Stochastic Inverse Modeling in a Mass Transport Problem

Transcription

1 Stochatc Invere Modelng n a Ma Tranport Problem Amr H Hoen and Clayton V Deutch The central dea n th paper to develop an nvere modelng approach for characterzaton of uncertanty n redual doluton rate and frt-order bodegradaton rate by talorng the etmaton of thee parameter to dtrbuton of uncertanty n ource ze and hydraulc conductvty feld. Such modelng can be ued a a creenng tool for etmatng plume length, total ma, and tme of remedaton n feld applcaton. The modelng technque baed on equental elf-calbraton approach, dtance-functon approach, and a gradent-baed optmzaton. It oberved that tyng the etmaton of tranport parameter to jont realzaton of tranmvty feld and ource geometry can effectvely characterze the uncertanty n thee parameter under feld condton and reduce the uncertanty n the tate varable. It alo oberved that rankng and creenng the realzaton baed on ther objectve functon value can effectvely reduce the uncertanty n the ource ze. Introducton Etmatng the length of tme requred for natural procee to remove a partcular contamnant from a groundwater ytem a ma balance problem termed tme of remedaton problem. There gnfcant uncertanty aocated wth ource properte and wth contrbuton and effcency of concentraton reducng mechanm. The mportant ource properte are the ource geometry and the doluton rate of contamnant pece nto groundwater. The concentraton reducng mechanm are advecton, dperon, dluton, orpton, volatlzaton, and aerobc and anaerobc bodegradaton. Many bodegradaton model that mulate complex knetc have been developed. It evdent that many of the requred knetc parameter for thee model can not be meaured or etmated by routne natural attenuaton protocol. Smpler approache wth lmted number of parameter are often preferred a they can be upported by the avalable data (Ead et al. 2003, Rfa and Rttaler 2005). Applcaton of frt-order reacton model common n natural attenuaton tude, partcularly at creenng level. Baed on the concentraton meaured at montorng locaton, the feld-cale frt-order rate are etmated by tral and error calbraton (Borden et al. 1986), by nvere modelng technque (Medna and Carrera 1996) or by feld approache uch a ma-flux (Borden et al. 1997) and concentraton-dtance relatonhp (Bucheck and Alcantar 1995). The parameter etmate by tral and error calbraton are modeler dependent and a meaure of uncertanty not often avalable. Among the nvere modelng technque, none of them quantfe the confdence n the etmated frt-order rate under uncertanty of ource properte and hydraulc conductvty dtrbuton. In the cae of feld etmaton technque, the etmated frt-order rate are affected by heterogenety. Undertandng of the ource doluton rate another mportant factor when nvetgatng dfferent apect of a TOR problem. A number of expermental (Imhoff et al. 1994), poreand feld-cale numercal (Dllard et al. 2001, Chrt et al. 2006) and nvere modelng tude (Scortno et al. 2000) have been reported to etmate doluton rate. None of the above nvere modelng technque deal wth characterzaton of ource properte when the reacton rate are uncertan. Smultaneou characterzaton of uncertanty n rate-lmted doluton and feld-cale bodegradaton mportant. Ead et al. (2003) mplemented an nvere modelng n an optmal ene to etmate doluton rate and ndvdual frt-order bodegradaton rate for BTEX compound a well a other parameter uch a recharge rate, hydraulc conductvty, and tranvere dpervty. They only acheved convergence when they etmated a ngle doluton rate for all BTEX compound through multaneou ue of oxygen durng aerobc bodegradaton (cro-over effect). In other word, they faled to etmate ndvdual doluton rate and frt-order bodegradaton rate contant for each BTEX compound due to hgh correlaton between thee parameter that reult n parameter non-unquene. The ue of parameter non-unquene dcued n Hoen and Deutch (2009). For groundwater management purpoe n the feld-cale, the worth of montorng data can be ued to etmate thee parameter through nvere modelng. Thee etmate, however, wll be affected by uncertanty n model tructure (ource ze) and other flow and tranport parameter, uch a 115-1

