Journl of Hyrology mnusript No. (will e inserte y the eitor) Stohsti Susurfe Dt Integrtion Assessing Aquifer Prmeter n Bounry Conition Unertinty Dongong Wng Ye Zhng Reeive: te / Aepte: te Astrt This reserh suessfully extens eterministi, physilly-se inverse theory tht is ple of simultneous prmeter n ounry onition estimtion to unertinty quntifition in inverting stey-stte grounwter flow in two-imensionl quifer with fies heterogeneity. Using fies n ynmi flow mesurements smple t wells s oservtions, stohsti susurfe t integrtion tehnique is propose: (1) Sequentil Initor Simultion integrtes fies t y hrterizing its geosttistil prmeters (experimentl iretionl vriogrms n smple fies proportions) to generte orrelte fies moels; (2) for eh fies moel, hyruli onutivities n flow fiel (inluing the unknown ounry onitions) re estimte vi iret inversion metho; (3) unertinty in inversion, inluing oth unertinties of the estimte hyruli onutivities n the flow fiel, is evlute y ssessing the inversion outome for ll fies moels. To test the propose integrtion tehnique, referene forwr moel provies oth fies hrteriztion n ynmi mesurements t inresing smpling ensities (i.e., t quntity) n mesurement errors (i.e., t qulity). Vi smoothing n gri orsening, lterntive hyruli onutivity prmeteriztion is lso teste in inversion. Unertinty in the estimte onutivities n ounry onitions is then quntifie ginst the referene moel to evlute the qulity of inversion. Results suggest tht for the rnges of teste vrition in t quntity, qulity, n inverse onutivity prmeteriztion, (1) t quntity hs the strongest impt on oth inversion ury n preision; (2) t qulity influenes inversion ury; (3) inverse prmeteriztion hs the wekest influene on inversion s long s the overll fies pttern is pture (i.e., suffiient t quntity). A lne n thus e hieve etween prmeteriztion, omputtionl effiieny, n inversion performne. For Dongong Wng Deprtment of Geologil Sienes, University of North Crolin, Chpel Hill, North Crolin, USA E-mil: wng1986@live.un.eu Ye Zhng 1 University Ave., University of Wyoming, Lrmie, Wyoming, USA E-mil: yzhng9@uwyo.eu
2 Dongong Wng, Ye Zhng the heterogeneity pttern investigte herein, y efining n eptle mrgin of unertinty for either onutivity or flow fiel estimtion, optiml well sping in reltion to the hrteristi length of heterogeneity n e etermine uner unknown ounry onitions. Finlly, inversion omin shoul e losely efine y the mesurement lotions in orer to minimize extrpoltion errors. Keywors Hyrogeologil Moeling Unertinty quntifition Physillyse inversion Bounry onitions 1 Introution Prmeterizing hyruli onutivities (Ks) of n quifer moel is hllenging topi in mny susurfe investigtions. Grounwter moeling often suffers from signifint unertinty in epiting oth the K vlues n its sptil istriution in the susurfe, s extensive rilling n smpling to ollet hyrogeologil t is imprtil. However, oth quifer mngement n susurfe ontminnt lenup opertions require the evelopment of urte n ppropritely prmeterize grounwter moels. Deision-mking further requires sientifilly informe ssessment of the unertinty in the moel preitions, whih often rises from the unertinty in inferring susurfe hyruli onutivities n the ssoite flow fiel. In nturl quifers, onutivity istriution is heterogeneous t multiple sles n its iret mesurement often suffers from the well-known sle effet, i.e., Ks mesure y ifferent instruments hnge with the support volumes eing teste [1][2][3]. On the other hn, y eveloping numeril quifer moel, Ks of the moel n e lirte using inverse theory se on mesurements of the hyrogeologil stte vriles (e.g., hyruli hes, flow rtes). In prtiulr, inverse methos hve een evelope to not only filitte prmeter estimtion, ut lso to quntify the ssoite estimtion unertinty, whih n le to n ssessment of the preition unertinty when moels re use for eision-mking. Generlly, inverse methos in hyrogeology n e tegorize into iniret n iret methos. With the iniret inversion metho, n ojetive funtion, typilly efine s mismth etween the mesurement t n the orresponing moel simulte vlues, is minimize [4]. During inversion, prmeters re upte itertively y running forwr moel tht provies link etween the prmeters n the t. The upte prmeters then le to n upte forwr moel, with whih the ojetive funtion is reevlute. This proess is repete until user-efine fitting riterion is rehe. The iniret metho is foun flexile n effiient in lirting mny susurfe moels, lthough numer of issues hve een ientifie in its pplitions. Among them, ill-poseness is well-known prolem, whih is mnifeste y instility (sensitivity of the estimte prmeters to smll hnges in the oservtion t n their errors), nonuniqueness (more thn one set of prmeters n lirte the moel, with results epening on the strting points in the prmeter spe), n filure to onverge to resonle solution. Beuse forwr moel is neee to optimize the ojetive funtion, ounry onitions (BC) of the moel re either ssume known, or re lirte s prt of the inversion proess. However, BC of quifers re often unknown or unertin. As emonstrte y [5], ifferent omintions
Stohsti Dt Integrtion 3 of prmeters n BC n le to the sme ojetive funtion vlues, thus results of iniret inversion my eome non-unique. (In trnsient prolems, oth quifer initil n BC re unknown this topi is resse elsewhere [6] n is not evlute in this work.) For quifer prmeter estimtion, iret inversion methos hve lso een explore owing to their mthemtil strightforwrness n omputtionl effiieny. However, iret methos suffer from issues suh s high sensitivity to oservtion errors t stremline lotions [7][8][9], stringent error ouns on mesurements [1], reue ury with inresing prolem imensions [11], et. As reviewe in [12], these methos lso simulte the forwr moel requiring the speifition of BC, they therefore n suffer from instility n non-uniqueness in inversion n re not ommonly use in grounwter moel lirtion. Reently, se on the Stress Trjetories Element Metho evelope for solving ertin geophysil n flui mehnis prolems [13], new iret metho ws propose for stey-stte quifer inversion tht llows the simultneous estimtion of quifer onutivities, soures/sinks (e.g., rel rehrge, pumping n injetion), n stte vriles (e.g., hyruli he n Dry flux) [5][14][15][16]. Beuse the stte vriles re solutions of the inversion, the unknown quifer ounry onitions n thus e estimte. The iret metho honors the physis of flow using ontinuity priniples, while it iretly inorportes noisy oserve t t the mesurement lotions in single step, without using forwr simultions n itertive prmeter uptes to optimize ny ojetive funtion. Beuse it oes not simulte the