Upload
others
View
3
Download
0
Embed Size (px)
Citation preview
AN Approach to stimulation candidate selection
and optimization
A
Research Thesis
Presented to the Department of Petroleum Engineering,
African University of Science and Technology,
Abuja
in Partial Fulfillment of the Requirements for the Award of Master of
Science (MSc)
in
Petroleum Engineering
By
BENSON OGHENOVO UGBENYEN
Abuja, Nigeria November, 2010
An Approach to Stimulation Candidate Selection and Optimization
By
Benson Oghenovo Ugbenyen
RECOMMENDED: ________________________________
________________________________
________________________________
________________________________
APPROVED: ________________________________ Supervisor: Prof. (Emeritus) David O. Ogbe
________________________________
________________________________
________________________________
Date
iii | An Approach to Stimulation Candidate Selection and Optimization
ABSTRACT
Well stimulation consists of several methods used for enhancing the natural producing ability of
the r eservoir when p roduction rate declines. A de tailed l iterature r eview of s ome of t he well
published stimulation models are discussed in this research. This d iscussion wa s preceded wi th
an introduction t o f ormation damage concepts and an o verview o f well stimulation m ethods.
Production decline curve analysis is combined with economic discounting concepts to develop a
model that can be used for optimizing stimulation decisions. The model is presented in the form
of a no n-linear programming pr oblem subject t o t he constraints imposed by t he p roduction
facilities, reservoir productivity and the stimulation budget approved by management. Production
data from four stimulation candidate wells, o ffshore Niger Delta was used to validate the model
developed by s etting up a maximization problem. Solution to the p roblem was ob tained using
non-linear o ptimization software. The r esult o btained was v erified u sing Wolfram R esearchβs
Mathematica 7.0. The results s how that the o ptimization m odel c an be c ombined w ith
stimulation t reatment modules, de veloped f rom i ndustry w ide models, t o q uantify s timulation
benefits. C andidate w ells w ere t hen r anked ba sed on stimulation c ost, p ayout t ime a nd
stimulation b enefit. Hence, th e m odel i s valid f or stimulation ca ndidate s election; and i s
therefore recommended for use in optimizing stimulation decisions.
iv | An Approach to Stimulation Candidate Selection and Optimization
DEDICATION
This research is dedicated to my Lord Jesus Christ who has been, and will ever be the best role
model anyone could find. And also, to the good people of the Niger Delta.
v | An Approach to Stimulation Candidate Selection and Optimization
ACKNOWLEDGEMENT I wish t o sincerely a ppreciate G od Almighty for H is l ove, c are a nd wonderful works t hat a re made m anifest i n m y life each da y. Also, m y s incere thanks go to my supervisor, Prof. (Emeritus) David O. Ogbe for guiding me to success in this work, Dr. Samuel Osisanya and Prof. Peters Ekwere f or s erving in m y thesis committee, and m y m other, M rs. G race Ugbenyen f or being there always for me. The following persons, among others, who contributed in no small measure to the success of this work deserved to be acknowledged. My friends: Lymmy B ukie O gbidi, Akpana Paul, R aymond Agav, Habibatu Ahmed, and Christopher Mudi who paid m e several v isits a t A UST t o c heer me u p. T he members o f H ope Hall Parish, Redeemed Christian Church of God, Galadimawa, Abuja, who have always been a warm family to me. Nature will not forgive me if I fail to thank Miss Esther Akinyede who was kind to provide me with a laptop to continue this work when lightning storm damaged my laptop on 14th
July 2010 a t Julius Nyerere Hall, AUST, Abuja, and I got no help from the University even t hough I pl eaded f or assistance. I w ill n ot f ail to m ention Mr. Alfred Emakpose who assisted me in no small measure to keep things straight when the odds were against me. Finally, I would like to thank my wonderful new friends, who would be mad at me if I fail to mention their names; Hatem, Adel, Amar, Fauzan and Andrew, who are here with me as I write these lines at The Beaches Hotel, Prestatyn, North Wales, where I neglected some of my schedule to put most parts of this work together.
vi | An Approach to Stimulation Candidate Selection and Optimization
TABLE OF CONTENTS
ABSTRACTβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦β¦.....iii
DEDICATIONβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦β¦....iv
ACKNOWLEDGEMENTβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦...v
TABLE OF CONTENTβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...vi
LIST OF FIGURESβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦β¦.....x
LIST OF TABLESβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦xi
CHAPTER ONE: INTRODUCTION 1.1 The Near Wellbore Conditionβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦β¦β¦β¦.β¦....1
1.1.1 The Composite Skin Effectβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦.....1
1.2 Well Stimulation: Definition and Objectivesβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦.β¦β¦..1
1.2.1 Well Stimulation Objectivesβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦1
1.3 Well Stimulation Methodsβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦...2
1.3.1 Matrix Stimulationβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦....2
1.3.1.1 Matrix Acidizing Fluid Selection and Treatment Additives β¦β¦β¦β¦β¦....3
1.3.1.2 Benefits and Limitations of Matrix Acidizing Processesβ¦β¦β¦β¦β¦β¦......4
1.3.2 Fracture Acidizingβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦.......4
1.3.3 Hydraulic Fracturingβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦....6
1.3.4 Recompletionβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦...7
1.4 Gravel Packingβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...β¦7
1.5 Stimulation Economics and Candidate Selectionβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦....8
1.6 Objective and Procedure of the Studyβ¦β¦β¦β¦..β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦8
1.7 Limitation of the Studyβ¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...β¦9
CHAPTER TWO: LITERATURE REVIEW 2.1 Review of Formation Damage Mechanismβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦.........10
2.1.1 Definitionβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦.β¦10
2.1.2 Causes of Formation Damageβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦.10
2.1.3 Quantifying Formation Damageβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦.11
2.1.3.1 Skin Factorβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦...β¦11
vii | An Approach to Stimulation Candidate Selection and Optimization
2.1.3.2 Depth of Damageβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦13
2.1.3.3 Damage Ratioβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦..14
2.1.3.4 Flow Efficiencyβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦....16
2.1.3.5 Permeability Variation Indexβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦..16
2.1.4 Economic Impact of Formation Damage on Reservoir Productivity
β¦β¦β¦β¦β¦β¦β¦β¦β¦...β¦β¦β¦β¦β¦.β¦.17
2.2 Matrix Acidizing Modelsβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦β¦.β¦..17
2.2.1 Sandstone Acidizing Modelsβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦..18
2.2.2 Carbonate Acidizing Modelsβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦..22
2.3 Acid Fracturing Modelsβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦.26
2.4 Hydraulic Fracturing Modelsβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦.β¦β¦28
2.5 Literatures on Stimulation candidate Selectionβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦...30
CHAPTER THREE: METHODOLOGY 3.1 Well Screening Techniqueβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦β¦β¦.β¦33
3.2 Design of Stimulation Treatment Modelsβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦...34
3.2.1 Matrix Acidizing Design Modelβ¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦38
3.2.1.1 Summaryβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦.38
3.2.2 Recompletion Design Modelβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦..38
3.2.3 Gravel-Pack Design Modelβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦.β¦40
3.3 Development of a Model for Optimizing Stimulation Decisionsβ¦β¦β¦β¦β¦β¦β¦.β¦..44
3.3.1 Optimization Model Assumptionsβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦.45
3.3.2 Stimulation Productivity Ratioβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦...46
3.3.3 The Present-value Discount Factorβ¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦β¦β¦β¦.β¦....46
3.3.4 Defining the Objective Function, QD
3.4 Optimization Model Constraintsβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...β¦β¦β¦.β¦β¦β¦β¦β¦.β¦β¦50
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦.46
3.4.1 Constraint 1: Break-even Requirementβ¦β¦β¦β¦β¦β¦β¦β¦...β¦β¦β¦β¦β¦.β¦β¦51
3.4.2 Constraint 2: Remaining Reserve Limitationβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦.51
3.4.3 Constraint 3: Flow String capacityβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦.52
3.4.4 Constraint 4: Budget Allocationβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦.53
3.4.5 Constraint 5: Maximum Formation Productivity ratioβ¦β¦...β¦β¦β¦β¦β¦.β¦β¦53
3.4.6 Constraint 6: Productivity Improvementβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦54
viii | An Approach to Stimulation Candidate Selection and Optimization
3.5 Stimulation Cost and Productivity Ratio Relationshipβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦..54
3.6 Summary of the Optimization Modelβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦β¦.β¦β¦55
3.7 Solution to the Optimization Modelβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦..56
CHAPTER FOUR: MODEL VALIDATION, RESULTS AND DISCUSSION 4.1 Sensitivity Analysisβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...β¦β¦β¦β¦β¦.β¦.58
4.1.1 Effect of Price of Oilβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦β¦β¦...58
4.1.2 Effect of Discount Rateβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦.β¦.58
4.1.3 Effect of Decline Rateβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦58
4.1.4 Effect of Pre-Stimulation Production rateβ¦β¦β¦β¦β¦β¦β¦......β¦β¦β¦β¦.........63
4.1.5 Effect of Abandonment Rateβ¦β¦β¦β¦β¦..β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦63
4.1.6 Effect of Stimulation Timeβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦.66
4.2 Model Validation: Case Study 1 β¦β¦β¦β¦β¦β¦β¦β¦...β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦.β¦...66
4.2.1 Formulation of the Bestfield Modelβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦....66
4.2.2 Solution of the Well BU 3 Modelβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦...72
4.2.3 Discussion of the Well BU 3 Model Resultβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦73
4.2.4 Application of the Model Result in Candidate Selectionβ¦β¦β¦β¦β¦β¦β¦..β¦..74
4.2.5 Effect of Price of Oil on Well BU 3 Model Resultβ¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦.74
4.3 Model Validation: Case Study 2β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦77
4.3.1 Formulation of Well BU 5 Modelβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦...77
4.3.2 Solution of the Well BU 5 Modelβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦80
4.3.3 Discussion of the Well BU 5 Model Resultβ¦β¦.β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦...81
4.3.4 Effect of Oil Price on Well BU 5 Model Resultβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...82
4.3.5 Using Case Study 2 Model Result in Candidate Selectionβ¦β¦β¦β¦β¦..β¦β¦β¦82
CHAPTER FIVE: CONCLUSION AND RECOMMENDATION
5.1 Conclusionβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...β¦84
5.2 Recommendationβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦85
REFERENCESβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦.87
NOMENCLATUREβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦.95
APPENDIX A: A SIMPLE WELL SCREENING FLOW CHARTβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...98
ix | An Approach to Stimulation Candidate Selection and Optimization
APPENDIX B: STIMULATION COST AND PERFORMANCEβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.....99
APPENDIX C: SOLVER RESULTSβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.100
APPENDIX D: WHATβS BEST 10.0 RESULTSβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...115
APPENDIX E: MATHEMATICA 7.0 RESULTSβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..120
APPENDIX F: DERIVATION OF THE OBJECTIVE FUNCTION FOR OTHER
DECLINE CASESβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦..124
x | An Approach to Stimulation Candidate Selection and Optimization
LIST OF FIGURES
3.1 Production Decline Profile for a Stimulated Well.β¦β¦β¦β¦β¦β¦β¦ β¦β¦β¦β¦β¦β¦.β¦45
4.1 Effect of oil price on the objective function β¦β¦β¦β¦.β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦.60
4.2 Effect of discount rate on the objective function β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦β¦β¦β¦.β¦.61
4.3 Effect of decline rate on the objective function β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦62
4.4 Effect of pre-stimulation production rate on the objective functionβ¦β¦β¦β¦β¦β¦.β¦...64
4.5 Effect of abandonment rate on the objective function β¦β¦β¦β¦β¦β¦...β¦β¦β¦β¦.........65
4.6 Effect of stimulation timeβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦β¦β¦β¦β¦.β¦...67
4.7 Cost Versus Productivity Ratio Plot for Well BU 3β¦..β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...β¦71
4.8 Effect of oil price on Well BU 3 model resultβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦..76
4.9 Cost Versus Productivity Ratio Plot for Well BU 5 β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦79
4.10 Effect of oil price on Well BU5 model resultβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.83
xi | An Approach to Stimulation Candidate Selection and Optimization
LIST OF TABLES
Table 4.1: Input Data for Sensitivity Analysisβ¦β¦β¦β¦β¦.β¦β¦β¦..59
Table 4.2: Bestfield Model Dataβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦.68
Table 4.3: Bestfield Model Summaryβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..75
Table 4.4: Well BU 5 Model Dataβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...78
1 | An Approach to Stimulation Candidate Selection and Optimization
Chapter One
Introduction
1.1 The Near Wellbore Condition
Permeability reduction i n t he r egion near t he wellbore in a producing zo ne i s r eferred t o a s
βdamageβ. The damaged region i s c alled s kin z one w hile the term βskin e ffectβ refers t o a
dimensionless parameter used to quantify the extent of damage. Reduction in permeability in the
near-wellbore region results in lower productivity due to increased pressure drop, hence damage
is not desirable.
1.1.1 The Composite Skin Effect
The skin effect can be o btained from a well te st. I t measures t he extent of damage in t he near-
wellbore zone. The total skin effect obtained from the well test is a composite parameter which
consists of s kin c omponents d ue to mechanical c auses β a di sturbance of t he fluid f low
streamline n ormal t o t he w ell, o r formation damage - alteration o f t he natural r eservoir
permeability. It is very important to be able to identify the formation damage component of the
skin s ince t his c an b e r educed by b etter operational practices, or possibly, b e r emoved or
bypassed by stimulation treatments. Formation damage can result from many different operations
such a s dr illing, cementing, perforating, completion/gravel pa cking, production, i njection,
workover, stimulation, etc.
1.2 Well Stimulation: Definition and Objectives
Well stimulation is a way of increasing well productivity by removing (or bypassing) formation
damage in t he near-wellbore r egion or by superimposing a highly conductive structure onto the
formation.
1.2.1 Well Stimulation Objectives
The objectives of w ell s timulation can be di vided into technical ob jectives and e conomic
objectives.
2 | An Approach to Stimulation Candidate Selection and Optimization
β’ Technical Objectives
Remove, reduce or b ypass t he f ormation damage, reduce sand production and cl eaning-
up the perforations.
β’ Economic Objectives
Increase flow rate and optimize production from the reservoir.
1.3 Well Stimulation Methods
Several stimulation t echniques e xist bu t t he commonly u sed methods i nclude matrix a cidizing,
fracture a cidizing, fracpack, ex treme o verbalance operations and hy draulic fracturing. These
methods h elp t o optimally increase well or reservoir productive c apacity by providing a net
increase in the productivity index. This increase in productivity index can then be used either to
increase t he p roduction r ate o r t o d ecrease the dr awdown pressure differential. Increase i n
production rate will eventually increase productivity. A decrease in drawdown can help prevent
sand pr oduction and water or gas coning and/or shift the phase equilibrium in the near-wellbore
region t owards s maller f ractions of condensate. Some of the m ost c ommon s timulation
techniques are discussed in the following sections.
1.3.1 Matrix Stimulation
Matrix stimulation is injecting an acid/solvent into the formation at below the fracturing pressure
of t he formation to d issolve/disperse materials th at im pair well production i n sandstone
reservoirs or to create new, unimpaired flow channels in carbonate reservoirs. Mineral acids are
most c ommonly us ed in matrix s timulation hence t his t echnique is f requently ca lled ma trix
acidizing. Matrix acidizing is a near-wellbore treatment, with all of the acid reacting within a few
to perhaps as much as 10 ft of the wellbore in carbonates. Matrix a cidizing lower permeability
limit is 10mD for oil wells and 1mD for gas wells.
In sandstone, only a small f raction o f the m atrix i s soluble hence r elatively s low r eacting acid
dissolves the permeability-damaging minerals. Carbonate formations are different in that a large
fraction of the matrix is soluble (usually > 50%), hence acid will react rapidly with flow channels
and pores and creates new flow paths by dissolving the formation rock.
3 | An Approach to Stimulation Candidate Selection and Optimization
As a rule of thumb, matrix acidizing i s a pplied only in situations where a well has a large skin
effect t hat cannot b e attributed t o mechanical, o peration o r surface p roblems. The r emoval of
damage by matrix a cidizing r equires t hat t he t ype ( or c ause) a nd location of t he damage be
identified before its removal is attempted. The damage identification process involves:
β’ Examining t he well r ecords to i dentify operations t hat might ha ve r esulted in formation
damage
β’ Carrying out specific laboratory testing, such as a reservoir core flushing, to determine if
the identified operations did indeed lead to core damage for the particular combination of
the fluids in question and the reservoir formation
β’ Examining t he da maged core with sophisticated a nalytical techniques s uch a s t he
scanning electron microscope to confirm the damage type and the damage location and
hence develop ideas on how to remove it.
1.3.1.1. Matrix Acidizing Fluid Selection and Treatment Additives
The t ype of a cid u sed for a s timulation j ob i s a function of t he da mage t ype. Generally, a cid
selection guidelines are based on temperature, mineralogy and petrophysics. The most common
acids u sed a re h ydrochloric a cid ( HCl) a nd a m ixture o f hydrochloric a nd h ydrofluoric a cids
(HF/HCl) usually known a s mud acid. HCl is suitable f or li mestone, d olomite, formation w ith
iron m aterials a nd C aSO4
Additives help make acid treatments more e ffective. They are mixed with the treating fluids to
modify a pr operty of t he fluid (e.g., corrosion, p recipitation, emulsification, s ludging, s caling, f ines
migration, clay swelling tendency, surface tension, flow per layer, friction pressure). The treating fluid
is d esigned t o e ffectively r emove or b ypass t he damage, whereas a dditives a re u sed t o prevent
excessive c orrosion, p revent s ludging and e mulsions, pr event iron pr ecipitation, improve
cleanup, improve coverage of the zone and pr event pr ecipitation of reaction products. Additives
. H F i s mostly us ed i n s andstone, c lay, f eldspar, s and (spent on
material, not quartz or sand), and it is not used in carbonate formations. Acid mixtures such as
acetic-hydrochloric a nd formic-hydrochloric a cids a re u sed i n high temperature ca rbonate
formation w hile t he formic-hydrofluoric a cid mixture i s us eful i n high t emperature sandstone
formation.
4 | An Approach to Stimulation Candidate Selection and Optimization
are a lso u sed i n preflushes a nd overflushes t o stabilize clays a nd di sperse pa raffins a nd
asphaltenes. Types of additives include: acid corrosion inhibitors, aromatic solvents, Iron stabilizers,
surfactants, mutual solvents, diverters, scale i nhibitors, clay stabilizers, aluminum stabilizer, retarders,
nitrogen and alcohols.
1.3.1.2. Benefits and Limitations of Matrix Acidizing Processes
Matrix a cidizing is usually very economically a ttractive (low c ost), because r elatively s mall
treatments may improve the well performance considerably.
Some pr oblems a ssociated with matrix a cidizing a re: difficulty to i dentify the type of damage,
multiple damages with completing remedies, detrimental by-products o f stimulation, frequently,
ineffective o r p artially e ffective treatments. It involves complex chemical a nd t ransport
phenomena t hat, w hile effective i n r emoving one k ind o f damage, may cr eate a nother o ne.
HCL/HF blends can create early damage in formations, however the lower the HF concentration
in t he b lend t he l ess chance there i s for damage creation. Acid placement and damage removal
from l aminated f ormations w here some perforations penetrate very h igh-permeability la yers is
especially problematic.
Successful m atrix treatments r equire correct c hoice of fluid t o a ttack damage an d u niform
placement o f the s elected treating f luid. Improper f luid pl acement i ncreases reservoir
heterogeneity. Misapplied stimulation treatments a re costly and ineffective, o ften creating more
problems than they solve.
It is important to note that not all da mage can be removed by matrix acidizing. Whenever there
are insoluble scales (e.g. BaSO4) or acid s ensitive sandstones, other s timulation methods (such
as acid fracturing to bypass scales) are considered.
1.3.2 Fracture Acidizing
In this method of acidizing, acid is injected into the formation at a rate high enough to generate
the pressure required t o fracture t he formation. T he r apid i njection produces a buildup i n the
5 | An Approach to Stimulation Candidate Selection and Optimization
wellbore pressure until it is large enough to overcome compressive earth stresses and the rockβs
tensile strength. At this p ressure, t he r ock fails, a llowing a c rack ( fracture) t o be formed.
Continued fluid injection increases the fracture length and width. The injected acid differentially
etches t he formation fracture faces as it r eacts, r esulting i n t he formation of h ighly c onductive
etched channels that remain open after the fracture closes. Two procedures are commonly used.
Acid alone i s injected, or a fluid ( called a pad) that will create a long, wide fracture is injected
and followed by an a cid. A conventional fracture acidizing treatment involves pumping an acid
system after fracturing. It may be preceded by a nonacid preflush and usually is overflushed with
a nonacid fluid.
Acid s olubility of th e f ormation is a key f actor i nfluencing w hether f racture acidizing or
proppant treatments should be employed. If the formation is less than 75% acid soluble, proppant
treatments should be used. For acid solubilities between 75 and 85%, special lab work can help
define w hich approach should be used. Above 85% acid solubility, fracture acidizing would b e
the most effective approach.
Treatment v olumes for fracture a cidizing a re much l arger t han for matrix acidizing t reatments,
being as high as 1,000 to 2,000 gal/ft of perforated interval.
As a general guideline, f racture a cidizing i s us ed on formations with > 80% hydrochloric a cid
solubility. Low-permeability carbonates (>20 md) a re t he best candidates for t hese t reatments.
Fluid loss to the matrix and natural fractures can also be better controlled in lower permeability
formations.
The su ccess of t he acid f racturing treatment depends on two ch aracteristics o f t he etched
fracture: effective fracture length (which is a function of the rate of acid consumption, acid fluid
loss ( wormhole formation) a nd acid convection a long t he fracture) a nd e ffective fracture
conductivity (a function of the etched pattern, vo lume of r ock di ssolved, r oughness of etched
surface, rock strength and closure st ress). The acidized fracture length and fracture conductivity
are therefore controlled largely by the treatment design and formation strength.
6 | An Approach to Stimulation Candidate Selection and Optimization
1.3.3 Hydraulic Fracturing
Hydraulic Fracturing consists of pumping a viscous fluid at a sufficiently high pressure (greater
than the formation fracture pressure) into the completion interval so that a two winged, hydraulic
fracture is formed. This fracture is then filled with a high conductivity, proppant which holds the
fracture open (maintains a high conductivity path to the wellbore) after the treatment is finished.
Propped hydraulic fracturing is aimed at raising the well productivity by increasing the e ffective
wellbore radius f or w ells c ompleted i n low p ermeability c arbonate or clastic f ormations.
Hydraulic fracturing i s t o improve productivity i n l ow-permeability f ormations, or to pe netrate
near-wellbore damage or for sand control in higher permeability formations.
Hydraulic fracturing is a mechanical process hence it is only necessary to know that formation
damage is present when designing such a treatment. When a well is hydraulically fractured, most
pre-treatment skin e ffects such a s f ormation da mage, perforation skins a nd s kins d ue t o
completion and partial penetrations are bypassed and have no effect on the post-treatment w ell
performance. Phase-and r ate-dependent skins effects a re either eliminated or contributes i n the
calculation of the fracture skin effects. Generally pre-treatment skin effects are not added to post-
fracture skin effects.
Hydraulic fracturing differs from fracture acidizing in that hydraulic fracturing fluids usually are
not c hemically r eactive, a nd a pr oppant i s placed i n the f racture t o keep the f racture open and
provide conductivity.
The Inflow Performance of a Fracture Stimulated well i s controlled by a quantity known as t he
dimensionless fracture conductivity which depends on the fracture permeability conductive
fracture w idth, f ormation permeability and the conductive fracture single wing length. The
fracture c onductivity i s i ncreased by an i ncreased fracture width, a n i ncreased proppant
permeability ( large, more spherical p roppant grains ha ve higher permeability), a nd m inimizing
the permeability damage to the proppant pack from the fracturing fluid.
Propped hy draulic f racture w ell s timulation s hould onl y be c onsidered when the: well i s
connected to adequate produceable reserves; reservoir pressure is h igh enough to maintain flow
7 | An Approach to Stimulation Candidate Selection and Optimization
when producing t hese r eserves ( or i t i s economically ju stifiable to i nstall a rtificial li ft);
production s ystem can pr ocess t he e xtra pr oduction; professional, experienced p ersonnel are
available for t reatment de sign, e xecution a nd supervision t ogether with h igh quality pu mping,
mixing and blending equipment.
1.3.4 Recompletion
For wells with certain t ypes of da mage such a s pa rtially or t otally p lugged p erforations,
insufficient perforation density o r low depth of perforation, it may b e sufficient t o r ecommend
recompletion technique. Hence the idea of recompletion is to increase the perforation density or
to increase the depth of perforations. The overall aim of this method is to increase production by
bypassing t he da mage. R ecompletion i s a lso u sed effectively in reducing water p roduction. I n
this approach t he w ell i s re-perforated at a new hi gher z one w hile t he pe rforations i n t he wa ter
zone are plugged off.
1.4 Gravel Packing
Gravel packing is used in weak formations that have been producing sand or have the tendency
of producing s and. The gr avel m ixed in a ba se f luid is pu mped as sl urry to f ill all p erforation
tunnels and t he s creen/casing a nnulus. Productivity a nd l ife of t he gravel pack depends on
packing t he perforations w ith gr avel. If not pa cked, f ormation f ines c an invade t he tunnels
impairing productivity and also reducing the area open to flow. Re-completions in low pressure
reservoirs w here formation s and ha s be en pr oduced, can accept l arge volumes o f additional
gravel.
