Page 313 - Fundamentals of Reservoir Engineering

REAL GAS FLOW: GAS WELL TESTING 248

Fig. 8.3 Real gas pseudo pressure, as a function of the actual pressure, as derived in
table 8.1; (Gas gravity, 0.85; Temperature 200°F)

∆p

2p
µ Z ∆ (Area) = ∆ m (p) = (2p / µZ) p
∆
2p
µ Z

m (p)
p Pressure

Fig. 8.4 2p/µZ as a function of pressure

()
∂ mp ∂ m () p ∂ p
Since = ⋅
r ∂ ∂ p r ∂
()
∂ mp 2p
and =
∂ p µΖ
()
∂ mp 2p ∂ p
Then = (8.8)
r ∂ µΖ r ∂

()
∂ mp 2p ∂ p
and similarly = (8.9)
t ∂ µΖ t ∂

These relations are evident from fig. 8.4 and, substituting for ∂p/∂r and ∂p/∂t in
equ. (5.1), using equs. (8.8) and (8.9) gives

()
1 ∂ kρ µΖ ∂ mp µΖ ∂ m () p
c
r = φρ (8.10)
rr ∂ µ 2p r ∂ 2p t ∂

Finally, using the equation of state for a real gas
Mp
ρ =
ZRT

and substituting this expression for ρ in equ. (8.10) leads, after some cancellation of
terms, to the simplified expression

()
1 ∂ r ∂ mp = φµ c ∂ m () p (8.11)
r ∂ r r ∂ k t ∂

Equation (8.11) has precisely the same form as the diffusivity equation, (5.20), except
that the dependent variable has been replaced by m(p).