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Effect of Gas Injection Rate on Oil Production Rate: Details of
Operating Mechanism
Asekhame U. Yadua, Nigerian Petroleum Development Company
(NPDC)
Abstract
It is well known that, during gas lift operations, as the gas
injection rate increases, the operating oil
production rate increases, gets to a peak, then begins to
decline resulting in the parabolic shape of the
gas lift performance curve. In this work, the mechanism behind
this phenomenon is unravelled and
clearly explained, with the aid of mathematics and MS Excel. It
is shown that, as gas injection rate
increases, the gravitational pressure drop in a producing oil
well will keep decreasing while the frictional
pressure drop will keep increasing. During gas injection, oil
production rate increases when the modulus
of the change in gravitational pressure drop is greater than the
modulus of the change in frictional
pressure drop; and oil production rate declines when the modulus
of the change in frictional pressure
drop is greater than the modulus of the change in gravitational
pressure drop.
Keywords: Gaslift, production optimisation, well
performance.
1. Introduction
At some point during the life of a well, the oil production rate
may be less than what is desired, hence,
necessitating an artificial lift technique. Gaslift, the only
artificial lift technique that does not require the
installation of a downhole pump is widely used in the industry
because it is relatively more reliable, simpler
and more flexible in terms of production rates and depth of lift
(Bellarby 2009). Gas lift entails the injection
of compressed gas into the lower section of the tubing, to
enhance well productivity. The injected gas does
this in two ways:
It mixes with the liquid column, reduces the density and
viscosity of the column, thereby
making it easier for the liquid to get to the surface.
It expands and displaces the liquid to the surface (Takacs 2005;
Guo et al. 2007a).
It is well known that, as gas injection rate increases, oil
production rate increases, gets to a peak, then
begins to decline. In this paper I present a detailed
explanation of this phenomenon, with the aid of
mathematics. Numerical simulation with MS Excel was carried out
to buttress and validate the analytical
model.
2. Well performance
The performance of a well is determined by the combination of
the inflow performance relationship (IPR)
curve of the reservoir and the outflow performance relationship
(OPR) curve of the wellbore, also known as
the Tubing Performance relationship (TPR). The point of
intersection of the IPR and the TPR curve is the
operating point of the well.
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2.1. IPR
Darcys Law for steady-state radial flow with formation damage
will be used in this work. The equation is as
follows (Ahmed 2006; Bedrikovetsky et al. 2012):
..(1)
2.2. TPR
Considering the fact that flow properties vary in the three
Cartesian coordinates and are unsteady, flow in
an oil well is an extremely complex problem. To develop some
understanding of tubing performance, it is
convenient to simplify the flow to single-phase, one-dimensional
flow (flow properties only vary along the
length of the tubing).
Consider oil flowing from the bottom to the top (wellhead) of a
single-diameter tubing string of measured
depth and true vertical depth (see Fig. 1). The law of
conservation of energy yields the equation for
pressure drop along a tubing string. The total pressure drop in
a tubing string is the sum of gravitational
pressure drop, acceleration pressure drop, and frictional
pressure drop. The general form of the equation is
. ...(2)
The explicit formula for the total pressure drop in the tubing
is (Guo et al. 2007b)
. ....................................(3)
The first, second and third terms of the right hand side of Eq.
3 are the gravitational pressure drop,
accelerational pressure drop, and frictional pressure drop
respectively.
Assuming the flow is steady, homogeneous and turbulent;
substituting for u and for A in the
third term of the right hand side of Eq. 3; and rearranging
yields
.
Simplifying the above equation yields
. (4)
Rearranging Eq. 4 yields
And
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, .............................................(5)
where is the water cut and is the fractional flow for gas in the
well.
. (6)
Converting the unit to barrels per day, Eq. 5 becomes
. .(7)
Eq. 7 is the TPR used for the simulation.
2.2.1. Effect of gas injection on TPR
When gas is injected into a producing oil well, the nature of
the well fluid changes, resulting in a new TPR
curve. For example, the density of the liquid column changes
from to
. ...............................(8)
where .
