Calibration and uncertainty analysis of a regional fault zone groundwater flow and solute transport model using Nonlinear Least-Square Regression

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1 Calbraton and uncertanty analyss of a regonal fault zone groundwater flow and solute transport model usng Nonlnear Least-Square Regresson Xuyan Wang Coffey Geotechncs Pty Ltd, Brsbane, Australa Detlef Brngemeer*) Golder Assocates Pty Ltd, Brsbane, Australa Lng L School of Cvl Engneerng, Unversty of Queensland, Brsbane, 407, QLD Australa ABSTRACT The Nonlnear Least-Square Regresson (NLSR) approach was used to estmate model parameter values n the calbraton of a transent groundwater flow and solute transport model of a heterogeneous fractured rock aqufer system. Regresson statstcs ncludng senstvty, relatve composte senstvty and correlaton coeffcents were calculated for analysng the parameter uncertanty. Wth the presented case study we demonstrated that the NLSR technque provdes a smple way to quantfy parameter uncertanty and to mprove calbraton of a groundwater flow and solute transport model of a complex fault zone system. RÉSUMÉ La méthode des mondres carrées non-lnéares (MCN) est utlsée afn d estmer les valeurs des paramètres de calbraton pour la modélsaton du passage de la nappe phréatque et du transport d un soluté à travers une roche fracturée de manère hétérogène, dans un système aqufère souterran. Cette régresson statstque, ncluant le coeffcent de sensblté, la sensblté relatve composée et le coeffcent de corrélaton, fut employée pour détermner le paramètre d ncerttude. Avec le cas d étude d une nappe phréatque d un ste mner, nous démontrons que la technque MCN est un moyen smple pour quantfer le paramètre d ncerttude d une part et d amélorer la calbraton du model d une nappe phréatque et d un soluté transporté à travers un système de falle complexe, d autre part. 1 INTRODUCTION Parameter calbraton n groundwater flow and solute transport modelng of heterogeneous fractured rock systems s stll a challenge due to the ncomplete knowledge of the flow pathways and the lmtatons n quantty and qualty of the data avalable for charactersng the fractured rock aqufer. In the last decade a varety of numercal approaches have been used to mprove the calbraton of models to smulate groundwater flow n fractured rock (Beven and Freer, 001; Selroos, et al, 00; Beauhem and Roberts, 00; Kavetsk and Kuczera, 007). Though the computatonal aspects of such approaches are reasonably well developed, the queston of confdence n parameter estmates has not been completely resolved (Seber and Wld, 003; Hassan, 004). Model calbraton s tradtonally accomplshed by manual tral-and-error approach wth teratve selecton of parameter values to mprove the model results usng ntuton about responses of modellng results to changes n parameter values. Consequently, a calbraton obtaned usng a tral-and-error approach alone does not guarantee the statstcally best soluton. Inverse model lke lnear least-squares regresson has been wdely utlsed n model calbraton n whch the parameter values are adusted automatcally to match ste observatons as closely as possble (Seber and Wld, 003; Hll and Tedeman, 007, Fogla et al, 009). Lnear nverse modelng can statstcally gve the most approprate soluton for the gven nput parameters, quantfes the uncertanty n parameter estmates and yeld results that are not readly avalable through tral-and-error calbraton efforts (Poeter and Hll, 1996). However, t rarely leads to optmal estmates of model parameters values n modellng of a heterogeneous groundwater system snce lnear nverse models frequently obtan local nstead of global mnma durng the error estmaton process, especally wth small data sets of the flow observatons n a fractured rock aqufer (Hassan, 004; Moore and Doherty, 006). Bates and Watts (1988) stated: we hasten to warn the reader that lnear approxmaton regons can be extremely msleadng. The nonlnear regresson technque, usng a much larger and more general class of obectve functons to overcome the once-drect soluton of Lnear Least- Squares Regresson (LLSR), has been used n groundwater modellng n the last decade (Yobb, 000; Zhang et al. 006; Mathod et al, 008). Ths approach enhances data representaton n space (wth weghts) and optmzes the best-ft parameter values n the calbraton, especally when the estmated parameter errors are uncorrelated but have dfferent uncertantes that generally occur n the modellng of heterogeneous systems (Bate and Watts, 1988). The purpose of ths paper s to present the applcaton of nonlnear leastsquares regresson (NLSR) to the groundwater flow and solute transport model calbraton around a fault zone n a mnng area. The NLSR method s brefly descrbed n the subsequent sectons. A numercal framework ncludng senstvty, relatve composte senstvty and *) Formerly wth Coffey Geotechncs, Brsbane, Australa

