pressure and temperature transient analysis during ... · turing problem, in addition to pressure...

PRESSURE AND TEMPERATURE TRANSIENT ANALYSIS

DURING HYDRAULIC FRACTURING

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF DEPARTMENT OF

ENERGY RESOURCES ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Priscila Magalhaes Ribeiro

March 2014

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/sh975bx8254

© 2014 by Priscila Magalhaes Ribeiro. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/sh975bx8254

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Roland Horne, Primary Adviser


Hamdi Tchelepi


Mark Zoback

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost for Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

Recent developments in bottomhole data acquisition techniques, such as distributed

temperature sensing systems (DTS), have brought attention to the potential increase

of information that can be obtained from temperature data. Studies have shown

the application of temperature surveys to estimate flowrate profiles, resolve the kind

of damage around the well, and improve the robustness of the history matching,

among others. Nonetheless, Temperature Transient Analysis (TTA) is not a mature

technique and its capabilities have not been explored fully yet.

In order to investigate the application temperature analysis to the hydraulic frac-

turing problem, in addition to pressure analysis, a numerical model was developed to

calculate pressure and temperature responses. Regarding the fracture and reservoir

fluid flow, a general approach can be adopted, where the formation permeability and

fracture characteristics dictate how the fluids flow during and after fracture growth.

We developed a comprehensive model, which accounts for the pressure effect on the

temperature response, as well as a dynamic fracture that grows and eventually is

allowed to close during falloff.

In this work we analyzed the temperature and pressure responses during and

immediately after hydraulic fracturing in order to improve our knowledge of this

complicated physical problem. Based on this study, we can better understand not

only the fracture properties, but also the reservoir itself. In addition, sensitivity

analysis shows how reservoir permeability can impact final fracturing performance, as

well as pressure and temperature responses. The developed model was also applied

to simulate minifrac analysis, and a field example is presented that shows a good

agreement with the simulated behavior during fracture closure.

v

One of the main contributions of this research is related to the creation of multiple

fractures along a horizontal well. This type of well and completion technique have

become the key factors for success in development of unconventional resources. Both

sequential and simultaneous fracture growth was studied. The presence of reservoir

permeability heterogeneity was investigated and the capability of temperature data

to identify the existence of such reservoir structure was explored. Capabilities and

limitations of information carried by temperature data are presented through different

geometry analyses.

Also related to horizontal multifractured wells, a case was considered in which

one of the fractures interconnects different zones vertically. This study is motivated

by microseismic evidence of activity captured out of the target reservoir location.

The temperature data analysis during the beginning of production life of the well

was shown to be very effective to identify the existence of such interconnection. The

difference in temperature due to geothermal gradient allows a very clear temperature

signature, where the pressure analysis would not reveal the connection.

vi

“Learn from yesterday, live for today, hope for tomorrow. The important thing is

not to stop questioning.”

Albert Einstein

vii

Acknowledgements

First of all I would like to thank God for so many blessings he provided in my life.

Specially this great experience that was my PhD at Stanford.

I would like to express my eternal gratitude and admiration to my adviser, Pro-

fessor Roland Horne. Thank you for all your guidance, patience and support. You

are a great example to be followed. You inspire all of us with your love for science

and constant optimism. I am extremely proud to be one of your students.

I am also thankful to Dr. Hamdi Tchelepi, Dr. Mark Zoback, Dr Sally Benson,

and Dr. Jerry Harris for serving as members of my PhD defense committee. Thank

you for all your contributions, and your precious time for reading and evaluating this

thesis.

Many thanks go to all the faculties in the Department of Energy Resources En-

gineering at Stanford. I learn so much from them throughout these years. A special

acknowledgment goes to ERE staff, always kind and ready to help.

There are so many people that helped me throughout this journey. I would like

to thank Alvaro Peres and the well testing group from Petrobras Research Center

for allowing me to pursue my PhD and trusting I would be able to accomplish this

important task. My former and current managers, Mauro Becker and Flavia Pacheco,

thank you very much. I would like to express my gratitude to Petrobras for providing

the financial support to my PhD. I am also thankful to my former Masters adviser, Dr.

Adolfo Puime Pires, for all his encouragement and guidance to initiate my interest

for research.

These years that I spent at Stanford would not be the same without the incredible

people I had the opportunity to meet. So many bright people from whom I learned a

ix

lot. I had the privilege to make really good friends that I will carry in my heart forever.

Special thanks goes to my office-mates Sara Farshidi, Karine Lenovian, Yangyang

Liu, and Sandy Ahn. Many other friends were present in my everyday life: Christin

Strandli, Amir Saheli, Ekin Ozdogan, and Ognjen Grujic made this experience even

more enjoyable. I am also thankful to all SUPRI-D friends. Thank you for every

comment, every suggestion, every word of incentive you all gave me.