forwr moel, the iret metho is omputtionlly effiient, n it utilizes n inversion gri with highly flexile isretiztion n prmeteriztion. However, previous works hve severl limittions tht restrit the metho to solving simple prolems with smll gri sizes. First of ll, onutivity istriution n ptterns were ssume to e known when formulting the inversion equtions. In rel quifers, K istriutions re typilly unknown or highly unertin. On the other hn, quifer hrteriztion n provie informtion on K istriution t orehole lotions. Seonly, inversion equtions hve so fr een evelope n teste for smll prolems: the mximum size of the inversion gri teste is 16 16 (two-imensionl prolems) or 14 14 14 (threeimensionl prolems), whih limits the inversion to resolving only orse heterogeneity fetures in the K moel. Conutivity in nturl quifers often exhiits irregulr sptil vrition, whih will require the evelopment of refine inversion gri with more gri ells. Thirly, euse the K ptterns were eterministi n ssume fully known, inversion outomes, inluing oth the estimte K vlues n the flow fiel, were eterministi. Unertinty in inversion thus oul not e quntifie. To emonstrte wier ppliility of the new iret metho, the ove issues nee to e resse y: (1) inorporting iret or iniret informtion out quifer heterogeneity, e.g., fies oservtions t orehole lotions; (2) inverting lrger gri sizes to etter resolve etile heterogeneities; (3) ounting for unertinty in inversion, inluing oth the unertinty of the estimte prmeters n the flow fiel. For stey-stte quifer flows, this reserh ims to ress the ove issues y eveloping n testing stohsti susurfe integrtion tehnique tht omines geosttistis (i.e., stti t integrtion) with iret inversion (i.e., ynmi t integrtion) into single estimtion n unertinty nlysis frmework. The previous inversion will thus e extene to inlue quifer pro-
4 Dongong Wng, Ye Zhng lems with relisti K heterogeneity, while the ssoite estimtion unertinty, inluing the unertinty in the inferre flow fiel, n e quntifie. To evlute the ppliility of the integrtion tehnique to fiel onitions with limite n noisy oservtions, this reserh first investigtes the impt of t quntity (i.e., numer of smpling wells) n qulity (i.e., mgnitue of mesurement errors) on the ury n stility of inversion, with results leing to set of smpling strtegies tht n rener the propose integrtion more ost-effetive. In prtiulr, y efining n eptle mrgin of unertinty in either K or flow fiel estimtion, optiml well sping in reltion to the hrteristi length of heterogeneity n e reommene. Here, the term optiml mens suffiient stti n ynmi mesurements n e otine from fewest wells (or lrgest well sping) tht n still provie onstrins to inversion, resulting in n eptle level of estimtion unertinty. Moreover, euse inverse K prmeteriztion is flexile n, s emonstrte y previous upsling stuies [17], high-resolution K heterogeneity is not inispensle for preiting ulk flow in quifers, this reserh will prmeterize the inversion gri with heterogeneity t vrying egrees of smoothness. In prtiulr, y orsening the inversion gir, lne etween omputtionl effiieny n estimtion ury/unertinty is explore. In the reminer of this rtile, the stohsti integrtion tehnique is introue first, followe y esription of syntheti forwr moel tht will e use s referene moel. The smpling esign is isusse, whih will provie inversion with fies n hyrogeologil t orehole lotions. Stti t integrtion vi geosttistil simultion is expline riefly, efore ynmi t integrtion vi the iret metho is introue. Beuse lrger inverse prolems re solve, mtrix onitioning tehniques re opte to improve the omputtionl effiieny of inversion. The results re presente in whih the effets of t qulity, quntity, n inverse prmeteriztion on inversion ury n unertinty re evlute. Finlly, implitions of these results for fiel implementtion of the propose integrtion tehnique is isusse n future reserh inite. 2 Metho In this work, stohsti integrtion tehnique is propose, onsisting of 3 steps: (1) Sequentil Initor Simultion (SIS) integrtes the fies t (i.e., experimentl iretionl vriogrms n smple fies proportions ompute from orehole mesurements) to generte rnom moels of orrelte fies; (2) for eh fies moel, hyruli onutivities n flow fiel (inluing the unknown BC) re estimte vi iret inversion; (3) unertinty in inversion inluing unertinties of the estimte Ks n the flow fiel is evlute y ssessing the outomes for ll fies moels. Beuse forwr moel is use s referene moel to test the qulity of inversion (i.e., estimtion ury n preision), elow, the forwr moel is esrie first. 2.1 Forwr (Referene) Moel To test n verify the propose integrtion tehnique, referene forwr moel, representing two-imensionl syntheti quifer trnset, ws rete with known
Stohsti Dt Integrtion 5 Fig. 1 The referene moel flow fiel ontining two fies (rk gry: silty sn with K1; light gry: len sn with K2). Given the speifie BC n fies Ks, hyruli he istriution is lso shown. Four ounry segments re lele: - enotes the inflow ounry, n - enotes the outflow ounry. fies ptterns n K vlues (Fig. 1). The grounwter flow eqution of the forwr moel, representing onfine quifer without soures/sinks, n e written s: (q) = (1) q = K(x, z) h (x, z) Ω where Ω is the sptil omin (ssume to e the sme for the forwr moel n for inversion), x n z represent the horizontl n vertil xes in Crtesin oorinte, is the grient opertor, h is hyruli he, q is Dry flux, n K(x, z) enotes lolly isotropi hyruli onutivity whih is prmeterize s fies in this work. Note tht though prior reserh hs resse inversion with soure/sink effets [14][16], this topi is eyon the sope of the urrent stuy n will e left for nother tretment. To rive stey-stte flow through the quifer, no-flow ounries re speifie to the moel top n ottom ounries, while onstnt hyruli hes re speifie on the left n right sies of the moel, i.e., feet n feet, respetively (Fig. 1). The mp of fies K is opte from [18], onsisting of 1 pixels in oth the horizontl n vertil iretions. The fies (true) onutivities were set to K 1 = 1 ft/yr (silty sn) n K 2 = 1 ft/yr (len sn) oring to lithology type [19]. Note tht the horizontl fies orreltion rnge of the len sn (light grey regions in Fig. 1) is pproximtely the lterl omin prolem size, i.e., 1 pixels. The forwr moel is isretize with uniform fine gri (N x = 5 n N z = 5; eh fies pixel is thus represente y 5 5 gri ells) n is solve with the finite ifferene metho (FDM) with MODFLOW [2]. (The forwr moel is lso referre to herein s the FDM.) Mss lne reports from MOD- FLOW suggest negligile errors on the orer of 1 7, thus the simulte flow fiel is onsiere error-free.