1.5 Stimulation Economics and Candidate Selection
The evaluation of the economics of stimulation treatment must consider many factors including:
treatment cost, initial increase in production rate, additional reserve that may be produced before
the well reaches i ts economic l imit, rate of pr oduction d ecline b efore and a fter s timulation, and
reservoir and mechanical problems that could cause the treatment to be unsuccessful.
Selection of the optimum size of a stimulation treatment is based primarily on economics. The
most c ommonly used m easure of e conomic e ffectiveness is t he n et present v alue (NPV). The
8 | An Approach to Stimulation Candidate Selection and Optimization
NPV is the difference between the present value of all receipts and costs, both current and future,
generated a s a r esult of t he stimulation treatment. Future r eceipts and costs a re converted i nto
present va lue u sing a discount rate and taking i nto a ccount the year in which t hey will a ppear.
Another measure of t he economic e ffectiveness i s t he payout period (PO); t hat is, t he t ime i t
takes for the cumulative present value of the net well revenue to equal the treatment costs. Other
indicators i nclude i nternal rate of return (IRR), profit-to-investment ratio (PIR) and gr owth rate
of return (GRR). The NPV (as well as other indicators) is sensitive to the discount rate and to the
predicted future hydrocarbon pr ices. A s with a lmost a ny other e ngineering a ctivities, costs
increase almost linearly with the size of the stimulation tr eatment but (after a certain point) the
revenues increase only marginally or may even decrease. This suggests that there is an optimum
size of t he t reatment t hat will maximize t he N PV. Hence it i s i mportant to select stimulation
candidate wells that have potentials for maximum benefit.
Candidate Selection (Recognition) is the process of identifying and selecting wells for treatment
which have the capacity for higher production and better economic return. Hence in stimulation
candidate w ell s election, t he w ell s timulation treatment yielding the hi ghest di scounted rate o f
return is the treatment which, in principle, should be carried out first.
1.6 Objective and Procedure of the Study
The goal o f t his r esearch i s to present a model for i dentifying s timulation candidates,
recommending stimulation treatment option and optimizing the stimulation process selected. The
model i s a lso used to rank stimulation candidates ba sed on economics. Hence this research will
attempt to answer the question: βgiven the need to stimulate several wells in a field, how do we
rank the wells ba sed on s timulation benefit and what stimulation approach to use in or der to get
the highest economic returns?β To answer these questions, a merit function is developed based
on production decline curve analysis and economic discounting concepts. In combination with a
good stimulation treatment module, the model can be used for ranking stimulation candidates.
The research procedure begins i n chapter one with an introduction to the concept o f skin factor
and w ell s timulation methods. S everal lit eratures o n f ormation da mage a nd s timulation models
9 | An Approach to Stimulation Candidate Selection and Optimization
are r eviewed in chapter t wo. Chapter t hree c ontains a w ell s creening m odule, design o f s ome
selected stimulation modules and an optimization model which consists of an objective function
with constraint. The optimization model combines the concept of production decline curves with
economic d iscounting. The m odel de veloped i n chapter three is va lidated in chapter f our using
actual field data from the Niger Delta.
1.7 Limitation of the Study
This research is intended for stimulation candidate selection in the Niger Delta. Matrix acidizing
technique is the main stimulation technique that has been used up to date in the Niger Delta due
to t he g ood permeability of t he N iger D elta formation. Hence only matrix a cidizing t echnique,
recompletion and gravel packing are considered in the methodology presented in chapter three of
this research. Acid fracturing and hydraulic fracturing are not considered.
10 | An Approach to Stimulation Candidate Selection and Optimization
Chapter Two
Literature Review
In or der t o properly select s timulation candidate w ells, i t i s n ecessary t o first ha ve a n i n-depth
understanding of t he c oncepts of f ormation d amage and w ell s timulation. A lot of researches
conducted on formation da mage and well s timulation methods can be found in literatures. We ll
stimulation i s c onsidered a m ajor ke y t o proper r eservoir m anagement, he nce several a uthors
made valid contributions.
2.1 Review of Formation Damage Mechanism
2.1.1 Definition Civan1 defined formation d amage a s a g eneric t erminology r eferring t o t he i mpairment o f t he
permeability of petroleum bearing formations by various adverse processes. It is an undesirable
operational a nd e conomic problem t hat c an o ccur du ring t he va rious p hases of oi l a nd ga s
recovery f rom s ubsurface r eservoirs including d rilling, production, hydraulic f racturing, and
workover operations. Bennion2 viewed formation damage as any process that causes a reduction
in t he natural inherent pr oductivity of an o il and ga s pr oducing formation, or a reduction i n the
injectivity o f a water or gas in jection well. Bennion also pointed out that the formation damage
issue is often overlooked because of ignorance and apathy. In many cases, the operators are not
seriously c oncerned w ith f ormation d amage because of t he b elief t hat i t can be circumvented
later o n, simply b y a cidizing a nd/or h ydraulic fracturing. B ut Porter3 and M ungan4
argued t hat
because formation damage is usually nonreversible, it is better to avoid formation damage rather
than deal with it later on using expensive and complicated procedures.
2.1.2 Causes of Formation Damage Amaefule et al.5
classified the various factors causing formation damage as following:
β’ Invasion of f oreign f luids, s uch as w ater and c hemicals used for i mproved
recovery, drilling mud invasion, and workover fluids;
11 | An Approach to Stimulation Candidate Selection and Optimization
β’ Invasion o f foreign particles and mobilization of indigenous particles, such a s
sand, mud fines, bacteria, and debris;
β’ Operation conditions s uch a s w ell flow r ates a nd wellbore pr essures a nd
temperatures;
β’ Properties of the formation fluids and porous matrix.
Amaefule et al.5
further grouped these factors in two categories:
β’ Alteration of formation properties by various processes, including permeability reduction,
wettability a lteration, lithology c hange, r elease of mineral p articles, pr ecipitation of
reaction-by products, and organic and inorganic scales formation
β’ Alteration of fluid properties by various processes, including viscosity alteration by
emulsion block and effective mobility change.
2.1.3 Quantifying Formation Damage Terms used in quantifying formation damage as presented by various authors include:
2.1.3.1
Van Everdingen and Hurst
Skin Factor 6 defined skin effect or skin factor as a mathematically dimensionless
number which r eflects t he altered permeability d ue to damage ππππ , at a d istance rd, causing a
steady-state pressure difference. A relationship between the skin effect, s, reduced permeability,
ππππ R and altered zone radius, rd
may be expressed as:
π π = οΏ½ ππππππβ 1οΏ½ ππππ οΏ½ππππ
πππ€π€οΏ½ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.....β¦β¦.2.1
Equation 2.1 is known as Hawkins7
formula. From the equation i t can be deduced that If ππππ <
ππ the well is damaged and π π > 0; conversely, if ππππ > ππ, then π π < 0 and the well is stimulated. For π π = 0,
the near-wellbore permeability is equal to the original reservoir permeability.
Generally, certain well logs may enable calculation of the damaged radius, rd , whereas pressure
transient analysis may provide the skin effect, s, and reservoir permeability, k. Equation 2.1 may
then be used to calculate the value of the altered permeability ππππ .
12 | An Approach to Stimulation Candidate Selection and Optimization
In the absence of production log data, Frick and Economides8
postulated that, an elliptical cone
is a more plausible shape of damage distribution along a horizontal well. They developed a skin
effect expression, analogous to the Hawkins formula:
π π ππππ = οΏ½ ππππππβ 1οΏ½ ππππ οΏ½ 1
πΌπΌππππππ +1οΏ½οΏ½4
3οΏ½ππππππ ,ππππππ
2
πππ€π€2+ ππππππ ,ππππππ
πππ€π€+ 1οΏ½ β¦β¦.β¦.β¦..2.2
where π π ππππ is the equivalent skin effect, πΌπΌππππππ is t he i ndex of a nisotropy a nd ππππππ ,ππππππ is the
horizontal axis of the maximum ellipse, normal to the well trajectory. The maximum penetration
of d amage is n ear t he vertical section of t he well. T hey stated t hat the shape of t he el liptical
cross-section will depend greatly on t he i ndex of a nisotropy. The i ndex of anisotropy πΌπΌππππππ is
defined as:
πΌπΌππππππ = οΏ½πΎπΎπππΎπΎππ
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦..2.3
with πΎπΎππ being the horizontal permeability and πΎπΎππ is the vertical permeability.
Piot and Lietard9 expressed the total skin of a well as a sum of the pseudoskin of flow lines from
the f ormation face to t he pi peline and the true skin du e to f ormation da mage. Economides and
Nolte10
The total skin effect may be written as:
shown t hat t he t otal skin effect i s a composite of a number of factors, most of which
usually cannot be altered by conventional matrix treatments.
π π π‘π‘ = π π ππ+ππ + π π ππ + π π ππ + βπππ π πππππππππ π πππππππ π β¦β¦β¦β¦β¦β¦β¦...............2.4
The last term in the right-hand side of Eq. 2.3 represents an array of pseudoskin factors, such as
phase-dependent a nd r ate-dependent e ffects that c ould b e altered b y hy draulic f racturing
treatments. The other three terms are the common skin factors. The th ird term π π ππ refers to the
damage skin e ffect as defined in equation 2.1. The fi rst term π π ππ+ππ is the skin effect caused by
partial completion and slant. Cinco-Ley et al.11 documented a detailed approach of estimating the
skin f actor du e t o partial completion a nd slant. T he pa rameters needed for t he estimation a re:
completion t hickness, r eservoir thickness, elevation, a nd penetration r atio. An e xample t o
13 | An Approach to Stimulation Candidate Selection and Optimization
illustrate the c alculation o f this s kin e ffect is do cumented b y Economides and Nolte10. The
second term π π ππ represents the skin e ffect resulting from perforations. I t is described by Harris12
and also expounding the concept, Karakas and Tariq13
have shown that:
π π ππ = π π ππ + π π ππ + π π π€π€π€π€ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦β¦β¦.2.5
In e quation 2.5, t he horizontal ps eudoskin factor, π π ππ is a f unction of t he pe rforation ph asing
angle and the wellbore radius. The vertical pseudoskin factor π π ππ and the wellbore skin effect π π π€π€π€π€
are functions of some dimensionless v ariables. A us eful definition of t hese v ariables a nd t he
application of equation 2.5 are also documented by Economides and Nolte14
.
Karakas and Tariq13
also shown that a combination of the damage and p erforation skin e ffects
(π π ππ)ππ can be approximated, for a case where the perforations terminate inside the damaged zone,
by:
(π π ππ)ππ = οΏ½ ππππππβ 1οΏ½ οΏ½ππππ ππππ
πππ€π€+ π π πποΏ½ = (π π ππ)ππ + ππ
πππππ π ππ β¦β¦β¦β¦β¦β¦β¦β¦β¦....2.6
ππππ is the damaged zone radius, and (π π ππ)ππ is the equivalent openhole skin effect (Eq. 2.1)
According to Economides and Nolte10
, it is of extreme importance to quantify the components of
the s kin e ffect in o rder to e valuate t he e ffectiveness of s timulation tr eatments. I n fact, t he
pseudoskin effects can overwhelm the skin effect caused by damage. They explained that it is not
inconceivable to obtain extremely large skin effects after matrix stimulation. This may be
attributed to the usually irreducible configuration skin factors.
2.1.3.2
Yan et al.
Depth of Damage 15
correlated t he depth of invasion of drilling a nd completion f luids by regression
analysis of e xperimental data o btained by means of the s lice cutting of d amaged c ore plugs.
Their empirical correlation is given by:
ππ = 1.612ππ0.521 οΏ½ππππ β οΏ½ οΏ½0.271
ππππππ(0.043πΎπΎ) β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.2.7
14 | An Approach to Stimulation Candidate Selection and Optimization
where ππ is the invasion depth in cm, p is the pressure in MPa, ππππ is the cumulative filtrate loss
in ππππ3, β is porosity in percentage, and πΎπΎ is permeability in ππππ2 (~ Darcy).
McLeod a nd C oulter16
used t he a pproximate s olution t o t he diffusivity e quation for
dimensionless time,π‘π‘π·π· greater than 100,
ππ(ππ, π‘π‘) = ππππ + 162.6ππππππππβ
(ππππππ πππ‘π‘ππππππ ππ2 β 3.23) β¦β¦......β¦β¦β¦β¦β¦.β¦β¦β¦2.8
to obtain an expression that can be used to estimate the damaged radius, ππππ ,
ππππ = οΏ½ πππ‘π‘ππ1690ππππππ
οΏ½1
2οΏ½ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦2.9
In equation 2.9, π‘π‘ππ is the t ime at which the two straight l ines representing the damage zone and
undamaged formation intersect on a plot of ππ π£π£π π log π‘π‘.
Appendix B of t he pa per pr esented b y Raymond and Hudson17
also contained a detailed
approach of estimating the radius of the damaged zone.
2.1.3.3
Damage Ratio
Amaefule et al18
π·π·π·π· = οΏ½ππβπππππποΏ½ = 1 β ππππ
ππ β¦.β¦.β¦β¦β¦β¦β¦..β¦β¦β¦β¦β¦β¦β¦β¦2.10
expressed the damage ratio (DR) as a change in production due to the effect of
the damage.
where ππππ and ππ the undamaged and damaged standard flow rates, respectively.
Using Muskat19
equation for the undamaged flowrate:
ππ = 2πππΎπΎβ(ππππβπππ€π€ )
ππππππππ οΏ½ππππ πππ€π€οΏ½ οΏ½ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦..2.11
and, also, Amaefule et al18 equation for the damaged flowrate:
15 | An Approach to Stimulation Candidate Selection and Optimization
ππ = 2πππΎπΎβ(ππππβπππ€π€ )
πππποΏ½πππποΏ½ππππ πππποΏ½ οΏ½+οΏ½ππ πππποΏ½ οΏ½πππποΏ½ππππ πππ€π€οΏ½ οΏ½οΏ½ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦....β¦β¦β¦β¦2.12
Civan20
expressed equation 2.10 in terms of 2.11 and 2.12 as:
π·π·π·π· =οΏ½ππ πππποΏ½ β1οΏ½πππποΏ½ππππ πππ€π€οΏ½ οΏ½
οΏ½ππ πππποΏ½ οΏ½πππποΏ½ππππ πππποΏ½ οΏ½+πππποΏ½ππππ πππποΏ½ οΏ½ β¦β¦β¦β¦β¦β¦.β¦β¦β¦.β¦β¦β¦β¦β¦β¦.β¦β¦β¦β¦..2.13
where ππ and ππ in Equations 2.11 and 2.12 are the fluid viscosity and formation volume factor. ππ
and ππππ are t he u ndamaged a nd damaged effective permeabilities, β is t he thickness of t he
effective pay zone, πππ€π€ and ππππ are the wellbore and reservoir drainage boundary fluid pressures,
πππ€π€ and ππππ are t he wellbore and reservoir drainage r adii, and ππππ is the r adius of t he d amaged
region.
Combining equation 2.1 and 2.13, t he damage ratio can be expressed i n t erms o f the effective
skin factor π π , as:
π·π·π·π· = π π
π π +πππποΏ½ππππ πππ€π€οΏ½ οΏ½ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦..β¦β¦β¦.β¦2.14
π π is as defined in equation 2.1. Equation 2.14 gives the production loss by alteration of formation
properties. Leontaritis21
ππ = ππππππ
= ππππππππ
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦β¦β¦2.15
stated t hat r apid flow o f o il a nd water i n t he near-wellbore r egion
promote mixing a nd e mulsification. T his causes a r eduction in t he hy drocarbon e ffective
mobility Ξ», because emulsion viscosity is several fold greater than oil and water viscosities. The
mobility Ξ» is defined by:
ππ and ππππ are respectively the absolute and relative permeabilities. High viscosity emulsion forms
a stationary block which resists flow. It is usually called βemulsion blockβ. If ππ and ππππ represent
the v iscosities of oil a nd e mulsion, r espectively, a nd a s teady-state and i ncompressible r adial
flow i s considered, t he t heoretical u ndamaged and damaged flow rates a re g iven, r espectively,
by:
ππ = 2πππΎπΎβ(ππππβπππ€π€ )ππππππππ οΏ½ππππ πππ€π€οΏ½ οΏ½
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦β¦...2.16
and,
16 | An Approach to Stimulation Candidate Selection and Optimization
ππππ = 2πππΎπΎβ(ππππβπππ€π€ )ππππππππ οΏ½ππππ πππποΏ½ οΏ½+πππππππππππποΏ½
ππππ πππ€π€οΏ½ οΏ½ β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦.β¦β¦β¦β¦β¦β¦β¦β¦.2.17
where ππππ represents the formation volume factor of the emulsion.
Civan22
π·π·π·π· =οΏ½ππ ππππππππ ππ
β1οΏ½πππποΏ½ππππ πππ€π€οΏ½ οΏ½
οΏ½ππ ππππππππ πποΏ½πππποΏ½ππππ πππ€π€οΏ½ οΏ½+πππποΏ½ππππ πππποΏ½ οΏ½
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦.β¦.2.18
substituted Equations 2.16 and 2.17 into Eq. 2.10 to obtain the following expression for
the damage ratio:
Equation 2.18 gives a means to calculate the production loss by alteration of fluid properties.
The viscous skin effect is also expressed similar to Zhu et al23
as:
π π ππ = οΏ½ππππππππππππ
β 1οΏ½ πππποΏ½ππππ πππ€π€οΏ½ οΏ½ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦2.19
2.1.3.4
Flow efficiency ( FE) i s defined a s the r atio o f t he damaged t o u ndamaged formation flow
(production or injection) indices.
Flow Efficiency
πΉπΉπΉπΉ = πΉπΉπΌπΌπππΉπΉπΌπΌ
= ππβπππ€π€ππ ββπππ π ππβπππ€π€ππ
......β¦β¦β¦β¦β¦β¦β¦..........β¦.β¦β¦2.20
where ππ and πππ€π€ππ denote t he a verage reservoir fluid and flowing well bottom hole pressures,
respectively, and βπππ π is the additional pressure loss by the skin effect.
Mukherjee a nd Economides24
presented the f low ef ficiency o f v ertical w ells f or radial and
incompressible fluid flow at a steady-state condition as:
πΉπΉπΉπΉπ£π£ =ππππ οΏ½ππππ πππ€π€οΏ½ οΏ½
π π +πππποΏ½ππππ πππ€π€οΏ½ οΏ½ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..2.21
Where π π , the effective skin factor is as defined by Hawkins7
in equation 2.1.
2.1.3.5
Civan
Permeability Variation Index 25 presented a n i ndex which can be u sed t o express t he variation i n pe rmeability due t o
near-wellbore damage. This index known as permeability variation (or reduction) index can be
expressed mathematically as:
17 | An Approach to Stimulation Candidate Selection and Optimization
πππππΌπΌ = πΎπΎβπΎπΎπππΎπΎ
= 1β πΎπΎπππΎπΎ
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦2.22
where πΎπΎ and πΎπΎππ denote the formation permeabilities before and after damage, respectively.
2.1.4 Economic Impact of Formation Damage on Reservoir Productivity
Amaefule et al.18
πΉπΉπ·π·$πΏπΏ = οΏ½365 πππππππ π πππππππππ π
οΏ½ οΏ½ππ π€π€π€π€ππππππππ
οΏ½ οΏ½ππ $π€π€π€π€πποΏ½ οΏ½π·π·π·π· π€π€π€π€ππ ππππππππππππππππππππ
π€π€π€π€ππ π‘π‘βπππππππππ‘π‘πππππππποΏ½ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.2.23
presented a model that can estimate the economic impact of formation damage
on r eservoir pr oductivity, ππ in t erms o f t he a nnual r evenue l oss by formation da mage per well
(FD$L) at a given price of oil, p, as:
Li e t al26 and a lso L ee a nd Kasap27
stated t hat b ecause t he d egree o f damage variation in t he
near-wellbore region, i t is more appropriate to express t he total skin, π π used in any of the
equations above as a sum of t he individual skins over consecutive c ylindrical s egments of t he
formation as:
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..2.24
where ππ is the number of cylindrical segments considered.
2.2
Matrix Acidizing Models
The optimal volume of acid for a particular acidizing job may be selected based on a laboratory
acid response curve or an acidizing model28. These models consider both the modification of the
pore structure as it dissolves and the change in acid concentration as a function of both time and
position within the pore system.
29
Dullien30 presented a c omprehensive literature r eview of t he models a nd the methods us ed t o
determine pore-size d istributions i n a porous medium. Scheidegger31 reviewed capillary models
and concluded that to predict quantities that relate to the geometric structure of a porous medium,
such as permeability and capillary pressure, an empirical correlation factor called tortuosity must
be introduced. Scheschter and Gidley32
π π = οΏ½π π ππ
ππ
ππ=1
= οΏ½οΏ½ππππππππ
β 1οΏ½ ππππ οΏ½ππππππππβ1
οΏ½ππ
ππ=1
proposed a capillary model to describe matrix acidizing.
18 | An Approach to Stimulation Candidate Selection and Optimization
In their model pores are assumed to be interconnected so that a fluid can flow through the matrix
under the influence of a p ressure g radient, and as the acid reacts with the matrix the pores
increase in size.
2.2.1 Sandstone Acidizing Models
Very many models of the sandstone acidizing pr ocess have been pr esented ov er t he y ears. The
models o nly differ i n t he d etail in w hich they d escribe the chemical interactions b etween t he
acids and the formation minerals and the extent to which they handle or model complexities such
as multiple reservoir zones, diversion methods, wellbore flow e ffects, and other factors. T he
acidizing m odels c an be di vided i nto equilibrium models a nd kinetic models. The equilibrium
models33-35 assume a ll c hemical r eactions a re a t e quilibrium a nd have been u sed p rimarily t o
study t he t endencies f or precipitation r eactions t o occur in a cidizing. T he ki netic models36-
40
consider the kinetics of the relatively slow reactions occurring in sandstones.
β’
The two-mineral model
The t wo-mineral m odel l umps all m inerals i nto on e of t wo c ategories: f ast reacting and s low
reacting species; a nd i t i s t he most common model i n use today. 36, 41 -42 Schechter43 categorizes
fieldspars, a uthogenic clays, a nd a morphous silica a s fast-reacting, while d etrital c lay p articles
and qu artz gr ains are the pr imary s low-reacting mi nerals. This model a s presented by
Economides a nd N olte44
consists o f material b alances ap plied t o t he H F a cid a nd r eactive
minerals, which for linear flow, such as in core-flood, can be written as:
πΏπΏ(β πΆπΆπππΉπΉ )πΏπΏπ‘π‘
+ ππ πΏπΏπΆπΆπππΉπΉπΏπΏππ
= βοΏ½πππΉπΉβπππΉπΉπΉπΉππ,πΉπΉ + ππππβπππππΉπΉππ,πποΏ½(1β β )πΆπΆπππΉπΉ β¦β¦.β¦β¦β¦β¦β¦β¦β¦β¦.2.25
πΏπΏπΏπΏπ‘π‘
[(1 β β )πππΉπΉ] = βπππππππΉπΉ πππΉπΉβπππΉπΉπ½π½πΉπΉπΉπΉππ ,πΉπΉπΆπΆπππΉπΉπππΉπΉ
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.....2.26
πΏπΏπΏπΏπ‘π‘
[(1 β β )ππππ] = βπππππππΉπΉ ππππβπππππ½π½πππΉπΉππ ,πππΆπΆπππΉπΉππππ
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦β¦β¦β¦2.27
where πΆπΆπππΉπΉ is the concentration of hydrofluoric acid (HF) in solution and πππππππΉπΉ is its molecular
weight, ππ is t he a cid flux, π π is th e d istance, πππΉπΉβ and ππππβ are the s pecific s urface a reas p er unit
19 | An Approach to Stimulation Candidate Selection and Optimization
volume of solids, πππΉπΉ and ππππ are the volume fractions, πΉπΉππ ,πΉπΉ and πΉπΉππ ,ππ are the reaction rate constants
(based on the rate of consumption of HF), πππππΉπΉ and ππππππ are the molecular weights, π½π½πΉπΉ and π½π½ππ
are t he dissolving powers of 100% H F, and πππΉπΉ and ππππ are t he densities of t he fast- and s low-
reacting minerals, respectively, denoted by the subscripts F and S.
When t he equations above are made d imensionless f or a c ore-flood of l ength πΏπΏ with constant
porosity, two dimensionless groups were observed for each mineral: the Damkohler number π·π·ππ
and the acid capacity number π΄π΄ππ. These two groups describe the kinetics and the stoichiometry of the
HF-mineral reactions. The shape of the acid reaction front depends on t he DamkΓΆhler number π·π·ππ. The
acid ca pacity n umber π΄π΄ππ regulates h ow m uch l ive acid reaches t he f ront, in ot her w ords, it
affects the frontal propagation rate directly.
The DamkΓΆhler number is the ratio of the rate of acid consumption to the rate of acid convection,
which for the fast-reacting mineral is:
π·π·ππ(πΉπΉ) =(1ββ 0)πππΉπΉ
0πΉπΉππ(πΉπΉ)πππΉπΉ
βπΏπΏ
ππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦.2.28
The acid capacity number is the ratio of the amount of mineral dissolved by the acid occupying a
unit vol ume o f rock por e s pace to the amount o f m ineral present in the u nit vol ume o f rock,
which for the fast-reacting mineral is:
π΄π΄ππ(πΉπΉ) = β 0π½π½πΉπΉπΆπΆπππΉπΉ
ππ πππππππΉπΉ(1ββ 0)πππΉπΉ
0πππΉπΉ β¦.β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...β¦β¦β¦β¦2.29
In equation 2.29, the acid concentration πΆπΆπππΉπΉππ is in weight fraction (not moles/volume).