2 dtrbuton of hydraulc conductvty and dpervte. In th paper, the non-lnear confdence nterval of frt-order bodegradaton rate contant and doluton rate contant are etmated under uncertan ource geometry and aqufer tranmvty through a mple nvere modelng approach. Talorng the etmaton of doluton rate and frt-order bodegradaton rate to dtrbuton of uncertanty n the ource geometry and tranmvty feld through Monte Carlo type nvere modelng help to (1) characterze the nherent uncertanty n the value of thee parameter, (2) reduce the uncertanty n the tate varable and ze and hape of the plume, and (3) pobly reduce the uncertanty n the ource ze by rankng the condtonal realzaton baed on the value of the objectve functon. Methodology A decoupled approach ha been adopted n th work: frt, equental-elf calbraton approach mplemented to generate multple realzaton of tranmvty feld condtoned to tranmvty and head data, thee realzaton are then combned wth realzaton of ource ze to create multple realzaton of ource/tranmvty, doluton rate and frt-order bodegradaton rate contant are then etmated for each jont realzaton to characterze the uncertanty n thee parameter and reduce the uncertanty n the tate varable. The forward teady-tate flow and tranport problem are repreented by: h k = qr x x [1] and, ( C) t = C D j x x j x eq ( vc) + [ 0, kd( C C )] λc max [2] where, k, h, q r, C, D j, v, k d, C eq, and λ repreent hydraulc conductvty, hydraulc head, dperon coeffcent, eepage velocty, doluton rate contant, equlbrum concentraton, and frt-order bodegradaton rate contant. The equlbrum concentraton expreed by: eq ol C = f. [3] C ol where, C the olublty lmt for pure ubtrate n water, and f mole fracton of the pece n the mxture of organc and ntert/non-bodegradable materal and can be calculated by (Parker et al. 1991): S / ω f = [4] S / ω + Tt / ωt where, S the ma of ubtrate per unt ma of dry ol, and T t repreent the equvalent ma of all nert and non-bodegradable materal t per unt ma of dry ol, ω the molecular weght of ubtrate, and ω t the equvalent molecular weght of mxture of all non-bodegradable and nert (noluble) materal. The fracton T T t ω t = where, I l I TR NI NT l lt + l= 1 ωl lt= 1 ωlt, TR lt t / ω can be calculated by: t, ω l and ω lt repreent each nert and tracer (non-bodegradable) materal and ther aocated molecular weght, repectvely. The ma of ubtrate per unt ma of dry ol (S ) decreae a the doluton occur. Th proce can be repreented by (Waddll and Wdowon 1998): ds dt θ = R [6] ρ b where, ρ b the bulk denty of the porou medum and R repreent the ma tranfer rate gven by: R eq [ 0, k ( C C )] = max [7] d Thu, due to doluton of nto groundwater, ol concentraton decreae and aqueou concentraton ncreae. In Equaton [2], the only term on the left hand de account for change n concentraton wth tme (a tranent problem); the frt term on the rght hand de repreent the hydromechancal dperon; the econd term repreent advecton; the thrd term repreent doluton; [5] 115-2

3 and the forth term repreent frt-order bodegradaton. Th equaton hould be olved numercally to fnd the dtrbuton of dolved pece concentraton n pace and tme. A mple teady-tate groundwater flow mulator, flm2d ha been developed for groundwater flow; and a Lagrangan- Euleran approach, the method of charactertc, ha been programmed nto the code nam to olve Equaton [1] and [2]. The uncertanty n the hydraulc conductvty feld characterzed by equental-elf calbraton approach (SSC). The detal of th approach can be found n (Gomez-Hernandez et al. 1997). The uncertanty n the ource geometry baed on the extng well arrangement can be modeled by the dtance functon approach. The uncertanty n areal lmt addreed by: DF ( u ) N = * DF ( u DF ) ( u ) DFID ( u0) = α λid ( u ) DF( u ) + β [8] 1 DF( u ) where, α and β are calng and eparaton factor that defne a unque uncertanty band and mut be calbrated agant a large number of ynthetc realzaton. The objectve functon for ma tranport nvere problem defned by: nc m 2 