Download - Texas A&M University Lecture 04: Diffusivity Equations …050131).pdf · PETE 613 (2005A) Slide —1 Diffusivity ... Texas A&M University Lecture 04: Diffusivity Equations for Flow

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 — [email protected]

Petroleum Engineering 613Natural Gas Engineering

Texas A&M University

Lecture 04:Diffusivity Equations

for Flow in Porous Media

PETE 613(2005A)


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

Lecture: Diffusivity Equations

PETE 613(2005A)


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)


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)


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)


Diffusivity Equations for Solution-Gas Drive: (p<pb)Oil Pseudopressure Form: (Accounts for o and Bo)

Oil Pseudopressure Definition: (pn is any reference pressure)

t

p

kc

p potpo

2

dpB

pp

Bpoobase

poopo n 1

Diffusivity Equation: Solution Gas Drive (p<pb)

PETE 613(2005A)


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)


(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)


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<pb.pwf = constant — but probably valid for any production/pressure scenario.

C

amac

ho

-V.,

R.G

.an

dR

agh

avan

,R.:

"Bo

un

dar

y-D

om

inat

edF

low

inS

olu

tio

n-G

as-D

rive

Res

ervo

irs,

"S

PE

RE

(No

vem

ber

1989

)50

3-51

2.

Diffusivity Equation: Soln Gas Drive (ct/t) vs. tD

PETE 613(2005A)


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)


t

p

k

cp ptg

p

2

tp

zp

kc

pz

p t

g

][

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)


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

tcttgntga )()(

1

0

Diffusivity Equation: Pseudotime (gcg vs. p)

PETE 613(2005A)


Dry Gas — p2 Relations

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

p2 Form — Approximation:

)()()][ln()( 2222

22 ptk

cpz

pp tg

g

)()( 222 ptk

cp tg

Diffusivity Equation: p2 Relations

PETE 613(2005A)


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<2000 psia.Even with numerical solutions, p2 formulation would not be appropriate.

dpz

pppp

zp

gbasep

gpg

n

Diffusivity Equation: p2 Relations (gz vs. p)

PETE 613(2005A)


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)


"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)


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 w

w

g

g

o

ot

kkk

Diffusivity Equation: Multiphase Relations

PETE 613(2005A)


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

Texas A&M UniversityCollege Station, TX 77843-3116

+1.979.845.2292 — [email protected]

Petroleum Engineering 613Natural Gas Engineering

Texas A&M University

Lecture 04:Diffusivity Equations

for Flow in Porous Media(End of Lecture)

Download - Texas A&M University Lecture 04: Diffusivity Equations …050131).pdf · PETE 613 (2005A) Slide —1 Diffusivity ... Texas A&M University Lecture 04: Diffusivity Equations for Flow

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