An analytical model for heat transfer process in steam assisted gravity drainage

Transcription

1 ENERGY EXPLORATION & EXPLOITATION Vlume 33 Numbe 015 pp An analytical mdel f heat tanfe pce in team aited gavity dainage Xia Tian 1,, Lngxin Mu 1, Xianghng Wu 1 and Feng Xu 1,3 1 Reeach Intitute f Petleum Explatin & Develpment, PetChina, Beijing , China Depatment f Petleum and Geytem Engineeing, The Univeity f Texa at Autin, Autin 78703, USA 3 Schl f Eath and Space Science, Peking Univeity, Beijing , China Auth f cepnding. ita.xiatian@gmail.cm (Received 1 Nvembe 014; Accepted 16 Febuay 015) Abtact A new analytical mdel i develped and lved t eveal heat tanfe mechanim duing team aited gavity dainage (SAGD). The new mdel i tanfmed t a bunday value pblem (BVP) f ecnd de dinay diffeential equatin (ODE) t be lved. Meve, a new elatinhip between il atuatin and dimeninle eevi tempeatue i etablihed and utilized t lve the mdel. The eult eveal thee main cncluin. Fitly, althugh the pecific pptin f cnvectin t cnductin duing heat tanfe pce i influenced by ppetie f ck and fluid in the eevi, cnvective heat flux i me imptant than cnductive heat flux nea team chambe edge (aund 0 m). Secndly, cnvective heat flux deceae apidly away fm the chambe edge and quickly becme negligible cmpaed t cnductive heat flux. A a eult, the influence f cnvectin n eevi tempeatue i maginal duing the whle heat tanfe pce. Thidly, claic elatinhip between eevi tempeatue and ditance meaued fm the team-chambe edge, which i develped unde cnductive heat tanfe aumptin, i till applicable when cnideing bth cnductive and cnvective heat tanfe. eywd: SAGD, Heat tanfe mdel, Cnductin, Cnvectin, Bunday value pblem 1. INTRODUCTION Steam aited gavity dainage (SAGD) ha been uccefully utilized in many heavy il eeviupe heavy (Liu et al., 011; Wang et al., 011). In SAGD, team i injected int the injectin well and fm a team chambe that uund the well and eleae it latent heat t the eevi. The heated il dain t the hizntal pductin well lcated a few mete belw the injectin well. Since the heating f il

2 170 An analytical mdel f heat tanfe pce in team aited gavity dainage and at the edge f the team chambe i a key fact influencing il pductin ate in SAGD (Butle, 1994), ppe undetanding f heat tanfe mechanim duing thi pce i vey imptant. Theetically, heat tanfe mechanim duing thi pce include bth cnductin and cnvectin heat flux. Mt eeache (Butle, 1994; Rei, 199; Azad and Chalatunyk, 010; Iani and Ghannadi, 013) aumed that cnvectin i negligible cmpaed t cnductin when building mathematical mdel in de t implify the mdel. On the the hand, me eeache (Fauq-Ali, 1997; It and Suzuku, 1999) believe that cnvectin culd be the pedminant tanfe mechanim. Accding t thee eeache, ince the heat capacity f wate i uually t 5 time that f il and and thee plenty f cndenate flwing in the eevi. Shama and Gate (011) etablihed a heat tanfe mdel cnideing bth cnductin and cnvectin in SAGD pce. Hweve, thee ae thee cntveial iue in thi mdel endeing the eult diputable. Fitly, the lving pce f thi mdel adpted expein f eevi tempeatue (T) v. ditance fm team chambe edge (ξ) built by Butle (1994). Hweve, Butle mdel wa built unde the aumptin f cnductive heat tanfe pce. Secndly, thi mdel ubtituted the themal diffuivity α by the appaent diffuivity α withut any detailed explanatin. Thidly, the auth dn t give any theetical fundatin egading the expein f il atuatin (S ) v. dimeninle eevi tempeatue (T) adpted in the mdel. Iani et al. (013) al invetigated the influence f cnvective heat flux duing SAGD pce. Hweve, lutin methd f thee mdel al adpted expein f eevi tempeatue (T) v. ditance fm team chambe edge (ξ) built by Butle in In thi pape, a new mdel wa etablihed cnideing bth cnductive and cnvective heat tanfe. The new mdel wa finally tanlated t a BVP f ecnd de ODE and lved cmbining with implicit Runge-utta and ODE45 aithmetic in MATLAB.. NUMERICAL MODEL In il field pductin, bth injectin and pductin well ae nmally me than 500 m lng (Shama and Gate, 011). A a eult, in hmgeneu eevi, the tempeatue gadient paallel t the wellbe tajecty i elatively negligible cmpaed t that alng the wellbe tajecty. What me, becaue the team chambe expand in the pependicula diectin, the tempeatue gadient paallel t the team chambe i much malle cmpaed t that pependicula t the team chambe. A a eult, nly tempeatue gadient in ne diectin i cnideed in a hmgeneu eevi. On the the hand, when heat tanfe pce eache quaiteady tate accding t Butle (1985; 1991), heat ahead f the fnt i un ve by the advancing team chambe fnt and the tempeatue at ditance ξ fm the fnt i cnideed cntant with time (Fig. 1). In cncluin, fu aumptin ae made in ectin.1 t pecify the cpe f applicatin and facilitate lving the new mdel withut impaiing the eliability f the mdel.

