Page 1
WELL PERFORMANCE IN SOLUTION GAS DRIVE RESERVOIRS
A
THESIS
Presented to the Graduate Faculty
Of the African University of Science and Technology
In Partial Fulfillment of the Requirements
For the Degree of
MASTER OF SCIENCE IN PETROLEUM ENGINEERING
By
Dorcas Karikari
Abuja-Nigeria
December 2010.
Page 2
WELL PERFORMANCE IN SOLUTION GAS DRIVE RESERVOIRS
A THESIS APPROVED BY THE PETROLEUM ENGINEERING DEPARTMENT
RECOMMENDED: ................................................... Chair, Dr. Alpheus Igbokoyi
.......................................................... Professor Tiab Djebbar
........................................................... Professor Godwin A. Chukwu
APPROVED: ...................................................... Chief Academic Officer
.................................................
Date
Page 3
Copyright by DORCAS KARIKARI 2010
All Rights Reserved
Page 4
i
ACKNOWLEDGEMENT
My utmost gratitude and appreciation goes to the Most High God, without whom I am nothing.
Indeed He who has begun a good work in me would surely bring me to an expected end.
My greatest measure of gratitude goes to my supervisor Dr. Alpheus Igbokoyi for his tireless
effort in ensuring that this thesis work becomes a reality. To my thesis committee members, Prof.
Djebbar Tiab, Prof. Godwin Chukwu and Dr. Calin Gheorghiu, I say may the good Lord bless all
your endeavors, I thank you all for serving and also appreciate all your assistance.
I am very much indebted to my parents, Nana Karikari Antwi and Mrs. Mary Karikari Antwi for
their prayers, substance and support. To my siblings, Daniel, Beatrice, Diana and Mina, I say
thank you for all you have been doing for me.
I wish to express my indebt gratitude to the Lecturers and staff of the Petroleum Engineering
Department. To Azeb Demisi, Titus Ofei and Babatunde Ayeni, I couldn’t have done much
without your support, thanks guys.
To the Petroleum Engineering class of 2010, I say your support has been great. My gratitude
goes to Joseph Akrong, Ebenezer Sekyi Parker, Lowestein Odai, Mark Owusu, Auphedeous
Dang-I, Richard Botah and all my group members for their unfailing support. To Dorothy
Maduagwu and all the ladies in my class, I appreciate all your efforts.
If I have seen farther than others, it’s because I was standing on the shoulders of giants.
-: Sir Isaac Newton:-
Page 5
ii
TABLE OF CONTENT
ACKNOWLEDGEMENT ........................................................................................................ i
TABLE OF CONTENT ........................................................................................................... ii
ABSTRACT............................................................................................................................ vii
CHAPTER 1: INTRODUCTION .............................................................................................1
1.1 PROBLEM STATEMENT ...........................................................................................1
1.2 STUDY APPROACH ...................................................................................................1
1.3 OBJECTIVES ...............................................................................................................2
1.4 OUTLINE OF THIS WORK .........................................................................................2
CHAPTER 2: LITERATURE REVIEW .................................................................................3
2.1 RESERVOIR NATURAL DRIVE MECHANISMS .....................................................3
2.1.1 Solution Gas Drive Mechanism ..............................................................................4
2.2 MATERIAL BALANCE EQUATION FOR DRIVE MECHANISMS ..........................7
2.2.1 Material Balance for Solution Gas Drive Reservoirs ..............................................8
CHAPTER 3: INFLOW PERFORMANCE .......................................................................... 12
3.1 INFLOW PERFORMANCE RELATIONSHIP (IPR) ................................................. 12
3.1.1 Saturated IPR (Depletion below the Bubblepoint Pressure) .................................. 14
3.1.2 Saturated Future IPR ............................................................................................ 14
3.1.3 Undersaturated IPR (Depletion above the Bubblepoint Pressure) ......................... 15
3.1.4 Future Undersaturated IPR ................................................................................... 16
3.1.5 Beggs and Brill Correlation .................................................................................. 16
3.1.6 Generation of IPR Curves .................................................................................... 16
3.1.6.1 Methodology for IPR development ............................................................... 17
Page 6
iii
3.1.6.2 Factors Affecting Well Performance ............................................................. 18
3.2 OUTFLOW PERFORMANCE RELATIONSHIP (OPR) ............................................ 22
3.3 WHEN TO APPLY ARTIFICIAL LIFT ..................................................................... 25
3.3.1 Methods of Artificial Lift ..................................................................................... 25
3.3.2 Larger tubings with Gas Lift ................................................................................ 27
3.3.3 Use of Velocity String (smaller tubings) with Gas Lift ......................................... 28
CHAPTER 4: IPRs FOR NATURALLY FRACTURED RESERVOIR .............................. 30
4.1 NATURALLY FRACTURED RESERVIORS ............................................................ 30
4.2 IPRS FOR A NFR VERTICAL WELL ....................................................................... 32
4.3 IPRS FOR A NFR HORIZONTAL WELL ................................................................. 40
CHAPTER 5: ESTIMATION OF PRODUCTIVITY INDEX OF HORIZONTAL WELL
IN RESERVOIR WITH PARTIAL PRESSURE SUPPORT ............................................... 43
5.1 INTRODUCTION ...................................................................................................... 43
5.2 MATHEMATICAL FORMULATION ....................................................................... 44
5.2.1 Method 1 ............................................................................................................. 44
Field Application ............................................................................................................... 46
5.2.2 Method 2 ............................................................................................................. 50
CHAPTER 6: CONCLUSIONS AND RECOMMENDATIONS .......................................... 62
6.1 CONCLUSIONS ........................................................................................................ 62
6.2 RECOMMENDATIONS ............................................................................................ 62
NOMENCLATURE ................................................................................................................ 64
REFERENCES ........................................................................................................................ 65
APPENDIX A - EQUATIONS................................................................................................ 70
Page 7
iv
LIST OF FIGURES
Figure 2.1 Schematic of the production history of a solution gas drive reservoir ..........................5
Figure 2.2 Individual well pressure declines displaying equilibrium in the reservoir ....................7
Figure 2.3 Solution gas drive reservoir; (a) above bubble point pressure; liquid oil, (b) below
bubble point; oil plus liberated solution gas .................................................................................9
Figure 3.1 Present and Future IPRs of the reservoir ................................................................... 18
Figure 3.2 (a) Water-Oil relative permeability functions. (b) Gas-Oil relative permeability
functions. .................................................................................................................................. 20
Figure 3.3 Behavoir of Reservoir Fluid Properties ..................................................................... 21
Figure 3.4 All IPRs with various tubing strings ......................................................................... 23
Figure 3.5 Performance of 2 7/8” tubing .................................................................................... 24
Figure 3.6 Performance of Larger tubing at different GLRs ....................................................... 27
Figure 3.7 Performance of 2 7/8” tubing with the natural flow and gas lift ................................ 28
Figure 3.8 Performance of 1 ½” tubing at different GLRs .......................................................... 29
Figure 4.1 Double-Porosity reservoir (=0.001) ........................................................................ 34
Figure 4.2 Double-Porosity Reservoir (=0.01) ........................................................................ 35
Figure 4.3 Double-Porosity Reservoir (=0.1) .......................................................................... 36
Figure 4.4 Double-Porosity Reservoir (=0.5) .......................................................................... 37
Figure 4.5 Single-Porosity Reservoir (Vertical Well) ................................................................ 38
Figure 4.6 ∆P and pressure derivative plot for an NFR .............................................................. 39
Figure 4.7 Semilog pressure behavior of an NFR ...................................................................... 39
Figure 4.8 Single-Porosity Reservoir (Horizontal Well) ............................................................ 41
Figure 5.1 Downhole Pressure Gauge data from the field example ............................................ 48
Page 8
v
Figure 5.2 Model for case 1 ....................................................................................................... 48
Figure 5.3 Model for Case 2 ...................................................................................................... 49
Page 9
vi
LIST OF TABLES
Table 3.1: Equilibrium flow capacities and Pwfs of larger tubing strings ................................... 22
Table 4.1: Summary of PI Calculation (Vertical Well) .............................................................. 40
Table 4.2: Summary of PI Calculation (Horizontal Well) .......................................................... 42
Table 5.1: Results of the field application using the first model ................................................. 49
Table 5.2: Results of the field application using the second model............................................. 50
Table 5.3: Results of the field application using the third model ................................................ 53
Table 5.4: Results of the field application using the fourth model .............................................. 54
Table 5.5: Results of the field application using the fifth model ................................................. 58
Table 5.6: Summary of the pressure drops and skins computed for the five cases ...................... 59
Page 10
vii
ABSTRACT
The practical application of various production parameters and relations to predict the well
performance analysis of a solution gas drive reservoir is the primary objective of this study.
These parameters include: IPR (inflow performance relation), OPR (outflow performance
relation) and PI (productivity index).
Theoretical data was used to predict the performance behavior of a solution gas drive reservoir
from start of production till its abandonment. IPRs and OPRs were developed during the
forecasting, over the life of the reservoir.
IPRs of a naturally fractured reservoir were also developed for both vertical and horizontal wells.
Hagoort equation was used to develop the NFR IPRs. The storativity ratio was varied to
investigate its effect on the productivity index equation in transforming single-porosity reservoir
into a double-porosity reservoir productivity index. It was observed that there is no significant
difference in the productivity index obtained with single porosity and that of double porosity
developed by Hagoort.
A new method of estimating productivity index in a horizontal well using the shut in pressure
data in the absence of bottom-hole flowing pressure was developed. The available method of
estimating productivity index in a horizontal is based on steady or pseudo-steady state flow
regime. The reservoir pressure drop may exhibit limited aquifer or pressure support which makes
the reservoir’s flow regime to behave like neither steady nor pseudo-steady state. Therefore, the
steady or pseudo-steady state equations developed for estimating productivity index are not
applicable
In this work, historical shut in pressure data acquired were used as the average reservoir pressure
to compute the pressure drop due to a particular production rate at any time. The productivity
index was then computed. Field data were used to test the model and good results were obtained.
Page 11
1
CHAPTER 1: INTRODUCTION
1.1 PROBLEM STATEMENT
Well Performance is the measurement of a well’s production of oil or gas as related to the well’s
anticipated productive capacity, pressure drop or flow rate (Anon, 2010). Productivity Index of a
well is a direct measurement of a well’s performance. A solution gas drive reservoir is one in
which the principal drive mechanism is the expansion of the oil and its originally dissolved gas.
The increase in fluid volumes during the process is equivalent to the production (Dake, 1978).
This is due to the fact that no extraneous fluids or gas caps are available to replace the gas and oil
withdrawals. This rapid and continuous decline in reservoir pressure has an immense effect on
the reservoir performance at the early stages of the life of the reservoir. Ultimate oil recovery of
a solution gas drive reservoir is less than 5% to about 30% (Tarek, 2001). This low recovery
suggests that large quantities of oil remain in the reservoir and, therefore, solution gas drive
reservoirs are usually considered the best candidates for secondary recovery applicants.
This work looks at the well performance analysis of a solution gas drive reservoir which involves
inflow performance, outflow performance and productivity index determination during the life of
the reservoir. However, due to the low recovery of solution gas drive reservoir, artificial lift
technologies such as gas lift may be employed for continuous production of the reservoir.
Another challenge is to know when to change tubing for optimum production. In this study, I
used IPR-OPR to determine the time of tubing change or gas-lift installation.
