Modern Time-Rate Relations

Transcription

1 Modern Tme-Rae Relaons Slde 1

2 Orenaon Tme-Rae Relaons: New me-rae relaons whch ulze he followng componens: Hyperbolc and modfed-hyperbolc relaons, Power-law/sreched exponenal relaons, and Exponenal relaons (e.g., he Fulford model). The bass for he proposed relaons are daa dagnoscs/characerscs. Model bass: Power-law componen for approxmang early-me behavor. Hyperbolc and exponenal componens for represenng lae me behavor. D-parameer: [(d/d)/] Bass: Based on he defnon of loss-rao. Power-law behavor for almos all gh gas/lud-rch shale reservors. Modfed-Hyperbolc model vald for mos gas shales. Power-law behavor of he D-parameer yelds he sreched exponenal funcon. Conclusons: Modelng me-rae behavor wh dfferen funconal forms reduces uncerany. Power-law exponenal has a very srong correlaon for gh gas/shale ol cases. Sreched exponenal should be consdered a vald model for gh gas/shales. Modfed-hyperbolc model remans prmary "currency" n me-rae analyss.,cp -dervave funcon s prmarly dependen on well compleon and geology. Slde 2

3 SPE Exponenal vs. Hyperbolc Declne n Tgh Gas Sands Undersandng he Orgn and Implcaons for Reserve Esmaes Usng Arps' Declne Curves D. Ilk, Texas A&M Unversy A.D. Perego, Anadarko Peroleum Corp. J.A. Rushng, Anadarko Peroleum Corp. T.A. Blasngame, Texas A&M Unversy Deparmen of Peroleum Engneerng Texas A&M Unversy College Saon, TX dlhan@amu.edu Slde 3

4 SPE Defnons of Rae Funcons Rae Funcon Defnons: Loss Rao: 1 D d/ d Dervave of Loss Rao: b d d 1 D d d d/ d or Exponenal and Hyperbolc Rae Relaons: D con or D b con d dq d dq (Exponenal Declne) (Hyperbolc Declne) exp[ D ]; or b 1 D [1 bd ] (1/ b) Dscusson: Hyperbolc relaon s ms-appled o ransen daa. Wha s he "characersc behavor" of he D and b-parameers? Evaluae connuously usng daa. Slde 4

5 SPE "Power-Law Exponenal" Rae Resul Observed Behavor of he "Declne" Parameer [D()]: D d 1 n D ndˆ (1 ) d D A B Solvng for Flowrae [()] Usng he D() Relaon: ˆ exp[ D Dˆ n ] Solvng for he "Hyperbolc" Parameer [b()]: b ndˆ (1 n) n [ ndˆ D (1 n) ] 2 Slde 5

6 SPE D-b Plo Small WF Tgh Gas Well Dscusson: Small "Waerfrac" Gas Well Lud loadng effecs are obvous n he laer poron of he flowrae daa. The onse of he boundary-domnaed flow regme s observed. We observe a very good mach of he flowrae daa usng D =0. Slde 6

7 SPE D-b Plo Large WF Tgh Gas Well Dscusson: Large "Waerfrac" Gas Well Errac rae behavor caused by lud loadng s seen a lae mes. Ousandng maches of he compued D- and b-parameers wh he power-law exponenal model are observed. Slde 7

8 SPE Rae Declne Model Type Curves We conver he "power-law exponenal" rae declne model no a dmensonless form. ˆ exp[ D Dˆ n ] ~ exp[ D n ] Slde 8

9 SPE Rae Declne Model Type Curves We develop ype curves usng he dmensonless form of he "power-law exponenal" rae declne model. ~ exp[ D n ] Slde 9

10 SPE Tgh Gas Well Dscusson: [Tgh Gas Well (Bosser)] Excellen mach of he daa wh he ype curve for n=0.2 hs yelds an upper bound for he reserves ( 5.34 BSCF). The lower bound for he reserves (G p,max ) s esmaed by he second ~ ype curve mach D 10 Slde 10

11 SPE A Smple Mehodology for Drec Esmaon of Gas-n-place and Reserves Usng Rae-Tme Daa N.L. Johnson, Texas A&M Unversy S.M. Curre, Texas A&M Unversy D. Ilk, Texas A&M Unversy T.A. Blasngame, Texas A&M Unversy Deparmen of Peroleum Engneerng Texas A&M Unversy College Saon, TX nahale.johnson@pe.amu.edu Slde 11

12 Quadrac rae-cumulave producon relaon can be rearranged o yeld a plong funcon as: g G p g D SPE g -G p Relaon 1 2 D G G p The plong funcon ( g - g )/G p versus G p yelds an nercep n he x-axs of 2G.e., use o esmae G. Slde 12