6 Dongong Wng, Ye Zhng GTM With Smple Hyruli He Lotions (12Wells) he lotion 2 2 GTM with Smple Flux Lotions (12Wells) flux lotion 4 4 Rows 6 Rows 6 8 8 1 2 4 6 8 1 Columns 1 2 4 6 8 1 Columns Fig. 2 Lotions of 12 smpling wells ple uniformly ross the quifer trnset. Along eh well, he n flux smpling points re shown. The referene fies mp (1 y 1 pixels) is shown in the kgroun. To provie the mesurements for inversion, set of syntheti oservtion wells, eh with with of one FDM gri ell (1/5 pixel with in the fies mp), ws smple for fies, hyruli hes, n point-sle Dry fluxes. Fies ws smple long the full length of eh well, representing orehole mesurements. Hyruli he n flux smplings were me t isrete lotions orresponing to multilevel onfigurtion, e.g., 1 hyruli he n 5 flux mesurements per well. A shemti igrm with 12 uniformly spe smpling wells is shown in Fig. 2. Moreover, if the lterl omin size is use to pproximte the horizontl fies orreltion rnge (λ H ), the lterl well sping of Fig. 2 is thus 1/12 of λ H. As prt of the nlysis on smpling ensity, the numer of smpling wells will e reue, i.e., 6 wells (well sping is 1/6 of λ H ) n 3 wells (well sping λ H well sping. is 1/3 of λ H ). A horizontl smpling inex (SI) n e efine s: For the ifferent smpling shemes, SI = 12 (12 wells), 6 (6 wells), n 3 (3 wells). A lrger SI vlue suggests higher smpling ensity, with orresponingly smller well sping in reltion to the lterl hrteristi length of the fies. Furthermore, twie n four times s mny mesurements were smple when SI is 12, ompre to when SI=6 n SI=3, respetively. For eh smpling ensity, the referene moel fies istriution, the fies Ks, n flui flow BC will e lter reovere using the propose stohsti integrtion tehnique. 2.2 Stti Dt Integrtion Initor geosttistis is use to ssimilte the oserve fies long the welloles [21]. Given the oe fies t (e.g., for silty sn, 1 for len sn), experimentl vriogrms re ompute long the horizonl n vertil iretions. These vriogrms re fitte with n exponentil moel, whih is foun to est pture the fies orreltion struture. When 12 wells were smple, Fig. 3 (top row) presents the experimentl fies vriogrms for len sn n the fitte
Stohsti Dt Integrtion 7 moels, whih re lso presente elow: γ( l ) = {.35 +.24[1 e ( lx 6 ) ].35 +.24[1 e ( lz 4 ) ] (2) where γ is vriogrm n l is lg istne in pixels (l x is horizontl lg; l z is vertil lg). In fitting the moel, geometri nisotropy is ssume, thus the moele horizontl sill (not rehe in Fig. 3; top row) is the sme s the moele vertil sill. The fitte horizontl fies orreltion rnge is 15 pixels, whih is greter thn the fitte vertil fies orreltion rnge ( 15 pixels). Both prmeters re onsistent with the verge lterl extent n thikness of strtifition s exhiite y len sn (Fig. 1). Given the fitte vriogrm moels n the smple initor histogrms, Sequentil Initor Simultion (SIS) ws use to generte 1 fies reliztions, whih were onitione to the oserve wellore fies (Fig. 4). Eh fies reliztion hs the sme resolution s the originl fies mp, i.e., 1 1 pixels. When well ensity is hnge, the ove nlysis, inluing oth vriogrm moeling n SIS, is repete. Fig. 3 lso presents the vriogrm moels fitte when fewer wells were smple: not only will the fitte moels e less urte, the numer of onitioning fies t for SIS is signifintly reue. As result, s the numer of wells is reue, the SIS reliztion of the fies hs eome less urte (Fig. 4). For given smpling ensity, 1 fies reliztions will e rete. Using iret inversion, 1 flow fiels n 1 sets of fies K vlues will e etermine, one set for eh reliztion. To rive inversion, ynmi oservtions (i.e., hyrogeologil t) will e smple from the sme wells in the forwr moel.
8 Dongong Wng, Ye Zhng horizontl vriogrm (12 wells) vertil vriogrm (12 wells) experimentl vriogrm.2.1 1 2 3 4 5 lg istne horizontl vriogrm (6 wells) experimentl vriogrm.2.1 1 2 3 4 5 lg istne vertil vriogrm (6 wells) experimentl vriogrm experimentl vriogrm.2.1 1 2 3 4 5 lg istne.2.15.1.5 horizontl vriogrm (3 wells) 1 2 3 4 5 lg istne experimentl vriogrm experimentl vriogrm.2.1.2.1 1 2 3 4 5 lg istne vertil vriogrm (3 wells) 1 2 3 4 5 lg istne Fig. 3 Vriogrm nlysis of the len sn fies when ifferent numer of wells ws smple. The otte lines enote the experimentl vriogrms n the smooth urves re the fitte exponentil moels. To eliminte ege effet, the mximum lg in oth horizontl n vertil iretions is hlf of the orresponing omin length sles, i.e., 5 pixels. 2.3 Dynmi Dt Integrtion For given fies reliztion, whih elinetes the K istriution, this reserh opte the iret metho of [5] to estimte fies K vlues n he istriution throughout the omin. To pply the iret metho, set of funmentl solutions of inversion is fitte lolly to the oserve ynmi t, while flow ontinuity is honore over ll inversion gri ells. The funmentl solutions of inversion re erive y solving the stey-stte grounwter flow eqution to otin set of lol nlytil solutions ssuming tht lol K of suomin or single inversion gri ell is homogeneous. However, unlike [5], whose funmentl solutions yiele only single K vlue (i.e., rtios etween fies Ks were ssume known s set of prior informtion onstrints for inversion), the funmentl so-
Stohsti Dt Integrtion 9 SIS Reliztion (12 Wells) 2 Rows 4 6 8 1 2 4 6 8 1 Columns SIS Reliztion (6 Wells) 2 Rows 4 6 8 1 2 4 6 8 1 Columns SIS Reliztion (3 Wells) 2 Rows 4 6 8 1 2 4 6 8 1 Columns Fig. 4 SIS reliztions given 12, 6, n 3 smpling wells. For eh smpling ensity, one fies reliztion (out of 1 reliztions) is shown.
