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.

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

Copyright by DORCAS KARIKARI 2010

All Rights Reserved

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:-

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

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

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

v

Figure 5.2 Model for case 1 ....................................................................................................... 48

Figure 5.3 Model for Case 2 ...................................................................................................... 49

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

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.

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

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.

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

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

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

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

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.

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

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

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:

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

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

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

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)

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

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

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

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.

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)

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)

21

Figure 3.3 Behavoir of Reservoir Fluid Properties

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

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”

24

Figure 3.5 Performance of 2 7/8” tubing

GLR of 600 scf/stb

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

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

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

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

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

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

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).

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

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.

34

Figure 4.1 Double-Porosity reservoir (=0.001)

35

Figure 4.2 Double-Porosity Reservoir (=0.01)

36

Figure 4.3 Double-Porosity Reservoir (=0.1)

37

Figure 4.4 Double-Porosity Reservoir (=0.5)

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

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

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

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.

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.

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

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:

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,

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

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.

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

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

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.

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:

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.

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:

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

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:

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,

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:

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

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

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:

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.

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.

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.

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

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.

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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,

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& D Library Translation, Bartlesville, Oklahoma, 1984.

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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.

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2001.

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-

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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)

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)