Vertical nonreactive solute transport data collected in three laboratory soil columns (made out of sediment samples from the Pampean aquifer located southeast of the Buenos Aires province) are contrasted with the explicit solutions of two model 1D linear PDEs: the classical advection–dispersion equation (ADE), and a fractional advection–dispersion equation (FADE) which has proven to be a useful modeling tool for highly inhomogeneous media exhibiting nontrivial scaling laws. Whereas two of the samples turn out to be quite homogeneous (thus requiring a fractional-derivative order γ → 2), the third one is best described by a FADE with fractional-derivative order γ = 1.68. This example illustrates the FADE’s ability to reveal self-similar geometric structures inside the sample.Fil: Benavente, Micaela Andrea. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Deza, Roberto Raul. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales. Departamento de Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata. Instituto de Investigaciones Físicas de Mar del Plata. Universidad Nacional de Mar del Plata. Facultad de Cs.exactas y Naturales. Instituto de Investigaciones Físicas de Mar del Plata; ArgentinaFil: Grondona, Sebastian. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales. Instituto de Geología de Costas y del Cuaternario. Provincia de Buenos Aires. Gobernación. Comisión de Investigaciones Científicas. Instituto de Geología de Costas y del Cuaternario; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Mascioli, S.. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales. Instituto de Geología de Costas y del Cuaternario. Provincia de Buenos Aires. Gobernación. Comisión de Investigaciones Científicas. Instituto de Geología de Costas y del Cuaternario; ArgentinaFil: Martinez, Daniel Emilio. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales. Instituto de Geología de Costas y del Cuaternario. Provincia de Buenos Aires. Gobernación. Comisión de Investigaciones Científicas. Instituto de Geología de Costas y del Cuaternario; Argentin

Benavente, Micaela Andrea

Deza, Roberto Raul

Grondona, Sebastian

Martinez, Daniel Emilio

Mascioli, S.