1 Dongong Wng, Ye Zhng lutions hve een moifie to llow the simultneous estimtion of oth Ks of the referene moel (this pproh is extenle to ny numer of fies): h(x, z) = + 1 x + 2 z + 3 xz + 4 (x 2 z 2 ) q x (x, z) = K( 1 + 3 z + 2 4 x) q z (x, z) = K( 1 + 3 x 2 4 z) (x, z) Ω i (3) where h enotes the pproximte hyruli he, ( q x, q z ) enote the pproximte grounwter fluxes, i (i =,..., 4) enote set of oeffiients tht lolly efine these pproximte solutions, K is lol hyruli onutivity: K (K 1, K 2,...) of the fies, n Ω i is suomin of the prolem, here orresponing to n inversion gri ell. The ontinuity equtions, whih penlize the mismth etween the funmentl solutions t the interfe etween jent inversion gri ells, n e written s: δ(p j (x j, z j ) ɛ)(k 1 h(k) (x, z) K 1 h(l) (x, z)) =, (k,l) K 1 δ(p j (x j, z j ) ɛ)(k 2 h(k) (x, z) K 2 h(l) (x, z)) =, (k,l) K 2 δ(p j (x j, z j ) ɛ)(k m h(k) (x, z) K m h(l) (x, z)) =, (k) K 1, (l) K 2, m (1, 2) δ(p j (x j, z j ) ɛ)( q n (k) (x, z) q n (l) (x, z)) =, K (k) K (l) δ(p j (x j, z j ) ɛ)( q (k) t (x, z) q (l) t (x, z)) =, K (k) = K (l) δ(p j (x j, z j ) ɛ)( q n (k) (x, z) q n (l) (x, z)) =, K (k) = K (l) (4) where p j (x j, z j ) enotes the jth ollotion point, whih lies on the interfe etween gri ells (k) n (l), q n is norml flux t p j, q t is tngentil flux t p j, δ(p j (x j, z j ) ɛ) is Dir elt weighting funtion [5] tht smples the mismth etween the funmentl solutions t p j (x j, z j ). The reltion etween ( q n, q t ) n ( q x, q z ) n e etermine using the ngles etween the interfe n the glol oorinte xes. Inversion further stisfies set of t onstrints whih n e written s: δ(p t ɛ)(k m h(k) (x t, z t ) K m h o (x t, z t )) = m (1, 2) δ(p t ɛ)( q (k) n (x t, z t ) q o n(x t, z t )) = δ(p t ɛ)( q (k) t (x t, z t ) q o t (x t, z t )) = (5) where δ(p t ɛ) is the Dir elt weighting funtion, whih reflets onfiene in the oserve t (e.g., it n e inversely proportionl to the mesurement error vrine), (x t, z t ) enotes the lotion where n oserve tum ws smple, n h o, qn, o qt o re the oservtions, K m enotes the onutivity of the fies whih ontins the oservtions. Flux mesurements re use here to provie flow rte relte informtion for inversion, euse onutivity nnot e uniquely ientifie from hyruli he oservtions lone. If susurfe flow rte mesurements re ville, however, the flux onitioning equtions n e integrte to enfore onitioning y flow rtes [5]. As ws isusse in [15], mesurements of in-situ fluxes n flow rtes n e me with vriety of orehole n wter lne tehniques.
Stohsti Dt Integrtion 11 Bse on the propose funmentl solutions [Eqn. (3)], y ssemling the ontinuity [Eqn. (4)] n the t equtions [Eqn. (5)], system of liner equtions ws evelope: A x (6) where A is the oeffiient mtrix, is the right hn sie vetor, n x is the solution vetor, where x = [K 1, K 2, (k), (k) 1, (k) 2, (k) 3, (k) 4,...], k = 1, 2,..., M (numer of ells in the inversion gri). Previous reserh [14] suggests tht if ynmi oservtions re too few resulting in uner-etermine inversion systems of equtions, stility of inversion n suffer. In this work, suffiient numer of he n flux oservtions ws smple (Fig. 2), thus Eqn. (6) is over-etermine n is solve with lest-squres minimiztion tehnique s implemente y the LSQR lgorithm [22]. For smller prolems reporte in previous reserh, LSQR ws foun to e n effiient solver. However, prolems inverte here re lrger with mny more unknowns, for whih the spee of onvergene ws foun to e sensitive to the onition numer of A. To improve mtrix onitioning, preonitioning tehniques inluing mtrix sling n Gussin Noise Perturtion were use efore the mtrix solve [23]. Depening on the prolem, pre-onitioning hs improve the solver spee y up to 1 times (generlly, greter speeup is hieve with lrger mtries), without suffering ill-effets in inversion ury. After pre-onitioning, inversion of single reliztion with the SIS gri will tke 1 min. The hyruli he fiel n then e reovere pieewise from the oeffiients efining the funmentl solutions t eh gri ell; from the estimte Ks, Dry flux fiel n e similrly reovere. The hyruli he BC n e otine y smpling the pproprite h(x, z) t the ounry lotions. Similrly, flux BC n e otine y smpling the pproprite q x (x, z) n q z (x, z) t the ounries. In this work, the inverte BC re presente s hyruli he vlues. Beuse forwr moel is not solve, the iret metho n employ flexile gris with vrious prmeteriztions for representing onutivity. In generl, the SIS-generte fies reliztions exhiit smll-sle rtifts whih le to pthiness in the simulte fies istriutions (Fig. 4). This my in turn impt the qulity of inversion, s greter egree of pthiness will le to fewer ontinuity equtions (i.e., the numer of flux ontinuity onstrints will e reue). Three K prmeteriztions re teste in this work: (1) inversion iretly opts the noisy SIS prmeteriztion with n ientil gri (N x = 1, N z = 1); (2) inversion opts the sme SIS gri (N x = 1, N z = 1), ut the SIS-simulte fies re smoothe y simulte nneling (SA)[24]; (3) inversion opts orsene gri (N x = 5, N z = 5) whose fies prmeteriztion is rete y upsling the SA fiel. For ese of referene, these prmeteriztions re lele s SIS, SA, n orsene inversion gris (exmple of eh gri n e seen in [25]). For the ltter two prmeteriztions, SIS reliztions re first smoothe, n then orsene, efore ynmi integrtion is rrie out to estimte the K n flow fiel ensemles. For given smpling ensity, y ompring the inversion outomes using ll gris, the impt of inverse K prmeteriztion on the qulity of inversion is ssesse. Moreover, if reue resolution in the prmeteriztion oes not signifintly egre inversion performne (i.e., ury n unertinty), lower resolutions will e preferre ue to their greter omputtionl effiieny.
