1. Reservoir Engineering 1 Course (1st Ed.)

2. 1. Future IPR Approximation
2. Generating IPR for Oil Wells
A. Wiggins’ Method
B. Standing’s Method
C. Fetkovich’s Method
3. Horizontal Oil Well Performance
4. Horizontal Well Productivity

3. 1. Vertical Gas Well Performance
2. Pressure Application Regions
3. Turbulent Flow in Gas Wells
A. Simplified Treatment Approach
B. Laminar-Inertial-Turbulent (LIT) Approach (Cases A.
& B.)

5. IPR for Gas Wells
Determination of the flow capacity of a gas well
requires a relationship between the inflow gas rate
and the sand-face pressure or flowing bottom-hole
pressure.
This inflow performance relationship may be established
by the proper solution of Darcy’s equation.
Solution of Darcy’s Law depends on the conditions of the flow
existing in the reservoir or the flow regime.
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Reservoir Engineering 1 Course: Gas Well Performance
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6. Gas Reservoir Flow Regimes
When a gas well is first produced after being shutin for a period of time, the gas flow in the reservoir
follows an unsteady-state behavior until the
pressure drops at the drainage boundary of the
well.
Then the flow behavior passes through a short transition
period, after which it attains a steady state or
semisteady (pseudosteady)-state condition.
The objective of this lecture is to describe the empirical as well
as analytical expressions that can be used to establish the
inflow performance relationships under the pseudosteady-state
flow condition.
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Reservoir Engineering 1 Course: Gas Well Performance
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7. Exact Solution of Darcy’s Equation for
Compressible Fluids under PSS
The exact solution to
the differential form of
Darcy’s equation for
compressible fluids
under the
pseudosteady-state
flow condition was
given previously by:
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Where
Qg = gas flow rate,
Mscf/day
k = permeability, md
ψ–r = average reservoir
real gas pseudopressure, psi2/cp
T = temperature, °R
s = skin factor
h = thickness
re = drainage radius
rw = wellbore radius
Reservoir Engineering 1 Course: Gas Well Performance
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8. Productivity Index for a Gas Well
The productivity index J for a gas well can be
written analogous to that for oil wells as:
With the absolute open flow potential (AOF), i.e.,
maximum gas flow rate (Qg)max, as calculated by:
Where J = productivity index, Mscf/day/psi2/cp
2013 H. AlamiNia
Reservoir Engineering 1 Course: Gas Well Performance
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9. Steady-State Gas Well Flow
In a linear relationship as:
Above Equation indicates that a plot of ψwf vs. Qg
would produce a straight line with a slope of (1/J)
and intercept of ψ–r, as shown in next slide.
If two different stabilized flow rates are available,
the line can be extrapolated and the slope is
determined to estimate AOF, J, and ψ–r.
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Reservoir Engineering 1 Course: Gas Well Performance
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10. Steady-State Gas Well Flow (Cont.)
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Reservoir Engineering 1 Course: Gas Well Performance
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11. Darcy’s Equation for Compressible
Fluids under PSS Regime
Darcy’s equation for compressible fluids under the
PSS regime can be alternatively written in the
following integral form:
Note that (p/μg z) is directly proportional to (1/μg
Bg) where Bg is the gas formation volume factor
and defined as:
2013 H. AlamiNia
Reservoir Engineering 1 Course: Gas Well Performance
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13. Typical Plot
of the Gas Pressure Functions vs. P
Figure shows a
typical plot of
the gas pressure
functions
(2p/μgz) and
(1/μg Bg) versus
pressure.
The integral in
previous
equations
represents the
area under the
curve between
p–r and pwf.
Gas PVT data
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Reservoir Engineering 1 Course: Gas Well Performance
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14. Pressure Regions
As illustrated in Figure, the pressure function
exhibits the following three distinct pressure
application regions:
Region III. High-Pressure Region
Region II. Intermediate-Pressure Region
Region I. Low-Pressure Region
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Reservoir Engineering 1 Course: Gas Well Performance
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15. Region III. High-Pressure Region
When both pwf and p–r are higher than 3000 psi, the
pressure functions (2p/μgz) and (1/μg Bg) are nearly
constants.
This observation suggests that the pressure term (1/μg
Bg) in Equation can be treated as a constant and
removed outside the integral, to give the following
approximation to Equation:
Where
Qg = gas flow rate, Mscf/day
Bg = gas formation volume factor, bbl/scf
k = permeability, md
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Reservoir Engineering 1 Course: Gas Well Performance
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16. Region III.
High-Pressure Region, P Method
The gas viscosity μg and formation volume factor Bg should
be evaluated at the average pressure pavg as given by:
The method of determining the gas flow rate by using below
Equation commonly called the pressure-approximation
method.
It should be pointed out the concept of the productivity
index J cannot be introduced into above Equation since it is
only valid for applications when both pwf and p–r are above
3000 psi.
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Reservoir Engineering 1 Course: Gas Well Performance
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17. Region II.
Intermediate-Pressure Region
Between 2000 and 3000 psi, the pressure function
shows distinct curvature.