13 SPE g -G p Relaon Boundary-domnaed flow regme can be denfed usng he -parameer hrough he modfcaon of he rae-cumulave producon relaon: G p G g g G G p 2 The plong funcon, versus G p /G has a dagnosc value n esablshng he boundary-domnaed flow regme (.e., = 2 as g 0 and G p G). Slde 13

14 SPE g -G p Relaon The plong funcons g / g versus G p /G and g versus G p are used n conjuncon wh he prevous plong funcons o yeld he bes esmae for G. Dscusson: g, D, G parameers are calbraed usng he plong funcons. We erae on all plos unl he bes mach s obaned. Slde 14

15 SPE Tgh Gas Well a. Plong Funcon 1: (Tgh Gas Well) ( g - g )/G p vs G p Plo (Caresan scale). b. Plong Funcon 2: (Tgh Gas Well) "" Dagnosc Plo reverse soluon for he -parameer (Caresan scale). c. Plong Funcon 3: (Tgh Gas Well) Model Valdaon Plo g / g versus G p /G (Caresan scale). d. Plong Funcon 4: (Tgh Gas Well) Model Valdaon Plo g (daa and model) versus G p (log-log forma). Slde 15

16 SPE Declne Curve Analyss for HP/HT Gas Wells: Theory and Applcaons D. Ilk, Texas A&M Unversy J.A. Rushng, Anadarko Peroleum Corp. T.A. Blasngame, Texas A&M Unversy Deparmen of Peroleum Engneerng Texas A&M Unversy College Saon, TX Slde 16

17 SPE Rae-Tme Relaon Rae-Tme Relaon: ((1 p 4p 2 ) (1 p exp[ p )exp[ p ] ]) 2 From: Knowles R.S Developmen and Verfcaon of New Sem-Analycal Mehods for he Analyss and Predcon of Gas Well Performance. M.S Thess, Texas A&M Unversy, College Saon, Texas. Ansah, J., Knowles, R.S., and Blasngame, T.A A Sem-Analyc (p/z) Rae- Tme Relaon for he Analyss and Predcon of Gas Well Performance. SPEREE. 3 (6): Dmensonless D-funcon (D D ): D D D D 1 b-funcon (b): b p(1 p (1 ( p 1 (1 p d d d d p)exp[ p )exp[ p ["Loss-Rao"] ["Dervave of Loss-Rao"] ( d/ d ) 2exp[ p b (1 p (1 p ] (1 p )exp[ p 2 ) ]) 2 ]) ]) Dscusson: Rae-Tme Gas Flow Relaon (Knowles e al.) Bass s he lnearzaon of he nonlnear " g c g " erm (Ansah, e al.). D-funcon and b-funcon are formulaed usng he defnons for lossrao and he dervave of he loss-rao. Slde 17

18 SPE Rae-Cumulave Producon Relaon Rae-Cumulave Producon Relaon: 1 G 2 pd G 2 pd 2 2/ (1 p ) Dmensonless D-funcon (D D ): D D d dg pd DD ( 1 GpD) ["Loss-Rao"] From: Knowles R.S Developmen and Verfcaon of New Sem-Analycal Mehods for he Analyss and Predcon of Gas Well Performance. M.S Thess, Texas A&M Unversy, College Saon, Texas. Ansah, J., Knowles, R.S., and Blasngame, T.A A Sem-Analyc (p/z) Rae- Tme Relaon for he Analyss and Predcon of Gas Well Performance. SPEREE. 3 (6): b-funcon (b): b ["Dervave of Loss-Rao"] d 1 dgpd ( d/ dgpd) GpD G pd b 2 2 ( GpD 1) Dscusson: Rae-Cumulave Gas Flow Relaon Defnon of he loss-rao can be re-cas n erms of rae and cumulave producon. A uadrac relaonshp exss beween rae and cumulave producon. Slde 18

19 SPE Analyss Mehodology Dscusson: Mehodology The man goal s o mach he daa wh he model usng he defnons for he -D-b funcons durng he boundary-domnaed flow regme. b-funcon 0.5 for hgh drawdown cases (almos consan behavor). Slde 19

20 SPE HP/HT Tgh Gas Well (p = psa and T R = 260 o F) Feld Example: Applcaon of he Mehodology 3.5 years of daly daa are avalable for a hydraulcally fracured well compleed n a HP/HT gas reservor. Well clean-up effecs, lud-loadng, and operaonal changes are observed n he daa rends. The flowrae daa are revewed pror o analyss; and any erroneous/ redundan daa pons are removed. The half-slope rend s evden n he rae-negral dervave funcon. Slde 20

21 Peroleum Engneerng 648 Pressure Transen Tesng SPE HP/HT Tgh Gas Well a. g versus Gp (Caresan plo). b. D-funcon versus (Caresan plo). c. b-funcon versus (Caresan plo). d. g versus (Semlog plo). e. D-funcon versus (Semlog plo). f. b-funcon versus (Semlog plo). Feld Example: Applcaon of he Mehodology For he compuaon of D- and b-parameer daa funcons we remove he oulyng daa pons; hen we perform he numercal dfferenaon. Our analyss usng he proposed sem-analycal relaon provdes a gas-n-place esmae of approxmaely 8.0 BSCF. Slde 21