12 Dongong Wng, Ye Zhng 2.4 Unertinty Quntifition Unertinty in the propose stohsti integrtion erives from oth stti n ynmi t unertinty, whih omes from limite orehole smpling n mesurement errors (if impose). For fies smpling long oreholes, no mesurement errors re impose, i.e., oserve fies type is ssume error-free. For smpling the ynmi t (hes, fluxes) from the sme wells, rnom mesurement errors re impose. Beuse the FDM is the referene moel, mesurements smple from this moel re error-free, tht is, true hes or true fluxes. To impose errors, the true hes re orrupte y rnom noises: h m = h F DM ± δh, where h m is mesure he provie to inversion, h F DM is true he, n δh is hyruli he mesurement error. The highest δh is ±1% of the totl hyruli he vrition in the forwr moel, with n solute error up to ±1 feet. This is resonle onsiering tht moern mesurement tehniques n etermine wter levels with preision s fine s 1 m [26]. Flux mesurements, though lso menle to errors, re ssume error-free. The effet of imposing errors on fluxes is similr to imposing errors on hes, while imposing oth errors t the sme time n le to inversion outomes tht re iffiult to interpret, e.g., positive error in oserve he grient n e nele y negtive error in nery flux mesurement. Given given fies prmeteriztion (SIS, SA, or orsene), 1 inversion systems of equtions were ssemle n solve, resulting in set of ensemle solutions inluing the inverte Ks, flow fiels (i.e., hes n fluxes), n BCs. The ury of eh solution n e ssesse y ompring the estimte prmeters n flow fiels to those of the referene moel. Beuse given flow fiel n e uniquely etermine from K istriution n BC, only the estimte Ks n the re ompre to those of the forwr moel. The inverte ensemle flow fiels will not e presente. Unertinty in the inverte onutivity of eh fies is evlute y set of ensemle error sttistis or performne metris: E[K inv ] K true ε K = 1% σ K = K true K inv mx K inv min 4K true 1% where K inv is the inverte fies onutivity, K true is the orresponing true onutivity, n E[ ] is expettion, tken here s rithmeti men. E[K inv ] is thus the ensemle men of the inverte fies onutivity. Kmx inv n Kmin inv enote the mximum n minimum inverte fies onutivity from the ensemle. To evlute the unertinty in the inverte BC, two performne metris were opte: n ε BC = (1/n) i=1 σ BC = n (1/n) i=1 (E[h inv i ( (h mx i ] h true h true i h min 4h true i i ) i ) 1% ) 2 1% where h inv i is the inverte hyruli he t ounry ell lotion i (n is the numer of ells long the omin ounry), h true i represents the true he t the (7) (8)
Stohsti Dt Integrtion 13 sme lotion, h mx i n h min i enote the mximum n minimum inverte hes t lotion i from the ensemle solution, respetively, n E[h inv i ] is the ensemle men of the reovere he t lotion i. Note tht y summing n verging over ll ounry ells, ε BC n σ BC represent the verge BC estimtion error n spre surrouning the ensemle men. In the following, inversion ury is efine y ε K n ε BC : higher vlues inite is in the ensemle men, thus poorer ury [4]. On the other hn, σ K n σ BC reflet the spre of the ensemle from the true prmeters n flow fiel, respetively, n re thus linke to the preision of inversion [4]. Higher preision is reflete y lower σ K n σ BC, n thus lower estimtion unertinty. The qulity of inversion is onsiere high if ε n σ re oth smll for onutivity estimtion n BC reovery. 3 Results In the unertinty stuy, the ury n preision of the ensemle inverse solutions re exmine when mesurement quntity, qulity, n inverse prmeteriztion re vrie. (1) Dt quntity ws vrie y reuing the numer of smpling wells from 12 to 6, n then to 3 wells, from whih stti n ynmi t were smple from the FDM. The oserve hyruli hes were then sujet to inresing mesurement errors (±1, ±2, ±5, ±1%), while inverse prmeteriztion opte the SIS (noisy) fies reliztions with 1 1 gri ells. (2) From the ove, t qulity ws exmine losely to evlute the effet of mesurement errors on inversion. (3) Using error-free mesurements from the 12 wells, inverse prmeteriztion ws then vrie, i.e., ynmi integrtion ws performe using the sme oserve hes n fluxes, ut with fies ptterns emoie y the SIS, SA, n orsene gris. (4) A o-effet stuy ws rrie out vrying two ftors t the sme time. In the following, units of the relevnt quntities re: K in ft/yr (1 ft/yr =.35 m/yr), q in ft/yr (1 ft/yr =.35 m/yr), h in ft (1 ft =.35 m), flow rte in ft 3 /yr (1 ft 3 /yr =.28 m 3 /yr). Alterntively, onsistent set of units n e ssume n ll unit informtion n e remove [27]. 3.1 Dt Quntity When 12 wells were smple from the forwr moel, stti n ynmi t from these wells were provie to the propose t integrtion: (1) given the oserve fies t the wells, vriogrm moel ws fitte, with whih 1 fies reliztions were generte with SIS; (2) for eh reliztion, ynmi t, smple t the sme wells, were inverte to otin fies Ks n the ssoite flow fiel; (3) for ll reliztions, n ensemle of inversion outomes ws generte, n Eqn.(7) n Eqn.(8) were use to ssess the qulity of inversion. The ynmi t were initilly error-free, efore inresing mesurement errors ±1, ±2, ±5, ±1% were impose (for the 5 levels of mesurement errors teste, 5 inversions were performe). When the smpling ensity ws reue to 6, n then to 3, the ove steps were repete (n itionl 1, inversions were performe).
14 Dongong Wng, Ye Zhng Reliztions 6 4 2 12 wells 6 wells 3 wells Inverte K1 Histogrm (SIS Error Free).6.7.8.9 1 1.1 1.2 1.3 Vlue Inverte K2 Histogrm (SIS Error Free) Reliztions 6 4 2 12 wells 6 wells 3 wells 4 6 8 1 12 14 Vlue Fig. 5 Inverte istriutions of K1 n K2 otine with ifferent smpling ensities when mesurements were error-free. For eh fies, rrow points to the true onutivity vlue. Inverse prmeteriztion is SIS. The results for onutivity estimtion re summrize in Tle 1. Severl oservtions n e me: (1) for given mgnitue of the mesurement error, the qulity of inversion is highest when 12 wells were smple n lowest when 3 wells were smple (Fig. 5). Clerly, when smpling ensity is reue, fewer stti n ynmi t re ville for the integrtion n, oringly, inversion outome worsens. (2) for given smpling ensity, the qulity of inversion worsens with inresing mesurement errors (Tle 1). When the oserve hes were impose with ±1% errors, onutivity estimtion error eomes very high suggesting strong ises: when 12 wells were smple, ε K is 76% for K1 n 27% for K2; when 6 wells were smple, ε K is 12% for K1 n 4% for K2; when 3 wells were smple, ε K is 156% for K1 n 34% for K2. When 6 n 3 wells were smple, E[K1] even eomes negtive. This ours when he grients yiel the wrong sign ue to noisy mesurements, ut the orret flux mesurements were still use in inversion. (3) n optiml smpling ensity for urte K estimtion epens on the mgnitue of the mesurement error (tht is expete in fiel prolem) n n eptle level of estimtion error (tht is user efine). For exmple, if