3 ENERGY EXPLORATION & EXPLOITATION Vlume 33 Numbe Steam Chambe, T Cndenate Cnvectin Reevi, T Cnductin Figue 1. Cnductin and cnvectin at the edge f a team chambe..1. Baic they Baic hypthei f heat tanfe mdel: 1. Bth cnvective and cnductive heat flux exit duing SAGD heat tanfe pce;. Hmgeneu eevi cnditin; 3. Heat tanfe eached quai-teady tate; 4. Cndenate flw velcity in the hizntal diectin depend n the wate elative pemeability and vicity vaiatin in that diectin. Heat tanfe ahead f team chambe accunting f bth cnductin and cnvectin can be etablihed baed n the they f Calaw and Jaege ρ + + T T T x y z V c T + T ρ + T x y z c TH c c pc p T t (1) TH themal cnductivity, J/( m ) T tempeatue, x ditance meaued in the hizntal diectin, m y ditance meaued in the vetical diectin, m z ditance meaued paallel t the well diectin, m V c cndenate-flw velcity meaued in hizntal diectin, m/ ρ c denity f cndenate at team tempeatue, kg/m 3 ρ denity f il and, kg/m 3 c P vlumetic heat capacity f il and, J/m 3 c Pc vlumetic heat capacity f cndenate, J/m 3 t time, Baed n the ecnd aumptin, in a hmgeneu eevi, tempeatue gadient paallel t the chambe inteface and -alng well tajecty ae much malle than that pependicula t chambe inteface. Eq. 1 can be implified t T ρ x V c T ρ c x TH c c pc p T t ()

4 17 An Analytical Mdel f Heat Tanfe Pce in Steam Aited Gavity Dainage Auming that the mving velcity nmal t the chambe edge f team chambe inteface i U x, hizntal cdinate axi i x, hizntal efeence cdinate ξ can be expeed a: ξ x Ux. t (3) U x ξ Since inteface velcity meaued in the hizntal diectin, m/ ditance meaued fm the team-chambe edge in the diectin nmal t it, m dt ( ξ, t) ξ Td + T dt ξ dt t dt dt (4) T x T ξ ξ ξ ξ + T ξ + T ξ T x y z ξ (5) Intduce Eq. 4 and Eq. 5 int Eq., yield T ρ V c T ρ ξ + c U T T t TH c c pc p x (6) Baed n the thid aumptin, heat tanfe ha aived at the quai-teady tate accding t Butle (1985; 1991): T 0 t Subtitute Eq. 7 int Eq. 6, Eq. 6 can be tanfmed t T ρ ρ + T ( U c V c ) 0 TH x p c c pc (7) (8) Baed n the fth aumptin, cndenate velcity in the hizntal diectin V c can be detemined: V U c x w wint / μw / μ wint (9) k w k wint μ w μ wint elative pemeability f wate, dimeninle elative pemeability f wate at team chambe inteface, dimeninle dynamic vicity f wate, mpa dynamic vicity f wate at team chambe inteface, mpa Subtitute Eq. 9 int Eq. 8, yield T U ρ c ρ c + TH x p c pc w wint / μw / μ wint T 0 (10)