1 F C, j = q jw ( C C ) wth w = [9] 2 2 cv C = 1 m where, C, C, q and cv are mulated concentraton, meaured concentraton, ource ze quantle and coeffcent of varaton aocated wth each obervaton. In order to olve the nvere problem and etmate the value of k d and λ, one need to calculate entvty coeffcent that can be calculated by entvty equaton: ( S α ) α ( α ) S R / n + vs D = [10] j t x x x j α where, α repreent ether k d or λ. The mnmzaton of objectve functon (Equaton [9]) and optmzaton of the tranport parameter can be mplemented through modfed Gau-Newton approach (Cooley and Naff 1990): T T 1 T T ( C XrωX rc + Imr ) C dr = C Xrω( y y( b r )) [11] where, C the dagonal calng matrx, X r the matrx of entvte, ω the matrx of weght, y the vector of oberved concentraton, y(b r ) the vector of mulated concentraton, and m r the Marquardt parameter. In each teraton of the Modfed Gau-Newton approach, the vector of etmated parameter updated by addton of an updatng vector d r multpled by a dampng parameter ρ r : b r+ 1 = ρ rdr + b [12] r where, b r and b r+1 are the vector of the etmated parameter n two conecutve teraton. The dampng parameter ρ r preerve the drecton of d r and enure that the change n the parameter reman le than the maxmum allowed change pecfed by the uer and ha a dampng effect on lkely ocllaton that may occur due to oppote drecton n conecutve updatng vector (d r and d r-1 ). In the modfed Gau-Newton method, the updatng vector d r calculated by Equaton [11]. Synthetc Example Error free obervaton A ynthetc example preented to nvetgate the performance of the Monte Carlo type decoupled nvere modelng n characterzaton of uncertanty n the doluton rate and frt-order bodegradaton rate and to tudy the effect of error n oberved data n the modelng outcome. A ynthetc hydraulc conductvty dataet, two dfferent head obervaton dataet wth two dfferent level of meaurement error, and four concentraton dataet are ampled from the reference tudy te. Applyng the SSC approach, the ampled hydraulc conductvty and head data are ued to create multple realzaton of hydraulc conductvty feld condtoned to both hydraulc conductvty and head meaurement. The dtance functon approach ued to create multple realzaton of areal extent of the ource zone. Invere modelng then mplemented to etmate the value of doluton rate and frt-order bodegradaton rate contant for the et of jont realzaton of ource geometry and hydraulc conductvty feld. The performance of the methodology nvetgated through tudyng the dtrbuton of the etmated parameter and ource zone ze and comparng the varaton of the tate 115-3

4 varable (e.g. plume length and ma loaded nto the aqufer) through tme wth thoe of the reference tudy te. The mulated tate varable are alo compared to the reult of a et of Monte Carlo mulaton performed ung k d and λ dtrbuton that repreent the range of varablty that may be oberved under realtc feld condton. The effect of head and concentraton meaurement error on the etmaton of k d and λ and the predcton of the tate varable are alo nvetgated. Fgure 1 how the reference tudy te wth the amplng locaton, the upected ource zone area, the reference hydraulc conductvty feld and the aocated head repone. The reference hydraulc conductvty feld hown n Fgure 1-b ha a Gauan dtrbuton n natural logarthmc unt wth a mean of log e m/, tandard devaton of 1.2log e m/, and a patal correlaton defned by a phercal varogram wth a nugget effect equal to 0.1 and a range of 32.0 m. The modelng doman 250m by 160m, whch decrtzed by 2.0m 2.0 m quared hape grd cell. The flow boundary condton nvolve fxed head boundary condton at the north and the outh of the te equal to 4.0m and 2.0m, repectvely. At the eat and wet of the te, no-flow boundary condton are agned. A hown n Fgure 1-a, there are a total of 40 obervaton well where pezometrc head (teady-tate) and concentraton are ampled. There are 11 of thee well (hown by blue crcle), wth hydraulc conductvty meaurement. The old black well ndcate the obervaton well where redual ha been oberved. Fgure 2-a how the calbrated uncertanty band