Substituting value in Eq. 7 yields
. ..(9)
The above equation was used to calculate the various TPR curves.
The fractional flow for gas is directly
proportional to gas injection rate, as shown below.
.
Rearranging the above equation yields
.(10)
But
Gas/liquid ratio ,
....................................(11)
As gas injection rate increases, the gas occupies more space in
the well, resulting in increasing gas/liquid
ratio. When , . As ,
.
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. .(12)
Therefore, as gas/liquid ratio tends to infinity, fractional
flow for gas tends to unity. So, as the gas injection
rate increases, the gas/liquid ratio increases and the
fractional flow for gas approaches unity. And as the
fractional flow for gas approaches unity (as ), the well
effectively becomes a gas well and liquid
production rate declines. For a given gas injection rate there
is a corresponding value of gas/liquid ratio and
fractional flow for gas. And a given value of fractional flow
for gas has a corresponding TPR curve, given
that all other factors remain constant.
So, sensitizing on bottomhole flowing pressure (BHFP) will yield
corresponding values of oil production
rate . The plot of BHFP versus oil production rate produces the
TPR curve for a given value of fractional
flow for gas as shown in Fig. 2.
2.2.2. Effect of gas injection rate on gravitational pressure
drop.
Consider the equation for gravitational pressure drop
. ..(13)
Since the acceleration due to gravity and the true vertical
depth of the tubing are constant, the critical
factor here is the mixture density .
Eq. 8 can be rewritten as
.
At all times, the fractional flow for gas falls in the range and
. Therefore, the
gravitational pressure drop will keep reducing as gas injection
rate increases ( ).
2.2.3. Effect of gas injection rate on frictional pressure
drop.
Consider the equation for frictional pressure drop
. .......................................(14)
To compare scenarios, we keep constant. Since other parameters
(f, , and water cut) are kept
constant as well, the critical factor is:
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. ....(15)
The minimum value of is 0 and the maximum value is 1. Using
limits to sensitize on yields
and
. ....(16)
Therefore, as the fractional flow for gas increases, the
critical factor also increases. This shows that the
frictional pressure drop will keep increasing as more gas is
injected into the well.
2.2.4. Effect of gas injection rate on operating point
Now it is clear that, as gas injection rate increases the
gravitational pressure drop decreases, while the
frictional pressure drop increases. And it has been established
that a given value of fractional flow for
gas will result in a unique TPR curve, given that all other
factors remain constant. When increases, the
TPR changes position it either moves westward or eastward (see
Eq. 9 and Fig. 2). When the TPR
moves westward, the TPR-IPR point of intersection also moves
westward, resulting in lower oil production
rate; and when the TPR moves eastward, the TPR-IPR point of
intersection also moves eastward, resulting
in a higher oil production rate. When the TPR moves westward, it
shows that a higher value of is
required for a given value of and and when it moves eastward, it
shows that a lower value of is
required for a given value of and . In other words, an increase
in the required due to increase in
, for a given and indicates a decline in oil production rate;
while a decrease in the required due
to increase in , for a given and indicates a boost in oil
production rate (see Fig. 3).
3. How exactly does change as increases?
Consider the well pressure drop equation under steady-state flow
and constant wellhead pressure at a
given value of oil production rate :
Starting from point 1;
, .(17)
at point 2,
. .(18)
Subtracting Eq. 17 from Eq. 18 yields
. .................................(19)
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. ...(20)
As gas injection rate increases, will always be less than and
will always be greater
than , as aforementioned. Therefore, and .
To have a boost in oil production rate, the TPR curve must move
eastward (i.e. under constant
and a given value of must be less than zero). For this to
happen, the following condition must be
fulfilled:
That is, the modulus of the change in gravitational pressure
drop must be greater than the modulus of the
change in frictional pressure drop. In other words, the
reduction in gravitational pressure drop must
dominate the increase in frictional pressure drop when gas
injection rate increases.
And to have a decline in oil production rate, the TPR curve must
move westward (i.e. under constant
and a given value of must be greater than zero). For this to
happen, the following condition must be
fulfilled:
That is, the modulus of the change in gravitational pressure
drop must be less than the modulus of the
change in frictional pressure drop. In other words, the increase
in frictional pressure drop must dominate
the reduction in gravitational pressure drop when gas injection
rate increases.