1.2 STUDY APPROACH
Inflow performance relation (IPR) in conjunction with the outflow performance relation (OPR)
for the whole life of the well is designed in accordance with the material balance equation
prediction. This design is done with regards to the available gas lift and maximum production
constraints. Production forecast is made based on Fetkovich’s model (for present IPR) and
Eickmeier's model (for future IPRs) to know the time when tubing strings will be replaced for
optimum production. Also, IPRs of a naturally fractured reservoir is also developed for both
Page 12
2
vertical and horizontal wells. Finally, a new productivity index equation for a horizontal well in a
reservoir with partial pressure support is developed.
1.3 OBJECTIVES
The objectives of this work are to:
Forecast the production plan of the oil well.
Develop IPRs for vertical and horizontal wells in naturally fractured reservoirs.
Develop new productivity index equations for horizontal wells in a reservoir with partial
pressure support.
1.4 OUTLINE OF THIS WORK
This work is made up of six (6) chapters. Chapter 1 defines the problem, the objectives and the
methodology used in solving the problem. Chapter 2 presents the literature review of this topic.
Chapter 3 introduces the design of the IPRs and OPRs for the production of the reservoir as well
as the artificial lift technology taking into consideration to set constraints. Chapter 4 presents the
IPRs for vertical and horizontal wells in naturally fractured reservoirs. Chapter 5 presents the
new productivity index equation for horizontal well in a reservoir with partial pressure support.
Chapter 6 gives the conclusions and recommendations for future research work in this area.
Page 13
3
CHAPTER 2: LITERATURE REVIEW
2.1 RESERVOIR NATURAL DRIVE MECHANISMS
Reservoirs can be classified on the basis of the boundary type, which determines the drive
mechanisms (Boyun et al., 2007). The source of pressure energy that causes hydrocarbons flow
into the wellbore has a substantial effect on both the performance of the reservoir and the total
production system (Beggs, 2003). To really understand reservoir behaviour and predict future
performance, it is of a necessity to have knowledge of the drive mechanisms that control the
behavior of fluids within the reservoirs (Tarek, 2001). In performance prediction of a
hydrocarbon reservoir under different drive mechanisms, different conditions arise during the
exploitation of the reservoirs (Ambastha & Aziz, 1987). These conditions could either be one of
the following:
Internal gas drive mechanism
External gas drive mechanism or
Gravity segregation.
With internal gas drive mechanism, volumetric undersaturated reservoirs are produced by liquid
expansion and rock compressibility. As the reservoir pressure declines, oil phase contracts due to
release of solution gas. With external gas drive mechanism, saturated reservoirs are produced by
depletion drive mechanism (Ambastha & Aziz, 1987). In many cases, reservoir pressure is
maintained by gas injection and oil is displaced by injected gas, thus making it an external gas
drive mechanism (Ambastha & Aziz, 1987). With the gravity segregation, high relief reservoirs
with good along-dip permeability give favourable conditions for gravity segregation of injected
gas or gas released from solution (Ambastha & Aziz, 1987). There are basically six driving
mechanisms that provide the natural energy necessary for oil recovery (Tarek, 2001):
Liquid and rock expansion drive
Depletion drive
Gas cap drive
Water drive
Gravity drainage drive
Page 14
4
Combination drive
Usually, one or more of the first five drive mechanisms are the predominant ones and each
reservoir is associated with one or two dominant primary drive mechanisms. This work considers
solution gas drive mechanism also known as the depletion drive mechanism.
2.1.1 Solution Gas Drive Mechanism
This is a drive mechanism that occurs in undersaturated oil reservoirs. It is also known as:
Depletion drive
Dissolved gas drive
Internal gas drive
This drive mechanism is characterized by the expansion of hydrocarbon with the gas remained in
solution when the reservoir pressure is above bubble point pressure. Therefore dissolved gas
drive is the drive mechanism where the reservoir gas is held in solution in the oil (Boyun et al,
2007). Thus a dissolved gas drive reservoir is closed from any outside source of energy, such as
water encroachment. Its pressure is initially above the bubble point pressure, and therefore, no
free gas exists. The only source of fluid to replace the produced fluids is the expansion of the
fluids remaining in the reservoir (Beggs, 2003). However, a closed saturated oil reservoir with
negligible gas cap will exhibit solution gas drive mechanism at the beginning.
When the pressure falls, the gas phase is generated. The gas phase being compressible helps in
maintaining the reservoir pressure and hence provides a secondary gas cap driving force for
primary production (Kumar et al., 2000).
Gas does not flow as an independent phase until it reaches certain saturation known as the
critical gas saturation (Kumar et al., 2000). Critical gas saturation is defined as the gas
saturation at which a steady, although intermittent, gas flow can be sustained (Kumar et al.,
2000). The lower the critical gas saturation, the more rapidly the gas will be mobilized and
produced, thus accelerating the depletion and impairing the final recovery (Consentino et al.,
2005). After this gas saturation, free gas flows as an independent phase, resulting in rapid decline
in reservoir pressure. To improve oil recovery in the solution gas drive reservoir, early pressure
Page 15
5
maintenance is usually preferred by re-injecting the gas produced. This is the minimum amount
of gas that should be injected to minimize decline in reservoir pressure.
Solution gas drive reservoir, in general is characterized by rapid and continuously declining
reservoir pressure. There is little or no water production, due to the absence of water drive, with
the oil during the entire producing life of the reservoir. It is also characterized by a rapidly
increasing gas-oil ratio (GOR) from the wells when producing below bubble point pressure,
regardless of their structural position (Tarek, 2001). Once the gas saturation exceeds the critical
gas saturation, free gas begins to flow toward the wellbore and GOR increases. Oil production by
this drive mechanism is usually the least efficient recovery method. This is as a result of the
formation of gas saturation throughout the reservoir. Ultimate oil recovery from solution gas
drive reservoirs may vary from less than 5% to about 30% (Tarek, 2001).
A typical producing history of a solution gas drive reservoir under primary producing conditions
is shown in Figure 2.1. As can be seen, the instantaneous or producing gas oil ratio R will greatly
exceed Rsi for pressures below bubble point and the same is true for the value of Rp. The pressure
will initially decline rather sharply above bubble point pressure because of the low
compressibility of the reservoir system but this decline will be partially arrested once free gas
starts to accumulate (Dake, 1994) as secondary gas cap.
Figure 2.1 Schematic of the production history of a solution gas drive reservoir
Page 16
6
All the methodologies that have been developed to predict the future reservoir performance are
essentially based on employing and combining the relationships that include the following
(Tarek, 2001):
MBE
Saturation equations
Instantaneous GOR
Equation relating the cumulative gas-oil ratio to the instantaneous GOR
Using the above information, it is possible to predict the field primary recovery performance
with declining reservoir pressure. There are three methodologies that are widely used in the
petroleum industry to perform a reservoir study based on material balance (Tarek, 2001). These
are:
Tracy's method
Muskat's method
Tarner's method
Page 17
7
2.2 MATERIAL BALANCE EQUATION FOR DRIVE MECHANISMS
The material balance equation (MBE) has long been recognized as one of the basic tools of
reservoir engineers for interpreting and predicting reservoir performance. The MBE, when
applied properly, can be used to (Tarek, 2001):
Estimate initial hydrocarbon in place
Predict and plan future reservoir performance
Predict ultimate hydrocarbon recovery under various types of natural drive mechanisms
It is commonly believed that rapid pressure equilibration is a prerequisite for successful
application of material balance but this is not the case; the necessary condition is that an average
pressure decline trend can be defined which is possible even if there are large pressure
differentials across the accumulation under normal producing conditions. All that is necessary is
to devise some means of averaging individual well pressure declines to determine a uniform
trend for the reservoir as a whole (Dake, 1994).
Figure 2.2 Individual well pressure declines displaying equilibrium in the reservoir
In its simplest form, the MBE can be expressed on volumetric basis as the initial hydrocarbon
volume in place equals the sum of the volume removed and the volume remaining (Tarek, 2001).
Upon various considerations, the MBE can be expressed mathematically in a more convenient
form as shown below.
Page 18
8
p
SccS
mBBB
mBBRRBB
BWBGBWWBRNGBNN
wi
fwwioi
gi
goigssioio
winjginjinjwpegsppp
111
0 (2.1)
There are essentially three unknowns in equation 2.1:
a) The original oil in place N.
b) The cumulative water influx We.
c) The original size of the gas cap as compared to the oil zone size m.
In developing a methodology for determining the above three unknowns, Havlena and odeh
(1963) expressed Eqn. 2.1 in the following form:
ginjinjwinjewi
fwiwoi
gi
goigssioiowpgspop
BGBWWpS
cScBmN
BB
mNBBRRBBNBWBRRBN
11
1
(2.2)
Havlena and Odeh (1963) further expressed this equation into a more condensed for as:
ginjinjwinjewfg BGBWWEmEENF ,0 (2.3)
Considering a solution gas drive reservoir, for the purpose of simplicity, with no pressure
maintenance by gas or water injection, the above equation can be further simplified and
expressed as:
WEmEENF wfgo , (2.4)
2.2.1 Material Balance for Solution Gas Drive Reservoirs
Havlena and Odeh (1963) examined several cases of varying reservoir types with equation 2.4
and showed that the relationship can be arranged into a form of a straight line. Solution gas drive
reservoirs are assumed to be volumetric due to the absence of water influx and gas caps. In
Page 19
9
determining the material balance for this type of drive mechanism, two phases can be
distinguished, as shown in Figure 2.2 (a) when the reservoir oil is undersaturated and (b) when
the pressure is fallen below the bubble point and a free gas phase exists in the reservoir (Dake,
1978).
Figure 2.3 Solution gas drive reservoir; (a) above bubble point pressure; liquid oil, (b)
below bubble point; oil plus liberated solution gas
I) Above bubble point pressure (undersaturated oil): Depletion above the bubble point is
what makes the reservoir undersaturated. For a solution gas drive reservoir, it is assumed
that there is no water influx, We, (making it a volumetric reservoir), and no gas cap, thus,
m=0 (making it undersaturated). Since all produced gas is dissolved in the oil, Rs=Rsi=Rp
(Tarek, 2001). Applying all the above listed conditions on equation 2.1 gives:
pS
ccSBBB
BNN
wi
fwwioioio
op
1
(2.5)
with
∆p = pi - p
where pi = initial reservoir pressure, psi
Page 20
10
p = current reservoir pressure, psi
Hawkins (1955) introduced the oil compressibility co into the MBE to further simplify the
equation. Oil compressibility is therefore defined as:
pBB
Bc oio
oio
1 Rearranging, gives: pBcBB oiooio (2.5a)
Combining the above expression with equation 2.5 gives:
pS
ccSBpBc
BNN
wi
fwwioioio
op
1
(2.6)
The denominator of the above equation can be expressed as:
pS
cScScB
wi
f
wi
wwiooi
11
(2.7)
Since there are only two fluids in the reservoir, oil and water, then: 1 wioi SS
Equation 2.7 can then be expressed as:
pS
ccScSB
wi
fwwiooioi
1 (2.8)
The term in the brackets of equation 2.8 is called the Effective compressibility and
defined by Hawkins (1955) as:
wi
fwwiooie S
ccScSc
1 (2.9)
Combining equations 2.6, 2.7 and 2.9, the MBE above the bubble point pressure
becomes:
Page 21
11
PPcBBN
pcBBN
Nieoi
op
eoi
op
(2.10)
Rearranging and solving for the cumulative oil production Np gives:
pBBNcN
oi
oep
(2.11)
The calculation of future reservoir production, therefore, does not require a trial-and-error
procedure, but can be obtained directly from the above expression (Tarek, 2001).