22 SPE HP/HT Tgh Gas Well Feld Example: Applcaon of he Mehodology Reasonable maches of he D-funcon wh he daa usng he semanalycal model s acheved (pos-ransen flow only). The maches of he b-funcon daa wh he sem-analycal model are problemac daa ndcae no unue characersc behavor. Compuaon of he b-parameer daa funcon s severely affeced by facors such as lud loadng. Slde 22

23 SPE HP/HT Tgh Gas Well Feld Example: Applcaon of he Mehodology We observe a good mach of he flowrae daa wh he model (excep for he early me daa affeced by "cleanup"). The "power-law exponenal" model yelds G p,max 8.0 BSCF. Gas-n-place esmaes are conssen comparng he mehods we used. Slde 23

24 SPE Hybrd Rae-Declne Models for he Analyss of Producon Performance n Unconvenonal Reservors D. Ilk*, Texas A&M U./DeGolyer and MacNaughon S.M. Curre, Texas A&M U./Devon Energy Corp. D. Symmons, Consulan J.A. Rushng, Apache Corp. T.A. Blasngame, Texas A&M Unversy *DeGolyer and MacNaughon Dallas, TX dlk@demac.com Slde 24

25 SPE Sreched Exponenal Funcon [Kohlrausch (1854)] Observed Behavor of Declne Parameer (D): D d 1 (1 n) d Solvng for Flowrae: ˆ exp[ Dˆ ndˆ Leraure: Kohlrausch (1854). Phllps (1996). Ksslnger (1993) Decays n randomly dsordered, chaoc, heerogeneous sysems (e.g. relaxaon, afershock decay raes, ec.). n ] Valkó (2009) ( ) ˆ exp[ ( / ) Jones (1942) and Arps (1945) ( ) o n Do exp 100 ( m ] 1 1) m (Sreched Exponenal Funcon) Slde 25

26 SPE Sreched Exponenal Funcon ( ) ( ) ˆ n 1 exp[ Dˆ n exp[ a ] ] (Sreched Exponenal Funcon) Dscusson: Sreched Exponenal Funcon Sngle, double and four exponenals are used o approxmae he daa usng lnear leas suares. Sreched exponenal funcon can be descrbed as a lnear super-poson of exponenal decays. Slde 26

27 SPE Theorecal Consderaons Rae-Tme Relaon: ((1 p 4p 2 ) (1 p exp[ p )exp[ p ] Conclusons: Theorecal jusfcaon for hyperbolc declne relaon for gas flow? b = 0.5 for hgh drawdown cases (p wf /p 0.05). ONLY vald for BOUNDARY- DOMINATED FLOW REGIME. Exponenal declne a very lae mes. ]) 2 (Theorecal Consderaons) Dscusson: Rae-Tme Gas Flow Relaon (Knowles e al.) Bass s he lnearzaon of he nonlnear " g c g " erm (Ansah e al.). D-funcon and b-funcon are formulaed usng he defnons for loss-rao and he dervave of he loss-rao. See Ansah e al. (2000), Knowles e al. (1999), and Ilk e al. (2009) for more deals. Slde 27

28 SPE ,cp -Dervave ()-Dervave: Well Tes Analyss (Hossenpour-Zonooz e al. 2006) p d d ln( p) 1 dp ( ) d ln( ) p d ()-Dervave: Modfcaon for hs work (for consan pressure) ( -cp -dervave), cp ( ) d ln( ) d ln( ) d d Dscusson: Srong dagnosc characer of he,cp -dervave funcon. Holly Branch gh gas feld producon daa exhb smlar characersc behavor. Early me daa are affeced by "non-reservor" effecs. Slde 28

29 SPE ,cp -Dervave Shale Gas Feld C Shale Gas Feld A Shale Gas Feld D Shale Gas Feld B ( -cp -dervave) Slde 29

30 SPE Mexco Gas Well (Mexco Gas Well) Dscusson: Fracured vercal gas well wh 43 years of producon. Slde 30

31 SPE Mexco Gas Well (Mexco Gas Well) Dscusson: Boundary-domnaed flow regme s apparen a lae mes. Slde 31

32 SPE Shale Gas Well (Feld D) (Shale Gas Well) Dscusson: Horzonal well wh mulple fracures wh 340 days of producon. Slde 32

33 SPE Shale Gas Well (Feld D) (Shale Gas Well) Dscusson: Ousandng daa ualy provdes remarkable characer. Slde 33

34 SPE Comparson [Knowles A-1] (All Models) Dscusson: Rae-Tme Models Rae-me models decrease he uncerany n reserves esmaes. Slde 34