Most specially, I want to thank my wonderful family. My parents Neiva and

Ruben, and my sisters Simoni and Juliana: your love and support are the reasons

for any accomplishment I had in my life. My beloved nephews and nieces are an

incredible source of happiness and motivation of mine, they make want to be a better

person and overcome challenges to be a good example for them.

Lastly, my deepest love and gratitude goes to my husband Carlos Eduardo (Cadu).

I am thankful for your endless love, patience, support, and encouragement. I could

not have done this without you and your love. Thank you for believing in this dream

with me, for every trip to Stanford, for everything you represent in my life. I love

you!

x

Contents

Abstract v

Acknowledgements ix

1 Introduction 1

1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Hydraulic fracturing modeling and characterization . . . . . . 5

1.1.2 Classical well testing fractured well pressure solutions . . . . . 7

1.1.3 Temperature modeling and analysis . . . . . . . . . . . . . . . 9

1.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Hydraulic Fracturing Forward Model 17

2.1 Mass Balance and Fracture Growth . . . . . . . . . . . . . . . . . . . 19

2.1.1 Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.2 Fracture growth and closure . . . . . . . . . . . . . . . . . . . 23

2.1.3 Coupling between fracture, well and reservoir . . . . . . . . . 26

2.2 Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3 Coupling Mass Balance, Fracture Growth and Energy Balance . . . . 35

2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4.1 PKN fracture geometry . . . . . . . . . . . . . . . . . . . . . . 40

2.4.2 KGD fracture geometry . . . . . . . . . . . . . . . . . . . . . 43

2.4.3 Effect of contact stress on the fracture closure . . . . . . . . . 44

2.4.4 Temperature response . . . . . . . . . . . . . . . . . . . . . . 46

xi

2.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Results: Single Vertical Fracture 53

3.1 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.1.1 Sensitivity to reservoir permeability . . . . . . . . . . . . . . . 55

3.1.2 Sensitivity to injection rate . . . . . . . . . . . . . . . . . . . 60

3.1.3 Sensitivity to reservoir porosity . . . . . . . . . . . . . . . . . 69

3.1.4 Sensitivity to closure pressure . . . . . . . . . . . . . . . . . . 73

3.1.5 Sensitivity to Young’s Modulus . . . . . . . . . . . . . . . . . 73

3.1.6 Sensitivity to asperity width . . . . . . . . . . . . . . . . . . . 79

3.2 Temperature Transient Derivative . . . . . . . . . . . . . . . . . . . . 80

3.3 Impact of Reservoir Heterogeneity . . . . . . . . . . . . . . . . . . . . 82

3.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4 Minifrac 93

4.1 Low Permeability Reservoirs . . . . . . . . . . . . . . . . . . . . . . . 100

4.2 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5 Horizontal Multifractured Well 105

5.1 Horizontal Well Model . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.1.1 Fracture representation . . . . . . . . . . . . . . . . . . . . . . 109

5.2 Sequential Fracturing . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.2.1 Reservoir heterogeneity . . . . . . . . . . . . . . . . . . . . . . 117

5.3 Simultaneous Fracturing . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.3.1 Simultaneous fracture growth in presence of heterogeneities . . 124

5.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6 Fracture Interconnecting Different Zones 145

6.1 Geometry Description . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.2 Effect of Position of Fracture Connection . . . . . . . . . . . . . . . . 149

6.3 Effect of Distance Between Connected Zones . . . . . . . . . . . . . . 157

6.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

xii

7 Fracture Crossing Multiple Fractures 163

7.1 Drawdown Solution for Multiple Crossing Fractures . . . . . . . . . . 164

7.1.1 Uniform flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

7.1.2 Infinite conductivity . . . . . . . . . . . . . . . . . . . . . . . 169

7.2 Wellbore Storage Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 175

7.2.1 Constant wellbore storage coefficient . . . . . . . . . . . . . . 175

7.2.2 Variable storage coefficient: multiple fracture closure . . . . . 178

7.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

8 Conclusions and Final Remarks 189

8.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

A Summary of Equations 199

A.1 Vertical Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

A.1.1 Mass balance and fracture growth . . . . . . . . . . . . . . . . 199

A.1.2 Energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . 200

A.2 Horizontal Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

A.2.1 Mass balance and fracture growth . . . . . . . . . . . . . . . . 202

A.2.2 Energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . 204

B Numerical Discretization of Flow Model 207

B.1 Reservoir Discretized Equation . . . . . . . . . . . . . . . . . . . . . 207

B.2 Fracture Discretized Equation . . . . . . . . . . . . . . . . . . . . . . 209

