Transcript Document
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:
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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
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Steady state flow (mass flow rate, qg,sc , is constant)
Gas density is described by real gas law, g=(pgMair)/(zRT)
Horizontal flow path (dZ/ds=0 =p)
A(0s 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:
qg
k
vs
A
μg
dp
ds
kA
q g ds
dp
μg
A
qg
L
2
1
• q12 > 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
• (zg )=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
(zg )=Constant
• Assumption that (zg ) is a constant function of pressure
is valid for pressures < 2,500 psia, across the range of
interest, for reservoir temperature and gas gravity
(zg )=Constant
• At other temperatures in the range of interest
Reservoir Temperature Gradient
dT/dZ 0.01 F/ft
(zg )=Constant, Linear Flow
• If (zg )=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
– (zg )=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