Stohsti Dt Integrtion 15 ±2% is onsiere relisti mesurement error (solute mesurement error is ±2 ft), 3 wells (SI=3) will provie suffiient ury for inversion if K estimtion error of 12% is onsiere eptle. However, for the sme level of mesurement error, 6 wells (SI=6) will e neee if K estimtion error of 5% is onsiere eptle. (4) n optiml smpling ensity for preise K estimtion lso epens on the mgnitue of the mesurement error n n eptle level of estimtion error. For exmple, if ±2% is onsiere relisti mesurement error, 6 wells will provie suffiient preision if σ K of 7% is onsiere eptle. However, 12 wells will e neee if σ K of 5% is onsiere eptle. The results for BC estimtion re summrize in Tle 2, where ensemle sttistis re ompile for 4 ounry segments (Fig. 1):,,, n. Along the sme segment, the inverte ounry hes re ompre to the true hes (Fig. 6 presents results given error-free mesurements). Similr to K estimtion, for given mesurement error, the qulity of BC estimtion (i.e., ury n preision) is highest when 12 wells were smple n lowest when 3 wells were smple. For given smpling ensity, the qulity of inversion worsens with inresing mesurement errors (more on this lter). For BC estimtion, the optiml numer of smpling wells lso epens on the mgnitue of the mesurement error n n eptle level of BC estimtion error. For eh ounry segment, istint ehvior in the ensemle sttistis is lso oserve. For exmple, he reovery long ifferent segments is sensitive to well ensity to ifferent egrees. Both ury n preision of he reovery long the inflow ounry ( ) re not signifintly ffete y the eresing well ensity (Fig. 6). However, inversion long the outflow ounry ( ) egres ppreily when the numer of smpling well is reue. Inversion qulity is sptilly non-uniform, espite the ft tht the smpling wells re uniform. This suggests the importne of ientifying optiml smpling lotions. To unerstn this issue, future work will omine inversion with sensitivity nlysis [4][28]. For given smpling ensity, hyruli he reovery long the inflow ( ) n outflow ( ) ounries re generlly less urte n preise, when ompre to he reovery long the moel top n ottom ounries (, ). This n e expline y extrpoltion of the funmentl solutions in the moel omins etween the inflow ounry n the first smpling well, n etween the outflow ounry n the lst smpling well (Fig. 2). Beuse no mesurements were me there, the inverse solution, whih is onitione t the well lotions, is extrpolte towrs the ounries. Inversion qulity long moel top n ottom is higher, euse oth ounries re losely onitione y well t (Fig. 2). This suggests tht (1) inversion omin shoul e efine y the well lotions to reue extrpoltion errors, n (2) smpling long n, in effet proviing (prtil) BC to inversion, will likely improve inversion t these ounry lotions. 3.2 Dt Qulity Results from the previous setion re reexmine to ssess the effet of mesurement errors on inversion. Agin, s reporte in Tle 1 n Tle 2, qulity of the inverte Ks n BCs egres with inresing he mesurement errors n eresing smpling ensity. Fig. 7 presents the istriutions of the estimte K1
16 Dongong Wng, Ye Zhng Tle 1 Inversion ury n preision for the estimte fies onutivities. SIS gri 12 wells 6 wells 3 wells Aury Preision Aury Preision Aury Preision K1 K2 K1 K2 K1 K2 K1 K2 K1 K2 K1 K2 Error ε(%) ε(%) σ(%) σ(%) ε(%) ε(%) σ(%) σ(%) ε(%) ε(%) σ(%) σ(%)..3 3.8 1.2 15. 4.2 7. 1.9 18. 8.3 1.3 1.7 ±1% 3. 1.3 4.3 1.5 13. 3.6 5.8 2. 17. 8.4 11.5 1.9 ±2% 7. 2.9 4.8 1.5 4. 1.6 6.3 1.8 12. 7.1 13.8 2.2 ±5% 29. 1.3 6.5 2.2 42. 12.1 8.3 3.4 42. 4.7 24.5 3.6 ±1% 76. 27.4 7.3 2.5 12. 4. 15.5 4.4 156. 33.7 31.5 4.1 Error-free Oserve Dt SIS SA Corsene Aury Preision Aury Preision Aury Preision K1 K2 K1 K2 K1 K2 K1 K2 K1 K2 K1 K2 Wells ε(%) ε(%) σ(%) σ(%) ε(%) ε(%) σ(%) σ(%) ε(%) ε(%) σ(%) σ(%) 12..3 3.8 1.2 3..2 5.75 1.65 6. 3.9 7.5 2.8 6 15. 4.2 7. 1.875 27. 6.9 8.75 2.725 3..6 8. 1.725 3 18. 8.3 1.25 1.725 6. 4.2 21. 3.45 34. 21.6 2.75 5.75 12 wells (Error-free) Aury Preision setion - setion - setion - setion - setion - setion - setion - setion - Gris ε(%) ε(%) ε(%) ε(%) σ(%) σ(%) σ(%) σ(%) SIS 2.17.924.335.44 1.52 1.821 1.11.772 SA 1.63.84.165.18 1.6 1.948 1.145.874 Corsene 3.39.352.6.772.89.95.79.669
Stohsti Dt Integrtion 17 Tle 2 Inversion ury n preision for the estimte hyruli he BC. The ol lphets enote ifferent ounry segments (Fig. 1). 12 wells (SIS gri) Aury Preision setion - setion - setion - setion - setion - setion - setion - setion - Error ε(%) ε(%) ε(%) ε(%) σ(%) σ(%) σ(%) σ(%) 2.17.924.335.44 1.52 1.821 1.114.772 ±1% 2.187.792.26.8 1.568 1.91 1..733 ±2% 2.183.66.195.6 1.632 1.98 1.488.721 ±5% 2.2.26.55.28 1.899 2.295 2.64.973 ±1% 1.273.412.2.452 2.958 3.627 3.111 1.648 6 wells (SIS gri) Aury Preision setion - setion - setion - setion - setion - setion - setion - setion - Error ε(%) ε(%) ε(%) ε(%) σ(%) σ(%) σ(%) σ(%) 3.433 1.396.565.736 1.591 1.96 1.263.794 ±1% 2.79.8.12.792 1.57 1.891 1.441.834 ±2% 2.4..285.844 1.793 2.164 2.19.868 ±5%.77 1.62 1.27.976 2.443 3.47 3.884.948 ±1% 6.79 4.74 2. 1.184 4.526 6.689 7.616 1.588 3 wells (SIS gri) Aury Preision setion - setion - setion - setion - setion - setion - setion - setion - Error ε(%) ε(%) ε(%) ε(%) σ(%) σ(%) σ(%) σ(%) 3.647 3.692.855.724 1.462 1.759 1.34 1.796 ±1% 3.893 3.828.85.4 1.431 1.716 1. 1.642 ±2% 4.57 3.956.925.4 1.46 1.675 1.365 1.589 ±5% 4.4 4.272 1.635 1.6 1.922 1.666 1.74 1.797 ±1% 2.383 4.6 4.295 5.516 3.598 2.782 2.84 2.863
18 Dongong Wng, Ye Zhng Hyruli Hes Hyruli Hes Hyruli Hes Oserve hes vs. Reovere hes (Error free 12Wells SIS) 5 1 15 35 4 Bounry Cell Numer Oserve hes vs. Reovere hes (Error free 6Wells SIS) 5 1 15 35 4 Bounry Cell Numer Oserve hes vs. Reovere hes (Error free 3Wells SIS) 5 1 15 35 4 Bounry Cell Numer Fig. 6 Inverte istriutions of ounry hes otine with ifferent smpling ensities when mesurements were error-free. The re lso shown. The ol lphets enote the ounry lotions (Fig. 1).
