, D. E., Philip, J. R., Knight, J. H. and Elrik, D. E. (1978). Hydrodynami dispersion during absorption of water by soil. Soil Siene Soiety Of Ameria Journal 42(2), pp Spanier, J. and Oldham, K. B. (1987). An Atlas of Funtions. New York, Hemisphere. Sposito, G. (1989). The hemistry of soils. New York, Oxford University Press. [A very useful text on soil hemistry] Stagnitti, F., Parlange, J.-Y., Steenhuis, T. S., Boll, J., Pivetz, B. and Barry, D. A. (1995). Chapter 7. Transport of moisture and solutes in the unsaturated zone by preferential flow. Environmental Hydrology. V. P. Singh. Dordreht, Kluwer Aademi Publishers. Water Siene and Tehnology Library, Vol. 15, pp Stagnitti, F., Sherwood, J., Allinson, G., Evans, L., Allinson, M., Li, L. (Ling), and Phillips, I. (1998). An investigation of loalised soil heterogeneities on solute transport using a multisegement perolation system. New Zealand Journal of Agriultural Researh, 41(4), pp Starr, J., L., Broadbent, F. E. and Nielsen, D. R. (1974). Nitrogen transformations during ontinuous leahing. Soil Siene Soiety of Ameria Proeedings 38, pp Starr, J. L. and Parlange, J. Y. (1975). Nonlinear denitrifiation kinetis with ontinuous-flow in soil olumns. Soil Siene Soiety of Ameria Journal 39(5), pp Starr, J. L. and Parlange, J. Y. (1979). Dispersion in Soil Columns - Snow Plow Effet. Soil Siene Soiety of Ameria Journal 43(3), pp Starr, J. L., Parlange, J. Y. and Braddok, R. D. (1982). Dispersion in Soil Columns - the Preursor Effet. Soil Siene 133(2), pp Steenhuis, T. S. and Walter, W. F. (198). Closed form solution for pestiide loss in runoff water. Transations of the ASAE 23(3), pp Stephens, D. B. (1996). Vadose Zone Hydrology. NY, Lewis Publishers. Stephenson, J. (1995). Some non-linear diffusion equations and fratal diffusion. Physia A 222, pp Su, N. (1995). Development of the Fokker-Plank equation and its solutions for modelling transport of onservative and reative solutes in physially heterogeneous media. Water Resoures Researh 31, pp Su, N. H., Sander, G. C., Liu, F. W., Anh, V. and Barry, D. A. (25). Similarity solutions for solute transport in fratal porous media using a time- and sale-dependent dispersivity. Applied Mathematial Modelling 29(9), pp

14 van Genuhten, M. T. and Alvers, W. J. (1982). Analytial solutions to the one-dimensional onvetiondispersion solute transport equation. USDA Teh. Bull [An exellent soure of numerous analytial solutions to one-dimensional transport equations.] van Genuhten, M. T. and Dalton, F. N. (1986). Models for simulating salt movement in aggregated field soils. Geoderma 38, pp van Genuhten, M. T. and Parker, J. C. (1984). Boundary-onditions for displaement experiments through short laboratory soil olumns. Soil Siene Soiety of Ameria Journal 48(4), pp Wallah, R., Jury, W. A. and Spener, W. F. (1988). Transfer of hemials from soil solution to surfae runoff - a diffusion-based soil model. Soil Siene Soiety Of Ameria Journal 52(3), pp Wallah, R. and Parlange, J. Y. (1998). Modeling transport in a single rak by the dual-porosity onept with a boundary layer at the interfae. Journal of Contaminant Hydrology 34(1-2), pp Wallah, R. and Parlange, J. Y. (2). Applying the boundary layer onept to model transport of dissolved hemials in preferential flow paths. Water Resoures Researh 36(1), pp Wallah, R. and Steenhuis, T. S. (1998). Model for non reative solute transport in strutured soils with ontinuous preferential flow paths. Soil Siene Soiety of Ameria, Journal. 62, pp Wallah, R., Steenhuis, T. S. and Parlange, J.-Y. (1998). Modeling the movement of water and solute through preferential flow paths. Handbook of groundwater engineering. J. Delleur, CRC Press, pp. 21. Walter, M. T., Kim, J. S., Steenhuis, T. S., Parlange, J. Y., Heilig, A., Braddok, R. D., Selker, J. S. and Boll, J. (2). Funneled flow mehanisms in a sloping layered soil: Laboratory investigation. Water Resoures Researh 36(4), pp Walter, M. T., Parlange, J. Y., Walter, M. F., Xin, X. and Sott, C. A. (21). Modeling pollutant release from a surfae soure during rainfall runoff. Journal of Environmental Quality 3(1), pp Wooding, R. A. (1969). Growth of fingers at an unstable diffusing interfae in a porous medium or hele-shaw ell. Journal of Fluid Mehanis 39, pp [A lassi experimental desription of the soil finger phenomenon] Biographial Skethes Frank Stagnitti, a mathematiian, soil physiist and environmental sientist, is an expert in the study of agriultural and hydrologial systems, partiularly in the fields of solute and ontaminant transport, bioremediation, eotoxiology, groundwater flow, oastal proesses