6 Oil Sale Symposium 16-19 Ocober, 006 Te forward and inverse problems in oil sale modeling Ne-Zeng Sun (email: nezeng@ucla.edu) Civil & Environmenal Engineering Deparmen, UCLA, USA Alexander Y. Sun (email: alexsunda@gmail.com) Sunda Environmenal Tecnology, LLC, USA Brian Gallager (email: bjgla@sbcglobal.ne) Ecoonics Environmenal Scieniss, USA Absrac Tis paper presens a preliminary ouline for modeling e in-siu oil sale eaing process in e field scale. Te proposed model consiss of a number of coupled and igly nonlinear parial differenial equaions. Numerical meods for solving ese equaions are described sorly. Some parameers in e model mus be idenified by solving e inverse problem wi lab and field daa. Te paper gives an algorim for idenifying e process dependen parameers, suc as e porosiy and permeabiliy. Finally, e model reducion and reliabiliy problems are considered wi a new concep of consrucing objecive-oriened models. Inroducion Maemaical modeling is a very useful ool for oil sale developmen researc. A reliable model can be used no only for feasibiliy and mecanism sudies, bu also for opimal engineering design and process managemen. In recen years, new in-siu reoring ecnologies ave been repored (RAND, 005). To model an in-siu oil sale developmen process we need ree models: a dewaering model, a reoring model, and an environmenal recovering model. Te ending condiion of e firs model provides e iniial condiion for e second model, and en e ending condiion of e second model provides e iniial condiion for e ird model. Te dewaering model and e environmenal recovering model are relaively simple and ave been considered in groundwaer ydrology and environmenal engineering. Developing a reoring model, owever, is a very callenging opic because of is ig complexiy. Tis ind of model as no been sudied very well in peroleum engineering. A model a can simulae an in-siu reoring process may consis of: A se of mulipase flow equaions for e oil, waer and gas pases; A ea ranspor equaion in porous media; A deformaion equaion for oil sale; A se of convecion-dispersion-reacion equaions wen solvens are added. Tere are difficulies in consrucing suc a model. Firs, e governing equaions of e model are coupled and igly non-linear parial differenial equaions. Effecive meods for solving suc a complex sysem are needed. Second, mos parameers in e model are unnown. Tey may vary no only wi locaion bu also wi ime and sae variables, suc as emperaure and sauraion. Tird, e soluions and parameers are scale dependen. Teir values obained in e lab scale canno be ransferred direcly in e field scale. To overcome ese difficulies, we mus develop effecive numerical meods for solving bo e forward and inverse problems. Te former gives e resuls of model predicion, wile e laer gives e resuls of model calibraion. Wiou e 1
6 Oil Sale Symposium 16-19 Ocober, 006 daa colleced from field experimens for model calibraion, i is impossible o consruc a useful oil sale model. Before a model is used for design and managemen purposes of oil sale developmen, is reliabiliy mus be sudied. Tis paper is a preliminary guideline o e oil sale modeling. A disribued parameer model is presened based on e pysical process of in-siu eaing ecnologies. Numerical meods for solving e forward problem are described sorly. A coupled inverse problem is formulaed for parameer idenificaion and model calibraion. Our discussion is concenraed on e idenificaion of e inrinsic permeabiliy. During e eaing process, permeabiliy is a variable even for a omogeneous sale formaion and is dependen on e emperaure disribuion. Based on e recen sudy of e firs auor on consrucing objecive oriened models, e paper also discusses ow o reduce e complexiy of an oil sale model wile e accuracy requiremen of model applicaion can sill be guaraneed. A Proposed In-siu Oil Sale Heaing Model Afer e dewaering process is compleed and e eaing process is sared, fluid oil and gas will be generaed from e sale gradually. Mulipase flow is governed by a se of parial differenial equaions, one for eac pase: ( θρ ) ρ ( ρ ) ρ S r = P + g z + q μ (1) were e subscrip denoes a pase. Several differen immiscible fluid and gas pases may exis during e reoring process, suc as waer, vapor, reored oil, gas and oer compounds wi differen densiies and viscosiies. Le e oal number of pases be n. Solid can also be considered as a pase bu i does no flow. Te variables and parameers in equaion (1) are defined as follows: S is e sauraion, P e pressure, θ e porosiy, ρ e densiy, μ e viscosiy, e inrinsic permeabiliy, e relaive permeabiliy, g e r acceleraion of graviy, and q e source erm. For example, for