Stohsti Dt Integrtion 19 n K2 uner inresing mesurement errors. Compre to K2 (len sn), estimtion of K1 (silty sn) egres more quikly with inresing errors. For the mesurement errors of, ±1, ±2, ±5, n ±1% of the totl he vrition, ε K for K1 is., 3., 7., 29., 76.%, n for K2, it is.3, 1.3, 2.9, 1.3, 27.4%; σ for K1 is 3.8, 4.3, 4.8, 6.5, 7.3, n for K2, it is 1.2, 1.5, 1.5, 2.2, 2.5%. When mesurement errors re low, inversion performne in estimting oth onutivities is similr, ut when errors grow higher, K2 estimtion is more urte n preise. Thus, onutivity of low-k fies my eome more iffiult to ientify when signifint errors exist in mesurements. This effet my e ttriute to the ft tht flow in low-k fies is more sensitive to vrition in the hyruli he n its sptil grient. The inverte ounry hes uner inresing mesurement errors re presente in Fig. 8. When 12 wells were smple, inresing mesurement errors only les to slight egrtion of the inverte ounry hes. For exmple, when mesurement errors re progressively inrese, slightly inrese σ BC is oserve long the inflow ounry ( ). When 6 wells were smple, this egrtion, Inverte K1 Histogrm (SIS 12wells Inresing Errors) Reliztions 6 4 2 error error ±1% error ±2% error ±5% error ±1% 1.5 1.5.5 1 1.5 2 Vlue Inverte K2 Histogrm (SIS 12wells Inresing Errors) Reliztions 6 4 2 error error ±1% error ±2% error ±5% error ±1% 4 6 8 1 12 14 Vlue Fig. 7 Distriution of the estimte K for eh fies uner inresing mesurement errors. (top) inverte K1 histogrms; (ottom) inverte K2 histogrms. Arrows point to the true vlues.
2 Dongong Wng, Ye Zhng 5 1 15 Bounry Cell Numer 35 4 5 1 15 Bounry Cell Numer 35 4 5 1 15 Bounry Cell Numer 35 4 5 1 15 Bounry Cell Numer 35 4 5 1 15 Bounry Cell Numer 35 4 35 5 1 15 Bounry Cell Numer 35 4 1 15 Bounry Cell Numer 35 4 5 1 15 Bounry Cell Numer 35 35 4 5 1 15 Bounry Cell Numer 35 4 5 1 15 Bounry Cell Numer 35 4 Oserve hes vs. Reovere hes (Error±5% 3Wells SIS) 4 15 Bounry Cell Numer 1 Oserve hes vs. Reovere hes (Error±2% 3Wells SIS) 5 5 Oserve hes vs. Reovere hes (Error±1% 3Wells SIS) 4 Oserve hes vs. Reovere hes (Error±1% 6Wells SIS) 15 Bounry Cell Numer Hyruli Hes Hyruli Hes 1 Oserve hes vs. Reovere hes (Error±1% 12Wells SIS) Oserve hes vs. Reovere hes (Error±5% 6Wells SIS) 4 Hyruli Hes Hyruli Hes 5 Oserve hes vs. Reovere hes (Error±5% 12Wells SIS) 35 Oserve hes vs. Reovere hes (Error±2% 6Wells SIS) Hyruli Hes Hyruli Hes 15 Bounry Cell Numer Oserve hes vs. Reovere hes (Error±2% 12Wells SIS) 1 Oserve hes vs. Reovere hes (Error±1% 6Wells SIS) 5 Oserve hes vs. Reovere hes (Error free 3Wells SIS) Hyruli Hes Hyruli Hes Hyruli Hes Oserve hes vs. Reovere hes (Error±1% 12Wells SIS) Hyruli Hes Hyruli Hes Oserve hes vs. Reovere hes (Error free 6Wells SIS) Hyruli Hes 5 1 15 Bounry Cell Numer 35 4 Oserve hes vs. Reovere hes (Error±1% 3Wells SIS) Hyruli Hes Hyruli Hes Hyruli Hes Oserve hes vs. Reovere hes (Error free 12Wells SIS) 5 1 15 Bounry Cell Numer 35 4 Fig. 8 Reovere ounry he ensemles uner eresing smpling ensity: (first olumn) 12 wells, (seon olumn) 6 wells, (thir olumn) 3 wells. In eh olumn, ounry hes re inverte uner inresing mesurement errors: (first row) %, (seon row) ±1%, (thir row) ±2%, (fourth row) ±5%, (fifth row) ±1%. in terms of inversion ury n preision, is more pronoune. When 3 wells were smple, similr egree of egrtion s tht of the 6 wells is oserve. Overll, for the prolem investigte here, BC reovery is not very sensitive to the impose rnom mesurement errors when suffiient numer of wells (12 wells) were smple. On the other hn, for prtil sitution where mesurement errors n e quntifie (e.g., ±2%), optiml smple sping n gin e ientifie y setting eptle tolernes for εbc n σbc. 3.3 Inverse Prmeteriztion Given error-free mesurements from 12 wells, inversion outome is presente when fies is prmeterize lterntively with SIS, SA, n the orsene gris (Fig. 9, Fig. 1). When the SIS gri is use, men onutivity preition is extremely lose to the true vlue, while σk remins very smll. When the SA gri is use, oth ury n preision of the inversion suffer slightly. When the orsene gri is use, inversion qulity is omprly the worst. The verge CPU time for solving n inverse prolem ws 55 s (SIS), 45 s (SA), n 33 s (orsene), thus speeup ue to heterogeneity smoothing n further orsening is roun 18% n
Stohsti Dt Integrtion 21 4%, respetively. When BC reovery is ompre, however, inversion performne oes not vry signifintly with the vrition in prmeteriztions (Fig. 1). The ove finings suggest tht K estimtion is sensitive to its prmeteriztion (i.e., smoothness in heterogeneity), while hyruli he reovery is not. It is wellknown tht hyruli he preition is not sensitive to smll-sle K vrition euse the grounwter flow eqution ts s filter: the high wve numer omponents orresponing to smll-sle heterogeneity re generlly filtere out. Moreover, if 1% K estimtion error is onsiere eptle, inversion using the orsene gri is optiml, s the spee of solver with this smller gri is fster y pproximtely 4% (ompre to the SIS gri) n 27% (ompre to the SA gri). 3.4 A Co-Effet Stuy In the previous setions, inversion ury n preision were nlyze seprtely ginst hnging oservtion quntity, qulity, n inverse prmeteriztion. To Reliztions 6 4 2 SIS SA Corsening Inverte K1 Histogrm (12wells Error Free).6.7.8.9 1 1.1 1.2 1.3 Vlue Inverte K2 Histogrm (12wells Error Free) Reliztions 6 4 2 SIS SA Corsening 4 6 8 1 12 14 Vlue Fig. 9 Distriution of the estimte K for eh fies uner ifferent inverse prmeteriztions. (top) inverte K1 histogrms; (ottom) inverte K2 histogrms. Arrows point to the true vlues.
