Diffusivity Equation
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1 Perleum Enineerin 34 Reservir Perfrmane Diffusiviy Equain 13 February 008 Thmas A. lasiname, Ph.D., P.E. Dilhan Il Dearmen f Perleum Enineerin Dearmen f Perleum Enineerin Texas A&M Universiy Texas A&M Universiy Cllee Sain, TX (USA) Cllee Sain, TX (USA) blasiname@amu.edu dilhan@amu.edu Perleum Enineerin 34 Reservir Perfrmane Dearmen f Perleum Enineerin Texas A&M U. Diffusiviy Equain Leure 05 T.A. lasiname/dilhan Il (13 February 008) Slide 1/17
2 Perleum Enineerin 34 Reservir Perfrmane Diffusiviy Equain Develmen Perleum Enineerin 34 Reservir Perfrmane Dearmen f Perleum Enineerin Texas A&M U. Diffusiviy Equain Leure 05 T.A. lasiname/dilhan Il (13 February 008) Slide /17
3 Diffusiviy Equain: Cne (1/3) Gvernin Equain: Mass Cninuiy Equain (Mass alane f he sysem) Cnrl Vlume (Sysem) Mass in Mass u Rae f mass fl in he sysem durin he inerval, Δ Rae f mass fl Rae f mass aumulain u f he sysem in he sysem durin durin he inerval, Δ he inerval, Δ Disussin: Diffusiviy Equain Cneually, he diffusiviy equain is bained by alyin mass balane ver a nrl vlume. Equain f min (Dary's La) and equain f sae (PVT relains) are hen mbined ih he mass balane equain bain he final frm f he diffusiviy equain. Perleum Enineerin 34 Reservir Perfrmane Dearmen f Perleum Enineerin Texas A&M U. Diffusiviy Equain Leure 05 T.A. lasiname/dilhan Il (13 February 008) Slide 3/17
4 Diffusiviy Equain: Cne (/3) Gvernin Equains: (Mahemaial desriins) Mass ninuiy equain: Equain f min (Dary's La): Equain f sae (ishermal mressibiliy fr a fluid): ( φρ) ( ρ r ν ) r r ν ( + ρ ) 1 ρ ρ f (Gradien) f ( x, y, z) x, f y, f z (in Caresian rdinaes) Disussin: Diffusiviy Equain Definiins fr he veliy and densiy erms are insered in he mass ninuiy equain. Perleum Enineerin 34 Reservir Perfrmane Dearmen f Perleum Enineerin Texas A&M U. Diffusiviy Equain Leure 05 T.A. lasiname/dilhan Il (13 February 008) Slide 4/17
5 Diffusiviy Equain: Cne (3/3) Diffusiviy equain fr he fl f a sinle hase fluid in a rus media ih rese ime and disane: Slihly Cmressible Liquid: (General Frm) ( ) + φ φ Slihly Cmressible Liquid: (Small and frm) Disussin: Diffusiviy Equain Diffusiviy equain is a arial differenial equain. Sluin f he diffusiviy equain requires an INITIAL CONDITION and TWO OUNDARY CONDITIONS. Diffusiviy equain (abve) alies fr any emeri nfiurain. Diffusiviy equain (abve) requires he fllin assumins hmeneus, isri frmain and 100% sauraed re sae. Perleum Enineerin 34 Reservir Perfrmane Dearmen f Perleum Enineerin Texas A&M U. Diffusiviy Equain Leure 05 T.A. lasiname/dilhan Il (13 February 008) Slide 5/17
6 Diffusiviy Equain: Reservir Mdels Frm: Kaa Enineerin (Dynami Fl Analysis, 007) Disussin: Reservir Mdels Fraured ell mdel Hriznal ell ih ransverse verial fraures Hriznal ell mdel Fully enerain slaned ell A mdel is a mahemaial arximain f he real sysem mbined ih he rules f hysis. The suess f a mdel deends n rerduibiliy f he main drives f he sysem. We use ANALYTICAL mdels fr he arriae reresenain f he hysial drive mehanisms f he sysem. ANALYTICAL mdel requires ha he hysial sysem shuld be relaed by a se f relaively simle LINEAR equains hih mdel he diffusin ress ihin a iee f r, assume a simle shae fr he reservir and ive he iniial sae f he sysem. Perleum Enineerin 34 Reservir Perfrmane Dearmen f Perleum Enineerin Texas A&M U. Diffusiviy Equain Leure 05 T.A. lasiname/dilhan Il (13 February 008) Slide 6/17
7 Diffusiviy Equain: Radial Fl Diffusiviy equain in ylindrial rdinaes (radial fl ms nvenien fr reservir enineerin aliains): Slihly Cmressible Liquid: r + 1 r r φ φ,, and are assumed be nsans (n a funin f ressure). Disussin: Diffusiviy Equain If he ermeabiliy () is larer, he ressure hane ill be smaller. If he vissiy () is larer, he ressure hane ill be smaller. The rai, / is alled he mbiliy rai. If he rsiy (φ) is larer, he ressure hane ill be smaller. If he al mressibiliy ( ) is larer, he ressure hane ill be smaller. Perleum Enineerin 34 Reservir Perfrmane Dearmen f Perleum Enineerin Texas A&M U. Diffusiviy Equain Leure 05 T.A. lasiname/dilhan Il (13 February 008) Slide 7/17
8 Diffusiviy Equain: la Oil and vs. ehavir f he and variables as funins f ressure fr an examle bla il ase. Ne behavir fr > b bh variables shuld be nsidered be "arximaely nsan" fr he sae f develin fl relains. Suh an assumin (i.e., and nsan) is n an abslue requiremen, bu his assumin is fundamenal fr he develmen f "liquid" fl sluins in reservir enineerin. Perleum Enineerin 34 Reservir Perfrmane Dearmen f Perleum Enineerin Texas A&M U. Diffusiviy Equain Leure 05 T.A. lasiname/dilhan Il (13 February 008) Slide 8/17
9 Diffusiviy Equain: la Oil vs. ehavir f he variable as a funin f ressure examle bla il ase. Ne he "jum" a b, his behavir is due he as exansin a he bubblein. Perleum Enineerin 34 Reservir Perfrmane Dearmen f Perleum Enineerin Texas A&M U. Diffusiviy Equain Leure 05 T.A. lasiname/dilhan Il (13 February 008) Slide 9/17
10 Diffusiviy Equain: Sluin Gas Drive < b 1 [ ] d n base "Sluin-Gas Drive" Pseudressure Cndiin: (1/( ) vs. ) Cne: IF 1/( ) nsan, THEN il seudressure NOT required. 1/( ) is NEVER "nsan" bu des n vary sinifianly ih. Oil seudressure alulain sraihfrard, bu rbably n neessary. Perleum Enineerin 34 Reservir Perfrmane Dearmen f Perleum Enineerin Texas A&M U. Diffusiviy Equain Leure 05 T.A. lasiname/dilhan Il (13 February 008) Slide 10/17
11 Diffusiviy Equain: Dry Gas Relains Diffusiviy Equains fr a "Dry Gas:" General Frm fr Gas: [ ] z φ Diffusiviy Relains: Pseudressure/Time: Pseudressure/Pseudime: Definiins: Pseudressure: z φ n z base z φ ( ) Pseudime: Perleum Enineerin 34 Reservir Perfrmane Dearmen f Perleum Enineerin Texas A&M U. Diffusiviy Equain Leure 05 T.A. lasiname/dilhan Il (13 February 008) n d [ ] a d n 0 ( ) ( ) 1 a Slide 11/17
12 Diffusiviy Equain: (Pseudime Cndiin) a n 0 1 ( ) [ ] d ( ) "Dry Gas" Pseudime Cndiin: ( vs. ) Cne: IF nsan, THEN seudime NOT required. is NEVER nsan seudime is alays required (fr liquid eq.). Hever, an enerae numerial sluin fr as ases (n seudime). Perleum Enineerin 34 Reservir Perfrmane Dearmen f Perleum Enineerin Texas A&M U. Diffusiviy Equain Leure 05 T.A. lasiname/dilhan Il (13 February 008) Slide 1/17
13 Diffusiviy Equain: Cndiin z n base z d "Dry Gas" PVT Preries: ( z vs. ) Cne: IF ( z) nsan, THEN -variable valid. ( z) nsan fr <000 sia. Even ih numerial sluins, frmulain uld n be arriae. Perleum Enineerin 34 Reservir Perfrmane Dearmen f Perleum Enineerin Texas A&M U. Diffusiviy Equain Leure 05 T.A. lasiname/dilhan Il (13 February 008) Slide 13/17
14 Diffusiviy Equain: Cndiin Diffusiviy Equains fr a "Dry Gas:" Relains Frm Full Frmulain: ( ) [ln( Frm Arximain: ( φ ) ( z)] ( ) ) φ ( ) Perleum Enineerin 34 Reservir Perfrmane Dearmen f Perleum Enineerin Texas A&M U. Diffusiviy Equain Leure 05 T.A. lasiname/dilhan Il (13 February 008) Slide 14/17
15 Diffusiviy Equain: Relains z n base z d "Dry Gas" PVT Preries: (/( z) vs. ) Cne: IF /( z) nsan, THEN -variable is valid. /( z) is NEVER nsan seudressure required (fr liquid eq.). frmulain is never arriae (even if eneraed numerially). Perleum Enineerin 34 Reservir Perfrmane Dearmen f Perleum Enineerin Texas A&M U. Diffusiviy Equain Leure 05 T.A. lasiname/dilhan Il (13 February 008) Slide 15/17
16 Diffusiviy Equain: Relains Diffusiviy Equains fr a "Dry Gas:" Relains Frm Full Frmulain: ln z ( ) Frm Arximain: φ Perleum Enineerin 34 Reservir Perfrmane Dearmen f Perleum Enineerin Texas A&M U. Diffusiviy Equain Leure 05 T.A. lasiname/dilhan Il (13 February 008) Slide 16/17
17 Slide 17/17 Perleum Enineerin 34 Reservir Perfrmane Dearmen f Perleum Enineerin Texas A&M U. Diffusiviy Equain Leure 05 T.A. lasiname/dilhan Il (13 February 008) Diffusiviy Equain: Mulihase Case Gas Equain: Oil Equain: Waer Equain: Mulihase Equain: s s s s S R S R S R R φ S φ S φ λ φ Cmressibiliy Terms: d dr d d s + 1 d dr d d s + 1 d d 1 f S S S λ + +
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