e oil pase, q o is e volume of oil generaed from a uni volume of sale during uni ime. All n sae variables S and P can be solved from e n equaions (1) wi combining e following n consrains: n S = 1 = 1 () P Pβ = Pβ (3) were and β are wo pases, and P β is e capillary pressure beween e wo pases. Noe a Equaion (3) conains only n-1 independen equaions because e summaion of all ese equaions is equal o zero. Te mulipase flow equaions are similar o ose used in e peroleum engineering, bu e values of some parameers, suc as e porosiy and permeabiliy, will cange wi e progress of e eaing process as more oil and gas are produced from e sale. Even for a omogeneous sale srucure, e porosiy will depend on ime and emperaure of eaing. Terefore, e following ea ranspor equaions for all pases, including e solid pase, are needed: ρθsct = ρθsd T ρθsctv + I + Q (4) ( ) ( ) ( ) [ ρs (1 θ) ct s s] = [ ρs(1 θ) D s Ts] + Is + Q (5) s were e subscrip in Equaion (4) denoes a fluid or a gas pase, D is e ea dispersion coefficien, V is e velociy, I is e ea excange beween pase and oer pases, and Q is e ea source erm. Te subscrip s in Equaion (5) denoes e solid pase. Tese ea
6 Oil Sale Symposium 16-19 Ocober, 006 ranspor equaions are also igly nonlinear. For example, e porosiy θ is a coefficien in ese equaions bu is dependen on e eaing process. Te soluions of e flow equaions depend on e emperaure disribuion, and in urn, e flow velociies in e ea ranspor equaions are dependen on e soluions of e flow equaions roug Darcy s law: r V = ( P + z) (6) θμ For a cemical compound β, suc as a solven added o a pase or generaed from e oil sale, we also need e following reacive mass ranspor equaion: β β β ( θsc ) = ( θsd C ) ( θsc V) β β β + I + R + Q (7) were C β is e concenraion of e compound β in pase ; D is e ydrodynamic dispersion coefficien; I β and R β are e mass incremens of compound β in pase caused by mass excanges and reacions, respecively, beween pase and oer pases, and Q β is e erm of mass source. Tis equaion is coupled wi flow and ea ranspor equaions roug e sae equaions for eac pase: and ρ = ρ ( P, T, C ) (8) μ = μ ( P, T, C ) (9) Te deformaion equaion is no explicily involved in e model. Insead, is effec is represened by e variable porosiy, permeabiliy, and densiy. Te densiy is deermined by e following equaion: [ ρs (1 θ)] = ρoq0 ρgqg (10) ρ s Summarily, n mulipase flow equaions wi n consrains, n Darcy s law equaions (in vecor form), (n+1) ea ranspor equaions, m mass ranspor equaions for m cemical compounds, n sae equaions, and one solid densiy equaion, wi appropriae subsidiary condiions ogeer form a complee in-siu oil sale reoring model. Tis model can be significanly simplified by assuming a e ea ranspor process is muc faser en e oil sare reoring. In oer words, e emperaure disribuion becomes seady sae before e oil and gas sar o generae from e sale. In is case, e model reduces o a model of muli-componen ranspor in mulipase flow, excep a e porosiy, inrinsic permeabiliy, and solid densiy are ime dependen. Numerical Soluion Before solving e forward problem for predicion, we ave o deermine e parameer values in all equaions of e model. Due o e nonlineariy naure, mos of em are funcions of locaion, ime and sae variables (pressure, sauraion, emperaure, and concenraion). We assume a e following funcional relaionsips can be obained based on e resuls of lab experimens: Te volume of oil and gas produced per uni volume of oil sale as a funcion of ime and emperaure: q (, T ) 0 and q (, ) g T. Figure 1 is a scemaic figure of e funcion q () o for a given emperaure T. Te porosiy of oil sale as a funcion of ime and emperaure: θ (, T). Figure is a scemaic figure of e funcion θ () for a given emperaure T. Te permeabiliy depends on ow e pores are conneced. I is a macroscopic propery of porous media and canno be measured direcly in e lab. During e reoring process, i ends o increase depending on e ime period of eaing and e eaing emperaure, i.e., = (, T). Figure 3 is a scemaic figure of e funcion () for a given emperaure T. In e nex secion, we will discuss ow o idenify suc a 3