for the gven well arrangement and upected ource area. The aocated optmal value of calng factor α and eparaton factor are 3.56 and 15.86, repectvely. To nvetgate the performance of the methodology when the actual ource ze devate from the average ource ze that characterzed by the dtance functon approach, three ource ze correpondng to lower quartle, medan and upper quartle of the calbrated uncertanty band are condered a reference cae for ource geometry. Fgure 2-b how the CDF of the ource ze aocated wth the calbrated uncertanty band n Fgure 2-a and the elected quartle. Accordng to Fgure 2-b, the reference ource ze (p 25, p 50 and p 75 quartle) have area equal to 643 m 2, 938 m 2, and 1395 m 2. Fgure 3 how the mulated plume for the maller ource ze. For mplcty, t ha been aumed that the dtrbuton of redual (ol concentraton) wthn the areal lmt of the ource zone unform. Varablty wthn areal lmt can ealy be ncorporated. The unform ol concentraton of et to 10gr/Kg. The ntal ma fracton of the ubtrate (e.g. Benzene) n equal to The ubtrate olublty, ubtrate and nert molecular weght are et equal to gr/cm3, 78.1 and 101gr/mole, repectvely. Dry ol denty, total poroty and effectve poroty are et equal to 1.6gr/cm3, 0.35 and 0.3, repectvely. The longtudnal and tranvere dpervte, doluton rate and frt-order bodegradaton rate are equal to 1.5m, 0.3m, day -1 and 0.006day -1, repectvely. Zero dperve flux boundary condton are agned at all boundare. Two ynthetc oberved dataet for pezometrc head are created by amplng from the reference pezometrc head dtrbuton and ubequent addton of Gauan noe. The frt et of head obervaton condered to be error-free. The econd head dataet condered to be noy by addton of Gauan noe wth a tandard devaton of σ nh =0.20m. Applyng the SSC technque, two et of 300 realzaton of hydraulc conductvty feld condtoned to both hydraulc conductvty and head meaurement are contructed form two level of head meaurement error (σ nh =0.0m and σ nh =0.20m), and combned wth realzaton of ource geometry to create two et of 300 jont realzaton that are ued n ubequent etmaton of k d and λ. In term of the ynthetc concentraton dataet, an error-free concentraton dataet ampled from the mulated plume (a total of 520 ample at 40 obervaton well over a perod of two year from 5 to 7 year from the tart of the mulaton). To nvetgate the effect of error n meaured concentraton on the modelng outcome, Gauan noe added to the ynthetc concentraton dataet ampled from the frt reference cae wth maller ource ze. The added Gauan noe ha a coeffcent of varaton equal to cv nc =0.35. To tudy the mportance of talorng the etmaton of frt-order bodegradaton rate contant and doluton rate to realzaton of ource geometry and hydraulc conductvty, the reult of the decoupled nvere modelng ncludng the mulated tate varable hould be compared to the avalable feld-cale parameter etmaton technque. Due to the fact that the propoed methodology amed to be an advanced creenng tool for characterzaton of uncertanty n the feld-cale parameter, t outcome hould be compared to the outcome of mlar creenng tool commonly appled to the feld. For th purpoe, a et of Monte Carlo mulaton (MCS) are performed wth (1) realzaton of 115-4

5 hydraulc conductvty condtoned to conductvty and head data by SSC, (2) realzaton of ource extend characterzed by the DF algorthm n Fgure 6-2-a, (3) the value of frt-order bodegradaton rate drawn from a dtrbuton reported by Bauer et al. (2006), and (4) the value drawn from a dtrbuton of doluton rate contant repreentng the uncertanty n a realtc feld condton. Bauer et al. (2006) howed that the feld-cale method of normalzaton to a recalctrant co-contamnant (Wedemeer et al. 1996) that correct for the effect of uncertanty n the value of longtudnal dpervty gve the cloet etmate to the true value of the frt-order bodegradaton rate contant. For an aqufer wth a lognormal hydraulc conductvty dtrbuton wth a mean of log e m/ and a tandard devaton of 1.3 logem/, Bauer et al. (2006) howed that the method of normalzaton to a recalctrant co-contamnant overetmate