The dimensionless form of equations 2.25 through 2.27 can only be solved numerically in their
general f orm, th ough a nalytical s olutions a re p ossible for certain simplified situations.
Schechter43 presented an approximate solution to these equations that is valid for relatively high
DamkΓΆhler number ( π·π·ππ(πΉπΉ) > 10). Numerical m odels providing solutions t o t hese equations,
such as that presented by Taha et al.36
are frequently used for sandstone acidizing design.
20 | An Approach to Stimulation Candidate Selection and Optimization
β’
The two-acid, three-mineral model
Bryant45, and also, da Motta et al.46
shown that at elevated temperatures the sandstone acidizing
process i s not well described by t he two-mineral m odel. These studies suggest that the r eaction
of fluosilicic acid with aluminosilicate (fast-reacting) minerals may be quite significant. Thus, an
additional acid and mineral must be considered to accommodate the following reaction, which is
added to the two-mineral model:
H2SiF6 + fast-reacting mineral π£π£ Si(OH)4
+ Al fluorides β¦β¦β¦β¦...2.30
The practical implications of the s ignificance o f this reaction a re th at le ss H F is required to
consume the fast-reacting minerals with a given volume of acid because the fluosilicic acid also
reacts with t hese m inerals a nd t he r eaction product of silica gel ( Si(OH)4) p recipitates. T his
reaction allows live HF to penetrate farther into the formation; however, there is an added risk of
a possibly damaging precipitate forming. An example presented by Sumotarto47
shows improved
performance with t he t wo-acid, t hree-mineral model when compared with t he one -acid, two-
mineral model. This is an example of a kinetic model.
β’
Precipitation Models
Though t he t wo-acid, t hree-mineral model c onsiders th e p recipitation o f silica g el i n it s
description of t he a cidizing process, yet o ther numerous r eaction pr oducts t hat may precipitate
were not considered.
Walsh et al.33
described a local equilibrium model, a common type of geochemical model (that
considers a large number of possible r eactions) u sed t o study sandstone a cidizing. This model
assumes that all reactions are in local equilibrium; i.e., all reaction rates are infinitely fast.
Sevougian et al.34 presented a geochemical model that includes kinetics for both dissolution and
precipitation r eactions. T his model shows t hat precipitation damage will be l essen i f either the
21 | An Approach to Stimulation Candidate Selection and Optimization
dissolution or the precipitation reactions are not instantaneous (i.e. i f the reaction rate decreases,
the amount of precipitate formed will also decrease).
β’
Permeability Models
Predicting permeability change as acid dissolves some of the formation minerals and precipitate
is f ormed i s a necessary s tep n eeded to predict the f ormation response to acidizing. The
permeability increases a s t he pores a nd pore t hroats a re enlarged by mineral dissolution. At the
same t ime, small particles ar e r eleased a s c ementing m aterial i s dissolved, a nd some of t hese
particles lodge (perhaps temporarily) in pore throats, reducing the permeability. Any precipitates
formed a lso t end t o d ecrease the permeability. T he formation of carbon d ioxide ( CO2) a s
carbonate mi nerals a re dissolved m ay a lso cause a t emporary r eduction i n t he r elative
permeability t o li quids.48The complex n ature o f the p ermeability response h as m ade its
theoretical pr ediction f or r eal sandstones impractical. For t his r eason empirical correlations
relating the permeability increase to the porosity change during acidizing are used. Guin et al.49
however a chieved s ome s uccess when a more i deal systems su ch a s si ntered disks was
considered. Labrid50
presented the following useful relationship:
ππππππ
= πποΏ½β ππβ οΏ½ππ
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..................2.31
The correlation presented by Lambert51
is:
ππππππ
= ππππππ[45.7(β ππ β β )] β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦2.32
Lund and Fogler52
correlation is:
ππππππ
= ππππππ οΏ½ππ οΏ½ β ππββ ββ ππππππ
οΏ½οΏ½β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦2.33
In Eq. 2.31 through 2.33, ππ and β are the initial permeability and porosity and ππππ and β ππ are the
permeability and porosity after acidizing. ππ and ππ are empirical constants. In Eq. 2.33, ππ and ππ
are reported to be 1 a nd 3 for Fontainbleau sandstone. In Eq. 2 .32, ππ = 7 .5 and ββ ππππππ = 0.08
best fit data f or pha coides s andstone. The b est a pproach i n u sing t hese correlations i s t o select
22 | An Approach to Stimulation Candidate Selection and Optimization
the e mpirical c onstants based o n c ore f lood responses, if such ar e available; and a lso, lacking
data for a particular formation, equation 2.31 will yield the most conservative design.
48
2.2.2 Carbonate Acidizing Models
Mcleod53
shown t hat t he fundamental di stinguishing f eature of a r ock t reatment i s t he H Cl
soluble fraction; and that for formation rocks largely soluble i n HCl, carbonate acidizing u sing
HCl (without H F) is recommended. For rocks with H Cl solubility less than 20%, sandstone
acidizing using mud acid is recommended.
Shaughnessy a nd K unze54, a nd a lso, Schechter43 have shown t hat he c hemistry of c arbonate
acidizing processes is much simpler than that of sandstone acidizing because there is no tendency
of precipitate being formed (the reaction products CO2 and CaCl2 are both quite water soluble).
But the physics i s complex because t he surface r eaction r ates i n carbonates a re very high, so
mass t ransfer o ften l imits the overall r eaction r ate, l eading t o hi ghly n on-uniform d issolution
pattern. Hofefner and Fogler55
have shown that due to the non-uniform dissolution of limestone
by HCl, a few large channels called wormholes are created. This unstable wormholing process is
not completely understood, but the knowledge of the depth of penetration of wormholes and the
physics o f wormhole growth i s n eeded t o predict t he effectiveness o f c arbonate a cidizing
processes.
β’
Schechter and Gidley
Pore Level Model 32
used a model o f pore growth and collision to study the natural tendency
for wormholes to form when r eaction i s mass transfer l imited. I n t his model, t he change i n the
cross-sectional area of a pore is expressed as:
πππ΄π΄πππ‘π‘
= πππ΄π΄1βππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦2.34
where π΄π΄ is the pore cross-sectional area, π‘π‘ is the time, and ππ is a pore growth function that does
depend on t ime. If ππ > 0, s maller pores gr ow faster than l arger p ores a nd wormhole cannot
form; when ππ < 0, larger pores grow faster than smaller pores and wormhole will develop. They
23 | An Approach to Stimulation Candidate Selection and Optimization
showed that if ππ = 1 2β , surface reaction rate controls the overall reaction rate, and if ππ = β1,
diffusion controls the overall reaction rate. This model does not give a complete picture of the
wormholing process because it does not include the effect of fluid loss from the pores.
β’
Mechanistic Models
Hung et al.56
considered fluid loss in their cylindrical model of the wormhole gr owth, and also
took i nto a ccount a number o f factors, i ncluding t he c ontributions of both a cid diffusion a nd
convection resulting from fluid l oss t o t he walls of t he wormhole where t he acid reacts. They
found t hat the w ormhole velocity i ncreases linearly w ith the i njection rate i nto the w ormhole,
implying that t he v olume of a cid needed to pr opagate a wormhole a gi ven distance i s
independent of injection rate. The model also predicts that wormhole velocity will be constantly
decreasing because t he a cid flux t o t he end of t he wormhole i s de creasing a s t he wormhole
length increases ( grows). The w ormhole ve locity is e xpressed in t erms o f the acid ca pacity
number π΄π΄ππ(which had been defined for a fast-reacting mineral in Eq. 2.29) as:
πππΏπΏπππ‘π‘
= οΏ½ππππβ οΏ½ οΏ½πΆπΆππ
πΆπΆπποΏ½ π΄π΄ππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦..2.35
where ππ and πΆπΆ are the flux and a cid concentration ( mass fraction), t he subscript o refers to th e
initial condition, the subscript e refers to conditions evaluated at the end or tip o f the wormhole,
and L is the length of the wormhole.
β’
Network Models
Hofefner and Fogler55
presented n etwork m odels in which the porous medium is approximated
as a collection of i nterconnected capillaries. T o model wormhole b ehavior, t he a cid
concentration i n each capillary is calculated a nd the radii of the capillaries are i ncreased as
dissolution occurs. These models a ppear t o give t he b est r epresentation o f w ormhole b ehavior
over a wide range of conditions, but they are difficult to generalize for treatment design.
24 | An Approach to Stimulation Candidate Selection and Optimization
β’
Stochastic Models
Daccord et al.57
πππΏπΏπππ‘π‘
= πππ΄π΄πππ΄π΄β
οΏ½πππ·π·οΏ½
2 3ββ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦β¦β¦2.36
recognized t he importance of propagating the wormhole to the fullest extent
possible; hence, ba sed o n laboratory experiments they p roposed a m odel of w ormhole
propagation that c onsidered the s tructures o f w ormhole ob served w hen f luid loss-limited
behavior occurs. Daccord et al.βs model for the rate o f wormhole propagation in l inear systems
is:
where a is a constant determined experimentally, D is the molecular diffusion coefficient, A is
the cr oss-sectional a rea o f t he wormhole and ππ is the injection rate. This model considers t he
influence of acid diffusion but does not take into account fluid loss; therefore, this equation does
not indicate a plateau value as the wormhole lengthens. Thus, the equation is only applicable to
short wormholes where fluid loss i s not a factor, and it should not be u sed for t he pr ediction of
wormhole penetration l ength. For a c onstant i njection r ate, t he skin e ffect pr edicted b y t he
Daccord et al.βs model is:
If there is a damaged zone,
π π = β ππππππππππ οΏ½πππ€π€
ππππ+ οΏ½ π€π€π΄π΄ππππ
ππππππππππππβ
π·π·β2 3β οΏ½ππβοΏ½β1 3β
οΏ½1 ππππβ
οΏ½ β ππππ πππππππ€π€
β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦.2.37
If there is no damaged zone or if the wormholes penetrated beyond the damaged region,
π π = βππππ οΏ½1 + οΏ½ π€π€π΄π΄ππππ
πππππ€π€ππππππβ
π·π·β2 3β οΏ½ππβοΏ½β1 3β
οΏ½1 ππππβ
οΏ½ β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦β¦β¦β¦β¦.2.38
where b is a constant, ex perimentally reported t o be 1.5 Γ 10β5 in S I un its, ππππ is th e fractal
dimension equal to about 1.6 and ππ is the cumulative volume of acid injected. Eq. 2.37 and 2.38
do not apply if the injection rate is changing during the treatment because of the dependence of
the wormhole velocity on injection rate in the Daccord et al.βs model.
25 | An Approach to Stimulation Candidate Selection and Optimization
Pichler et al.58
presented a stochastic m odel of wormhole growth b ased on diffusion-limited
kinetics and included pe rmeability anisotropy, permeability h eterogeneity a nd na tural f ractures.
This model predicts the branched wormhole structures found in carbonate acidizing.
β’
Volumetric Model
Economides et al.59
proposed a n empirical volumetric model t o predict t he volume of a cid
required t o pr opagate wormholes a gi ven distance, a ssuming t hat a cid will di ssolve a c ertain
fraction of the r ock penetrated. F or r adial flow, the r adius of wormhole pe netration πππ€π€β is
expressed as:
πππ€π€β = οΏ½πππ€π€2 + π΄π΄ππππππππππ β
β¦β¦.β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦β¦β¦.β¦..β¦β¦2.39
where ππ, the w ormholing e fficiency, is de fined as the fraction of r ock d issolved in the r egion
penetrated by the acid, mathematically expressed as:
ππ = π΄π΄πππππππ€π€π‘π‘ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦2.40
where πππππ€π€π‘π‘ is the number of pore volumes of acid injected at the time of wormhole breakthrough
at the end of the core. The skin effect during injection is expressed as:
If there is a damaged zone,
π π = β ππ2ππππ
ππππ οΏ½οΏ½πππ€π€πππποΏ½
2+ π΄π΄ππππ
ππππππππ2ππβ
οΏ½ β ππππ πππππππ€π€
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦...2.41
If there is no damaged zone or if the wormholes penetrated beyond the damaged region,
π π = β 12ππππ οΏ½1 + π΄π΄ππππ
ππππ ππππ2ππβ
οΏ½ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦2.42
26 | An Approach to Stimulation Candidate Selection and Optimization
β’
Generalized Carbonate Dissolution Model
In or der t o p resented a generalized d escription o f carbonate d issolution process which a ccount
for the various transport and reaction processes that may influence the rate of dissolution, Fredd
and Fogler60
modeled the overall carbonate dissolution mechanism as three sequential processes
of the mass transfer of reactants to the surface, reversible surface reactions and mass transfer of
products a way from t he surface. In t he generalized m odel, t he rate of reactant consumption πππ΄π΄
can then be expressed as:
πππ΄π΄ = ππ οΏ½πΆπΆ β πΆπΆππ1βπ£π£πΎπΎππππ
οΏ½ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦2.43
Where π£π£ is the s toichiometric ratio of reactants consumed to pr oducts pr oduced, πΎπΎππππ is th e
effective equilibrium constant, πΆπΆππ is the initial reactant concentration a nd ππ is t he o verall
dissolution rate constant which depends on the sum of resistances in series, i.e.
ππ =1+ 1
π£π£πΎπΎππππ1πΎπΎ1
+ 1π£π£πΎπΎππ
+ 1π£π£πΎπΎππππ πΎπΎ3
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦....2.44
Kr is the effective surface reaction constant. K1 and K3
are the mass transfer coefficients for the
reactants a nd products, r espectively. Eq. 2 .43 and 2.44 can be u sed t o determine t he r ate of
carbonate dissolution in any flow geometry, provided that an appropriate expression for the rate
of mass transfer is available.
2.3 Acid Fracturing Models
The f ollowing e quations d escribed linear flow of a cid down a fracture, with fluid l eakoff a nd
acid diffusion to the fracture walls.
πππΆπΆπππ‘π‘
+ ππ(πππππΆπΆ)ππππ
+ πποΏ½πππππΆπΆοΏ½ππππ
β πππππποΏ½π·π·ππππππ
πππΆπΆπππποΏ½ = 0 β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦2.45
27 | An Approach to Stimulation Candidate Selection and Optimization
πΆπΆ(ππ,ππ, π‘π‘ = 0) = 0 β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦2.46
πΆπΆ(ππ = 0,ππ, π‘π‘) = πΆπΆππ(π‘π‘) β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.2.47
πΆπΆππππ β πΆπΆπΏπΏπππΏπΏ β π·π·πππππππππΆπΆππππ
= πΉπΉπππΆπΆππ(1β β ) β¦β¦β¦...β¦β¦.2.48
where πΆπΆ is the acid concentration, ππππ is the flux along the fracture, ππππ is the transverse flux due
to fluid loss, π·π·ππππππ is an effective diffusion coefficient, πΆπΆππ is the injected acid concentration, πΉπΉππ is
the r eaction rate co nstant, ππ is t he or der of the r eaction, a nd β is porosity. Ben-Naceur a nd
Economides61, Lo and Dean62, and Settari63 provided complex nu merical solutions t o t he a bove
equations considering c omplications s uch as t he temperature d istribution along the f racture,
viscous fingering of l ow-viscosity acid through a vi scous pad, the effect of the a cid on leak-off
behavior, a nd various fracture geometries. Neerode and Williams64
also pr esented a solution t o
the a bove e quations by a ssuming a steady state, laminar flow of a N ewtonian fluid between
parallel plates with constant fluid loss flux along the fracture. They presented the solution for the
concentration p rofile as a f unction of t he leakoff P eclet n umber. At l ow Peclet n umbers,
diffusion controls a cid propagation, while a t hi gh P eclet numbers, fluid l oss i s t he c ontrolling
factor.
The conductivity (πππππ€π€) of an acid fracture depends on a stochastic process. Nierode and Kruk65
presented the following correlation for the acid fracture conductivity based on the ideal fracture
width π€π€οΏ½ππ ,
πππππ€π€ = πΆπΆ1ππβπΆπΆ2ππππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.2.49
where
πΆπΆ1 = 1.47 Γ 107π€π€ππ2.47 β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦2.50
and for
ππππππππππ < 20,000 psi: πΆπΆ2 = (13.9 β 1.3ππππππππππππππ ) Γ 10β3 β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.2.51
28 | An Approach to Stimulation Candidate Selection and Optimization
ππππππππππ > 20,000 psi: πΆπΆ2 = (13.9β 1.3ππππππππππππππ ) Γ 10β3 β¦β¦β¦β¦β¦β¦β¦β¦β¦....2.52 In Eq. 2. 49 t hrough 2. 52, ππππ is the f racture closure s tress and ππππππππππ is the r ock e mbedment
strength. The average ideal fracture width is defined as:
π€π€οΏ½ππ = ππππ2(1ββ )βππππππ
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦2.53
where ππ is the volumetric dissolving power of the acid, ππ is the total volume of acid injected, βππ
is t he fracture height, a nd ππππ is the f racture h alf-length. The conductivity varies a long t he
fracture; hence Bennet66
defined an average conductivity (πππππ€π€οΏ½οΏ½οΏ½οΏ½οΏ½) that can be used to estimate the
productivity of the acid fracture well.
πππππ€π€οΏ½οΏ½οΏ½οΏ½οΏ½ = 1ππππβ« ππππππππ
0 π€π€ππππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦.2.54
For lower values of Peclet number (< 3), this average overestimate the well productivity, hence
Ben-Naceur and Economides67
presented a harmonic a verage which better a pproximates the
behavior of the fractured well as:
πππππ€π€οΏ½οΏ½οΏ½οΏ½οΏ½ = ππππ
β« ππππ/ππππππππ
0 π€π€ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..2.55
Ben-Naceur and Economides67
also presented a series of performance type curves for a cid-
fractured wells producing at a constant bottomhole flowing pressure of 500 psi.
2.4 Hydraulic Fracturing Models
Hydraulics fractures c an b e c lassified a ccording to one of three m odels: infinite conductivity
model (assuming no pressure loss in the fracture), uniform flux model (assumes a slight pressure
gradient i n t he fracture), a nd finite c onductivity m odel (assumes co nstant a nd l imited
permeability i n the fracture f rom proppant crushing o r p oor pr oppant distribution). Every
hydraulic fracture i s characterized by i ts l ength, conductivity a nd r elated equivalent skin effect.
29 | An Approach to Stimulation Candidate Selection and Optimization
The fracture length, which is the conductive length and not the hydraulic length, is assumed to be
consisting of t wo e qual half-lengths, ππππ in e ach s ide of the w ell. Prats68
provided p ressure
profiles in a fractured r eservoir as a function of t he f racture h alf-length ππππ and t he relative
capacity, a, which he defined as:
ππ = ππππππππ2πππππ€π€
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦β¦β¦β¦2.56
where ππ is the r eservoir p ermeability, ππππ is t he fracture permeability, a nd π€π€ is t he propped
fracture w idth. A rgawal et al.69 and C inco-Ley and Samaniego70
introduced t he dimensionless
fracture conductivity, πΉπΉπΆπΆπ·π· which is defined as:
πΉπΉπΆπΆπ·π· = πππππ€π€ππππππ
.. β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦.2.57
The dimensionless fracture conductivity πΉπΉπΆπΆπ·π· is related to the relative capacity ππ by:
πΉπΉπΆπΆπ·π· = ππ2ππ
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦...2.58
Prats68
πππ€π€π·π·Μ = πποΏ½ΜοΏ½π€ππππ
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...β¦β¦β¦β¦2.59
showed t hat for a s teady-state f low, a fracture affects productivity t hrough t he
dimensionless equivalent (effective) wellbore r adius πππ€π€π·π·Μ which i s related t o the fracture half-
length or penetration ππππ by the dimensionless fracture conductivity πΉπΉπΆπΆπ·π· .
where πποΏ½ΜοΏ½π€ is expressed in terms of the equivalent skin effect π π ππ and the wellbore radius πππ€π€ as:
πποΏ½ΜοΏ½π€ = πππ€π€ππβπ π ππ β¦...................................................................................2.60
For infinite conductivity fractures, Prats68
showed that:
πποΏ½ΜοΏ½π€ = 0.5ππππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦β¦2.61
30 | An Approach to Stimulation Candidate Selection and Optimization
Cinco-Ley et al.71
integrated t his i nto a full description of r eservoir r esponse by i ncluding
transient f low and pseudoradial flow ( where t he pressure-depletion r egion >> ππππ but i s not
affected by e xternal boundaries). Cinco-Ley et al.βs descriptions presented in form of charts can
be used a s powerful reservoir engineering tools to assess p ossible post-fracture p roductivity
benefits from propped fracturing. The productivity index π½π½ in the pseudosteady state flow regime
is expressed as:
π½π½ = 2ππππβππππ
Γ 1
ln 0.472ππππ+0.5ππππ βππππππππππ
+οΏ½0.5πππππΉπΉπΆπΆπ·π· +π π ππ+πππππππππππ€π€οΏ½ β¦....................................2.62
πΉπΉπΆπΆπ·π· = 1.6, is t he optimum value of the dimensionless fracture conductivity for which the
productivity index π½π½ is maximum.
2.5 Literatures on Stimulation Candidate Selection
Several techniques for stimulation candidate selection exist in l iteratures a nd a lso i n practice i n
the i ndustries. Stimulation jobs ha ve w itnessed bot h s uccesses and f ailures, and in s ome c ases
yield less than the expected result. Stimulation failure is usually due to poor candidate selection,
inaccurate treatment de sign or improper f ield pr ocedures72. Nnanna et al.73
cautioned t hat
applying t he b est t reatment d esign a nd field pr ocedures t o t he wrong candidate w ill r esult i n a
failure, while a poor treatment design and good field procedures on the right candidate will also
result in a failure. They a dded that t hough treatment design and field pr ocedures a re fairly well
understood, candidate selection ha s been approached in different ways by various operators and
service companies.
Nitters et al.74
presented a structured a pproach t o stimulation candidate selection and treatment
design. T hey i solated t he r eal skin caused b y da mage ( the p ortion o f t he t otal skin t hat can b e
removed by matrix treatment) from the total skin as follows:
π π ππππππ = π π π‘π‘πππ‘π‘ β οΏ½π π ππππππππ + π π π‘π‘πππππ€π€ + π π πππππ£π£ + π π πππππππ£π£ ππππ + π π ππππππππ π π πππ π ππ οΏ½ β¦β¦β¦..β¦.2.63
31 | An Approach to Stimulation Candidate Selection and Optimization
where π π ππππππ is the skin due to formation damage, π π π‘π‘πππ‘π‘ is the total skin factor (Eq. 2 .1), π π ππππππππ is
the skin resulting from limited perforation height, π π π‘π‘πππππ€π€ is the skin due to turbulent (non-Darcy)
flow, π π πππππ£π£ is t he skin du e t o wellbore deviation, π π πππππππ£π£ππππ is the skin due to gravel packing, and
π π ππππππππ π π πππ π ππ is the skin resulting from a small perforation. Nitters et al then suggested the ranking of
stimulation candidates based on the magnitude of the damage skin factor.
Jones75
presented a nalytical r elationship which i s convenient t o estimate productivity
improvement achievable by skin removal. At equal pressure and also approximating ππππ(ππππ πππ€π€β ) to
7, Jones defined the ratio of rates before and after stimulation (the stimulation ratio, πΉπΉπ π ) as:
πΉπΉπ π = ππ2
ππ1= 7+π π 1
7+π π 2 β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.2.64
where ππ is flow ra te, π π is the skin factor, and t he subscripts 1 and 2 refer t o before and a fter
stimulation.
To properly interpret t he skin and t herefore determine t he appropriate r emedial action r equires
analysis of t he contributing factors. Nnanna and Ajienka76 used the simplified approach for
determining the c ompletion s kin f actor as developed b y A l Qahtani a nd A l Shehri77 in
combination w ith t he non-linear summation r elationship between the pseudoskins and the t otal
skin as demonstrated by Yildiz78 to present a method for stimulation candidate selection. Nnanna
and Ajienka expressed the removable skin factor in the form presented by Lee79
as:
ππππ = βππβ
(ππππ + ππππ+ππ )β ππππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦2.65
where ππππ+ππ is the skin factor due to partial penetration and deviation, ππππ is the total skin
factor as d eternmined f rom a w ell t est. ππππ is t he perforation skin factor. hp is th e perforation
interval thickness and h is the thickness of the oil sand. They used the stabilized inflow equation,
approximating the natural logarithm of t he ratio of drainage radius t o wellbore radius as 8 , a nd
the cu t-off of O nyekonwu80
to define a simplified R -factor which c an b e used for c andidate
selection. The factor is defined as:
32 | An Approach to Stimulation Candidate Selection and Optimization
π·π· = ββππβ ππππ
8+ππ β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦............................................ 2.66
They concluded that if R β₯ 0.6, then the well is a good stimulation candidate in the Niger Delta.
Afolabi et al.81
also presented candidate selection criterion that is based on minimum economic
reserve, productivity Index (PI) of less than 10bpd/psi, flow efficiency of less than 0.5 and the PI
decline rate that is greater than 30%.
