Gas Well Production Optimization Using Dynamic Nodal Analysis

i

THE UNIVERSITY OF TULSA

THE GRADUATE SCHOOL

GAS WELL PRODUCTION OPTIMIZATION USING DYNAMIC NODAL

ANALYSIS

BY

ARSENE BITSINDOU

A THESIS

APPROVED FOR THE DISCIPLINE OF

PETROLEUM ENGINEERING

By Thesis Committee

, Chairperson

ii

ABSTRACT

Bitsindou Arsene (Master of Science in Petroleum Engineering)

Gas well Production Optimization using Dynamic nodal Analysis.

Directed by Dr. Mohan Kelkar

(130 words)

This work presents a numerical algorithm that permits the production optimization

of gas wells using the concept of dynamic nodal analysis. By combining the desirable

features of nodal analysis, material balance technique and decline curve analysis, the

method is able to match the historical performance of the well data. It is also able to predict

the future performance of the gas well under the existing condition as well as altered

conditions. The proposed technique, which has several advantages over the classical nodal

analysis, can be used for the selection of the timing and capacity of surface compressor, the

evaluation of the economic viability of a well stimulation, and the understanding of the

effect of individual production component on the productivity of a gas well over the life of

that well.

iii

ACKNOWLEDGEMENT

I would like to take this opportunity to thank Dr. Mohan Kelkar for his invaluable

guidance and support during the course of my Master’s study. I also express my gratitude to

Dr Leslie G. Thompson of the University of Tulsa, and Stuart Cox of Marathon Co. for their

comments and suggestions and for serving on my dissertation committee.

I am grateful to Marathon Co. for providing the field data used during the test of the

computer program.

I would like to express my appreciation to all the other faculty members who

contributed to my education as a TU graduate student. I would also like to thank my

graduate student colleagues who made my life easier at TU, especially Harun Ates with who

I shared the office during the preparation of this thesis.

This dissertation is dedicated to my family whose support and encouragement will

always be appreciated.

iv

TABLE OF CONTENTS

TITLE PAGE .............................................................................................................................i

ABSTRACT .............................................................................................................................ii

ACKNOWLEDGEMENTS ................................................................................................... iii

TABLE OF CONTENTS ........................................................................................................iv

LIST OF TABLES ............................................................................................................... viii

LIST OF FIGURES ..................................................................................................................x

CHAPTER I INTRODUCTION...............................................................................1

CHAPTER II PROCEDURE ...................................................................................11

2.1 Mathematical Modeling ................................................................................11

2.1.1 History Match ....................................................................................11

2.1.2 Future Performance Prediction..........................................................15

2.2 Regression Analysis .......................................................................................16

2.2.1 Parameter constraints.........................................................................21

2.2.1.1 Imaging Extension Method.............................................22

2.3 Nodal Analysis Technique .............................................................................23

2.4 Summary ........................................................................................................28

v

CHAPTER III IMPLEMENTATION .......................................................................29

3.1 Computer Program.........................................................................................29

3.2 Models and Correlations ................................................................................30

3.1.1 Reservoir ...........................................................................................30

3.1.2 Perforations........................................................................................33

3.1.3 Gravel Pack........................................................................................37

3.1.4 Tubing String ....................................................................................41

3.1.5 Subsurface Device (Subsurface Restriction).....................................43

3.1.6 Subsurface safety valve .....................................................................44

3.1.7 Well Head Choke...............................................................................45

3.1.8 Surface Pipeline .................................................................................45

3.1.9 Fluid Properties..................................................................................45

3.3 Sensitivity Studies With Respect to Input Parameters .................................47

3.3.1 Sensitivity With Respect to Pressures Decrement ...........................47

3.3.2 Sensitivity With Respect to Tolerance .............................................47

3.3.3 Sensitivity with Respect to Input Parameters in Order to

Get the Match .............................................................................48

CHAPTER IV RESULTS/VALIDATION................................................................49

vi

4.1 Synthetic Data ...............................................................................................49

4.1.1 History Match ....................................................................................53

4.1.2 Sensitivity Analysis ...........................................................................56

4.1.2.1 Sensitivity of History Match Results With

Respect to Pressure Decrements

Values .......................................................................56

4.1.2.2 Sensitivity of History Match Results With

Respect to Tolerance Values ...................................59

4.1.2.3 Verification of the Robustness With

Respect to Errors.......................................................62

4.1.3 Future Performance Simulations.......................................................72

4.1.3.1 Future Performance Simulations for

Different Well Head Pressure Values ......................72

4.1.3.2 Future Performance Simulations for

Different Skin Values ...............................................75

4.2 Field Data .......................................................................................................78

4.2.1 Case #1: Dry Gas Well Producing at a Constant Well

Head Pressure ..............................................................................78

4.2.1.1 History Match..................................................................81

4.2.1.2 Future Performance Predictions .....................................88

vii

4.2.1.2.1 Future Performance Prediction

Using Different Well head

Pressure Values ............................................. 88


Using Different Skin Values ..................... 91

4.2.1.2.3 Future Performance Prediction for

Different Density of

Perforation .................................................... 94

4.2.2 Case #2: Conversion of Original Data from Constant

Flow Rate to Constant Well Head Pressure................................97

4.2.2.1 History Match................................................................104

4.2.2.2 Future Performance Predictions ...................................108



Pressure Values ........................................... 108


Using Different Skin Values ................... 111


Using Different Perforated

Interval Values............................................ 114

viii

4.2.3. Case #3: Conversion of Original Data from Constant

Flow Rate to Constant Well Head Pressure..............................117

4.2.3.1 History Match................................................................124




Pressure Values ........................................... 127

4.2.3.2.2. Future Performance Prediction for

Different Perforation Density

Values ........................................................... 130


Using Different Perforated

Interval Values............................................ 133

4.2.4 Case #4: Use of the Last Two Years of Production Only...............136

4.2.4.1 History Match................................................................138


4.2.4.2.1 Reduction in Well Head Pressure ................. 143

4.2.4.2.2 Reduction in Tubing Size............................... 146

4.2.4.2.3 Choke Installation............................................ 149

CHAPTER V CONCLUSIONS .............................................................................152

ix

RECOMMENDATIONS......................................................................................................154

NOMENCLATURE .............................................................................................................155

REFERENCES......................................................................................................................158

ix

LIST OF TABLES

3.1 Gas Reservoir Inflow Performance Relationship ...............................................32

3.2 Correlations for Multiphase Flow in Pipes ..........................................................42

3.3 Correlations for Flow across Chokes and Restrictions .......................................43

3.4 Correlations for Multiphase Subcritical Flow in Subsurface

Safety Valves..................................................................................................44

3.5 Correlations for Fluid Physical Properties ...........................................................46

4.1.1 Synthetic Data: Input Parameters.........................................................................50

4.1.2 Production Synthetic Data....................................................................................52

4.1.3 History Match for Synthetic data .........................................................................53

4.1.4 History Match for Synthetic data #2 ....................................................................65

4.1.5. System description Data for Synthetic Data #2 ...................................................68

4.1.6. Well Performance and Reservoir Pressure Data for Synthetic

Data #2............................................................................................................70

4.2.1.1 System description Data for Case #1 ...................................................................78

4.2.1.2 Well Performance and Reservoir Pressure Data for Case #1 ..............................80

4.2.1.3 History Match for Case #1 ...................................................................................84

4.2.2.1 System Description Data for Case #2 ..................................................................97

4.2.2.2 Original Field Production Data for Case #2 ......................................................100

x

4.2.2.3 Converted Production Data for Case #2 ............................................................101

4.2.2.4 History Match for Case #2 .................................................................................104

4.2.3.1 System Description Data for Case #3 ................................................................117

4.2.3.2 Original Field Production Data for Case #3 ......................................................119

4.2.3.3 Converted Production Data for Case #3 ............................................................120


4.2.4.1 System Description Data for Case #4 ................................................................136

4.2.4.2 Production Data for Case #4 ..............................................................................138


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LIST OF FIGURES

1.1 System Description and Pressure Losses. ..............................................................8

2.1 Typical Inflow and Outflow Curves ....................................................................26

2.2 Example of an Unstable Production Condition (Liquid Loading) ......................27

3.1 Structure of the Computer Program.....................................................................34

3.2 Typical Perforated Hole .......................................................................................35

3.3 Perforated Hole Turned 90*.................................................................................36

3.4 Gravel Pack Schematic.........................................................................................38

3.5 Details of L ...........................................................................................................40

4.1.1 Synthetic Data: Production History Match ..........................................................54

4.1.2 Synthetic Data: Reservoir Pressure History Match .............................................55

4.1.3 Synthetic Data: Sensitivity of Production History Match with

Respect to Pressure Decrement......................................................................57

4.1.4 Synthetic Data: Sensitivity of Reservoir Pressure History Match

with Respect to Pressure Decrement .............................................................58


Respect to Tolerance ..................................................................................60


with Respect to Tolerance..........................................................................61

xii


Respect to Errors in the Rate Data.............................................................63


with Respect to Errors in the Rate Data.....................................................64

4.1.9 Synthetic Data #2: Sensitivity of Rate History Match with

Respect to Errors in the Rate Data.............................................................66

4.1.10 Synthetic Data #2: Sensitivity of Reservoir Pressure History

Match with Respect to Errors in the Rate Data .............................................67

4.1.11 Synthetic Data: Sensitivity of Rate with Respect to Well Head

Pressure...........................................................................................................73

4.1.12 Synthetic Data: Sensitivity of Reservoir Pressure with Respect to

Well Head Pressure........................................................................................74

4.1.13 Synthetic Data: Sensitivity of Rate with Respect to Skin Factor ........................76

4.1.14 Synthetic Data: Sensitivity of Reservoir Pressure with Respect to

Skin Factor .....................................................................................................77

4.2.1.1 Case #1: Production History Match .....................................................................82

4.2.1.2 Case #1: Reservoir Pressure History Match ........................................................83

4.2.1.A Case #1: Production History Match .....................................................................86

4.2.1.B Case #1: Reservoir Pressure History Match ........................................................87

4.2.1.3 Case #1: Sensitivity of Rate with Respect to Well Head Pressure......................89

xiii

4.2.1.4 Case #1: Sensitivity of Reservoir Pressure with Respect to Well

Head Pressure.................................................................................................90

4.2.1.5 Case #1: Sensitivity of Rate with Respect to Skin Factor ...................................92

4.2.1.6 Case #1: Sensitivity of Reservoir Pressure with Respect to Skin .......................93

4.2.1.7 Case #1: Sensitivity of Rate with Respect to Perforation Density ......................95

4.2.1.8 Case #1: Sensitivity of Reservoir Pressure with Respect to

Perforation Density ........................................................................................96

4.2.2.1 Case #2: Original Field Data..............................................................................102

4.2.2.2 Case #2: Converted Rate....................................................................................103

4.2.2.3 Case #2: Production History Match ...................................................................106

4.2.2.4 Case #2: Reservoir Pressure History Match ......................................................107

4.2.2.5 Case #2: Sensitivity of Rate with Respect to Well Head Pressure....................109


Head Pressure...............................................................................................110

4.2.2.7 Case #2: Sensitivity of Rate with Respect to Skin Factor .................................112

4.2.2.8 Case #2: Sensitivity of Reservoir Pressure with Respect to Skin .....................113

4.2.2.9 Case #2: Sensitivity of Rate with Respect to Perforated Interval......................115

4.2.210 Case #2: Sensitivity of Reservoir Pressure with Respect to

Perforated Interval........................................................................................116

4.2.3.1 Case #3: Original Field Data..............................................................................122

xiv

4.2.3.2 Case #3: Converted Rate....................................................................................123





Head Pressure ..............................................................................................129

4.2.3.7 Case #3: Sensitivity of Rate with Respect to Density of

Perforation ....................................................................................................131


Density of Perforation ............................................................................132

4.2.3.9 Case #3: Sensitivity of Rate with Respect to Perforated Interval......................134

4.2.310 Case #3: Sensitivity of Reservoir Pressure with Respect to

Perforated Interval........................................................................................135





Head Pressure ..............................................................................................145

4.2.4.5 Case #4: Sensitivity of Rate with Respect to Tubing Size ................................147

xv


Tubing Size .................................................................................................148

4.2.4.7 Case #4: Sensitivity of Rate with Respect to Choke Size .................................150

4.2.4.8 Case #4: Sensitivity of Reservoir Pressure with Respect to Choke

Size............................................................................................................................151

1

CHAPTER I

INTRODUCTION

The production optimization of a gas well requires an appropriate selection of the

individual components in the production system. Currently nodal analysis is used to

accomplish this task. Nodal analysis involves calculating the pressure drop in individual

components within the production system so that pressure value at a given node in the

production system (e.g., bottom hole pressure) can be calculated from both ends (separator

and reservoir) [See Figure 1.1]. The rate at which pressure is calculated at the node from

both ends must be the same. This is the rate at which the well produces. Once the rate under

existing conditions is obtained, by adjusting individual components, the sensitivity of

individual components on the overall production can be investigated; Hence an optimum

selection of components can be obtained at a given time. The major drawback of the

conventional nodal analysis is that it only provides the user with a snapshot picture of the

well production. It does not provide any information as to how the production will change

as a function of time. For example, if tubing size is changed, the nodal analysis may provide

the best tubing size at present time; however, it may not be able to indicate which tubing

size is the best over the life of the well based on the future production. Even generating

future inflow performance curves (which characterize how the reservoir will behave in the

2

future at discrete times) may not help since we will not be able to estimate how the rate has

changed over the time intervals.

