Tom BLASINGAME Texas A&M U. Slide 1
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1 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl Slide 1
2 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl Slide
3 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl Slide 3
4 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl Slide 4
5 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl Derivain f he Perrine-Marin Diffusiviy Equains fr Individual Phases frm Dearmen f Perleum Enineerin Curse Nes (1997) Slide 5
6 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Derivain f he Perrine-Marin Diffusiviy Equains fr Individual Phases) Slide 6
7 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Derivain f he Perrine-Marin Diffusiviy Equains fr Individual Phases) Slide 7
8 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Derivain f he Perrine-Marin Diffusiviy Equains fr Individual Phases) Slide 8
9 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl Phase ehavir f Oil and Gas Sysems Slide 9
10 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Phase ehavir f Oil and Gas Sysems) "lac Oil" Fluid Preries Varius "lac Oil" PVT Preries: (eneral behavir, b =5000 sia) Ne he dramaic influence in reries a he bubblein ressure. The il cmressibiliy is he ms affeced variable (ee his in mind). Slide 10
11 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Phase ehavir f Oil and Gas Sysems) "Sluin-Gas Drive" Preries 1/( ) fr < b "Sluin-Gas Drive" PVT Preries: (1/( ), < b, b =5000 sia) Aem illusrae ha 1/( ) cnsan fr < b. This uld all us arximae behavir usin "liquid" equains. Slide 11
12 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Phase ehavir f Oil and Gas Sysems) "Sluin-Gas Drive" Preries c fr < b "Sluin-Gas Drive" PVT Preries: ( c, < b, b =5000 sia) Aem illusrae ha c is NEVER cnsan. CAN NOT arximae behavir usin "liquid" equains (r s i seems). Slide 1
13 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Phase ehavir f Oil and Gas Sysems) "Dry Gas" z vs. "Dry Gas" PVT Preries: ( z vs. ) asis fr he "ressure-squared" arximain (i.e., use f variable). Cnce: ( z) = cnsan, valid nly fr <000 sia. Slide 13
14 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Phase ehavir f Oil and Gas Sysems) "Dry Gas" /( z) vs. "Dry Gas" PVT Preries: (/( z) vs. ) asis fr he "ressure" arximain (i.e., use f variable). Cnce: (/ z) = cnsan (never valid). Slide 14
15 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Phase ehavir f Oil and Gas Sysems) "Dry Gas" c vs. "Dry Gas" PVT Preries: ( c vs. ) Cnce: If c cnsan, seudime NOT required. Readily bserve ha c is NEVER cnsan, seudime required. Slide 15
16 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Phase ehavir f Oil and Gas Sysems) Summary Frmain Vlume Facr Frmain Vlume Facr:,, Fluid vlume a reservir cndiins,, = Fluid vlume a sandard cndiins,, is defined as a vlume cnversin fr il, as, r aer and is defined n a mass (r densiy) basis. The Frmain Vlume Facr "cnvers" surface vlumes dnhle cndiins. Tyical values: Oil: 1..4 R/ST Gas: rcf/scf Slide 16
17 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Phase ehavir f Oil and Gas Sysems) Summary Fluid Viscsiy Viscsiy:,, Is a measure f a fluid's inernal resisance fl... he rrinaliy f shear rae shear sress, a sr f inernal fricin. Fluid viscsiy deends n ressure, emeraure, and fluid cmsiin. Tyical values: Oil: c Gas: c Waer: c Slide 17
18 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Phase ehavir f Oil and Gas Sysems) Summary Fluid Cmressibiliy Fluid Cmressibiliy: c,, c 1 d d dr d s c 1 d d c 1 d d dr d s Tyical values: Oil: 5 0 x10-6 si -1 (> b ) x10-6 si -1 (< b ) Gas: x10-6 si -1 Waer: 3 5 x10-6 si -1 Frmain Cmressibiliy: c f 1 d c f d Tyical values: Nrmal: 10 x10-6 si -1 Abnrmal: x10-6 si -1 Slide 18
19 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl Varius Diffusiviy Equains Slide 19
20 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Varius Diffusiviy Equains) Diffusiviy Equains Perinen Cases Diffusiviy Equains: "lac Oil" (> b ) "Sluin-Gas Drive" (valid fr all, referenced fr < b ) "Dry Gas" (> d ) Mulihase Fl Slide 0
21 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Varius Diffusiviy Equains) Diffusiviy Equains fr a lac Oil: Slihly Cmressible Liquid: (General Frm) c( ) lac Oil > b c Slihly Cmressible Liquid: (Small and c frm) c Slide 1