5 ENERGY EXPLORATION & EXPLOITATION Vlume 33 Numbe Appaent themal diffuivity α i impted t implify Eq. 10 and dente cmpehenive themal diffuin ate cnideing cexitence f cnductin and cnvectin in heat tanfe pce TH ρ c p α ρccpc 1 ρcp w w int / μw / μ wint (11) α appaent themal diffuivity, m / Subtitute Equ. 11 int Equ. 10 yield T α + U x T 0 (1).. Relatinhip between il atuatin S and dimeninle eevi tempeatue T It and Suzuki (1999) gave ut the definitin f dimeninle eevi tempeatue T which wa defined in Eq. 13. It and Suzuki (1999) al built the elatinhip between il atuatin S and dimeninle eevi tempeatue T (Fig. ). T T T T T (13) T initial eevi tempeatue, T team tempeatue, T dimeninle tempeatue, f Oil Satuatin, S m belw tp f eevi 0. 18m belw tp f eevi Dimeninle tempeatue T Figue. Relatinhip between S and T.

6 174 An Analytical Mdel f Heat Tanfe Pce in Steam Aited Gavity Dainage Fm Figue, when eevi dimeninle tempeatue lie between 0 0.5, il atuatin emain at the level f initial il atuatin; when eevi dimeninle tempeatue lie between , il atuatin deceae linealy with dimeninle eevi tempeatue. Relatinhip between il atuatin and eevi dimeninle tempeatue can be expeed a S Si T [0, 0.5] S A T + B T [0.5, 1] (14) S i initial il atuatin, dimeninle S il atuatin, dimeninle T dimeninle tempeatue, f A and B can be lved baed n the imulatin eult f It and Suzuki (1999), when T 0.5, S S i ; when T 1.0, S S. Subtitute value f A and B int Eq. 15, yield S S Si T [0, 0.5] S ( S S ) T + S S T [0.5, 1] i i eidual il atuatin, dimeninle (15).3. Relatinhip between appaent heat diffuivity a and dimeninle eevi tempeatue T Cey Equatin i intduced t expe the elatinhip between il/wate elative pemeability and atuatin: S wd Sw Swc 1 S S wc (16) ( S ) w w wd b (17) b S wc S w S wd k w Cey cefficient, dimeninle cnnate wate atuatin, dimeninle wate atuatin, dimeninle cnnate wate atuatin, dimeninle elative pemeability f wate at eidual il atuatin, dimeninle Subtitute Eq. 16 and Eq. 17 int Eq. 11, yield α ρcc 1 ρ c α themal diffuivity, m / pc p w wint α / μ w Sw S wc / μ 1 S S wint wc ^ b (18)

7 ENERGY EXPLORATION & EXPLOITATION Vlume 33 Numbe Eq. 15 can be tanfmed t S S S S S S i S S i i i 0 T [0, 0.5] T 1 T [0.5, 1] (19) S wc S w S wc S w cnnate wate atuatin, dimeninle wate atuatin, dimeninle cnnate wate atuatin, dimeninle wate atuatin, dimeninle Since Sw Swc 1 S S wc 1 S (1 Si) S S 1 S S S S wc i i (0) Subtitute Eq. 0 int Eq. 19, yield Sw Swc 1 Swc S Sw Swc 1 S S wc 0 T [0, 0.5] T 1 T [0.5, 1] (1) Subtitute Eq. 1 int Eq. 11, btain the elatinhip between appaent themal diffuivity α and dimeninle eevi tempeatue α α T [0, 0.5] α α T [0.5, 1] ρccpc w / μw b 1 (T 1)^ ρcp w / μ int wint ().4. Relatinhip between eevi tempeatue T and ditance fm team chambe edge ξ Subtitute Eq. int Eq. 1 and implify, get the elatinhip between eevi tempeatue T and ditance fm team chambe edge ξ. Intduce bunday cnditin int thi mdel and the pblem i tanfmed t a BVP f ecnd de ODE. T T T T A B T F T + (1 ( ) ( )) 0 T, T ξ ξ + T T A 0 T T, T T + ξ ξ + (3)

