Measurement of Thennal Diffusivity, Optical Transmission ...
Texas A&M University Lecture 04: Diffusivity Equations …050131).pdf · PETE 613 (2005A) Slide...
Transcript of Texas A&M University Lecture 04: Diffusivity Equations …050131).pdf · PETE 613 (2005A) Slide...
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)
Slide — 2Diffusivity Equationsfor Flow in Porous Media
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)
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<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)
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<pb.pwf = constant — but probably valid for any production/pressure scenario.
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Diffusivity Equation: Soln Gas Drive (ct/t) vs. tD
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 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)
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
tcttgntga )()(
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()( 2222
22 ptk
cpz
pp tg
g
)()( 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<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)
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 w
w
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 — [email protected]
Petroleum Engineering 613Natural Gas Engineering
Texas A&M University
Lecture 04:Diffusivity Equations
for Flow in Porous Media(End of Lecture)