II) Below bubble point pressure (saturated oil): Saturated reservoir is one that originally
exists at its bubble point pressure. Once the pressure falls below the bubble point solution
gas is liberated from the oil leading, in many cases, to a chaotic and largely
uncontrollable situation in the reservoir, which is the characteristic of what is referred to
as the solution gas drive process. Assuming that the water and rock expansion term Ef,w =
0 or negligible in comparison with the expansion of solution gas, the general MBE may
be expressed by:
gssioio
gsppop
BRRBBBRNGBN
N
(2.12)
The above MBE contains two unknowns, which are:
Cumulative oil production Np
Cumulative gas production Gp
In predicting the primary recovery performance of a solution gas drive reservoir in terms
of these unknowns, the following reservoir and PVT data must be available (Tarek,
2001):
Original Oil in Place N
Hydrocarbon PVT data
Initial fluid saturations
Relative permeability data
Page 22
12
CHAPTER 3: INFLOW PERFORMANCE
3.1 INFLOW PERFORMANCE RELATIONSHIP (IPR)
Inflow Performance Relationship (IPR) of a well is the relation between the production rate and
flowing bottom-hole pressure (Jahanbani et al., 2009). During production, the fluids will flow
inside the porous media losing pressure as they flow towards the perforations. The driving force
for the fluids to move inside the porous media is the pressure drop in the reservoir. Therefore, the
rate of the inflow of fluids from the reservoir into the well is a function of the bottom-hole
flowing pressure, Pwf, at the midpoint of the perforations. This function is referred to as IPR.
The IPR represents the pressure available in front of the perforations for the fluids to flow inside
the porous media at certain flow rate. A commonly used measure of the ability of the well to
produce is called the Productivity Index (J). Productivity index is defined as the ratio of the total
liquid flow rate to the pressure drawdown (Tarek, 2001). In monitoring the productivity index
during the life of a well, it is possible to determine if the well has become damaged due to
completion, work over, production, injection operations, or mechanical problems. Evinger and
Muskat (1942), based on multi-phase flow equations showed that a curved relationship existed
between flow rate and pressure, when two phase flow occurs in the reservoir (i.e. saturated oil).
For oil wells, it is frequently assumed that fluid inflow rate is proportional to the difference
between reservoir pressure and wellbore pressure. This assumption leads to a straight line
relationship that can be derived from Darcy's law for steady state flow of an incompressible,
single phase fluid. However, this assumption is valid only above the bubble point pressure. The
single phase IPR can be represented as:
wfPPJq (3.1)
Aw
e
CrrB
khJ657.2lnln
100708.0
(3.2)
where q = Flow rate, STB/D
Page 23
13
J = Productivity Index, STB/D/psi
wfPP = Pressure Drawdown, psi
P = Average Reservoir Pressure, psi
Pwf = Bottom-hole Flowing Pressure, psi
The absolute open flow (AOF) is defined as the maximum flow rate the reservoir can produce
when the bottom-hole flowing pressure is zero.
PJq max (3.3)
PP
qq wf1max
(3.4)
The incompressible single phase or straight line IPR/Linear IPR is valid when the fluids flowing
inside the reservoir are in single phase incompressible conditions. As the pressure inside the
reservoir goes below the bubble point value, gas comes out of solution increasing the oil
viscosity. In case of two phase flow conditions, oil productivity is reduced, since the driving
force for fluid movement is spent moving the liquid and gas phases. The flow rate under these
conditions for a certain pressure gradient is smaller than the flow rate under single phase flow
conditions for the same pressure gradient.
There are several empirical methods that are designed to predict the non-linearly behavior of the
IPR for solution gas drive reservoirs. The following empirical methods are designed to generate
the current and future inflow performance relationships (Tarek, 2001):
Vogel's Method
Fetkovich’s Method
Wiggin's Method
Standing's Method
The Klins-Clark Method
Linear Method
Eickmeier's Method
Page 24
14
3.1.1 Saturated IPR (Depletion below the Bubblepoint Pressure)
The most commonly used correlations for saturated IPR are Vogel and Fetkovich methods.
Vogel used a numerical reservoir simulator to generate the IPR. One of the most achievements of
Vogel's work is the recognition that the inflow performance is a strong function of the average
reservoir pressure and AOF potential. The value of the productivity index J, needs to be
redefined for the case of saturated IPR.
2
max
11
PP
bP
Pb
qq wfwf (3.5)
The best value for b according to Vogel's numerical results is -0.2. Fetkovich following an
analytical approach also proposed an IPR with a b value of 0. Several other investigators like
Wiggins, Klins-Clark, Linear, and Standing obtained different values for the b. Among these
investigators, Fetkovich’s method is the most conservative. The difference between the flow
capacities calculated using Vogel and Fetkovich is about 11%. The difference between the flow
capacities calculated using Linear IPR and Vogel/Fetkovich can be as high as 80% to 100%.
3.1.2 Saturated Future IPR
The prediction of future IPR is very important to forecast future well production. There are many
approximate methods to simulate the effects of depletion on productivity index for saturated
conditions. Usually, these methods provide an equation relating changes in the productivity
index J* as a function of the average reservoir pressure. In essence, the methods for future
reservoir prediction express changes in J* as a function of changes in average reservoir pressure.
Also, effects of changes in average reservoir pressure over AOF (qmax) can be determined.
1
2*
*
1
2
PPF
JJ
P
P (3.6)
1
2
1
2
max
max
1
2
PPF
PP
qq
P
P
(3.7)
Page 25
15
In determining the F to use when predicting the future IPR, Eickmeier has been proved to be
most conservative with its function as:
1
2
PP simplifying equations 3.6 & 3.7 to:
2
1
2*
*
1
2
PP
JJ
P
P (3.8)
3
1
2
max
max1
2
PP
qq
P
P
(3.9)
3.1.3 Undersaturated IPR (Depletion above the Bubblepoint Pressure)
An undersaturated reservoir is a reservoir that has an average reservoir pressure higher than the
bubble point pressure. When the bottom-hole flowing pressure is higher than the bubble point
pressure, the flow in the reservoir is single phase and the Linear IPR is valid. As a result, in
designing the IPR for a reservoir, if Pwf is higher than the bubble point pressure, Linear IPR is
used up to the bubble point pressure, then Fetkovich’s IPR is used to the AOF (qmax) of the
reservoir. Fetkovich’s equation to be used for the saturated IPR below the bubble point can
therefore be expressed as:
2
max
11
b
wf
b
wf
b
b
PP
bPP
bqq
qq (3.10)
bb PPJq (3.11)
where, b = 0 (Fetkovich)
qb = Flow rate at the bubblepoint
Equation for the Linear IPR section can be expressed as: wfPPJq
Page 26
16
3.1.4 Future Undersaturated IPR
The future undersaturated IPR is just a combination of the future IPR behaviour for the single
phase and saturated cases. For the Linear part, the productivity index is almost constant with
reservoir depletion. For saturated part, the productivity index will decline with average reservoir
pressure as described by several correlations for this case.
3.1.5 Beggs and Brill Correlation
The Beggs and Brill program (Prado, 2009) is a spreadsheet program developed to obtain the
IPR and OPR curves. This correlation enables the calculation of the pressure gradient as a
function of other production variables like pipe diameter, GLR and flow rates. This correlation is
applicable to inclined wells with or without water cut. It also predicts pressure drop for upward
and downward fluid flow with accuracy.
3.1.6 Generation of IPR Curves
Data used for the class project were used for the development of the IPRs and OPRs for this
work. The purpose of this example is to predict the performance of a well as the reservoir
pressure declined. The best tubing string was determined using the IPRs and OPRs developed.
Fetkovich correlation was used in developing the current IPR and Eickmeier for the future IPRs.
Excel spreadsheet was used in building the model for the IPRs while Beggs and Brill pressure
traverses correlation was used in developing the values for the OPRs. The effect of artificial lift
mechanism was also reviewed to determine the point where it would be introduced in the life of
the well.
Available Well Data
API Gravity – 25
Gas Specific Gravity – 0.7
Average Temperature – 170 F
Reservoir Depth – 7500ft
Page 27
17
Wellhead Pressure – 150 psi
Tubing diameters – ½'', 1'', 1 ½'', 2 3/8'', 2 7/8'', 3 ½''
Average Reservoir Pressure , Pr – 3500 psi
Water Cut, WC – 0%
Gas Liquid Ratio, GLR – 600 SCF/STB
Productivity Index, J* - 1.5 bpd/psi
Using Beggs and Brill Pressure Traverses correlation, the bubble point pressure of the reservoir
was determined as 3401psi.
Basic Characteristics of the reservoir include the following:
Undersaturated reservoir due to bubble point pressure of 3401psi which is below the
reservoir pressure of 3500psi.
The reservoir is a solution gas reservoir with no water production
Reservoir fluid is Black Oil due to the API gravity of 25.
3.1.6.1 Methodology for IPR development
As mentioned earlier, bubble point pressure of 3401 psi was determined. Using equation 3.11,
the bubble point flow rate, qb was calculated as 148.5 psi. The Absolute Open Flow, qmax1 was
then calculated using equation 3.5 to be 2699.25 bpd considering Fetkovich correlation (b=0).
At the bubble point, taking b = 0, qmax2 was calculated to be 2550.75 bpd. For the undersaturated
part of the IPR, the Pr, Pb, qmax1 and qmax2 were the only parameters needed for the model
building. But for the saturated part of the IPR, equation 3.10 was used to calculate the flow rate.
Inputting Pwf, Pb, qmax1, qmax2, qb and b=0 into the excel spreadsheet created, the current IPR was
generated. Fetkovich correlation was used in all the calculations made for the current IPR.
For the future IPRs, Eickmeier correlation was used. As discussed earlier, the productivity index
for the future IPRs – undersaturated part, remained constant while that of the saturated changed.
The average reservoir pressure was assumed to be declining by 250 psi gradually. These
Page 28
18
pressures and flow rates calculated were inputted into the model built and various data points
were generated which were used in generating the IPR graphs. A plot of the current and future
IPRs is shown in Figure 3.1 below.
Figure 3.1 Present and Future IPRs of the reservoir
3.1.6.2 Factors Affecting Well Performance
The following factors affect the productivity of a well, and in effect, affect the well performance.
Reservoir pressure: With a high reservoir pressure, given the good bottom-hole flowing
pressure will correspond to a higher productivity.
Page 29
19
Pay zone thickness and permeability: With a good reservoir pay thickness and high
permeability, a high productivity is assured.
Reservoir boundary type and size (Drive Mechanism): As discussed earlier in Chapter 2,
each reservoir drive mechanism gives an idea of the ultimate recovery of the reservoir
and this affects the well productivity.
Wellbore Radius: How large the wellbore radius is affects the productivity. This is due to
the fact the productivity index of a well is computed inversely proportional to the
wellbore radius.
Near-wellbore conditions: The presence or absence of a near-wellbore damage (Skin
damage) goes in high way to affect the well productivity.
