Notes on Implementation of Colloid Transport and Colloid-Facilitated Solute Transport into HYDRUS-1D

Transcription

1 Note on Implementation of Colloid Tranport and Colloid-Failitated Solute Tranport into HYDRUS-1D (a text implified from van Genuhten and Šimůnek (24) and Šimůnek et al. (26)) Jirka Šimůnek Department of Environmental Siene Univerity of California Riveride 1. Colloid tranport Colloid fate and tranport model are ommonly baed on ome form of the advetiondiperion equation, but modified to aount for olloid filtration. In the abene of olloid inativation and degradation, the olloid tranport equation i then given by θc S C qc + ρ = θd t t x x x (1) where C i the olloid onentration in the aqueou phae (nl -3 ), S i the olid phae olloid onentration (nm -1 ), θ i the volumetri water ontent (L 3 L -3 ), D i the diperion oeffiient for olloid (L 2 T -1 ), and q i the volumetri water flux denity (LT -1 ). The olloid ma tranfer term between the aqueou and olid phae i traditionally given a: ρ t θ ψ ρ S = k C - a k S d (2) in whih k a and k d are firt-order olloid attahment and detahment oeffiient (T -1 ), repetively, and ψ i a dimenionle olloid retention funtion (-). To imulate redution in the attahment oeffiient due to filling of favorable orption ite, ψ i ometime aumed to dereae with inreaing olloid ma retention. Langmuirian dynami equation ha been propoed for ψ to deribe thi bloking phenomenon: S ψ = (3) 1 max S in whih S max i the maximum olid phae olloid onentration (nm -1 ). Notie that Eq. (2) lump the effet of a variety of phyial and hemial proee into a ingle attahment oeffiient parameter. Bradford et al. (22, 23) hypotheized that the influene of training and attahment mehanim on olloid retention hould be eparated into two ditint omponent: attahment and detahment per e, and training (being the entrapment of olloid in pore throat that are too mall to allow paage). Bradford et al. (23) propoed the following depth-dependent training oeffiient of the form ψ tr d5 + z = d 5 -β (4) where d 5 i the median grain ize of the porou media (L), β i a fitting parameter (-), and z i 1

2 ditane from the porou medium inlet (L). Data from Bradford (22, 23) for different olloid diameter (d p ) and porou media median grain ize howed an optimal value of.43 for β. 2. Colloid-failitated olute tranport Colloid-failitated tranport require knowledge of olloid tranport, diolved ontaminant tranport, and olloid-failitated ontaminant tranport. Tranport and/or ma balane equation mut therefore be formulated for the total ontaminant, for ontaminant orbed kinetially or intantaneouly to the olid phae, and for ontaminant orbed to mobile olloid, to olloid attahed to the oil olid phae, and to olloid aumulating at the air-water interfae. A an illutration of the omplexitie involved, we review here the equation for olloid-failitated tranport that we reently inorporated in the HYDRUS oftware pakage. Ma balane equation for the total ontaminant. The ombined diolved and olloidfailitated ontaminant tranport equation (in one dimenion) i given by: θc Se Sk θcsm SSi JT + ρ + ρ + + ρ = - R t t t t t x J C qc C qc S = θ + θ x x x x x x x T - D Sm D m (5) where θ i the volumetri water ontent (L 3 L -3 ), C i the diolved ontaminant onentration in the aqueou phae (ML -3 ), S e and S k are ontaminant onentration orbed intantaneouly and kinetially, repetively, to the olid phae (MM -1 ); S m and S i are ontaminant onentration orbed to mobile and immobile olloid (Mn -1 ), repetively; J T i the total flux of the diolved ontaminant (ML -3 T -1 ), D i the diperion oeffiient for ontaminant in olution (L 2 T -1 ), q i the volumetri water flux denity for the ontaminant (LT -1 ), and R repreent variou hemial and biologial reation, uh a degradation and prodution (ML -3 T -1 ), diued below. Note that the left ide um the ma of ontaminant aoiated with the different phae (ontaminant in the liquid phae, ontaminant orbed intantaneouly and kinetially to the olid phae, and ontaminant orbed to mobile and immobile olloid), while the right ide onider variou patial ma fluxe (diperion and advetive tranport of the diolved ontaminant, and diperion and advetive tranport of ontaminant orbed to mobile olloid). Ma balane equation for ontaminant orbed to the olid phae. Equation (13) invoke the onept of two-ite orption for modeling nonequilibrium adorption-deorption reation. The two-ite orption onept aume that total orption, S, an be divided into two fration: S = S + S (6) e k with orption S e (MM -1 ) on one fration of the ite (type-1 ite) aumed to be intantaneou, and orption S k (MM -1 ) on the remaining ite (type-2 ite) being time-dependent aording to S ρ t k [(1 ) ] = ω f KC S R d k (7) 2