Jennings82
presented a methodology for candidate selection ba sed o n w ell c apacity a nd
concluded that well stimulation tr eatments in high-productivity wells a llow better r eservoir
management through sustained productivity and more uniform reservoir depletion throughout the
life of the well, and that good wells make better candidates for matrix stimulation.
Kartoatmodjo et al.83 presented a risk-based c andidate selection a pproach by c onsidering the
range of probability of all the possible outcomes in a stimulation campaign using Monte Carlo
simulation technique. They concluded that decision risk analysis is a valuable tool for candidate
selection. Stimulation c andidate selection c ampaign ba sed on highest expected ga in a nd/or
lowest expected risk has also been reported.
84
The published literatures reviewed did not consider a detailed and efficient optimization process
for s timulation candidate selection, especially i n t he N iger D elta, a nd hence t he n eed f or this
study.
33 | An Approach to Stimulation Candidate Selection and Optimization
Chapter Three
Methodology This methodology is a modification of the modular approach to stimulation decisions presented
by S inson et al.85
The m odels pr esented are de rived f rom i ndustry-wide a ccepted well
stimulation procedures and techniques.
3.1 Well Screening Technique
It i s a ssumed that from well t est da ta, t he well pr oblem could b e diagnosed a nd then matched
with either of acidizing, gravel-packing or re-completion. It is also assumed that all wells can be
acidized, recompleted or gravel-packed successfully if necessary.
Diagnose each well pr oblem. For w ells w ith s kin va lues s howing formation da mage problems,
acidizing i s t he r ecommended t reatment. Wells with m echanical pr oblems s uch a s pa rtially or
totally plugged perforations, i nsufficient perforation density, l ow depth of perforation o r water
production, r e-completion i s r ecommended. I f t he pr oblem i s sand production, t hen gravel
packing i s r ecommended. A s imple screening module flow chart f or t his s ection i s s hown i n
Appendix A.
3.2 Design of Stimulation Treatment Models
The treatment m odels p resented in t his s ection are to b e used f or the s timulation t reatment
design. The choice of which model to use is dependent on the nature of well problem diagnosed
and the result of the screening module.
3.2.1 Matrix Acidizing Design Model
The extent t o which acid will penetrate a rock is dependent on both the rock properties and the
local acid reaction rate. The reaction rate in turn depends on matrix properties and other variables
like temperature, pressure, and composition of the reacting fluids.
34 | An Approach to Stimulation Candidate Selection and Optimization
The m odel p resented here i s a c ombination o f t he a pproaches presented b y S chechter a nd
Gidley32 and E conomides a nd N olte86
. I n t his model, pores ar e a ssumed to be i nterconnected
such that the acid can flow through the matrix under the influence of a pressure gradient.
The Niger Delta formation is c hiefly made up of sandstone. S andstone formations are of ten
treated with a mixture of hydrochloric a cid (HCl) and hydrofluoric a cid ( HF) commonly called
mud acids. T he t reatment is done at l ow injection rate to prevent fracturing. The mud acid,
chosen because of its ability to dissolve the clay found in drilling mud, also will react with most
constituent of naturally occurring sandstones, including silica, feldspar, and calcareous materials.
The following steps are presented for sandstone acidizing design:
β’ Determine the present fracture gradient for the well. If the instantaneous shut-in
pressure value is not available, use the following equation to calculate the fracture
gradient:
ππππ = πΌπΌ + (ππππππ β πΌπΌ) πππππ·π·
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦.3.1
where:
ππππ = fracture gradient, psi/ft
πΌπΌ = 0.33 to 0.50 psi/ft
ππππππ = overburden gradient (1.0 psi/ft for formation depth π·π· less than 10,000ft or 1.2 psi/ft
for depth greater than 10,000ft)
ππππ = reservoir pressure, psi
π·π· = depth of formation, ft
β’ Predict the maximum possible injection rate that does not fracture the formation
using:
ππππ ,ππππππ = 4.917Γ10β6ππβοΏ½οΏ½ππππΓπ·π·οΏ½ββπππ π πππππ π βπππποΏ½
ππππ οΏ½πππποΏ½πππ π πππ€π€οΏ½+π π οΏ½
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.3.2
where:
ππππ,ππππππ = injection rate, bbl/min
35 | An Approach to Stimulation Candidate Selection and Optimization
ππ = effective permeability of the undamaged formation, md
β = net pay thickness, ft
βπππ π πππππ π = safety margin for the pressure, psi (usually 200 to 500 psi)
ππ = viscosity of the injected fluid, cp
πππ π = drainage radius, ft
πππ€π€ = wellbore radius, ft
π π = skin factor, dimensionless
ππ = formation volume factor, bbl/STB (it has a value of 1 for incompressible fluids)
Using Equation 3.2 with zero value for the skin effect π π gives the maximum pump rate during the
treatment.
β’ Estimate the pipe or coil tubing friction pressure gradient
If the injection fluid is Newtonian, and at pumping rates that are less than 9 bbl/min, t he coil
tubing friction pressure can be calculated using:
ππππππππππππππππππ = 0.518πΎπΎ0.79ππ1.79ππ0.207
ππ4.79 β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦.β¦β¦.3.3
where:
ππππππππππππππππππ = frictional pressure, psi/ft
πΎπΎ = specific gravity of the acid (or density of acid in g/cc)
ππ = pump rate, bbl/min
ππ = diameter of pipe, inches
This friction pressure component should be ignored if the pumping rate is greater than 9 bbl/min.
β’ Predict maximum surface pressure.
If p ipe or c oil tubing f riction pressure is co nsidered, the maximum s urface p ressure f or w hich
fluids can be injected without fracturing the formation is calculated using:
36 | An Approach to Stimulation Candidate Selection and Optimization
πππ π ,ππππππ = οΏ½ππππ + ππππππππππππππππππ β ππππππππππ οΏ½π·π· β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...3.4
where:
ππππππππππ = acid hydrostatic gradient, psi/ft
If pipe or coil tubing friction pressure is ignored, then
πππ π ,ππππππ = οΏ½ππππ β ππππππππππ οΏ½π·π· β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦.β¦β¦β¦β¦β¦3.5
β’ Determine the volume of mud acid to use
It i s a ssumed that t he a cid volume r equired is equal to the pore volume of t he damaged zone.
Also, i t i s a ssumed that a cid flows t hrough the porous media with a front t hat i s u niform a nd
stable, then the acid injection is piston-like and the first acid in is the last acid out. The mud acid
volume is estimated using:
ππππ = 7.48[ππβ (πππ π 2 β πππ€π€2)] β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...3.6
where:
ππππ = volume of mud acid, gal/ft
β = porosity, fraction
πππ π = damaged radius (displaced section), ft
In formations where the HCl solubility is moderate to high more HCl is necessary. The following
equation is used to calculate this volume and address the HCl-soluble materials:
πππ»π»π»π»ππ = 7.48 ππ(1ββ )πππ»π»π»π»ππ [πππ π 2βπππ€π€2 ]π½π½
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..3.7
where:
πππ»π»π»π»ππ = volume of HCl required, gal/ft
37 | An Approach to Stimulation Candidate Selection and Optimization
πππ»π»π»π»ππ = fraction of the bulk rock dissolved by HCl
π½π½ = dissolving coefficient, expressed as amount of rock dissolved per gallon of acid
β’ Specify the acid treatment
a. Preflush
Normally, inject 50 gallons of regular acid per foot of perforation interval.
b. Mud Acid
Inject the volume of mud acid calculated from Equation 3.6.
c. Afterflush
In oil wells, inject a volume of diesel oi l or hydrochloric acid equal to the mud acid
volume.
β’ Calculate cost of sandstone matrix acidizing
π»π»π π = πππ π ππ Γ ππβ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..3.8
where:
πππ π ππ = cost of acid used per unit volume, $/gal
β’ Calculate the maximum productivity ratio
In sandstone it is difficult to increase the permeability above the natural state because of reaction
kinetics li mitations, r eaction stoichiometry a nd economics. In th is th esis, the maximum
formation productivity r atio for sandstone acidizing, given some set of reservoir parameters, is
defined by the reciprocal of the flow efficiency, and is approximated from Equation 2.21, using
the semi-steady state definition:
38 | An Approach to Stimulation Candidate Selection and Optimization
πΉπΉππππππ =ππ πποΏ½0.472πππ π
πππ€π€οΏ½+π π
ππ πποΏ½0.472πππ π πππ€π€
οΏ½β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦3.9
where:
πΉπΉππππππ = maximum productivity ratio, dimensionless
π π = skin factor (defined in Equation 2.1), dimensionless
3.2.1.1 Summary
1. Determine the present fracture gradient for the well.
2. Predict the maximum possible injection rate that does not fracture the formation.
3. Estimate the pipe or coil tubing friction pressure gradient.
4. Predict maximum surface pressure.
5. Determine the volume of mud acid to use.
6. Specify the acid treatment.
7. Calculate cost of sandstone matrix acidizing
8. Calculate the maximum productivity ratio.
3.2.2 Recompletion Design Model
The approach considered in this section assumes that the well is already completed. The concept
of recompletion is either t o increase the perforation density or increase the depth of perforation
penetration in order to increase production. The procedure presented below is based on the works
of Strubhar et al.87
β’ Calculate the skin due to perforation geometry
π π ππ = οΏ½ ββππβ 1οΏ½οΏ½ππππ β
πππ€π€β 2οΏ½ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.3.10
where:
β = total formation thickness, ft
βππ = perforated interval length, ft
πππ€π€ = wellbore radius, ft
39 | An Approach to Stimulation Candidate Selection and Optimization
β’ Calculate the perforation damage
π π ππππ = οΏ½ βπΏπΏππππ
οΏ½ οΏ½ππππ πππππππππποΏ½ οΏ½ ππππ
ππππππβ ππππ
πππποΏ½ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦..3.11
where:
ππππππ = ππππ + 0.5 β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦3.12
ππππππππππ
= 10.03
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..3.13
and:
πΏπΏππ = depth of penetration in rock, ft
ππ = number of perforations
ππππππ = radius of compacted zone around the perforations, ft
ππππ = radius of perforation in rock, ft
ππππ = reservoir permeability, md
ππππππ = permeability of compacted zone around perforation in rock, md
β’ Calculate the total skin
π π = π π ππ + π π ππππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦3.14
β’ Calculate cost of recompletion
π»π»π π = πππππ π ππππ Γ πππππ π ππππ β¦β¦β¦β¦β¦.β¦β¦β¦.β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦3.15
where:
πππππ π ππππ = cost per perforation, $
πππππ π ππππ = number of perforations
40 | An Approach to Stimulation Candidate Selection and Optimization
β’ Calculate the maximum productivity ratio
The productivity index for a semi-steady state condition is used to define the productivity ratio,
and hence, defining the productivity ratio as the reciprocal of the flow efficiency, it is expressed
as:
πΉπΉππππππ =ππ πποΏ½0.472πππ π
πππ€π€οΏ½+π π
ππ πποΏ½0.472πππ π πππ€π€
οΏ½ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...β¦β¦β¦β¦β¦β¦β¦3.16
where:
π π = total skin factor calculated from Equation 3.14
3.2.3 Gravel-Pack Design Model
The f ollowing gravel pa ck d esign m odule is modified from Schlumbergerβs gravel pack design
and c alculation m anual. The vo lume of g ravel r equired i s de pendent o n t he formation
permeability, to tal l ength o f t he in terval a nd t he c ondition of th e well ( i.e. whether it i s a n ew
well or an old well). The ideal situation is that all perforation tunnels and screen casing annulus
be filled with gravel. The gravel pack design considered is for re-completion of zones that have
produced sands. The following steps are considered in the design.
β’ Calculate the blank/casing annular volume
100% or less of t his volume m ay be c onsidered as the e xcess g ravel. T his volume e nsures
complete screen/formation coverage by the gravel.
ππ1 = ππ4Γ144
οΏ½πΌπΌπ·π·πππ π ππ2 βπππ·π·ππππππππππ2 οΏ½πΏπΏππππππππππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦3.17
where:
ππ1 = blank/casing annular volume, ft3
πΌπΌπ·π·πππ π ππ = inner diameter of the casing string, inches
πππ·π·ππππππππππ = outer diameter of the blank string, inches
41 | An Approach to Stimulation Candidate Selection and Optimization
πΏπΏππππππππππ = length of the blank string, ft
β’ Calculate the screen/casing annular volume
This volume must be filled up completely with gravel in order to have an efficient pack.
ππ2 = ππ4Γ144
οΏ½πΌπΌπ·π·πππ π ππ2 β πππ·π·π π πππππ π π π ππ2 οΏ½πΏπΏπ π πππππ π π π ππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦3.18
where:
ππ2 = screen/casing annular volume, ft3
πππ·π·π π πππππ π π π ππ = outer diameter of the screen, inches
πΏπΏπ π πππππ π π π ππ = length of the screen, ft
β’ Calculate the volume of gravel to be injected into perforations
This is the volume of gravel required to pack the perforations.
ππ3 = π»π»ππ β βπππ π ππππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..3.19
where:
ππ3 = volume of gravel injected into perforations, ft
π»π»ππ = 0.5 β 1.5 ft
3 3
βπππ π ππππ = vertical height of perforated interval, ft
/ft (for the zones that have produced sands).
β’ Calculate total volume of gravel needed
ππππ = ππ1 β (ππ) + ππ2 + ππ3 β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦3.20
where:
ππππ = total volume of gravel needed, ft
ππ = fraction of the blank/casing annulus needed to be filled (in this thesis, taken to
3
be 60% - 90%)
42 | An Approach to Stimulation Candidate Selection and Optimization
β’ Calculate the weight of gravel needed
ππππ = 7.48ππππ β ππππππππππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦3.21
where:
ππππ = weight of gravel, lbs
ππππππππππ = bulk density of gravel, ppg. (It is the density of the bulk that includes the air
between the grains).
β’ Calculate the carrier fluid volume
ππππππ = ππππ
42βππππππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦..β¦.3.22
where:
ππππππ = volume of the carrier fluid (base fluid), bbls
ππππππ = pounds of proppant (gravel) per gallon added (i.e. pounds of gravel in 1
gallon of the carrier fluid).
β’ Calculate the slurry volume
πππ π ππ = ππππππ β πππππ π ππππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦.β¦β¦3.23
where:
πππππ π ππππ = 1 + πππππππππππππ π
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦.β¦.β¦β¦3.24
πππ π ππ = slurry (gravel + carrier fluid) volume, bbls
πππππππ π = absolute proppant (gravel) density, ppg. (πππππππ π of pacsan β 22.1ppg)
β’ Calculate the slurry density
πππ π ππ = ππππππ+πππ π πππππππ π ππππ
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦.3.25
43 | An Approach to Stimulation Candidate Selection and Optimization
where:
πππ π ππ = slurry density, ppg
ππππππ = density of the carrier fluid (base fluid), ppg
β’ Calculate the gravel-pack skin factor
π π ππππ =96οΏ½ ππ
πππππποΏ½βπΏπΏππ
ππππ2βππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..3.26
where:
π π ππππ = skin factor due to Darcy flow through the gravel-pack, dimensionless
β = net pay thickness, ft
ππππππ = permeability of the gravel-pack gravel, md
ππ = reservoir permeability, md
πΏπΏππ = length of flow path through gravel pack, inches
ππ = number of perforations open
ππππ = diameter of perforation tunnel, inches
β’ Calculate cost of gravel packing
π»π»ππππ = π»π»ππππππ + π»π»ππππππππππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.3.27
where:
π»π»ππππ = cost of gravel packing, $
π»π»ππ = cost of gravel, $/ft
π»π»ππππ = cost of carrier fluid (base fluid), $/bbl
3
44 | An Approach to Stimulation Candidate Selection and Optimization
β’ Calculate the maximum productivity ratio
πΉπΉππππππ =ππ πποΏ½0.472 πππ π
πππ€π€οΏ½+π π ππππ
ππ πποΏ½0.472πππ π πππ€π€
οΏ½ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.3.28
3.3 Development of a Model for Optimizing Stimulation Decisions
At s ome point du ring the pr oducing l ife o f a w ell, t he pr oduction rate may become s o low and
well diagnosis may result in th e need f or well stimulation. Figure 3. 1 s hows the production
profile (production rate vs. t ime) of a well t hat a t some point during its producing l ife was
profitably stimulated. This figure shall serve as the theoretical basis for the model developed in
the following sections.
Consider F igure 3.1. The curve ABC r epresents t he well pre-stimulation decline curve profile.
The w ell initial p roduction r ate is ππππ. At point B, t he w ell is c onsidered f or s timulation. T he
curve DEF i s t he r esulting pos t-stimulation production pr ofile. The p roduction r ate ππππ is the
abandonment rate of the w ell. The t ime ππππ is th e a bandonment t ime o f th e well if i t is n ot
considered f or a s timulation t reatment. The s timulation treatment i s in itiated at ti me ππππ
corresponding to the production rate ππππ . At time πππ π the stimulated well i s open for production.
Thus, t he difference between t he t imes ππππ and πππ π is the d uration of t he stimulation job. The
production l oss du e t o t he duration of the stimulation j ob i s r epresented by the shaded a rea
BCHI. The i nitial production r ate a fter stimulation i s r epresented b y πππ π which c orresponds t o
point D in Figure 3.1. The well is now produced along the curve DEF until the abandonment rate
ππππ is r eached a t t ime πππππ π corresponding t o point F i n t he figure shown. T he a rea DEFH
represents the incremental production due to stimulation treatment.
The model formulated in the following section uses the production profile described above and
an e xponential decline curve analysis with e conomic concept of c ontinuous discounting. The
derivation of the model for several cases of hyperbolic decline is presented in Appendix F.
45 | An Approach to Stimulation Candidate Selection and Optimization
Fig. 3.1 Production decline profile for a stimulated well.
3.3.1 Optimization Model Assumptions
The following assumptions are used in the development of the model.
1. The stimulation will result in improved productivity.
2. The well could be operated profitably if stimulated.
3. The factors t hat control p roduction i n t he past will continue to control pr oduction i n the
future.
4. The well production-rate versus time decline profile is exponential.
5. The well pre-stimulation decline profile will be the same as the post-stimulation profile.
Time, days
Prod
uctio
n R
ate,
stb/
day
ππππ
πππ π
ππππ
ππππ
ππππ πππ π ππππ πππππ π
A
B
C
D
F G H I
E
46 | An Approach to Stimulation Candidate Selection and Optimization
6. The n ominal d ecline r ate c onstant π·π· is the s ame f or bo th t he pr e-stimulation and post-
stimulation profile.
7. The abandonment rate of the well is the same for both the p re-stimulation and post-
stimulation profile.
3.3.2 Stimulation Productivity Ratio
The stimulation productivity ratio πΉπΉ is defined as the ratio of the initial (maximum) production
rate obtained a fter stimulation t o t he p roduction r ate a t which t he well w as considered for
stimulation. From Figure 3.1,
πΉπΉ = πππ π ππππ
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦.β¦β¦β¦.3.29
3.3.3 The Present-Value Discount Factor
The pr esent va lue i nterest factor (πππππΌπΌπΉπΉ) for c ontinuous or daily c ompounding i s u sed i n the
following derivations and it is defined as:
πππππΌπΌπΉπΉ = π π βπΌπΌππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..3.30
where πΌπΌ is t he effective i nterest (discount) rate per day, and ππ is t he t ime period considered in
days.
3.3.4 Defining the Objective Function, πΈπΈπ«π«
The u ltimate g oal o f well stimulation i s to exploit t he r eservoir p rofitably. I n optimizing well
stimulation processes, the measure of effectiveness is the net i ncremental post-stimulation
production subject to the limitations imposed by the system. Therefore an objective function will
be defined to maximize the net post-stimulation production. The objective function is defined as:
47 | An Approach to Stimulation Candidate Selection and Optimization
πππ·π· = πππ·π·π·π· β πππ·π·πππΏπΏ β πππ·π·π»π» β¦............................................................................3.31
where πππ·π·π·π· is the discounted production from stimulation, πππ·π·πππΏπΏ is the discounted production loss
from stimulation, and πππ·π·π»π» is the discounted production equivalent to total stimulation cost.
The exponential decline curve analysis shall be used to derive the mathematical expressions for
each of the components of Equation 3.31. Before proceeding with the derivation, it is necessary
to define some of the variables in Figure 3.1. First, let us shift the time axis such that the time at
the start of the stimulation job ππππ is set to zero. Then let πππ π be the duration of the stimulation job
and πππππ π be the abandonment time of the post-stimulation production profile.
A. Discounted Incremental Post-Stimulation Production, πΈπΈπ«π«π«π«
The discounted incremental production resulting from the stimulation process is derived from the
area enclosed by DEFH in Figure 3.1 by:
πππ·π·π·π· = β« πππ π π π βπ·π·(ππβπππ π )πππππ π πππ π
β (πππππΌπΌπΉπΉ)ππππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦3.32
where π·π· is the exponential decline rate per day. Substituting for πππ π and πππππΌπΌπΉπΉ from Equations
3.29 and 3.30, Equation 3.32 is expressed as:
πππ·π·π·π· = β« πΉπΉπππππ π βπ·π·(ππβπππ π )πππππ π πππ π
π π βπΌπΌππππππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦.β¦3.33
Evaluating the integral on the right hand side of equation 3.33 yields:
πππ·π·π·π· = πΉπΉπππππ π π·π·πππ π
(βπ·π·βπΌπΌ)οΏ½π π (βπ·π·βπΌπΌ)ππ οΏ½πππ π
πππππ π
= πΉπΉπππππ π π·π·πππ π
(βπ·π·βπΌπΌ)οΏ½π π (βπ·π·βπΌπΌ)πππππ π β π π (βπ·π·βπΌπΌ)πππ π οΏ½ β¦.β¦β¦.β¦β¦.β¦β¦β¦β¦β¦β¦β¦β¦β¦3.34
The abandonment production rate ππππ for the post-stimulation production forecast is given by:
48 | An Approach to Stimulation Candidate Selection and Optimization
ππππ = πππ π π π βπ·π·(πππππ π βπππ π ) β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.3.35
Substituting for πππ π from E quation 3. 29 and r earranging Equation 3 .35, the economic l ife
resulting from the stimulation treatment πππππ π can be expressed as:
πππππ π = β 1π·π·ππππ οΏ½ ππππ
πΉπΉπππποΏ½+ πππ π
= 1π·π·πππππΉπΉ β 1
π·π·ππππ οΏ½ππππ
πππποΏ½+ πππ π β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦3.36
Putting Equation 3.36 into 3.34 gives:
πππ·π·π·π· = πΉπΉπππππ π π·π·πππ π
(βπ·π·βπΌπΌ)οΏ½π π οΏ½
(βπ·π·βπΌπΌ)π·π· πππποΏ½πΉπΉππππππππ
οΏ½+(βπ·π·βπΌπΌ)πππ π οΏ½ β π π (βπ·π·βπΌπΌ)πππ π οΏ½ β¦β¦β¦β¦β¦β¦β¦..3.37
Using mathematical i ndices t ransformation of t he f orm: π π ππ+π¦π¦ β π π π¦π¦ = π π π¦π¦(π π ππ β 1) and
rearranging the terms, Equation 3.37 can be written as:
πππ·π·π·π· = πΉπΉπππππ π π·π·πππ π βπ π (βπ·π·βπΌπΌ)πππ π
(βπ·π·βπΌπΌ)οΏ½πΉπΉοΏ½
βπ·π·βπΌπΌπ·π· οΏ½ οΏ½ππππ
πππποΏ½π·π·+πΌπΌπ·π· β 1οΏ½
= πΉπΉβπΌπΌπ·π· πππππ π π·π·πππ π βπ π (βπ·π·βπΌπΌ)πππ π
(βπ·π·βπΌπΌ)οΏ½πππππππποΏ½π·π·+πΌπΌπ·π· β πΉπΉπππππ π π·π·πππ π βπ π (βπ·π·βπΌπΌ)πππ π
(βπ·π·βπΌπΌ) β¦β¦β¦β¦β¦β¦.β¦3.38
B. Discounted Production Loss Due to Stimulation, πΈπΈπ«π«π«π«π«π«
The concept of production loss is similar to the idea of opportunity cost. The production loss is
an essential component of the ob jective f unction that t akes care of the z ero-production time
during stimulation.
49 | An Approach to Stimulation Candidate Selection and Optimization
The discounted production loss during the stimulation pr ocess is derived from the area enclosed
by BCHI in Figure 3.1 by:
πππ·π·πππΏπΏ = β« πππππ π βπ·π·πππππ π
0 β (πππππΌπΌπΉπΉ)ππππ
= β« πππππ π βπ·π·πππππ π
0 π π βπΌπΌππππππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.3.39
Evaluating the integral gives:
πππ·π·πππΏπΏ = ππππ(βπ·π·βπΌπΌ)
οΏ½π π (βπ·π·βπΌπΌ)ππ οΏ½0πππ π
= ππππ(βπ·π·βπΌπΌ)
οΏ½π π (βπ·π·βπΌπΌ)πππ π β 1οΏ½ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦3.40
C. Discounted Stimulation Cost, πΈπΈπ«π«π«π«
The t otal stimulation c ost, which includes site preparation cost, equipment mobilization &
demobilization cost a nd the stimulation tr eatment cost, can be c onverted to i ts e quivalent
discounted production as:
πππ·π·π»π» = π»π»ππβ π π βπΌπΌπππ π β¦.......................................................................................3.41
where π»π» is the total cost of the stimulation treatment in dollars, and ππ is the price (in dollars) per
barrel of oil.