6 Oil Sale Symposium 16-19 Ocober, 006 funcion by solving e inverse problem wi e field daa. Noe a q, θ, may also depend on locaion wen (1) e oil sale srucure is eerogeneous, or () e oil sale srucure is omogeneous bu e emperaure disribuion is inomogeneous. Tese parameers may ave differen values a differen locaions because of e difference in e saring ime of reoring. Afer all parameers are deermined, a finie difference or a finie elemen based numerical meod (Sun, 1996; Helmig, 1997; Cen e al., 006) can be used o solve e oil sale model. Due o e nonlineariy, e governing equaions of e model mus be solved ieraively and e values of parameers mus be updaed in eac sep of ieraion unil a convergen crierion is saisfied. Te soluion procedure for eac ime sep consiss of e following nine seps: Sep 1. Solve emperaures T and Ts from e ea ranspor equaions (4) and (5) wi given ea sources and iniial parameer values. Sep. Updae e values of producion volume, porosiy, inrinsic permeabiliy and solid densiy wen e reoring process sars. Sep 3. Solve pressure P and sauraion S from e mulipase flow equaions (1) and e consrains in equaions () and (3) wi e updaed values of parameers. Sep 4. Use e P-S- consiuive relaions o updae e values of relaive permeabiliy. Sep 5. Use Darcy s law o calculae e velociy V disribuion for eac pase. Sep 6. Use e sae equaions o updae e densiy ρ and viscosiy μ of eac flow pase. Sep 7. Use e updaed parameer values o solve e ea ranspor equaions in (5) o obain updaed emperaure disribuionst. Sep 8. Cec e convergence of e calculaion by comparing e emperaure disribuions calculaed in Sep 1 and Sep 7. Sep 9. Move o e nex ime sep wen a convergence crierion is saisfied. Oerwise, reurn o Sep 1 for ieraion. Because of e ig nonlineariy of e sysem, generally, e compuaional effor is uge and e convergence procedure is slow. Te developmen of more effecive numerical meods for oil sale modeling is an imporan researc opic of ineres. Parameer Idenificaion We ave menioned in e las secion a a number of parameers in e in-siu oil sale reoring model need o be deermined before e forward problem is solved. Some of em can be measured in e lab, suc as e parameers in e sae equaions, e ea capaciies c, and e volumes q (, T) of oil and gas generaed from a uni of volume of oil sale ( a funcion of ime and emperaure). Te consiuive relaionsips can also be obained in e lab bu need o be calibraed wi e field daa. Tis secion concenraes on e idenificaion of e inrinsic permeabiliy because i canno be measured direcly in e lab scale. I depends on e emperaure and ime of eaing and varies wi locaion even for omogeneous sale srucures. In is paper, is idenified by solving a coupled inverse problem (Sun and Ye, 1990) based on e values of sae variables measured in e observaion wells during e field scale in-siu reoring researc. Te iniial value of e inrinsic permeabiliy, 0, can be obained by well esing and calibraed using e daa of ead values 4
6 Oil Sale Symposium 16-19 Ocober, 006 measured in e dewaering process. Te inverse problem may be formulaed as: * 0 cal obs pri { λ } = arg min ( ) ( ) + * 0 0 0 0 0 0 (11) cal were is e idenified permeabiliy, obs and are calculaed and observed ead values, respecively, λ is e regularizaion pri facor, and 0 is e permeabiliy esimaed by prior informaion. During e eaing process, e oil and gas sar o generae from a ime, depending on e eaing emperaure. Afer ime, e porosiy of oil sale increases and a causes e increase of pore connecion and e increase of permeabiliy. For a fixed emperaure T, e permeabiliy can be regarded as a funcion of ime, i.e., (). Tis funcion can be parameerized by a series of imes 0 < 1 < < < n < f called e basis poins of parameerizaion (Figure 4). Te unnown vecor = [ ( 1), ( ),, ( n )] (1) en can be idenified by solving a coupled inverse problem (Sun and Ye, 1990). Assume e observed values of pressure, sauraion, and emperaure of eac pase obs obs obs are P, S and T, respecively. Te coupled inverse problem can be formulaed as follows: * cal obs = arg min{ [ wp, P ( ) P + cal obs cal obs ws, S S + wt, T T ( ) ( ) ]} 0 0 (13) were e summaion is for all pases, and wp,, ws,, wt, are weiging coefficiens. Tis problem can be solved by a numerical opimizaion algorim wi appropriae consrains (Sun, 1994). Te number n and locaions v n of e basis poins of parameerizaion can also be opimized by solving e following exended inverse problem (Sun and Sun, 00): * * cal obs n) arg minmin{ [ wp, (, n) vn (, v = P v P + w cal obs cal obs S, S vn S + wt, T vn T (, ) (, ) ]} Algorims for solving is ind of combinaorial opimizaion problem are considered by many researcers as reviewed in Sun (005). Finding effecive algorims for idenifying ime dependen permeabiliy is also a very imporan researc opic of ineres. Model Reducion and Parameer Upscaling (14) Teoreically, an in-siu oil sale model sould be a 3-D model involving mulipase flow, deformaion, ea ranspor, and mass ranspor. To solve e forward and inverse problems for suc a complex sysem is ime consuming and e reliabiliy of e model predicion is no guaraneed. Design of an effecive observaion sysem for model calibraion is also a very callenging