When the bottom-hole flowing pressure and
average reservoir pressure are both between 2000
and 3000 psi, the pseudopressure gas pressure
approach (i.e., below Equation) should be used to
calculate the gas flow rate.
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Reservoir Engineering 1 Course: Gas Well Performance
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18. Region I. Low -Pressure Region,
P2 Method
At low pressures, usually less than 2000 psi, the
pressure functions (2p/μgz) and (1/μg Bg) exhibit a
linear relationship with pressure.
Golan and Whitson (1986) indicated that the product
(μgz) is essentially constant when evaluating any
pressure below 2000 psi.
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Reservoir Engineering 1 Course: Gas Well Performance
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19. Region I. Low -Pressure Region,
P2 Method (Cont.)
Implementing above observation gives
(pressure-squared approximation method):
Where
Qg = gas flow rate, Mscf/day
k = permeability, md
T = temperature, °R
z = gas compressibility factor
μg = gas viscosity, cp
It is recommended that the z-factor and gas
viscosity be evaluated at the average pressure pavg
as defined by:
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Reservoir Engineering 1 Course: Gas Well Performance
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20. Region I. J Calculation
If both p–r and pwf are lower than 2000 psi, the
equation can be expressed in terms of the
productivity index J as:
With
Where
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Reservoir Engineering 1 Course: Gas Well Performance
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22. Laminar Vs. Turbulent Flow
All of the mathematical formulations presented
thus far in this lecture are based on the assumption
that laminar (viscous) flow conditions are observed
during the gas flow.
During radial flow, the flow velocity increases as the
wellbore is approached.
This increase of the gas velocity might cause the development
of a turbulent flow around the wellbore.
If turbulent flow does exist, it causes an additional pressure
drop similar to that caused by the mechanical skin effect.
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Reservoir Engineering 1 Course: Gas Well Performance
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23. PSS Equations Modification
(Turbulent Flow)
As presented earlier, the semisteady-state flow
equation for compressible fluids can be modified to
account for the additional pressure drop due the
turbulent flow by including the rate-dependent skin
factor DQg.
The resulting pseudosteady-state equations are
given in the following three forms:
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Reservoir Engineering 1 Course: Gas Well Performance
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24. PSS Equations Modification
(Turbulent Flow) (Cont.)
First Form: Pressure-Squared Approximation Form
Second Form: Pressure-Approximation Form
Third Form: Real Gas Potential (Pseudopressure) Form
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Reservoir Engineering 1 Course: Gas Well Performance
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25. Empirical Treatments to Represent
the Turbulent Flow in Gas Wells
The PSS equations, which were given previously in
three forms, are essentially quadratic relationships
in Qg and, thus, they do not represent explicit
expressions for calculating the gas flow rate.
Two separate empirical treatments can be used to
represent the turbulent flow problem in gas wells.
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Reservoir Engineering 1 Course: Gas Well Performance
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26. Empirical Treatments to Represent the
Turbulent Flow in Gas Wells (Cont.)
Both treatments, with varying degrees of approximation,
are directly derived and formulated from the three
forms of the pseudosteady-state equations. These two
treatments are called:
Simplified treatment approach
Laminar-inertial-turbulent (LIT) treatment
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Reservoir Engineering 1 Course: Gas Well Performance
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29. The Simplified Treatment Approach
Based on the analysis for flow data obtained from a large
member of gas wells, Rawlins and Schellhardt (1936)
postulated that the relationship between the gas flow rate
and pressure can be expressed as:
Where Qg = gas flow rate, Mscf/day
p –r = average reservoir pressure, psi
n = exponent
C = performance coefficient, Mscf/day/psi2
The exponent n is intended to account for the additional
pressure drop caused by the high-velocity gas flow, i.e.,
turbulence.
 Depending on the flowing conditions, the exponent n may vary from
1.0 for completely laminar flow to 0.5 for fully turbulent flow.
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Reservoir Engineering 1 Course: Gas Well Performance
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30. Deliverability or
Back-Pressure Equation
The performance coefficient C in the equation is
included to account for:
Reservoir rock properties
Fluid properties
Reservoir flow geometry
The Equation is commonly called the deliverability
or back-pressure equation.
If the coefficients of the equation (i.e., n and C) can be
determined, the gas flow rate Qg at any bottom-hole
flow pressure pwf can be calculated and the IPR curve
constructed.
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Reservoir Engineering 1 Course: Gas Well Performance
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31. Deliverability or Back-Pressure
Equation (Logarithmic Form)
Taking the logarithm of both sides of the Equation
gives:
This equation suggests that a plot of Qg versus (p–r2 −
p2wf) on log-log scales should yield a straight line having
a slope of n.
In the natural gas industry the plot is traditionally
reversed by plotting (p–r2 − p2wf) versus Qg on the
logarithmic scales to produce a straight line with a slope
of (1/n).
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Reservoir Engineering 1 Course: Gas Well Performance
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32. Well Deliverability Graph or
the Back-Pressure Plot
This plot as
shown
schematically
in Figure is
commonly
referred to as
the
deliverability
graph or
the backpressure
plot.