2 correlaton coeffcents was developed for analysng the parameter uncertanty. The groundwater flow model utlzed for ths study s based on the data and nformaton presented by Coffey (010a) for a mne ste hydrogeologcal nvestgaton. A short descrpton of the hydrogeologcal model and regresson procedure follows. The optmal set of parameter values and assocated statstcs fnally determned by the developed NLSR framework s presented. NONLINEAR LEAST-SQUARES REGRESSION The process of model calbraton s essentally the same usng ether nverse models or the tral-and-error approach: parameter values and other aspects of the model are adusted untl the dependent varables (water levels, flows, concentratons) match feld observatons. The nonlnear least-squares regresson method extends lnear regresson for use wth a larger class of functons that can be wrtten n a closed form..1 NLSR Approach Nonlnear regresson attempts to ft a model functon that may depend nonlnearly on the parameter vector pˆ to a set of data wth y-values at the correspondng x- values y f ( x, pˆ ) [1] where f s the nonlnear model functon, and s a random error term. The nonlnear model functons are very commonly used n groundwater flow and solute transport modellng. For example flow through fractures used to be conceptualzed as cubc law (Naver- Stokes equatons) and the exponental functon represents a sem nfnte lne dpole of decayng strength dstrbuton (Anderson, 000). The goal of nonlnear regresson s to fnd the optmal values of parameter vector pˆ that mnmze the least squares functon whch s the sum-of-squared dfferences between observed and modelled quanttes and s defned as n S ( p) [( f ( x, pˆ) y )] [] 1 ˆ where f ( x, p) s the th modelled value wth the parameter vector pˆ used n the model functon, y s the th measured value, and n s the number of observatons. Nonlnear least squares optmzaton algorthm has been desgned to explot the structure of a sum of the squares obectve functon (Fogla et al, 009) where mn r n S ( p) [ w r ( pˆ)] [3] 1 pˆ) f ( x, pˆ ) y ( s the th least squares resdual and w s the weghtng factor. Optmzaton method used n ths study s a modfed Gauss-Newton method (Yobb, 000), the numercal framework of whch s descrbed n the followng secton.. NLSR Framework The NLSR framework for estmatng parameter values, ncludng parameter senstvtes and correlaton n groundwater model calbraton, s developed n the sx smplfed stages outlned below. (1) The parameter-estmaton process s to perform one executon of the model to establsh the ntal dfferences (resduals) between smulated and measured water levels. The resduals are squared and summed to produce the sum-of-squared resduals obectve functon (eq. []), whch s used by the regresson to quantfy the model ft to the observatons. () Composte senstvty (CS) ndcates the cumulatve amount of nformaton that measurements contan toward the estmaton of the parameters and s quantfed wth the least squares resdue ( r n equaton [3] usng the equaton: CS k N k 1 r / p [4] The senstvty coeffcents r p are gven by the Jacoban (senstvty) matrx of the least-squares resdue vector rˆ and serve as an approxmate ndcator of the senstvty of the N k observatons n the k th zone wth respect to the th parameter estmated. (3) Correlaton coeffcents between parameters, ndcates the degree of lnear dependency n the senstvty matrx and s calculated wth CS vector values Cov( CSk, CSkl ) R k 1 1 ( ) ( ) [5] CS k CS kl where Cov(CS k, CS kl) s covarance of composte senstvty of estmated and l parameters The CS k and CS are varances of composte senstvty of kl estmated and l parameters, respectvely. (4) The relatve composte senstvty (RCS), whch s the square root of the man dagonal value of the Jacoban matrx of the least-squares resdue vector dvded by the maxmum of the man dagonal value for each parameter, s quantfed for parameter uncertanty analyss. The RCS for the th parameter s qualfed as: CS, RCS [6] max(, ) CS (5) After the resduals and senstvtes are calculated, a sngle parameter-estmaton teraton s performed. The arrays of senstvty coeffcents and resduals are used by a quas-newton procedure (Bate and Watts, 1988) to compute the parameter change for mprovng the model. 1