Reservoir relative permeability: The relative permeability to oil, gas and water recorded
during well testing also in a way affect the well performance. This is illustrated in the
equations and curves below.
Relative Permeability to Oil
P
oooro Pt
Bkh
qk
'6.70 (3.12)
Relative Permeability to Gas
Poo
ggrosrg B
BkRGORk
(3.13)
Relative Permeability to Water
Poo
wwro
o
wrw B
Bkqqk
(3.15)
Page 30
20
Relative Permeability Curves (Dake, 1978)
Figure 3.2 (a) Water-Oil relative permeability functions. (b) Gas-Oil relative
permeability functions.
From the above Figures, it is observed that, below the bubble point pressures in a reservoir,
relative permeabilities to gas increases as that to oil decreases. When water production starts, the
relative permeability will to water will increase which will also lead to decrease in oil relative
permeability, thus decreasing the amount of oil produce and the well productivity.
The behavior of fluid properties at various reservoir pressures is also illustrated in Figure 3.4
below. With reservoir pressure above the bubble point pressure, the oil dynamic viscosity is
barely constant but rapidly increases below the bubble point pressure. This is due to the
liberation of free gas from the solution gas drive reservoir. This free gas produced causes a
decrease in the solution GOR whiles increasing the produced GOR.
(a) (b)
Page 31
21
Figure 3.3 Behavoir of Reservoir Fluid Properties
Page 32
22
3.2 OUTFLOW PERFORMANCE RELATIONSHIP (OPR)
This process is used to determine the most preferable tubing size for any well system (Beggs,
2003). The size or diameter of the production tubing can play an important role in the
effectiveness with which a well can produce liquid (Lea et al., 2008). Smaller tubing sizes have
higher frictional losses and higher gas velocities which provide better transport for the produced
fluids. Larger tubing sizes, on the other hand, tend to have lower frictional losses due to lower
gas velocities and in turn lower the liquid carrying capacity (Lea et al., 2008). Tubing sizes too
large will cause a well to load up too quickly with liquid and lead to faster depletion of the
reservoir. In designing tubing string, it becomes important to balance the effects over the life of
the field (Lea et al., 2008).
Figure 3.2 below is a plot of the outflow performance (OPR) of the various tubing strings
superimposed on the IPR curves. It is observed that the smaller sized tubings (0.5”, 1” and 1.5”)
have excessive frictional losses with low production rates thereby restricting production. For this
reason, only the larger sized tubings (2 3/8”, 2 7/8” and 3 ½”) are considered much better
candidates to start producing the well. However, the 3 ½” tubing exhibits the lowest frictional
loss which might cause the well to load up with liquids and die too early. The 2 7/8” tubing gives
a more reasonable frictional loss as compared to that of 3 ½” and 2 3/8” tubings with an
equilibrium production rate of about 2060 bpd and an equilibrium bottom-hole flowing pressure
of about 1700 psi (Figure 3.2). As such, the 2 7/8” tubing is most preferable at the start of
production of the reservoir.
The producing capacities and equilibrium bottom-hole flowing pressures were recorded from the
intersections of the IPRs and OPRs in Figure 3.2
Table 3.1: Equilibrium flow capacities and Pwfs of larger tubing strings
Tubing sizes, inches Equilibrium Flow rates, bpd Equilibrium Pwf, psi
2 3/8 1780 2050
2 7/8 2060 1700
3 1/2 2227 1450
Page 33
23
Figure 3.4 All IPRs with various tubing strings
At lower flow rates, effect of gravity is dominant. However, this effect is observed at almost a
common bottom-hole flowing pressure point, thus, about 1100 psi for the three larger tubing
strings. This suggests that the effect of gravity is the same irrespective of the tubing size selected.
The 2 7/8” tubing produces the reservoir at an average reservoir pressure of 3500 psi and a GLR
of 600 scf/stb up to a Pwf of about 1100 psi and an equilibrium flow rate of 550 bpd as shown in
Figure 3.3 below.
3 ½”
2 7/8”
2 3/8”
1 ½” 1”
½”
Page 34
24
Figure 3.5 Performance of 2 7/8” tubing
GLR of 600 scf/stb
Page 35
25
3.3 WHEN TO APPLY ARTIFICIAL LIFT
An IPR curve describes the effects of flowing pressures on production rates. The goal in
producing a well efficiently should be to produce at the lowest possible flowing pressure. When
the reservoirs pressure is insufficient to sustain the flow of oil to the surface at adequate rates,
natural flow must be aided by artificial lift. The rate-pressure relationship (IPR) of a well is used
for investigating the need to introduce artificial lift, selecting the most suitable lift system, and
determining its size and capacity (Golan and Whitson, 1995). The only way to obtain a high
production rate of a well is to increase production pressure drawdown by reducing the bottom-
hole pressure with artificial lift methods (Boyun et al, 2007). Approximately 90% of wells
worldwide produce by some form of artificial lift systems (Prado, 2009).
As the pressure in a reservoir declines from depletion, the producing capacity of the wells will
decline. The decline is caused by both a decrease in the reservoir's ability to supply fluid to the
wellbore, and, in some cases, an increase in the pressure required to lift the fluids to the surface.
That is, both inflow and outflow conditions may change (Beggs, 2003). The only way in which
the inflow can be kept high, once the well has been stimulated to reduce reservoir pressure drop
to a minimum, is by pressure maintenance or secondary recovery. This will eventually be
initiated in most oil reservoirs, but methods are available to reduce the flowing wellbore pressure
by artificial means, that is, to modify the outflow performance of the well (Beggs, 2003).
All the methods presented earlier for generating IPRs, apply equally well to either flowing or
artificial lift wells. The reservoir inflow performance depends on Pwf and is completely
independent of what methods are employed to obtain a particular value of Pwf. Therefore, no new
procedures are required for reservoir performance in analyzing artificial lift wells (Beggs, 2003).
3.3.1 Methods of Artificial Lift
There are two basic forms of artificial lift: downhole pumping and gas lift. Downhole pumping
is accomplished by operating a pump at the bottom of the well. Gas lift is accomplished by
injecting gas into the lower part of the production tubing. Downhole pumps boost the transfer of
liquid from the bottom hole to the wellhead, eliminating backpressure, caused by the fluid
Page 36
26
flowing in the tubing. Injection of gas into the production string aerates the flowing fluid,
reducing the pressure gradient and lowering backpressure at the formation (Golan and Whitson,
1995).
For both lift methods, the production rate is increased by reducing wellbore flowing pressure. In
principle, both methods achieve the same result as lowering wellhead pressure or increasing
tubing size in naturally flowing wells, but, because artificial lift consumes significant amounts of
generated energy, it is introduced only after all adjustments in natural flow systems are
exhausted (Golan and Whitson, 1995).
The commonly used downhole pumping methods include the following (Boyun et al, 2007):
Sucker rod pumping or Beam pumping
Electrical submersible pumping
Hydraulic piston pumping
Hydraulic jet pumping
Plunger lift
Progressive cavity pumping
The commonly used gas lift methods also include the following:
Continuous gas lift
Intermittent gas lift
For deep water conditions, it may be more convenient to install the artificial lift device outside
the production well at some point of the seabed. Those methods are called Boosting Methods.
The most common ones are:
Subsea multiphase pumping
Riser gas lift
Subsea separation and pumping
Page 37
27
3.3.2 Larger tubings with Gas Lift
2 7/8” tubing used for the start of production had to be aided with gas lift in order to produce at
the lower pressures, thus, below 1100psi (equilibrium Pwf). For the preferable amount of gas to
be used for the artificial lift, this tubing was investigated at varying GLRs ranging from 1000,
1500, 2000, 2500 scf/stb. This was analyzed and plotted in the Figure below.
Figure 3.6 Performance of Larger tubing at different GLRs
From Figure 3.4 above, it is observed that the 2 7/8” tubing produced the reservoir to a Pwf of
1100 psi and production flow rate of 550 bpd. Thus, the preferable GLR to be used should be
able to produce below the above-mentioned Pwf and flow rate. GLRs of 2000 and 2500 virtually
give the same flow capacities up to 320 bpd and pressures up to 500 psi. This may be as a result
of gas saturation reaching its critical point in the reservoir. In addition, higher frictional loss is
observed for these two GLRs at higher flow rates, making them undesirable for use. GLR of
1500 exhibits reasonable frictional loss with an equilibrium Pwf up to 600 psi and flow rate up to
600
1000
20001500
2500
Page 38
28
300 bpd. GLR of 1000 shows frictional loss same as 1500 GLR but can only produce the
reservoir up to a Pwf of 900 psi and flow rate up to 250 bpd. This analysis makes 1500 GLR most
preferable for gas lifting the reservoir. This is illustrated in the Figure below.
Figure 3.7 Performance of 2 7/8” tubing with the natural flow and gas lift
Producing at lower pressures below Pwf of 600 psi with a 2 7/8” tubing will not be profitable
even with gas lift as shown in Figure 3.5 above. At the latter stage in the life of the reservoir,
velocity strings (smaller sized tubings) are considered.
3.3.3 Use of Velocity String (smaller tubings) with Gas Lift
Smaller sized tubing of ½”, 1”and 1 ½” tubings were analyzed for equilibrium Pwf and flow rates
below 600 psi and 300 bpd respectively. 1 ½” tubing was observed to be the most preferred for
this threshold. This tubing was then gas lifted with varying GLRs as used for the 2 7/8” tubing as
shown in Figure 3.6.
By natural flow, 1 ½” tubing could produce the reservoir from an equilibrium flow rate of 320 to
160 bpd with a Pwf up to 1300 psi. This equilibrium Pwf is higher than what the 2 7/8” tubing
600
1500
Page 39
29
produced up to, thus the need for gas lifting so as to produce below the threshold of 600 psi. As
illustrated in Figure 3.6 below, with the increment of gas in the reservoirs with the various
GLRs, not a single one of the GLRs showed the ability to produce the reservoir below 600 psi
(equilibrium Pwf).
Figure 3.8 Performance of 1 ½” tubing at different GLRs
This shows that, introduction of the velocity strings would not be economically viable since the 1
½” tubing does not perform any better than the 2 7/8” tubing. Therefore, the reservoir is
produced using the 2 7/8” tubing with GLRs of 600 and 1500 scf/stb from an average reservoir
pressure of 3500 psi to Pwf of 600 psi. At equilibrium Pwf and flow rate of 600 psi and 300 bpd,
pumping may be the option to consider.
2500
2000
600
1000
1500
Page 40
30
CHAPTER 4: IPRs FOR NATURALLY FRACTURED RESERVOIR
4.1 NATURALLY FRACTURED RESERVIORS
A Naturally fractured reservoir can be defined as a reservoir that contains fractures (Planar
discontinuities) created by natural processes like diastrophism and volume shrinkage, distributed
as a consistent connected network throughout the reservoir (Anon, 2010). A naturally fractured
reservoir is composed of a heterogeneous system of vugs, fractures and matrix which are
randomly distributed. Such type of system is modeled by assuming that the reservoir is formed
by discrete matrix block elements separated by an orthogonal system of continuous and uniform
fractures which are oriented parallel to the principal axes of permeability (Tiab, Restrepo and
Igbokoyi, 2006).
Naturally fractured reservoirs are anisotropic systems with flow characteristics that depend on
the fracture network. Their permeability variation is not stratigraphic in nature, but also depends
on the fractures’ distribution, orientation and permeability impairment caused by pressure
solution within the fractures. The reservoir becomes more complex when both the matrix and
fracture exhibit anisotropy and have the capability to flow into the wellbore as in double-
permeability case (Igbokoyi and Tiab, 2010).