8 176 An Analytical Mdel f Heat Tanfe Pce in Steam Aited Gavity Dainage ξ Bunday Value Cnditin: 0 T T (4) ξ T T whee A U x α ρccpc w / μw BT ( ) ρ c / (5) p w μ int wint b FT ( ) (T 1)^ 3. CASE STUDY Implicit Runge-utta methd and ODE algithm in MATLAB ae ued t get lutin f the new mdel. The pecific paamete ued in the mdel (Table 1) ae taken fm T + T Shama and Gate (011). Numeical lutin f T, T ( ξ [0, ξ ]) ae btained and expeed a Eq. 6. Meanwhile, analytical lutin f T T T T +, ( ξ [ ξ, ]) ae calculated and expeed a Eq. 7. When T + T ξ ξ, T ( ξ) T.155ξ ξ+ 533 ξ [0, ξ ] Ux α ξ T ( T T ) e + T ξ [ ξ, ] (6) (7) Cuve f Eq. 6 and Eq. 7 ae dawn in Figue 3. Accding t Figue 3, cuve f Eq. 6 and Eq. 7 almt velap with each the. Calculate the eevi tempeatue uing Eq. 6 and Eq. 7 epectively and define the eult a T 1 and T. Lit the T 1 and T in Table and make cmpain. The dipaity between calculated tempeatue at ditance ξ fm team chambe edge i le than 1.5. A a eult, expein f the piecewie functin can be appximated t a ingle expein uing Eq. 8: Ux α ξ T ( T T ) e + T ξ [0, ] (8)

9 ENERGY EXPLORATION & EXPLOITATION Vlume 33 Numbe Table 1. Rck and fluid ppetie. Paamete Value T, 83 T, 533 α, m / wint, f 0.0 w, f 0.0 S, f 0.14 S wc, f 0.16 S i, f 0.84 b, f 3.5 μ w, mpa U x, m/ Shama and Gate (011) lgμ w 658.(1/T-1/83.16) 550 TT 500 T.155ξ^ ξ+533 R T() 400 T(T+T)/ 350 ξξ T(T-T)exp(-Uxξ/α )+T ξ(m) Figue 3. Relatinhip between tempeatue and ξ. 4. DISCUSSION Expein f tempeatue at ditance ξ fm team chambe edge f the new mdel (Eq. 8) i almt the ame a the expein f tempeatue develped by Butle cnideing nly cnductin duing hea tanfe pce. Influence f cnductin n eevi tempeatue i negligible in heat tanfe mechanim duing SAGD pce (Table ), jutifying mt eeache aumptin f nly cnductin heat tanfe when etablihing diffeent mathematical SAGD mdel.

10 178 An Analytical Mdel f Heat Tanfe Pce in Steam Aited Gavity Dainage Table. Dipaity between calculated tempeatue. T1( ) T( ) dipaity( ) T1( ) T( ) dipaity( ) Appaent themal diffuivity f the new mdel can be btained cmbining Eq. 8 with Eq.. α α ρ μ c Ux c pc w / w α 1 ρ μ e ξ 1 ^b cp w / int wint α α ξ [ ξ, ] ξ [0, ξ ] (9) α appaent themal diffuivity accunting f bth cnvective and cnductive heat tanfe α themal diffuivity cnideing imply cnductive heat tanfe The ati f (α α)/α exhibit the ptin f cnvectin cmpaed t cnductin in the whle heat tanfe pce. Rck and fluid ppetie ae taken fm Table 1 and elatinhip f (α α)/α v. ξ (Fig. 4) ae etablihed by the new mdel (Eq. 9) and Shama mdel (Eq. 30) epectively. α ρcc 1 ρ c pc p α w wint / μw / μ wint e b U x α ξ (30)

