Page 319 - Fundamentals of Reservoir Engineering

REAL GAS FLOW: GAS WELL TESTING 254

e r 2p q 2
() nD
∆ mp = βρ dr (8.20)
µΖ 2 π rh
w r
Also since ρ = γ g × density of air at s.c. × E

p
= constant × γ ×
g
ΖΤ
where γ g is the gas gravity (air=1)

Then equ. (8.20) can be written as

e r pq 2 β Tγ
∆ m () p = constant × g dr (8.21)
nD 22
w r ΖΤ µ rh
pq pq
and since = sc sc = constant × q sc
ΖΤ T sc

then for isothermal reservoir depletion equ. (8.21) becomes

2
q
Τ
βγ gsc e r dr
()
∆ mp = constant × (8.22)
nD h 2 r µ 2
w r
Since non-Darcy flow is usually confined to a localised region around the wellbore
where the flow velocity is greatest, the viscosity term in the integrand of equ. (8.22) is
usually evaluated at the bottom hole flowing pressure p wf and hence is not a function of
position. Integrating equ. (8.22) gives

2
q
βγ gsc 1 1
Τ
()
∆ mp = constant × − (8.23)
nD h 2 r w r e
-1
If equ. (8.23) is expressed in field units (Q - Mscf/d, β - ft ) and assuming
1 1 ,then
r w >> r e

βγ g Q 2
Τ
∆ m () p = 3.161 × 10 − 12 = FQ 2 (8.24)
nD
2
hr
µ wp w
2
2
where F is the non-Darcy flow coefficient psia /cp/(Mscf/d) .
Since non-Darcy flow is only significant very close to the wellbore, two assumptions are
commonly made in connection with equ. (8.24), these are
- the value of the thickness h is conventionally taken as h p, the perforated interval
of the well
2
- the pseudo pressure drop ∆m(p) nD = FQ can be considered as a perturbation
which readjusts instantaneously after a change in the production rate.