There are many naturally fractured reservoirs in the world (Yang, Zhang & Gu, 2001). It is
undeniable that more than 60% of the world's proven reserves lie in naturally fractured reservoirs
(Anon, 2010). However, it is difficult to characterize naturally fractured reservoirs and predict
their hydrocarbon production (Yang, Zhang & Gu, 2001). Thus, estimating reserves and
predicting production in naturally fractured reservoirs is a difficult task. Production may be
dependent on fractures, either assisted by them or inhibited by their presence. Due to the high
degree of reservoir heterogeneity, geologists and engineers face several challenges in the
appraisal and management of naturally fractured reservoirs (Anon, 2010). Thus, the initial high
oil rates seen in these reservoirs have misled petroleum engineers, in many instances, to
overestimate their future production performance (Yang, Zhang & Gu, 2001).
Natural fractures play an important role in the technical and economic performance of all
hydrocarbon producing reservoirs. Analysis reveals that production may induce up to 7-percent
permeability reduction which can affect the production rate (Anon, 2010). Thus, accounting for
Page 41
31
reservoir compaction facilitates future production forecast and/or stimulation planning. For a
vertical well, horizontal fractures orientation show higher degree of compaction and permeability
damage than vertically oriented. The presence of extensive networks of natural fractures creates
a number of challenges for evaluating and optimizing recovery from naturally fractured
reservoirs. The use of dual porosity or dual permeability approaches is often necessary,
providing the basis for both analytical models (such as used for pressure transient analysis) as
well as for reservoir simulation (Anon, 2010). Appropriate application of dual porosity and dual
permeability models, however, rely on: a) accurate representation of the fracture system as an
equivalent porous and permeable medium, and b) accurate determination of the rates of fluid
transport between matrix blocks and the fracture system (Anon, 2010).
According to Nelson (2001), naturally fractured reservoirs can be divided into four categories;
based on the extent the fractures have altered the reservoir matrix porosity and permeability
(Igbokoyi and Tiab, 2008):
In Type 1 reservoirs, fractures provide the essential reservoir storage capacity (porosity)
and permeability.
In Type 2 naturally fractured reservoirs, fractures provide the essential permeability, and
the matrix provides the essential porosity.
In Type 3 naturally fractured reservoirs, the matrix has an already good primary
permeability. The fractures assist the permeability in an already producible reservoir and
can result in considerably high flow rates.
In Type 4 naturally fractured reservoirs, the fractures are filled with minerals and provide
no additional porosity or permeability. These types of fractures create significant
reservoir anisotropy (barriers) due to mineral filled.
In deep naturally fractured reservoirs, fractures and the stress axis on the formation generally
are vertically oriented. Thus when the pressure drops due to reservoir depletion, the fracture
permeability reduces at a lower rate than one would expect. In Type-2 naturally fractured
reservoirs, where matrix porosity is much greater than fracture porosity, as the reservoir
pressure drops, the matrix porosity decreases in favor of fracture porosity. This is not the
case in Type-1 naturally fractured reservoirs, particularly if the matrix porosity is very low or
negligible (Tiab, Restrepo and Igbokoyi, 2006).
Page 42
32
In naturally fractured reservoirs, the matrix pore volume, therefore the matrix porosity is reduced
as a result of large reservoir pressure drop due to oil production. This large reservoir pressure
drop causes the fracture pore volume, therefore fracture porosity, to increase. This behaviour is
observed particularly in reservoir where matrix porosity is much greater than fracture porosity.
The behavior of naturally fractured reservoirs (NFR) at pseudo-steady state is similar to that of a
homogenous reservoir. However, because of the double porosity nature of NFR, the transient
behaviour is quite different (Igbokoyi and Tiab, 2006).
4.2 IPRS FOR A NFR VERTICAL WELL
Hagoort (2008) developed a simple analytical formula for the stabilized productivity index (PI)
of an arbitrary well in an arbitrary enclosed naturally fractured reservoir that can be modeled as a
double-porosity reservoir. The formula relates the PI of a double-porosity reservoir (NFR) to the
PI of a well in a single-porosity reservoir with permeability equal to the effective fracture
permeability of the double-porosity reservoir, as shown in equation 4.1.
sp
em
sp
wfm
scdp
Jhrk
BJ
PPqJ
2
211
(4.1)
where, = storativity ratio
fmP = average pressure in the dual-porosity reservoir
wP = the well pressure
= shape factor given by: =60/L2 (Hagoort, 2008)
km= matrix permeability, md
The data used for the IPR development in chapter 3 were used with the NFR data below to
develop the IPRs for a vertical well in a naturally fractured reservoir. The storativity ratios of the
Page 43
33
NFR were varied so as to determine its effect on the PI when acting as a double-porosity
reservoir as compared to a single-porosity reservoir.
Available NFR Data
re=3000ft
Gas Specific Gravity = 0.7
L=200 ft
=0.001, 0.01, 0.1, 1
h=100 ft
Jsp=1.5 STB/D/psi
km=0.01 md
µ=0.3620 cp
B=1.8235 RB/STB
Temperature = 170oF
Assumptions made:
A closed boundary reservoir was assumed, thus, a solution gas drive reservoir.
All other conditions used for the single-porosity reservoir in chapter 3 were used here.
was calculated to be 0.0015. Using the equation 4.1, Jdp were calculated at different storativity
ratios. Using Beggs and Brill program developed in chapter 3, the bubble point pressure was
estimated to be 3401 psi which is the same as that of the single-porosity reservoir. With the
methodology for developing IPRs outlined in chapter 3, the IPRs for NFR were developed using
Fetkovich and Eickmeier correlations for the current and future IPRs. Plots of IPR curves
developed for the single-porosity reservoir and that for the NFRs are illustrated in the Figures
below.
Page 44
34
Figure 4.1 Double-Porosity reservoir (=0.001)
Page 45
35
Figure 4.2 Double-Porosity Reservoir (=0.01)
Page 46
36
Figure 4.3 Double-Porosity Reservoir (=0.1)
Page 47
37
Figure 4.4 Double-Porosity Reservoir (=0.5)
Page 48
38
Figure 4.5 Single-Porosity Reservoir (Vertical Well)
Discussion
Comparing Figure 4.5 to all the other four NFR Figures, it is observed that there is no significant
difference in the plots. This could be due to the fact that the late time behavior of naturally
fractured reservoirs and homogeneous system are similar. During the early time flow regime of a
naturally fractured reservoir, the flow is from the fractures only. This flow regime corresponds to
the first radial flow in the log-log and Horner’s plots (Figures 4.6 and 4.7). As the flow
continues, the pressure differential between the fracture face and the matrix becomes significant
to cause the matrix flow into the fracture. This is characterized by the transition and the usual
pseudo steady state nature of the matrix flow. This second flow regime corresponds to the entire
trough period in Figure 4.6 and the inflexion region in Figure 4.7. As the flow between the
matrix and fractures stabilizes, the pressure response will be due to the entire system. During this
flow regime, the behavior of a naturally fractured reservoir is similar to that of homogeneous
Page 49
39
system. Therefore, the equation used for computing the productivity index for single porosity is
applicable to double-porosity system.
Figure 4.6 ∆P and pressure derivative plot for an NFR
Figure 4.7 Semilog pressure behavior of an NFR
Page 50
40
Table 4.1: Summary of PI Calculation (Vertical Well)
Type of Reservoir Productivity Index Computed (J) Double-Porosity (=0.001) 1.499965065 Double-Porosity (=0.01) 1.499965692 Double-Porosity(=0.1) 1.499971646 Double-Porosity(=0.5) 1.499991249 Single-Porosity (=1) 1.5
From the above table, it is observed that there is no significant difference in the productivity indexes computed at the different storativity ratios.
4.3 IPRS FOR A NFR HORIZONTAL WELL
For developing the IPRs, the following equation can be used to calculate the pseudo-steady state
productivity index for a horizontal well for single-phase flow (Joshi, 1991).
DqcsssArrIn
BkhPP
qJ
CAhmiw
e
oo
wfRh
'''/007078.0
(4.2)
where,
ftAre ,/' (4.3)
and
sm = mechanical skin factor, dimensionless
si = skin factor of an infinite-conductivity, fully penetrating fracture of length, L
si = -In[L/(4rw)]
sCAh = shape-related skin factor
c’ = shape factor conversion constant = 1.386
Page 51
41
Furthermore, A’ = 0.750, for a circular drainage area.
With the available data below, the productivity index for a horizontal well was estimated.
Area =649 acres Bo = 1.8235 RB/STB rw = 0.365ft sm = 0
h = 100ft zw = 25 ft kv/kh = 0.1 D = 0
µo = 0.3680 cp kh=kx=ky = 1 md kf = km*1000 L = 4000 ft
Using equation 4.2, the PI of a horizontal well in a single porosity reservoir was then calculated
to be 10.87 STB/D/psi. The IPR was then developed as shown in the Figure below.
Figure 4.8 Single-Porosity Reservoir (Horizontal Well)
Using the same conditions in transforming this Single-Porosity reservoir into an NFR by Hagoort
model (equation 4.1); it was observed that the same effect experienced for the vertical well
occurred.
Page 52
42
Table 4.2: Summary of PI Calculation (Horizontal Well)
Type of Reservoir Productivity Index Computed (J)
Double-Porosity (=0.001) 10.868166
Double-Porosity (=0.01) 10.868199
Double-Porosity(=0.1) 10.868511
Double-Porosity(=0.5) 10.86954
Single-Porosity (=1) 10.87
Nevertheless, comparing the PI of the single porosity vertical well to that of the horizontal well
indicated that, a horizontal well gives a higher productivity than vertical well. In this case, about
ten times (1.5 and 10.87) that of the vertical well. As the thickness of the reservoir increases, this
difference in productivity will also decrease.
Page 53
43
CHAPTER 5: ESTIMATION OF PRODUCTIVITY INDEX OF
HORIZONTAL WELL IN RESERVOIR WITH PARTIAL PRESSURE
SUPPORT
5.1 INTRODUCTION
Application of horizontal well is presently unlimited. Its initial successful application as a
producer has led to other varieties such as injection, multi-lateral, hydraulic fracturing, heavy oil
recovery and many others. The understanding of horizontal well behavior in oil and gas
exploitation is a key to its successful application. The key important aspect of any well, either
producer or injector is its productivity index. This is a function of reservoir rock and fluid
properties coupled with the completion methodology. Proper evaluation is therefore necessary at
the planning, drilling and completion, and production stages. To this end, many authors have
developed methods of evaluation the productivity index in steady and pseudo-steady state flow
regimes. In most cases, the reservoir behavior is neither pseudo-steady nor steady state. This is
the case with a well producing from a reservoir with partial pressure support. It is therefore
necessary to develop a method of estimating productivity index in a reservoir with partial
pressure support.
Babu and Odeh’s (1989) presented a rigorous method of estimating productivity index of a
horizontal in a closed system. This method used the Green’s function of a plane source in the
three coordinates x-y-z to solve for the pseudo-steady state problem. Mutalik et al’s (1988)
method for a closed system is similar to a radial solution but involved horizontal well length
relative to the dimension of the drainage area with a skin factor related to the infinite
conductivity behavior of a horizontal well. Economides et al (1994) solution obtains
dimensionless pressure for a point source of unit length in a no-flow boundary rectangular box.