To include the effect of time on the production performance, the most commonly

used technique is the decline curve analysis. Decline curve analysis involves matching the

prior production data using one of the decline types (exponential, hyperbolic or harmonic),

and using the estimated decline parameters, predicting the future performance under

existing conditions. Decline curve analysis is a very powerful tool, and has been used

extensively to predict the future performance by ignoring the effects of tubing size, choke,

surface pipeline or other components in the production system. In addition, although it is

true that decline curve analysis can predict the future performance under existing

conditions, it may not predict how the well will behave in future if the production

conditions are altered. These alterations include, for example, changing skin factor,

changing choke size, or changing the surface compressor.

Conventional material balance techniques which uses diagnostic plots have also

been proven to be useful in understanding the behavior of the gas wells. These plots, for

example, include P/Z (reservoir pressure over compressibility factor) versus gas production

to predict how much gas the well will eventually produce. These techniques can also

account for, through a trial and error procedure, the presence of water influx. The drawback

of the material balance technique is that it does not account for time. It can predict the

production as a function of reservoir pressure, but not as a function of time. Further, it also

only accounts for reservoir component, and not for any other component of the production

system. The effect of alterations on the gas well performance cannot be predicted using the

3

material balance technique. The inclusion of time in terms of predicting the future

performance is critical from economic point of view. This cannot be accomplished using

this technique.

To overcome the drawbacks presented in the above methods, we need a technique

which can:

♦ Predict the future performance as a function of time in the presence of various

production components including the reservoir.

♦ Match the prior production data in the presence of various production

components so that the appropriate parameters can be assigned for future

production prediction. This is similar to decline curve analysis except that we

need to include the production components in the system.

♦ Quantify the uncertainties with respect to various parameters ( e.g., reservoir

permeability, skin factor, tubing roughness, drainage area, the type of pressure

drop correlation) by generating alternate possibilities of parameters which can

match the production data.

♦ Predict the future performance under existing conditions as well as altered

conditions to compare the production scenarios in the future.

♦ Quantify the uncertainty in predicting the future performance which can be

combined with the price of gas to conduct a risk analysis.

4

♦ Optimize the producing well configuration so that the net profit over the life of

the well is maximized.

Some specific examples where the proposed technique can be applied are:

♦ Effect of Installing the Gas Compressor: As the well head pressure declines, there

may be a need to install a gas compressor at the well head. The compressor allows the

reduction of well head pressure, and hence increase in production. Various installation

alternatives that can be considered are the timing (when it will be installed), and what

capacity. Nodal analysis may indicate the possible rate of production at the existing

condition, but it does not indicate how the well will perform in the future. Installation

of the compressor will allow the operator to accelerate the production and increase the

reserves by lowering the abandonment pressure. However, for the cost benefit analysis,

we need to know how the gas production rate will change as a function of the

installation as well as the capacity of the compressor. Currently, no method is available

to evaluate the effect of compressor installation on the gas production as a function of

time.

♦ Fracturing or Stimulating a Gas Well: A service company will always compare the

production with and without stimulation to sell a particular stimulation procedure.

However, stimulation, typically, does not increase the reserves. It only accelerates the

production. Therefore, after stimulation, the gas well will decline faster then at the

current conditions. For proper economic evaluation, it is critical that we examine the

incremental gas production. – difference between production with stimulation minus

5

production without stimulation (which is positive at the beginning but will become

negative at later times) as a function of time.

♦ Changing the Production Components: The prediction of the gas well performance

in the future is critical under existing as well as modified conditions. For example, for a

condensate gas reservoir, we would like to know when the gas well will start loading

under existing conditions so that appropriate production components can be changed

before the actual loading occurs. These alterations include changing choke size,

changing the tubing size or reducing the well head pressure. Based on the production

scenarios under existing as well as altered conditions, a proper method can be selected

for continued gas production.

♦ History Matching of Prior Production Data: To instill confidence in the predictive

ability of any program, the user should be able to match the prior production from the

same gas well. Decline curve analysis essentially matches the prior production data by

using a specific model and then predicts the future performance based on prior data. In

reality, we know that significant uncertainties exist with respect to the input parameters

used for predicting the past performance. For example, the same prior production data

can be matched by either altering the permeability or skin factor, or by changing the

tubing correlation or the roughness factor. Changing the drainage area or thickness or

the porosity or saturation can all alter the possible reserves the well is capable of

producing. However, of these four components, the productivity of the well can only be

significantly affected by the thickness of the reservoir. If we want to quantify the

uncertainties in predicting the future performance, we need to develop alternate

6

scenarios – all matching the prior performance. Subsequently, these scenarios can be

used to predict the future performance of a gas well under existing as well as modified

conditions. This type of information is extremely useful in economic risk and

uncertainty analysis.

In our approach, we will assume that the operator has already conducted a decline curve

analysis using many of the commercial programs readily available. Therefore, the type of

decline (exponential, hyperbolic or harmonic) is already known. If the information is

unavailable, we can use the recommended values by Fetkovich et al26,27. For example,

Fetkovich et al. recommend exponential decline for high pressure gas wells (>5000 psia),

Hyperbolic decline with b value between 0.4 and 0.5 for typical gas wells, and a value

greater than 0.5 and less than 1.0 for multiple layered reservoirs.

The system considered in this work is shown in figure 1.1. It represents a single well

producing from a gas reservoir up to the separator. This system is divided into the following

completion and piping components:

§ reservoir

§ perforations

§ gravel pack

§ tubing

§ bottom hole device

§ subsurface safety valve (SSSV)

§ well head choke

7

§ surface pipeline

§ separator

8

Figure 1.1: System Description and Pressure Losses (after Brown et al.)1

9

Assumptions

The major assumptions made with respect to the flow of gas in the reservoir and the

piping system are:

§ The production system operates under pseudo-steady state conditions. The well

is flowing at a steady flow rate for a fixed average reservoir pressure and

separator pressure. This implies that the gas well produces with a fixed

liquid/gas ratio.

§ The drainage mechanism of the reservoir is assumed to be natural depletion

mechanism.

§ The production exhibits a certain type of decline during the period of time

considered in the history match computations. That decline can be exponential,

hyperbolic or harmonic according to the behavior of the reservoir under

consideration. This behavior is assessed by using the decline curve analysis

theory and the Fetkovich type curve.

§ For wet gas reservoir, it is assumed that the reservoir pressure is above the dew

point pressure. This assumption implies that the flow is single-phase gas in the

reservoir.

§ The well head pressure is reasonably constant throughout the period of time

considered for the history match.

10

§ It is assumed that the gas flows from the reservoir into the well only through a

tubing consisting of a constant inside pipe diameter. The pressure drop between

the tubing shoe and the producing interval is assumed to be negligible.

Other limitations involved in this work depend on the type of correlation selected to

compute the pressure losses across the individual component in the system. These

limitations are presented in Chapter III.

This thesis is divided into several chapters. After this introduction chapter, Chapter

II describes the algorithm for the dynamic nodal analysis technique and details the

mathematical models as well as the regression analysis used in this technique. Chapter III

discusses the implementation of this technique into a computer program and provides

sensitivity studies with respect to input parameters. Chapter IV presents the results of the

application of the computer program to several field cases and validates the dynamic nodal

analysis technique. Finally, in Chapter V, conclusions and recommendations are provided.

11

CHAPTER II

PROCEDURE

2.1 Mathematical Modeling

The mathematical scheme used to perform dynamic nodal analysis for gas reservoirs

can be summarized in two different parts: the history match and the forecast analysis.

2.1.1 History Match

The procedure used to compute the history match is summarized in the following

steps:

1. Assume that the production history is known. Thus, for each observed

production time Tobs1, Tobs2,…, Tobs j,…, Tobs n, the corresponding observed rate

Qobs1, Qobs2, …, Qobsj, …, Qobsn is known.

2. Assume that at time Tj the following data are known:

§ reservoir pressure Pj.

§ fluid properties as a function of pressure and temperature.

§ The type of decline (harmonic, hyperbolic or exponential) as well as the rate

of decline. If these are not known, assume exponential decline.

12

§ The pressure drop correlations as a function of rate for each Q.

3. The gas in place at this time Tj is computed as:

gj

gbj B

SVG

**φ=

(2.1)

where

scj

scRjgj TP

PTZB

*

**= . (2.2)

4. Calculate the rate Qj at which the well will produce under the existing conditions. This

is done by using the nodal analysis technique. As stated earlier, in this study the node is

chosen at the bottom hole. The nodal analysis technique is presented in section 2.3 of

this chapter.

5. Assume a small decrement in reservoir pressure ∆Pj. The new reservoir pressure is then

Pj+1 = Pj- ∆Pj . At this reservoir pressure , calculate the new gas in place Gj+1 :

1

**1

+

=+

jg

gbj B

SVG

φ. (2.3)

The total amount of gas produced when the reservoir pressure decreases from Pj to Pj+1

is:

1+−=∆ jj GGG (2.4)

6. Calculate the rate Qj+1 at which the well will produce under the present reservoir

pressure Pj+1. This is done by nodal analysis at bottom hole.

13

7. Knowing the total amount of gas produced (∆G) and the gas flow rate Qj and Qj+1 at

reservoir pressures Pj and Pj+1, we can calculate the elapsed time ∆T required to reach

that production.

• For exponential decline:

1

11

+

++

−

−=

∆

−=

jj

jjjj

GG

QQ

G

QQD (2.5)

1

ln1

+

=∆j

j

Q

Q

DT (2.6)

• For harmonic decline:

1

ln+∆

=j

jj

Q

Q

G

QD (2.7)

[ ]1

11

+

+−=∆

j

jj

Q

QQ

DT (2.8)

• For hyperbolic decline:

−

∆−=

−

+

b

j

jj

Q

Q

Gb

QD

1

11**)1(

(2.9)

+−=∆

−

+

b

j

j

Q

Q

DbT 11*

*1

(2.10)

The total calculated time when the reservoir pressure is Pj+1 can be calculated as:

TTT jj ∆+=+1 . (2.11)

14

8. Assume a new reservoir pressure Pj+1 :

Pj+1 = Pj-∆P where ∆P is the pressure decrement.

Repeat the process from step 4 to step 7 until the total calculated time Tj+k is greater or

equal to the observed production time.

9. At this point, we have the model predicted times

T1, T2, …, Tj, …Tj+k, …

and the corresponding rates:

Q1, Q2, …, Qj, …, Qj+k, …

For each observed time Tobs j, we calculate the corresponding model predicted rate Q’j

by interpolating the model predicted rates.

At this point, we check how the calculated flow rate Q’j compares with the historical

observed production rate Qobs j at the same time Tobs j. This check represents the history

match of the observed data.

If significant differences exist between the calculated and the observed production, then

some selected reservoir parameters have to be adjusted in order to match the historical

performance.

In order to match the historical observed performance, a non-linear regression

calculation is performed to minimize the difference between calculated and observed

production. This regression analysis is discussed in section 2.2 of this chapter.

15

Once a satisfactory match between the predicted and the observed performance is

obtained, we can proceed with forecast of future performance calculations.

2.1.2 Future Performance Prediction

1. The future performance of the well under the existing conditions as well as under

altered conditions can be calculated. The procedure is the same as described from step 2

to step 8 in the History Matching section. Repeat the steps till an abandonment rate is

reached.

2. Consider different scenarios for variations in production procedures. These include, for

example, changing the number of perforations, stimulating the well, fracturing the well,

installing the compressor at the surface.

3. Predict the future performance under the new operating conditions using the same

procedure as explained in step 1.

4. Repeat step 3 for alternate combinations of input parameters to quantify uncertainties in

the prediction of future performance.

5. Compare the performance under the new scenario with the base case to calculate the

incremental gas production as a function of time.

6. Repeat step 5 for different input configuration.

7. Use information generated in step 5 and step 6 to study the economic feasibility of

making the changes in the production configuration.

16

2.2 Regression Analysis

The basic objective of using the non-linear regression in this problem is to determine the

optimum set, α, of reservoir/completion parameters such that the observed data match as

closely as possible to the calculated data from the model.

In this study, the parameters on which the regression is performed consist of any set of 3

independent variables chosen among the following parameters: permeability, skin, radius of

drainage, pay, perforated interval, radius of perforations, diameter of perforations, porosity,

water saturation, and density of perforations. For example, one can choose α such that

α={permeability, skin, radius of drainage }. In this case the regression calculations will be

performed on the following variables: permeability, skin and radius of drainage.

In this study, the Levenberg-Marquardt algorithm LMDIF133, has been used. This algorithm

has been selected because it does not require to provide the derivatives of the functions to

minimize.

The purpose of LMDIF133 is to minimize the sum of the squares of m non-linear functions

in n variables. This is done by the more general least square solver LMDIF. The user must

provide the subroutines that compute the functions. The jacobian is then calculated by a

forward-difference approximation.

As stated earlier, in this work, the variables on which to regress are any set of 3 independent

variables chosen by the user among the following parameters: permeability, skin, radius of

drainage, pay, perforated interval, radius of perforations, diameter of perforations, porosity,

water saturation, and density of perforations.

17

So, n is equal to 3.

The m non-linear functions F1(α), F2(α), …, Fm (α) can be considered as the components

of a vector FVEC. The objective function is then computed as the square of the euclidian

norm of FVEC, that is:

Objective function = ΣFj2 .

The functions Fj are chosen such that the computation is more resistant to errors in the

observed data and is less sensitive to outliers. The definition of the functions Fj is presented

below.

Function F1

This function compares observed data with the predicted data. Ideally the correlation

coefficient between the observed and model predicted performance is equal to 1.