4. Simulation, results and discussions
Eqs. 1 and 9 were used for the IPR and TPR calculations
respectively. MS Excel was used to run the
simulations. Apart from the density of water, other input data
were arbitrarily chosen (see Tables 1 and 2).
Each TPR curve plotted corresponds to a given value of
fractional flow for gas (see Fig. 4). All other
parameters in the TPR formula were kept constant. To determine
the optimum fractional flow for gas , and
consequently the optimum gas injection rate, the operating oil
production rate derived from Fig. 4 was
plotted against the corresponding value of (see Fig. 5).
From Fig. 4, it can be seen that as increases from 0 to 0.3, the
TPR curve keeps moving eastward,
resulting in higher production rates. When was increased to 0.5,
the TPR curve moved westward and this
trend continued as was increased to 1, resulting in lower
production rates. Fig. 5 clearly illustrates the
explanation in the preceding section. At , the oil production
rate is 2,340 bbl/day. As increases, the
oil production rate increases (when reduction in gravitational
pressure drop dominates the increase in
frictional pressure drop), gets to the peak point = 0.26, =
2,475 bbl/day, then begins to decline to the
point = 1, = 0 bbl/day (as the increase in frictional pressure
drop starts dominating the reduction in
gravitational pressure drop).
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5. Conclusions
1. Gas injection into a producing oil well changes the TPR
curve, resulting in new operating point(s).
2. As gas injection rate increases, the gravitational pressure
drop keeps decreasing while the
frictional pressure drop keeps increasing.
3. When the modulus of the change in gravitational pressure drop
is greater than the modulus of the
change in frictional pressure drop, oil production rate
increases; and when the modulus of the
change in frictional pressure drop is greater than the modulus
of the change in gravitational
pressure drop, oil production rate decreases.
4. On the gas lift performance curve (Fig. 5), the area to the
left of the abscissa of the optimum point
is the area where reduction in gravitational pressure drop
dominates the increase in frictional
pressure drop; and the area to the right of the abscissa of the
optimum point is the area where
increase in frictional pressure drop dominates reduction in
gravitational pressure drop.
5. The optimum fractional flow for gas is always in the range
.
Nomenclature
Roman letters
Dt = tubing internal diameter, L, ft
fF = Fanning friction factor
g = acceleration due to gravity, L , ft/s2
h = payzone thickness, L, ft
kO = effective permeability to oil, , mD
Lmd = measured depth of tubing, L, ft
Lv = true vertical depth of tubing, L, ft
pA = accelerational pressure drop, m , psi
pe = pressure at drainage radius, m , psi
pF = frictional pressure drop, m , psi
pG = gravitational pressure drop, , psi
pT = total pressure drop in tubing string, , psi
pwf = bottomhole flowing pressure, , psi
pwh = wellhead flowing pressure, , psi
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qO = oil flow rate in the reservoir, , ft3/s [bbl/day]
QG = gas flow rate in the well, , ft3/s
QL = liquid flow rate in the well, , ft3/s [bbl/day]
QO = oil flow rate in the well, , ft3/s [bbl/day]
QT = total flow rate in the well, , ft3/s [bbl/day]
QW = water flow rate in the well, , ft3/s [bbl/day]
re = drainage radius, L, ft
rw = wellbore radius, L, ft
s = skin factor
u = velocity, L , ft/s
VG = volume of gas in the well, , ft3
VL = volume of liquid in the well, , ft3
Greek letters
= fractional flow for gas
= gas/liquid ratio
= change
= viscosity of oil, m , cp
= pi
= density, m , lbm/ft3
= gas density, m , lbm/ft
3
= liquid density, m , lbm/ft
3
= gas-liquid mixture density, m , lbm/ft
3
= oil density, m , lbm/ft
3
= water density, m , lbm/ft
3
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References
(1) Bellarby, J. 2009. Artificial Lift. In Developments in
Petroleum Science, Vol. 56, 303 369. Elsevier.