3 where f i the fration of orption ite (type-1 ite) at equilibrium with the olution, K d i the ditribution oeffiient (L 3 M -1 ), and R repreent variou hemial and biologial reation of the kinetially orbed ontaminant (ML -3 T -1 ). Ma balane equation for ontaminant orbed to mobile olloid. The ma balane equation for ontaminant orbed to mobile olloid an be written a θwcs m C qcs m = θwsmd + t x x x tr att + θk ψ C θ k C S θ ( k ψ + k ψ ) C S + ρk S S R am m w dm m w a tr m d i m (8) where k am i the adorption rate to mobile olloid (T -1 ), k dm i the deorption rate from mobile olloid (T -1 ), and R m repreent variou hemial and biologial reation for ontaminant orbed to mobile olloid (ML -3 T -1 ). The parameter ψ m adjut the orption rate to the number of mobile olloid preent, i.e., C ψ m = (9) C ref where C ref i the referene onentration of olloid for whih the orption rate k am i valid (ML -3 ). In equation (15), the firt two term on the right ide repreent diperion and advetive tranport, repetively, of ontaminant orbed to mobile olloid; the third and fourth term aount for orption and deorption of ontaminant to/from mobile olloid, repetively; the fifth and ixth term aount for the attahment (inluding training) and detahment of mobile olloid ontaining orbed ontaminant, repetively; while the eventh term repreent degradation or other reation involving ontaminant orbed to mobile olloid. Ma balane equation for ontaminant orbed to immobile olloid. The ma balane equation for ontaminant orbed to immobile olloid an be written a follow SS i att ρ = θkaiψi C ρkdissi + θkaψ CSm ρkds Si Ri t (1) where k ai i the adorption rate to immobile olloid (T -1 ), k di i the deorption rate from immobile olloid (T -1 ), and R i repreent variou reation for ontaminant orbed to immobile olloid (ML -3 T -1 ). The parameter ψ i adjut the orption rate to the number of immobile olloid preent: S ψ i = (11) S ref where S ref i the referene onentration of immobile olloid for whih the orption rate k ai i valid (ML -3 ). In equation (17) the firt two term on the right ide repreent adorption and deorption of ontaminant to/from immobile olloid, repetively; the third and fourth term deribe the attahment (inluding training) and detahment of immobile olloid with orbed ontaminant, repetively; and finally the fifth term repreent reation of ontaminant orbed to 3