Mathematically,

1),()( mod1 −= QQF obsρα (2.12)

mod

*),(

),( modmod

QQ

obsobs

obs

QQCOVQQ

σσρ = (2.13)

)()1( 1 αFFVEC = . (2.14)

The advantage of using the correlation coefficient is that it is resistant to noise in the data. It

is not sensitive to outliers. It should be noted that high correlation coefficient does not

necessarily mean that the values are similar.

18

The basic assumption here is that we are modeling the measured data correctly that the

errors in the measured data are normally distributed with mean zero, and the errors are not

correlated.

Function F2

This function is chosen to represent the fact that ideally the plot of Qmod versus Qobs is a

straight line of slope one ( with intercept equal to zero).

1)()2( 2 −== SLOPEFFVEC α (2.15)

obs

obs QQCOVSLOPE 2

mod ),(σ

= (2.16)

where COV is the covariance between the observed and model predicted rates. So,

1),(

)2( 2mod −=

obs

obs QQCOVFVEC

σ. (2.17)

Function F3

This function is chosen to represent the fact that ideally the intercept of the straight line

Qmod versus Qobs is equal to zero.

0*mod =−= obsQSLOPEQINTERCEPT (2.18)

1)3( mod −=obsQ

QFVEC (2.19)

because ideally the slope is equal to 1: SLOPE=1.

)3(3 FVECF = . (2.20)

19

Function F4

This function compares observed reservoir pressure with the predicted reservoir pressure.

Ideally the correlation coefficient between the observed and model predicted reservoir

pressure is equal to 1. Mathematically,

1),()(mod,,1 −= robsr PPF ρα (2.21)

mod,,

*

),(),( mod,,

mod,,

robsr PP

robsrrobsr

PPCOVPP

σσρ = (2.22)

)()4( 1 αFFVEC = . (2.23)

The advantage of using the correlation coefficient is that it is resistant to noise in the data. It

is not sensitive to outliers. It should be noted that high correlation coefficient does not

necessarily mean that the values are similar.

The basic assumption here is that we are modeling the measured data correctly that the

errors in the measured data are normally distributed with mean zero, and the errors are not

correlated.

Function F5

This function is chosen to represent the fact that ideally the plot of Pmod versus Pobs is a

straight line of slope one ( with intercept equal to zero).

1)()5( 5 −== SLOPEFFVEC α (2.24)

20

obsP

robsr

r

PPCOVSLOPE 2

mod,, ),(

σ= (2.25)

where COV is the covariance between the observed and model predicted rates. So,

1),(

)5( 2mod,, −=

obsP

robsr

r

PPCOVFVEC

σ. (2.26)

Function F6

This function is chosen to represent the fact that ideally the intercept of the straight line

Pmod versus Pobs is equal to zero.

0* ,mod, =−=obsrr PSLOPEPINTERCEPT (2.27)

1)6(,

mod, −=obsr

r

PP

FVEC (2.28)

because ideally the slope is equal to 1: SLOPE=1.

)6(6 FVECF = . (2.29)

Also, the user specifies the tolerance FTOL which is used in the regression. The program

terminates when the algorithm estimates either that the relative errors in the sum of squares,

ΣFj2, is at most FTOL or that the relative error between in the regression variables is at most

FTOL. On termination, the regression algorithm output an integer variable INFO whose

value means the following.

INFO = 0: improper input parameters.

21

INFO = 1: algorithm estimate that the relative error in the sum of squares is at most

FTOL .

INFO = 2: algorithm estimates that the relative error between the calculated values of the

regression parameters and the ideal solution is at most FTOL.

INFO = 3: condition for info =1 and info = 2 both hold.

INFO = 4: FVEC is orthogonal to the columns of the jacobian to machine precision.

INFO = 5: number of calls to the function that compute FVEC has reached or exceed

200*(n+1).

INFO = 6: FTOL is too small. No further reduction in the sum of squares is possible.

INFO = 7: FTOL is too small. No further improvement in the approximate solution is

possible.

2.2.1 Parameter Constraints

The Levenberg –Marquardt algorithm33 that we use is “ unconstrained “ : i.e., variables can

be chosen to minimize the objective function with value between ± infinite. Obviously, for

our problem, we need to ensure that the values of the variables lie in the predefined interval

of uncertainty and that these values are meaningful. For example we may want the regressed

permeability value to be between Kmax and Kmin.

In order to keep the values of the regression variables in certain predefined intervals, we can

use a couple of methods. It has been shown that the use of the penalty function improves the

22

convergence of the iterative procedure; however, it is also reported that the penalty function

method may not prevent the values of the regression variables to be out of the predefined

domain when the initial estimates of the regression variables are far from the solution. In

this study, the imaging extension19 procedure is used.

2.2.1.1 Imaging Extension Method19

The idea behind the method is to extend the objective function in such a way that the new

objective function is defined everywhere (i.e., unconstrained) and that the solution of this

new unconstrained problem is related to the solution of the original constrained problem.

The procedure for translating the unconstrained variable estimate ξLMDIF1 calculated by the

regression algorithm LMDIF133 to the corresponding physically constrained value of the

parameter ξc is the following:

§ For ξLMDIF1 > ξmax , compute :

−

−=

minmax

min1intξξ

ξξLMDIFN . (2.30)

§ For ξLMDIF1 < ξmin , compute :

1intminmax

min1 −

−

−=

ξξξξLMDIFN . (2.31)

After calculating N, ξc can be obtained as:

§ For N odd:

23

1minmaxmaxmin )( LMDIFc N ξξξξξξ −−++= . (2.32)

§ For N even

)( minmax1 ξξξξ −−= NLMDIFc . (2.33)

For more details about the imaging extension method, the reader is referred to the

reference 19.

2.3 Nodal Analysis Technique

Nodal analysis provides a method to determine the rate at which a producing system will

perform under certain applied conditions. In order to evaluate that producing rate, the

production system is divided into two parts at a fixed node and the performance curves of

each part are compared. These two performance curves are denoted as inflow (flow into the

node) and outflow (flow out of the node) performance curves. For convenience, the node is

chosen at the bottom hole16. This choice does not affect the results of the performance

computations.

With the node at bottom hole, the inflow performance curve represents the pressure loss

across the reservoir, the perforations and the gravel pack. It can be mathematically

expressed in dimensionless form as:

maxQQ

versusPP

Ir

WF

(2.34)

where

∆−

∆−

−−=

−−

−−=

r

gp

r

perf

r

WFSr

r

WFWFS

r

WFSr

Ir

WF

P

P

P

P

PPP

PPP

PPP

PP

11 (2.35).

24

Qmax is the maximum flow rate at which the well can flow.

The outfow performance curve describes the pressure loss in the tubing, the bottom hole

restriction (subsurface device), the safety valve, the well head choke and the surface

pipeline. It can be mathematically expressed in dimensionless form as:

MAXOr

WF

QQ

versusP

P

(2.36)

where

∆+

∆+

∆+

∆+

∆=

r

PIPELINE

r

CHOKE

r

SV

r

REST

r

TBG

Or

WF

P

P

P

P

P

P

P

P

P

P

P

P. (2.37)

A typical plot of the inflow curve as well as the two commonly observed outflow curves is

shown in Figure 2.1.

The overall performance of the producing system is obtained when the inflow and outflow

curves intercept. This implies that the flow rate and the bottom hole flowing pressure are

obtained by solving the equation:

Or

WF

Ir

WF

PP

PP

=

(2.38)

This equation is solved numerically using the secant method16.

As can be seen on Figure 2.1 and Figure 2.2, this equation can have two different roots or

one single root.

25

If the equation has two different roots, the root corresponding to the lower flow rate

represents an unstable production condition while the root corresponding to the higher flow

rate represents a stable production condition. This situation is typical of system producing in

two-phase flow with high gas velocity.

If the equation has a single root, one of the following situations can happen:

• The derivative of the outflow curve at the root is positive. In this case the system

produces under a stable condition. This is typical of systems close to single-phase flow.

• The derivative of the outflow curve at the root is negative. In this case the system

produces under an unstable condition (liquid loading).

26

Figure 2.1. Typical Inflow and Outflow Curves16.

27

Figure 2.2 Example of an Unstable Production Condition (Liquid Loading)

C

Inflow curve

Outflow curve

Unstable rate

P

Pwf

r

QQmax

0

28

2.4 Summary

In this chapter, the dynamic nodal analysis procedure has been presented. The mathematical

models used in the history match and forecast algorithms have been detailed. In addition,

the regression analysis method used in the computer program has been discussed. Finally, a

brief description of the conventional nodal analysis technique has been reviewed.

29

CHAPTER III

IMPLEMENTATION

3.1 Computer Program

A computer program has been developed that implements the mathematical procedures

discussed in the previous section. After all the data describing all the components of the

system has been provided, the computer program can conduct the dynamic nodal analysis

calculations (history match, sensitivity analysis) as well as classic static nodal analysis. The

computer program described in this section is tested with synthetic as well as field data; the

results of these test are presented in Chapter IV. The general structure of the computer

program is presented in Figure 3.1. An important consideration in the computer program

development has been to provide a user-friendly environment and an algorithmic

architecture which is easy to maintain and expand. This is realized by providing a flexible

and interactive procedure to input, modify and view the data describing each component of

the system as well as allowing to save the results in the restart files which can be used for

future sensitivity analysis and forecast studies. The program is easy to maintain because of

its modularity which allows each specific problem to be handled by specific subroutines.

30

Figure 3.1 Structure of the computer program

Select option-Dynamic nodal analysis-New well-Conventional analysis

Dynamic nodal analysis

Select option-Input/Display data-Modify data-History match -Forecast-Conventional nodal analysis

Conventional nodal analysis

Select option-Input/Display data-Modify data-Conventional nodal analysis

New well

Select option-Input/Display data-Modify data-Forecast-Conventional nodal analysis

Results

Do you want to save results in a restart file?

End

Enter a restart file name

Start

No Yes

31

In addition, an error file is included which contains eventual error messages if the ranges or

the limitations of the correlations and model selected are surpassed.

3.2 Models and Correlations

In this section the models and correlations used in the computer program to compute the

pressure drops in each components of the system are presented. A special consideration is

given to the limitations involved in these models and correlations.

3.2.1 Reservoir

The flow in the reservoir is considered to be single phase gas. This assumes that the

reservoir pressure is above the dew point throughout the well production time in the case of

wet gas reservoirs. The pressure drops across the reservoir porous media are computed by

an inflow performance relationship (IPR) using Darcy’s law modified by Jones, Blount and

Glazes12 and expressed in terms of pseudo-real pressure. This equation which takes into

account the turbulent effect as well as the damage effect (skin), relates the reservoir pressure

to the sand-face pressure.

QbQaPmPm wfsR **)()( 2 +=− (3.1)

where Q is in MMscf/D. The coefficients a and b are defined as,

µ

γβ

**

***10*166.32

6

wp

g

RH

Ta

−

= (3.2)

32

+−= S

RR

HKT

bw

e

43

ln*

*10*424.1 6

(3.3)

where β is defined as,

201.1

1010*33.2K

=β . (3.4)

The pseudo-real pressure is defined as follows:

dPZ

PPm

P

P gbase

**

*2)( ∫=

µ. (3.5)

It should be noted that, in general, the IPR calculated with data obtained from well test

analysis usually gives a better description of the reservoir performance.

33

IPR Range of Requirements AdvantagesApplicability

Darcy's law single phase flow Properties describing the May be expressed in terms of modified by reservoir pseudo-real pressure. DamageJones et al. and high velocity effects are

included.

Table 3.1Gas reservoir inflow performance relationship used

34

3.2.2 Perforations

The computer program computes the pressure drop across the perforations using Mc-Leod’s

method11. This equation takes into account the pressure losses across the compacted zone. It

does not account for the converging effect of the flow near the well bore.

Several assumptions are made in this method such as:

1. The permeability of the crushed zone or compacted zone is:

§ 10 % of the formation permeability if the well is perforated under overbalanced

conditions.

§ 40 % of the formation permeability if the well is perforated under underbalanced

conditions.

2. The thickness of the crushed zone is ½ inch.

3. The small perforation hole is producing under steady state conditions.

Figure 3.2 and Figure 3.3 show a typical perforated hole.

The equation for the pressure losses across the perforations is:

QbQaPmPm wfwfs **)()( 2 +=− (3.6)

where Q = flow rate/perforation (Mscf/D). The coefficients a and b are defined as,

µ

γβ

*

11****10*16.3

2

12

Lp

RRT

a CP

−

=

−

(3.7)

35

PP

P

C

LKRR

Tb

*

ln**10*424.1 3

= (3.8)

where

201.1

1010*33.2

pK=β . (3.9)

36

Figure 3.2: Typical Perforated Hole (after Brown et al.)1

37

Figure 3.3: Perforated Hole Turned 90* (after Brown et al)1

38

3.2.3 Gravel Pack

The pressure drop across the gravel pack is computed using the Jones, Blount and Glazes

equation modified by Brown for single-phase gas. This simple model takes into account the

pressure losses from the perforation tunnel to the liner. It also accounts for the turbulent

flow regime (high velocity flow). In addition, Brown provides some guidelines about the

estimation of the gravel pack effective permeability as a function of the gravel size. Figure

3.4 displays the typical gravel pack schematic.

The equation is:

QbQaPmPm wfwfs **)()( 2 +=− (3.10)

where Q is in Mscf/D. The coefficients a and b are defined as,

µ

γβ

*

****10*247.12

10

A

LTa g

−

= (3.11)

39

Figure 3.4. Gravel Pack Schematic (after Brown et al)1

40

AK

LTb

G ***10*93.8 3

= (3.12)

where

55.0

710*47.1

GK=β . (3.13)

Figure 3.5 provides the details to calculate the linear flow path L.