There are many solutions for the steady states behavior. They are Giger’s (1983), Borisov’s
(1984), Giger, Reiss and Jourdan (1984), Joshi’s (1988), and Shedid et al (1996).
Even though these solutions may not be applicable to the partial pressure support case, they have
been used to evaluate horizontal well performance in a reservoir with such pressure support. In
Page 54
44
all practical purposes, they produce usable results. In this work, we developed models which can
be used to evaluate a behavior that is neither steady nor pseudo-steady state.
5.2 MATHEMATICAL FORMULATION
5.2.1 Method 1
Case 1
This Method uses the Duhamel’s theorem provided by Carslaw and Jaeger (1959). The solution
for the temperature distribution in a rectangular parallelepiped whose surface is maintained at a
temperature φ(t) is given by the following equation.
t
nmlnml
l m n
nmlnml
tc
znb
yma
xl
nmlt)z,y,T(x,
0,,,,
0 0 0
,,3
expexp2
12cos2
12cos2
12cos
121212164
(5.1)
The domain of this solution is -a<x<a, -b<y<b and –c<z<c. The initial condition is zero
temperature within the domain. The diffusivity equation for a horizontal well in anisotropic
system can be written as:
tPc
zPk
yPk
xPk t
zyx
006329.02
2
2
2
2
2 (5.2)
This equation is transformed with the following parameters
xkxx 1' ,
ykyy 1' and
zkzz 1' (5.3)
to give:
Page 55
45
tPc
zP
yP
xP t
006329.0''' 2
2
2
2
2
2 (5.4)
Equation 5.4 is the analogue equation to the heat equation with the solution in equation 5.1. The
expression for αl,m,n will be:
2
2
2
2
2
22
,,121212
4006329.0
cn
bm
al
ctnml
(5.5)
Therefore the solution for point source can be written as:
t
nmlnml
l m n
nmlnml
bd
tc
znb
yma
xl
nmlPt),z,y,P(x
0,,,,
0 0 0
,,3
expexp2
'12cos2
'12cos2
'12cos
121212164'''
(5.6)
Pbd is the pressure at the boundary. For practical purposes, Pbd can be replaced by the static
reservoir pressure at any time. For the horizontal line source, we obtained:
t
nmlnml
ww
l m n
nmlnml
Hstaticwf
t
czn
bym
bym
axl
nmlLbPt),z,y,(xP
0,,,,
''1
'2
'
0 0 02
,,4
expexp
212cos
212sin
212sin
212cos
121212164'''
(5.7)
Equation 5.7 represents the pressure due to the line source well. aH, bH and h are the dimensions
of the rectangular box in x, y and z directions with 2a = aH, 2b = bH and 2c = h. y1 and y2 are the
positions of the line source. These dimensions must be transformed before used in the equations
above and have been defined in Appendix B.
Equation 5.7 can further be simplified as:
cH
staticwf LbPtzyxP
656.0,,, ''' (5.8)
where,
Page 56
46
t
nmlnml
ww
l m n
nmlnml
C
t
czn
bym
bym
axl
nml
0,,,,
''1
'2
'
0 0 02
,,
expexp
212cos
212sin
212sin
212cos
1212121
(5.9)
Taking into consideration pressure drop due to skin damage at the wellbore,
sLk
BqPh
skin2.141
(5.10)
where,
zyh kkk (5.11)
The total pressure drop in the well becomes:
skinwf PPP (5.12)
The Productivity Index equation then becomes:
skinwf PPqJ
(5.13)
Field Application
The following data are obtained from actual field test data.
Compressibility 0.00002 psi-1
Porosity 0.3
Oil viscosity 0.422 cp
Formation volume factor 1.38 rb/stb
L 1300 ft
Page 57
47
Formation thickness 50 ft
Reservoir length aH 2050 ft
Reservoir width bH 3096 ft
Well coordinate xw 900 ft
Well coordinate zw 6 ft
Well coordinate y1 and y2 -1350 ft and -50 ft
kx = ky 2500 md
kz 60 md
Oil rate at time t =301, 407, 803 days 4746, 5731 and 5634 stb/d
Initial reservoir pressure 2854 psi
Static reservoir pressure @ t = 407 and 803 days 2800 and 2755 psi
Bottom-hole flowing pressure @ t = 407 and 803 days 2701 and 2653 psi
Note that aH, bH, h, xw, zw, y1 and y2, and L must be transformed by the equations in Appendix B
before using them in equations 8 to 10. Function φ(t) was obtained from the static data to be
2854+0.000024t2-0.142329t. This is obtained by curve fitting the static pressure data with time.
Figure 5.1 is the downhole pressure gauge data recorded during the production history of the
well. The pressure data recorded for the last 200 days of production show severe near wellbore
damage. The productivity index was modeled by varying the mechanical damage skin S to
account for the progressive damage. Figure 5.2 represents the schematic diagram of the reservoir
as used above. There are no-flow boundaries in the North, East and West of the reservoir. The
results obtained for case 1 are tabulated in Table 5.1 below.
Page 58
48
2000
2200
2400
2600
2800
3000
01/09/2002 28/06/2003 23/04/2004 17/02/2005 14/12/2005 10/10/2006
Pre
ssu
re -
psi
a
DHG FBP DHG Static
All pressure data@ datum
Figure 5.1 Downhole Pressure Gauge data from the field example
Figure 5.2 Model for case 1
0
y
x
z
3096 ft
2050 ft
50 ft
Date
Page 59
49
Table 5.1: Results of the field application using the first model
Time (Days)
Actual PI (stb/d/psi)
Computed PI
(stb/d/psi) Skin used
Actual static reservoir pressure
(psi)
Computed pressure drop
∆Pwf (psi)
Computed pressure drop
due to skin
∆Pskin (psi)
301 63 62.8 82 2813 12.0 63.6
407 58.4 58.3 92 2800 12.2 86.1
803 55.2 55.2 98 2755 11.9 88.3
Case 2
Due to the structural position of the oil water contact, water movement could be from South and
West flanges within the faults, coupled with bottom water movement. The reservoir can then be
modeled as no-flow boundary at faces x = 0, y = 0 and z = 0 as shown in Figure 5.3. A similar
solution exists in section 3.5, equation 3, on page 104 of Carslaw and Jaeger (1959). By using a
product solution method, equation 5.7 can be obtained in the domain 0 < x < a, 0 < y < b and 0 <
z < c.
Figure 5.3 Model for Case 2
x
3096 ft
50 ft
2050 ft
y
z
0
Page 60
50
In this case a = aH, b = bH and c = h. These dimensions should be transformed before using in
calculating the PI. With this domain, the pressure drop at the well becomes:
cH
staticwf LbPtzyxP
312.1,,, ''' (5.14)
The Productivity Index remains as expressed in equation 5.13 and skinP as in equation 5.10.
With this new domain, the well position becomes:
Well coordinate xw 100 ft
Well coordinate zw 19 ft
Well coordinate y1 and y2 1600 ft and 2900 ft
The results obtained for case 2 are tabulated in Table 5.2 below.
Table 5.2: Results of the field application using the second model
Time (Days)
Actual PI (stb/d/psi)
Computed PI
(stb/d/psi) Skin used
Actual static reservoir pressure
(psi)
Computed pressure drop
∆Pwf (psi)
Computed pressure drop
due to skin
∆Pskin (psi)
301 63 63.0 90 2813 5.6 69.8
407 58.4 58.7 98 2800 5.9 91.7
803 55.2 55.6 104 2755 5.6 95.7
5.2.2 Method 2
The Green’s functions and source functions for the instantaneous plane source solution are
obtained from the solutions provided by Carslaw and Jaeger (1959), and Gringarten and Ramey
(1973). The general expression for the pressure drop at the well ∆Pwf at any arbitrary point (x, y,
z) may be obtained by integrations of the appropriate point sink functions. The well, or line sink,
is parallel to the y-axis, and is located along the line x = xw, yw1 ≤ y ≤ yw2, z = zw. Three cases
were considered depending on the boundary conditions.
Page 61
51
Case 3
Case 3 considers two mixed boundary conditions with a no- flow boundary at the East and West
flange and no-flow boundary at the top with water movement from the bottom. The North and
South flanks are mixed boundaries. The solution for these boundary conditions is:
dydtSSSabhLc
qBtzyxPPiPt y
yzyx
t
0
2
1
615.5),,,(
(5.15)
Here, Sx, Sy, and Sz are the instantaneous point sink functions (Green’s Functions) located at (x,
y, z) and satisfying the zero flux boundary conditions at x = 0, a; y = 0, and z = 0. These Green’s
Functions for this case can be expressed as:
12
22
12
22
12
22
12cos12cos412exp2
12cos12cos412
exp2
coscosexp211
n
wzz
m
wyy
l
wxx
hzn
hzn
htn
hS
bym
bym
btm
bS
axl
axl
atl
aS
(5.16)
with,
2
2
2
2
2
22
,,12124
4006329.0
hkn
bkm
akl
czyx
tnml
(5.17)
2
2
2
22
,1212
4006329.0
hkn
bkm
czy
tnm
(5.17a)
and L = (y2-y1) = Length of well, ft.
Carrying out integrations, we get the expressions for the pressure drop at any point (x, y, z)
inside the reservoir at time t.
For the horizontal line source at the well, the pressure drop can be expressed as:
Page 62
52
2
12
1 1
2
,,1
12
2
1 1 ,
12cos12sin12sin
12
12
coscos11615.58
12sin12sin
12cos12
12
cos11615.54),,,(
,,
,
hzn
bym
bym
mb
ym
axle
kahLc
qB
bym
bym
hzn
mb
ym
ek
ahLcqBtzyxP
www
l m
w
wt
nmln
y
t
ww
w
m n
w
t
nm
y
twf
nml
nm
(5.18)
Equation 5.18 can then be simplified as:
SCt
ywf ahLc
kqBtzyxP 5.0
3.14),,,(
(5.19)
where,
2
12
1 1
2
,,1
12
2
1 1 ,
12cos12sin12sin
12
12
coscos11
12sin12sin
12cos12
12
cos11
,,
,
hzn
bym
bym
mb
ym
axle
bym
bym
hzn
mb
ym
e
www
l m
w
wt
nmlnC
ww
w
m n
w
t
nmS
nml
nm
(5.20)
The Productivity Index remains as expressed in equation 5.13 and skinP as in equation 5.10.
The results obtained for case 3 are tabulated in Table 5.3 below.
Page 63
53
Table 5.3: Results of the field application using the third model
Time (Days)
Actual PI (stb/d/psi)
Computed PI
(stb/d/psi) Skin used
Actual static reservoir pressure
(psi)
Computed pressure drop
∆Pwf (psi)
Computed pressure drop
due to skin
∆Pskin (psi)
301 63 63.2 66 2813 23.9 51.2
407 58.4 58.4 74 2800 28.9 69.3
803 55.2 55.2 80 2755 28.4 73.6
Case 4
This case considers mixed boundary conditions with no-flow boundary at the East and North
flanks and the top of the reservoir. Bottom water movement is also considered from the top.
