PERMEABILITY
Gas Flow in Porous Media
Gas Flow vs. Liquid Flow
• Gas density is a function of pressure (for isothermal
reservoir conditions)
 pV 
 pV 
– Real Gas Law
R 



 znT  reservoir  znT standard conditions
– We cannot assume gas flow in the reservoir is incompressible
• Gas density determined from Real Gas Law
ρg 
p γ g M air
zRT
– Darcy’s Law describes volumetric flow rate of gas flow at
reservoir conditions (in situ)
k  dp 
vs 

A
μ g  ds 
qg
Gas Flow vs. Liquid Flow
– Gas value is appraised at standard conditions
• Standard Temperature and Pressure
• qg,sc[scf/day] is mass flow rate (for specified g)
t2
Income [$]   Price [$/Mscf]  q g,sc[Mscf/day] dt [days]
t1
– For steady state flow conditions in the reservoir, as
flow proceeds along the flow path:
•
•
•
•
Mass flow rate, qg,sc , is constant
Pressure decreases
Density decreases
Volumetric flow rate, qg , increases
Gas Formation Volume Factor
– Given a volumetric gas flow rate at reservoir
conditions, qg, we need to determine the mass flow
rate, qg,sc
• Bg has oilfield units of [rcf/scf]
– scf is a specified mass of gas (i.e. number of moles)
– reservoir cubic feet per standard cubic foot
– (ft3)reservoir conditions / (ft3)standard conditions
Vres (znRT/p) res p scTz
Bg 


; z sc  1
Vsc (znRT/p) sc
pTsc
q g  q g,scBg
Linear Gas Flow
• 1-D Linear Flow System Assumptions
•
•
•
•
•
•
•
•
Steady state flow (mass flow rate, qg,sc , is constant)
Gas density is described by real gas law, g=(pgMair)/(zRT)
Horizontal flow path (dZ/ds=0  =p)
A(0s  L) = constant
Darcy flow (Darcy’s Law is valid)
k = constant (non-reactive fluid)
single phase (Sg=1)
Isothermal (T = constant)
A
qg
L
1
2
Linear Gas Flow
• Darcy’s Law:
k  dp 
vs 

A
μ g  ds 
qg
kA
q g ds  
dp
μg
A
qg
L
2
1
• q12 > 0, if p1 > p2
kA
q g,sc ds  
dp
Bg μ g
 Tsc   p 
 
q g,sc ds   k A 
 dp
 T p sc   z μ g 
 Tsc 

q g,sc  ds  k A 
 T p sc 
0
L
p2
p
p z μ g dp
1
Integral of Pressure Dependent Terms
• Two commonly used approaches to the evaluating the integral of the pressure
dependent terms:
p2
• (zg )=Constant approach, also called “p2 Method”
• valid when pressure < 2,500 psia
• for Ideal Gas a subset of this approach is valid
– z = 1; valid only for low pressures
– g depends on temperature only
p
I 
dp
z μg
p1
– Pseudopressure approach
• “real gas flow potential” - Paul Crawford, 2002 (see notes view).
• the integral is evaluated a priori to provide the pseudopressure function,
m(p)
– specified gas gravity, g
– specified reservoir temperature, T
– arbitrary base pressure, p0
– valid for any pressure range
(zg )=Constant
• Assumption that (zg ) is a constant function of pressure
is valid for pressures < 2,500 psia, across the range of
interest, for reservoir temperature and gas gravity
(zg )=Constant
• At other temperatures in the range of interest
Reservoir Temperature Gradient
dT/dZ  0.01 F/ft
(zg )=Constant, Linear Flow
• If (zg )=Constant
p2
p
1
1 p 
I 
dp 
p dp 
 

z
μ
(z
μ
)
(z
μ
)
g
g p1
g  2  p1
p1
p2
p2
2
• Gas Flow Rate (at standard conditions)
q g,sc
k A  Tsc 



L  T p sc 
 1

 2zμ
g

 2
 p1  p 22




Real Gas Pseudopressure
• Recall piecewise integration:
c
b
c
a
a
b
 f(x) dx   f(x) dx   f(x) dx
the ordering (position along x-axis) of the integral limits a,b
and c is arbitrary
2p
m(p)  
dp
zμ g
p0
p
• pseudopressure, m(p), is defined as:
• Piecewise Integration of the pressure dependent terms:
p
p1
 1
p
1  2 2p
2p
I 
dp   
dp - 
dp  m(p 2 )  m(p 1 )
z μg
2  p 0 z μ g
z μg
 2
p1
p0
p2
Real Gas Pseudopressure, Linear Flow
• Recalling our previous equation for linear gas flow
 Tsc 

q g,sc  ds  k A 
 T p sc 
0
L
p2
p
p z μ g dp
1
• And substituting for the pressure integral
q g,sc
k A  Tsc  1 

 m(p 1 )  m(p 2 )

L  T psc  2 
Radial Gas Flow
• The radial equations for gas flow follow from the previous
derivation for liquid flow and are left as self study
– (zg )=Constant
q g,sc
2 π k h  Tsc 



ln(r e /rw )  T p sc 
 1

 2zμ
g

 2
 p e  p 2w




– Pseudopressure
q g,sc
2 π k h  Tsc  1 

 m(p e )  m(p w )

ln(r e /rw )  T psc  2 