41

Figure 3.5: Details of L (after Brown et al)1

42

3.2.4 Tubing String

The pressure drop across the tubing string is computed with commonly used multiphase

flow correlations in the literature. Table 3.2 summarizes the correlations2 used in the

computer program. Also shown, in that table, are the ranges of applicability of each

correlation.

It should be noted that for a given production system, the choice of the appropriate

correlation for tubing pressure drop computations is generally based on field experience and

on the correlation limitations. However, in the absence of any information, Brown1 gives

the following suggestions:

§ Poettman and Carpenter correlation and Beggs and Brill correlation for dry gas and

§ Gray’s correlation9 for wet gas.

In the computer program the temperature gradient across the tubing is assumed to be

constant.

43

Correlation Considerations Recommended rangesof slip conditions and flow regime

Hagedorn and Considers slip conditions All pipe sizes, all fluidsBrown and no flow regimeBeggs and Brill Considers slip conditions All pipe sizes, all fluids

and flow regime All angles of inclinationsGray Considers slip conditions Pipe size <=3.5 in.

and no flow regime Condensate <= 50 BBL/MMscfWater <=350 BBL/MMscf

Beggs and Brill Considers slip conditions All pipe sizes, all fluidsand flow regime All angles of inclinations

Dukler Considers slip conditions All pipe sizes, all fluids(with Eaton et al. and no flow regimeholdup correlation)

Beggs and Brill Considers slip conditions All pipe sizes, all fluidsand flow regime All angles of inclinations

Inclined flow

Table 3.2Correlations for multiphase flow in pipes

Vertical flow

Horizontal flow

44

3.2.5 Subsurface Device (Subsurface Restriction)

The pressure drop across the subsurface restriction is calculated using one of the

correlations listed in Table 3.3. The choice of the appropriate correlation for the subsurface

restriction depends on the type of gas phase flowing across the component.

Correlation Recommended range

Sachdeva Critical-subcritical flow boundary determined by model.Uses discharge coefficient equal to 0.85 or 0.75 in the presence of an upstream elbow

Adiabatique expansion Critical-subcritical flow boundary determined fromequation a specific heat ration

Table 3.3Correlations for flow across chokes and restrictions

Two-phase flow

Single phase gas flow

45

3.2.6 Subsurface Safety Valve

The pressure losses across the subsurface safety valve are computed with the correlations

listed in Table 3.4. These correlations take into account the subcritical two-phase flow

regime under which the subsurface safety valves are normally operated.

3.2.7 Well Head Choke

Correlation Discharge coefficient limitations

API 14B Discharge coefficient calculated by no-slipweighting average of specified liquid and gas

single phase discharge coefficients Tulsa university Empirical relations for discharge coefficient,Model No. 2 for Otis J valves (8/64 in. to 32/64 in.)

Table 3.4Correlations for multiphase subcritical flow

in subsurface safety valves

46

The pressure losses across the wellhead choke are computed by using one of the

correlations listed in Table 3.3 depending upon the type of flow regime (subcritical or

critical) and the type of phase ( dry gas or wet gas.)

3.2.8 Surface Pipeline

The pressure losses across the surface pipeline are computed using the multiphase

correlations in Table 3.2. As in the case of tubing string, the selection of pressure drop

correlation is usually based on field experience and the limitations of the correlations.

However, in absence of any field information, Brown and Lea recommend the use of Beggs

and Brill correlation or Dukler correlation for horizontal and inclined pipeline.

3.2.9 Fluid Properties

The correlations used in the computer program to estimate the physical properties of the

fluids are listed in Table 3.5. These experimental correlations are function of temperature,

pressure, type of fluid (gas, oil or water), densities of the different phase which are present

in the flow. It should be noted that fluid properties obtained from direct measurement on

fluid sample should be preferred if available. However, when used properly, fluid

correlations are generally good enough for well performance estimations.

47

Fluid property Correlation Validity considerations

Solution Gas-Oil Lasater Suggested for crude with API>=15ratio Standing Suggested for crude with API<=15

Vasquez and Two correlations for crudes with APIBeggs above and below 30

Formation volume Standing For black oils below bubble-pointfactor pressure

Vasquez and Beggs Two correlations for crudes with APIabove and below 30

Glaso Developed for North Sea oils.May be valid for other crudes after correction for CO2,N2 and H2S

Surface tension Baker and Swerdoff empirical data interpolationOil viscosity Beggs and Robinson Correlates dead and live oil viscosity

Vasquez and Beggs Correlation for viscosity abovebubble point pressure.

Glaso Developed for North Sea oils.May be valid for other crudes after correction for CO2,N2 and H2S

Compressibility Yarborough and Hall Fitting to Standing and Katz reduced pressure - reduced temperature curves

Viscosity Lee et al. Empirical correlation. Good for wide ranges of pressure and temperature (from 100 to 8000 psia, and from 100 to 340 F)

Formation volume Gould Correlates value of pure water and gas saturated factor water.Viscosity Beggs and Brill Expression to fit temperature effect on viscosity

Water

Table 3.5Correlations for fluid physical properties.

Two-phase flow

Gas

48

3.3 Sensitivity Studies With Respect to Input Parameters

3.3.1 Sensitivity With Respect to Pressure Decrement

The pressure decrement ∆P is used to calculate the reservoir pressures at which the

calculations are performed during the history match and the forecast computations.

Typically the value of ∆P can be chosen between 50 to 500 psia. This choice depends on

the magnitude of the reservoir pressure decrease during the period of time of the history

match and/or the forecast. In general, a pressure decrement of 50 psia can be used in most

cases.

It should be noted that the computations are longer when using small reservoir pressure

decrement values and this may provide an improvement of the results. However it has been

noticed that for some production systems, the use of smaller increments does not improve

the match. For example, the results obtained were the same for pressure decrement of 100

psia as for a decrement of 300 psia but the computation was more intensive for 100 psia

decrement. A numerical example of this problem is shown in section 4.1.2.1, chapter 4.

3.3.2 Sensitivity with Respect to Tolerance

The tolerance used to calculate the pressure drops across each component of system as well

as to converge the regression procedure, is very important. The results obtained as well as

the duration of the computations are directly affected by the tolerance. In general, a

tolerance value of 0.001 to 0.000001 will be required to get a good match . Although it is

advisable to choose the highest precision possible, it should be noted that for some

49

production system it is still possible to get results which are good enough for engineering

purposes at lower precision. Section 4.1.2.2 of Chapter 4 gives a numerical example of this

problem.

3.3.3 Sensitivity with Respect to Input Parameters in Order to Get the Match.

If the model predicted performance does not match with the observed data, it may be

advisable to modify certain reservoir parameters. The following guidelines can be used in

an attempt to improve the match.

If Qobs (T j) / Qmod (T j) is sensibly constant for all time Tj, the reservoir productivity needs

to be modified. This can be accomplished by changing either the permeability or the pay

range.

If the decline rate is predicted to be greater than the observed value, pore volume can be

increased. If the decline rate is smaller than the observed value, the pore volume can be

decreased.

If the changes do not show consistent behavior, for fine tuning purposes only, we can

change the pressure drop correlations, the relative roughness of the pipes, the fluid

properties and the correlations for individual components in the system.

50

CHAPTER IV

RESULTS/VALIDATION

In this chapter the dynamic nodal analysis described in Chapter II is applied to several

production systems. Those production systems include synthetic data as well as actual field

data. The results obtained from the computer program that are presented in this chapter

validate the dynamic nodal analysis technique.

4.1 Synthetic Data

Synthetic data represent an ideal production system which is used to verify the robustness of

the computer program. These synthetic data have been generated using the results from a

simulation of an actual field well. They represent a gas condensate well which was open to

production for five years. The characteristics of the reservoir as well as the description of

the completion are summarized in Table 4.1.1.

51

Table 4.1.1 Synthetic data: input parameters. Type of decline = exponential Pressure decrement [psia] = 50 Optimization tolerance = 0.000001 Reservoir Initial pressure [psia] = 5010.98 Initial temperature [F] = 212 Pay [ft] = 64.454 Skin = 116.454 Drainage radius [ft] = 9107.852 Permeability [md] = 11.057 Porosity [fraction] = 0.06 Water saturation [fraction] = 0.533 Fluid properties Specific gravity of produced gas = 0.646 Oil density [API] = 51.080 Specific gravity of produced water = 1.0 Completion Hole diameter [in] = 8.496 Casing diameter [in] = 5 Perforated interval [ft] = 17 Perforation diameter [in] = 0.36 Perforation tunnel length [in] = 12.33 Perforation density [SPF] = 4 Mode of perforation = overbalance Tubing inside diameter [in] = 1.945 Tubing roughness [ft] = 0.00015 Tubing length [ft] = 8688.0 Hole inclination angle [degree] = 90 Pressure drop correlation: Beggs and Brill Production Oil/Gas ratio, [SBBLO/MMscf] = 145.0 Water/Gas ratio, [SBBLW/MMscf] = 0.0 Well head pressure, [psia] = 2250.0 Well head temperature, [F] = 111.0 Reference separator pressure, [psia] = 14.7 Reference separator temperature, [deg F] = 60.0

52

Limits of regression parameters PERMIN [md] = 0.0 PERMAX [md] = 100.0 SMIN = -5.0 SMAX = 175.0 REMIN [ft] = 2500.0 REMAX [ft] = 10000.0

The production and the reservoir pressure as functions of time are presented in table 4.1.2.

Notice that reservoir pressure is not available at each time step.

53

Table 4.1.2

Production synthetic data

Time Rate Reservoir Time Rate Reservoir Pressure pressure [days] [Mscf/D] [psia] [days] [Mscf/D] [psia] 0 1987.263 5010.98 1064 1819.304 4811.30 31 1982.147 5004.82 1095 1814.686 4806.01 61 1977.195 4998.85 1126 1810.070 4800.73 92 1972.079 4992.69 1156 1805.603 4795.62 122 1967.128 4986.72 1187 1800.987 4790.34 153 1962.012 4980.56 1297 1796.520 4785.23 184 1956.895 4974.40 1248 1791.904 4779.95 212 1952.274 4968.83 1279 1787.289 4774.67 243 1947.158 4962.67 1307 1783.119 4769.90 273 1942.254 4956.86 1338 1778.504 4764.62 304 1937.206 4950.92 1368 1774.088 4759.57 334 1932.321 4945.18 1399 1769.659 4754.51 365 1927.273 4939.24 1429 1765.373 4749.61 396 1922.225 4933.30 1460 1760.943 4744.54 426 1917.34 4927.56 1491 1752.227 4739.47 457 1912.292 4921.62 1551 1747.941 4729.67 518 1902.405 4909.98 1581 1743.654 4724.77 549 1897.619 4904.26 1611 1739.368 4719.86 577 1893.296 4899.10 1642 1734.938 4714.80 608 1888.510 4893.39 1672 1730.681 4709.94 638 1883.879 4887.86 1703 1726.390 4705.08 669 1879.093 4882.14 1734 1722.098 4700.23 699 1874.462 4876.61 1763 1718.083 4695.69 730 1869.676 4870.90 1794 1713.791 4690.83 761 1864.890 4865.19 1824 1709.638 4686.13 791 1860.289 4859.71 822 1855.635 4854.21 852 1851.131 4848.89 883 1846.477 4843.40 914 1841.823 4837.90 942 1837.620 4832.93 973 1832.966 4827.44 1003 1828.462 4822.12 1034 1823.808 4816.62

54

4.1.1 History Match

The computer program was run with the regression parameters selected to be radius of

drainage, skin and permeability. The result of the history match is shown in figure 4.1.1 and

figure 4.1.2 and table 4.1.3

Table 4.1.3

History Match for Synthetic Data

Calculated value Calculated

value

Initial

value

Comment

Permeability [md] 11.054 10.8 From well test

Skin 116.454 101 From well test

Radius of drainage [ft] 9107.852 2500.0 Estimated

The parameter INFO equals 2 when the program terminates.

Several runs of the program were conducted in order to assess the sensitivity of the history

match with respect to the following input parameters: pressure decrement, tolerance and

eventual errors in the input historical production data.

55

1650

1700

1750

1800

1850

1900

1950

2000

2050

0 500 1000 1500 2000

Time [days]

Observed rate Predicted rate

Figure 4.1.1 Synthetic data : production history match

56

4650

4700

4750

4800

4850

4900

4950

5000

5050

0 500 1000 1500 2000

Time [days]

Observed reservoir pressure predicted reservoir pressure

Figure 4.1.2. Synthetic data: reservoir pressure history match.

57

4.1.2 Sensitivity Analysis

4.1.2.1 Sensitivity of History Match Results with Respect to Pressure Decrements

Values.

Pressure decrements values of 50 psia, 100 psia, 200 psia, and 300 psia were used to

perform history match calculations. The results are shown in Figure 4.1.3 and Figure 4.1.4.

As it can be seen, the production history match as well as the pressure history match are

excellent for all the pressure decrement values used. So, the use of a lower value for

pressure decrement does not necessarily improve the history match results but may instead

increase the computational intensity compared to the use of greater pressure decrement

value. For this particular synthetic well, the computational time for all these decrement

values is very small. However, in general the computational intensity may notably increase

when the pressure decrement value decreases.

58

Figure 4.1.3. Synthetic data: sensitivity of production history match with respect to

pressure decrement.