11 ENERGY EXPLORATION & EXPLOITATION Vlume 33 Numbe New Mdel Shama Mdel (α-α)/α ξ(m) Figue 4. Relatinhip between (α α)/α and ξ. Thee eult can be cncluded fm the cmputing eult f new mdel and Shama mdel. Fitly, cnvectin i a me imptant heat tanfe mechanim nea team chambe edge (Fig. 4). Hweve, the pptin f cnductin deceae apidly nea the team chambe edge. Secndly, the acting ange f cnvectin in fnt f team chambe depend n ck and fluid ppetie in the eevi. Accding t the eult f cae tudy, calculated acting ange f cnvectin i aund 3 mete f Shama mdel and mete f the new mdel; the new mdel indicate a much fate deceae f cnvective heat flux in fnt f team chambe. Thee ae tw ean f the diffeent eult: (1) The lutin pce f Shama mdel adpted expein f T v. ξ built by Butle unde the aumptin f cnductin heat tanfe pce and ubtitute the themal diffuivity α by the appaent diffuivity α withut theetical fundatin. While the lutin methd f the new mdel i t tanfm the pblem t a BVP f ecnd de ODE. Bth numeical and analytical lutin ae calculated f the new mdel; () A me accuate S v. T elatinhip i etablihed and utilized in the new mdel. 5. CONCLUSIONS In thi pape, a heat tanfe mdel wa built t tudy the le f cnductin and cnvectin in heat tanfe mechanim duing SAGD. The elatinhip between il atuatin and eevi tempeatue i etablihed and ued t lve the heat tanfe mdel. It i futhe pved that cnvectin i negligible cmpaed t cnductin duing heat tanfe pce. It i eanable t cnide nly heat cnductin when building heat tanfe mdel duing SAGD pductin. Reult ae cmpaed between the mdel in thi pape and Shama mdel. The cmputed functin ange f cnvectin i 3 mete f Shama mdel while mete uing the new mdel. The new mdel exhibited a much malle ptin f cnvectin in SAGD heat tanfe pce.

12 180 An Analytical Mdel f Heat Tanfe Pce in Steam Aited Gavity Dainage ACNOWLEDGMENTS The auth ae gateful f the uppt f the Natinal Science and Technlgy Maj Pject f China: Effective develpment technlgie f Heavy il eevi and il and (011ZX ). REFERENCES Azad A. and Chalatunyk R.J., 010. A mathematical impvement t SAGD uing gemechnical mdelling. Junal f Canadian Petleum Technlgy 49(10), Butle R.M., A new appach t the mdelling f Steam-Aited Gavity Dainage. Junal f Canada Petleum Technlgy 4(3), Butle R.M., Steam-aited gavity dainage: cncept, develpment, pefmance and futue. Junal f Canada Petleum Technlgy 33(), Butle R.M., Themal Recvey f Oil and Bitumen. Old Tappan, NJ (United State) and Pentice Hall Inc. United State. pp. 17. Fauq-Ali S.M., I thee life afte SAGD? Junal f Canada Petleum Technlgy 36(6), 0 4. It Y. and Suzuku S., Numeical imulatin f the SAGD pce in the Hangingtne il and eevi. Junal f Canada Petleum Technlgy 38(9), Liu Z.B. et al., 011. Pductin featue f team and ga puh: Cmpaative analyi with team aited gavity dainage. Petleum Explatin and Develpment 38(1), Iani M. and Ghannadi S., 013. Undetanding the Heat-Tanfe Mechanim in the Steam-Aited Gavity-Dainage (SAGD) Pce and Cmpaing the Cnductin and Cnvectin Flux in Bitumen Reevi. SPE Junal 18(1), Iani M. and Gate I.D., 013. Undetanding the Cnvectin Heat-Tanfe Mechanim in the Steam-Aited-Gavity-Dainage Pce. SPE Junal 18(6), Rei J.C., 199. A Steam-aited gavity dainage mdel f ta and: linea gemety. Junal f Canada Petleum Technlgy 31(10), Shama J. and Gate I.D., 011. Cnvectin at the Edge f a Steam-Aited-Gavity- Dainage Steam Chambe. SPE Junal 16(3), Wang B.J., Zhang X. and Ma D.S., 011. Dicete element methd mathematical mdel f and-caying cld heavy il pductin. Petleum Explatin and Develpment 38(4), 1.