Page 106 - Petroleum Production Engineering, A Computer-Assisted Approach

Guo, Boyun / Computer Assited Petroleum Production Engg 0750682701_chap08 Final Proof page 98 20.12.2006 10:36am

8/98 PETROLEUM PRODUCTION ENGINEERING FUNDAMENTALS
ð t
c
p
8.1 Introduction kh( p t p ) c t N i
wf
h i dt ¼ ( p 0 p t ): (8:5)
p
p
Production decline analysis is a traditional means of 141:2B o m ln 0:472r e þ s B o
identifying well production problems and predicting 0 r w
well performance and life based on real production Taking derivative on both sides of this equation with
data. It uses empirical decline models that have little respect to time t gives the differential equation for reser-
fundamental justifications. These models include the voir pressure:
following: c
kh( p t p ) d p t
p
p
h wf i ¼ c t N i (8:6)
. Exponential decline (constant fractional decline) 141:2m ln 0:472r e þ s dt
. Harmonic decline r w
. Hyperbolic decline Because the left-hand side of this equation is q and Eq.
(8.2) gives
Although the hyperbolic decline model is more general, the
p
other two models are degenerations of the hyperbolic dq ¼ h kh i d p t , (8:7)

decline model. These three models are related through dt 141:2B 0 m ln 0:472r e þ s dt
the following relative decline rate equation (Arps, 1945): r w
1 dq Eq. (8.6) becomes
d
¼ bq , (8:1) h i
q dt 141:2c t N i m ln 0:472r e þ s dq
q ¼ r w (8:8)
where b and d are empirical constants to be deter- kh dt
mined based on production data. When d ¼ 0, Eq. (8.1) or the relative decline rate equation of
degenerates to an exponential decline model, and
when d ¼ 1, Eq. (8.1) yields a harmonic decline model. 1 dq ¼ b, (8:9)
When 0 < d < 1, Eq. (8.1) derives a hyperbolic decline q dt
model. The decline models are applicable to both oil and where
gas wells.
kh
b ¼ h i : (8:10)
141:2mc t N i ln 0:472r e þ s
r w
8.2 Exponential Decline
The relative decline rate and production rate decline equa-
tions for the exponential decline model can be derived 8.2.2 Production rate decline
from volumetric reservoir model. Cumulative production Equation (8.6) can be expressed as
expression is obtained by integrating the production rate d p t
p
c
p
decline equation. b( p t p ) ¼ dt : (8:11)
wf
By separation of variables, Eq. (8.11) can be integrated,
8.2.1 Relative Decline Rate ð t p p ð t
p
Consider an oil well drilled in a volumetric oil reservoir. bdt ¼ d p t , (8:12)
c
p
Suppose the well’s production rate starts to decline when a ( p t p )
wf
critical (lowest permissible) bottom-hole pressure is 0 p p 0
reached. Under the pseudo–steady-state flow condition, to yield an equation for reservoir pressure decline:
the production rate at a given decline time t can be
p
expressed as p p t ¼ p c wf þ p 0 p c wf e bt (8:13)
c
kh( p t p )
p
q ¼ h wf i , (8:2) Substituting Eq. (8.13) into Eq. (8.2) gives the well pro-
141:2B 0 m ln 0:472r e þ s duction rate decline equation:
r w
c
kh( p 0 p )
p
wf
where q ¼ h i e bt (8:14)
141:2B o m ln 0:472r e þ s
p p t ¼ average reservoir pressure at decline time t, r w
p c wf ¼ the critical bottom-hole pressure maintained during or
the production decline.
bc t N i c bt
q ¼ ( p 0 p ) e , (8:15)
p
The cumulative oil production of the well after the B o wf
production decline time t can be expressed as which is the exponential decline model commonly used for
ð t c production decline analysis of solution-gas-drive reser-
p
kh( p t p )
wf
N p ¼ h i dt: (8:3) voirs. In practice, the following form of Eq. (8.15) is used:
141:2B o m ln 0:472r e þ s bt
0 r w q ¼ q i e , (8:16)
The cumulative oil production after the production de- where q i is the production rate at t ¼ 0.
b
cline upon decline time t can also be evaluated based on It can be shown that q 2 ¼ q 3 ¼ .. ... . ¼ q n ¼ e . That
q 1 q 2 q n 1
the total reservoir compressibility: is, the fractional decline is constant for exponential
decline. As an exercise, this is left to the reader to prove.
c t N i
N p ¼ ( p 0 p t ), (8:4)
p
p
B o
8.2.3 Cumulative production
where
Integration of Eq. (8.16) over time gives an expression for
c t ¼ total reservoir compressibility, the cumulative oil production since decline of
N i ¼ initial oil in place in the well drainage area, ð t ð t
p p 0 ¼ average reservoir pressure at decline time zero. bt
N p ¼ qdt ¼ q i e dt, (8:17)
Substituting Eq. (8.3) into Eq. (8.4) yields 0 0