Diffusivity Equations for Flow in Porous Media
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1 Peroleum Engineering 324 Reservoir Performance Texas A&M Universiy T.A. Blasingame, Texas A&M U. Dearmen of Peroleum Engineering Texas A&M Universiy College Saion, TX Slide 1
2 Lecure: : "Black Oil" (> b ) "Soluion-Gas Drive" (valid for all, referenced for < b ) "Dry Gas" (> d ) Mulihase Flow Slide 2
3 Diffusiviy Equaion: Black Oil (> b ) for a Black Oil: Slighly Comressible Liquid: (General Form) 2 2 c c ( ) k 2 c k Slighly Comressible Liquid: (Small and c form) Slide 3
4 Diffusiviy Equaion: Black Oil o and B o vs. Behavior of he o and B o variables as funcions of ressure for an examle black oil case. Noe behavior for > b boh variables should be considered o be "aroximaely consan" for he sake of develoing flow relaions. Such an assumion (i.e., o and B o consan) is no an absolue requiremen, bu his assumion is fundamenal for he develomen of "liquid" flow soluions in reservoir engineering. Slide 4
5 Diffusiviy Equaion: Black Oil c o vs. Behavior of he c o variable as a funcion of ressure examle black oil case. Noe he "jum" a = b, his behavior is due o he gas exansion a he bubbleoin. Slide 5
6 Diffusiviy Equaion: Soluion Gas Drive (< b ) for Soluion-Gas Drive: (< b ) Oil Pseudoressure Form: (Accouns for o and B o ) 2 o c k o Oil Pseudoressure Definiion: ( n is any reference ressure) 1 B B d o o o n base o o Slide 6
7 Diffusiviy Equaion: Soln Gas Drive 1/( o B o ) vs. 1/( o B o ) vs. ( b =5000 sia, Various T) "Soluion-Gas Drive" PVT Proeries: (1/( o B o ), < b, b =5000 sia) Aem o illusrae ha 1/( o B o ) consan for < b. This would allow us o aroximae behavior using "liquid" equaions. Slide 7
8 Diffusiviy Equaion: Soln Gas Drive 1/( o B o ) vs. 1/( o B o ) vs. ( b =5000 sia, T=175 Deg F) 1 o o Bo d n B base o o "Soluion-Gas Drive" Pseudoressure Condiion: (1/( o B o ) vs. ) Conce: IF 1/( o B o ) consan, THEN oil seudoressure NOT required. 1/( o B o ) is NEVER "consan" bu does no vary significanly wih. Oil seudoressure calculaion sraighforward, bu robably no necessary. Slide 8
9 Diffusiviy Equaion: Soln Gas Drive ( oco) vs. ( oco) vs. (b=5000 sia, Various T) "Soluion-Gas Drive" PVT Proeries: ( oco, <b, b=5000 sia) Aem o illusrae ha oco is NEVER consan. CAN NOT aroximae behavior using "liquid" equaions (or so i seems). Slide 9
10 Diffusiviy Equaion: Soln Gas Drive ( oco) vs. ( oco) vs. (b=5000 sia, T=175 Deg F) 1 a,o o c d n 0 o ( )c ( ) "Soluion-Gas Drive" Pseudoressure Condiion: (( oco) vs. ) Conce: IF ( oco) consan, THEN oil seudoime NOT required. ( oco) is NEVER "consan" BUT, oil seudoime would be very difficul. Oher evidence suggess ha ignoring ( oco) variance is acceable. Slide 10
11 Diffusiviy Equaion: Soln Gas Drive (c/ ) vs. D Dominaed Flow in Soluion-Gas-Drive Reservoirs," SPERE (November 1989) Camacho-V., R.G. and Raghavan, R.: "Boundary- Soluion-Gas Drive Mobiliy/Comressibiliy (Camacho) "Soluion-Gas Drive" Behavior: ((c/ ) vs. ime) Observaion: (c/ ) consan for >b and laer, for <b. wf = consan bu robably valid for any roducion/ressure scenario. Slide 11
12 Diffusiviy Equaion: Soln Gas Drive (2-hase ) Hisorical Noe Evinger-Muska Conce (1942) Why no use liquid seudoressure? Evinger and Muska (1942) noe ha: The indefinie inegral may be evaluaed, as was done for he wo-hase sysem, and he ressure disribuion may be deermined. However, i will be sufficien for he calculaion of he roduciviy facor o consider only he limiing form... (i.e., he consan roery liquid relaion). Slide 12
13 Diffusiviy Equaion: Dry Gas Relaions for a "Dry Gas:" General Form for Gas: c [ ] g z k z Diffusiviy Relaions: Pseudoressure/Time: g c k 2 Pseudoressure/Pseudoime: ( g c ) n k a 2 Definiions: Pseudoressure: g z g Pseudoime: d base g z n 1 a g c d n 0 g ( )c ( ) Slide 13
14 Diffusiviy Equaion: Pseudoime ( gcg vs. ) Dry Gas Pseudoime Condiion ( gcg vs., Various T) "Dry Gas" PVT Proeries: ( gcg vs. ) Conce: If gcg consan, seudoime NOT required. Readily observe ha gcg is NEVER consan, seudoime required. Slide 14
15 Diffusiviy Equaion: Pseudoime ( gcg vs. ) Dry Gas Pseudoime Condiion ( gcg vs., T=200 Deg F) 1 a g c d n 0 g ( )c ( ) "Dry Gas" Pseudoime Condiion: ( gcg vs. ) Conce: IF gcg consan, THEN seudoime NOT required. gcg is NEVER consan seudoime is always required (for liquid eq.). However, can generae numerical soluion for gas cases (no seudoime). Slide 15
16 Diffusiviy Equaion: 2 Relaions Dry Gas 2 Relaions for a "Dry Gas:" 2 Relaions 2 Form Full Formulaion: g c 2 ( ) 2 [ln( g z )] ( ) ( ) k Form Aroximaion: g c 2 ( ) ( ) k 2 2 Slide 16
17 Diffusiviy Equaion: Gas 2 ( gz vs., Various T) Dry Gas 2 Condiion ( gz vs., Various T) "Dry Gas" PVT Proeries: ( gz vs. ) Basis for he "ressure-squared" aroximaion (i.e., use of 2 variable). Conce: ( gz) = consan, valid only for <2000 sia. Slide 17
18 Diffusiviy Equaion: Gas 2 ( gz vs., T=200 Deg F) Dry Gas 2 Condiion ( gz vs., T=200 Deg F) g z g d base g z n "Dry Gas" PVT Proeries: ( gz vs. ) Conce: IF ( gz) = consan, THEN 2-variable valid. ( gz) consan for <2000 sia. Even wih numerical soluions, 2 formulaion would no be aroriae. Slide 18
19 Diffusiviy Equaion: Relaions Dry Gas Relaions for a "Dry Gas:" Relaions Form Full Formulaion: g z 2 g c ln ( ) k 2 Form Aroximaion: g c k 2 Slide 19
20 Diffusiviy Equaion: Gas (/( gz) vs., Various T) Dry Gas Condiion (/( gz) vs., Various T) "Dry Gas" PVT Proeries: (/( gz) vs. ) Basis for he "ressure" aroximaion (i.e., use of variable). Conce: (/ gz) = consan (never valid). Slide 20
21 Diffusiviy Equaion: Gas (/( gz) vs., T=200 Deg F) Dry Gas Condiion (/( gz) vs., T=200 Deg F) g z g d base g z n "Dry Gas" PVT Proeries: (/( gz) vs. ) Conce: IF /( gz) = consan, THEN -variable is valid. /( gz) is NEVER consan seudoressure required (for liquid eq.). formulaion is never aroriae (even if generaed numerically). Slide 21
22 Diffusiviy Equaion: Mulihase Relaions Mulihase Case -Form Relaions (Perrine-Marin) Gas Equaion: kg ko k w S g So Sw Rso Rsw Rso Rsw B B B B B B o o w w o w g g g Oil Equaion: k S o o B o o Bo Waer Equaion: k S w w w Bw Bw Mulihase Equaion: c ko k g k w o g w 2 Comressibiliy Terms: 1 dbo B g drso co Bo d Bo d 1 dbw B g drsw cw Bw d Bw d 1 db g c g B g d c co So cw S w c g S g c f Slide 22
23 Peroleum Engineering 324 Reservoir Performance Texas A&M Universiy (End of Lecure) T.A. Blasingame, Texas A&M U. Dearmen of Peroleum Engineering Texas A&M Universiy College Saion, TX Slide 23
Tom BLASINGAME Texas A&M U. Slide 1
Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl Slide 1 Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl Slide Fundamenal Fl Lecure 7 The Diffusiviy Equain fr Mulihase Fl Slide 3
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