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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

Darcy’s 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/ft3

= liquid density, m , lbm/ft3

= gas-liquid mixture density, m , lbm/ft3

= oil density, m , lbm/ft3

= water density, m , lbm/ft3

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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 1—DATA 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 2—INPUT 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. 1—Flow along a tubing string (adapted from Guo et al. 2007b).

Fig. 2—Effect of gas injection rate on TPR curve.

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Fig. 3—Effect of gas injection rate on operating oil production rate.

Fig. 4—Calculated 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. 5—Gas 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: aseyadua@gmail.com and yadua.au@npdc-nigeria.com

Office address: NPDC, 62/64 Sapele Road, Benin City, Nigeria