IntroductionInflow performance relationship, IPR is crucial to petroleum engineers in the forecasting of well behavior, designing artificial lift equipment, simulation treatments and in general production system optimization. IPR curves show a relationship between a well’s producing bottomhole pressures and corresponding production rates under given conditions. Simply put, IPR gives a representation of a well’s ability to give up fluids and the curves are different for different reservoir types and conditions. The rate of production represented against pressures on the IPR curve depends largely on the reservoir type and drive mechanism and other variables such as reservoir pressure, permeability etc.

Darcy’s law helps to determine the IPR relationship for a single phase reservoir where the reservoir pressure is greater than the bubble point pressure as well as the bottomhole pressure which shows a straight line IPR.

QO=7.08∗10−3KOh(Pr−Pwf )

μo Bo[ ln rerw−0.75+St+Dqo ](1)

Where,

Qo= oil flow rate into the well (stb/d)

Ko = permeability of the formation to oil (mD)

h = net thickness of the formation (ft)

Pr= average reservoir pressure (psia)

Pwf= bottomhole flowing pressure (psia)

μo= viscosity of oil (cp)

Bo= formation volume factor of oil (bbl/stb)

re= radius of drainage (ft)

rw= wellbore radius

St = total skin

Dqo = pseudo skin due to turbulence which is insignificant in oil wells due to low permeability in some reservoirs.

Darcy’s law simplifies to (eq. 2) when re = 1466ft, rw= 0.583 and there is no turbulence.

Qo=khμoBo

(Pr−Pwf ) (2)

(Where k is in Darcy)

The IPR relationship based on Darcy’s law is a straight line relationship for steady state, radial flow of slightly compressible fluid and it is shown below:

This straight line IPR assumes that the inflow rate into a given well at constant reservoir conditions is directly proportional to the pressure drawdown.

q∝∆ P

q=J (∆ P)

where ∆ P=Pr−Pwf

J is the constant of proportionality and it is expressed as the productivity index which is chosen as the slope of any two points on the IPR curve which is constant for the straight line IPR and it is the first differential of this IPR.

J=Qo∆ P

=0.00708KOh

μo Bo[ ln rerw−0.75+St ] (Bpd/psi)

Factors affecting productivity index

Productivity index changes with time, cumulative oil recovery as well as different drawdown rates at a particular time in life of well, but there are essentially three factors that affect productivity index and they are oil viscosity, oil formation volume factor and relative permeability to oil. These three factors are basically described in the phase behavior of the reservoir which simultaneously affects these factors.

1) Phase behavior of reservoirs: In explaining phase behaviors of reservoirs, it is imperative for us to understand the concept of bubble point pressure. Bubble point pressure, Pb is the pressure at which the first gas forms and is evolved in the reservoir as pressure is reduced at constant temperature. At initial reservoir pressure Pi > Pb, no free gas would exist anywhere in the reservoir but with decline in pressure at any point in the reservoir which is less than Pb, free gas would evolve and the relative permeability to oil will be reduced, and with oil production rate less than the required bottomhole pressure less than bubble point pressure, the relative permeability to oil will be reduced as well as the productivity index.

2) Relative permeability behavior: As gas evolves in the reservoir below bubble point, the liquid phase has a reduced ability to flow even if gas has not reached its critical saturation, but the space occupied by the gas reduces the liquid phase effective flow area.

relative permeability=effective permeability ¿a fluid ¿permeability of oil

3) Oil viscosity: At constant temperature, the viscosity of oil which is saturated with gas in the reservoir decreases with decline in reservoir pressure to bubble point pressure, and below Pb, oil viscosity will increase as gas comes out of solution hereby leaving heavier molecules of the liquid phase.

4) Oil formation volume factor: As pressure decreases in the reservoir, the liquid will expand until bubble point is reached where below it, the gas evolves from the liquid solution hereby causing the oil to shrink.

It should be noted that below bubble point pressure, IPR curve deviates from the straight line and it is shown below.

There are different empirical relationships used to explain this phenomenon by various authors such as Vogel, Fetkovich, Standing etc which is explained further in the next chapter.

Factors affecting inflow performance

a) Drive mechanism

For a solution gas drive which has constant volume with no water influx, there is 2 phase flow at pressures below bubble point pressure and gas comes out of solution but there is no gas cap because gas bubbles remaining in the oil and flow simultaneously with the oil and the oil production is due to volumetric expansion of oil.

In early production life of this reservoir type, oil is replaced by gas on equal volume basis and as pressure declines larger gas phase develops and more gas expansion is required per unit volume of oil produced, which creates excessive drawdown hence increased permeability to gas and reduced permeability to oil. With increase in gas oil ratio, there is rapid decrease in pressure and further decline in productivity index.

Considering a water drive reservoir type, the volume of oil does not remain not remain constant and there is water encroachment which changes the initial volume of the reservoir which displaces the oil hereby generating optimum production rate because of the presence of a gas phase which may result into a combination water depletion drive mechanism. The pressure decline may be small or relatively constant which remains above bubble point and the productivity index remains constant throughout the well’s life due to relatively constant GOR and the PI can be extrapolated linearly for drawdown necessary to give the desired production but in some cases whereby there is increase in water-cut, this may reduce the productivity index of the system.

Finally, considering gas cap expansion drive mechanism, which is otherwise known as gravity drainage or segregation whereby the oil zone is overlain by gas cap and it can be further differentiated by whether gas comes out of the solution or not. In a case whereby gas comes out of solution and migrates to the gas cap causing its expansion and movement downwards, the permeability of the formation determines the occurrence of counter-flow and this only occurs when the permeability is less than 100mD. Here, pressure declines fairly rapidly as well

as the productivity index.

In consideration of combination drive mechanism, in which very often we find an oil reservoir which is saturated and is n contact with an aquifer which may exhibit all the three drive mechanisms explained above contributing to the reservoir drive. As oil is produced, both gas cap and aquifer will expand and the gas-oil contact will drop with simultaneous rise in oil-water contact which may arise into complex production problems.

b) Drawdown/ Producing rate

Productivity index is a function of oil mobility and below Pb , there is evolution of gas from the oil which further decreases productivity index. In a case whereby Pr > Pb, it may be necessary to reduce the bottom-hole flowing pressure below bubble point pressure to cause a zone of reduced permeability to oil around the wellbore out to the radius at which pressure in reservoir is equal to bubble point pressure. The pressure profile of the reservoir depends on the damage area of the well which is a function of the skin factor.

c) Effect of depletion

As pressure declines below bubble point pressure, gas saturation increases causing decreased permeability to oil which further increases the slope of the IPR curve. In order to maintain constant inflow rate, it is necessary to increase drawdown as Pr declines from depletion.