the true frt-order rate (on average) by a factor of two, whle the tandard devaton of the normalzed rate equal to 2. Smlar reult were found by Bauer et al. (2007) for the mproved method of Stenback et al. (2003) wth off-centerlne meaurement. Fgure 4-b how a dtrbuton of frt-order bodegradaton rate mlar to the dtrbuton oberved by Bauer et al. (2006) baed on normalzaton to a recalctrant co-contamnant. To nvetgate the effect of varablty n the doluton rate, a unform dtrbuton wth an order of magntude varablty (whch eem to be a lower bound to varablty n th parameter baed on the obervaton n Dllard et al and Chrt et al. 2006) and a mean equal to day -1 (computed by Ead et al. (2003) for Bemdj te) condered (Fgure 4-a). Fgure 5 how the varaton of the ma loaded nto the aqufer and the plume length n tme for the reference cae a well a the mean and quartle of the tate varable baed on the reult of the Monte Carlo mulaton wth 100 jont realzaton of hydraulc conductvty feld and ource geometry and the value of k d and drawn from the dtrbuton n Fgure 4-a and 4-b. Fgure 6 how the probablty map for the concentraton to exceed a threhold value of mg/l (water qualty tandard for benzene). Fgure 5 and 6 how that the MCS may reult n large uncertante n the dmenon of the mulated plume a well a the ma loaded nto the aqufer. The dtrbuton of the parameter hown n Fgure 4 and the MCS reult hown n Fgure 5 and 6 wll be compared to the reult of nvere modelng. Fgure 7 how the htogram of the etmated k d and λ for the 100 jont realzaton that are calbrated to concentraton meaurement from the maller reference cae. It ha been aumed that the head and concentraton meaurement are error-free. Fgure 8 how the varaton of the ma loaded nto the aqufer and the plume length through tme for three et of 100 realzaton correpondng to the maller reference cae. Fgure 9 how the probablty map for concentraton to exceed a threhold value of mg/l for the maller reference cae. A expected, the doluton rate contant lghtly under-etmated for the cae wth the maller reference ource zone. The propoed approach gnfcantly reduce the uncertanty n the tate varable (comparng to the reult of the Monte Carlo mulaton). Although the enemble of realzaton on average over/under-etmate the reference value, for both tate varable, the reference curve fall wthn the 90% non-lnear confdence nterval. Comparng Fgure 9 to Fgure 6, t alo evdent that the etmaton of doluton rate and frt-order bodegradaton rate for jont realzaton of hydraulc conductvty and ource geometry ung concentraton data can gnfcantly reduce the uncertanty n the dmenon of the plume. The oberved over-etmaton and under-etmaton of the tate varable partally due to unreolved uncertante n the ource ze whch can not be fully handled by adjutng the value of k d and λ by the model. Thu, a rankng-baed creenng can be appled (Smlar to the work of Poeter and McKenna 1995) to chooe from the et of realzaton baed on the value of the modfed objectve functon and to decreae the uncertanty n the ource zone ze prevouly characterzed by the dtance-functon approach. To nvetgate the effectvene of rankng on the reducton of uncertanty and to have enough realzaton to explore the pace of uncertanty, 300 jont realzaton of hydraulc conductvty (condtoned to head data wth σ nh =0.0 m) and ource geometry are contructed and the concentraton ampled from the three reference cae (wth cv nc =0.0) are ued to etmate the value of doluton rate contant and frt-order bodegradaton rate. Fgure 11 how the CDF of the ource ze for 100 realzaton (out of 300 realzaton) havng mallet value of modfed objectve functon defned by Equaton [9]. Comparng Fgure 11 to Fgure 2-b, one can oberve that rankng and creenng the realzaton can effectvely reduce the uncertanty n the ource zone ze for each reference cae. To further nvetgate the effect of rankng, one may alo look at Fgure 10 where the cro-plot of doluton rate 115-5