22 Dongong Wng, Ye Zhng Hyruli Hes Oserve hes vs. Reovere hes (Error free 12Wells SIS) 5 1 15 35 4 Bounry Cell Numer Hyruli Hes Oserve hes vs. Reovere hes (Error free 12Wells SA) 5 1 15 35 4 Bounry Cell Numer Hyruli Hes Oserve hes vs. Reovere hes (Error free 12Wells Corsening) 5 1 15 35 4 Bounry Cell Numer Fig. 1 Reovere ensemles of ounry hes long ---- uner ifferent inverse prmeteriztions: (top) SIS, (mile) SA, (ottom) orsene gri. unerstn the reltive importne of eh ftor to the performne of inversion, o-effet nlysis ws onute. To etermine the reltive rnking, three seprte two-imensionl experiments vrying two ftors t time were esigne, i.e., t quntity versus prmeteriztion, t quntity versus t qulity, t qulity versus prmeteriztion. For exmple, when the o-effet of t quntity n prmeteriztion is of interest, three ifferent prmeteriztions were employe for given smpling ensity, resulting in 3 sets of stohsti inversions (i.e., ensemle solutions). For the 3 smpling ensities (12 versus 6 versus 3 wells), 9 inversion equtions were thus solve. The inversion outomes ensemle mens n stnr evition of K or BC estimtion, were then nlyze in light of the joint vrition of the two ftors. For given smpling ensity, this nlysis revels tht prmeteriztion with the orsene gri lwys les to less ury n lower preision in inversion, s expete. The sme n e si when fewer wells re
Stohsti Dt Integrtion 23 provie to inversion uner fixe prmeteriztion. To unerstn the reltive importne of eh ftor on inversion, mtries of the ensemle outomes n e rete n nlyze [25]. For the rnges of teste vrition in t quntity n inverse prmeteriztion, t quntity is foun to the ominnt ftor tht influenes oth the ury n preision of K n BC estimtions. Similr two-imensionl experiments were rrie out to evlute the reltive importne of t quntity versus t qulity, n t qulity versus prmeteriztion (n itionl 1,8 ensemle solutions were otine n nlyze). Detil out these experiments n the o-effet stuy is provie in [25], n will not e presente. Here, we summrize the insights gine from this nlysis. For the rnges of teste vrition in t quntity, qulity, n inverse prmeteriztion, t quntity plys the ominnt role in etermining the ury n preision of the inverte Ks. Dt qulity is importnt to the ury of K inversion, ut hs limite influenes on its preision. Compre to t quntity n qulity, the impt of inverse prmeteriztion on the qulity of inversion is generlly smller, suggesting the viility of using upsle prmeter fiels with orse gris to hieve greter omputtion effiieny. Tht is, resolution-ury tre-off exists for prmeter estimtion. For BC n thus flow fiel estimtion, the most importnt ftor tht influenes the qulity of inversion is t quntity, while t qulity n heterogeneity resolution hve lesser influenes. An, regrless of the omintions exmine, regions of the inversion omin ner the inflow n outflow ounries, when ompre to the interior regions, lwys orrespon to lower inversion ury n preision. This is result of extrpoltion errors s no mesurements exist long these ounry segments. This is onsistent with the erlier results with eterministi inversion [16]. 4 Conlusions In this reserh, stohsti susurfe t integrtion tehnique is propose y omining geosttistil simultion with iret inverse metho to filitte prmeter n flow fiel estimtion in stey-stte quifer inversion while quntifying the ssoite estimtion unertinty. The oservtion t re otine from set of smpling wells, n inlue stti t (i.e., fies types) n ynmi t (i.e., hyruli hes n flux mesurements). The stti t re ssume error-free, while inresing mesurement errors re impose onto the oserve hyruli hes. First, sequentil initor simultion is use to integrte the stti t, whih les to set of orrelte fies reliztions with ifferent hyruli onutivity istriutions. This fies ensemle is then provie to iret inversion whih utilizes the ynmi t smple from the sme wells to estimte set of ensemle prmeters n flow fiels. With these ensemle inversion outomes, the men (expete) prmeters n flow fiels n their estimtion unertinty n then e etermine. Beuse the ove stohsti integrtion requires ensemle inversions on mny fies reliztions, omputtionl effiieny is of interest. Vi smoothing n gri orsening, fies prmeteriztion y SIS n e moifie y reuing the resolution of the fies moel prior to inversion. The smoothing of the fies n improve mtrix onitioning in inversion, while smoothing omine
24 Dongong Wng, Ye Zhng with gri orsening n le to reue eqution size. Both pprohes n reue the omputtionl time neee to solve the inversion systems of equtions. To test the propose integrtion tehnique, referene forwr moel provies oth the fies hrteriztion n ynmi mesurements t inresing smpling ensities (i.e., t quntity) n mesurement errors (i.e., t qulity). Beuse flow fiel n e uniquely etermine if onutivities n BC re known, the estimte BC ensemle is exmine in lieu of the flow fiels. Unertinty in the estimte onutivities n BC is quntifie ginst the referene moel to evlute the qulity of inversion. Results suggest tht for the rnges of teste vrition in t quntity, qulity, n inverse prmeteriztion, (1) t quntity hs the gretest impt on oth inversion ury n inversion preision; (2) t qulity impts inversion ury; (3) inverse prmeteriztion hs the wekest influene on inversion s long s the overll fies pttern is pture (i.e., suffiient t quntity). A lne n thus e hieve etween prmeteriztion, omputtionl effiieny, n inversion performne. Moreover, for the heterogeneity investigte herein, y efining n eptle mrgin of unertinty for either onutivity or flow fiel estimtion, optiml well sping in reltion to the hrteristi length of heterogeneity n e etermine uner unknown quifer BC. Finlly, results of this stuy suggest tht inversion omin shoul e losely efine y the mesurement lotions in orer to minimize extrpoltion errors. Results n insights of this work n le to fiel implementtion of the propose stohsti integrtion tehnique for prolems with unknown informtion out quifer s ounry onitions. Along with prmeter estimtion, the integrtion tehnique n expliitly quntify ounry onitions inluing their men vlues n their unertinty, thus proviing powerful tool for hrterizing eep, t-poor environments whih re inresingly eing exploite for wste isposl (e.g., geologil ron storge n sequestrtion) s well s for energy proution (e.g., hyruli frking, geotherml, oil/gs) opertions. Using the integrtion tehnique, y ssuming resonle level of mesurement errors in the hyrogeologil t, well sping rnging from 1/3 to 1/6 of the lterl fies orreltion length n le to equte inversion outomes with eptle estimtion unertinties. Future work will exten the methos of this stuy to three-imensionl prolems, while stti t integrtion will inorporte not only iret orehole informtion ut other uxiliry t (e.g., seismi fies nlysis) tht n yiel itionl informtion of susurfe heterogeneity in etween wells. Aknowlegements This work is supporte y the University of Wyoming Shool of Energy Resoures Center for Funmentls of Susurfe Flow (WYDEQ49811ZHNG), n NSF CI- WATER (Cyerinfrstruture to Avne High Performne Wter Resoure Moeling), n NSF EPSCoR (EPS 12899). Referenes 1. Rihr L. Chmers, Jeffrey M. Yrus, n Kirk B. Hir. Petroleum Geosttistis for Nongeosttistiins. The Leing Ege, 19(5):474 479,. 2. C. Deutsh n P. Cokerhm. Prtil onsiertions in the ppliton of simulte nneling to stohsti simultion. Mthemtil Geology, 26(1):67 82, 1994.
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