1. PRODUCTION ENGINEERING FUNDAMENTALS I 21
2.3 Inflow Performance of Gil Wells
2.3.1/ntroduction
The proper design of any artificiallift system requires an accurate knowledge of the fluid rates that can be produced
from the reservoir through the given well. Present and also future production rates are needed to accomplish the
following basic tasks of production engineering:
• selection of the right type of lift
• detailed design of production equipment
• estimation of future well performance
The production engineer, therefore, must have a clear understanding of the effects governing fluid inflow into a
well. Lack of information may lead to over-design of production equipment or, in contrast, equipment limitations may
restrict attainable liquid rates. Both of these conditions have an undesirable impact on the economy of artificiallifting
and can cause improper decisions as well.
A well and a productive formation are interconnected at the sandface, the cylindrical surface where the reservoir
is opened. As long as the well is kept shut in, sandface pressure equals reservo ir pressure and thus no inflow occurs to
me well. It is easy to see that, in analogy to flow in surface pipes, fluids in the reservoir flow only between points having
different pressures. Thus, a well starts to produce when the pressure at its sandface is decreased below reservoir pressure.
Auid particles in the vicinity of the well then move in the direction of pressure decrease and, after an initial period, a
Stabilized rate develops. This rate is controlled mainly by the pressure prevailing at the sandface, but is also affected by
iI. multitude of parameters such as reservoir properties (rock permeability, pay thickness, etc.), fIuid
properties (viscosity, density, etc.) and well completion effects (perforations, well damage). These latter parameters
being constant for a given well, at least for a considerable length of time, the only means of controlling production rates
isthe control of bottomhole pressures. The proper description of well behavior, therefore, requires that the relationship
between bottomhole pressures and the corresponding production rates be established. The resulting function is called
the well's inflow performance relationship (IPR) and is usually obtained by running well tests.
This section discusses the various procedures available for the description of well inflow performance. Firstly, the
most basic terms with relevant descriptions are given.
23.2 Basic concepts
Darcy's Law
The equation describing filtration in porous media was originally proposed by Darcy and can be written in any
rnnsistent set of units as:
2.34
This formula states that the rate of liquid flow,q, per cross-sectional area, A, of a given permeable media is directly
puportional to permeability, k, the pressure gradient, dp/dl, and is inversely proportional to liquid viscosity. The
megative sign is included because flow takes place in the direction of decreasing pressure gradients. Darcy's equation
iiiII55UITlesa steady state, linear flow of a single-phase fluid in a homogeneous porous media saturated with the same fluido
Almough these conditions are seIdom met, all practical methods are based on Darcy's work.

2. 22 I GAS LIFr MANUAL
Drainage Radius, re
Consider a welI producing a stable fluid rate from a homogeneous formation. Fluid particles from alI directions
around the welI flow toward the sandface. ln idealized conditions, the drainage area, i.e. the area where fluid is moving
to the welI, can be considered a circle. At the outer boundary of this circle, no flow occurs and undisturbed reservoir
conditions prevail. Drainage radius, fe, is the radius of this circle and represents the greatest distance of the given welI's
influence on the reservoir under steady-state conditions.
Average Reservoir Pressure, PR
The formation pressure outside the drainage area of a welI equals the undisturbed reservoir pressure, PR, which can
usualIy be considered a steady value over longer periods of time. This is the same pressure as the bottomhole pressure
measured in a shut-in welI, as seen in Figure 2-3.
Pressure
Sondfoce
Well shut -in
Orawdown =
= SBHP -FBHP
FBHP
Distonce
Formation
Fig. 2-3 Pressure distribution around a well in the formation.
Flowing Bottomhole Pressure (FBHP),Pwf
Figure 2-3 shows the pressure distribution in the reservoir around a producing welI. In shut-in conditions, the
average reservoir pressure, PR prevails around the welIbore and its value can be measured in the welI as SBHP.After flow
hasstarted, bottomhole pressure is decreased and pressure distribution at intermediate times is represented by the
dashed !ines. At steady-state conditions, the welI produces a stabilized liquid rate and its bottomhole pressure attains a
stable value, PwI- The solid line on Figure 2-3 shows the pressure distribution under these conditions.
Pressure Drawdown
The difference between static and FBHP is calIed pressure drawdown. This drawdown causes the flow of formation
fluids into the welI and has the greatest impact on the production rate of a given welI.