Substituting Equations 3.38, 3.40 and 3.41 into 3.31 gives:
πππ·π· =πΉπΉοΏ½
βπΌπΌπ·π· οΏ½πππππ π π·π·πππ π βπ π (βπ·π·βπΌπΌ)πππ π
(βπ·π·βπΌπΌ)οΏ½πππππππποΏ½οΏ½1βοΏ½βπΌπΌπ·π· οΏ½οΏ½ β πΉπΉπππππ π π·π·πππ π βπ π (βπ·π·βπΌπΌ)πππ π
(βπ·π·βπΌπΌ)
50 | An Approach to Stimulation Candidate Selection and Optimization
β ππππ(βπ·π·βπΌπΌ)
οΏ½π π (βπ·π·βπΌπΌ)πππ π β 1οΏ½ βπ»π»ππβ π π βπΌπΌπππ π β¦β¦β¦β¦...β¦β¦3.42
Let
ππ = βπΌπΌπ·π·
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.3.43
πΌπΌ1 = ππππ(βπ·π·βπΌπΌ)
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.3.44
πΌπΌ2 = π π π·π·πππ π β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦3.45
πΌπΌ3 = π π (βπ·π·βπΌπΌ)πππ π β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...3.46
πΌπΌ4 = π π βπΌπΌπππ π
ππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦β¦β¦β¦.β¦3.47
πΌπΌ5 = οΏ½πππππππποΏ½
(1βππ) β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦......3.48
Therefore the objective function as defined in Equation 3.42 can be expressed in the form:
πππ·π· = πΌπΌ1πΌπΌ2πΌπΌ3πΌπΌ5πΉπΉππ β πΌπΌ1πΌπΌ2πΌπΌ3πΉπΉβ πΌπΌ4π»π»β πΌπΌ1(πΌπΌ3 β 1) β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.3.49
3.4 Optimization Model Constraints
To o btain a p ractical s olution t o t he ob jective function, t he f ormulation m ust i nclude some
constraints. In this study, a budgetary constraint is imposed such that the cost of the stimulation
does not exceed the budget as determined by top management. Also, a break-even condition is
imposed such t hat t he r evenue obtained from the stimulation is a t least equal to the stimulation
cost. The reservoir sets a limit on the maximum cumulative production. Existing facilities, both
51 | An Approach to Stimulation Candidate Selection and Optimization
in the sub-surface and surface, limit production rates that can be obtained from the choice of the
stimulation treatment. These constraints are developed mathematically below.
3.4.1 Constraint 1: Break-even Requirement
The discounted revenue from any stimulation decision should be greater than or at least equal to
the discounted cost of the project. That is:
πππ·π·π·π· β₯ πππ·π·πππΏπΏ + πππ·π·π»π» β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦3.50
Using the definitions of Equations 3.33, 3.39 and 3.41; Equation 3.50 can be expressed as:
β« πΉπΉπππππ π βπ·π·(ππβπππ π )πππππ π πππ π
π π βπΌπΌππππππ β₯ π»π»ππβ π π βπΌπΌπππ π + β« πππππ π βπ·π·ππ
πππ π 0 π π βπΌπΌππππππ β¦β¦β¦β¦β¦..β¦3.51
By examining Equations 3.42 through 3.49, this constraint can be expressed as:
πΌπΌ1πΌπΌ2πΌπΌ3πΌπΌ5πΉπΉππ β πΌπΌ1πΌπΌ2πΌπΌ3πΉπΉ β₯ πΌπΌ4π»π» + πΌπΌ1(πΌπΌ3 β 1) β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦....3.52
In a practical sense, this constraint is satisfied if and only if the value of the objective function
πππ·π· is positive, that is:
πππ·π· β₯ 0 β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦3.53
3.4.2 Constraint 2: Remaining Reserve Limitation
The recovery from the stimulation should not e xceed the r emaining produceable oil i n pl ace
(reserve). Mathematically, this constraint can be expressed as:
β« πΉπΉπππππ π βπ·π·(ππβπππ π )πππππ π πππ π
ππππ β€ π π πππΌπΌππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.3.54
52 | An Approach to Stimulation Candidate Selection and Optimization
where π π πππΌπΌππ is the remaining oil reserve in place during stimulation. Solving Equation 3.54 we
get:
πΉπΉπππππ π π·π·πππ π
βπ·π·[π π βπ·π·πππππ π β π π βπ·π·πππ π ] β€ π π πππΌπΌππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦3.55
Applying the definition of πππππ π given in Equation 3.36 to Equation 3.55 gives:
πΉπΉπππππ π π·π·πππ π
βπ·π·οΏ½π π βπ·π·οΏ½
1π·π·πππποΏ½
πΉπΉππππππππ
οΏ½+πππ π οΏ½ β π π βπ·π·πππ π οΏ½ β€ π π πππΌπΌππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦....3.56
Simplifying,
πΉπΉπππππ·π·β ππππ
π·π·β€ π π πππΌπΌππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦....β¦3.57
Let
β 1 = πππππ·π·
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦.β¦.3.58
β 2 = πππππ·π·
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...β¦3.59
Substituting Equations 3.58 and 3.59 into 3.57, this constraint can be written as:
β 1πΉπΉ β β 2 β€ π π πππΌπΌππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...3.60
3.4.3 Constraint 3: Flow String Capacity
The pr oduction r ate a fter s timulation should n ot e xceed t he maximum d esign capacity o f t he
flow string. In the case of gas wells, this constraint is imposed by the gas pipeline capacity.
53 | An Approach to Stimulation Candidate Selection and Optimization
The exponential decline equation for the post-stimulation production rate ππππ can be expressed as:
ππππ = πππ π π π βπ·π·(ππβπππ π )
= πΉπΉπππππ π βπ·π·(ππβπππ π ) β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..3.61
The m aximum production rate is obtained w hen the well is opened f or production just after
stimulation, i.e. at time ππ = πππ π (see Fig. 3.1). Using this substitution in Equation 3.61, constraint
3 can then be formulated as:
πΉπΉππππ β€ ππππππππ therefore:
πΉπΉ β€ ππππππππππππ
β¦β¦.β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...3.62
where ππππππππ is the maximum design capacity (flow rate) for the well tubing string.
3.4.4 Constraint 4: Budget Allocation
The total cost of s timulation should not exceed the maximum budget allocated by management
for the job. This constraint is formulated mathematically as:
π»π» β€ π»π»ππππππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦β¦β¦β¦β¦β¦β¦3.63
where π»π»ππππππ is the maximum budget allocated by management for stimulation.
3.4.5 Constraint 5: Maximum Formation Productivity Ratio
Given a s et of r eservoir a nd t reatment pa rameters, t he r eservoir c ould o nly be stimulated t o a
certain maximum extent.
54 | An Approach to Stimulation Candidate Selection and Optimization
πΉπΉ β€ πΉπΉππππππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦3.64
where πΉπΉππππππ is the m aximum productivity ratio t hat c an be o btained given the reservoir a nd
treatment pa rameters. I t i s t he productivity ratio obtained from t he d esign module pr esented i n
section 3.2.1 through 3.2.3.
3.4.6 Constraint 6: Productivity Improvement
The stimulation must, at l east, result i n a n i mprovement i n t he productivity r atio and must not
itself cause m ore damage to the formation. This constraint is imposed on the productivity ratio
such that it must not be less than one or negative. It can be formulated mathematically as:
πΉπΉ β₯ 1 β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦3.65
3.5 Stimulation Cost & Productivity Ratio Relationship
From t he design module pr esented i n section 3 .21 t hrough 3.23, i t c ould be observed t hat t he
input design parameters determine the stimulation cost (C) a nd t he maximum pr oductivity ratio
(F). For e xample, f rom the acidizing d esign m odule in s ection 3.21, it c ould b e seen that t he
stimulation c ost depends on t he vo lume of acid pumped, and also the volume of a cid pumped
will d etermine t he extent o f damage r emoval ( productivity r atio). T his discussion shows t hat a
relationship can be formulated between the stimulation cost and the productivity ratio based on
the design module. Hence, in order to use the model presented in section 3 .3 as an optimization
model, i t i s necessary t o develop a stimulation cost versus productivity ratio r elationship ba sed
on the design module presented.
The combined effects of the treatment and reservoir variables are lumped into a stimulation cost
versus productivity equation of the form:
π»π» = πππΉπΉππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦β¦β¦β¦β¦..3.66
55 | An Approach to Stimulation Candidate Selection and Optimization
where ππ and ππ are obtained from the power equation of the trend l ine of a log-log plot of
stimulation cost v ersus p roductivity r atio. It is this equation that in corporates the stimulation
option into the optimization model. Hence we must substitute Equation 3.66 into Equation 3.49
in order to use the model.
3.6 Summary of the Optimization Model
Combining the objective function and the constraints, the optimization model formulated can be
summarized as:
Maximize:
πππ·π· = πΌπΌ1πΌπΌ2πΌπΌ3πΌπΌ5πΉπΉππ βπΌπΌ1πΌπΌ2πΌπΌ3πΉπΉβπΌπΌ4 β πππΉπΉππ β πΌπΌ1(πΌπΌ3 β 1) β¦β¦β¦β¦β¦β¦β¦β¦.3.49
subject to:
1. Break-even Requirement:
πΌπΌ1πΌπΌ2πΌπΌ3πΌπΌ5πΉπΉππ β πΌπΌ1πΌπΌ2πΌπΌ3πΉπΉ β₯ πΌπΌ4 β πππΉπΉππ + πΌπΌ1(πΌπΌ3 β 1) β¦β¦β¦β¦β¦β¦β¦β¦....3.52
2. Remaining Reserve Limitation:
β 1πΉπΉ β β 2 β€ π π πππΌπΌππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...3.60
3. Flow String Capacity:
πΉπΉ β€ ππππππππππππ
β¦β¦.β¦β¦β¦β¦β¦.β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.3.62
4. Budget Allocation:
π»π» β€ π»π»ππππππ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦β¦β¦β¦β¦.β¦β¦3.63
5. Maximum Formation Productivity Ratio:
πΉπΉ β€ πΉπΉπΉπΉ β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦3.64
56 | An Approach to Stimulation Candidate Selection and Optimization
6. Productivity Improvement:
πΉπΉ β₯ 1 β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦3.65
The constants ππ, πΌπΌ1 , πΌπΌ2, πΌπΌ3, πΌπΌ4 and πΌπΌ5 are as defined in Equation 3.43 through 3 .48; β 1 and β 2
are defined in Equation 3.58 and 3.59; ππ and ππ are from Equation 3.66.
It is important to note that the optimization model is a non-linear programming (NLP) problem.
The o bjective function consists o f t wo variables, na mely productivity r atio πΉπΉ and total
stimulation cost π»π». The two variables are related based on the discussion presented in section 3.5.
This research investigated the matrix stimulation cost and performance data presented by Vogt et
al.88
An a ttempt was ma de to obtain a relationship between total stimulation cost π»π» and
productivity ratio πΉπΉ. The data is presented in Table 1 of Appendix B. A regression analysis on the
data using Microsoft Excel shows a trend between pr oductivity r atio a nd t otal stimulation c ost
similar to the form presented in Equation 3.66. Therefore, to use this model, the stimulation cost
versus productivity ratio constants ππ and ππ must be obtained from the design module. The design
and op timization m odel included i n t he a ccompany compact di sk ( CD) of t his t hesis only
requires the input, stimulation design parameters, to generate the constants.
3.7 Solution to the Optimization Model
In this research, the model was solved using the Solver in Microsoft Excel and also Whatβs Best
10.0 LINDO S ystems optimization so ftware. The r esults obtained w ere v erified b y c omparing
the solution with t hat ob tained by us ing Mathematica 7.0 β software developed by Wolfram
Research. The Solver i mplemented i n E xcel ( developed by Frontline Systems) u ses numerical
iterative methods (generalized reduced gr adient m ethod) to s olve e quations a nd t o o ptimize
linear and n onlinear functions with e ither c ontinuous or i nteger variables. But Solver has some
limitations hence t he need t o verify t he r esults. Wolframβs Mathematica 7.0 on t he ot her ha nd
uses several numerical algorithms for constrained no nlinear optimization. T he a lgorithms ar e
categorized into gradient-based methods and direct search methods. Gradient-based methods use
first d erivatives ( gradients) or second d erivatives ( Hessians). Examples a re t he sequential
57 | An Approach to Stimulation Candidate Selection and Optimization
quadratic programming ( SQP) m ethod, t he a ugmented Lagrangian method, a nd t he ( nonlinear)
interior point method. D irect search methods do no t us e derivative i nformation. E xamples a re
Nelder Mead, genetic algorithm and d ifferential e volution, and s imulated a nnealing. The most
general m ethod used b y Mathematica 7.0 for e xact c onstrained op timization problems is based
on the cylindrical algebraic decomposition (CAD) algorithm. Mathematica 7.0 can automatically
choose a lgorithm based on t he nature o f t he problem - a quality which makes i t the preferred
choice for verification of the Solverβs results.
Based on the stimulation modules presented in this chapter, a computer model is developed using
Microsoft Excel. This model is intended for use in the Niger Delta and as such it is assumed that
all w ells can either be acidized, g ravel-packed or r e-completed. T he screening module i s n ot
included in the computer model because it is assumed that prior to the use of this model, the well
must have been matched with one of acidizing, gravel-packing or re-completion. Also, hydraulic
fracturing is not considered in this m odel because t he N iger D elta formation is made up o f
sandstone with good permeability.
58 | An Approach to Stimulation Candidate Selection and Optimization
Chapter Four
Model Validation, Results and Discussion
To s tudy t he b ehavior o f t he o ptimization model t o changes i n i nput parameters, a s ensitivity
analysis was carried out on the acidizing model. Some published data from the Niger Delta were
also used to validate the model. The results obtained are discussed in the following section.
4.1 Sensitivity Analysis
The parameters i n t he following section were va ried a nd the values of t he o ptimal objective
function obtained are plotted against the productivity ratio f or each parameter va lue. The input
data used for the sensitivity analysis are presented in Table 4.1.
4.1.1 Effect of Price of Oil
The p rice o f o il d etermines the amount of revenue derived from the s timulation. T herefore an
increase in the price of oil is accompanied with an increase in the optimal point of the objective
function as shown in Fig. 4.1. The price of oil is purely an economic input to the optimization
model. T he decision t o perform well s timulation depends on t he current price o f oil. H ence t he
higher t he p rice of oil, t he greater t he benefit derived from s timulation. It is important to note
that below a pr oductivity ratio o f about 3 .2, the di scounted pr oduction will not change with the
price of oil, but the overall monetary benefit will reduce when the price of oil falls.
4.1.2 Effect of Discount Rate
Fig. 4.2 illustrates the effect of the interest rate on t he objective f unction. The value of t he
discount rate was varied from 5% to 20%. The discounted production decreases with an increase
in t he discount r ate. The di scount rate can be viewed a s an additional cost of stimulation. The
higher the discount rate, the higher the cost of money and well stimulation, and consequently, the
lower the benefit to be derived from the stimulation job.
4.1.3 Effect of Decline Rate
The effect of the exponential decline rate on the objective function is shown in Fig. 4.3. For this
analysis, the value of the decline constant was varied between 0.032/yr and 0.32/yr. It is noticed
that the smaller the exponential decline rate, the higher the stimulation benefit. The exponential
59 | An Approach to Stimulation Candidate Selection and Optimization
Table 4.1 Input Data for Sensitivity Analysis
Average Reservoir Pressure, Pr 2200 psi Drainage Radius, r 1053 ft e Wellbore Radius, r 0.3 ft w Net Pay Thickness, h 20 ft Depth of Formation 12000 ft Damaged Zone Radius r 7 ft d Undamaged Reservoir Permeability, k 200 md Damaged Zone Permeability, k 20 md d
Porosity 25% Formation Volume Factor 1 bbl/stb Acid Hydrostatic Gradient 0.45 psi/ft Specific Gravity of Acid 1.04 Viscosity of Injected Acid 0.57 cp Pump Rate 2 bbl/min
Safe Margin for Injection Pressure 200 psi Diameter of Coil Tubing 1.75 inches Cost of Acid Per Unit Volume $ 38 per gal
πΌπΌ 0.4 psi/ft Current Production Rate, q 1000 stb/d o Abandonment Rate, q 200 stb/d a
Exponential Decline Rate, D 0.32 per day Duration of Stimulation, t 2 days s Remaining Recoverable Reserve, ROIP 3 MM stb Price Per Barrel of Oil, P 80 $/stb Effective Discount Rate Per Day, I 10% Tubing Maximum Design Flowrate, q 10000 stb/d max
Maximum Stimulation Budget, C 1.2 MM $ max
60 | An Approach to Stimulation Candidate Selection and Optimization
Figure 4.1 Effect of oil price on the objective function
0
1000
2000
3000
4000
5000
6000
0 1 2 3 4 5 6
Dis
coun
ted
Prod
uctio
n, b
bl
Productivity Ratio (F)
Oil Price = $50/bbl
Oil Price = $60/bbl
Oil Price = $70/bbl
Oil Price = $80/bbl
Oil Price = $90/bbl
61 | An Approach to Stimulation Candidate Selection and Optimization
Figure 4.2 Effect of discount rate on the objective function
0
1000
2000
3000
4000
5000
6000
7000
8000
0 1 2 3 4 5 6
Dis
coun
ted
Prod
uctio
n, b
bl
Productivity Ratio (F)
I = 5%
I = 10%
I = 15%
I = 20%
62 | An Approach to Stimulation Candidate Selection and Optimization
Figure 4.3 Effect of decline rate on the objective function
0
5000
10000
15000
20000
25000
30000
0 1 2 3 4 5 6 7 8
Dis
coun
ted
Prod
uctio
n, b
bl
Productivity Ratio (F)
D = 0.032/yr
D = 0.16/yr
D = 0.32/yr
63 | An Approach to Stimulation Candidate Selection and Optimization
decline rate is the parameter that controls the concavity of the objective function. The smaller the
value of the exponential decline constant for a w ell pr oduction pr ofile, the more the benefit we
could get if such well i s considered f or s timulation. In pr actice, w e ha ve no c ontrol ov er the
value of the decline rate constant. H owever, i t gives us a direct i nsight i nto candidate selection
for stimulation decisions. Smaller decline rate is desirable for profitable stimulation decisions.
4.1.4 Effect of Pre-Stimulation Production Rate
The effect of pre-stimulation production rate on t he objective function is i llustrated i n F ig. 4 .4.
The value of the pre-stimulation production rate was varied from 500stb/d to 1500stb/d. A higher
pre-stimulation pr oduction r ate indicates a higher r eservoir energy dr ive. T he main goal of
stimulation i s t o i ncrease pr oduction u sing t he r eservoir e nergy a s t he driving force in moving
the oil from th e r eservoir i nto th e wellbore. If t he r eservoir has litt le or n o e nergy, stimulation
benefit w ill b e small. Th is is clearly illustrated in the figure. Since a higher pre-stimulation
production will give a higher optimal point in the objective function, therefore, from the figure, a
higher p re-stimulation production rate will give a higher optimal stimulation benefit. This
suggests that as production declines during production, there should be an optimal time in which
it is best to initial stimulation jobs. Because of the huge impact of this pre-stimulation production
rate on t he ob jective function, this parameter must be given a major attention in the selection of
stimulation candidates
4.1.5 Effect of Abandonment Rate
The e ffect of t he a bandonment r ate o n t he stimulation d ecision i s shown i n F ig. 4. 5. The
abandonment r ate i s v aried b etween 100stb/d a nd 50 0stb/d. I t i s observed t hat i ncreasing t he
abandonment r ate r esults i n decrease i n t he overall s timulation b enefit. The a bandonment r ate
can be i nterpreted in terms of t he r emaining r ecoverable o il i n t he r eservoir. A hi gher
abandonment r ate m eans a h igher a mount of r ecoverable oil r emaining i n t he r eservoir. B ut a
reduced incremental production is expected because when the abandonment rate is set high, the
incremental p roduction w ill be reduced since w e h ave a l imit t o w hich w e ca n produce.
Consequently, a reduced incremental production will eventually decrease the optimal point of the
objective function. Therefore, the abandonment rate is a major factor that influences the choice
of stimulation candidate selection.
64 | An Approach to Stimulation Candidate Selection and Optimization
Figure 4.4 Effect of pre-stimulation production rate on the objective function
0
2000
4000
6000
8000
10000
12000
0 1 2 3 4 5 6
Dis
coun
ted
Prod
uctio
n, b
bl
Productivity Ratio (F)
qo = 500stb/d
qo = 1000stb/d
qo = 1500stb/d
65 | An Approach to Stimulation Candidate Selection and Optimization
Figure 4.5 Effect of abandonment rate on the objective function
0
1000
2000
3000
4000
5000
6000
0 1 2 3 4 5 6
Dis
coun
ted
Prod
uctio
n, b
bl
Productivity Ratio (F)
qa = 100stb/d
qa = 300stb/day
qa = 500stb/d
66 | An Approach to Stimulation Candidate Selection and Optimization
4.1.6 Effect of Stimulation Time
The stimulation time represents the duration of time the stimulation job is performed. The effect
of the s timulation t ime on the optimal point of the objective function is shown in Fig. 4.6. The
stimulation time is varied between 1 day and 3 days. The optimal point of the objective function
lowers as the stimulation time increases. This means that if more time is spent on the stimulation
job, t he p roduction l oss du ring t he du ration of s timulation will i ncrease, a nd hence, l owers t he
overall benefit d erivable f rom the s timulation j ob. Therefore i t i s b eneficial i f t he duration of
stimulation is reduced to possibly a day in order to get a higher return from stimulation.
4.2 Model Validation: Case Study 1
In this section, the optimization model is applied with the acidizing treatment model to quantify
stimulation benefit derivable from four typical acidizing jobs, and also, to rank the wells for the
stimulation process. Production data from four wells: Well BU 1, Well BU 2, Well BU 3 and Well
BU 4 were u sed t o va lidate t he model. The four w ells completed i n May 2004 are l ocated in
Bestfields, offshore Niger Delta. This high permeability field is located in a water depth of 200m.
The a verage pe ak p roduction recorded i n J anuary 2006 from e ach of t he four wells i s
7000stb/day. Production d ecline s tarts a fter a 3 -year peak pr oduction period. The a vailable
production data f or each o f t he four wells shows t hat t he d ecline pr ofile for e ach w ell i s
exponential. The wells are being considered as potential candidates for acidizing after a well test
confirms the presence of acid removable damage. The field data is presented in Table 4.2. These
data served a s input da ta for t he acidizing design a nd optimization model. Additional da ta used
were t aken from published l iteratures by Ofoh a nd H eikal89, Nnanna et al.73, Nnanna an d
Ajienka76, and Onyekonwu80
The data in Table 4.2 are used to formulate the Bestfield Model, which gives an insight into how
the model can be used to optimize acidizing candidate well selection process in the Niger Delta.
The design and optimization model is available in the included CD.
.
4.2.1 Formulation of the Bestfield Model
In combination with the data provided in Table 4.2, letβs assume that the remaining recoverable
reserve is 500 MM bbls, and the tubing maximum design flow rate for each well is 12500stb/d.
67 | An Approach to Stimulation Candidate Selection and Optimization
Figure 4.6 Effect of stimulation time on the objective function
0
1000
2000
3000
4000
5000
6000
7000
8000
0 2 4 6
Dis
coun
ted
Prod
uctio
n, b
bl
Productivity Ratio (F)
Stim. Time = 1 day
Stim. Time = 2 days
Stim. Time = 3 days
68 | An Approach to Stimulation Candidate Selection and Optimization
Table 4.2 Bestfield Model data
Well BU 1 Well BU 2 Well BU 3 Well BU 4 Average Reservoir Pressure, Pr (psi) 3200 3200 3200 3200 Drainage Radius, re 1000 (ft) 1000 1000 1000 Wellbore Radius, rw 0.3 (ft) 0.3 0.3 0.3 Net Pay Thickness, h (ft) 120 85.6 148.29 68.2 Depth of Formation, Df 9000 (ft) 9060 8950 9000 Damaged Zone Radius, rd 6.8 (ft) 6.4 6 6.5 Undamaged Reservoir Permeability, k (md) 3500 3300 4600 3500 Damaged Zone Permeability, kd 900 (md) 450 510 580
Porosity (%) 25 25 25 25 Formation Volume Factor (bbl/stb) 1.159 1.159 1.159 1.159 Acid Hydrostatic Gradient (psi/ft) 0.45 0.45 0.45 0.45 Specific Gravity of Acid 1.04 1.04 1.04 1.04 Viscosity of Injected Acid (cp) 0.57 0.57 0.57 0.57 Pump Rate (bbl/min) 3.5 3.5 3.5 3.5
Safe Margin for Injection Pressure (psi) 200 200 200 200 Diameter of Coil Tubing (inches) 1.75 1.75 1.75 1.75 Cost of Acid Per Unit Volume ($ per gal) 30 30 30 30 Ξ± (psi/ft) 0.4 0.4 0.4 0.4 Current Production Rate, qo 4000 (stb/d) 4100 3900 5200 Abandonment Rate, qa 250 (stb/d) 250 250 250
Exponential Decline Rate, D (per day) 0.000519 0.000568 0.000547 0.000533 Duration of Stimulation, ts 1 (day) 1 1 1
69 | An Approach to Stimulation Candidate Selection and Optimization
Also letβs assume an average oil price of $80/bbl, effective discount rate of 10% and a maximum
acidizing budget of $1200000 per well.