6 and ource ze quantle hown. The color-cale repreent the rank of each realzaton baed on the value of the modfed objectve functon (black how lower value of the modfed objectve functon, hgher rank and therefore accepted realzaton). For th example, Fgure 10 how that (1) there a negatve correlaton between the ze of the ource the etmated doluton rate; and (2) rankng of the realzaton can effectvely dentfy the jont realzaton that have not properly converged n optmzaton (old crcle) and the jont realzaton that have ource ze that devate from the reference ource ze (dahed crcle). Fgure 11 and 12 how the cro-plot between the doluton rate and frt-order bodegradaton rate contant and the cro-plot between the ource ze quantle and frt-order bodegradaton rate (after rankng). Accordng to Fgure 13, there ext a potve correlaton between frt-order bodegradaton rate and doluton rate. Lookng at Fgure 14, one oberve that there lttle correlaton between the value of ource quantle and frt-order bodegradaton rate. The obervaton n Fgure 10, 12 and 13 jutfy the mportance of multaneou characterzaton of uncertanty n ource areal extent, ource doluton rate and frt-order bodegradaton rate. Fgure 14 to 16 how the etmated parameter, the aocated tate varable and the probablty map after rankng and creenng baed on the value of the modfed objectve functon. Accordng to Fgure 10 to 16, one can conclude that (1) reducton n the uncertanty of the ource zone ze appear to be achevable by rankng and creenng the realzaton baed on the value of the modfed objectve functon (Equaton [9]); (2) for th purpoe, an approprate number of realzaton hould be elected; and (3) by reducng the uncertanty n the ource zone ze, there wll be reducton n the aocated uncertanty n the etmated parameter value and the tate varable. Synthetc Example Erroneou obervaton In practce, t qute rare to conder the obervaton data a error-free. Due to the fact that the propoed methodology a decoupled approach, one may generate hydraulc conductvty realzaton honorng a partcular level of error n head obervaton (meaure of ft cloe to one); and then etmate the rate contant for the jont realzaton, whle calbratng to concentraton wth a partcular level of error n the data value. In the ubequent analy, Gauan noe (wth a relatvely large tandard devaton/coeffcent of varaton) ha been added to the head obervaton and concentraton meaurement aocated wth the maller reference cae. It aumed that (1) a good knowledge of magntude of error ext n the obervaton, (2) the error n obervaton Gauan noe wth a mean equal to zero (no ytematc ba ntroduced), and (3) error at dfferent locaton and for head and concentraton are ndependent of each other. A t can be oberved n the followng example, uncertanty n the etmated parameter and the predcted tate varable ncreae wth an ncreae n the meaurement error. Comparng Fgure 17, 19 and 18 to Fgure7, 8 and 9, one can oberve that ntroducng meaurement error to head and concentraton reult n (1) conderable ncreae n the tandard devaton of the etmated parameter, (2) larger devaton of the average bodegradaton rate contant from the reference value, (3) ntroducng more uncertanty and ba n the etmaton of the length (and wdth) of the plume and (4) ncreang the uncertanty n the etmaton of ma loaded nto the aqufer. Concluon Th paper preented a modelng approach and a ynthetc example to nvetgate the performance of the decoupled nvere modelng approach n characterzng the uncertanty n the doluton rate and frtorder bodegradaton rate and reducng the uncertanty n the aocated tate varable. A reference cae wth maller ource ze wa condered. Frt, a et of Monte Carlo mulaton were mplemented whoe reult were ubequently compared to the reult of the nvere modelng methodology. Comparng the reult of the Monte Carlo mulaton to the reult of nvere modelng for the maller reference cae, t wa oberved that talorng the etmaton of frt-order bodegradaton rate and doluton rate to dtrbuton of uncertanty n the ource geometry and hydraulc conductvty feld reult n characterzaton of uncertanty n thee parameter and gnfcant reducton of uncertanty n the tate varable, beng ma loaded nto the aqufer and the dmenon of the plume. Although the reference value alway fell wthn 90% confdence nterval, a ba wa oberved n the predcton of the reference tate varable by the enemble of mulated realzaton. Th ba wa deemed to be partally due to 115-6