3 (6) The model s updated to reflect the latest parameter estmates and a new set of resduals s calculated. The entre process of changng a parameter n the model, calculatng new resduals, and computng a new value for the parameter s contnued teratvely untl the model error change s reduced to a specfed level or untl a specfed number of teratons are made. The crtera of uncertanty analyss based on statstcal parameters from the above descrbed framework are:. (1) The CS values are used for assessng the relatve senstvty of the model (ether as a whole or locally) to each parameter. () Correlaton between parameters ndcates whether or not the parameter estmates are unque wth the gven model constructon and observatons. It s an ndcator of the degree of lnear dependency n the senstvty matrx and reflects the redundancy of the problem. Correlaton coeffcents greater than 0.95 usually ndcate a par of parameters that are hghly correlated (Hll and Tedeman, 007). These hghly correlated parameters are not desrable because they cannot be ndependently estmated. (3) The most senstve parameter has an RCS equal to 1.00 and the RCS of all other estmated parameters s less than one. The larger the value of the RCS, the more senstve the model s to that parameter, as a whole. Parameters wth smaller RCS values also tend to have hgher parameter uncertanty and broader confdence ntervals. very short dstance. The conceptual ste model s shown n Fgure 1. Based on the fracture densty and frequency of fault structures observed n exploraton boreholes the central zone was subdvded nto two zones wth Zone 1 representng ntensely fractured and faulted rocks and Zone charactersed by numerous faults but less ntensely fractured rock. The back fll n the pts were conceptualsed by a zone of Rock Fll n the South of the southern pt and a zone flled wth process resdue (RSF). 3 CASE STUDY In Australa mned out ore pts have been used ncreasngly for the storage of mnng process resdue. As part of approval requrements, a three-dmensonal FEFLOW fnte element groundwater flow and solute transport model was developed for smulatng the effects of n-pt fllng on the groundwater envronment at a mne ste n Vctora, Australa. Fault zones, dsruptng Ordovcan meta-sandstone and meta-sltstones, are understood to be the controllng hydrogeologcal structures at the ste. Abandoned and partally flooded mne workngs are algned along these fault zones wth the actve underground mnng followng the fault zones to depth n the south of the workngs. The country rock s overlan by a thn cover of alluval sedments wth cover thckness ncreasng to the east towards a rver valley. Groundwater n the alluval sedments s consdered to be relatvely fresh wth a low salnty of 50 to 100 mg/l Total Dssolved Solds (TDS). In contrast, the salnty of groundwater n the fractured country rock ranges from,000 to 10,000 mg/l TDS. The groundwater envronment across the ste was conceptualzed as consstng of three zones wth relatvely small groundwater head gradents n broad areas to the West and the East of the pts and an area between fault zones, where water levels drop over a Fgure 1 Locaton of study area and parameter zonaton. Numbers at the fgure frame represent mne coordnates n metres. The transent groundwater flow and solute transport model was calbrated wth groundwater level recorded n 009 over ½ months at 0 montorng bores and wth concentratons of selected solutes obtaned from groundwater and n-pt water sampled over the same tme perod. Calbraton was conducted usng both methods: LLSR and NLSR, and results obtaned wth both methods were compared. The general results from the analyss descrbed n the followng sectons ndcate that there s nsuffcent observaton data to ndependently estmate all model parameter values gven the present zonaton. 3.1 Relatve Composte Senstvtes for zone specfc hydraulc conductvtes Changes to horzontal hydraulc conductvty (K h) and to the ansotropy factor for hydraulc conductvty K h/k v that defnes the rato of horzontal hydraulc conductvty and