4 immobile olloid. Reation term. The reation term R in (12) may be ued to aount for a variety of hemial and biologial reation and tranformation, inluding degradation and prodution, not already expliitly inorporated in the main total ontaminant ma tranport equation. Conitent with urrent apabilitie of the HYDRUS oftware pakage to imulate equential firt-order deay hain, R may inlude proviion for two firt-order degradation reation: one whih i independent of other olute and one whih provide the oupling between olute involved in equential firt-order deay reation. A diued earlier in Setion 2, problem of olute tranport involving equential firt-order deay reation frequently our in oil and groundwater ytem. The reation term R in (12) for olloid-failitated tranport enario i now given by: R = -μ θc- μ ρ ( S + S )- μ ( θc S + ρs S ) (12) w b e k m i where μ w, μ, and μ are firt-order rate ontant (T -1 ) for olute in the liquid, olid, and olloid phae (T -1 ), repetively;. The reation term R, R m, and R im for reation in the kinetially orbed phae, on mobile olloid, and olloid aoiated with the olid phae, repetively, are a follow: R R R = μ ρs m m i i k = μθcs = μ ρss (13) The above mathematial development how that a omplete deription of olloid-failitated tranport require a total of ix oupled partial differential equation involving ix unknown variable (C, S, C, S k, S m, S i ). 3. Example appliation for olloid-failitated tranport Typial feature of olloid and olloid-failitated tranport are demontrated here for a hypothetial olumn experiment. The olumn wa aumed to have a length, L, of 1 m, while the experiment lated 6 minute. Water flowed through the olumn at full aturation (the aturated water ontent wa equal to.5) at a flux denity, q, of.1 m min -1. Both olloid and the ontaminant were aumed to be applied at unit (relative) onentration at the top of the olumn during a time period of 6 min. The oil bulk denity, ρ, wa et equal to 1.5 g m -3, while the diperivity, λ, for both the olloid and the ontaminant wa aumed to be.1 m. The olloid attahment and detahment oeffiient, k a and k d, were taken to be.1 and.5 min -1, repetively. Solute orption to oil wa onidered intantaneou with the ditribution oeffiient, K d, equal to 2 L kg -1. Solute adorption wa aumed to be the ame for both the mobile and immobile olloid, but with different rate oeffiient of k am = k ai =.1 min -1 and k dm = k di =.2 min -1, repetively, and with C ref = S ref =1. For thee ondition, one pore volume, T, i equal to 5 minute [t = TL/v = TLθ /q = 1 * (1 m) *.5 / (.1 m/min) = 5 min], while the retardation fator for the ontaminant equal 7 (R = 1 + ρk d /θ = *2/.5 = 7). Figure 1 how olloid and total olute onentration at depth of 5 and 1 m. The main onentration front for both olloid and olute arrived, a expeted, at 1 and 7 pore volume (i.e., after 5 and 35 minute), repetively. However, a ignifiant amount of olute arrived muh earlier than at 7 pore volume. Thi earlier arrival i due to the aelerated movement of olute orbed to mobile olloid. 4

5 Figure 2 preent olloid and olute fluxe at the bottom of the olumn. Notie that the olute flux (Fig. 2b) ha two onentration peak. The firt peak orrepond to ontaminant arriving orbed to olloid, whih have a retardation fator, R, equal to 1 (no orption or anion exluion), while the eond peak orrepond to olute arriving diolved in water (R=7, t=35 min). Alo notie that the initial ontaminant fluxe (Fig. 2b) are maller relative than the olloid fluxe, whih how a plateau of ertain duration. Thi i beaue ontaminant orption onto the olloid wa aumed to be a kineti proe whih require a ertain time period for orption to be omplete. The ontaminant flux for thi reaon keep inreaing until it uddenly drop at the end of the olloid pule. Hene, the larget ontaminant fluxe during the firt peak oured jut before the end of the olloid pule (Fig. 2b). Colloid keep arriving at the bottom of the olumn after the main olloid pule due to kineti olloid detahment from the olid phae (Fig. 2a). By omparion, oniderable olute fluxe are preent between the two olute peak a a reult of olute both orbed to olloid and, inreaingly, diolved in water. The diolved olute fluxe at the end of the olumn were initially alo due to enhaned tranport by the olloid in epeially the upper part of the olumn, and ubequent deorption into the liquid phae (Fig. 2b). A expeted, the bulk of the ontaminant arrived at approximately 35 minute, onitent with a olute retardation fator, R, of a) 5 m 1 m.4.3 b) 5 m 1 m Conentration [-].6.4 Conentration [-] Figure 1. Colloid (a) and ontaminant (b) onentration at a depth of 5 and 1 m..8.7 a).3.25 b).6 Solute Flux [-/min] Solute Flux [-/min] Figure 2. Calulated olloid (a) and ontaminant (b) flux denitie at the bottom of the 1-m long oil olumn. 5