22 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Varius Diffusiviy Equains) lac Oil and vs. ehavir f he and variables as funcins f ressure fr an examle blac il case. Ne behavir fr > b bh variables shuld be cnsidered be "arximaely cnsan" fr he sae f develin fl relains. Such an assumin (i.e., and cnsan) is n an abslue requiremen, bu his assumin is fundamenal fr he develmen f "liquid" fl sluins in reservir enineerin. Slide
23 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Varius Diffusiviy Equains) lac Oil c vs. ehavir f he c variable as a funcin f ressure examle blac il case. Ne he "jum" a = b, his behavir is due he as exansin a he bubblein. Slide 3
24 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Varius Diffusiviy Equains) Diffusiviy Equains fr Sluin-Gas Drive: (< b ) Oil Pseudressure Frm: (Accuns fr and ) c Oil Pseudressure Definiin: ( n is any reference ressure) Sluin-Gas Drive < b 1 d n base Slide 4
25 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Varius Diffusiviy Equains) Sluin-Gas Drive 1/( ) vs. ( b =5000 sia, T=175 De F) 1 d n base "Sluin-Gas Drive" Pseudressure Cndiin: (1/( ) vs. ) Cnce: IF 1/( ) cnsan, THEN il seudressure NOT required. 1/( ) is NEVER "cnsan" bu des n vary sinificanly ih. Oil seudressure calculain sraihfrard, bu rbably n necessary. Slide 5
26 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Varius Diffusiviy Equains) Sluin-Gas Drive c vs. ( b =5000 sia, T=175 De F) a, 1 n 0 ( ) c c d ( ) "Sluin-Gas Drive" Pseudressure Cndiin: (( c ) vs. ) Cnce: IF ( c ) cnsan, THEN il seudime NOT required. ( c ) is NEVER "cnsan" UT, il seudime uld be very difficul. Oher evidence suess ha inrin ( c ) variance is acceable. Slide 6
27 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Varius Diffusiviy Equains) Sluin-Gas Drive Mbiliy/Cmressibiliy (Camach) Camach-V., R.G. and Rahavan, R.: "undary-dminaed Fl in Sluin-Gas-Drive Reservirs," SPERE (Nvember 1989) "Sluin-Gas Drive" ehavir: ((c / ) vs. ime) Observain: (c / ) cnsan fr > b and laer, fr < b. f = cnsan bu rbably valid fr any rducin/ressure scenari. Slide 7
28 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Varius Diffusiviy Equains) Hisrical Ne Eviner-Musa Cnce (194) Why n use liquid seudressure? Eviner and Musa (194) ne ha: The indefinie ineral may be evaluaed, as as dne fr he -hase sysem, and he ressure disribuin may be deermined. Hever, i ill be sufficien fr he calculain f he rduciviy facr cnsider nly he limiin frm... (i.e., he cnsan rery liquid relain). Slide 8
29 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Varius Diffusiviy Equains) Diffusiviy Equains fr a "Dry Gas": General Frm fr Gas: [ ] z Diffusiviy Relains: Pseudressure/Time: c c Definiins: Pseudressure: Dry Gas Relains ase Relains z Pseudressure/Pseudime: ( c ) Pseudime: n z d 1 a c d base z n 0 ( ) c ( ) n a Slide 9
30 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Varius Diffusiviy Equains) Dry Gas Pseudime Cndiin c vs., T=00 De F a n 0 1 ( ) c c d ( ) "Dry Gas" Pseudime Cndiin: ( c vs. ) Cnce: IF c cnsan, THEN seudime NOT required. c is NEVER cnsan seudime is alays required (fr liquid eq.). Hever, can enerae numerical sluin fr as cases (n seudime). Slide 30
31 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Varius Diffusiviy Equains) Diffusiviy Equains fr a "Dry Gas": Relains Frm Full Frmulain: ( ) Frm Arximain: Dry Gas Relains [ln( z)] ( ) c ( ) ( ) c ( ) Slide 31
32 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Varius Diffusiviy Equains) Dry Gas Cndiin z vs., T=00 De F z n base z d "Dry Gas" PVT Preries: ( z vs. ) Cnce: IF ( z) = cnsan, THEN -variable valid. ( z) cnsan fr <000 sia. Even ih numerical sluins, frmulain uld n be arriae. Slide 3
33 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Varius Diffusiviy Equains) Diffusiviy Equains fr a "Dry Gas": Relains Frm Full Frmulain: Dry Gas Relains ln Frm Arximain: z ( ) c c Slide 33
34 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl (Varius Diffusiviy Equains) Dry Gas Cndiin /( z) vs., T=00 De F z n base z d "Dry Gas" PVT Preries: (/( z) vs. ) Cnce: IF /( z) = cnsan, THEN -variable is valid. /( z) is NEVER cnsan seudressure required (fr liquid eq.). frmulain is never arriae (even if eneraed numerically). Slide 34
35 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl Gas Equain: Oil Equain: Waer Equain: Mulihase Equain: s s s s S R S R S R R S S c Cmressibiliy Terms: d dr d d c s 1 d dr d d c s 1 d d c 1 f c S c S c S c c Mulihase Case -Frm Relains (Perrine-Marin) (Varius Diffusiviy Equains) Slide 35
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