1. Reservoir Engineering 1 Course (2nd Ed.)

2. 1. PSS Regime
A.
B.
C.
D.
Average Reservoir Pressure
PSS regime for Radial Flow of SC Fluids
Effect of Well Location within the Drainage Area
PSS Regime for Radial Flow of C Fluids
2. Skin Concept
3. Using S for Radial Flow in Flow Equations
4. Turbulent Flow

3. 1. Superposition
A. Multiple Well
B. Multi Rate
C. Reservoir Boundary
2. Productivity Index (PI)
3. Inflow Performance Relationship (IPR)

5. Flash Back: Solutions
to the Radial Diffusivity Equation
The solutions to the radial diffusivity equation
appear to be applicable only for describing the
pressure distribution in an infinite reservoir that
was caused by a constant production from a single
well.
Since real reservoir systems usually have several
wells that are operating at varying rates, a more
generalized approach is needed to study the fluid
flow behavior during the unsteady state flow
period.
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
5

6. Superposition Theorem
The principle of superposition is a powerful
concept that can be applied to remove the
restrictions that have been imposed on various
forms of solution to the transient flow equation.
Mathematically the superposition theorem states
that any sum of individual solutions to the
diffusivity equation is also a solution to that
equation.
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
6

7. Superposition Concept Applications
Superposition concept can be applied to account
for the following effects on the transient flow
solution:
Effects of multiple wells
Effects of rate change
Effects of the boundary
Effects of pressure change
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
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8. Effects of Multiple Wells
Frequently, it is desired to account for the effects of
more than one well on the pressure at some point
in the reservoir.
The superposition concept states that the total
pressure drop at any point in the reservoir is the
sum of the pressure changes at that point caused
by flow in each of the wells in the reservoir.
In other words, we simply superimpose one effect upon
the other.
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
8

9. Appling Superposition:
Effects of Multiple Wells
Figure shows three
wells that are
producing at different
flow rates from an
infinite acting reservoir,
i.e., unsteady-state flow
reservoir. The principle
of superposition shows
that the total pressure
drop observed at any
well, e.g., Well 1, is:
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
9

10. Appling Superposition:
Effects of Multiple Wells (Cont.)
 The pressure drop at Well 1 due to
its own production is given by the
log-approximation to the Ei-function
solution presented by: (Qo1=oil flow
rate from well 1)
 The pressure drop at Well 1 due to
production at Wells 2 and 3 must be
written in terms of the Ei-function
solution. The log-approximation
cannot be used because we are
calculating the pressure at a large
distance r from the well, i.e., the
argument x > 0.01, or:
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It should also be noted
that if the point of
interest is an operating
well, the skin factor s
must be included for
that well only.
Reservoir Engineering 1 Course (2nd Ed.)
10

12. Effects of Rate Change
All of the mathematical expressions presented
previously require that the wells produce at a
constant rate during the transient flow periods.
Practically all wells produce at varying rates and,
therefore, it is important that we be able to predict
the pressure behavior when the rate changes.
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
12

13. Superposition: Effects of Rate Change
For predicting the pressure behavior when the rate
changes, the concept of superposition states:
“Every flow rate change in a well will result in a pressure
response which is independent of the pressure
responses caused by other previous rate changes.”
Accordingly, the total pressure drop that has
occurred at any time is the summation of pressure
changes caused separately by each net flow rate
change.
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
13

14. Production and
Pressure History of a Multi-Rate Well
Consider the
case of a shutin well, i.e., Q
= 0, that was
then allowed
to produce at
a series of
constant rates
for the
different time
periods
shown in
Figure.
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
14

15. Pressure Drop of Multi-Rate Well
To calculate the total pressure drop at the sand
face at time t4, the composite solution is obtained
by adding the individual constant-rate solutions at
the specified rate-time sequence, or:
The above expression indicates that there are four
contributions to the total pressure drop resulting
from the four individual flow rates.
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
15

16. Pressure Drop of Multi-Rate Well:
1st Contribution
The first contribution results from increasing the rate
from 0 to Q1 and is in effect over the entire time period
t4, thus:
It is essential to notice the change in the rate, i.e., (new
rate − old rate), that is used in the above equation.
It is the change in the rate that causes the pressure
disturbance.
Further, it should be noted that the “time” in the
equation represents the total elapsed time since the
change in the rate has been in effect.
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
16

17. Pressure Drop of Multi-Rate Well:
Other Contributions
Second contribution results from decreasing the
rate from Q1 to Q2 at t1, thus:
Note, however, the above approach is valid only if the
well is flowing under the unsteady-state flow condition
for the total time elapsed since the well began to flow at
its initial rate.
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
17

19. Effects of the Boundary
The
superposition
theorem can
also be
extended to
predict the
pressure of a
well in a
bounded
reservoir.
 Figure, which
shows a well
that is located
at distance r
from the nonflow boundary,
e.g., sealing
fault.
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
19