6 Referene Bradford, S. A., S. R. Yate, M. Bettahar, and J. Šimůnek, Phyial fator affeting the tranport and fate of olloid in aturated porou media, Water Reour. Re., 38(12), 1327, doi:1.129/22wr134, 22. Bradford, S.A., J. Šimůnek, M. Bettahar, M. Th. van Genuhten, S. R. and Yate, Modeling olloid attahment, training, and exluion in aturated porou media, Environ. Si. Tehnol., 37, , 23. Šimůnek, J., Changming He, J. L. Pang, and S. A. Bradford, Colloid-failitated tranport in variably-aturated porou media: Numerial model and experimental verifiation, Vadoe Zone Journal, 5, , 26. van Genuhten, M. Th., and J. Šimůnek, Integrated modeling of vadoe zone flow and tranport proee, Pro. Unaturated Zone Modelling: Progre, Challenge and Appliation, Ed. R. A. Fedde, G. H. de Rooij, and J. C. van Dam, Wageningen UR Fronti Serie, Vol. 6, Chapter 2, pp , x-xi, Kluwer Aademi Publiher, Dordreht, The Netherland, 24. 6

7 Implementation of olloid tranport and olloid-failitated olute tranport: Input Parameter: a) Colloid tranport (Solute 1) a in the tandard HYDRUS-1D: 6 d diameter of the and grain [L], d 5 11 μ l firt-order immobilization/deay in the liquid phae 12 μ firt-order immobilization/deay in the olid phae 13 ipi2 type of olloid bloking for the eond ite (not for filtration theory) 14 ipi1 type of olloid bloking for firt ite (not for filtration theory) 15 S max2 maximum retention apaity of the olid phae [MM -1 ] on the eond ite 16 k a2 depoition (attahment) oeffiient [T -1 ] on the eond ite 17 k d2 entrainment (detahment) oeffiient [T -1 ] on the eond ite 18 S max1 maximum retention apaity of the olid phae [MM -1 ] on the firt ite 19 k a1 depoition (attahment) oeffiient [T -1 ] on the firt ite 2 k d1 entrainment (detahment) oeffiient [T -1 ] on the firt ite b) Solute tranport (Solute 2) (thee parameter are reinterpreted differently than in the tandard HYDRUS-1D): 11 μ l firt-order deay in the liquid phae 12 μ firt-order deay in the olid phae (both attahed to oil and to olloid) 13 Omega kineti orption to olid phae [T -1 ] 14 GamL empty 15 S max2 referene onentration of immobile olloid for whih orption rate k a2 i valid [ML -3 ] 16 k a2 orption rate to immobile olloid [T -1 ] 17 k d2 deorption rate from immobile olloid [T -1 ] 18 S max1 referene onentration of mobile olloid for whih orption rate k a1 i valid [-] 19 k a1 orption rate to mobile olloid [T -1 ] 2 k d1 deorption rate from mobile olloid [T -1 ] Only olloid haraterized with parameter 18-2 (only one orption ite) are onidered for olute tranport! Filtration theory i not onidered for olloid tranport in the olloid failitated tranport model. Invere Solution: See example Colloid1i and Colloid1j. In the Colloid1i example, the objetive funtion i defined uing olloid (Poition=2) and diolved olute (Poition=-2) onentration at the eond obervation node. In the Colloid1j example, the objetive funtion i defined uing olloid (Poition=2) and olute (Poition=-2) fluxe at the bottom of the profile, and diolved olute (Poition=12) onentration at the eond obervation node. Output: The program print intead of the hydrauli ondutivity the onentration of ontaminant orbed to mobile olloid (C S m ) and intead of the hydrauli apaity the onentration of ontaminant orbed to immobile olloid (S S i ). 7