Petroleum Engineering 613
Natural Gas Engineering
Texas A&M University
Lecture 04:
Diffusivity Equations
for Flow in Porous Media
T.A. Blasingame, Texas A&M U.
Department of Petroleum Engineering
Texas A&M University
College Station, TX 77843-3116
+1.979.845.2292 — t-blasingame@tamu.edu
PETE 613
(2005A)
Diffusivity Equations
for Flow in Porous Media
Slide — 1
Lecture: Diffusivity Equations
 Diffusivity Equations:




"Black Oil" (p>pb)
"Solution-Gas Drive" (valid for all p, referenced for p<pb)
"Dry Gas" (p>pd)
Multiphase Flow
PETE 613
(2005A)
Diffusivity Equations
for Flow in Porous Media
Slide — 2
Diffusivity Equation: Black Oil (p>pb)
Diffusivity Equations for a Black Oil:
 Slightly Compressible Liquid: (General Form)
2
2
c(p)   p 
ct p
k t
 Slightly Compressible Liquid: (Small p and c form)
2 p 
PETE 613
(2005A)
ct p
k t
Diffusivity Equations
for Flow in Porous Media
Slide — 3
Diffusivity Equation: Black Oil — o and Bo vs. p
 Behavior of the o and Bo variables as functions of pressure for
an example black oil case. Note behavior for p>pb — both variables should be considered to be "approximately constant" for
the sake of developing flow relations. Such an assumption (i.e.,
o and Bo constant) is not an absolute requirement, but this
assumption is fundamental for the development of "liquid" flow
solutions in reservoir engineering.
PETE 613
(2005A)
Diffusivity Equations
for Flow in Porous Media
Slide — 4
Diffusivity Equation: Black Oil — co vs. p
 Behavior of the co variable as a function of pressure — example
black oil case. Note the "jump" at p=pb, this behavior is due to
the gas expansion at the bubblepoint.
PETE 613
(2005A)
Diffusivity Equations
for Flow in Porous Media
Slide — 5
Diffusivity Equation: Solution Gas Drive (p<pb)
Diffusivity Equations for Solution-Gas Drive: (p<pb)
 Oil Pseudopressure Form: (Accounts for o and Bo)
2
 ppo 
ct ppo
t
k
 Oil Pseudopressure Definition: (pn is any reference pressure)
ppo  o Bo  p
PETE 613
(2005A)
n

p
1
dp
pbase o Bo
Diffusivity Equations
for Flow in Porous Media
Slide — 6
Diffusivity Equation: Soln Gas Drive 1/(oBo) vs. p
1/(oBo) vs. p (pb=5000 psia, T=175 Deg F)
p
1
ppo  o Bo  p
dp
n p
base o Bo

 "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.
PETE 613
(2005A)
Diffusivity Equations
for Flow in Porous Media
Slide — 7
Diffusivity Equation: Soln Gas Drive (oco) vs. p
(oco) vs. p (pb=5000 psia, T=175 Deg F)
1
ta ,o  o ct n t
dt
0 o ( p)ct ( p)

 "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.
PETE 613
(2005A)
Diffusivity Equations
for Flow in Porous Media
Slide — 8
Diffusivity Equation: Soln Gas Drive (ct/lt) vs. tD
Dominated Flow in Solution-Gas-Drive Reservoirs,"
SPERE (November 1989) 503-512.
 Camacho-V., R.G. and Raghavan, R.: "Boundary-
Solution-Gas Drive — Mobility/Compressibility (Camacho)
 "Solution-Gas Drive" Behavior: ((ct/lt) vs. time)
 Observation: (ct/lt)  constant for p>pb and later, for p<pb.
 pwf = constant — but probably valid for any production/pressure scenario.
PETE 613
(2005A)
Diffusivity Equations
for Flow in Porous Media
Slide — 9
Diffusivity Equation: Soln Gas Drive (2-phase pp)
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 for
the two-phase system, and the pressure distribution may
be determined. However, it will be sufficient for the
calculation of the productivity factor to consider only the
limiting form ... (i.e., the constant property liquid relation).
PETE 613
(2005A)
Diffusivity Equations
for Flow in Porous Media
Slide — 10
Diffusivity Equation: Dry Gas Relations
Diffusivity Equations for a "Dry Gas:"
 General Form for Gas:
 [
p
g z
p ] 
ct p p
k z t
 Diffusivity Relations:
Pseudopressure/Time:
2
 pp 
 g ct pp
t
k
Pseudopressure/Pseudotime:

pp
 p p  (  g ct ) pn
k
ta
2
 Definitions:
Pseudopressure:
Pseudotime:
 g z 
p
p
ppg  
dp

p
 p  pn base  g z

PETE 613
(2005A)