1650

1700

1750

1800

1850

1900

1950

2000

2050

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Time [days]

Observed rate Predicted rate [DP=50 psia] Predicted rate [DP=100 psia]Predicted rate [DP=200 psia] Predicted rate [DP=300 psia]

59

Figure 4.1.4. Synthetic data: sensitivity of pressure history match with respect to

pressure decrement

4650

4700

4750

4800

4850

4900

4950

5000

5050

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Time [days]

Predicted reservoir pressure [DP=50 psia] Predicted reservoir pressure [DP=100 psia]Predicted reservoir pressure {DP=200 psia] Predicted reservoir pressure [DP=300 psia]

60

4.1.2.2 Sensitivity of History Match Results with Respect to Tolerance Values.

In order to verify the sensitivity of the history match results with respect to the tolerance

used in the calculations, different runs of the program have been conducted using tolerance

values of 0.1, 0.001, 0.000001. The results obtained are shown in Figure 4.1.5 and Figure

4.1.6.

As it can be noted, the production history match as well as the pressure history match are

excellent for tolerance values of 0.000001 and 0.001. Instead, the results obtained using the

tolerance value of 0.1 are clearly not acceptable. This particular example shows that the

result of the history match calculations may be affected by the tolerance value used for the

computations. In general, the history match results are better when using a lower tolerance

value. However, as it can be noted in this particular synthetic case, the quality of the history

match is acceptable for both tolerance values of 0.001 and 0.000001 but the computational

intensity is greater for the tolerance value of 0.000001.

For practical purposes, a tolerance value of 0.000001 can be used for most cases. A value of

tolerance lower than 0.000001 is generally not necessary to achieve an acceptable history

match results.

61

Figure 4.1.5. Synthetic data: sensitivity of production history match with respect to

tolerance

500

700

900

1100

1300

1500

1700

1900

2100

0 500 1000 1500 2000

Time [days]

Observed rate Predicted rate [Tolerance=10^-6]Predicted rate [Tolerance= 10^-1] Predicted rate [Tolerance=10^-3]

62

Figure 4.1.6. Synthetic data: sensitivity of reservoir pressure with respect to tolerance

3500

3700

3900

4100

4300

4500

4700

4900

5100

0 500 1000 1500 2000

Time [days]

Observed pressure [psia] Predicted reservoir pressure [Tolerance=10̂ -6]Predicted rate [Tolerance = 10̂ -1] Predicted reservoir pressure [Tolerance = 10̂ -3]

73

4.1.3 Future Performance Simulations

4.1.3.1 Future Performance Simulations for Different Well Head Pressure Values

Different runs of the program were conducted at various well head pressures to simulate the

effect of the installation of a compressor on the future performance of the producing system.

Well head pressure values of 2250 psia, 1000 psia, 500 psia and 100 psia were used for the

forecasting. The well was producing at a well head pressure of 2250 psia. The results of

these simulations are shown in figure 4.1.9 and figure 4.1.10.

As can be seen, the decrease of the well head pressure from 2250 psia to 1000 psia provided

an increase in flow rate of about 1000 Mscf/D. However a further decrease of well head

pressure from 1000 psia to 500 psia provided an increase of only 200 Mscf/D. Moreover,

the gain resulting from an eventual decrease of the well head pressure from 500 psia to 100

psia can be considered to be negligible. This sensitivity analysis can be used in deciding

whether or not to install a compressor and under what optimum conditions it can be

operated.

74

Figure 4.1.11. Synthetic data: sensitivity of rate with respect to well head pressure

0

500

1000

1500

2000

2500

3000

3500

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Time [days]

Observed rate [well head pressure = 2250 psia] Predicted rate [well head pressure = 2250 psia]Well head pressure = 2250 psia Well head pressure = 1000 psiaWell head pressure=500 psia Well head pressure=100 psia

75

Figure 4.1.12. Synthetic data: sensitivity of reservoir pressure with respect to well head

pressure.

0

1000

2000

3000

4000

5000

6000

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Time [days]

Observed reservoir pressure Predicted rate [well head pressure = 2250]Well head pressure = 2250 psia] Well head pressure = 1000 psiaWell head pressure = 500 psia Well head pressure = 100 psia

76

4.1.3.2 Future Performance Simulations for Different Skin Values

In order to simulate the effect of a stimulation job (acidizing, fracturation,..) on the

performance of the well, the program has been run with different skin factors. The skin of

134.459, 50 and 0.0 has been used in the forecast computations. The results of this

sensitivity analysis are shown in Figure 4.1.11 and Figure 4.1.12.

The improvement of the well performance as the skin factor is reduced is clearly displayed

on the graph. The forecast performance declines faster as the skin is lower. For example the

decline rate corresponding to skin 0.0 is greater than the one corresponding to skin 134.459.

This is due to the fact that the removal of the skin does not increase the reserves, but

accelerates the gas recovery.

77

Figure 4.1.13. Synthetic data: sensitivity of rate with respect to skin factor.

0

500

1000

1500

2000

2500

3000

3500

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Time [days]

Observed rate History match [skin=134.459] Skin=134.459 Skin=50.0 Skin=0.0

78

Figure 4.1.14. Synthetic data: sensitivity of reservoir pressure with respect to skin.

0

1000

2000

3000

4000

5000

6000

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Time [days]

Observed rate History match [skin=134.459] Skin=134.459 Skin=50 Skin=0.0

79

4.2 Field Data

Four cases of actual field well data are presented in this section. Each of those cases exhibits

a specific problem and gives the solution to overcome it.

4.2.1 Case #1: Dry Gas Well Producing at a Constant Well Head Pressure.

Case #1 represents a dry gas well, open to production since 1989 in Beluga reservoir

(Alaska), which has been produced at a constant well head pressure. The initial reservoir

pressure is estimated to be 2083 psia. The characteristics of the reservoir as well as the

description of the reservoir are summarized in Table 4.2.1.1.

Table 4.2.1.1

System description data for case #1

Type of decline = exponential Pressure decrement [psia] = 50 Optimization tolerance = 0.000001

Reservoir

Initial pressure [psia] = 2083 Reservoir temperature [F] = 110 Pay [ft] = 51.589 Skin = 4.101 Drainage radius [ft] = 4141.645 Permeability [md] = 26.858 Porosity [fraction] = 0.19 Water saturation [fraction] = 0.15

Fluid properties

Specific gravity of produced gas [fraction] = 0.7 Oil density [API] = 52.0 Specific gravity of produced water [fraction] = 1.0

80

Solution gas/oil ratio correlation: Lasater Oil formation volume factor: Standing Oil viscosity correlation: Robinson Dranchuck and Purvis. Z-factor correlation: Hall and Yarborough

Completion

Hole diameter [in] = 12.240 Casing diameter [in] = 9.625 Perforated interval [ft] = 80.0 Perforation diameter [in] = 0.720 Perforation tunnel length [in] = 12.33 Perforation density [SPF] = 4 Mode of perforation = overbalance Tubing inside diameter [in] = 2.992 Tubing roughness [ft] = 0.00015 Well inclination [degree] = 90.0 Tubing length [ft] = 7682.0 Pressure drop correlation: Beggs and Brill

Production

Oil/Gas ratio [SBBLO/MMscf] = 0.0 Water/Gas ratio [SBBLW/MMscf] = 0.0 Well head pressure [psia] = 870.0 Well head temperature [F] = 70.0 Reference separator pressure, [psia] = 14.7 Reference separator temperature, [deg F] = 60.0

Limit of regression parameters

PERMIN [md] = 20.0 PERMAX [md] = 45.0 SMIN = 4.0 SMAX = 10.0 REMIN [ft] = 500.0 REMAX [ft] = 5000.0

The production performance and the reservoir pressure in function of time are presented in

Table 4.2.1.2.

81

Table 4.2.1.2

Well performance and reservoir pressure data for case #1

Time Rate Reservoir Time Rate Reservoir

Pressure pressure

[days] [Mscf/D] [psia] [days] [Mscf/D] [psia]

0 8520.0 2108 960 12107.5 30 12977.235 990 12030.484 60 12243.839 1020 11663.323 90 10463.684 1050 11203.964 120 12625.0 1080 10776.935 150 12812.0 1110 10335.3 180 14953.5 1140 10276.645 210 14521.097 1170 10089.167 240 12580.733 1200 10076.548 270 14147.193 1230 10279.580 300 13874.323 1260 9733.178 1663.75 330 13317.464 1290 9544.839 360 12657.258 1320 9423.3 390 12736.467 1350 9389.742 420 12773.258 1380 9550.581 450 12267.2 1410 9688.966 480 11836.5 1440 9458.452 510 12250.516 1470 9285.633 540 12038.933 1500 9111.167 570 10934.935 1560 8700.742 600 10815.8 1590 8573.148 1569 630 11196.548 1620 8147.767 660 10794.645 1650 8063.193 690 10499.0 1680 8262.1 720 10513.645 1710 8069.097 750 9867.133 1880 1740 8073.806 780 10665.096 1770 8202.214 810 12686.1 1800 8060.129 840 12639.709 1830 7873.7 870 12529.870 1860 7979.613 900 12300.1 1890 8035.333 930 12339.097 1920 7898.516

82

1950 7551.258065 2460 5508.742 1980 7509.655172 1492 2490 5575 2010 7321.322581 2520 5672.839 2040 7206.733333 2550 5634.4 2070 7055.903226 2580 5472.548 2100 7036.16129 2610 5275.6 2130 7113.928571 2640 5222.580 2160 7058.064516 2670 5073.742 2190 6821.466667 2700 4975.867 2220 6480.16129 2730 5953.963 1326 2250 6382.206897 1355 2760 5169.0 2280 6570.419355 2790 4805.581 2310 6478.387097 2820 4523.387 2340 5897.2 2850 4208.345 2370 5593.677419 2880 4043.032 2400 5925.5 2910 3523.633 2430 5662.903226 2940 3302.588

4.2.1.1 History Match


drainage, skin and permeability. The results of the history match are shown in Figure 4.2.1.1

and Figure 4.2.1.2.

83

Figure 4.2.1.1. Case #1: production history match

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

0 500 1000 1500 2000 2500 3000 3500

Time [days]


84

Figure 4.2.1.2. Case #1: reservoir pressure history match.

0

500

1000

1500

2000

2500

0 500 1000 1500 2000 2500 3000 3500Time [days]

Observed reservoir pressure Predicted reservoir pressure

85

An excellent production history match is obtained. The reservoir pressure history match is

also in very good agreement with the observed field data.

The calculated values of the regressed parameters as well as the observed values of those

parameters are shown in Table 4.2.1.3.

Table 4.2.1.3

History Match for case #1


value

Initial

value

Comment

Permeability [md] 32.974 24 From well test

Skin 4.148 6.4 From well test

Radius of drainage [ft] 4532.738 1000 Estimated


The skin exhibits a good agreement between the observed value and the calculated value.

The permeability and radius of drainage calculated from the program are higher than the

corresponding observed values. This is due to the fact that the actual reservoir drive

mechanism may not be exactly natural depletion. Some other mechanism such as

compaction drive may contribute to the actual reservoir mechanism.

86

The change in rate observed at time 750 days is simply due to a well head pressure

perturbation that was very limited in time.

Note

Figure 4.2.1.A and Figure 4.2.1.B represent the history match results obtained when the

objective function used is simply the sum of squares of differences between the observed

rates and the predicted rates and differences between the observed reservoir pressure and

predicted reservoir pressure.

As can be seen, the rate history match is very good. However the reservoir history match is

not very good. This is due to the fact that this objective function assigns the same weight to

each rate and reservoir pressure data point. As the number of rate data points is greater than

the number of reservoir pressure data points, the rate history match is better than the

reservoir pressure history match.

87

Figure 4.2.1.A Case #1: production history match

0

2000

4000

6000

8000

10000

12000

14000

16000

0 500 1000 1500 2000 2500 3000 3500

Time [days]


88

Figure 4.2.1.B. Case #1: reservoir pressure history match.

4.2.1.2 Future Performance Predictions

0

500

1000

1500

2000

2500

0 500 1000 1500 2000 2500 3000 3500

Time [days]

Predicted reservoir pressure Observed reservoir pressure

89

4.2.1.2.1 Future Performance Prediction Using Different Well Head Pressure Values.



Well head pressure values of 870 psia, 700 psia, 500 psia, 300 psia, and 100 psia were used

in the forecast computations. The well was producing at a well head pressure of 870 psia.

The results of these simulations are shown in Figure 4.2.1.3 and Figure 4.2.1.4.

As can be seen, the well performance improves as the well head pressure decreases.

However the increase in flow rate is not linearly related to the decrease in the well head

pressure. For example, the gain in flow rate obtained from reducing the well head pressure

from 870 psia to 700 psia is about 2400 Mscf/D, whereas the increase in the well

performance is only 500 Mscf/D when the well head pressure is reduced from 300 to 100

psia. This sensitivity analysis is useful to the engineer in the process of deciding whether or

not to install a compressor and under what optimum conditions it can be operated.

90

Figure 4.2.1.3. Case #1: sensitivity of rate with respect to well head pressure.

0

2000

4000

6000

8000

10000

12000

14000

16000

0 1000 2000 3000 4000 5000 6000 7000

Time [days]

Observed rate [Well head pressure = 870 psia] Predicted rate [Well head pressure = 870 psia]]Well head pressure= 870 psia Well head pressure = 700 psiaWell head pressure = 500 psia Well head pressure = 300 psiaWell head pressure= 100 psia

91

Figure 4.2.1.4 Case #1: sensitivity of reservoir pressure with respect to well head

pressure

0

500

1000

1500

2000

2500

0 1000 2000 3000 4000 5000 6000 7000

Time [days]

Observed reservoir pressure [ well head pressure = 870 psia]Predicted reservoir pressure [well head pressure = 870 psia]Well head pressure = 870 psiaWell head pressure = 700 psiaWell head pressure = 500 psiaWell head pressure = 300 psiaWell head pressure = 100 psia

92

4.2.1.2.2 Future Performance Prediction Using Different Skin Values.

In order to simulate the effect of a stimulation job (acidizing, fracturation, ..) on the

performance of the well, the program was run with different skin factors. The skin values of

4.101, 0.0 and -5 were used in the forecast computations. The results of this sensitivity

analysis are shown in Figure 4.2.1.5 and Figure 4.2.1.6.