3. PRODUCTION ENGINEERING FUNDAMENTALS I 23
2.3.3 The productivity index concept
The simplest approach to describe the infIow performance of oil weIls is the use of the productivity index (PI)
concept. It was developed using the foIlowing simplifying assumptions:
• fIow is radial around the weIl,
• a single-phase liquid is fIowing,
• permeability distribution in the formation is homogeneous, and
• the formation is fuIly saturated with the given liquido
For the previous conditions, Oarcy's equation (Equation 2.34) can be solved for the production rate:
O.00708kh (PR - Pwf)
q = r ),uB ln (r:
2.35
where: q = liquid rate, STB/d
k = effective permeability, mO
h = pay thickness, ft
/l = liquid viscosity, cP
B = liquid volume factor, bbI/STB
re = drainage radius of weIl, ft
rw = radius of weIlbore, ft
Most parameters on the right-hand side are constant, which permits coIlecting them into a single coefficient caIled PI:
2.36
This equation states that liquid infIow into a weIl is directly proportional to pressure drawdown. lt plots as a
straight line on apressure vs. rate diagram, as shown in Figure 2-4. The endpoints of the PI line are the average reservoir
pressure, PR, at a fIow rate of zero and the maximum potential rate at a bottomhole fIowing pressure of zero. This
maximum rate is the weIl's absolute open fIow potential (AOFP) and represents the fIow rate that would occur if fIowing
bottomhole pressure could be reduced to zero. In practice, it is not possible to achieve this rate, and it is only used to
compare the deliverability of different weIls.
The use of the PI concept is quite straightforward. If the average
reservoir pressure and the PI are known, use of Equation 2.36 gives the fIow
Iate for any FBHP. The weIl's PI can either be caIculated from reservoir
parameters, or measured by taking fIow rates at various FBHPs.
SBHP
a..
Ia:l
LL
Líquid Rate
AOFP
Fig. 2~ Well performance with the PI concept.

4. 24 I GAS Lwr MANUAL
Example 2-9. A well was tested at Pwf = 1,400 psi (9.7 MPa) pressure and produced q = 100 bpd
(15.9 m3/d) of oil. Shut-in bottom pressure was Pws = 2,000 psi (13.8 MPa). What is the well's PI and what is
the oil production rate at Pwf = 600 psi (4.14 MPa).
Solution
Solving Equation 2.36 for PI and substituting the test data:
PI = q / ( PR - Pwf) = tOO / (2,000 - 1,400) = 0.17 bopd/psi (0.0039 m3lkPald)
The rate at 600 psi (4.14 MPa) is found from Equation 2.36:
q = PI ( Pws - Pwf ) = 0.17 ( 2,000 - 600 ) = 238 bopd (37.8 m3/d).
2.3.4/nflow performance relationships
2.304.1 Introduction. In most wells on artificial lift, bottomhole pressures below bubblepoint pressure are
experienced. Thus, there is a gas phase present in the reservo ir near the wellbore, and the assumptions that were used to
develop the PI equation are no longer valido This effect was observed by noting that the PI was not a constant as suggested
by Equation 2.36. Test data from such wells indicate a downward curving line, instead of the straight line shown in Figure
2-4.
The main cause of a curved shape of inflow performance is the liberation of solution gas due to the decreased
pressure in the vicinity of the wellbore. This effect creates an increasing gas saturation profile toward the well and
simultaneously decreases the effective permeability to liquido Liquid rate is accordingly decreased in comparison to
single-phase conditions and the well produces less liquid than indicated by a straight-line PI. Therefore, the constant
PI concept cannot be used for wells producing below the bubblepoint pressure. Such wells are characterized by their
IPR curves, to be discussed in the following section.
2.3 04.2 Vogel's IPR correlation. Vogel used a numerical reservoir simulator to study the inflow performance
of welIs depleting solution gas drive reservoirs. He considered cases below bubblepoint pressure and varied pressure
drawdowns, fluid, and rock properties. After running several combinations on the computer, Vogel found that all the
calculated IPR curves exhibited the same general shape. This shape is best approximated by a dimensionless equation
[18]:
q
qmax = 1-0.2 ;wf _ 0.8( Pwf )2R PR
2.37
where: q = production rate at bottomhole pressure Pwf, STB/d
qmax = maximum production rate, STB/d
PR = average reservoir pressure, psi
Equation 2.37 is graphically depicted in Figure 2-5.
Although Vogel's method was originally developed for solution gas drive reservoirs, the use of his equation is
generally accepted for other drive mechanisms as welI [19]. It was found to give reliable resuIts for almost any well with
a bottomhole pressure below the bubblepoint of the crude.
In order to use Vogel's method, reservoir pressure needs to be known along with a single stabilized rate and the
corresponding FBHP. With these data, it is possible to construct the well's IPR curve by the procedure discussed in the
following example problem.