1
t
ta   g ct
dt
n
0  g ( p )ct ( p )
Diffusivity Equations
for Flow in Porous Media
Slide — 11
Diffusivity Equation: Pseudotime (gcg vs. p)
Dry Gas Pseudotime Condition (gcg vs. p, T=200 Deg F)


1
t
ta   g ct
dt
n
0  g ( p )ct ( p )
 "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).
PETE 613
(2005A)
Diffusivity Equations
for Flow in Porous Media
Slide — 12
Diffusivity Equation: p2 Relations
Dry Gas — p2 Relations
Diffusivity Equations for a "Dry Gas:" p2 Relations
 p2 Form — Full Formulation:
2
2
 (p )

p
[ln(  g z )]( p ) 
2
2 2
 g ct 
k
t
( p2 )
 p2 Form — Approximation:
2
2
 (p ) 
PETE 613
(2005A)
 g ct 
k
t
( p2 )
Diffusivity Equations
for Flow in Porous Media
Slide — 13
Diffusivity Equation: p2 Relations (gz vs. p)
Dry Gas p2 Condition (gz vs. p, T=200 Deg F)
 g z 
p
p
ppg  
dp

p
 p  pn base  g z

 "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.
PETE 613
(2005A)
Diffusivity Equations
for Flow in Porous Media
Slide — 14
Diffusivity Equation: p Relations
Dry Gas — p Relations
Diffusivity Equations for a "Dry Gas:" p Relations
 p Form — Full Formulation:
    g z 
2  g ct p
 p  ln 
 (p) 

p   p  
k t
2
 p Form — Approximation:
2
 p
PETE 613
(2005A)
 g ct p
k
t
Diffusivity Equations
for Flow in Porous Media
Slide — 15
Diffusivity Equation: p Relations (p/(gz) vs. p)
Dry Gas p Condition (p/(gz) vs. p, T=200 Deg F)
 g z 
p
p
ppg  
dp

p
 p  pn base  g z

 "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).
PETE 613
(2005A)
Diffusivity Equations
for Flow in Porous Media
Slide — 16
Diffusivity Equation: Multiphase Relations
Multiphase Case — p-Form Relations (Perrine-Martin)
Gas Equation:
 k g
ko
kw      S g
So
Sw 
  
 Rso
 Rsw
 Rso
 Rsw
p    


B

B

B

t
B
B
B
o o
w w 
o
w  
  g g

  g
Oil Equation:
Compressibility Terms:
 k
  S 
   o p    o 
 o Bo
 t  Bo 
1 dBo Bg dRso
co  

Bo dp Bo dp
Water Equation:
 k
  S 
   w p    w 
  w Bw
 t  Bw 
1 dBw Bg dRsw
cw  

Bw dp Bw dp
Multiphase Equation:
1 dBg
cg  
Bg dp
ct p
ko k g k w
lt 


 p 
o  g  w
lt t
2
PETE 613
(2005A)
ct  co So  cw S w  c g S g  c f
Diffusivity Equations
for Flow in Porous Media
Slide — 17
Petroleum Engineering 613
Natural Gas Engineering
Texas A&M University
Lecture 04:
Diffusivity Equations
for Flow in Porous Media
(End of Lecture)
T.A. Blasingame, Texas A&M U.
Department of Petroleum Engineering
Texas A&M University
College Station, TX 77843-3116
+1.979.845.2292 — t-blasingame@tamu.edu
PETE 613
(2005A)
Diffusivity Equations
for Flow in Porous Media
Slide — 18