The improvement of the well performance as the skin factor is reduced is clearly displayed

on the graph. The forecast performance declines faster as the skin is lower. For example the

decline rate corresponding to skin 0.0 is greater than the one corresponding to skin 4.101.

This is due to the fact that the removal of the skin does not increase the reserves, but

accelerates the gas recovery.

93

Figure 4.2.1.5. Case #1: sensitivity of rate with respect to skin factor

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

0 1000 2000 3000 4000 5000 6000

Time [days]

Observed rate Predicted rate [S=4.148] Skin=4.148 Skin=0.0 Skin=-5

94

Figure 4.2.1.6. Case #1: sensitivity of reservoir pressure with respect to skin.

0

500

1000

1500

2000

2500

0 1000 2000 3000 4000 5000 6000

Time [days]

Observed reservoir pressure Predicted reservoir pressure [S=4.148]Skin=4.148 Skin=0.0Skin= -5

95

4.2.1.2.3 Future Performance Prediction for Different Density of Perforation.

In order to assess the sensitivity of the density of perforations on the well performance, the

program was run with different values of perforation densities. Perforation densities of 4

spf, 8 spf and 12 spf are used in the forecast computations. The overbalanced perforation

mode is used. The well is actually perforated overbalanced with a perforation density of 4

spf. The results of the simulations are summarized in Figure 4.2.1.7 and Figure 4.2.1.8.

As it can be seen, the well performance improves slightly as the perforation density

increases. However the gain in flow rate remains marginal compared to those obtained by

reducing the skin (acidizing / fracturation) or by reducing the well head pressure (by

installing a compressor for example).

96

Figure 4.2.1.7. Case #1: sensitivity of rate with respect to perforation density.

0

2000

4000

6000

8000

10000

12000

14000

16000

0 1000 2000 3000 4000 5000 6000 7000

Time [days]

Observed rate [SPF=4] Predicted rate [SPF=4] SPF=4 SPF=8 SPF=12

97

Figure 4.2.1.8. Case #1: sensitivity of reservoir pressure with respect to perforation

density

0

500

1000

1500

2000

2500

0 1000 2000 3000 4000 5000 6000 7000

Time [days]

Observed pressure [SPF=4] Predicted pressure [SPF=4] SPF=4 SPF=8 SPF=12

98

4.2.2 Case #2: Conversion of the Original Data from Constant Flow Rate to Constant

Well Head Pressure.

Case #2 represents a condensate gas production system. The well, open to production since

1989, exhibits a very high condensate yield of 145 BBL/MMscf. The initial reservoir

pressure is 5010.98 psia. The PVT analysis estimates the dew point pressure at 5025 psia.

The decline curve analysis indicates that the well produces with exponential decline. The

characteristics of the reservoir as well as the description of the completion are summarized

in Table 4.2.2.1.

Table 4.2.2.1

System description data for case #2


Reservoir

Initial pressure [psia] = 5010.98 Reservoir temperature [F] = 212.0 Pay [ft] = 64.454 Skin = 116.441 Drainage radius [ft] = 9107.845 Permeability [md] = 11.053 Porosity [fraction] = 0.060 Water saturation [fraction] = 0.533

Fluid properties

Specific gravity of produced gas [fraction] = 0.646 Oil density [API] = 51.080 Specific gravity of produced water [fraction] = 1.0 Solution gas/oil ratio correlation: Lasater Oil formation volume factor: Standing

99

Oil viscosity correlation: Robinson Dranchuck and Purvis. Z-factor correlation: Hall and Yarborough

Completion

Hole diameter [in] = 8.496 Casing diameter [in] = 5.0 Perforated interval [ft] = 17.0 Perforation diameter [in] = 0 .73 Perforation tunnel length [in] = 12.33 Perforation density [SPF] = 4 Mode of perforation = overbalance Tubing inside diameter [in] = 1.945 Tubing roughness [ft] = 0.00015 Tubing length [ft] = 8688.0 Well inclination angle [degree] = 90.0 Pressure loss correlation: Gray

Production

Oil/Gas ratio [SBBLO/MMscf] = 145.0 Water/Gas ratio [SBBLW/MMscf] = 0.0 Well head pressure [psia] = 2250.0 Well head temperature[F] = 111.0 Reference separator pressure , [psia] = 14.7 Reference separator temperature, [deg F] = 60.0

Limits of regression parameters

PERMIN [md] = 1.0 PERMAX [md] = 100.0 SMIN = -5.0 SMAX = 175.0 REMIN [ft] = 2500.0 REMAX [ft] = 10000.0

100

In order to use the computer program presented in this work, it is required that the well

head pressure be reasonably constant during the period of time considered in the history

match computations. Case #2 does not satisfy this requirement as it is producing with

constant rate but not with constant well head pressure. For this well, the data were

converted from constant rate to equivalent constant well head pressure. The conversion

equation used is the following:

[ ][ ]2

22

21

2

2

1

WFR

WFR

PPPP

Q

Q

−−

= . (4.2.2.1)

Q1 is the actual constant flow rate corresponding to the flowing bottom hole pressure Pwf1.

Since Q1 and Pwf1 are known, the flow rate Q2 can be computed by assuming a fixed value

of the corresponding bottom hole pressure Pwf2.

This conversion technique works well if the total reservoir pressure decline is small during

the time period considered for history match calculations.

The production data before and after conversion are shown in Table 4.2.2.2, Table 4.2.2.3,

Figure 4.2.2.1 and Figure 4.2.2.2.

101

Table 4.2.2.2 Original field production data for case #2. Time Rate Reservoir Time Rate Reservoir pressure pressure [days] [mscf/D] [psia] [days] [mscf/D] [psia] 0 1000 5010.980 973 2000 4817.397 31 1000 5004.158 1003 2000 4812.156 61 1000 4997.165 1034 2000 4807.139 92 1000 4990.452 1064 2000 4802.011 122 2000 4983.572 1095 2000 4796.940 153 2000 4976.750 1126 1000 4792.089 184 2000 4970.637 1156 2000 4787.133 212 1000 4963.925 1187 2000 4782.391 243 1000 4957.484 1217 2000 4777.548 273 2000 4950.885 1248 1000 4772.762 304 2000 4944.554 1279 2000 4768.489 334 2000 4938.068 1307 2000 4763.814 365 2000 4931.640 1338 2000 4759.344 396 2000 4925.475 1368 2000 4754.781 426 2000 4919.160 1399 2000 4750.421 457 2000 4913.105 1429 2000 4745.972 518 1000 4900.760 1460 2000 4741.581 549 2000 4895.261 1491 2000 4737.388 577 2000 4889.228 1521 2000 4733.246 608 2000 4883.444 1551 2000 4729.159 638 2000 4877.524 1581 1000 4725.127 669 2000 4871.849 1611 2000 4721.017 699 2000 4866.043 1642 2000 4717.094 730 2000 4860.294 1672 2000 4713.097 761 2000 4854.785 1703 2000 4709.158 791 2000 4849.149 1734 2000 4705.525 822 2000 4843.751 1763 2000 4701.697 852 2000 4838.229 1794 2000 4698.048 883 2000 4832.765 1824 2000 4694.334 914 1000 4827.879 942 2000 4822.524 For the conversion computation, the bottom hole flowing pressure has been fixed to 3200

psia.

102

Table 4.2.2.3

Converted production data for case #2.

Time Rate Reservoir Time Rate Reservoir Pressure pressure [days] [Mscf/D] [psia] [days] [Mscf/D] [psia] 0 1986.358 5010.98 973 1830.879 31 1980.588 1003 1826.169 61 1974.662 1034 1821.655 92 1968.962 1064 1817.036 122 1976.169 1095 1812.463 153 1970.287 1126 1797.912 184 1965.088 1156 1803.601 212 1946.560 1187 1798.710 243 1941.056 1217 1794.287 273 1947.993 1248 1779.319 4772.76 304 1942.497 1279 1786.016 334 1936.857 1307 1781.735 365 1931.354 1338 1777.639 396 1925.976 1368 1773.455 426 1919.835 1399 1769.782 457 1914.602 1429 1765.722 518 1891.111 1460 1761.709 549 1899.131 1491 1757.872 577 1893.881 1521 1754.080 608 1888.841 4883.44 1551 1750.335 638 1883.673 1581 1736.368 4725.13 669 1878.711 1611 1742.858 699 1873.626 1642 1739.327 730 1868.582 1672 1735.530 761 1863.754 1703 1731.907 791 1858.791 1734 1728.546 822 1854.030 1763 1724.934 852 1849.153 1794 1721.685 883 1844.655 1824 1718.249 4694.33 914 1829.030 942 1835.480

103

Figure 4.2.2.1. Case #2: Original field data.

0

500

1000

1500

2000

2500

0 500 1000 1500 2000 2500 3000

Time [days]

Original field data

104

Figure 4.2.2.2. Case #2: converted rate

1700

1750

1800

1850

1900

1950

2000

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Time [days]

Observed rate

105



drainage, skin and permeability.

The results obtained are presented in Figure 4.2.2.3 and Figure 4.2.2.4.

An excellent production history match is obtained. The reservoir pressure history match is

also very good.


parameters are shown in Table 4.2.2.4.

Table 4.2.2.4

History Match for Case #2


value

Initial

value

Comment


Skin 147.0 101 From well test



106

The regressed value of the permeability agrees with the value obtained from well test. The

predicted radius of drainage is greater than the observed drainage radius. This is probably

due to the fact that the computer program uses volumetric drive mechanism and it has been

documented that the reservoir drive mechanism for case #2 is not volumetric.

107

Figure 4.2.2.3. Case #2: production history match.

1650

1700

1750

1800

1850

1900

1950

2000

2050

0 500 1000 1500 2000

Time [days]

Observed rate predicted rate

108

Figure 4.2.2.4. Case #2: reservoir pressure history match

4650

4700

4750

4800

4850

4900

4950

5000

5050

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Time [days]


109


4.2.2.2.1 Future Performance Prediction Using Different Well Head Pressure Values



Well head pressure values of 2250 psia, 1500 psia, 1000 psia, 500 psia, and 100 psia have

been used in the forecast computations. The well has been producing at a well head pressure

of 2250 psia. The results of these simulations are shown in Figure 4.2.2.5 and Figure

4.2.2.6.

As it can be seen, the well performance improves as the well head pressure decreases.



from 1500 psia to 1000 psia is about 350 Mscf/D, whereas the increase in the well

performance is only 200 Mscf/D when the well head pressure is reduced from 1000 psia to

500 psia. The gain in gas rate is almost negligible (about 50 Mscf/D) when the well head

pressure is decreased further from 500 psia to 100 psia. This sensitivity analysis is useful to

the engineer in the process of deciding whether or not to install a compressor and under

what optimum conditions it can be operated.

110

Figure 4.2.2.5. Case # 2: sensitivity of rate with respect to well head pressure

0

500

1000

1500

2000

2500

3000

3500

0 500 1000 1500 2000 2500 3000 3500 4000

Time [days]

Observed data [well head pressure=2250 psia] History match [well head pressure=2250 psia]well head pressure=2250 psia well head pressure=1500 psiawell head pressure=1000 psia well head pressure=500 psiawell head pressure=100 psia

111

Figure 4.2.2.6. Case # 2: sensitivity of reservoir pressure with respect to well head

pressure.

4200

4300

4400

4500

4600

4700

4800

4900

5000

5100

0 500 1000 1500 2000 2500 3000 3500 4000

Time [days]

Observed data [well head pressure = 2250 psia] History match [well head pressure =2250 psia ]Well head pressure = 2250 Well head pressure = 1500 psiaWell head pressure = 1000 psia Well head pressure=500 psiaWell head pressure=100 psia

112

4.2.2.2.2 Future Performance Prediction for Different Skin Values

In order to simulate the effect of a stimulation job (acidizing, fracturing,...) on the

performance of the well, the program has been run with different skin factors. The skin

values of 116.441, 50 and 0.0 has been used in the forecast computations. The results of this

sensitivity analysis are shown in Figure 4.2.2.7 and Figure 4.2.2.8.

The improvement of the well performance as the skin factor is reduced is clearly seen. The

forecast performance declines faster as the skin is lower. For example the decline rate

corresponding to skin 0.0 is greater than the one corresponding to skin 116.4. This is due to

the fact that the removal of the skin does not increase the reserves, but accelerates the gas

recovery.

113

Figure 4.2.2.7. Case #2: sensitivity of rate with respect to skin factor.

0

500

1000

1500

2000

2500

3000

3500

0 500 1000 1500 2000 2500 3000 3500 4000

Time [days]

Observed rate History match [S=116.44] Skin=116.4 Skin=50.0 Skin=0.0

114

Figure 4.2.2.8. Case #2: sensitivity of reservoir pressure with respect to skin.