problem because e model complexiy is difficul o deermine. Consrucing a more complex model requires more daa. How is a model s complexiy deermined? Can we find a simple model srucure wi represenaive parameers o replace e rue sysem wile eeping e model resuls reliable? Sun (005) and Sun and Ye (006a,b) developed a new meodology for consrucing so called objecive-oriened models. Tis meodology could be applied o e oil sale model consrucion o find a reliable model wi minimum cos. In is meodology, e model complexiy is deermined by e objecives of model applicaion; e model reliabiliy is guaraneed by e use of sufficien daa; and e sufficien daa are obained from a robus experimenal design. Tere are differen objecives a can be presened for consrucing an oil sale model. Te following is a possible saemen: To design a eaing sysem S wi minimum cos C=C(S) wile e oil producion P is maximized. 5
6 Oil Sale Symposium 16-19 Ocober, 006 To solve is problem, a model M is used o predic e oil producion P for a given design S, i.e., P=M(S). A represenaive r model M a can be used o replace M sould saisfy e following condiion: r PM [ ( S)] PM [ ( S)] < ε (15) were ε is a given accuracy requiremen. Te problem of finding a represenaive model under is condiion is called a generalized inverse problem (Sun and Sun, 00). I can be solved by consrucing a series of models, in wic e complexiy of model srucure is increased gradually: r r r r M1, M,, Mm, M m + 1, (16) Effecive algorim for consrucing a nonnesed model series is developed recenly in Sun and Ye (006a,b) for groundwaer modeling. A srucure in (16) can be considered as a global upscaling of a disribued parameer sysem. We expec a e objecive-oriened meod can also be used for reducing e srucure of an oil sale model under condiion (15). 6. Conclusions Maemaical models are imporan ools for opimally designing and operaing complex sysems. Tis paper presens a preliminary ouline for in-siu oil sale reoring modeling. Te proposed field scale model is based on e pysical process of in-siu oil sale eaing ecnologies repored recenly. I consiss of a number of coupled and igly nonlinear parial differenial equaions, including e mulipase flow equaions, movemen equaions, ea and mass ranspor equaions, and sae equaions. Tis model is differen from exising groundwaer and peroleum models in suc a way a e porosiy and permeabiliy become unnown funcions even for omogeneous sale srucure. As a resul, solving e inverse problem for idenifying ese ime and locaion dependen parameers in e field scale becomes e ey of successful oil sale modeling. Major seps of solving bo e forward and inverse problems are preliminary described in e paper. Finally, e paper considers e possibiliy of using e objeciveoriened meod o find a simplified bu reliable oil sale model. A more complicaed model may no be more reliable. To develop a useful model for in-siu oil sale reoring, ere are several imporan researc opics of ineres, including e inverse problem, scaling problem, reliabiliy and model reducion problems inroduced in is paper. Tese problems sould be sudied a e same ime wi e field experimens. A realisic oil sale model can be consruced only wen sufficien field daa become available. References Cen, Z., Huan G., and Ma Y., Compuaional Meods for Mulipase Flow in Porous Media, pp. 531, SIAM, 006. Helmig, R., Mulipase Flow and Transpor Processes in e Subsurface, pp. 367, Springer, 1997. RAND, Oil Sale Developmen in e Unied Saes, 005 Sun, N.-Z., Inverse Problems in Groundwaer Modeling, Kluwer Academic Publisers, e Neerlands, pp. 337, 1994. Sun, N.-Z., Maemaical Modeling of Groundwaer Polluion, Springer-Verlag, New Yor, pp. 377, 1996. Sun, N.-Z., and A. Y. Sun, Parameer idenificaion of environmenal sysems, Caper 9, in Environmenal Fluid Mecanics: Teories and Applicaions, Edied by H. Sen e al., ASCE, 00. Sun, N.-Z., Srucure reducion and robus experimenal design for disribued parameer idenificaion, Inverse Problems, 1(4), 739-758, 005. Sun, N.-Z., and W. W.-G. Ye, Developmen of Objecive-Oriened Groundwaer Models: 1. Robus Parameer Idenificaion, acceped for publicaion, Waer Resour. Res., 006. 6
Sun, N.-Z., and W. W.-G. Ye, Developmen of Objecive-Oriened Groundwaer Models:. Robus Experimenal Design, acceped for publicaion, Waer Resour. Res., 006. 6 Oil Sale Symposium 16-19 Ocober, 006 For a given emperaure T q For a fixed emperaure T 1 n Figure 4. A parameerizaion meod for idenifying e permeabiliy. Figure 1. Volume rae of oil producion as a funcion of ime θ For a fixed emperaure T Figure. Te porosiy as a funcion of ime For a given emperaure T Figure 3. Te permeabiliy as a funcion of ime 7