7 large varablte n the ze of the ource zone whch cannot be fully reolved by adjutng the value of doluton rate and frt-order bodegradaton rate. In th work, t wa oberved that rankng and creenng the condtonal realzaton baed on the value of modfed objectve functon (Chapter 5) can effectvely reduce the uncertante n the ze of the ource zone and uncertanty and ba n the predcton of the tate varable. The value of the modfed objectve functon deemed to be ndependent of overall level of concentraton n the modelng doman due to the fact that t a dmenon-le number and a normalzaton that take place by defnng the weght nvere proportonal to the value of mulated concentraton. On the other hand, the number of realzaton that are kept after rankng and creenng may be condered problem dependent for a large part. Depte th apparent ubjectvty, t wa oberved through a entvty analy that rankng the realzaton and keepng any number of realzaton can tll be ueful a t can gve a general dea about the ze of the ource, whle reducng t uncertanty. The mportance of multaneou characterzaton of uncertanty n the parameter wa nvetgated through cro-plot of the parameter, where t wa oberved that a potve correlaton ext between the value of doluton rate and frt-order bodegradaton rate, a negatve correlaton ext between the value of doluton rate and ource ze quantle and lttle correlaton ext between frt-order bodegradaton rate and ource ze quantle. To nvetgate the effect of error n obervaton data on the modelng outcome, relatvely large level of obervaton error (Gauan noe wth a mean of zero and pre-pecfed tandard devaton/coeffcent of varaton) were added to head and concentraton obervaton and the tranport parameter were etmated. Accordng to the reult, extence of Gauan noe n the data reulted n an ncreae n the uncertanty and ba of the etmated parameter and the predcted tate varable. Comparng thee reult to the reult of Monte Carlo mulaton ndcated that even f the oberved data are ubject to a relatvely large level of Gauan noe, the uncertanty n the predcted tate varable are tll maller than the reult of Monte Carlo mulaton. Reference Bauer, S., C. Beyer, and O. Koldtz (2006), Aeng meaurement uncertanty of frt order degradaton rate n heterogeneou aqufer, Water Reource Reearch, 42, W01420, do: /2004WR Borden, R.C., P.B., Bedent, M.D., Lee, C.H., Ward, J.T. Wlon (1986), Tranport of dolved hydrocarbon nfluenced by oxygenlmted bodegradaton: 2. Feld applcaton, Water Reource Reearch, 22(13), Bucheck, T.E., and C. M. Alcantar (1995), Regreon technque and analytcal oluton to demontrate ntrnc boremedaton, n Intrnc Boremedaton, edted by R.E. Hnchee, T.J. Wlon, and D. Downey pp , Battelle Pre, Columbu, Oho Chrt, J.A., C.A. Ramburg, K.D. Pennell, and L.M. Abrola (2006) Etmatng ma tranfer from dene nonaqueou phae lqud ource zone ung upcaled ma tranfer coeffcent: An evaluaton ung multphae numercal mulaton, Water Reource Reearch, 42(11), W11420 Cooley R.L., and R.L. Naff (1990), Regreon modelng of ground-water flow, Technque of Water-Reource Invetgaton of the US Geologcal Survey, 232 pp. Dllard, L.A. H.I. Ead, M.J. Blunt (2001), A functonal relaton for feld-cale nonaqueou phae lqud doluton developed ung a pore network model, Journal of Contamnant Hydrology, 48(1-2), Ead, H.I., I.M. Cozzarell, R.P. Eganhoue, W.N., Herkelrath, BA. Bekn, G.N. Deln (2003), Invere modelng of BTEX doluton and bodegradaton at the Bemdj, MN crude-ol pll te, Journal of Contamnant Hydrology, 67(1-4), Gomez-Hernandez, J.J., Stochatc mulaton of tranmvty feld condtonal to both tranmvty and pezometrc data 1.Theory (1997), Journal of Hydrology, 203(1-4): Hoen, A.H., C.V. Deutch (2009), Stablty of the nvere problem: a cae tudy, Center for Computatonal Geotattc, Report 11, paper 129 Imhoff, P.Y., P.J. Jaffe, and G.F. Pnder (1994), An expermental tudy of complete doluton of a non-aqueou phae lqud n aturated porou meda, Water Reource Re., 