4 vertcal hydraulc conductvty were found to have the most sgnfcant effect on computed groundwater heads. Except for the resdue fll n the pts (RSF zone n Fgure 1), RCS values computed for the remanng hydraulc conductvty zones are shown n Table 1. NLSR analyss results ndcate that the avalable groundwater level data provde suffcent nformaton to relably estmate hydraulc conductvty of the fractured country rock for two of the four parameter zones (RCS > 0.5). Relatve Composte Senstvty s hghest for the Zone 1 and s lowest for the Zone 3. The low senstvty for Zone 3 and the Rock Fll zone s prmarly due to the lack of montorng data wthn these zones. Table 1 Relatve Composte Senstvty (RCS) of horzontal hydraulc conductvty adopted for four parameter zones durng model calbraton. Zone K h RCS Number of bores (m/d) Zone Zone Rock Fll Zone The parameter zones are not hghly correlated to one another, except for Zone 1 and Zone that show hgh correlaton wth a correlaton coeffcent R of The hgh correlaton between Zone 1 and Zone ndcates that wth respect to the hydraulc conductvty these zones may not be separable. The lowest degree of correlaton occurred between Zone 3 and the Rock Fll (r = 0.8). Addtonal groundwater-level montorng data n these zones would be benefcal for mprovng the model calbraton. RCS values for the ansotropy factor of hydraulc conductvty are summarsed n Table. Wth RCS values for Zone 1 and Zone ansotropy factors of hydraulc conductvty exceedng 0.5 estmates are consdered to be relable. However, the RCS s very low for the Rock Fll zone (RCS<0.1) ndcatng low relablty of the calbraton result for the ansotropy factor of ths zone. Improvements of the relablty of the parameter estmate for ths zone could be acheved wth addtonal observaton locatons n the Rock Fll. Table Relatve Composte Senstvty (RCS) of ansotropy factor of hydraulc conductvty adopted for four parameter zones durng model calbraton. Zone K h/k v RCS Number of bores Zone Zone Zone Rock fll Assessment of zone specfc absorpton coeffcent estmates The RCS values of arsenc absorpton coeffcents (c a) computed for each parameter zone are shown n Table 3. The RCS values ndcate that the water qualty data used for model calbraton provdes suffcent nformaton to estmate the absorpton coeffcents for two of the four parameter zones (RCS > 0.5). The lowest RCS value computed for the Rock Fll zone s bascally due to the lack of water qualty data wthn the Rock Fll zone whle the second lowest senstvty for Zone 3 s possbly due to very low solute concentratons observed n Zone 3. Table 3 Relatve Composte Senstvty (RCS) of arsenc absorpton coeffcents adopted for four parameter zones durng model calbraton. Zone c a RCS Number of bores Zone Zone Rock Fll Zone Most of the estmated absorpton coeffcents are hghly correlated. As shown n Table 5, the most hghly correlated zones are Zone 1 and Zone wth a correlaton coeffcent (R ) of The lowest correlated par of zones s adacent Zones 3 and Rock Fll wth a correlaton coeffcent of These relatonshps reflect the less dependency of model results on the observaton data set and the hgh uncertanty wthn the parameter-zonaton of absorpton coeffcent. 3.3 Comparson of NLSR and LLSR model calbraton results Two dfferent approaches, LLSR and NLSR, were mplemented for the analyss of parameter uncertanty as part of the model calbraton. Tables 4 and 5 show the adopted values for hydraulc conductvtes and absorpton coeffcents for the two calbraton approaches. Table 4 Horzontal hydraulc conductvtes derved from ntal (LLSR) and optmal (NLSR) calbraton approaches. Zone Intal (m/d) Optmal (m/d) RCS The most correlated Zone Zone (.93) Zone Zone 1 (.93) Zone Zone (.87) Rock Fll Zone 1 (-.79) Table 5 Absorpton coeffcents derved from ntal and optmal calbraton approaches. Zone Intal Optmal RCS The most correlated Zone Zone (.97) Zone Zone 1 (.97) Zone Zone (.9) Rock Fll Zone 1 (-.85)