20. Method of Images
in Solving Boundary Problems
The no-flow boundary can be represented by the
following pressure gradient expression:
Mathematically, the above boundary condition can
be met by placing an image well, identical to that of
the actual well, on the other side of the fault at
exactly distance r.
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
20

21. Method of Images
Consequently, the effect of the boundary on the
pressure behavior of a well would be the same as
the effect from an image well located a distance 2r
from the actual well.
In accounting for the boundary effects, the
superposition method is frequently called the
method of images.
Thus, for a well that is located at distance r from
the non-flow boundary, the problem reduces to one
of determining the effect of the image well on the
actual well.
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
21

22. Method of Images (Cont.)
The total pressure drop at the actual well will be
the pressure drop due to its own production plus
the additional pressure drop caused by an identical
well at a distance of 2r, or:
Notice that this equation assumes the reservoir is
infinite except for the indicated boundary.
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
22

23. Extension of the Image Wells Concept
The effect of boundaries
is always to cause greater
pressure drop than those
calculated for infinite
reservoirs.
The concept of image
wells can be extended to
generate the pressure
behavior of a well located
within a variety of
boundary configurations.
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
23

24. Effects of Pressure Change
Superposition is also used in applying the constantpressure case.
Pressure changes are accounted for in this solution
in much the same way that rate changes are
accounted for in the constant rate case.
The superposition method to account for the
pressure-change effect is used in the Water Influx.
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
24

25. Transient Well Testing
Detailed reservoir information is essential to the
petroleum engineer in order to analyze the current
behavior and future performance of the reservoir.
Pressure transient testing is designed to provide
the engineer with a quantitative analysis of the
reservoir properties.
A transient test is essentially conducted by creating a
pressure disturbance in the reservoir and recording the
pressure response at the wellbore, i.e., bottom-hole
flowing pressure pwf, as a function of time.
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
25

26. Pressure Transient Tests
The pressure transient tests most commonly used
in the petroleum industry include:
Pressure drawdown
Pressure buildup
Multirate
Interference
Pulse
Drill stem
Fall off
Injectivity
Step rate
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
26

27. Information Available From a Well Test
It has long been recognized that the pressure
behavior of a reservoir following a rate change
directly reflects the geometry and flow properties
of the reservoir.
Information available from a well test includes:
Effective permeability
Formation damage or stimulation
Flow barriers and fluid contacts
Volumetric average reservoir pressure
Drainage pore volume
Detection, length, capacity of fractures
Communication between wells
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
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30. Well Performance
These lectures presents the practical reservoir
engineering equations that are designed to predict
the performance of vertical and horizontal wells.
Also describe some of the factors that are governing the
flow of fluids from the formation to the wellbore and
how these factors may affect the production
performance of the well.
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
30

31. Production Performance Analysis
The analysis of the production performance is
essentially based on the following fluid and well
characteristics:
Fluid PVT properties
Relative permeability data
Inflow-performance-relationship (IPR)
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
31

32. Productivity Index
A commonly used measure
of the ability of the well to
produce is the Productivity
Index.
Defined by the symbol J,
the productivity index is the
ratio of the total liquid flow
rate to the pressure
drawdown.
For a water-free oil
production, the
productivity index is given
by:

Fall 13 H. AlamiNia
Where
Qo = oil flow rate,
STB/day
J = productivity index,
STB/day/psi
p–r = volumetric
average drainage area
pressure (static
pressure)
pwf = bottom-hole
flowing pressure
Δp = drawdown, psi
Reservoir Engineering 1 Course (2nd Ed.)
32

33. Productivity Index Measurement
The productivity index is generally measured during
a production test on the well.
The well is shut-in until the static reservoir pressure is
reached.
The well is then allowed to produce at a constant flow rate of Q
and a stabilized bottom-hole flow pressure of pwf.
Since a stabilized pressure at surface does not necessarily
indicate a stabilized pwf, the bottom-hole flowing pressure
should be recorded continuously from the time the well is to
flow.
The productivity index is then calculated from:
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
33

34. Productivity Index Conditions
It is important to note that the productivity index is
a valid measure of the well productivity potential
only if the well is flowing at pseudosteady-state
conditions.
Therefore, in order to accurately measure the
productivity index of a well, it is essential that the well is
allowed to flow at a constant flow rate for a sufficient
amount of time to reach the pseudosteady-state.
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
34

35. Productivity Index during Flow
Regimes
The figure
indicates that
during the
transient flow
period,
the calculated
values of the
productivity
index will vary
depending
upon the time
at which the
measurement
s of pwf are
made.
Productivity index during flow regimes
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
35

36. Productivity Index Calculation
The productivity index can be numerically
calculated by recognizing that J must be defined in
terms of semisteady-state flow conditions.
Recalling below Equation:
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
36