Equation 5.15 was used for this case with the Green’s functions expressed as follows:
12
22
12
22
12
22
12cos12cos412exp2
12cos12cos412
exp2
12cos12cos412exp2
n
wzz
m
wyy
l
wxx
hzn
hzn
htn
hS
bym
bym
btm
bS
axl
axl
atl
aS
(5.21)
with,
2
2
2
2
2
22 1212124
006329.0h
knb
kma
klc
zyx
t (5.22)
Carrying out integrations, we get the expressions for the pressure drop at any point (x, y, z)
inside the reservoir at time t. For the horizontal line source at the well, the pressure drop can be
expressed as:
Page 64
54
2
12
1 1
2
,,1
12cos12sin12sin12
12
cos
12cos118615.5),,,( ,,
hzn
bym
bym
mb
ym
axle
kahLc
qBtzyxP
www
w
l m
wt
nmln
y
twf
nml
(5.23)
Equation 5.23 can then be simplified as:
Ct
ywf ahLc
kqBtzyxP
3.14),,,( (5.24)
where,
2
12
1 1
2
,,1
12cos12sin12sin
12
12
cos12cos11 ,,
hzn
bym
bym
mb
ym
axle
www
l m
w
wt
nmlnC
nml
(5.25)
The Productivity Index remains as expressed in equation 5.13 and skinP as in equation 5.10. The
results obtained using case 4 model is tabulated in Table 5.4 below.
Table 5.4: Results of the field application using the fourth model
Time (Days)
Actual PI (stb/d/psi)
Computed PI
(stb/d/psi) Skin used
Actual static reservoir pressure
(psi)
Computed pressure drop
∆Pwf (psi)
Computed pressure drop
due to skin
∆Pskin (psi)
301 63 63.0 82 2813 11.7 63.6
407 58.4 58.2 90 2800 14.2 84.2
803 55.2 55.1 96 2755 13.9 88.3
Page 65
55
Case 5
This case considers mixed boundary conditions with no-flow boundary at the East and West
flanks, and the top and bottom of the reservoir. The North flank is also considered no-flow
boundary with water influx from the South only. The Green’s functions for these conditions are:
12
22
12
22
12
22
coscosexp211
12cos12cos412
exp2
coscosexp211
n
wzz
m
wyy
l
wxx
hzn
hzn
htn
hS
bym
bym
btm
bS
axl
axl
atl
aS
(5.26)
with,
2
2
2
2
2
22
,,4124
4006329.0
hkn
bkm
akl
czyx
tnml
(5.27)
2
2
2
22
,412
4006329.0
hkn
bkm
czy
tnm
(5.27a)
2
2
2
22
,
1244
006329.0b
kma
klc
yx
tml
(5.27b)
2
22 124
006329.0b
kmc
y
tm
(5.27c)
Carrying out integrations, we get the expressions for the pressure drop at any point (x, y, z)
inside the reservoir at time t.
For the horizontal line source at the well, the pressure drop can be expressed as:
Page 66
56
212
1 1
2
,,1
212
1 1 ,
12
1 1
2
,
12
1
cos12sin12sin
12coscos112
1615.58
cos12sin12sin
12cos112
1615.54
12sin12sin
12coscos112
1615.54
12sin12sin
12cos112
1615.52),,,(
,,
,
,
hzn
bym
bym
bym
axle
mk
ahLcqB
hzn
bym
bym
byme
mk
ahLcqB
bym
bym
bym
axle
mk
ahLcqB
bym
bym
byme
mk
ahLcqBtzyxP
www
l m
wwt
nmln
y
t
www
m n
wt
nm
y
t
ww
l m
wwt
ml
y
t
ww
m
wt
m
y
twf
nml
nm
ml
m
(5.28)
Equation 5.28 can then be simplified as:
CSLMt
ywf ahLc
kqBtzyxP 5.05.025.0
3.14),,,(
(5.29)
where,
Page 67
57
212
1 1
2
,,1
212
1 1 ,
12
1 1
2
,
12
1
cos12sin12sin
12coscos112
1
cos12sin12sin
12cos112
1
12sin12sin
12coscos112
1
12sin12sin
12cos112
1
,,
,
,
hzn
bym
bym
bym
axle
m
hzn
bym
bym
byme
m
bym
bym
bym
axle
m
bym
bym
byme
m
www
l m
wwt
nmlnC
www
m n
wt
nmS
ww
l m
wwt
mlL
ww
m
wt
mM
nml
nm
ml
m
(5.30)
In this case, it is possible to get expression for the average pressure drop because not all the term in equation 5.28 will vanish when average over the entire volume. Therefore,
abh
PdxdydztzyxP
a b h
0 0 0),,,( (5.31)
The solution of equation 5.31 becomes:
bym
bym
byme
mk
ahLcqBtzyxP
ww
m
wt
m
y
t
m
12
1
12sin
12sin
12cos112
1615.52),,,(
(5.32)
Equation 5.32 can then be simplified as:
Page 68
58
1
57.3),,,( M
t
y
ahLckqB
tzyxP
(5.33)
where,
bym
bym
byme
m
ww
m
wt
mM
m
12
11
12sin12sin
12cos1
121
(5.34)
The total pressure drop in the well becomes:
skinwf PPPP (5.35)
The Productivity Index equation then becomes:
skinwf PPPqJ
(5.36)
The skinP remains as in equation 5.10
The results obtained for case 5 are tabulated in Table 5.5 below.
Table 5.5: Results of the field application using the fifth model
Time (Days)
Actual PI (stb/d/psi)
Computed PI
(stb/d/psi) Skin used
Actual static reservoir pressure
(psi)
Computed pressure drop
∆Pwf (psi)
Computed pressure drop
due to skin
∆Pskin (psi)
301 63 63.1 76 2813 16.4 58.9
407 58.4 58.3 84 2800 19.7 78.7
803 55.2 55.1 90 2755 19.4 82.8
Page 69
59
The table below shows the summary of all pressure drops computed for the various models.
Table 5.6: Summary of the pressure drops and skins computed for the five cases
Time (Days)
Actual PI (Stb/D/Psi)
Case 1 Case 2 Case 3 Skin Used
∆Pwf (psi)
∆Pskin (psi)
Skin Used
∆Pwf (psi)
∆Pskin (psi)
Skin Used
∆Pwf (psi)
∆Pskin (psi)
301 63 82 12.0 63.6 90 5.6 69.8 66 23.9 51.2 407 58.4 92 12.2 86.1 98 5.9 91.7 74 28.9 69.3 803 55.2 98 11.9 88.3 104 5.6 95.7 80 28.4 73.6
Time (Days)
Actual PI (Stb/D/Psi)
Case 4 Case 5 Skin Used
∆Pwf (psi)
∆Pskin (psi)
Skin Used
∆Pwf (psi)
∆Pskin (psi)
301 63 82 11.7 63.6 76 16.4 58.9 407 58.4 90 14.2 84.2 84 19.7 78.7 803 55.2 96 13.9 88.3 90 19.4 82.8
Babu and Odeh Joshi Steady State Skin Used ∆Pwf (psi) ∆Pskin (psi) Skin Used ∆Pwf (psi) ∆Pskin (psi)
82.5 11.4 63.9 61 28.1 47.3 90.1 13.8 84.3 69 33.6 64.6 96.2 13.5 88.5 75 33.1 69.0
Discussion of the results
The initial test in this well gave a damage skin of 62 (Company’s data). Based on this, case 3 is
considered the best model for monitoring the well performance. Results obtained from the Joshi
steady state also shows that, the reservoir flow regime can be modeled as steady state. This is
because the skin factor obtained is similar to the actual. The results of case 1, case 4, case 5 and
Babu and Odeh are similar. However, the skin factor used in the modeling after one year of
production is higher than the one obtained at the initial test. Case 5 is similar to Babu and Odeh
since the reservoir model is almost a closed boundary system like Babu and Odeh’s. Even
though case 2 is similar to case 4 in that both have the same boundary conditions, the pressure
drop at the well are not similar. This could be as a result of the difference in the mathematical
model. Cases 3, 4 and 5 converge faster and more stable than cases 1 and 2. Case 3 is
Page 70
60
recommended as the best model for this reservoir. The reservoir boundary conditions are thus
completely described as in case 3. Even though the Joshi’s steady state method gave similar
result as case 3, Joshi’s steady state method cannot be used to completely describe the reservoir
flow boundary conditions. On the other hand, the Joshi steady state method of estimating the
horizontal well productivity index is based on equivalent wellbore radius. If the pressure drop
due to skin damage is estimated based on formation thickness and average horizontal
permeability sqrt(kxky), then the matched skin factor will be about 15 which is much lower than
the actual skin estimated at the initial test. Therefore, only case 3 accurately model the actual
productivity index.
Analysis of the mathematical models
Cases 1 and 2 do not involve rate in computing the pressure drop at the wellbore. The variation
in the rate impulse has been captured in the rate of reservoir pressure decline. This eliminates the
complication in the rate variation.
Cases 3, 4 and 5 are developed for constant rate as in Babu and Odeh, and Joshi methods.
Practically speaking, no well is produced at constant rate throughout its life cycle. A pragmatic
way of handling the rate variation is to use average rate as the cumulative divided by the time of
production. The principle of superposition can also be used to handle the rate variation, in which
case equation 5.15 becomes:
n
t y
ynznynx
N
nnn
t
dydtStStSqqabhLc
BtzyxPPiP
0111
11
2
1
615.5),,,(
(5.37)
Further analysis needs to be done to simplify equation 5.18 and 5.23 as in Babu and Odeh’s case.
Equation 5.37 needs to be tested with practical examples and investigate the impact of constant
rate assumption on the estimation of productivity index. With the presence of bottom hole
flowing pressure from downhole pressure gauge, the pressure derivative of equation 5.37 can be
used to determine the type of external flow boundary conditions. Taking the pressure derivative
of the original instantaneous source equation for horizontal well will give:
Page 71
61
ySSStqtP
zyx
y
y
2
1
(5.38)
and,
ySSSttqtPt zyx
y
y
2
1
(5.39)
Equations 5.38 and 5.39 can then be used to model the actual bottom hole flowing pressure data
from downhole gauge to arrive at the external flowing boundary conditions.
Summary
The results obtained has proved that the productivity index in a well depend on the current
situation existing in the reservoir. Each reservoir should be modeled based on the perceived
boundary conditions rather using steady or pseudo-steady state to approximate the well
performance. The best practical approach to obtain an accurate performance of well and reservoir
is to install downhole pressure gauge in all the wells. Based on the behavior of the reservoir
deduced from the downhole pressure gauge coupled with the structural map, an analytical model
can be developed to monitor the well performance. The process used in the field example
provides a method of identifying the reservoir flow boundary conditions, most especially the
direction of the water influx or pressure support.
Page 72
62
CHAPTER 6: CONCLUSIONS AND RECOMMENDATIONS
6.1 CONCLUSIONS
1. There is always a specific quantity of gas to be injected for a specific tubing size during
gas lifting. Velocity strings are associated with high frictional losses which impede oil
flow at lower pressures. As such, pumping was a better option for producing at such
reservoir pressures.
2. From the naturally fractured IPRs developed for both vertical and horizontal wells, it was
observed that, the IPRs for the single-porosity and the NFR showed no significant
difference. This could be due to the fact that the late time behavior of naturally fractured
reservoirs and homogeneous system are similar.
3. The productivity index in a well depends on the current situation in existing in the
reservoir. Each reservoir should be modeled based on the perceived boundary conditions
rather than using steady or pseudo-steady state to approximate the well performance.