Using these data, the model is formulated as follows:
Step A:
The Cost versus Productivity-Ratio plots for each of the four wells was generated by the design
model. The a nalysis i n t he following section is for Well BU 3. The analysis for t he Well BU 1,
Well BU 2 and Well BU 4 is similar, h ence o nly t he results w ere discussed. T he C ost versus
Productivity-Ratio plot for the Well BU 3 data is shown in Fig. 4.7.
Enter each well data given into the Acidizing Design and Optimization Model and
generate the Cost versus Productivity-Ratio relationship for each well.
Step B:
For the Well BU 3 input data, the equation is obtained as:
Obtain the relationship between the stimulation cost (C) and productivity ratio (F) in
form of power equation of a trendline through a log-log regression of the data.
πΆπΆ = 87.15πΉπΉ6.499 ............β¦β¦.β¦.β¦β¦β¦.β¦β¦β¦..β¦β¦β¦4.1
The above relationship, as presented in Equation 4.1 was obtained from a regression analysis o f
the simulated data generated by the design model using Microsoft Excel. The design model used
the Well BU 3 input data to account f or c ost as s hown in Equation 4.1 based on t he damaged
radius of the well, which is one of the parameters with greatest influence on the acidizing design.
The acidizing design and optimization model will generate this equation once the data input step
is completed.
Step C:
The constants needed to define the objective function can be calculated easily using the equation
listed a bove. It i s i mportant t o point ou t that w hile u sing the a cidizing a nd d esign model
included in the CD, one does not need to calculate the objective function as presented below. The
program is designed to calculate the objective function, set up the constraints and then awaits the
user t o call a s olver pr ogram f or t he o ptimization step. Hence, S tep C i s only i ncluded for t he
purpose of proper understanding of how the model and its constraints were formulated.
Use Equations 3.49, 3.52, 3.60, 3.62, 3.63, 3.64 & 3.65 to formulate the objective
function and its constraints.
70 | An Approach to Stimulation Candidate Selection and Optimization
The non linear programming model formulated as a maximization problem using the Well BU 3
data is presented below.
ππππππππππππππππ:
πππ·π· = β1.7008 Γ 10β215πΉπΉβ182.82 + 3.5097 Γ 104πΉπΉ
β0.9857πΉπΉ6.499 β 3710.3427 .................................4.2
ππππππππππππππ ππππ:
1. Break-even Constraint:
β1.7008 Γ 10β215πΉπΉβ182.82 + 3.5097 Γ 104πΉπΉ
β₯ 0.9857πΉπΉ6.499 + 3710.3427 β¦β¦.β¦β¦4.3
2. Remaining Reserve Constraint:
7.1298πΉπΉ β 457038.391 β€ 500000000 β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦4.4
3. Flow String Capacity Constraint:
πΉπΉ β€ 3.21 β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...4.5
4. Budget Allocation Constraint:
87.15πΉπΉ6.499 β€ 1000000 β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦β¦β¦4.6
5. Productivity Improvement Constraint:
πΉπΉ β₯ 1 β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦4.7
6. Maximum Formation Productivity Ratio Constraint:
πΉπΉ β€ 4.20 β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦4.8
From the non-linear programming optimization problem presented a bove i t could b e seen t hat,
simply, we seek an optimum value for the productivity ratio which has a lower and upper bound
of 1 and 3.21 respectively. This is true because the limit sets by the facility constraints (Equation
4.5) is more binding than the maximum productivity r atio a ttainable given t he r eservoir a nd
treatment parameters (Equation 4.8).
71 | An Approach to Stimulation Candidate Selection and Optimization
Figure 4.7 (a) and (b): Cost versus Productivity Ratio plots for Well BU 3
(a)
(b)
0.00
500000.00
1000000.00
1500000.00
2000000.00
2500000.00
3000000.00
1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00
Cost
, $
Productivity Ratio, F
Cost-Productivity Ratio Plot(Cartesian Plot)
C = 87.15F 6.4991000.00
10000.00
100000.00
1000000.00
10000000.00
1.00 10.00
Cost
, $
Productivity Ratio, F
Cost-Productivity Ratio Plot(Log-Log Plot)
72 | An Approach to Stimulation Candidate Selection and Optimization
Step D:
In this research, two different solvers, which use different a lgorithms, were used to get an
optimum solution to the model. The solvers are: Frontline Systemβs Microsoft Excel Solver and
Whatβs Best 10.0 LINDO S ystems op timization s oftware. T he solution t o t he model was t hen
verified u sing t he W olfram R esearchβs Mathematica 7.0. The results a re discussed i n t he
following sections.
Find a solution to the non-linear programming model formulated in Step C above.
4.2.2 Solution of the Well BU 3 Model
The Well BU 3 Model was solved u sing t he Microsoft Excel Solver and Whatβs Best 10.0, the
results a re presented i n Appendix C and D . The solutions gave the same r esult for t he optimal
point.
The value of the objective function at optimal point is:
πππ·π· = 106868.12 ππππππππ
The v alue o f t he productivity r atio a t th e o ptimal p oint i s g iven a s t he f inal value o f t he
adjustable cell. The optimal productivity ratio is equal to 3.21.
The o bjective function b ehaviour is viewed in the vicinity of the lower and upper bound of its
constraint. Mathematica 7.0 is us ed t o ge nerate t he pl ot shown i n A ppendix E . Also,
Mathematica 7.0 is us ed t o s olve t he non -linear pr ogramming op timization problem. The
functions NMaximize and Maximize are u sed a nd t he r esult for t he objective function i s
107012.6.7 bbls, while the va lue of the pr oductivity ratio at the optimal point is 3.21. The input
syntax for Mathematica 7.0 is also included in Appendix E.
The slight variation i n t he r esults obtained is du e t o t he fact t hat approximated values a re
inputted i nto Mathematica 7.0 for the optimization. Also, Mathematica 7.0 uses the Cylindrical
Algebraic Decomposition algorithm to s eek i ts optimal point while EXCEL o r L INDO solver
uses the conjugate gradient method. This result is only used to know that the solution obtained is
in the correct range. It is important to note that both results are the same when corrected to three
significant figures. The plot in Appendix E gives a better picture of the optimal point.
73 | An Approach to Stimulation Candidate Selection and Optimization
The results obtained using the Microsoft Excel Solver and Whatβs Best 10.0, is taken as the actual
value of the objective function at the optimal point. Hence, the following discussions are based
on this result.
4.2.3 Discussion of the Well BU 3 Model Result
This optimal v alue of the objective f unction is 106868.12bbls, meaning that i f this well i s
considered for s timulation, given that t he a ssumptions c onsidered i n section 4 .2.1 a re binding,
the benefit derivable is 106868.12bbls of oi l. The l ife of the well is estimated to be 19.6 years.
The payout time on the acidizing cost is also estimated to be 0.94 day.
The A nswer R eport for Well BU 3 in Appendix C, Section 1, shows t hat five o ut of t he six
constraints are not binding. The only binding constraint is the tubing string capacity. Hence, the
optimal solution was f ound within th e lim its o f a ll t he constraints. That is t o s ay th at a ll
constraints are satisfied. No constraint is violated.
The tubing string capacity constraint is binding, meaning that if the tubing f low capacity is
increased, there will be more benefit from this project, but on the other hand, this extremely high
rate w ill kill o ur w ell s ooner than la ter. In g eneral, f or a c onstraint to b e bi nding m eans any
movement to the right would still give a better result to the objective function.
From the sensitivity report it could be seen that the value of the Lagrange Multiplier associated
with th e flow string c apacity c onstraint is 31221.9668. T his gi ves a n i dea of t he fractional
change of the ob jective f unction i f t he flow s tring c apacity c onstraint c hanges b y 1stb/day.
Hence, if the flow string capacity is increased by 1stb/day, the benefit derivable from stimulation
will increase by 31221.9668 bbls. Hence, the v alue o f t he Lagrange Multiplier will help th e
stimulation design engineer to know i f it is necessary to increase the stimulation benefit by
adjusting the constraints. It also gives the estimate of the derivable benefit.
Considering Well BU 1, t he u ltimate s olution obt ained is s hown in A ppendix C, Section 2 . All
constraints are n ot b inding, m eaning that t he optimum po int of the objective f unction w as
attained before any of the constraint bound was reached. Hence any shift to the right or left of the
optimum point will only decrease the stimulation benefit.
74 | An Approach to Stimulation Candidate Selection and Optimization
4.2.4 Application of the Model Result in Candidate Selection
This m odel can be u sed easily t o r ank stimulation candidates based on t he be nefits derivable
from t he stimulation operation and the pa yout t ime. Since the ultimate goal of stimulation is to
increase pr oduction, the well with th e h ighest stimulation benefit and shortest pa yout t ime is
considered f irst for s timulation. Hence wells are ranked first to last in t he decreasing order o f
their s timulation benefits (profits), provided the payout i s a cceptable by management. The w ell
ranked βfirstβ is then selected for stimulation before the one ranked βsecondβ and so on.
The t able below gives a s imple stimulation ca ndidate selection c harts for the four w ells i n
Bestfield, offshore N iger D elta. Table 4. 3 is generated u sing t he a cidizing de sign a nd
optimization model. Each s olution p oint as o btained by t he model i s shown i n t he figures i n
Appendix C, Section 1 & 2. It is important to note that the payout time calculated by the model is
based on the stimulation design cost, site preparation cost (including equipment mobilization and
demobilization cost). The lease operating costs, federal and state taxes should also be considered
in calculating the actual payout time for this project.
Assuming a lease operating cost (LOE) of $4000 per month, the summary table for the Bestfield
Model i s shown i n T able 4. 3. The choice of which well is selected first for s timulation,
considering the stimulation benefit and the payout time will depend on t he operating companyβs
guidelines and criteria for making reservoir management decisions. The payout time for the wells
in the Bestfield Model are fairly close, hence, in th is research, the stimulation benefit i s u sed to
rank the wells. Well BU 3 will be selected first for stimulation before selecting Well BU 4, then
Well BU 2, and finally Well BU 1.
4.2.5 Effect of Price of Oil on Well BU 3 Model Result
The price of oil is varied between $40 and $80 per barrel, and its effect on the objective function
is studied. Figure 4.8 shows the result obtained. From the result it is seen that the higher the price
of o il, the m ore the benefits derivable from the s timulation. However, w ith f acility c onstraint,
binding on the objective function, there is little or no difference in the benefit derivable from the
stimulation jobs. This suggests that if the price of oil increase, more benefits can be derived from
stimulation if the capacity of the production string is adequate. At productivity ratios less than 3,
the discounted production is insensitive to the price of oil.
75 | An Approach to Stimulation Candidate Selection and Optimization
Table 4.3 Besfield Model summary
Well Stimulation Budget ($)
Stimulation Cost ($)
LOE ($/month)
Life of Well
(years)
Discounted Production
(bbls)
Forcast Oil Price
($)
Payout (days)
Stimulation Benefit ($) Ranking
BU1 1200000 1096162.22 4000 18.6 64936.28 80 1.62 5194902.4 4th
BU2 1200000 736662.37 4000 18.9 106275.38 80 0.74 8502030.4 3rd
BU3 1200000 1058634.07 4000 19.6 106868.12 80 1.06 8549449.6 1st
BU4 1200000 626820.32 4000 20.1 106652.90 80 0.63 8532232.0 2nd
76 | An Approach to Stimulation Candidate Selection and Optimization
Figure 4.8 Effect of oil price on Well BU 3 model result
0
20000
40000
60000
80000
100000
120000
140000
160000
0 1 2 3 4 5 6
Dis
coun
ted
Prod
uctio
n, b
bl
Productivity Ratio, F
Oil Price = $40/bbl
Oil Price = $50/bbl
Oil Price = $60/bbl
Oil Price = $70/bbl
Oil Price = $80/bbl
77 | An Approach to Stimulation Candidate Selection and Optimization
4.3 Model Validation: Case Study 2
The data used for this case study was taken from published literatures by Nnanna et al.73, Nnanna
and A jienka76, a nd Onyekonwu80
4.3.1 Formulation of the Well BU 5 Model
. T hese data are used to formulate t he Well BU 5 Model. The
data for Well BU 5 is presented in Table 4.4.
Letβs assume that the remaining recoverable reserve in the drainage area of this well is 2.5 MM
bbls, and the tubing maximum design flow rate is 10000 stb/d. Also letβs assume that the price of
oil is $80/bbl, the effective discount rate is 10% and the maximum budget for the acidizing job is
set at $1000000.
The model is formulated as follows:
Step A:
The Cost versus Productivity-Ratio plot for Well BU 5 is shown in Fig. 4.9.
Enter t he da ta gi ven i nto t he A cidizing Design and Optimization Model and ge nerate
the Cost versus Productivity-Ratio data .
Step B:
For the input data, this equation is obtained as:
Obtain t he relationship be tween the s timulation c ost (C) a nd productivity ratio (F) in
form of power equation of a trendline through a log-log regression of the data.
πΆπΆ = 27.82πΉπΉ6.187 ............β¦β¦.β¦.β¦β¦β¦.β¦β¦β¦..β¦β¦β¦4.9
The above relationship, as presented in Equation 4.1 was obtained from a regression analysis of
the model generated cost data using Microsoft Excel.
78 | An Approach to Stimulation Candidate Selection and Optimization
Table 4.4 Well BU 5 Model data
Average Reservoir Pressure, Pr 3850 psi Drainage Radius, r 1000 ft e Wellbore Radius, r 0.3 ft w Net Pay Thickness, h 68.2 ft Depth of Formation 12100 ft
Damaged Zone Radius r 6 ft d Undamaged Reservoir Permeability, k 1050 md Damaged Zone Permeability, k 100 md d Porosity 25% Formation Volume Factor 1.159 bbl/stb Acid Hydrostatic Gradient 0.45 psi/ft
Specific Gravity of Acid 1.04 Viscosity of Injected Acid 0.57 cp Pump Rate 2 bbl/min Safe Margin for Injection Pressure 200 psi Diameter of Coil Tubing 1.75 inches Cost of Acid Per Unit Volume $ 30 per gal
Ξ± 0.4 psi/ft Current Production Rate, q 500 stb/d o Abandonment Rate, q 150 stb/d a Exponential Decline Rate, D 0.04 per day Duration of Stimulation, t 1 day s Price Per Barrel of Oil, P 80 $/stb
79 | An Approach to Stimulation Candidate Selection and Optimization
Figure 4.9 Cost versus Productivity Ratio plot for Well BU 5.
0.00
200000.00
400000.00
600000.00
800000.00
1000000.00
1200000.00
1400000.00
1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00
Cost
, $
Productivity Ratio, F
80 | An Approach to Stimulation Candidate Selection and Optimization
Step C:
ππππππππππππππππ:
Using Equations 3.49, 3.52, 3.60, 3.62, 3.63, 3.64 & 3.65, the non linear programming
model can be formulated as a maximization problem as presented below.
πππ·π· = β47.79πΉπΉβ2.5 + 3231.562πΉπΉ β 0.4114163πΉπΉ6.187 β 466.5777 ...........4.10
ππππππππππππππ ππππ:
1. Break-even Constraint:
β47.79πΉπΉβ2.5 + 3231.562πΉπΉ β₯ 0.4114163πΉπΉ6.187 + 466.5777 β¦β¦.β¦..4.11
2. Remaining Reserve Constraint:
12500πΉπΉ β 3750 β€ 2500000 β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦4.12
3. Flow String Capacity Constraint:
πΉπΉ β€ 20 β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.4.13
4. Budget Allocation Constraint:
27.82πΉπΉ6.187 β€ 1000000 β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦....4.14
5. Productivity Improvement Constraint:
πΉπΉ β₯ 1 β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦β¦4.15
6. Maximum Formation Productivity Ratio Constraint:
πΉπΉ β€ 4.79 β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..β¦4.16
In t his pr oblem, we s eek a n o ptimum value for t he productivity r atio with a l ower a nd upper
bound of 1 and 4.79 respectively. Equation 4.16 is more binding than Equation 4.13.
4.3.2 Solution of the Well BU 5 Model
The Well BU 5 Model was solved u sing t he Microsoft Excel Solver and Whatβs Best 10.0, t he
results are presented in Appendix C and D, Case Study 2. The solutions gave the same result for
the optimal point. The value of the objective function at optimal productivity ratio is:
πππ·π· = 10846.95 ππππππππ
81 | An Approach to Stimulation Candidate Selection and Optimization
The o ptimal p roductivity r atio i s e qual to 4 .18. The o bjective function be havior i s vi ewed
Mathematica 7.0. The plot generate i s shown i n Appendix E . Also, using Mathematica 7.0 to
solve t he n on-linear pr ogramming optimization pr oblem, the r esult for t he ob jective function i s
10276.7 bbls, while the value of the productivity ratio at the optimal point is 3.97. The solution
using Mathematica 7.0 is also shown in Appendix E.
The slight variation in the results obtained is due to same reasons as discussed in section 4.2.2.
4.3.3 Discussion of the Well BU 5 Model Result
This op timal v alue of t he o bjective f unction i s 10846.95 bbls, meaning th at i f th is well i s
considered f or s timulation, gi ven t hat the assumptions considered in s ection 4.3.1 a re binding,
the benefit derivable from the stimulation equals ($80/bbl Γ 10846.95 bbls ), i.e. $867,756.
The Answer Report for Case Study 2 βSection 1, in Solvers Result section of Appendix C shows
that t he six c onstraints a re n ot bi nding. T his m eans t hat t he o ptimal solution was f ound within
the limits of all the constraints. Hence no constraint is violated.
If we a ssume t hat t he stimulation budget a pproved by management i s $185,000 for t his well,
then t he ou tput o f t his model will be a s shown in Appendix C , Case Study 2 - Section 2 . T he
Answer Report Section shown that the budget allocation constraint is now binding. This means
that if management is willing to allocate more money to this project, there will be more benefit.
From the sensitivity report it could be seen that the value of the Lagrange Multiplier associated
with the budget allocation constraint is 0.0004047. Hence i f the stimulation budget is increased
by $1, the benefit derived from stimulation will increase by 0.0405%. This can be interpreted in
a much better sense a s $1 increase i n stimulation budget will r esult i n an additional production
benefit of 0.0004047 bbls.
The amount in dollars, X, needed to be added to the present budget in order to get an optimum
result can be roughly approximated with the following relationship (valid only for this case):
ππ β 100ππβππ1β100ππ
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦.β¦4.17
82 | An Approach to Stimulation Candidate Selection and Optimization
where M is t he s timulation budget in $ a nd ππ is th e Lagrange M ultiplier a ssociated w ith th e
budget allocation constraint. ππ is obtained from the optimization model sensitivity result.
For the above result the amount needed to be a dded i n order to get an op timum benefit i s
estimated as:
ππ β 100β0.000405β185000(1β100β0.000405)
β $7809
4.3.4 Effect of Price of Oil on Well BU 5 Model Result
The effect of t he price o f oi l on the o bjective function is s tudied by varying the pr ice o f oi l
between $40 and $80 per barrel (Figure 4.11). From the figure it is seen that if productivity ratio
is less t han 3 , t he di scounted production obtained from t he stimulation will b e independent o f
price o f o il. H owever, for pr oductivity r atios gr eater t han 3 , t he pr ice o f o il b ecomes a
determining factor, i.e. the higher the price of oil, the higher the discounted production, hence the
more the benefits derivable from the stimulation. From Figure 4.11, if the price of oil is $40/bbl,
the d iscounted production w ill be 9000bbls, a nd t he t otal benefit would equal ( $40/bbl Γ
9000bbls), i.e. $360,000.
4.3.5 Using Case Study 2 Model Result in Candidate Selection
As di scussed in pr evious s ections, this model can b e used t o r ank s timulation candidates based
on t he benefits derivable from the stimulation operation. The well with th e highest stimulation
benefit is considered first for stimulation. The knowledge of the stimulation benefit to be derived
if a constraint i s a djusted will a lso have gr eat i nfluence on t he c hoice of which candidate i s
selected first. As seen i n section 4 .3.3, ba sed o n t he v alue of t he Lagrange Multiplier,
management m ay be w illing t o allocate m ore m oney to the s timulation job, and this w ill ha ve
great in fluence o n which candidate i s selected first. But th e u ltimate decision will b e based on
the companyβs guidelines and criteria for making reservoir management decisions
83 | An Approach to Stimulation Candidate Selection and Optimization
Figure 4.10 Effect of oil price on Well BU 5 model result
0
2000
4000
6000
8000
10000
12000
0 1 2 3 4 5 6
Dis
coun
ted
Prod
uctio
n, b
bl
Productivity Ratio, F
Oil Price = $40/bbl
Oil Price = $50/bbl
Oil Price = $60/bbl
Oil Price = $70/bbl
Oil Price = $80/bbl
84 |An Approach to Stimulation Candidate Selection and Optimization
Chapter Five
Conclusion and Recommendations
5.1 Conclusion
This research seeks a method to quantify stimulation benefits derivable from different candidate
wells, a nd use the result t o rank economically profitable candidates. To achieve t his, a d esign
module was developed for a cidizing based o n t he works o f Schechter a nd G idley32, a nd
Economides and Nolte86
The o ptimization model derived i n t his r esearch c ombines the o utput from t he stimulation
treatment de sign m odule w ith production de cline c urve analysis and economic c ontinuous
discounting c oncepts. The o bjective f unction is f ormulated in the f orm of a no n-linear
programming pr oblem with some c onstraints. Hence, a constrained optimization p roblem is
presented. The s olution o f the objective function seeks a maximum di scounted production t hat
satisfies the constraints. The c onstraints considered i nclude those imposed b y the r emaining
recoverable oil i n p lace, t ubing string c apacity, maximum formation productivity a nd t he
stimulation budget approved by management.
. O ther design m odules were a lso developed f or gravel packing a nd
recompletion stimulation techniques.
To solve the objective function, a non-linear programming solver in Microsoft Excel and LINDO
Systemsβ What is Best 10 were used to get an optimum solution. In all similar cases considered,
the s ame o ptimum s olutions were ob tained us ing either of t he t wo s olvers. Wolframβs
Mathematica 7.0 was used to verify the solverβs results. They were found to be within acceptable
significant figures. Hence the results are correct and meaningful.
Field data ob tained f rom Bestfield offshore N iger D elta were u sed t o validate t he model. F our
candidate wells were selected for acidizing based on a well test data. The four wells are: Well BU
1, Well BU 2, Well BU 3 and Well BU 4. The application of the model to quantify the stimulation
benefits for each of t he four wells r eveals t hat t he Well BU 3 will h ave the g reatest e conomic
returns. Hence, Well BU 3 was ranked first for the stimulation treatment. In all cases, when using
85 |An Approach to Stimulation Candidate Selection and Optimization
this model, stimulation decisions should be based on the cost of the project, payout time and the
stimulation benefit.
A sensitivity a nalysis on t he r esults of t he m odel was a lso performed. B ased on t he s ensitivity
analysis and the results of the Bestfield Model, it can be concluded that:
1. The proposed methodology and models can be quantitatively used to estimate the benefits
derivable from stimulation options like: acidizing, gravel-packing and recompletion.
2. The mo del and non -linear o ptimization model can be u sed t o r ank c andidate w ells for
selective stimulation. Hence it can be used for stimulation candidate well selection.
3. Below a p roductivity r atio of 3, t he di scounted pr oduction from a cidizing does n ot
depend on the price of oi l. H owever, the ove rall m onetary be nefit derived fro m
stimulation depends on price of oil.
4. The optimization model can a lso be used to study t he effect o f t he t reatment parameters
on the objective function.
5.2 Recommendation
The following recommendations are presented to highlight areas of additional research to
improve the methodology and models developed in this research.
It is recommended that the model be used t o quantify stimulation be nefit derivable from a
stimulation decision o nce a well has been matched t o either of acidizing, gravel packing or
recompletion. F or e ffective use o f the model, i t i s r ecommended t hat t he l ease o perating cost
(LOE) and also, federal and state taxes be considered before ranking the wells for stimulation.
The effect of t he pr e-stimulation production rate on t he optimal point of t he objective function
(Fig. 4 .4) needs further investigations. S uch study will help to k now optimal t ime to initiate
stimulation jobs during the producing life of a well.
It i s al so r ecommended, for f urther s tudy, that a nother a pproach, other t hat t he pr oduction
decline curve analysis, that can be used to quantify the stimulation benefit be investigated. This
86 |An Approach to Stimulation Candidate Selection and Optimization
approach m ay combine inflow performance r elationship and other pressure da ta to obtain a
model that can be used to optimize the stimulation process.
It is further recommended that stimulation optimization correlations be developed, based on the
results of this model, that has a wide range of applications in the Niger Delta and other oil and
gas producing basins.
87 | An Approach to Stimulation Candidate Selection and Optimization
References
1. Civan, F .: Reservoir Formation Damage β Fundamentals, Modeling, Assesment and Mitigation, Gulf Publishing Company, Houston, Texas (2000) p.1
2. Bennion, B ., βFormation D amageβThe I mpairment o f th e I nvisible, b y the
Inevitable and Uncontrollable, R esulting i n an Indeterminate Reduction of t he Unquantifiable!β Journal of Canadian Petroleum Technology, V ol. 3 8, No. 2, February 1999, pp. 11-17.
3. Porter, K. E ., β An O verview of F ormation D amage,β Journal of Petroleum
Technology, Vol. 41, No. 8, 1989, pp. 780-786.