37. Application of Productivity Index
Since most of the well life is spent in a flow regime
that is approximating the pseudosteady-state, the
productivity index is a valuable methodology for
predicting the future performance of wells.
Further, by monitoring the productivity index during the
life of a well, it is possible to determine if the well has
become damaged due to completion, workover,
production, injection operations, or mechanical
problems.
If a measured J has an unexpected decline, one of the indicated
problems should be investigated.
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
37

38. Specific Productivity Index
A comparison of productivity indices of different
wells in the same reservoir should also indicate
some of the wells might have experienced unusual
difficulties or damage during completion.
Since the productivity indices may vary from well to well
because of the variation in thickness of the reservoir, it is
helpful to normalize the indices by dividing each by the
thickness of the well.
This is defined as the specific productivity index Js, or:
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
38

40. Qo vs. Δp Relationship
Assuming that the well’s
productivity index is
constant:
Where
Δp = drawdown, psi
J = productivity index
The Equation indicates
that the relationship
between Qo and Δp is a
straight line passing
through the origin with a
slope of J as shown in
Figure.
Qo vs. Δp relationship
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
40

41. Inflow Performance Relationship
Alternatively, productivity
Index Equation can be written
as:
 The above expression shows
that the plot pwf against Qo is
a straight line with a slope of
(−1/J) as shown schematically
in Figure.
This graphical representation
of the relationship that exists
between the oil flow rate and
bottom-hole flowing pressure
is called the inflow
performance relationship and
referred to as IPR.
Qo STB/day
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
41

42. Features of the Straight-Line IPR
Several important features of the straight-line IPR
can be seen in Figure:
When pwf equals average reservoir pressure, the flow
rate is zero due to the absence of any pressure
drawdown.
Maximum rate of flow occurs when pwf is zero. This
maximum rate is called absolute open flow and referred
to as AOF.
The slope of the straight line equals the reciprocal of the
productivity index.
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
42

43. Absolute Open Flow
Although in practice AOF may not be a condition at
which the well can produce,
It is a useful definition that has widespread applications
in the petroleum industry
(e.g., comparing flow potential of different wells in the field).
The AOF is then calculated by:
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
43

44. IPR For Below Pb
(Qo=JΔP) suggests that the
inflow into a well is directly
proportional to the
pressure drawdown and
the constant of
proportionality is the
productivity index.
Muskat and Evinger (1942)
and Vogel (1968) observed
that when the pressure
drops below the bubblepoint pressure, the IPR
deviates from that of the
simple straight-line
relationship as shown in
Figure.
IPR below pb
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
44

45. Pressure Dependent Variables
Affecting PI
Recalling following
Equation:
Treating the term
between the two
brackets as a constant c,
the above equation can
be written in the
following form:

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Above equation reveals
that the variables
affecting the
productivity index are
essentially those that
are pressure
dependent, i.e.:
Oil viscosity μo
Oil formation volume
factor Bo
Relative permeability to
oil kro
Reservoir Engineering 1 Course (2nd Ed.)
45

46. Schematically Illustration of the
Variables as a Function of P
Effect of pressure on Bo, μo, and kro
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kro/μoBo as a function of pressure
Reservoir Engineering 1 Course (2nd Ed.)
46

47. Behavior of Pressure Dependent
Variables
Above the bubble-point pressure pb
The relative oil permeability kro equals unity (kro = 1)
and the term (kro/μoBo) is almost constant.
As the pressure declines below pb:
The gas is released from solution, which can cause a
large decrease in both kro and (kro/μoBo).
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
47

48. Effect of Reservoir Pressure on IPR
Figure shows
qualitatively
the effect of
reservoir
depletion on
the IPR.
Effect of reservoir pressure on IPR
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
48

49. Empirical Methods
to Predict NL Behavior of IPR
Several empirical methods are designed to predict
the non-linearity behavior of the IPR for solution
gas drive reservoirs.
Most of these methods require at least one stabilized
flow test in which Qo and pwf are measured.
All the methods include the following two computational
steps:
Using the stabilized flow test data, construct the IPR curve at
the current average reservoir pressure p–r.
Predict future inflow performance relationships as to the
function of average reservoir pressures.
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
49

50. Empirical Methods to Generate IPR
The following empirical methods that are designed
to generate the current and future inflow
performance relationships:
Vogel’s Method
Wiggins’ Method
Standing’s Method
Fetkovich’s Method
The Klins-Clark Method
Fall 13 H. AlamiNia
Reservoir Engineering 1 Course (2nd Ed.)
50

51. 1. Ahmed, T. (2010). Reservoir engineering
handbook (Gulf Professional Publishing).
Chapter 6 and 7

52. 1. Generating IPR for Oil Wells
A. Vogel’s Method
B. Vogel’s Method (Undersaturated Reservoirs)
a.
Future IPR Approximation
C. Wiggins’ Method
D. Standing’s Method
E. Fetkovich’s Method