Thus, the horizontal well performance should include the features exhibits by the
structural map.
4. The method of using the instantaneous plane source function as a Green function to
evaluate the horizontal well productivity provides an excellent approach in modeling the
reservoir external flow boundary conditions.
6.2 RECOMMENDATIONS
Economic evaluation could be considered for further research work to be done for a real
life problem. Thus, the costs of tubings, injection gas, and pumps can be evaluated
critically to come out with the optimal production design for the reservoir.
Field data for a naturally fractured reservoir should be used to compare this work in order
to investigate if the IPRs that will be developed would be different from the ones
developed for this work.
The pressure and productivity index computed from field data should be used to model
the flow boundary conditions which can be used in reservoir simulation.
Page 73
63
Future research should be aimed at quantifying the contribution of the various “Sigma”
terms of the productivity indexes in Chapter 5, and developing practical correlations to
determine them.
Page 74
64
NOMENCLATURE
Symbol Description Units
Q flow rate stb/d
B formation volume factor rb/stb
P pressure psi
Pwf bottom hole flowing pressure psi
PI productivity index stb/psi/d
L horizontal well length ft
bH reservoir width ft
aH reservoir length ft
rw radius of the well ft
re radius of the reservoir ft
t time days
h formation thickness ft
k permeability md
φ surface temperature (time dependent)
ct compressibility factor psi-1
Porosity [fraction]
Π pi
Viscosity cp
Subscripts
x = x-direction
y = y-direction
z = z-direction
w = wellbore
sp = single-porosity
dp = double-porosity
m = matrix
f = fracture
Page 75
65
REFERENCES
1. Ambastha, A. K., and Aziz, K.: “Material Balance Calculations for Solution Gas Drive
Reservoirs with Gravity Sagregation,” paper SPE 16959 presented at the 62nd Annual
Technical Conference and Exhibition of the Society of Petroleum Engineers held in
Dallas. TX September 27-30, 1987.
2. Allan, C. G., and Henry, J. R.: “The Use of Source and Green’s Functions in Solving
Unsteady-Flow Problems in Reservoirs,” paper SPE 3818. Copyright 1973 American
Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. October, 1973, SPEJ.
3. Babu, D. K., and Odeh, A. S.: “Productivity of a Horizontal Well,” paper SPE 18334
presented at the 63rd Annual Technical Conference and Exhibition of the Society of
Petroleum Engineers held in Houston, Texas, October 2-5, 1988.
4. Beggs, D.: “Production Optimization Using nodal Analysis,” 2nd Edition, OGCI and
Petroskills Publications, Tulsa, Oklahoma, pp. 27, 150-153. 2003.
5. Borisov, Ju. P.: “Oil Production Using Horizontal and Multiple Deviation Wells,” Nedra,
Moscow, 1964. Translated by J. Strauss, S. D. Joshi (ed.), Phillips Petroleum Co., the R
& D Library Translation, Bartlesville, Oklahoma, 1984.
6. Boyun, G., Lyons, W. C., and Ghalambor, A.: “Petroleum Production Engineering,”
Elsevier Science and Technology Books, pp 17, 164. 2007.
7. Carslaw, H. S., and Jaegar, J. C.: “Conduction of Heat in Solids,” Oxford at the
Clarendon Press, 1959, 2nd Edition, pp. 185.
8. Consentino, L.: “Integrated Reservoir Studies,” Technip Editions, Paris, pp. 182-187.
2001.
Page 76
66
9. Craft, B. C., and Hawkins, M., (Revised by Terry, R. E.), “Applied Petroleum Reservoir
Engineering,” 2nd Edition Englewood Cliffs, NJ: Prentice Hall, 1991.
10. Dake, L. P.: “Fundamentals of Reservoir Engineering,” Elsevier, Amsterdam, The
Netherlands, pp. 78-79. 1978.
11. Dake, L. P.: “The Practice of Reservoir Engineering,” Revised Edition, Elsevier, pp. 78-
79. 1994.
12. Economides, M. J., Brand, C. W., and Frick, T. P.: “Well Configurations in Anisotropic
Reservoirs,” paper SPE 27980 presented at the 1994 University of Tulsa Centennial
Petroleum Engineering Symposium held in Tulsa, August 29-31, 1994.
13. Evinger, H. H., and Muskat, M.: “Calculation of Theoretical Productivity Factors,”
Trans., AIME (1942) 126-146.
14. Fetkovich, M. J.: “The Isochronal testing of Oil Wells,” paper SPE 4529 presented at the
SPE Annual Meeting, Las Vegas, Sept. 30-Oct 3.
15. Giger, F.: “Reduction du nombre de puits par l’utilisation de forages horizontaux,” Revue
de l’Institut Francais du Petrole, vol. 38, No. 3, May-June 1983.
16. Giger, F. M., Reiss, L. H., and Jourdan, A. P.: “The reservoir Engineering Aspect of
Horizontal Drilling,” paper SPE 13024 presented at the SPE 59th Annual Technical
Conference and Exhibition, Houston, Texas, Sept. 16-19, 1984.
17. Golan, M., and Whitson, C. H.: “Well Performance,” 2nd Edition, Prentice-Hall Inc., pp.
29. 1995.
Page 77
67
18. Hagoort, J.: “Stabilized Well Productivity in Double-Porosity Reservoirs,” paper SPE
110984-PA-P peer approved 23 February 2008.
19. Havlena, D., and Odeh, A, S.: “The Material Balance as an Equation of a Straight Line,”
JPT, August 1963, pp. 896-900.
20. Igbokoyi, A. O., and Afulukwe, C. R.: “Encouraging Experience in the Use of Permanent
Downhole Gauges for Well and Reservoir Management,” paper SPE 128344 presented at
the 33rd Annual SPE International Technical Conference and Exhibition in Abuja,
Nigeria, August 3-5, 2009.
21. Igbokoyi, A. O., and Tiab, D.: “New Method of Well Test Analysis in Naturally
Fractured Reservoirs Based on Elliptical Flow,” paper 2008-101 accepted for the
Proceedings of the Canadian International Petroleum Conference/SPE Gas Technology
Symposium 2008 Joint Conference (the Petroleum Society’s 59th Annual Technical
Meeting), Calgary, Alberta, Canada, 17-19 June 2008.
22. Igbokoyi, A. O., and Tiab, D.: “Estimation of Average Reservoir Pressure and Drainage
Area in Naturally Fractured Reservoirs – Tiab’s Direct Synthesis,” paper SPE 104060
presented at the First International Oil Conference and Exhibition in Mexico held in
Cancun, Mexico, 31 August-2 September 2006.
23. Igbokoyi, A. O., and Tiab, D.: “New Method of Well Test Analysis for Naturally
Fractured Reservoirs Based on Elliptical Flow,” JCPT, June 2010, Volume 49, No. 6.
24. Jahanbani, A., and Shadizadeh, S. R.: “Determination of Inflow Performance
Relationship (IPR) by Well Testing,” paper 2009-086 accepted for the proceedings of the
Canadian International Petroleum Conference (CIPC) 2009, Calgary, Alberta, Canada,
16-18 June, 2009.
Page 78
68
25. John, L., John, D. R., and John, P. S.: “Pressure Transient Testing,” SPE Textbook
Series, vol. 9, pp. 223-244, 2003.
26. Joshi, S. D.: “Augmentation of Well Productivity Using Slant and Horizontal Wells,”
Journal of Petroleum Technology, pp. 729-739, June 1988.
27. Joshi, S. D.: “A Review of Horizontal Well and Drain hole Technology,” paper SPE
16868, presented at the 1987 Annual Technical Conference, Dallas, Texas. A revised
version was presented at the SPE Rocky Mountain Regional Meeting, Casper, Wyoming,
May 1988.
28. Joshi, S. D: “Horizontal Well Technology,” Joshi Technologies International, Inc. Tulsa,
OK, USA, Penn Well Books, pp. 84-85, 221-227. 1991.
29. Klins, M. A., and Majher, m. W.: “Inflow Performance Relationships for Damaged or
Improved Wells Producing Under Solution Gas Drive,” (Paper SPE 19852) JPT, Dec.
1992, p. 1357-1363.
30. Kumar, R., Pooladi-Darvish, M., and Okawaza, T.: “An Investigation into Enhanced
Recovery under Solution Gas Drive in Heavy Oil Reservoirs,” paper SPE 59336
presented at the 2000 SPE/DOE Improved Oil Recovery Symposium held in Tulsa,
Oklahoma, 3-5 April 2000.
31. Lea, J. F., Nickens, H. V., and Mike, R.: “Gas Well Deliquification,” 2nd Edition, Gulf
Professional Publishing, 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA,
588 pp., 2008.
32. Mutalik, P. N., Godbole, S. P., and Joshi, S. D.: “Effect of Drainage Area Shapes on the
Productivity of Horizontal Wells,” paper SPE 18301 presented at the 63rd Annual
Technical Conference and Exhibition of the Society of Petroleum Engineers held in
Houston, Texas, October 2-5, 1988.
Page 79
69
33. Nelson, R.: “Geologic Analysis of Naturally Fractured Reservoir”. Gulf Professional
Publishing, 2nd Edition, 2001.
34. Prado, M.: “Two Phase Flow and Nodal Analysis & Production Engineering I,” Lecture
material delivered at the African University of Science and Technology, 2009.
35. Shedid, A. E., Samuel, O. O., and Tiab, D.: “A Simple Productivity Equation for
Horizontal Wells Based on Drainage Area Concept,” paper SPE 35713 presented at the
Western Regional Meeting held in Anchorage, Alaska, May 22-24, 1996.
36. Tarek, A.: “Reservoir Engineering Handbook,” 2nd Edition, Gulf Professional Publishing,
pp. 484-780, 2001.
37. Tiab, D., Restrepo, D. P., and Igbokoyi, A.: “Fracture Porosity of Naturally Fractured
Reservoirs,” paper SPE 104056 presented at the First International Oil Conference and
Exhibition in Mexico held in Cancun ,Mexico, 31August-2 September 2006.
38. Vogel, J. V.: “Inflow Performance Relationships for Solution Gas Drive Wells,” JPT,
January 1968 (SPE 1476).
39. Wiggins, M. L., “Generalized Inflow Performance Relationship for Three-Phase flow”
paper SPE 25458, Production Operations Symposium, Oklahoma City, OK, March 21-23
(1993).
40. Yang, D., Zhang, Q., and Gu, Y.: “Determination of Production Operation Methods in
Naturally Fractured Reservoirs,” paper 2001-012 JCPT peer reviewed paper.
Page 80
70
APPENDIX A - EQUATIONS
x
H
kaa 12
(B.1)
y
H
kbb 12
(B.2)
zkhc 12
(B.3)
xHH k
aa 1
(B.4)
yHH k
bb 1
(B.5)
yxh kkk (B.6)
v
h
kk
(B.7)
Babu and Odeh’s Equation for PI
dw
HpH
w
zxH
wf sLbsInC
rAInB
kkbPP
qJ75.0
00708.02/1
(B.8)
Page 81
71
Joshi’s Steady State Equation for PI
w
oohh
rhIn
Lh
L
LaaIn
BhkJ
22
2
/007078.0
2
22
(B.9)
Transformed Joshi’s Steady State Equation for PIskin
SkL
BJ
J
h
skin 2.14111
(B.10)
where,
zy kkk (B.11)