4200

4300

4400

4500

4600

4700

4800

4900

5000

5100

0 500 1000 1500 2000 2500 3000 3500 4000

Time [days]

Observed bottom hole reservoir pressure History match [S=116.44]Skin=116.4 Skin=50.0Skin=0.0

115

4.2.2.2.3 Future Performance Prediction Using Different Perforated Interval Values

In order to assess the sensitivity of the well performance with respect to the perforated

interval, the program is run with different values of perforated interval. Perforated interval

values of 17 ft and 64 ft are used in the forecast computations. The actual perforated interval

of the well is 17 ft. The results of the simulations are summarized in Figure 4.2.2.9 and

Figure 4.2.2.10.

As it can be seen, the well performance increases as the perforated interval increases.

116

Figure 4.2.2.9. Case #2: Sensitivity of rate with respect to perforated interval.

500

700

900

1100

1300

1500

1700

1900

2100

2300

0 500 1000 1500 2000 2500 3000 3500 4000Time [days]

Observed rate [Perforated interval= 17 ft] History match [Perforated interval= 17 ft]Perforated interval= 17 ft Perforated interval= 64 ft

117

Figure 4.2.2.10. Case #2: sensitivity of reservoir pressure with respect to perforated

interval.

4300

4400

4500

4600

4700

4800

4900

5000

5100

0 500 1000 1500 2000 2500 3000 3500 4000

Time [days]

Observed pressure History match [perforated interval=17ft]Perforated interval = 17 ft perforated interval = 64 ft

118

4.2.3 Case #3: Conversion of the Original Data from Constant Flow Rate to Constant

Well Head Pressure

Case #3 represents a condensate gas production system. The well, open to production since

1989, exhibits a very high condensate yield of 150 BBL/MMscf. The initial reservoir

pressure is 5164.3 psia. The PVT analysis estimates the dew point pressure at 5040 psia.

The decline curve analysis indicates that the well produces with exponential decline. The

characteristics of the reservoir as well as the description of the completion are summarized

in Table 4.2.3.1.

Table 4.2.3.1

System description data for case #3.


Reservoir

Initial pressure [psia] = 5164.7 Reservoir temperature [F] = 216.0 Pay [ft] = 174.48 Skin = 0.922 Drainage radius [ft] = 4936.265 Permeability [md] = 6.484 Porosity [fraction] = 0.086 Water saturation [fraction] = 0.288

Fluid properties

Specific gravity of produced gas = 0.66 Oil density [API] = 52.26 Specific gravity of produced water = 1.0

119

Solution gas/oil ratio correlation: Lasater Oil formation volume factor: Standing Oil viscosity correlation: Robinson Dranchuck and Purvis Z-factor correlation: Hall and Yarborough

Completion Hole diameter [in] = 6.0 CSG diameter [in] = 5.0 Perforated interval [ft] = 44.0 Perforation diameter [in] = 0.73 Perforation tunnel length [in] = 12.33 Perforation density [SPF] = 4

Mode of perforation = overbalance TBG inside diameter [in] = 1.945 TBG roughness [ft] = 0.00015 Tubing length [ft] = 8826 Well inclination angle [degree] = 90.0 Pressure loss correlation Gray

Production

Oil/Gas ratio [SBBLO/MMscf] = 143.0 Water/Gas ratio [SBBLW/MMscf] = 0.0 Well head pressure [psia] = 3200.0 Well head temperature [F] = 111.0 Reference separator pressure , [psia] = 14.7 Reference separator temperature , [deg F] = 60.0

Limits for regression variables

PERMIN [md] = 0.0 PERMAX [md] = 7.0 SMIN = -1.0 SMAX = 7.0 REMIN [ft] = 2500.0 REMAX [ft] = 7000.0 In order to use the computer program presented in this work, it is required that the well head

pressure be reasonably constant during the period of time considered in the history match

computations. Case #3 does not satisfy this requirement as it is producing with constant rate

but not with constant well head pressure. For this well, the data have been converted from

120

constant rate to equivalent constant well head pressure. Again, the conversion equation used

is the following:

[ ][ ]2

22

21

2

2

1

WFR

WFR

PPPP

Q

Q

−−

= . (4.2.3.1)

Q1 is the actual constant flow rate corresponding to the bottom hole pressure Pwf1. Since Q1

and Pwf1 are known, the flow rate Q2 can be computed by assuming a fixed value of the

corresponding bottom hole pressure Pwf2.

The production data before and after conversion are shown in Table 4.2.3.2, Table 4.2.3.3,

Figure 4.2.3.1 and Figure 4.2.3.2.

Table 4.2.3.2

Original field production data for case #3 Time Rate Reservoir Time Rate Reservoir Pressure pressure [days] [mscf/D] [psia] [days] [mscf/D] [psia] 0 1000 5168.100 1064 2000 4941.702 30 1000 5160.786 1095 2000 4936.124 61 1000 5153.285 1125 2000 4930.781 91 5000 5146.081 1156 2000 4925.317 122 6000 5138.693 1186 2000 4920.084 1 53 5000 5131.363 1217 3000 4914.732 183 4000 5124.325 1248 3000 4909.440 214 5000 5117.108 1278 3000 4904.371 244 6000 5110.179 1309 3000 4899.192 275 5000 5103.076 1339 3000 4894.234 306 3000 5096.031 1370 3000 4889.168 334 2000 5089.717 1401 2000 4884.159 365 3000 5082.781 1429 2000 4879.685 426 2000 5069.302 1460 2000 4874.786 456 2000 5062.755 1490 2000 4870.1

121

487 1000 5056.046 1521 1000 4865.314 518 1000 5049.395 1551 3000 4860.738 548 2000 5043.013 1582 2000 4856.066 579 1000 5036.476 1613 2000 4851.452 609 1000 5030.204 1643 2000 4847.041 640 1000 5023.78 1674 2000 4842.540 671 2000 5017.414 1704 3000 4838.240 699 2000 5011.713 1735 2000 4833.852 730 2000 5005.456 1766 2000 4829.522 760 3000 4999.456 1794 2000 4825.661 791 2000 4993.313 1825 1000 4821.441 821 1000 4987.423 1855 2000 4817.412 852 3000 4981.393 1886 2000 4813.306 883 3000 4975.421 1916 2000 4809.386 913 3000 4969.696 1947 2000 4805.393 944 3000 4963.837 1978 2000 4801.458 974 2000 4958.222 2008 2000 4797.704 1005 2000 4952.477 2038 2000 4794.004 1036 2000 4946.790 2068 2000 4790.359 For the conversion computation, the bottom hole flowing pressure has been fixed to 4400

psia.

Table 4.2.3.3

Converted production data for case #3.

Time Rate Reservoir Time Rate Reservoir Pressure pressure [days] Mscf/D] [psia] [days] Mscf/D] [psia] 0 4954.999 5164.3 1500 3490.987 30 4625.527 1530 3457.712 120 4869.370 1560 3425.802 180 4330.156 1590 3393.117 240 4205.933 1620 3353.956 270 4252.592 1650 3229.772 300 4313.037 1680 3199.266 4830 330 4177.194 1710 3170.008

122

360 4057.840 1740 3140.049 390 4102.713 1770 3111.324 420 4267.720 1800 3081.919 450 4346.216 1830 3142.278 810 4193.768 1860 3115.699 840 4237.993 1890 3086.544 870 4200.233 1920 3058.604 900 4313.851 1950 3129.203 930 4271.649 1920 3058.604 960 4085.903 1950 3129.203 990 4189.687 1980 2915.950 1020 4151.635 4960 2010 2974.621 1050 4109.156 2040 2946.907 1080 3936.359 2070 2920.370 1110 3904.164 2100 2893.242 1140 3868.797 2130 2782.703 1170 3722.768 2160 2840.745 1200 3795.545 2190 2814.517 1260 3878.464 2220 2791.091 1290 3619.289 2250 2861.306 1320 3584.966 2310 2740.927 1350 3552.015 2340 2715.895 1380 3518.240 4890 2370 2691.969 4775 1410 3589.217 2400 2667.556 1440 3555.106 2430 2643.458 1470 3521.284 2460 2620.441 2490 2597.723 2520 2575.306

123

Figure 4.2.3.1. Case #3: Original field data.

0

1000

2000

3000

4000

5000

6000

7000

0 500 1000 1500 2000 2500

Time [days]

Original field data

124

Figure 4.2.3.2. Case #3: converted rate.

0

1000

2000

3000

4000

5000

6000

0 500 1000 1500 2000 2500 3000

Time [days]

Rat

e [M

scf/

D]

Observed data

125




The results obtained are presented in Figure 4.2.3.3 and Figure 4.2.3.4.

An excellent production history match is obtained. The pressure history match deviates

slightly from the observed pressure data in the end. This is probably due to the fact that the

computer program uses volumetric drive mechanism and it has been reported that the

reservoir drive mechanism for case #3 is not volumetric.


parameters are shown in Table 4.2.3.4. The parameter INFO equals 2 when the program

terminates.

Table 4.2.3.4.

History Match for case #3

Calculated value Calculated value Initial value Comment




126

Figure 4.2.3.3. Case #3: Production history match

0

1000

2000

3000

4000

5000

6000

0 500 1000 1500 2000 2500 3000

Time [days]


127


0

1000

2000

3000

4000

5000

6000

0 500 1000 1500 2000 2500 3000

Time [days]

Observed data [perforated interval = 44 ft] Predicted reservoir pressure

128

The regressed value of the permeability agrees with the value obtained from well test. Also,

the skin agrees with the value obtained from well test.


4.2.3.2.1 Future Performance Prediction Using Different Well Head Pressure Values

Different runs of the program have been conducted at various well head pressures to

simulate the effect of the installation of a compressor on the future performance of the

producing system. Well head pressure values of 3200 psia, 2000 psia, 1000 psia, and 500

psia have been used in the forecast computations. The well has been producing at a well

head pressure of 3200 psia. The results of these simulations are shown in Figure 4.2.3.5 and

Figure 4.2.3.6.

As can be seen, the well performance improves as the well head pressure decreases.



from 3200 psia to 2000 psia is about 4000 Mscf/D whereas the increase in the well

performance is only 2000 Mscf/D when the well head pressure is reduced from 2000 to 500

psia. This sensitivity analysis is useful to the engineer in the process of deciding whether or

not to install a compressor and under what optimum conditions it can be operated.

129

Figure 4.2.3.5. Case # 3: sensitivity of rate with respect to well head pressure

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 1000 2000 3000 4000 5000

Time [days]

Observed rate [well head pressure= 3200 psia] Predicted rate [well head pressure= 3200 psiaWell head pressure = 2000 psia Well head pressure = 1000 psiaWell head pressure = 500 psia Well head pressure = 3200 psia

130

Figure 4.2.3.6. Case # 3: sensitivity of reservoir pressure with respect to well head

pressure.

0

1000

2000

3000

4000

5000

6000

0 1000 2000 3000 4000 5000Time [days]

Observed reservoir pressure [well head pressure= 3200 psia]History match [well head pressure=3200 psia]Well head pressure = 2000 psiaWell head pressure = 1000 psiaWell head pressure= 500 psiaWell head pressure = 3200 psia

131

4.2.3.2.2 Future Performance Prediction for Different Perforation Density Values

In order to simulate the effect of the perforation density on the performance of the well, the

program has been run with different perforation density values. The perforation density

values of 4 spf, 8 spf and 12 spf have been used in the forecast computations. The results

of this sensitivity analysis are shown in Figure 4.2.3.7 and Figure 4.2.3.8.

The improvement of the well performance as the density of perforation is increased is

clearly seen. The forecast performance declines faster as the density of perforation is higher.

For example the decline rate corresponding to the density of perforation 12 spf is greater

than the one corresponding to the density of perforation 4 spf. This is due to the fact that the

increase of the perforation density does not increase the reserves, but accelerates the

recovery.

Note that the production of the well stops at time 4290 days when the perforation density is

12 spf. This is due to liquid loading.

132

Figure 4.2.3.7. Case #3: sensitivity of rate with respect to density of perforation.

0

1000

2000

3000

4000

5000

6000

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Time [days]

Observed rate [perforation density=4 spf] Predicted rate [perforation density=4 spf]Perforation density = 4 spf Perforation density = 8 spfPerforation density =12 spf

133

Figure 4.2.3.8. Case #3: sensitivity of reservoir pressure with density of perforation.

0

1000

2000

3000

4000

5000

6000

0 1000 2000 3000 4000 5000

Time [days]

Observed pressure History match [perforation density = 4 spf]Perforation density = 4 spf Perforation density = 8 spfPerforation density = 12 spf

134

4.2.3.2.3 Future Performance Prediction Using Different Perforated Interval Values

In order to assess the sensitivity of the well performance with respect to the perforated

interval, the program is run with different values of perforated interval. Perforated interval

values of 44 ft and 106 ft are used in the forecast computations. The actual perforated

interval of the well is 44 ft. The results of the simulations are summarized in Figure 4.2.3.9

and Figure 4.2.3.10.

As it can be seen, the well performance increases as the perforated interval increases.

However, the well production stops at time 4290 days due to liquid loading when the

perforated interval is extended to 106 ft.

135

Figure 4.2.3.9. Case #3: Sensitivity of rate with respect to perforated interval.

0

1000

2000

3000

4000

5000

6000

0 1000 2000 3000 4000 5000

Time [days]

Observed data [perforated interval = 44 ft] History match [perforated interval= 44 ft]Perforated interval = 44 ft Perforated interval = 106 ft

136

Figure 4.2.3.10. Case #3: sensitivity of reservoir pressure with respect to perforated

interval.

0

1000

2000

3000

4000

5000

6000

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Time [days]

Observed data [perforated interval = 44 ft] History match [perforated interval = 44 ft]Perforated interval= 44 ft Perforated interval = 106 ft

137

4.2.4 Case #4: Use of the Last Two Years of Production Only

Case #4 provides one of the most difficult cases used to validate the computer program.