5. PRODUCTION ENGINEERING FUNDAMENTALS I 25
FBHP, Fraction of Reservior Pressure
•••••••••
••••••
•••••••••......
••••••••
.....•.•.••...
......
"""
......••
'"~"-"-'"'-I...
~.
"",,0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
o
O 0.2 0.4 0.6 0.8
Production Rate, Fraction of Maximum
Fig. 2-5 Vogel's dimensionless inflow performance curve.
Example 2-10. Using data of the previous example nnd the well's AOFP and construct its IPR curve, by
assuming multiphase flow in the reservoir.
SoIution
Substituting the test data into Equation 2.37:
1001 cfrnax = 1 - 0.2 (1,400/2,000) - 0.8 (1,400/2,000)2 = 0.468.
From the previous equation the AOFP of the well:
cfrnax= 10010.468 = 213.7bopd (34 m3/d)
Now nnd one point on the IPR curve, where Pwf = 1,800 psi (12.42 MPa) using Figure 2-5.
Pwjl PR = 1,800/2,000 = 0.9
From Figure 2-5:
cf 1 cfrnax = 0.17, and q = 213.70.17 = 36.3 bopd (5.8 m3/d).
The remaining points of the IPR curve are evaluated the same way.

6. 26 I GAS LIFT MANUAL
Figure 2-6 shows the calculated IPR curve along with a straight line valid for PI = 0.17 (0.0039 m3/kPald),
as found in Example 2-9. Calculated parameters for the two cases are listed as folIows:
Vogel
Constant PI
Max. Rate
213.7
340
Rate at 600 psi
185.5
238
.~ 2000
cD 1800
•....
:J 1600(/) (/)Q)
1400•.... c... 1200Q)
"'6
1000..c E 800o - 600Õ
CO
400Ol c 200
.~ o
O
Li: O
50100150200250300350
Fig.2-6 Comparison of IPR curves for Examples 2-7 and 2-8.
Comparison of the preceding results indicates that considerable errors can occur if the constant PI method
is used for conditions below the bubblepoint pressure.
2.3.4.3 Fetkovich's method. Fetkovich demonstrated that oil welIs producing below the bubblepoint pressure
and gas welIs exhibit similar inflow performance curves [20]. The general gas welI deliverability equation can thus also
be applied to oil wells:
q = C(p~- p~l
2.38
Coefficients C and n in this formula are usualIy found by curve-fitting of multipoint welI test data. Evaluation of
welI tests and especialIy isochronal tests is the main application for Fetkovich's method.
2.4 Single-phase Flow
2.4.1/ntroduction
In all phases of oil production, several different kinds of fluid flow problems are encountered. These involve vertical
or inclined flow of a single-phase fluid or of a multiphase mixture in welI tubing, as welI as horizontal or inclined flow
in flowlines. A special hydraulic problem is the calculation of the pressure exerted by the static gas column present in
a welI's annulus. AlI the problems mentioned require that the engineer be able to calculate the main parameters of the
particular flow, especialIy the pressure drop involved.
In this section, basic theories and practical methods for solving single-phase pipe flow problems are covered that
relate to the design and analysis of gas lifted welIs. As alI topics discussed have a common background in hydraulic
theory, a general treatment of the basic hydraulic concepts is given first. This includes detailed definitions of and
relevant equations for commonly used parameters of pipe flow.