5 The optmzed hydraulc conductvtes (K) shown n Tables 4 were ncreased by 0% to 100% from the ntal estmated parameter values whle the fnal arsenc absorpton coeffcents (c a) shown n Table 5 were adusted by 10% to 0% from the ntal estmated values. As ndcated prevously, parameters (K and c a) wth hgher RCS values for Zone 1 and Zone are lkely to have lower parameter uncertanty and the parameters wth smaller RCS values n Zone 3 and Rock Fll zone are lkely to have hgher parameter uncertanty. Most of the estmated K-values are not hghly correlated to one another. However, as ndcated by correlaton coeffcents n Table 4, the zones wth the most hghly correlated par are adacent fractured Zones 1 and wth a correlaton coeffcent of Smlarly the zones wth the most hghly correlated par n c a- values shown n Table 5 are also adacent fractured Zone 1 and Zone wth a correlaton coeffcent of These values reflect the hgh uncertanty n the parameter-zonaton of the model. Table 6 summarses statstcal parameters of the smulaton results n a perod of 4 months from the models calbrated usng LLSR and NLSR approaches. The results demonstrate that the mprovement of the model calbraton acheved by usng NLSR approach compared to the LLSR calbraton s essental wth the dmnuton of 30% of groundwater level resdual. Correlaton coeffcent between measured water level values and values computed wth the NLSR calbrated models ncreased by % when compared to the correlaton coeffcent computed wth results of the LLSR calbrated model. Table 6 Statstcal summary of groundwater flow model smulatons R +) MSE ++) Resdue Intal (m) Optmal (m) +) Correlaton coeffcent between measured and modelled water level; ++) Mean square error Statstcal parameters obtaned from arsenc concentraton smulatons usng absorpton coeffcents (c a) calbrated wth LLSR and NLSR approaches are shown n Table 7. The mprovement of the solute transport model calbraton made by NLSR compared wth the results from the LLSR approach s sgnfcant wth a 40% reducton of arsenc concentraton resdual and a 5% reducton of Mean Square Error (MSE). Also the correlaton coeffcent between measured and computed arsenc concentraton values slghtly ncreased by about 0.5% when compared to the LLSR calbrated model. Table 7 Statstcal summary of arsenc solute transport model smulatons R +) MSE ++) Resdue Intal (mg/l) Optmal (mg/l) +) Correlaton coeffcent between measured and modelled As concentraton; ++) Mean square error 3.4 Model valdaton The calbrated model was valdated wth montorng data recorded n 009 and 010 over a perod of 14 months durng the pre-fllng and the fllng of the resdue storage faclty (Coffey, 010b). For the model valdaton the modellng effcency ndex (IA) was used to analyse the correspondence between model results and observatons. The IA s commonly used as a goodnessof-ft measure and s defned as: Where are the modelled and observed varables, n s the number of data, and are the correspondng mean values. The IA s defned n a range between 0 and 1, and the closer t s to 1, the better s the ft between modelled and observed varables and the valdty of the model (Hassan, 003). Modellng effcency ndex between observed and computed water level and arsenc concentraton for each parameter zone were calculated and are summarsed n Table 6. Based on an IA threshold value of 0.8 whch was ntally adopted for ths case study, the parameters, hydraulc conductvty and ansotropy factor of hydraulc conductvty for all zones were assessed to be vald. However, the absorpton coeffcent of all materal zones requred re-calbraton. A comparson between the IA computed for parameter values obtaned wth the LLSR and the NLSR method showed that the IA values for parameters adopted durng calbraton usng the NLSR method are slghtly hgher. In partcular the IA computed for Zone 1 and Zone absorpton coeffcents are by 10% to 0% hgher for the NLSR method compared to the LLSR method (Table 6). Table 6 Statstcal summary of two model valdatons Zone IA for groundwater head IA for arsenc concentraton LLSR NLSR LLSR NLSR Zone Zone Zone Rock Fll CONCLUSION Ths paper presents the results of a study on the applcaton of nonlnear least-squares regresson to a groundwater flow and solute transport model of an n-pt resdue storage faclty located n a regonal fault zone. The man advantages of usng nonlnear least-squares regresson for model calbraton are the ablty to estmate parameter values more accurate when compared to parameter estmaton wth LLSR and to quantfy the qualty of model calbraton usng more [7]