30(2), Medna, A., and J. Carrera (1996), Coupled etmaton of flow and olute tranport parameter, Water Reource Reearch, 32(10): Poeter, E.P., S. McKenna (1995), Reducng uncertanty aocated wth ground water flow and tranport predcton, Ground Water, 33(6), Rfa, H. S., and T. Rttaler (2005), Modelng natural attenuaton of benzene wth analytcal and numercal model, Bodegradaton, 16, Scortno, A., T.C. Harmon and W.G. Yeh (2000), Invere modelng for locatng dene nonaqueou pool n groundwater under teady flow condton, Water Reource Reearch, 36(7): Stenback, G.A., S.K. Ong, S.W. Roger, and B.H. Kjartonon (2004), Impact of tranvere and longtudnal dperon on frt-order degradaton rate contant etmaton, Journal of Contamnant Hydrology, 73, 3-14, do: /j.jconhyd

8 (a) Fgure 1: (a) The reference tudy te wth montorng locaton and upected ource zone area (dahed box), (b) the reference hydraulc conductvty feld, and (d) the reference hydraulc head dtrbuton. Fgure 2: (a) The calbrated band of uncertanty for the contamnant ource zone, and (b) the CDF of the ource ze. Fgure 3: (a) The maller ource zone ze correpondng to p 25 of the calbrated uncertanty band, (b) the mulated plume after 550 day, (c) the mulated plume after 1281 day, and (d) mulated plume after 2562 day

9 Fgure 4: The dtrbuton of uncertanty n (a) the doluton rate contant and (b) the frt-order bodegradaton rate contant, ued n the ubequent MCS. Fgure 5: The varaton of mulated (p 05, p 25, p 50, p 75, and p 95 quantle of enemble of realzaton) and reference (p 50 ource ze) (a) total ma loaded nto the aqufer and (b) plume length baed on the reult of the MCS. The reference curve aocated wth the medan ource ze (Fgure 5-24) Fgure 6: The probablty of concentraton exceedng mg/l baed on the reult of the Monte Carlo mulaton Fgure 7: The htogram of (a) k d and (b) for the cae wth nh = 0.0 m and cv n = 0.0 and the maller reference ource ze

10 Fgure 8: The varaton of mulated (p 05, p 25, p 50, p 75, and p 95 quantle of enemble of realzaton) and reference (a) total ma loaded nto the aqufer and (b) plume length for the maller reference ource ze. Fgure 9: The probablty of concentraton exceedng mg/l after condtonng to concentraton for the maller reference cae Fgure 10: The cro-plot between the ource ze quantle and the etmated doluton rate. The old crcle how the realzaton that are lkely not converged and the dahed crcle how the realzaton that ther ource ze gnfcantly devate from the reference ource ze. The color cale how the rank of realzaton baed on ther modfed objectve functon value (black repreent lower value of the objectve functon) Fgure 11: The CDF of the ource ze of the 100 accepted realzaton after rankng baed on the modfed objectve functon value. The red arrow how the reference ource ze for each cae

11 Fgure 12: The cro-plot between the doluton rate contant and fr-order bodegradaton rate contant for the reference cae wth maller ource ze (p 25 ) wth a correlaton coeffcent equal to Fgure 13: The cro-plot between the ource ze quantle and frt-order bodegradaton rate contant for the reference cae wth maller ource ze (p 25 ) Fgure 14: The htogram of (a) k d and (b) for the accepted realzaton after rankng, baed on the reference cae wth the maller ource ze. Fgure 15: The varaton of mulated (p 05, p 25, p 50, p 75, and p 95 quantle of enemble of realzaton) and reference (a) total ma loaded nto the aqufer and (b) plume length for the maller ource ze after rankng

12 Fgure 16: The probablty of concentraton exceedng mg/l after condtonng to concentraton and rankng for the maller reference cae Fgure 17: The htogram of (a) k d and (b) for the cae wth nh = 0.2 m and cv n = 0.3 and the maller reference ource ze. Fgure 18: The probablty of concentraton exceedng mg/l for the cae wth nh = 0.2 m and cv n = 0.3 and the maller reference ource ze. Fgure 19: The varaton of mulated (p 05, p 25, p 50, p 75, and p 95 quantle of enemble of realzaton) and reference (a) total ma loaded nto the aqufer and (b) plume length for the cae wth nh = 0.2 m and cv n = 0.3 and the maller reference ource ze