4. Mungan, N ., βDiscussion o f an Overview o f F ormation D amage,β Journal of Petroleum Technology, Vol. 41, No. 11, Nov. 1989, p. 1224
5. Amaefule, J. O., Ajufo, A., Peterson, E., & Durst, K., "Understanding Formation
Damage Processes: An E ssential I ngredient for I mproved Measurement a nd Interpretation o f R elative P ermeability Data
," SPE 16232 paper, SPE Production Operations Symposium, 1987, Oklahoma City, Oklahoma
6. Van Everdingen, A.F. a nd H urst, W .: βThe Application o f the L aplace
Transformation to Flow Problems in Reservoirs,β Trans., AIME (1949) 186, 305β324.
7. Hawkins, M.F.: βA Note on the Skin Effect,β Journal of Petroleum Technology (December 1956) 8, 356β357.
8. Frick, T .P. a nd Economides, M .J.: β Horizontal We ll Damage Characterization and Removal,β SPE Production & Facilities (February 1993) 8, No. 1, pp.15β22.
9. Piot, B . M., & L ietard, O . M ., βNature of Formation Damage in Reservoir
Stimulation,β in Economides, M. J . & Nolte, K. S., (eds.), Reservoir Stimulation, Schlumberger Education Services, Houston, Texas, 1987.
10. Economides, M.J., and Nolte, K.G., Reservoir Simulation, Third Edition. Wiley, N.Y. (hardbound) 2000, Chapter One, p. 1-12
11. Cinco-Ley, H., Ramey, H.J. Jr. and Miller, F.G.: βPseudoskin Factors for Partially Penetrating D irectionally Drilled Wells,β paper SPE 5589, presented a t t he SPE Annual Technical C onference and Exhibition, Dallas, Texas, USA (September 28βOctober 1, 1975a).
88 | An Approach to Stimulation Candidate Selection and Optimization
12. Harris, M.H.: βThe Effect of Perforating on Well Productivity,β paper SPE 1236, Journal of Petroleum Technology (April 1966) 18, pp. 518β528.
13. Karakas, M. a nd Tariq, S .: βSemi-Analytical Productivity Models for Perforated Completions,β paper S PE 18247, p resented a t t he S PE Annual Technical Conference and Exhibition, Houston, Texas, USA (October 2β5, 1988).
14. Economides, M.J., and N olte, K.G., Reservoir Simulation, Third Edition. W iley, N.Y. (hardbound) 2000, Chapter One, p. 1-13
15. Yan, J ., J iang, G ., & Wu, X., βEvaluating o f F ormation D amage C aused by Drilling a nd Completion Fluids i n H orizontal W ells,β Journal of Canadian Petroleum Technology, Vol. 36, No. 5, 1997, pp. 36-42.
16. McLeod, Jr., H. O and Coulter, Jr., A. W.: βThe Stimulation Treatment Pressure Record β An Overlooked Formation Evaluation Toolβ Journal of Petroleum Technology (August, 1969) p. 952
17. Raymond, L . R . a nd H udson, J . L .: β Short-Term Well Te sting to D etermine Wellbore Damageβ, Journal of Petroleum Technology (Oct. 1966) 1363-1370
18. Amaefule, J. O., Kersey, D. G., Norman, D. L., & Shannon, P. M., βAdvances in Formation Da mage Assessment a nd C ontrol Strategies,β CIM Paper No. 88 -39-65, Proceedings of t he 39th Annual Technical Meeting o f P etroleum S ociety of CIM an d C anadian G as P rocessors Association, Ju ne 1 2-16, 1988, C algary, Alberta, 16 p. 65-2
19. Muskat, M., The Flow of Homogeneous Fluids Through Porous Media, McGraw-Hill Book Co., New York, New York, 1937
20. Civan, F .: Reservoir Formation Damage β Fundamentals, Modeling, Assesment and Mitigation, Gulf Publishing Company, Houston, Texas (2000) p.688
21. Leontaritis, K. J ., βAsphaltene Ne ar-Wellbore F ormation D amage Modeling,β SPE 39446 p aper, P roceedings of t he 1998 S PE Formation Damage C ontrol Conference, February 18-19, 1998, Lafayette, Louisiana, pp. 277-288.
22. Civan, F .: Reservoir Formation Damage β Fundamentals, Modeling, Assesment and Mitigation, Gulf Publishing Company, Houston, Texas (2000) p.688
23. Zhu, D ., H ill, A. D ., & Morgenthaler, L . N ., β Assessment o f M atrix Ac idizing Treatment R esponses i n G ulf of M exico W ells,β Proceedings of t he 199 9 S PE Mid-Continent Operations Symposium held in Oklahoma City, Oklahoma, USA, March 28-31, 1999.
89 | An Approach to Stimulation Candidate Selection and Optimization
24. Mukherjee, H ., & Economides, M. J ., βA Parametric Co mparison of Horizontal
and Vertical Well Performance,β SPE Formation Evaluation, June 1991, pp. 209-216.
25. Civan, F .: Reservoir Formation Damage β Fundamentals, Modeling, Assesment and Mitigation, Gulf Publishing Company, Houston, Texas (2000) p.687
26. Li, Y -H., Fambrough, J . D ., & Montgomery, C. T ., β Mathematical M odeling of Secondary P recipitation f rom Sandstone A cidizing,β SPE Journal, December 1998, pp. 393-401
27. Lee, J ., & Kasap, E ., βFluid Sampling from D amaged Formations,β SPE 39817 paper, P roceedings of t he 1998 S PE P ermian B asin O il a nd G as Recovery Conference, March 25-27, 1998, Midland, Texas, pp. 565-570.
28. Economides, M.J., Hill, A.D. and Ehlig-Economides, C.A.: Petroleum Production Systems, Englewood Cliffs, New Jersey, USA, Prentice-Hall (1994) p.350
29. Williams, B . B ., G idley, J . L . a nd S chechter, R . S .: Acidizing Fundamentals, Monograph Series vol. 6, SPE, Dallas, TX, (1979), p. 68
30. Dullien, F. A. L.: βPore Structure Analysis,β paper presented at the ACS State-of-the-Art Summer S ymposium, Div. Ind. Eng. C hem., Washington, D. C. (June, 1969).
31. Scheidegger, A. E: The Physics of Flow Through Porous Media, The Macmillian Co., New York (1960).
32. Schechter, R. S. and Gidley, J . L.: βThe Change i n Pore Size Distribution From Surface Reactions in Porous media,β AIChE Journal (May 1969) 15, pp. 339-350
33. Walsh, M. P ., L ake, L . W ., a nd S chechter, R . S .: β A D escription o f C hemical Precipitation Mechanism and Their Role in Formation Damage during Stimulation b y H ydrofluoric Acid,β Journal of Petroleum Technology, (September 1982) pp. 2097-2112.
34. Sevougian, S . D ., L ake, L . W ., a nd S chechter, R . S .: β A N ew G eochemical Simulator to Design More Effective Sandstone Acidizing Treatment,β SPE paper 24780, 1992.
35. Quinn, M . A., Lake, L arry W . a nd S chechter, R obert S.: "Designing E ffective Sandstone A cidizing Treatments Through Geochemical Modeling," S PE 38173 presented at the 1997 European Formation Damage Conference, The Hague, June 2-3, 1997.
90 | An Approach to Stimulation Candidate Selection and Optimization
36. Taha, R ., H ill, A. D ., a nd S epehmoori, K., β Sandstone Acidizing D esign with a Generalized Model,β SPE Production Engineering, (Feb. 1989), pp. 49-55.
37. Hekim, Y ., Fogler, H . S., a nd M cCune, C . C .: β The Radial M ovement of Permeability F ronts a nd M ultiple R eaction Z ones i n P orous Media,β Journal of Petroleum Technology, pp. 99-107, February 1982
38. da Motta, E . P ., P lavnik, B., and Schechter, R . S .: βOptimizing Sandstone acidizing,β SPE Reservoir Engineering, (February 1992a), pp. 149-153,.
39. Sumotarto, U ., Hill, A . D ., and S epehrnoori, K .: βAn Integrated S andstone Acidizing Fluid Selection a nd Simulation to Optimize Treatment D esign,β SPE paper 30520 presented at t he SPE Annual T echnical C onference & Exhibition held in Dallas, TX, U.S.A., (Oct., 1995) pp. 22-25.
40. Gdanski, R. D ., a nd S huchart, C . E.: β Advanced Sandstone Acidizing D esigns With Improved R adial M odels,β S PE paper 52397 p resented at the 199 7 SPE Annual T echnical Conference a nd E xhibition h eld i n S an A ntonio, T exas,(Oct. 1998), pp. 5-8.
41. Hill, A. D., Lindsay, D . M., Silberberg, I . H., and Schechter, R. S.: βTheoretical
and E xperimental S tudies of Sandstone A cidizing,β Journal of Petroleum Technology, (Feb. 1981), pp. 30-42.
42. Hekim, Y ., Fogler, H . S., a nd McCune, C . C., β The R adial Movement of Permeability Fronts a nd Multiple R eaction Zones i n Porous M edia,β Journal, (Feb. 1982), pp. 99-107.
43. Schechter, R. S., Oil Well Stimulation, Prentice Hall, Englewood Cliffs, NJ, 1992.
44. Economides, M.J., and N olte, K.G., Reservoir Simulation, Third Edition. W iley, N.Y. (hardbound) 2000, Chapter 13, p. 16-13.
45. Bryant, S.L.: βAn I mproved Model of Mud A cid/Sandstone Chemistry,β paper SPE 22855, presented a t the SPE Annual Technical C onference and Exhibition, Dallas, Texas, USA (October 6β9, 1991).
46. da M otta, E.P., Plavnik, B ., S chechter, R .S. a nd H ill, A.D.: β The Relationship Between Reservoir Mineralogy a nd O ptimum Sandstone Acid T reatment,β paper SPE 23802, presented at the SPE International Symposium on Formation Damage Control, Lafayette, Louisiana, USA (February 26β27, 1992b).
47. Sumotarto, U .: βSandstone A cidizing S imulation: Development of an E xpert System,β PhD dissertation, T he University of T exas a t Austin, Au stin, T exas, USA (1995).
91 | An Approach to Stimulation Candidate Selection and Optimization
48. Economides, M.J., Hill, A.D. and Ehlig-Economides, C.A.: Petroleum Production
Systems, Englewood Cliffs, New Jersey, USA, Prentice-Hall (1994) p.359
49. Guin, J.A., Schechter, R.S. and Silberberg, I.H.: βChemically Induced Changes in Porous Media,β Ind. & Eng. Chem. Fund. (February 1971) 10, No.1, pp. 50β54.
50. Labrid, J .C.: βThermodynamic and Kinetic Aspects of Argillaceous S andstone Acidizing,β paper SPE 5165, Journal of Petroleum Technology (April 1975), pp. 117β128.
51. Lambert, M.E.: βA Statistical Study of Reservoir Heterogeneity,β MS thesis, The University of Texas at Austin, Austin, Texas, USA (1981).
52. Lund, K. a nd Fogler, H .S.: β Acidization V. T he Prediction of t he M ovement o f Acid a nd P ermeability F ronts i n S andstone,β Chemical Engineering Science (1976) 31, No. 5, pp. 381β392.
53. McLeod, H .O. J r.: βMatrix Acidizing,β paper SPE 13752, Journal of Petroleum Technology (December 1984) 36, pp. 2055β2069.
54. Shaughnessy, C.M. and Kunze, K.R.: βUnderstanding Sandstone Acidizing Leads to Improved Field Practices,β paper SPE 9388, Journal of Petroleum Technology (July 1981), pp. 1196β1202.
55. Hoefner, M.L. and Fogler, H.S.: βPore Evolution and Channel Formation During Flow and Reaction in Porous Media,β AIChEJ. (Jan., 1988) 34, No.1, pp. 45β54.
56. Hung, K.M., Hill, A.D. and Sepehrnoori, K.: βA Mechanistic Model of Wormhole Growth i n Carbonate Matrix A cidizing a nd Acid Fracturing,β paper SPE 16886, Journal of Petroleum Technology, (January 1989) 41, No. 1, 59β66.
57. Daccord, G ., T ouboul, E . a nd L enormand, R .: β Carbonate A cidizing: T oward a Quantitative M odel of t he Wormholing Phenomenon,β p aper SPE 16887, SPE Production Engineering (February 1989), pp. 63β68.
58. Pichler, T., Frick, T.P., Economides, M.J. and Nittmann, J.: βStochastic Modeling of W ormhole Growth i n C arbonate A cidizing with B iased R andomness,β pa per SPE 25004, p resented a t t he S PE European P etroleum C onference, Cannes, France (November 16β18, 1992).
59. Economides, M.J., Hill, A.D. and Ehlig-Economides, C.A.: Petroleum Production Systems, Englewood Cliffs, New Jersey, USA, Prentice-Hall (1994) p.400
60. Fredd, C.N. and Fogler, H.S.: βInfluence of Transport and Reaction on Wormhole Formation in Porous Media,β AIChE J. (September 1998b), pp. 1933β1949.
92 | An Approach to Stimulation Candidate Selection and Optimization
61. Ben-Naceur, K., E conomides, M . J .: β The E ffectiveness of Acid Fractures an d
Their Production Behaviour,β Spe Paper 18536, 1988.
62. Lo, K. K., a nd Dean, R . H.: β Modeling of Acid Fracturing,β SPE Production Engineering, (May, 1989), pp. 194-200.
63. Settari, A.: βModelling of Acid Fracturing Treatment,β paper SPE 21870, 1991.
64. Nierode, D .E. a nd Williams, B.B.: β Characteristics of Acid R eactions in Limestone Formations,β paper SPE 3101, presented at the SPE Annual Meeting, Houston, Texas, USA (Oct. 4-7, 1971); also in Journal of Petroleum Technology (Dec., 1971), pp. 408-418.
65. Nierode, D.E. a nd Kruk, K.F.: βAn Evaluation of A cid F luid Loss A dditives, Retarded Acids and Acidized Fracture Conductivity,β paper SPE 4549, presented at the SPE Annual Meeting, Las Vegas, Nevada, USA (Sept. 30βOct. 3, 1973).
66. Bennett, C . O .: β Analysis Fractured W ells,β P h. D. t hesis, University of Tulsa, Tulsa, Oklahoma, 1982.
67. Ben-Naceur, K., an d E conomides, M. J .: β Acid Fracture Propagation an d Productionβ i n Reservoir Stimulation, M . J . E conomides a nd K. G . N olte, e ds., Prentice Hall, Englewood Cliffs, NJ, Chap. 18, 1989.
68. Prats, M.: βEffects of Vertical Fractures on Reservoir Behaviour β Incompressible Fluid Case,β paper SPE 1575-G, Journal of Petroleum Technology, (June 1961) 1, No. 1, pp. 105-118; also in Trans., AIME (1961) 222.
69. Agarwal, R . G., Carter, R. D., and Pollock, C. B.: βEvaluation a nd Prediction of Performance o f L ow-Permeability Gas We lls S timulated b y M assive Hydraulic Fracturing,β Journal of Petroleum Technology (March, 1979), Trans. AIME, 267, pp 362-372.
70. Cinco-Ley, H. and Samaniego-V., F.: βTransient Pressure Analysis for Fractured Wells,β paper S PE 7490, Journal of Petroleum Technology (September 1981b) 33, 1749β1766.
71. Cinco-Ley, H ., Samaniego-V., F . an d D ominguez, N .: β Transient P ressure Behavior for a W ell with a F inite-Conductivity V ertical F racture,β paper SPE 6014, p resented a t the SPE Annual T echnical C onference a nd E xhibition, N ew Orleans, L ouisiana, USA ( October 3 β6, 1976) ; a lso i n Journal of Petroleum Technology (August 1978) 18, 253β264.
93 | An Approach to Stimulation Candidate Selection and Optimization
72. Rae, P., and Di Lullo, G.: βAchieving 100 Percent Success in Acid Stimulation of Sandstone R eservoirs,β paper SPE 77808, presented a t the SPE Asia Pacific Oil Conference and Exhibition held in Melbourne, Australia (Oct. 8-10, 2002).
73. Nnanna E., O suagwu M., a nd Okereke O.: β Important C onsiderations i n Matrix Stimulation Candidate Selection in Niger Deltaβ, paper SPE 128604 presented at the 2009 SPE Annual International Conference and Exhibition, Abuja (Aug 3-5).
74. Nitters, G., Roodhart, L., Jongma, H., Yeager, V., Buijse, M., Fulton, D., Dahl, J., and J antz, E. : β Structured Approach t o Advanced C andidate Selection a nd Treatment Design of Stimulation Treatments,β paper SPE 63179, presented at the 2000 SPE Annual Technical Conference a nd Exhibition held in D allas, T exas (Oct. 1-4).
75. Jones, L . G.: c ourse no tes, Formation Damage School, Mobil, Duncan, OK (March 1989).
76. Nnanna, E; Ajienka, J : βCritical Success Factors for W ell Stimulationβ, p aper SPE 98823 p resented a t t he 2005 N igerian Annual C onference & E xhibition, Abuja (August 1-3).
77. Al Qahtani, A., and Al Shehri, D.: βThe Ec-Factor: A Correlation for Optimizing Completion Efficiency,β paper SPE 81490 presented at the 2003 SPE Middle East Oil Show, Bahrain (April 5-8).
78. Yildiz, Y .: β Assessment of T otal Skin Factor i n Perforated W ells,β p aper S PE 82249 presented a t t he 2003 SPE European Formation Damage Conference, The Hague (May 13-14).
79. Lee, W. J.: Well Testing, Society of Petroleum Engineers Textbook (1982).
80. Onyekonwu, M. O.: Principles of Bottomhole Pressure Testing, Laser Publishers Ltd, Port Harcourt. p 91.
81. Afolabi, A, O pusunju A , H enri J , O nyekwere C letus, O nyekwere C hris a nd Davalos J.: βIncreasing Production in a Brown Field with Heavy Crude and Fines Problem by Application of a N ew H F-Acid System: Case Historiesβ paper SPE 112558 p resented a t the 2008 S PE I nternational S ymposium on Formation Control, Lafayette, Louisiana (Feb 13-15).
82. Jennings, A. R .: β Good Wells Make th e B est Candidates for W ell Stimulation,β SPE Production Engineering (Nov., 1991), pp 371-376.
83. Kartoatmodjo, G ., C aretta, F ., F lew, S ., J adid, M .: β Risk-Based C andidate Selection W orkflow I mprove Acid S timulation S uccess R atio i n M ature F ield,β paper SPE 109278 presented at the 2007 SPE Asia Pacific Oil & Gas Conference, Jarkata, Indonesia (Oct. 30 to Nov. 1st).
94 | An Approach to Stimulation Candidate Selection and Optimization
84. Al-Araimi, M ., M ahajan, M.: βSuccessful Revival o f L ong-term C losed-in Ga s
Well by Right Matrix Stimulation Treatment,β paper SPE 96735 presented at the 2005 SPE Annual Technical Conference & Exhibition, Dallas, TX (Oct. 9-12).
85. Sinson, C . M ., O gbe, D . O ., D ehghani, K., β Optimization of W ell Stimulation Strategies in Oil and Gas Fields,β paper SPE 17792, 1988.
86. Economides, M.J., and N olte, K.G., Reservoir Simulation, Third Edition. W iley, N.Y. (hardbound) 2000, Chapter Eighteen, p. 24-25.
87. Strubhar, M. K., Blackburn, J. S., and Lee, J. W.: βProduction Operations Course II: Well Diagnosis,β Lecture Notes for a Video-Tape Course, SPE, Dallas (1972), pp. 525-544.
88. Vogt, T. C., Anderson, M. L.: βOptimizing the Profitability of Matrix Acidizing Treatments,β pa per SPE 4550 (preprint) presented at the SPE AIME 48th
Annual Fall Meeting, Las Vegas, Nev., Sept. 30- Oct. 3, 1973. (Revised manuscript, May 13, 1974).
89. Ofoh, E . P ., and Heikal S.: β Reservoir M anagement Optimization through a Systematic Removal of Formation D amage, a C ase Study of Okpoho F ield in Niger-Deltaβ, paper presented at the 30th Annual SPE I nternational Technical Conference and Exhibition in Abuja, Nigeria, July 31- August 2, 2006.
95 | An Approach to Stimulation Candidate Selection and Optimization
Nomenclature
πΆπΆππππ cost of gravel packing, $
πΆπΆππ cost of gravel, $/ft
πΆπΆππππ cost of carrier fluid (base fluid), $/bbl
3
Cmax
πππ π π π cost of acid used per unit volume, $/gal
Maximum Stimulation Budget, $
πΆπΆππππππππ cost of perforation, $
ππ diameter of pipe, inches
ππππ diameter of perforation tunnel, inches
π·π· decline rate, per day
πΉπΉπ π ππππ maximum productivity ratio, dimensionless
πΉπΉ stimulation productivity ratio, dimensionless
ππππ fracture gradient, psi/ft
qmax
h thickness of the oil sand, ft
tubing maximum design flowrate, stb/day
βππ perforated interval length, ft
I effective discount rate per day, %
πΌπΌππππππ index of anisotropy
πΌπΌπ·π·πππ π ππ inner diameter of the casing string, inches
πΎπΎπ»π» horizontal permeability, md
πΎπΎππ vertical permeability, md
ππππ reservoir permeability, md
ππππππ permeability of compacted zone around perforation in rock, md
96 | An Approach to Stimulation Candidate Selection and Optimization
ππππππ permeability of the gravel-pack gravel, md
ππ reservoir permeability, md
πΏπΏππ length of flow path through gravel pack, inches
πΏπΏπ π ππππππππππ length of the screen, ft
πΏπΏππππππππππ length of the blank string, ft
πΏπΏππ depth of penetration in rock, ft
ππ number of perforations
ππππππππππ number of perforations open
πππ·π·ππππππππππ outer diameter of the blank string, inches
πππ·π·π π ππππππππππ outer diameter of the screen, inches
πππ π ,π π ππππ maximum surface pressure, psi
Pr average reservoir pressure, psi
P price per barrel of oil, $
ππππ damaged flow rate, stb/day
ππ damaged standard flow rates, stb/day
ππππ initial production rate, stb/day
ππππ the production rate, stb/day
ππππ reservoir drainage radius, ft
ππππ damaged zone radius, ft
πππ€π€ wellbore radius, ft
ππππππ radius of compacted zone around the perforations, ft
ππππ radius of perforation in rock, ft
ROIP remaining recoverable reserve, bbl
π π ππ+ππ skin effect caused by partial completion and slant, dimensionless
97 | An Approach to Stimulation Candidate Selection and Optimization
π π skin effect, dimensionless
ππππππ equivalent skin effect, dimensionless
π π ππππππππππππ skin due to gravel packing
π π ππππ skin factor due to Darcy flow through the gravel-pack, dimensionless
π‘π‘π π duration of the stimulation job, days
π‘π‘ππ abandonment time, day
π‘π‘πππ π abandonment time of the post-stimulation production, days
πππ π volume of mud acid, gal/ft
ππππ total volume of gravel needed, ft
ππππππ volume of the carrier fluid (base fluid), bbls
3
πππ π ππ slurry (gravel + carrier fluid) volume, bbls
πππ»π»πΆπΆππ volume of HCl required, gal/ft
ππππ weight of gravel, lbs
πππ»π»πΆπΆππ fraction of the bulk rock dissolved by HCl
β porosity, fraction
πππ π ππ slurry density, ppg
ππππππ density of the carrier fluid (base fluid), ppg
β porosity, fraction
πΎπΎ specific gravity of the acid (or density of acid in g/cc)
98 | An Approach to Stimulation Candidate Selection and Optimization
Appendix A
A Simple Well Screening Flow Chart
Positive skin effect?
No
No Yes
gravel packing
Mechanical problems?
(e.g plugged perf.)
matrix acidizing
high sand production?
No Yes
high water production?
No Yes
Yes recompletion
recompletion
Benson Best Ugbenyen, 2010
Evaluate well problems
Re-evaluate well problem
99 | An Approach to Stimulation Candidate Selection and Optimization
Appendix B
(After Vogt et. al, 1973)
Stimulation Cost and Performance
100 | An Approach to Stimulation Candidate Selection and Optimization
Appendix C
Solver Results
CASE STUDY 1
Well BU 1 Model output result
101 | An Approach to Stimulation Candidate Selection and Optimization
Solver Answer Report for Well BU 1 Model
102 | An Approach to Stimulation Candidate Selection and Optimization
Solver Sensitivity Report for Well BU 1 Model
103 | An Approach to Stimulation Candidate Selection and Optimization
Solver Limits Report for Well BU 1 Model
106 | An Approach to Stimulation Candidate Selection and Optimization
Solver Answer Report for Well BU 3 Model
107 | An Approach to Stimulation Candidate Selection and Optimization
Solver Sensitivity Report for Well BU 3 Model
108 | An Approach to Stimulation Candidate Selection and Optimization
Solver Limits Report for Well BU3 Model
110 | An Approach to Stimulation Candidate Selection and Optimization
CASE STUDY 2
β Section 1
Solver answer report for Well BU 5 model
111 | An Approach to Stimulation Candidate Selection and Optimization
Solver sensitivity report for Well BU 5 model
112 | An Approach to Stimulation Candidate Selection and Optimization
Solver limit report for Well BU 5 model
113 | An Approach to Stimulation Candidate Selection and Optimization
CASE STUDY 2
β Section 2
(a)
115 | An Approach to Stimulation Candidate Selection and Optimization
Appendix D
Whatβs Best 10.0 Results
CASE STUDY 1
120 | An Approach to Stimulation Candidate Selection and Optimization
Appendix E
Mathematica 7.0 Results
CASE STUDY 1
Mathematica 7.0 plot of the objective function for Well BU 3 Model
121 | An Approach to Stimulation Candidate Selection and Optimization
Mathematica 7.0 input syntax and results for Well BU 3 Model
122 | An Approach to Stimulation Candidate Selection and Optimization
Mathematica 7.0 inpretation of input data and results for Well BU 3 Model
124 | An Approach to Stimulation Candidate Selection and Optimization
Appendix F
The derivation of the optimization model presented in this section is modified from the published work of Sinson et al.
