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5. L
A
q
P1
q
P2
a. Conditions
1) horizontal system,
dz
ds
= 0
2) linear system, A = constant
3) incompressible liquid, q = constant
4) laminar flow, can use Darcy's equation
5) non-reactive fluid, k = constant
6) 100% saturated with one fluid
7) constant temperature, µ, q
1
A. Flow of incompressible liquid
1. Horizontal, linear flow system
Derivation of Darcy,s Equa
t
i
o
n

6. b. derivation of flow equation
vs = - k
µ
dP
ds
-
ρg
1.0133 x 106
dz
ds
vs = - k
µ
dP
ds
=
q
A
q ds
0
L
= - kA
µ
dP
p
1
p2
q L - 0 = - kA
µ
P2 - P1
q = kA
Lµ
P2 - P1
Note: P
1
acts at L = 0
P
2
acts at L = L
q is + if flow is from L = 0 to L = L
2

7. h
rw
re
Pe
Pw
re rw
a. Conditions
1) horizontal system,
dz
ds = 0
2) radial system, A = 2πrh , ds = - dr, flow is inward
3) constant thickness, h = constant
4) incompressible liquid, q = constant
5) laminar flow, use Darcy equation
6) non-reactive fluid, k = constant
7) 100% saturated with liquid,
8) constant temperature, µ, q
8
4. Horizontal, radial flow sy s-
tem

8. b. Derivation of flow Equation
vs = - k
µ
dP
ds
-
ρg
1.0133 x 106
dz
ds
vs = + k
µ
dP
dr
=
q
A
=
q
2πrh
q
2πh
dr
r
rw
re
= k
µ
dp
pw
pe
q
2πh
1n(re) - 1n( rw) = k
µ
Pe - Pw
q = 2πhk
µ
L
A
q
P1
q
P2
a. Conditions
1) horizontal system,
dz
ds = 0
2) linear system, A = constant
3) compressible gas flow, q = f(p)
4) laminar flow, use Darcy equation
5) non-reactive fluid, k= constant
6) 100% saturated with one fluid
7) constant temperature
9
1n (re/rw)
Pe - Pw
Note: if q is + , flow is from re to rw
B. Flow of gas (compressible fluid)
1. horizontal, linear flow system

9. b. Assumptions
1) µ, Z = constant
2) Z(and µ ) can be determined at mean pressure
c. Derivation of equation for qsc
vs = - k
µ
dP
ds
-
ρg
1.0133 x 106
dz
ds
vs = - k
µ
dP
ds
=
q
Ads
but
q =
Psc qscz T
PTsc
thus
Psc T qsc
Tsc A
ds
o
L
= - k PdP
µz
p1
p2
Psc T qsc
Tsc A
L -0 = - k
µz
P2
2 - P1
2
2
qsc = kA
µL
Tsc
Tz Psc
P1
2
- P2
2
2
Note: real gas equation of state
Pq = Z n R T
where q = volumetric flow/time
n = mass flow/time
thus,
Pq
Pscqsc
= Z n R T
n R Tsc
q =
Psc qscz T
Tsc
1
P
where qsc is constant
Z is determined at P, T
10

10. d. Derivation of equation for q
qsc = kA
µL
Tsc
Tz Psc
P1
2
- P2
2
2
but
qsc =
P q Tsc
Z Psc T
= k A
µL
Tsc
T z Psc
P1
2 - P2
2
2
q = k A
µL
1
P
P1
2 - P2
2
2
q = k A
µL
2
P1 + P2
P1
2 - P2
2
2
q = k A
µL
P1 - P2
This equation is identical to the equation for horizontal, linear flow of incompressible liquid
thus
if gas flow rate is determined at mean pressure, P, the equation for incompressible liquid
can be used for compressible gas!
Note: real gas equation of state
Pq = Z n R T
thus
Psc qsc
P q
=
n R Tsc
z n R T
where
P =
P1 + P2
2
P = volumetric flow rate at P, T
z is determined at P, T
qsc =
P q Tsc
z Psc T
11

11. h
rw
re
Pe
Pw
re rw
a. Conditions
1) horizontal system
dz
ds = 0
2) radial system, A = 2πrL, ds = - dr,
inward flow
3) constant thickness, h = constant
4) compressible gas flow, q = f (P)
5) laminar flow, use Darcy equation
6) non-reactive fluid, k = constant
7) 100% saturated with one fluid
8) constant temperature
12
2.Horizontal, radial flow system

12. b. Assumptions
µz = constant
z (and µ ) can be determined at mean pressure
c. derivation of equation for qsc
vs = - k
µ
dP
ds
-
ρg
1.0133 x 106
dz
ds
vs = - k
µ
dP
ds
=
q
A
but
q =
Psc qsc z T
PTsc
and
A = 2πrh and ds = - dr
thus
Psc T qsc
2Tsc π h
dr
r
rw
re
= k
ρdP
µz
Pw
Pe
PscT qsc
2 Tsc π h
1n
re
rw
= k
µz
Pe
2 - Pw
2
2
qsc = 2 π h k
µ 1n re/rw
Tsc
Psc zT
Pe
2 - Pw
2
2
13

13. d. derivation of equation for q
qsc = 2 π h k
µ 1n re/rw
Tsc
Psc zT
Pe
2 - Pw
2
2
but
q =
P q Tsc
z Psc T
thus
P q Tsc
z Psc T
= 2 π h k
µ 1n re/rw
Tsc
Psc zT
Pe
2 - Pw
2
2
q = 2 π h k
µ 1n re/rw
1
P
(Pe
2 - Pw
2 )
2
q = 2 π h k
µ 1n re/rw
2
Pe + Pw
(Pe
2 - Pw
2 )
2
q = 2 π h k
µ 1n re/rw
Pe - Pw
Note: Equation for real gas is identical to equation for incompressible liquid when
volumetric flow rate of gas, q, is measured at mean pressure.
14

14. II - 8
Θ
-Z
P1
S
X
P2
a. Conditions
1) non-horizontal system,
dz
ds
= sinθ
= constant
2) linear system, A = constant
3) incompressible liquid, q = constant
4) laminar flow, use Darcy equation
5) non-reactive fluid, k = constant
6) 100% saturated with one fluid
7) constant temperature µ, q
4
2. Non-horizontal, linear sy stem
inclined