6 general statstcal measures for both model senstvty and uncertanty. The qualty of model calbraton and uncertanty n smulaton results has been demonstrated wth regresson statstcs lke Relatve Composte Senstvty (RCS), correlaton coeffcent and IA values. Wth calculated RCS, t could be demonstrated that the qualty of the calbraton hghly reled on the densty of montorng locatons for each parameter zone. Calbraton of hydraulc conductvtes for hghly fractured rock zones (Zone 1 and Zone ) showed a hgh value of RCS ndcatng that the senstvty of computed groundwater heads to changes n hydraulc conductvty s very hgh. However, the uncertanty of the calbrated parameters of these zones s not hgher than that of the others zones. The hgh correlaton between Zone 1 and Zone RCS suggest that the dfferentaton between the two zones s arbtrary and cannot be ustfed by the lmted montorng data used n the calbraton. The uncertanty of the calbrated parameters for Rock Fll and Zone 3 tends to be very hgh possbly due to lmted montorng data avalable for ths unts and less well defned unt boundares. The valdaton of the NLSR calbrated model showed an approxmately 0% reducton n groundwater level resdual and 40% reducton n solute concentraton resdual when compared to the LLSR calbrated model. The IA values for model parameters estmated wth the NLSR method were up to 0 % hgher when compared to IA values obtaned for the LLSR parameter values. These results suggest that the NLSR technque can mprove the qualty of groundwater flow and solute transport models and yeld results of a qualty that are not readly achevable wth lnear regresson calbraton. Consequently, NLSR approaches could be used to determne strengths and weaknesses of the model and measures of parameter uncertanty n a heterogeneous aqufer system. Furthermore, the model s lkely to be able to obtan a unque set of parameter values f the nverse model converges. The stablty of algorthm used n ths study for NLSR model calbraton are affected by the number of parameters and observatons wth some nstabltes encountered when the number of observaton s small compared to the model parameter numbers to be calbrated. Nevertheless the statstcal framework of the NLSR method provdes an effectve way to mprove the calbraton of groundwater flow and solute transport models. ACKNOWLEDGEMENTS The authors are grateful for the support from Northgate Australa Ventures, Fostervlle Gold Mne. The authors also would lke to acknowledge the contrbuton of Coffey employees who have been workng on the proect. REFERENCES Anderson E.I., 000, The method of mages for leaky boundares. Advances n Water Resources, 3: Bate, D.B. and Watts, D.G., 1988, Nonlnear Regresson analyss and ts applcaton, Wley Seres n probablty and Mathematcal statstcs, John Wley and Sons, New York. Beven, K.J., J. Freer, 001, Equfnalty, data assmlaton, and uncertanty estmaton n mechanstc modelng of complex envronmental systems usng the GLUE methodology, Journal of Hydrology, 49, Coffey, 010a, Groundwater flow and solute transport modellng for the proposed Fostervlle RSF desgn, Coffey report Coffey, 010b, Valdaton of groundwater flow and solute transport model for the Fostervlle Gold Mne, Coffey report Fogla, L., S.W. Mehl, M.C. Hll, and P. Burlando, 009, Senstvty analyss, calbraton, and testng of a dstrbuted hydrologcal model usng error-based weghtng and one obectve functon, Water Resources Research., 45, W0647 Hassan, A.E., 004, Valdaton of numercal ground water models used to gude decson makng, Ground Water, 4(): Hll, M.C. and C.R. Tedeman, 007, Effectve calbraton of groundwater models, wth analyss of data, senstvtes, predctons, and uncertanty, John Wley and Sons, New York. Kavetsk, D., and G. Kuczera, 007, Model smoothng strateges to remove mcroscale dscontnutes and spurous secondary optma n obectve functons n hydrologcal calbraton, Water Resources Research, 43, W03411 Mathon, B.R., Ozbek M.M. and Pnder, G,F., 008, Transmssvty and storage coeffcent estmaton by couplng the Cooper Jacob method and modfed fuzzy least-squares regresson, Journal of Hydrology 353: Moore, C. and Doherty, J The cost of unqueness n groundwater model calbraton. Advances n Water Resources 9: Poeter, E.P., and Hll, M.C., 1996, Inverse Models: A necessary next step n ground-water modelng: Ground Water, 35(): Seber G.A.F. and Wld, C.J., 003, Nolnear Regresson, Wley Seres n probablty and statstcs, John Wley and Sons, Hoboken Selroos, J.O., Walker, D.D. Strom A., Gyllng B. and Folln, S., 00 Comparson of alternatve modellng approaches for groundwater flow n fractured rock, Journal of Hydrology, 57(1-4): Yobb, D.Y., 000, Applcaton of nonlnear leastsquares regresson to groundwater flow modellng, West-Central Florda, Water Resources Investgaton Report, USGS Zhang D., Beven, K. and Mermoud, A A comparson of non-lnear least square and GLUE for model calbraton and uncertanty estmaton for pestcde transport n sols. Advances n Water Resources 9: