# texas a&m university lecture 04: diffusivity equations 050131).pdf · pete 613 (2005a) slide —1...     Post on 30-Jun-2018

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• PETE 613(2005A)

Slide 1Diffusivity Equationsfor Flow in Porous Media

T.A. Blasingame, Texas A&M U.Department of Petroleum Engineering

Texas A&M UniversityCollege Station, TX 77843-3116

+1.979.845.2292 t-blasingame@tamu.edu

Petroleum Engineering 613Natural Gas Engineering

Texas A&M University

Lecture 04:Diffusivity Equations

for Flow in Porous Media

• PETE 613(2005A)

Slide 2Diffusivity Equationsfor Flow in Porous Media

Diffusivity Equations: "Black Oil" (p>pb) "Solution-Gas Drive" (valid for all p, referenced for ppd)Multiphase Flow

Lecture: Diffusivity Equations

• PETE 613(2005A)

Slide 3Diffusivity Equationsfor Flow in Porous Media

Diffusivity Equations for a Black Oil:Slightly Compressible Liquid: (General Form)

Slightly Compressible Liquid: (Small p and c form)

tp

kc

p t

2

tp

kc

ppc t

22)(

Diffusivity Equation: Black Oil (p>pb)

• PETE 613(2005A)

Slide 4Diffusivity Equationsfor Flow in Porous Media

Behavior of the o and Bo variables as functions of pressure foran example black oil case. Note behavior for p>pb both vari-ables should be considered to be "approximately constant" forthe sake of developing flow relations. Such an assumption (i.e.,o and Bo constant) is not an absolute requirement, but thisassumption is fundamental for the development of "liquid" flowsolutions in reservoir engineering.

Diffusivity Equation: Black Oil o and Bo vs. p

• PETE 613(2005A)

Slide 5Diffusivity Equationsfor Flow in Porous Media

Behavior of the co variable as a function of pressure exampleblack oil case. Note the "jump" at p=pb, this behavior is due tothe gas expansion at the bubblepoint.

Diffusivity Equation: Black Oil co vs. p

• PETE 613(2005A)

Slide 6Diffusivity Equationsfor Flow in Porous Media

Diffusivity Equations for Solution-Gas Drive: (p

• PETE 613(2005A)

Slide 7Diffusivity Equationsfor Flow in Porous Media

1/(oBo) vs. p (pb=5000 psia, T=175 Deg F)

"Solution-Gas Drive" Pseudopressure Condition: (1/(oBo) vs. p)Concept: IF 1/(oBo) constant, THEN oil pseudopressure NOT required.1/(oBo) is NEVER "constant" but does not vary significantly with p.Oil pseudopressure calculation straightforward, but probably not necessary.

dpB

pp

Bpoobase

poopo n 1

Diffusivity Equation: Soln Gas Drive 1/(oBo) vs. p

• PETE 613(2005A)

Slide 8Diffusivity Equationsfor Flow in Porous Media

(oco) vs. p (pb=5000 psia, T=175 Deg F)

"Solution-Gas Drive" Pseudopressure Condition: ((oco) vs. p)Concept: IF (oco) constant, THEN oil pseudotime NOT required. (oco) is NEVER "constant" BUT, oil pseudotime would be very difficult.Other evidence suggests that ignoring (oco) variance is acceptable.

dtpcp

tctto

ntoo,a )()(1

0

Diffusivity Equation: Soln Gas Drive (oco) vs. p

• PETE 613(2005A)

Slide 9Diffusivity Equationsfor Flow in Porous Media

Solution-Gas Drive Mobility/Compressibility (Camacho)

"Solution-Gas Drive" Behavior: ((ct/t) vs. time)Observation: (ct/t) constant for p>pb and later, for p

• PETE 613(2005A)

Slide 10Diffusivity Equationsfor Flow in Porous Media

Historical Note Evinger-Muskat Concept (1942)Why not use liquid pseudopressure?

Evinger and Muskat (1942) note that:The indefinite integral may be evaluated, as was done forthe two-phase system, and the pressure distribution maybe determined. However, it will be sufficient for thecalculation of the productivity factor to consider only thelimiting form ... (i.e., the constant property liquid relation).

Diffusivity Equation: Soln Gas Drive (2-phase pp)

• PETE 613(2005A)

Slide 11Diffusivity Equationsfor Flow in Porous Media

t

p

k

cp ptgp

2

tp

zp

kc

pz

p tg

][

a

pptgp t

pc

kp

n

)(2

dpz

pppp

zp

gbasep

gpg

n

dt

pcptct

tgntga )()(

1

0

Diffusivity Equation: Dry Gas Relations

Diffusivity Equations for a "Dry Gas:"General Form for Gas:

Diffusivity Relations:Pseudopressure/Time: Pseudopressure/Pseudotime:

Definitions:Pseudopressure: Pseudotime:

• PETE 613(2005A)

Slide 12Diffusivity Equationsfor Flow in Porous Media

Dry Gas Pseudotime Condition (gcg vs. p, T=200 Deg F)

"Dry Gas" Pseudotime Condition: (gcg vs. p)Concept: IF gcg constant, THEN pseudotime NOT required.gcg is NEVER constant pseudotime is always required (for liquid eq.).However, can generate numerical solution for gas cases (no pseudotime).

dtpcp

tcttgn

tga )()(1

0

Diffusivity Equation: Pseudotime (gcg vs. p)

• PETE 613(2005A)

Slide 13Diffusivity Equationsfor Flow in Porous Media

Dry Gas p2 Relations

Diffusivity Equations for a "Dry Gas:" p2 Relationsp2 Form Full Formulation:

p2 Form Approximation:

)()()][ln()( 222222 p

tk

cpz

pp tgg

)()( 222 ptk

cp tg

Diffusivity Equation: p2 Relations

• PETE 613(2005A)

Slide 14Diffusivity Equationsfor Flow in Porous Media

Dry Gas p2 Condition (gz vs. p, T=200 Deg F)

"Dry Gas" PVT Properties: (gz vs. p)Concept: IF (gz) = constant, THEN p2-variable valid. (gz) constant for p

• PETE 613(2005A)

Slide 15Diffusivity Equationsfor Flow in Porous Media

Dry Gas p RelationsDiffusivity Equations for a "Dry Gas:" p Relationsp Form Full Formulation:

p Form Approximation:

tp

k

cp

p

z

pp tgg

22 )(ln

tp

k

cp tg

2

Diffusivity Equation: p Relations

• PETE 613(2005A)

Slide 16Diffusivity Equationsfor Flow in Porous Media

"Dry Gas" PVT Properties: (p/(gz) vs. p)Concept: IF p/(gz) = constant, THEN p-variable is valid.p/(gz) is NEVER constant pseudopressure required (for liquid eq.).p formulation is never appropriate (even if generated numerically).

Dry Gas p Condition (p/(gz) vs. p, T=200 Deg F)

dpz

pppp

zp

gbasep

gpg

n

Diffusivity Equation: p Relations (p/(gz) vs. p)

• PETE 613(2005A)

Slide 17Diffusivity Equationsfor Flow in Porous Media

Multiphase Case p-Form Relations (Perrine-Martin)Gas Equation:

Oil Equation:

Water Equation:

Multiphase Equation:

w

wsw

o

oso

g

g

ww

wsw

oo

oso

gg

g

BS

RBS

RB

S

tp

Bk

RB

kR

B

k

o

o

oo

oBS

tp

Bk

w

w

ww

wBS

tp

Bk

tpc

pt

t

2

Compressibility Terms:

dpdR

B

B

dpdB

Bc so

o

go

oo

1

dpdR

B

B

dpdB

Bc sw

w

gw

ww

1

dp

dB

Bc g

gg

1

fggwwoot cScScScc ww

g

g

o

ot

kkk

Diffusivity Equation: Multiphase Relations

• PETE 613(2005A)

Slide 18Diffusivity Equationsfor Flow in Porous Media

T.A. Blasingame, Texas A&M U.Department of Petroleum Engineering

Texas A&M UniversityCollege Station, TX 77843-3116

+1.979.845.2292 t-blasingame@tamu.edu

Petroleum Engineering 613Natural Gas Engineering

Texas A&M University

Lecture 04:Diffusivity Equations

for Flow in Porous Media(End of Lecture)

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