Derivation of the Objective Function for Other Decline Cases
85
F.1
Let u s start by deriving t he o ptimization model for the general decline curve analysis.
The general decline curve equation is given by:
General Decline Curve Optimization
ππ(π‘π‘) = ππππ
[1+πππ·π·ππ]1ππ (F.1)
Where:
ππ = hyperbolic constant
π·π·ππ = initial decline rate
The general form for the discounted production from stimulation,πππ·π·π·π· , is expressed as:
πππ·π·π·π· = β« πΉπΉππππππβπΌπΌπ‘π‘
[1+πππ·π·ππ(π‘π‘βπ‘π‘π π )]1ππ
π‘π‘πππ π π‘π‘π π
πππ‘π‘ (F.2)
If we denote the denominator of (F.2) by π₯π₯, and solving for π‘π‘:
π₯π₯ = 1 + πππ·π·ππ(π‘π‘ β π‘π‘π π ) (F.3)
π‘π‘ = 1πππ·π·ππ
(π₯π₯ β 1 + πππ·π·πππ‘π‘π π ) (F.4)
πππ‘π‘ = 1πππ·π·ππ
πππ₯π₯ (F.5)
Substituting (F.4) and (F.5) into the original equation (F.2):
πππ·π·π·π· = πΉπΉππππ β«ππβπΌπΌ[ 1
πππ·π·πποΏ½π₯π₯β1+πππ·π·πππ‘π‘π π οΏ½]
π₯π₯1ππ
π‘π‘πππ π π‘π‘π π
1πππ·π·ππ
πππ₯π₯ (F.6)
SinceπΌπΌ, ππ,π·π·ππ and π‘π‘π π are constants, we can express the equation as:
πππ·π·π·π· = πΉπΉπππππππ·π·ππ
ππ1
πππ·π·ππβπΌπΌπ‘π‘π π β« ππ
β 1πππ·π·ππ
π₯π₯
π₯π₯1ππ
π‘π‘πππ π π‘π‘π π
πππ₯π₯ (F.7)
Similarly the general form of the production loss component,πππ·π·π·π·π·π· , is:
125 | An Approach to Stimulation Candidate Selection and Optimization
πππ·π·π·π·π·π· = β« πΉπΉππππππβπΌπΌπ‘π‘
[1+πππ·π·ππ(π‘π‘βπ‘π‘π π )]1ππ
π‘π‘π π 0 πππ‘π‘ (F.8)
Using equation (F.3), (F.4), and (F.5) and simplifying, we obtain:
πππ·π·π·π·π·π· = πππππππ·π·ππ
ππ1
πππ·π·ππβπΌπΌπ‘π‘π π β« ππ
β 1πππ·π·ππ
π₯π₯
π₯π₯1ππ
π‘π‘π π 0 πππ₯π₯ (F.9)
Note that the general form of the solution for πππ·π·π·π· and πππ·π·π·π·π·π· using integral transformations is of the form:
β« πππππ₯π₯
π₯π₯πππππ₯π₯ = β1
πππππππ₯π₯
π₯π₯ππβ1 + ππππβ1 β«
πππππ₯π₯
π₯π₯ππβ1 πππ₯π₯ (F.10)
A closed form solution exists when:
ππ = 0 β« πππππ₯π₯ πππ₯π₯ = πππππ₯π₯
ππ (EXPONENTIAL DECLINE CASE) (F.11)
ππ = 1 β« πππππ₯π₯ πππ₯π₯ = πΈπΈππ(πππ₯π₯) (HARMONIC DECLINE CASE) (F.12)
ππ = 2 β« πππππ₯π₯
π₯π₯2 πππ₯π₯ = πππππ₯π₯
π₯π₯+ πππΈπΈππ(πππ₯π₯) (HYPERBOLIC DECLINE CASE) (F.13)
Note that equation (F.13) is the most common form of hyperbolic decline curve.
Let:
1ππ
= ππ, (F.14)
and π·π·ππ = ππππ (F.15)
F.2
F.2.1 Objective Function, πΈπΈπ«π«
Harmonic Decline Optimization
πππ·π· = πππ·π·π·π· β πππ·π·π·π·π·π· β πππ·π·π·π· (F.16)
Where:
πππ·π·π·π· = discounted production from stimulation
πππ·π·π·π·π·π·= discounted production loss
πππ·π·π·π· = discounted equivalent production cost
1. The discounted production from stimulation is:
126 | An Approach to Stimulation Candidate Selection and Optimization
πππ·π·π·π· = β« πΉπΉππππ1+πππ·π·ππ(π‘π‘βπ‘π‘π π )
π‘π‘πππ π π‘π‘π π
ππβπΌπΌπ‘π‘πππ‘π‘ (F.17)
Where:
ππππ = the production rate before stimulation
πΉπΉ = the productivity ratio
ππππ = the nominal decline rate before stimulation
Changing variables, we get:
π₯π₯ = 1β πππππ‘π‘π π + πππππ‘π‘ (F.18)
Such that: π‘π‘ = 1ππππ
(π₯π₯ β 1 + πππππ‘π‘π π ) (F.19)
and
πππ‘π‘ = πΌπΌπππππππ₯π₯ (F.20)
Substituting (F.19) and (F.20) into the original equation, (F. 17):
πππ·π·π·π· = πΉπΉππππππππππ
( πΌπΌππππβπΌπΌπ‘π‘π π )
β« ππβ πΌπΌπππππ₯π₯
π₯π₯π‘π‘πππ π π‘π‘π π
πππ₯π₯ (F.21)
Integrating:
πππ·π·π·π· = πΉπΉππππππππππ
( πΌπΌππππβπΌπΌπ‘π‘π π )
οΏ½πΈπΈππ οΏ½οΏ½βπΌπΌπππποΏ½ (1 β πππππ‘π‘π π + πππππ‘π‘)οΏ½οΏ½
π‘π‘π π
π‘π‘πππ π (F.22)
Simplifying:
πππ·π·π·π· = πΉπΉππππππππππ
( πΌπΌππππβπΌπΌπ‘π‘π π )
οΏ½πΈπΈππ οΏ½βπΌπΌππππ
+ πΌπΌπ‘π‘π π β πΌπΌπ‘π‘οΏ½οΏ½π‘π‘π π
π‘π‘πππ π (F.23)
and:
πππ·π·π·π· = πΉπΉππππππππππ
( πΌπΌππππβπΌπΌπ‘π‘π π )
οΏ½πΈπΈππ οΏ½βπΌπΌππππ
+ πΌπΌπ‘π‘π π β πΌπΌπ‘π‘πππ π οΏ½ β πΈπΈππ οΏ½βπΌπΌπππποΏ½οΏ½ (F.24)
Note that the time to reach the economic limit is:
π‘π‘πππ π = πΉπΉππππππππππππ
β 1ππππ
(F.25)
127 | An Approach to Stimulation Candidate Selection and Optimization
Where:
ππππ = abandonment production rate
Therefore equation (F.24) becomes:
πππ·π·π·π· = πΉπΉππππππππππ
( 1ππππβπΌπΌπ‘π‘π π )
οΏ½πΈπΈππ οΏ½πΌπΌπ‘π‘π π βπΌπΌπΉπΉππππππππππππ
οΏ½ β πΈπΈππ οΏ½β1πππποΏ½οΏ½ (F.26)
2. The discounted production lost during stimulation is:
πππ·π·π·π·π·π· = β« ππππππβπΌπΌπ‘π‘
1+πππππ‘π‘π‘π‘π π
0 πππ‘π‘ (F.27)
Using the same change of variable:
π₯π₯ = 1β πππππ‘π‘π π + πππππ‘π‘ (F.18)
Such that: π‘π‘ = 1ππππ
(π₯π₯ β 1 + πππππ‘π‘π π ) (F.19)
and
πππ‘π‘ = 1πππππππ₯π₯ (F.20)
We get:
πππ·π·π·π·π·π· = ππππππππππ
1ππππ β« ππ
β πΌπΌπππππ₯π₯
π₯π₯π‘π‘π π
0 πππ₯π₯ (F.28)
and
πππ·π·π·π·π·π· = ππππππππππ
πΌπΌππππ οΏ½πΈπΈππ οΏ½
βπΌπΌπππποΏ½ (1 + πππππ‘π‘)οΏ½
0
π‘π‘π π (F.29)
Therefore:
πππ·π·π·π·π·π· = ππππππππππ
πΌπΌππππ οΏ½πΈπΈππ οΏ½
βπΌπΌππππβπΌπΌπ‘π‘π π οΏ½οΏ½
0
π‘π‘π π (F.30)
πππ·π·π·π·π·π· = ππππππππππ
πΌπΌππππ οΏ½πΈπΈππ οΏ½
βπΌπΌππππβπΌπΌπ‘π‘π π οΏ½ β πΈπΈππ οΏ½
βπΌπΌπππποΏ½οΏ½ (F.31)
3. The discounted equivalent production from stimulation cost is:
πππ·π·π·π· = π·π·π·π·ππβπΌπΌπ‘π‘π π (F.32)
With the above derivations, πππ·π· can now be written as:
πππ·π· = π½π½1πΈπΈππ(π½π½2 β π½π½3πΉπΉ)πΉπΉ β π½π½1πΈπΈππ(π½π½4)πΉπΉ β π½π½6π·π· β π½π½5 (F.33)
128 | An Approach to Stimulation Candidate Selection and Optimization
Where:
π½π½1 = ππππππππππ
( πΌπΌππππβπΌπΌπ‘π‘π π )
(F.34)
π½π½2 = πΌπΌπ‘π‘π π (F.35)
π½π½3 = πΌπΌππππππππππππ
(F.36)
π½π½4 = βπΌπΌππππ
(F.37)
π½π½5 = ππππππππππ
πΌπΌππππ οΏ½πΈπΈππ οΏ½
βπΌπΌβπΌπΌππππ π‘π‘π π ππππ
οΏ½ β πΈπΈππ οΏ½βπΌπΌπππποΏ½οΏ½ (F.38)
π½π½6 = ππβπΌπΌπ‘π‘π π
π·π· (F.39)
F.2.2 Constraints
The same constraints formulated in the exponential case also applied here.
Constraint 1
The incremental revenue from any stimulation decision should be greater than or at least equal to
the cost of the project.
. Break-even point
β« πππΉπΉππππππβπΌπΌπ‘π‘
(1βπππππ‘π‘+πππππ‘π‘π π )π‘π‘πππ π π‘π‘π π
πππ‘π‘ β₯ π·π·π·π·ππβπΌπΌπ‘π‘π π + β« ππππππβπΌπΌπ‘π‘
1+πππππ‘π‘π‘π‘π π
0 πππ‘π‘ (F.40)
Performing the integration and using the definition of π‘π‘πππ π given in equation (F.25) and using the
constants above, we get:
π½π½1πΈπΈππ(π½π½2 β π½π½3πΉπΉ)πΉπΉ β π½π½1πΈπΈππ(π½π½4)πΉπΉ β₯ π½π½6π·π· β π½π½5 (F.41)
Constraint 2
The recovery from the stimulation cannot exceed the remaining oil in place.
. Recoverable oil in place
β« πΉπΉππππ(1βπππππ‘π‘+πππππ‘π‘π π )
π‘π‘πππ π π‘π‘π π
πππ‘π‘ β€ π π π π πΌπΌπ·π· (F.42)
The integral is equal to:
π·π·πΏπΏπ·π· = πΉπΉππππππππ
[ππππ π₯π₯]π‘π‘π π π‘π‘πππ π (F.43)
129 | An Approach to Stimulation Candidate Selection and Optimization
Using the definition of π‘π‘πππ π and substituting in the equation:
πΉπΉππππππππ
ln οΏ½πΉπΉππππβππππππππππππ
οΏ½ β πΉπΉππππππππππππ π‘π‘π π β€ π π π π πΌπΌπ·π· (F.44)
and:
π½π½7ππππ(π½π½8 β π½π½9)πΉπΉ β π½π½10 β€ π π π π πΌπΌπ·π· (F.45)
where:
π½π½7 = ππππππππ
(F.46)
π½π½8 = ππππππππππππ
(F.47)
π½π½9 = 1ππππ
(F.48)
π½π½10 = ππππππππππππ π‘π‘π π (F.49)
Constraint 3
The pr oduction r ate a fter s timulation should n ot e xceed the maximum d esign capacity o f t he
flow string, i.e.,
. Flow string capacity
πΉπΉ β€ πππππππ₯π₯ππππ
(F.50)
Where πππππππ₯π₯ is the maximum design capacity of the flowing string.
Constraint 4
The cost of the stimulation should not exceed the budget.
. Budget limitation
π·π· β€ π·π·πππππ₯π₯ (F.51)
Where π·π·πππππ₯π₯ is the budget for the stimulation job.
Constraint 5. Reservoir productivity ratio constraint
130 | An Approach to Stimulation Candidate Selection and Optimization
The maximum attainable pr oductivity ratio from stimulation depends on t he reservoir pr operties
and treatment parameters.
πΉπΉ β€ πΉπΉπππππ₯π₯ (F.52)
Where πΉπΉπππππ₯π₯ is the maximum attainable productivity ratio.
Constraint 6
The cost and productivity ratio relationship can be formulated into the following equation.
. Cost productivity ratio equation
π·π· = 10ππ0πΉπΉππ1 (F.53)
Where ππ0 and ππ1 are the intercept and slope of a regression line through the data.
F.2.3 Form of the NLP
Equations (F.33) and (F.53) define the NLP model for the harmonic case.
Maximize:
πππ·π· = π½π½1πΈπΈππ(π½π½2 β π½π½3πΉπΉ)πΉπΉ β π½π½1πΈπΈππ(π½π½4)πΉπΉ β π½π½6π·π· β π½π½5 (F.33)
Subject to:
π½π½1πΈπΈππ(π½π½2 β π½π½3πΉπΉ)πΉπΉ β π½π½1πΈπΈππ(π½π½4)πΉπΉ β₯ π½π½6π·π· β π½π½5 (F.41)
π½π½7ππππ(π½π½8 β π½π½9)πΉπΉ β π½π½10 β€ π π π π πΌπΌπ·π· (F.45)
πΉπΉ β€ πππππππ₯π₯ππππ
(F.50)
π·π· β€ π·π·πππππ₯π₯ (F.51)
πΉπΉ β€ πΉπΉπππππ₯π₯ (F.52)
π·π· = 10ππ0πΉπΉππ1 (F.53)
131 | An Approach to Stimulation Candidate Selection and Optimization
To develop a closed form solution to the model, we shall consider only the case when ππ = 2.
The solution form to the general case is given in equation (F.13).
F.3 Hyperbolic Decline Optimization
F.3.1 Objective Function, πΈπΈπ«π«
The objective function is formulated as:
πππ·π· = πππ·π·π·π· β πππ·π·π·π·π·π· β πππ·π·π·π· (F.16)
Where:
πππ·π·π·π· = discounted production from stimulation
πππ·π·π·π·π·π·= discounted production loss
πππ·π·π·π· = discounted equivalent production cost
1. The discounted production from stimulation is:
πππ·π·π·π· = β« πΉπΉππππ
οΏ½1+ππππ2 (π‘π‘βπ‘π‘π π )οΏ½
2π‘π‘πππ π π‘π‘π π
ππβπΌπΌπ‘π‘πππ‘π‘ (F.54)
Changing variables, we get:
π₯π₯ = 1β ππππ2π‘π‘π π + ππππ
2π‘π‘ (F.55)
Such that:
π‘π‘ = 2ππππ
(π₯π₯ β 1 + ππππ2π‘π‘π π ) (F.56)
and
πππ‘π‘ = 2πππππππ₯π₯ (F.57)
Substituting equations (F.55), (F.56) and (F.57) into (F. 54), then simplifying:
πππ·π·π·π· = 2πΉπΉππππππππ
πποΏ½2πΌπΌππππβπΌπΌπ‘π‘π π οΏ½ β« ππ
β2πΌπΌπππππ₯π₯
π₯π₯2π‘π‘πππ π π‘π‘π π
πππ₯π₯ (F.58)
132 | An Approach to Stimulation Candidate Selection and Optimization
The economic life is:
π‘π‘πππ π = 2πππποΏ½πΉπΉπππππππποΏ½
0.5β 2
ππππ (F.59)
Integrating equation (F.58), then substituting equation (F.59) and simplifying:
πππ·π·π·π· = βππ3πππΌπΌπ‘π‘π π β
2πππποΏ½πΉπΉππ ππππππ
οΏ½0.5
πΉπΉ
βππππ2 π‘π‘π π +οΏ½πΉπΉππ πππππποΏ½
0.5 + ππ1ππ3πΉπΉ β ππ2ππ3πΉπΉ (F.60)
where:
ππ1 = οΏ½β2πΌπΌπππποΏ½ πΈπΈππ οΏ½πΌπΌπ‘π‘π π β
2πΌπΌπππποΏ½πΉπΉπππππππποΏ½
0.5οΏ½ (F.61)
ππ2 = βππβ2πΌπΌππππ + οΏ½β2πΌπΌ
πππποΏ½ πΈπΈππ οΏ½
β2πΌπΌπππποΏ½ (F.62)
ππ3 = 2ππππππ2πΌπΌππππ πππΌπΌπ‘π‘π π
ππππ (F.63)
2. The discounted production lost during stimulation is:
πππ·π·π·π·π·π· = β« ππππππβπΌπΌπ‘π‘
1+ππππ2 π‘π‘
π‘π‘π π 0 πππ‘π‘ (F.64)
Using similar variable change and simplifying:
πππ·π·π·π·π·π· = 2ππππππππππ
2πΌπΌππππ οΏ½βππ
οΏ½β2πΌπΌππππ
οΏ½οΏ½1+ππππ2 π‘π‘οΏ½
οΏ½1+ππππ2 π‘π‘οΏ½
+ οΏ½β2πΌπΌπππποΏ½ πΈπΈππ οΏ½οΏ½
β2πΌπΌπππποΏ½ οΏ½1 + ππππ
2π‘π‘οΏ½οΏ½οΏ½
0
π‘π‘π π
(F.65)
Simplifying further:
πππ·π·π·π·π·π· = 2ππππππππππ
2πΌπΌππππ οΏ½βππ
β2πΌπΌβπΌπΌπ‘π‘π π
οΏ½1+ππππ2 π‘π‘π π οΏ½
+ οΏ½β2πΌπΌπππποΏ½πΈπΈππ οΏ½
β2πΌπΌππππβ πΌπΌπ‘π‘π π οΏ½ + ππ
β2πΌπΌππππ + οΏ½β2πΌπΌ
πππποΏ½ πΈπΈππ οΏ½
β2πΌπΌπππποΏ½οΏ½ (F.66)
This expression for πππ·π·π·π·π·π· is constant.
133 | An Approach to Stimulation Candidate Selection and Optimization
We can define:
ππ4 = 2ππππππππππ
2πΌπΌππππ οΏ½βππ
β2πΌπΌβπΌπΌπ‘π‘π π
οΏ½1+ππππ2 π‘π‘π π οΏ½
+ οΏ½β2πΌπΌπππποΏ½πΈπΈππ οΏ½
β2πΌπΌππππβ πΌπΌπ‘π‘π π οΏ½ + ππ
β2πΌπΌππππ + οΏ½β2πΌπΌ
πππποΏ½ πΈπΈππ οΏ½
β2πΌπΌπππποΏ½οΏ½ (F.67)
1. The discounted equivalent production from stimulation cost is:
πππ·π·π·π· = π·π·π·π·ππβπΌπΌπ‘π‘π π (F.32)
We can define:
ππ5 = ππβπΌπΌπ‘π‘π π
π·π· (F.69)
Therefore:
πππ·π·π·π· = ππ5π·π· (F.70)
The objective function πππ·π· can now be written as:
πππ·π· = βππ3ππππ8ππππ6πΉπΉ0.5πΉπΉππ9 +ππ7πΉπΉ0.5 + ππ1ππ3πΉπΉ β ππ2ππ3πΉπΉ β ππ4 β ππ5π·π· (F.71)
Where:
ππ6 = β2πΌπΌπππποΏ½ πππππππποΏ½
0.5 (F.72)
ππ7 = οΏ½πππππππποΏ½
0.5 (F.73)
ππ8 = πΌπΌπ‘π‘π π (F.74)
ππ9 = βπππππ‘π‘π π 2
(F.75)
F.3.2 Constraints
The same constraints formulation as in previous cases applies here.
Constraint 1
The incremental revenue from any stimulation decision should be greater than or at least equal to
the cost of the project.
. Break-even point
134 | An Approach to Stimulation Candidate Selection and Optimization
β« πΉπΉππππ
οΏ½1+ππππ2 (π‘π‘βπ‘π‘π π )οΏ½
2π‘π‘πππ π π‘π‘π π
ππβπΌπΌπ‘π‘πππ‘π‘ β₯ π·π·π·π·ππβπΌπΌπ‘π‘π π + β« ππππππβπΌπΌπ‘π‘
1+ππππ2 π‘π‘
π‘π‘π π 0 πππ‘π‘ (F.76)
Evaluating:
βππ3πππΌπΌπ‘π‘π π β
2πΌπΌπππποΏ½πΉπΉππ ππππππ
οΏ½0.5
πΉπΉβππππ π‘π‘π π
2 +οΏ½πΉπΉππ πππππποΏ½
0.5 + ππ1ππ3πΉπΉ β ππ2ππ3πΉπΉ β₯ ππ4 + ππ5π·π· (F.77)
Constraint 2
The recovery from the stimulation cannot exceed the remaining oil in place.
. Recoverable oil in place
β« πΉπΉππππ
οΏ½1+ππππ2 (π‘π‘βπ‘π‘π π )οΏ½
2π‘π‘πππ π π‘π‘π π
ππβπΌπΌπ‘π‘πππ‘π‘ β€ π π π π πΌπΌπ·π· (F.78)
Evaluating the left hand side:
π·π·πΏπΏπ·π· = 2πΉπΉππππππππ
οΏ½ β1
1+ππππ2 π‘π‘β
ππππ2 π‘π‘π π οΏ½π‘π‘π π
π‘π‘πππ π (F.79)
Simplifying using the definition of π‘π‘πππ π and the defined constants:
ππ10πΉπΉππ7πΉπΉ0.5βππ1 1ππ7πΉπΉ0.5βππ9
β€ π π π π πΌπΌπ·π· (F.80)
Where:
ππ10 = 2ππππππππ
(F.81)
Constraint 3
The pr oduction r ate a fter stimulation should n ot e xceed t he maximum d esign capacity o f t he
flow string, i.e.,
. Flow string capacity
πΉπΉ β€ πππππππ₯π₯ππππ
(F.82)
Where πππππππ₯π₯ is the maximum design capacity of the flowing string.
135 | An Approach to Stimulation Candidate Selection and Optimization
Constraint 4
The cost of the stimulation should not exceed the budget.
. Budget limitation
π·π· β€ π·π·πππππ₯π₯ (F.51)
Where π·π·πππππ₯π₯ is the budget for the stimulation job.
Constraint 5
. Reservoir productivity ratio constraint
The maximum attainable pr oductivity ratio from stimulation depends on t he reservoir pr operties
and treatment parameters.
πΉπΉ β€ πΉπΉπππππ₯π₯ (F.52)
Where πΉπΉπππππ₯π₯ is the maximum attainable productivity ratio.
Constraint 6
The cost and productivity ratio relationship can be formulated into the following equation.
. Cost productivity ratio equation
π·π· = 10ππ0πΉπΉππ1 (F.53)
Where ππ0 and ππ1 are the intercept and slope of a regression line through the data.
A.3.3
The equation (F.71), together with all the constraints considered, can be summarized as:
Form of the NLP
Maximize:
πππ·π· = βππ3ππππ8ππππ6πΉπΉ0.5πΉπΉππ9 +ππ7πΉπΉ0.5 + ππ1ππ3πΉπΉ β ππ2ππ3πΉπΉ β ππ4 β ππ5π·π· (F.71)
Subject to:
136 | An Approach to Stimulation Candidate Selection and Optimization
βππ3πππΌπΌπ‘π‘π π β
2πΌπΌπππποΏ½πΉπΉππ ππππππ
οΏ½0.5
πΉπΉβππππ π‘π‘π π
2 +οΏ½πΉπΉππ πππππποΏ½
0.5 + ππ1ππ3πΉπΉ β ππ2ππ3πΉπΉ β₯ ππ4 + ππ5π·π· (F.77)
ππ10πΉπΉππ7πΉπΉ0.5βππ1 1ππ7πΉπΉ0.5βππ9
β€ π π π π πΌπΌπ·π· (F.80)
πΉπΉ β€ πππππππ₯π₯ππππ
(F.82)
π·π· β€ π·π·πππππ₯π₯ (F.51)
πΉπΉ β€ πΉπΉπππππ₯π₯ (F.52)
π·π· = 10ππ0πΉπΉππ1 (F.53)
F.4
The s timulation optimization models developed i n this section of t he appendix can be used i n
place of the exponential model used in the thesis. The solution procedure is the same.
Summary