English

CONICET Digital

							 ! "#$%&'&()*$(+,#-(#,".!%(/%##&.)%(*																	MatemáticaAplicada,ComputacionaleIndustrialISSN:23143282DirectorCristinaMacielUniversidadNacionaldelSur,BahíaBlanca,ArgentinaComitéEditorial/EditorialBoardCarlosD’Attellis UniversidadFavaloroUniversidadNacionaldeSanMartín,BuenosAiresPabloJacovkis UniversidadNacionaldeTresdeFebreroUBA,BuenosAiresSergioPreidikman CONICETUniversidadNacionaldeCórdobaDianaRubio UniversidadNacionaldeSanMartín,BuenosAiresRubénSpies IMALCONICET,UNL,SantaFeJuanSantos CONICET,InstitutodelGasydelPetróleoUniversidaddeBuenosAiresDomingoTarzia CONICET,FacultaddeCienciasEmpresariales,UniversidadAustral,RosarioCristinaTurner CONICET,FAMAF–UniversidadNacionaldeCórdobaASAMACIAsociaciónArgentinadeMatemáticaAplicada,ComputacionaleIndustrialGüemes3450,(3000)SantaFe,Argentina.http://asamaci.org.ar/Email:asamaci@gmail.comNONREACTIVE SOLUTE TRANSPORT IN SOIL COLUMNS:CLASSICAL AND FRACTIONAL-CALCULUS MODELINGM.A. Benavente, R.R. Deza†, S.I. Grondona∗, S. Mascioli∗ and D.E. Martı´nez∗Dto. Matema´tica, FCEyN-UNMdP, Dea´n Funes 3350, B7602AYL Mar del Plata, Argentina, benavent@mdp.edu.ar†Dto. Fı´sica e IFIMAR, FCEyN-UNMdP, Dea´n Funes 3350, B7602AYL Mar del Plata, Argentina, deza@mdp.edu.ar∗CGCC, FCEyN-UNMdP, Dea´n Funes 3350, B7602AYL Mar del Plata, Argentina, demarti@mdp.edu.arAbstract: Vertical nonreactive solute transport data collected in three laboratory soil columns (made out of sed-iment samples from the Pampean aquifer located southeast of the Buenos Aires province) are contrasted with theexplicit solutions of two model 1D linear PDEs: the classical advection–dispersion equation (ADE), and a fractionaladvection–dispersion equation (FADE) which has proven to be a useful modeling tool for highly inhomogeneousmedia exhibiting nontrivial scaling laws. Whereas two of the samples turn out to be quite homogeneous (thus requir-ing a fractional-derivative order γ → 2), the third one is best described by a FADE with fractional-derivative orderγ = 1.68. This example illustrates the FADE’s ability to reveal self-similar geometric structures inside the sample.Keywords: Fractional partial differential equations - Smoothing, curve ﬁtting - Sensitivity, stability, parametricoptimization2000 AMS Subject Classiﬁcation: 35R11 - 65D10 - 90C311 INTRODUCTIONThe classical advection–dispersion equation (ADE) has been traditionally employed to model solutetransport in soils (in Hydrology and Soil Science, the term “dispersion” does not imply waves but is asynonym of diffusion). In fact, the ADE captures adequately the behavior of a solute in a homogeneousmedium. Its 1D version reads∂C∂t= v(α∂2C∂x2− ∂C∂x), (1)where C(x, t) is the solute concentration, v the average ﬂow speed through the pores, α := D/v thedispersivity, and D the longitudinal dispersion coefﬁcient (namely, along ﬂow lines). Dimensionally, [α] =l, hence its usefulness as a modeling parameter. The derivation of Eq. (1) in statistical physics assumesthe solute particles to undergo Brownian motion, which is the composition of successive increments drawnfrom independent normally distributed random variables with identical ﬁnite variance.We must recall however that a soil is a porous medium and will more likely exhibit heterogeneities atall scales, often leading to self-similar geometric structure. In 1990 some researchers [1] reported that thedispersivity showed scaling effects, tending to increase with distance. This type of process is often describedas anomalous dispersion. During the 90’s, transport models with fractional derivatives such as the fractionaladvection–diffusion equation (FADE) emerged, in which the scaling effects are accounted for by the orderof the fractional derivative. Its 1D version with symmetric dispersion reads∂C∂t= v[αf2(∂γC∂+xγ+∂γC∂−xγ)− ∂C∂x], (2)where 1 < γ ≤ 2 is the fractional-derivative order, αf := Df/v the “fractional dispersivity” ([αf ] = lγ−1),Df the fractional longitudinal dispersion coefﬁcient ([Df ] = lγt−1), and ∂± a shorthand for the fractionalderivatives to the left and to the right, deﬁned as usual through Riemann–Liouville integrals.Equation (2), which reduces to Eq. (1) for γ = 2, has been derived by assuming the solute particles toundergo the so-called Le´vy motion, which is the composition of iid random variables with inﬁnite variance.In Le´vy motion, the persistence of the solute-particle motion, as well as signiﬁcant deviations from theaverage, are much more likely than in Brownian motion. Hence there are particles that travel long distancesabove average speed, while others do so at lower speeds experiencing long waiting times before engagingthe ﬂuid motion. This has also been modeled by considering mobile and immobile zones (MIM) [2, 3].	  !At present, the articles which apply Eq. (2) to solute transport in porous media (and among them thosewhich compare the results yielded by the classical and fractional models) are not few. Some research [4, 5]has concluded that the beneﬁts of FADE with respect to ADE (which as already pointed out is a special caseof the former) are not so evident in saturated sandy substrates, or clay substrates at high ﬂow rates [4]. Inthe present study we too compare the goodness of both models, for sediment samples from the Pampeanaquifer located southeast of the Buenos Aires province.2 MATERIALS AND METHODS2.1 EXPERIMENTAL DATAThe experimental data O(t) come from column tests made on undisturbed sediment samples from thePampean aquifer, located southeast of the Buenos Aires province (mainly loessic, silt-clay and silt-sandysediments). The samples are represented in a barycentric diagram in Fig. 1:A: Taken on a site located along National Route 226, 15 km far from Mar del Plata city; average composition18.62% ﬁne sand, 65.29% silt and 16.08% clay [6].B1: Taken on cultivated soil near La Dulce town, in the basin of the Queque´n river; average composition66.5% sand, 20.4% silt and 13.5% clay [7].B2: Taken on adjacent natural soil near La Dulce town, in the basin of the Queque´n river; average compo-sition 76.5% sand, 10.4% silt and 12.5% clay [7].The column tests consist in injecting a ClNa solution on top of the vertical columns (previously saturatedwith distilled water), so that its ﬂow is due to gravity. Cl− was chosen as a tracer for being unreactive withthe medium. The conductivity of the eluted solution is then measured at regular intervals until it equalsthat of the injected one. Now, here enters another crucial hydraulic parameter: the porosity ρ. Differentsoils will differ in ρ values and that will lead to different ﬂow times. In order to compare them, the regularintervals must be chosen as fractions of the estimated pore volume of the sample (i.e. ρe × V , where ρe isan estimation since determining ρ is one of our goals) and V the volume of the column. For each case andeach time interval, several measurements are made and the average is taken.2.2 MATHEMATICAL MODELING PROCEDUREThe analytical solution of Eq. (1) for a column of length L and initial condition C(x, 0) = Ci for0 < x ≤ L is given by [8]C(L, t) = Ci +C0 − Ci2[erfc(L− vt√4vαt)+ exp(Lα)erfc(L+ vt√4vαt)], (3)where C0 = C(0, t), t ≥ 0 is the boundary condition at the plane of the source. Hereafter we take Ci = 0.For L/α ∼ 102, the input from the transient in the second term can be safely neglected [8].The analytical solution of Eq. (2) is obtained by Fourier transformation [9]. For the same conditions asin Eq. (3) it readsCf(L, t) = C0[1− Fγ(L− vtσ)], (4)where σ = [vαft cos(πγ/2)]1/γ and Fγ is the γ-stable Le´vy distribution function, which is evaluated nu-merically by means of a Matlab adaptation of the Fortran subroutine cfastd of Ref. [9]. The graphs of thissolution look similar to those of Eq. (3).Two optimization processes were separately conducted for each dataset [namely OA(t), OB1(t), andOB2(t)], choosing as objective functions the Euclidean distance between the experimental vectors and thecorresponding analytical ones. For each, the corresponding Matlab script was written, using the minimiza-tion package fmincon. As ﬁtness measures, the relative distances (the ratios between the absolute distancesand the norm of the experimental data vector)δ =minα,ρ(∑ni=1[C(L, ti)−O(ti)]2)1/2(∑ni=1[O(ti)]2)1/2, δf =minγ,αf(∑ni=1[Cf(L, ti)−O(ti)]2)1/2(∑ni=1[O(ti)]2)1/2(5)	  !"are quoted in the inset of the ﬁgures.2.3 NUMERICAL RESULTSFigures 1 and 2, which plot the Cl− concentration as a function of time (expressed in pore volumes)summarize the results of the optimization process, exhibiting together with the experimental data the ADEand FADE solutions [Eqs. (3) and (4)] which best ﬁt them. Quoted in the inset are the optimal parametersfor the ADE (α and ρ) and FADE (γ and αf ) solutions, together with the respective relative distances andthe initial concentration C0 for each case.For case A (Fig. 1), the best ﬁt (δf = 0.0167 vs. δ = 0.0532) is achieved by a FADE with γ = 1.68. Wenote moreover that the obtained values αf = 0.0294 m.68 and α = 0.00847 m are consistent with the onesreported elsewhere. The ﬂow speed for this case resulted to be v = 3.814× 10−6 m/sec.For cases B (Fig. 2), the differences between models are not remarkable (the optimal value of γ ≈ 2).The recorded ﬂow speeds (vb1 = 1.594 × 10−5 m/sec and vb2 = 1.100 × 10−5 m/sec) are higher than incase A, which is consistent with the observation at the end of Sec. 1.sandsilt clayAB1B2Figure 1: Parameter optimization results for case A. A FADE with γ = 1.68 better ﬁts the data.Figure 2: Parameter optimization results for cases B. Either an ADE or a FADE with γ ≈ 2 ﬁt the data.3 CONCLUSIONSIn this work, the classical (ADE) and fractional (FADE) advection–dispersion equations have been ap-plied to model nonreactive solute transport in soil columns, in three different cases. It is noteworthy that inall cases, the curve ﬁtting was excellent.	  !#Case A: The best ﬁt (δf = 0.0167 vs. δ = 0.0532) was achieved by a FADE with γ = 1.68, and themeasured ﬂow speed was v = 3.814× 10−6 m/sec. The optimal FADE (αf = 0.0294 m.68) and ADE(α = 0.00847 m) dispersivities are consistent with the ones reported before.The important differences between both theoretical breakthrough curves are those at the beginning andespecially at the end. An almost parallel deviation at the center is of lesser importance, and may bedue to a small difference in the optimal porosity, which could result in different advective componentsfor both models. We are now undertaking a sensitivity analysis of the results to the values of ρ.The fact that the transport of a solute in a porous medium is well described by a FADE is potentiallyof great importance, as it would allow to follow the plume of contaminant in the ﬁeld, using directlythe parameters found in laboratory soil columns.Cases B: In these cases, the initial and ﬁnal tails of the breakthrough curves appear superimposed. Togetherwith the fact that the optimal γ ≈ 2, this indicates that the differences between both models are notrelevant, and the FADE would not offer advantages. Again, we note that the ﬂow speeds are in thesecases higher than in case A, so we ﬁnd consistence with what was pointed out at the end of Sec. 1[4, 5].The explanation for the differences between cases A and B would reside primarily in the different com-positions of the respective sediments, since those of case B have an overwhelmingly higher proportion ofsand. Whereas this is conceivably relatively monodisperse, compact clay grains may be highly polydisperse,thus leading to statistically fractal geometric structures. We have also noted the difference in ﬂow speeds.Despite the existence of residual Cl− after washing the columns, we have chosen to model with zeroinitial condition in order not to introduce noise in the comparisons. This fact should be taken into accountwhen interpreting the results.Finally it should be added that these investigations are at an early stage and need to be validated by ﬁeldstudies.ACKNOWLEDGMENTSR.D. acknowledges support from UNMdP, through grant EXA544/11.REFERENCES[1] A.U.H. KHAN, AND W.A. JURY, A laboratory test of the dispersion scale effect, J. Contam. Hydrol., 5 (1990), pp.119-132.[2] D.R. NIELSEN, M.TH. VAN GENUCHTEN, AND J.W. BIGGAR, Water ﬂow and solute transport processes in the unsaturatedzone, Water Resour. Res., 22 (1986), pp.89S-108S.[3] O. SILVA, J. CARRERA, M. DENTZ, S. KUMAR, A. ALCOLEA, AND M. WILLMANN, A general real-time formulation formulti-rate mass transfer problems, Hydrol. Earth Syst. Sci., 13 (2009) pp.1399-1411.[4] Y.A. PACHEPSKY, D.A. BENSON, AND W. RAWLS, Simulating scale-dependent solute transport in soils with the fractionaladvective–dispersive equation, Soil Sci. Soc. Am. J., 64 (2000), 1234-1243.[5] F. SAN JOSE´, Y.A. PACHEPSKY, AND F.J. TAGUAS, Modelos fraccionarios para la descripcio´n del transporte de solutos encolumnas de suelo, Estudios de la Zona No Saturada del Suelo Vol. VIII, J.V. Gira´ldez Cervera y F.J. Jime´nez Hornero, eds.(2007), pp.355-360.[6] S. MASCIOLI, M.A. BENAVENTE, AND D.E. MARTI´NEZ, Estimation of transport hydraulic parameters in loessic sediment,Argentina: Application of column test, Hydrogeol. J., 13 (2005) pp.849-857.[7] S.I. GRONDONA, D.E. MARTI´NEZ, M.A. BENAVENTE, M. GONZA´LEZ, H.E. MASSONE, AND S.K.B. MIGLIORANZA,Determination of hydraulic parameters in experimental soil columns from the southeast of Buenos Aires province, under review(2013).[8] A. OGATA, AND R.B. BANKS, A solution of the differential equation of longitudinal dispersion in porous media, USGSprofessional paper 411-A. U.S. Gov. Print. Ofﬁce, Washington, DC (1961).[9] D.A. BENSON, The fractional advection–dispersion equation: Development and application, University of Nevada Reno Ph.D. Thesis (1998).	  !

Nonreactive solute transport in soil columns: classical and fractional-calculus modeling

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Nonreactive solute transport in soil columns: classical and fractional-calculus modeling

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