(2) Takacs, G. 2005. Gas Lift Manual. Oklahoma: PennWell
Corporation.
(3) Guo, B., Lyons, W.C., Ghalambor, A. 2007a. Gas Lift. In
Petroleum Production Engineering, Chap. 13,
181-206. Burlington, Massachusetts: Gulf Professional
Publishing/Elsevier.
(4) Ahmed, T. 2006. Reservoir Engineering Handbook, third
edition. Burlington, Massachusetts: Gulf
Professional Publishing/Elsevier.
(5) Bedrikovetsky, P., Vaz, A., Machado, F. et al. 2012. Skin
Due to Fines Mobilization, Migration, and
Straining During Steady-State Oil Production. Petroleum Science
and Technology 30 (15): 1539-1547.
http://dx.doi.org/10.1080/10916466.2011.653702
(6) Guo, B., Lyons, W.C., Ghalambor, A. 2007b. Wellbore
Performance. In Petroleum Production
Engineering, Chap. 4, 46-58. Burlington, Massachusetts: Gulf
Professional Publishing/Elsevier.
TABLE 1DATA OF IPR CALCULATION
ko (mD) h (ft) pe (psi) pwf (psi) re (ft) rw (ft) s qo (bbl/day)
120 120 5000 0 1.8 2932 0.3177 1.5 26641.36038
500 23977.22434
1000 21313.08831
1500 18648.95227
2000 15984.81623
2500 13320.68019
3000 10656.54415
3500 7992.408115
4000 5328.272077
4500 2664.136038
5000 0
TABLE 2INPUT DATA FOR TPR CALCULATIONS
Dt (ft) QW/QL pwf (psi) pwh (psi)
(lbm/ft3)
(lbm/ft3) Lv (ft) fF Lmd (ft)
0.1875 0.6 0 120 0.072 58 7391 0.0065 8900
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
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Fig. 1Flow along a tubing string (adapted from Guo et al.
2007b).
Fig. 2Effect of gas injection rate on TPR curve.
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Fig. 3Effect of gas injection rate on operating oil production
rate.
Fig. 4Calculated IPR and TPR curves for various values of
fractional flow for gas.
250
1250
2250
3250
4250
5250
0 500 1000 1500 2000 2500
Bo
tto
mh
ole
Flo
win
g P
ressu
re,
pw
f, p
si
Oil Production Rate, Qo, bbl/day
IPR
TPR 1 (beta = 0)
TPR 2 (beta = 0.1)
TPR 3 (beta = 0.2)
TPR 4 (beta = 0.3)
TPR 5 (beta = 0.5)
TPR 6 (beta = 0.7)
TPR 7 (beta = 0.9)
TPR 8 (beta = 1)
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Fig. 5Gas lift performance curve.
SI metric conversion factors
Bbl x 1.589873 E-01 = m3
cp x 1.0* E-03 = Pa.s
ft x 3.048* E-01 = m
lbm x 4.535924 E-01 = kg
psi x 6.894757 E+00 = kPa
*Conversion factor is exact.
Author
Asekhame U. Yadua is a graduate Facilities Engineer at the
Nigerian Petroleum Development Company
(NPDC), a subsidiary of the Nigerian National Petroleum
Corporation (NNPC). His research interests
include Petroleum Production Engineering, Process Engineering,
and Reservoir Engineering. He holds a
BEng degree in Chemical Engineering (First Class Honours) from
Covenant University, Nigeria, and an
MSc degree in Oil and Gas Engineering (Distinction) from the
University of Aberdeen. He is a member of
the Society of Petroleum Engineers (SPE) and Energy Institute
(EI).
Telephone numbers: +234 8183117508 and +234 8106853967
0
250
500
750
1000
1250
1500
1750
2000
2250
2500
2750
0 0.2 0.4 0.6 0.8 1
Oil
Pro
du
cti
on
Rate
, Q
o,
bb
l/d
ay
Optimum Point (0.26, 2475)
Fractional Flow for Gas,
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E-mail addresses: [email protected] and
[email protected]
Office address: NPDC, 62/64 Sapele Road, Benin City, Nigeria