and wetlands tehnology. Dr Stagnitti has established an international researh profile in the field of environmental systems modeling, is extensively published in this field and has attrated signifiant researh funding within Australia, mainly through the Australian Researh Counil. Dr Stagnitti has partiipated in a number of international projets, mainly funded by the EU under the 5th and 6th Framework agreements. Dr Stagnitti is a member of several sientifi advisory ommittees and boards inluding the Australian Researh Counil, Center of Exellene in Light Metals and the Vitorian Partnership in Advaned Computing. He urrently holds the position of Assoiate Dean Researh for the Faulty of Siene and Tehnology at Deakin University and is a full Professor and Chair in Aquati Siene in the Shool of Life and Environmental Siene. Dr Stagnitti is a Fellow of Australian Mathematis Soiety and a Fellow of Institute of Mathematis and its Appliations (UK). Jean-Yves Parlange, is a Professor of Agriultural and Biologial Engineering at Cornell University having previously been a Professor of Applied Mathematis at Griffith University, in Brisbane, Australia. Yves Parlange has been a Fellow of the Amerian Geophysial Union sine 1996, reeived the Hydrology award in 1996, Horton medal in 22 and eleted to the U.S. National Aademy of Engineering in 26. His researh interests are many and varied but entre on problems in environmental siene and inlude water movement in porous media, solute transport in soils, surfae and subsurfae hydrology and erosion and sediment transport. D. A. Barry, arries out researh on porous media, flow and transport proesses, partiularly the modeling of suh proesses. He has been involved in many ollaborative projets involving ombinations of laboratory, field and theoretial work, inluding infiltration modeling. For example, he has ollaborated in the development of a biogeohemial transport models for prediting transport and fate of ontaminants in

15 omplex subsurfae environments. Other omputer-modeling efforts inlude distributed athment modeling and modeling of on-shore/off-shore sediment transport on oean beahes. He is Editor of the journal Advanes in Water Resoures. Tammo Steenhuis, is a professor in the Department of Biologial and Environmental Engineering at Cornell University. He works with a group of 25 graduate students, postdotoral researhers and researh assoiates to see if it is possible to do researh in hydrology from the nano to watershed sale years. He has ollaborated extensively with Yves Parlange during the last 2 years on finding new ways for desribing spatial variability in flow fields both above and below the ground. Ling Li,is a Professor and Chair in Environmental Engineering in Shool of Engineering at the University of Queensland, Australia. His researh and teahing interests are in environmental siene and engineering with a partiular fous on modeling of environmental systems. His urrent researh work addresses interations between the oean and oastal aquifers, and pathways and fluxes of hemials to oastal waters via submarine groundwater disharge. He is on the editorial board for Advanes in Water Resoures and is Assoiate Editor of Hydrogeology Journal. David Lokington, is the Head of Environmental Engineering in the Shool of Engineering, as well as the Diretor of the Centre for Water Studies, at the University of Queensland, Australia. He is also the Diretor of the Engineering, Tehnology and Design Researh Program in the national Cooperative Researh Centre for Sustainable Tourism. His researh interests enter on modeling fluid flow in unsaturated porous media, variable density groundwater flow and ontaminant transport. A partiular interest is the use of these models in quantifying oastal athment proesses suh as: subsurfae estuary dynamis; submarine groundwater disharge; tidal marsh, wetlands and island hydrology; and seawater intrusion. He is a member of the editorial board of Advanes in Water Resoures and is a Fellow of the Institute of Mathematis and its Appliations. Graham Sander, is a Reader in Hydrology in the Department of Civil and Building Engineering at Loughborough University, England. Prior to his urrent position he was in the Faulty of Siene and the Faulty of Environmental Sienes at Griffith University, Australia. His researh and teahing interests are in environmental siene and engineering and over predominantly soil erosion modeling, water and solute transport in porous media and unsaturated two-phase flow. He is on the editorial board for Advanes in Water Resoures and is Assoiate Editor of Water Resoures Researh.