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Reservoir Engineering 1 Course: Gas Well Performance
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33. Calculation of N & C
The deliverability exponent n can be determined
from any two points on the straight line, i.e., (Qg1,
Δp12) and (Qg2, Δp22), according to the flowing
expression:
Given n, any point on the straight line can be used
to compute the performance coefficient C from:
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Reservoir Engineering 1 Course: Gas Well Performance
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34. Gas Well Testing
The coefficients of the back-pressure equation or
any of the other empirical equations are
traditionally determined from analyzing gas well
testing data.
Deliverability testing has been used for more than
sixty years by the petroleum industry to
characterize and determine the flow potential of
gas wells.
There are essentially three types of deliverability tests
and these are:
Conventional deliverability (back-pressure) test
Isochronal test
Modified isochronal test
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Reservoir Engineering 1 Course: Gas Well Performance
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35. Deliverability Testing
These tests basically consist of flowing wells at
multiple rates and measuring the bottom-hole
flowing pressure as a function of time.
When the recorded data are properly analyzed, it is
possible to determine the flow potential and establish
the inflow performance relationships of the gas well.
The deliverability test is out of scope of this course and
would be discussed later in well test course.
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Reservoir Engineering 1 Course: Gas Well Performance
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37. The Laminar-Inertial-Turbulent
(LIT) Approach
The three forms of the semisteady-state equation
as presented earlier in this lecture can be
rearranged
in quadratic forms for separating the laminar and
inertial-turbulent terms composing these equations as
follows:
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Reservoir Engineering 1 Course: Gas Well Performance
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38. Case A.
Pressure-Squared Quadratic Form
a. Pressure-Squared
Quadratic Form
With
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Where
 a = laminar flow coefficient
 b = inertial-turbulent flow
coefficient
 Qg = gas flow rate, Mscf/day
 z = gas deviation factor
 k = permeability, md
 μg = gas viscosity, cp
The term (a Qg) in
represents the pressuresquared drop due to
laminar flow while the term
(b Q2g) accounts for the
pressure squared drop due
to inertial-turbulent flow
effects.
Reservoir Engineering 1 Course: Gas Well Performance
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39. Case A.
Graph of the Pressure-Squared Data
Above equation can be linearized by dividing both
sides of the equation by Qg to yield:
The coefficients a and b can be determined by
plotting ((p–r^2-pwf^2)/Qg) versus Qg on a
Cartesian scale and should yield a straight line with
a slope of b and intercept of a.
Data from deliverability tests can be used to
construct the linear relationship.
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Reservoir Engineering 1 Course: Gas Well Performance
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40. Graph of the pressure-squared data
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41. Case A. Current IPR of the Gas Well
Given the values of a and b, the quadratic flow
equation, can be solved for Qg at any pwf from:
Furthermore, by assuming various values of pwf
and calculating the corresponding Qg from above
Equation,
The current IPR of the gas well at the current reservoir
pressure p–r can be generated.
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Reservoir Engineering 1 Course: Gas Well Performance
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42. Case A. Pressure-Squared Quadratic
Form Assumptions
It should be pointed out the following assumptions
were made in developing following Equation:
Single phase flow in the reservoir
Homogeneous and isotropic reservoir system
Permeability is independent of pressure
The product of the gas viscosity and compressibility
factor, i.e., (μg z) is constant.
This method is recommended for applications at
pressures below 2000 psi.
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Reservoir Engineering 1 Course: Gas Well Performance
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43. Case B. Pressure-Quadratic Form
The pressure-approximation equation, i.e., can be
rearranged and expressed in the following
quadratic form.
The term (a1 Qg) represents the pressure drop due
to laminar flow, while the term (b1 Q2 g) accounts
for the additional pressure drop due to the
turbulent flow condition. In a linear form, the
equation can be expressed as:
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Reservoir Engineering 1 Course: Gas Well Performance
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44. Case B.
Graph of the Pressure-Method Data
The laminar
flow
coefficient a1
and inertialturbulent flow
coefficient b1
can be
determined
from the
linear plot of
the equation
as shown in
Figure.
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Reservoir Engineering 1 Course: Gas Well Performance
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45. Case B. Gas Flow Rate Determination
Having determined the coefficient a1 and b1, the
gas flow rate can be determined at any pressure
from:
The application of following Equation is also
restricted by the assumptions listed for the
pressure-squared approach.
However, the pressure method is applicable at
pressures higher than 3000 psi.
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Reservoir Engineering 1 Course: Gas Well Performance
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46. 1. Ahmed, T. (2006). Reservoir engineering
handbook (Gulf Professional Publishing). Ch8

47. 1. Turbulent Flow in Gas Wells: LIT Approach
(Case C)
2. Comparison of Different IPR Calculation
Methods
3. Future IPR for Gas Wells
4. Horizontal Gas Well Performance
5. Primary Recovery Mechanisms
6. Basic Driving Mechanisms