This well is open to production since 1989. The initial reservoir pressure is 5149 psia. An

exponential decline behavior is assumed.

In order to use the computer program presented in this work, it is required that the well head

pressure be reasonably constant during the period of time considered in the history match

computations. Case #4 does not satisfy this requirement as it is not produced with constant

well head pressure throughout his past production life. For this well, only the data

corresponding to the last two years of production are considered in the history match.

During this period of time the well head pressure is reasonably constant. The reservoir

pressure at the beginning of this period of time is 576 psia. The characteristics of the

reservoir as well as the completion are summarized in Table 4.2.4.1

Table 4.2.4.1

System description data for case #4.

Type of decline = exponential Pressure decrement = 25.0 Optimization tolerance [FTOL] = 0.000001 Reservoir Initial pressure [psia] = 576.0 Initial temperature [F] = 177.0 Pay [ft] = 32.0 Drainage radius [ft] = 3015.0 Permeability [md] = 69.9 Porosity [fraction] = 0.16

138

Water saturation [fraction ] = 0.15 Fluid properties Specific gravity of produced gas = 0.65 Oil density [API] = 52.6 Specific gravity of produced water = 1.0 Solution gas/oil ratio correlation: Lasater Oil formation volume factor: Standing Oil viscosity correlation: Robinson Dranchuck and Purvis. Z-factor correlation: Hall and Yarborough Completion Hole diameter [in.] = 5.0 Casing diameter [in.] = 2.992 Perforated interval [ft] = 25.0 Perforation diameter [in.] = 0.73 Perforation tunnel length [in.] = 12.33 Perforation density [SPF] = 4 Mode of perforation = Overbalance TBG inside diameter [in.] = 2.992 TBG roughness [ft] = 0.00015 Length of tubing [ft] = 8187.0 Pressure drop correlation = Beggs & Brill Production Oil/Gas ratio [SBBLO/MMscf] = 3.555 Water/Gas ratio [SBBLW/MMscf] = 0.526 Well head pressure [psia] = 90.0 Well head temperature [F] = 77.0 Reference separator pressure , [psia] = 14.7 Reference separator temperature , [deg F] = 60.0 Limits for regression parameters PERMIN = 60.0 PERMAX = 80.0 SMIN = 4.0 SMAX = 50.0 REMIN = 1000.0 REMAX = 3000.0

The historical performance of the well as well as the observed reservoir pressure are

summarized in Table 4.2.4.2

139

Table 4.2.4.2

Production data for case #4.

Time Rate Reservoir pressure

[days] [Mscf/D] [psia]

0 1073.714 576 31 1071.461 60 1104.928 91 1035.828 121 928.006 152 899.623 517 182 1004.964 213 1086.213 244 1060.120 274 995.614 305 1015.550 335 948.457 366 1027.111 446 397 932.107 425 968.0 456 954.286 485 866.233




The results are shown in Figure 4.2.4.1. and Figure 4.2.4.2.

A satisfactory production history match is obtained. The predicted pressure deviates from

the observed data in the end. The predicted reservoir pressure remains higher than the

observed values. The difference in the drive mechanism between the model which assumes

140

natural depletion drive and the actual drive mechanism of the reservoir may be the cause of

that deviation in reservoir pressure. Also, this case is difficult because the observed rate

decline is small during the period of time considered for the history match.

141

Figure 4.2.4.1. Case #4: Production history match.

0

200

400

600

800

1000

1200

1400

0 100 200 300 400 500 600

Time [days]

Observed rate History match

142


0

100

200

300

400

500

600

700

0 100 200 300 400 500 600

Time [days]

Oberved reservoir pressure Predicted reservoir pressure

143

The regressed parameters values obtained are presented in Table 4.2.4.3.


Table 4.2.4.3

History Match for Case #4


value

initial

value

Comment




144

The permeability value exhibits a good agreement between the observed data and the

predicted value. The calculated skin value is higher than the observed value.


4.2.4.2.1 Reduction of Well Head Pressure

The well head pressure has been decreased from 90 psia to 50 psia and 15 psia in order to

simulate the possibility to avoid the liquid loading which occurs after 515 days when the

well head pressure remains 90 psia.

The results are shown on Figure 4.2.4.3 and Figure 4.2.4.4.

As it can be seen on Figure 4.2.4.3, the well produces at a flow rate of 875 Mscf/D when the

well head is reduced to 50 psia; the rate is 890 Mscf/D when the well head pressure is

reduced to 15 psia. The production stops for liquid loading after 515 days if the well head

pressure remains at 90 psia.

This kind of sensitivity analysis may be important when trying to optimize a gas well

production where a risk of loading exists.

145

Figure 4.2.4.3. Case #4: sensitivity of rate with respect to well head pressure.

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400

Time [days]

Observed rate [ Well head pressure = 90 psia ] Predicted rate [ well head pressure = 90 psia ]Well head pressure = 90 psia Well head pressure = 50 psiaWell head pressure = 15 psia

146

Figure 4.2.4.4. Case #4: sensitivity of reservoir pressure with respect to well head

pressure

0

100

200

300

400

500

600

700

0 200 400 600 800 1000 1200 1400 1600

Time [days]

Oberved reservoir pressure [well head pressure = 90 psia]Predicted reservoir pressure [ well head pressure = 90 psia]Well head pressure = 90 psiaWell head pressure = 50 psiaWell head pressure = 15 psia

147

4.2.4.2.2 Reduction of Tubing Size

The inside tubing diameter has been reduced from 2.992 in. to 1.995 in. and 1.049 in. in

order to simulate the possibility to avoid the liquid loading which occurs after 515 days

when the well produces through a 2.992 inches inside diameter.


As it can be seen on Figure 4.2.4.5, the rate is 665 Mscf/D when the well produces through

a 1.995 in. inside diameter tubing. The rate is 215 Mscf/D when the inside tubing diameter

is1.049 inch. If the tubing inside diameter remains 2.992 inches, the well production stops

for liquid loading after 515 days.


production where a risk of loading exists. This analysis will ultimately be integrated in an

economic evaluation .

148

Figure 4.2.4.5. Case #4: sensitivity of rate with respect to tubing size.

0

200

400

600

800

1000

1200

1400

0 500 1000 1500 2000 2500

Time [days]

Observed rate [ tubing inside diameter= 2.992 in.] Predicted rate [tubing inside diameter = 2.992 in. Tubing inside diameter = 2.992 in.Tubing inside diameter = 1.995 in.Tubing inside diameter = 1.049 in.

149

Figure 4.2.4.6. Case #4: sensitivity of reservoir pressure with respect to tubing size.

0

100

200

300

400

500

600

700

0 500 1000 1500 2000 2500

Time [days]

Obseved reservoir pressure [ tubing inside diameter = 2.992 in]Predicted reservoir pressure [tubing inside diameter= 2.991 in.]Tubing inside diameter = 2.992 in.Tubing inside diameter = 1.995 in.Tubing inside diameter = 1.049 in.

150

4.2.4.2.3 Choke Installation

In order to avoid the liquid loading which occurs after 515 days of production, a well head

choke is installed. The program is run with different choke inside diameter. The values of

0.5 inch (32/64) and 0.38 inch (24/64) have been used.


As it can be seen on Figure 4.2.4.7, the well production stops after 515 days if there is no

well head choke. However if a well head choke of 0.38 (24/64) inch is installed, the well

produces at a rate of about 773 Mscf/D and eventually will stop due to liquid loading at

time of 845 days. If a 0.5 inch (32/64) well head choke is installed, the well will produce at

about 744 Mscf/D without any risk of liquid loading for about 5 years.


production where loading problems exist.

151

Figure 4.2.4.7. Case #4: sensitivity of rate with respect to choke size.

0

200

400

600

800

1000

1200

1400

0 500 1000 1500 2000 2500

Time [days]

Observed rate [ no choke] Predicted rate [ no ckoke]No choke Choke inside diameter = 0.5 in. (32/64)Choke inside diameter = 0.38 in. (24/64)

152

Figure 4.2.4.8.Case #4: sensitivity of reservoir pressure with respect to choke size

0

100

200

300

400

500

600

700

0 500 1000 1500 2000 2500

Time [days]

Observed reservoir pressure [no choke] Predicted reservoir pressure [No choke]No choke Choke inside diameter= 0.5 in. (32/64)Choke inside diameter = 0.38 in. (24/64)

153

CHAPTER V

CONCLUSIONS

In this thesis, dynamic nodal analysis technique has been discussed. This technique

allows to perform sensitivity analysis of future performance for gas wells once a satisfactory

match of the past production performance is obtained. The major contribution of this work

is that it provides a tool to analyze the well performance changes as a function of time when

the production parameters are altered. The classic nodal analysis can only be used if the

production parameters remained unchanged.

The dynamic nodal analysis provides valuable means to help the engineer in

decisions making. Opening a gas well to production always involves considerable expenses

whereas a model can be run many times at lower cost to try many different possible

scenarios in order to make technical and economical decisions.

It should be noted that the prediction of the future performance based on history

match of well performance is not unique. There are many other sets of system parameters

that can match the past performance of the well. There is always some uncertainty

associated to the model used to arrive at a satisfactory historical performance match. Based

upon the history match results, the engineer can obtain a range of future performances, and

hence can make a decision in light of uncertainties.

154

The computer program presented in this study is capable of history matching the

production data as well as predicting the future performance under different scenarios. The

program has been validated with the help of both synthetic and field data. The program

definitely provides a logical improvement in conventional nodal analysis.

155

RECOMMENDATIONS

This thesis can be complimented by implementing the following features in the

computer program.

• Different drive mechanisms rather than natural depletion can be implemented. Water

drive as well as compaction drive can easily be added to the program.

• Horizontal gas wells inflow performance can be added to the computer program in order

to expand its use to gas wells that have this geometry.

• The dynamic nodal analysis method can be expanded to oil reservoir producing with a

reservoir pressure above the bubble point. In this condition, the single phase flow in the

reservoir can be easily described.

• The program can be expanded to production system where the flow in the reservoir is

two-phase flow. For example a condensate gas reservoir where the reservoir pressure is

well below the dew point or an oil reservoir with a reservoir pressure below the bubble

point pressure. However, these situations are more complex and difficult.

• Time increment ∆T can be used instead of pressure decrement ∆P in the Dynamic

Nodal Analysis algorithm. By doing so, the algorithm will be directly used for

production system where the well head pressure varies during the period of time

considered for the history match.

156

NOMENCLATURE

Symbol

A total area open to flow, ft2

Bg gas formation volume factor, cf/scf

COV covariance

Fk functions used in the definition of the objective function

FVEC function vector. Its components are the functions Fk

G gas in place, Mscf

H pay, ft

Hp perforated interval, ft

INFO convergence criteria under which the program terminates.

K reservoir permeability, md

KG gravel pack permeability, md

Kp permeability of compacted zone, md

L gravel pack linear flow path, ft

Lp perforation tunnel length, ft

P pressure, psia.

∆P pressure drop, psia

Qmod flow rate predicted from model, Mscf/D

Qobs observed flow rate, Mscf/D

157

Re drainage radius, ft

Rc radius of compacted zone, ft

Rp radius of perforation, ft

Rw well radius, ft

Sg gas saturation, fraction

S skin factor, dimensionless

T temperature, R

TR reservoir temperature, K

Tsc temperature at standard conditions, K

Tobs Observed production Time, days

∆T elapsed time, days

Z compressibility factor, dimensionless

α set of 3 independent regression variables

β turbulence factor, ft-1

γ specific gravity, dimensionless

µ viscosity, cp

σ standard deviation

φ porosity

ξc constrained variable estimate

ξLMDIF1 unconstrained variable estimate calculated by LMDIF1

ξmax maximum value of the variable

ξmin minimum value of the variable

158

Subscript

gp gravel pack

g gas

I inflow

O outflow

max maximum value of variable

min minimum value of variable

perf perforations

r reservoir

REST restriction

SV safety valve

TBG tubing

WF at bottom hole in well flowing conditions

WFS at sand face in well flowing conditions

159

REFERENCES

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Company, Tulsa (1984), volume 4

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Tech. (May 1973), 607-617.

3. Hagedorn, A.R. and Brown, K.E.: “Experimental Study of Pressure Gradients Occurring

During Continuous Two-Phase Flow in Small-Diameter Vertical Conduits,” J. Pet.

Tech. (April 1965), 475-484.

4. Pudjo Sukarno: “Inflow Performance Relationship Curves In Two-Phase and Three-

Phase Flow Conditions,” PHD Dissertation, University of Tulsa, Tulsa (1986)

5. Perez, G.. “Overall Performance of Oil and Gas Production Systems,” M.S. Thesis, The

University of Tulsa, 1988.

6. Standing, M.B.: “Volumetric and Phase Behavior of Oil Field Hydrocarbon Systems,”

8th Printing, Society of Petroleum Engineers, 1977, P.121

7. Vasquez, A.: “Correlations for Fluid Physical Property Prediction, “ M.S. Thesis, The

University of Tulsa.

160

8. Thomas, G.W.: “Principles of Hydrocarbon Reservoir Simulation,” International

Human Resources Development Corporation, Boston.

9. Gray, H.E. “Vertical Flow Correlation in Gas Wells,” In User Manual for API 14B,

Subsurface Controlled Safety Valve Sizing Computer Program, App. B. (June 1974)

10. Havlena, D. Odeh, A.S. “The material Balance as an Equation of a straight line,”

Transactions of the AIME (1963)

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