B.2.1 Mass balance . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

B.2.2 Stress balance equation . . . . . . . . . . . . . . . . . . . . . . 210

B.3 Jacobian Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

B.3.1 Reservoir grid-blocks . . . . . . . . . . . . . . . . . . . . . . . 213

B.3.2 Fracture grid-blocks . . . . . . . . . . . . . . . . . . . . . . . 213

C Numerical Discretization of Thermal Model 217

C.1 Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

C.2 Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

xiii

D Numerical Model Verification 221

D.1 Reservoir Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

D.1.1 Vertical well . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

D.1.2 Vertical fractured well . . . . . . . . . . . . . . . . . . . . . . 223

D.2 Fracture Creation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

D.2.1 PKN fracture geometry . . . . . . . . . . . . . . . . . . . . . . 225

D.2.2 KGD fracture geometry . . . . . . . . . . . . . . . . . . . . . 226

D.3 Temperature Response . . . . . . . . . . . . . . . . . . . . . . . . . . 228

D.3.1 Pressure effect in vertical well . . . . . . . . . . . . . . . . . . 228

D.3.2 Fracture temperature . . . . . . . . . . . . . . . . . . . . . . . 232

D.3.3 Reservoir temperature during fracturing . . . . . . . . . . . . 233

E Wellbore Temperature Analytic Solutions 237

E.1 Temperature Analytic Solution during Warmback . . . . . . . . . . . 237

E.1.1 Constant temperature around the well . . . . . . . . . . . . . 238

E.1.2 Variable temperature around the well . . . . . . . . . . . . . . 239

E.2 Temperature Analytic Solution during Flowback . . . . . . . . . . . . 241

xiv

List of Tables

2.1 Reservoir and fluid properties. . . . . . . . . . . . . . . . . . . . . . . 38

2.2 Rock and fluid thermal properties. . . . . . . . . . . . . . . . . . . . 38

3.1 Reservoir and fluid base case model parameters for sensitivity analyses. 54

3.2 Base case thermal properties for sensitivity analyses. . . . . . . . . . 55

3.3 Fracture half length (xf ) for different reservoir permeability. . . . . . 55

3.4 Fracture half length (xf ) for different injection rates. . . . . . . . . . 61

3.5 Fracture half length (xf ) for different reservoir porosities. . . . . . . . 69

3.6 Fracture half length (xf ) for different closure pressures. . . . . . . . . 73

3.7 Fracture half length (xf ) for different heterogeneous cases. . . . . . . 83

4.1 Comparison between input and interpreted parameters for minifrac test. 96

4.2 Fracture stiffness (Sf ) for PKN and KGD geometries. . . . . . . . . . 98

5.1 Base case reservoir and fluid properties for horizontal multifractured

well. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.2 Base case thermal properties for horizontal multifractured well. . . . . 113

5.3 Cases analyzed for simultaneous fracture growth. . . . . . . . . . . . 128

5.4 Input parameters for simultaneous fracture growth in a heterogeneous

medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.5 Flowback rates estimated from temperature profiles. . . . . . . . . . . 140

6.1 Input parameters for base case geometry. . . . . . . . . . . . . . . . . 150

xv

List of Figures

1.1 Real data of distributed pressure and temperature surveys along hori-

zontal well, extracted from Valiullin et al. (2009). . . . . . . . . . . . 4

1.2 DTS deployment schemes, extracted from Sierra et al. (2008). . . . . 12

2.1 Hydraulic fracture in vertical well. . . . . . . . . . . . . . . . . . . . . 18

2.2 Solution path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 PKN and KGD fracture geometries. . . . . . . . . . . . . . . . . . . . 20

2.4 Single fracture in a vertical well model representation. . . . . . . . . . 20

2.5 Mass balance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6 Asperities on fracture walls (modified from Danko (2013)). . . . . . . 24

2.7 Contact stress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.8 Two-dimensional reservoir, well and fracture volume discretization. . 27

2.9 Isotropic and homogeneous reservoir, well and fracture simplified grid

representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.10 Numerical problem structure: Jacobian, unknowns and residual. . . . 28

2.11 Smoothing function. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.12 Fracture growth algorithm. . . . . . . . . . . . . . . . . . . . . . . . . 31

2.13 Solution algorithm overview. . . . . . . . . . . . . . . . . . . . . . . . 37

2.14 Pressure maps (psi) at the end of injection (top) and falloff (bottom)

periods for PKN fracture geometry. . . . . . . . . . . . . . . . . . . . 39

2.15 Wellbore pressure history for PKN geometry. . . . . . . . . . . . . . . 41

2.16 Fracture length history for PKN geometry. . . . . . . . . . . . . . . . 41

2.17 Average width profiles during fracturing (top) and falloff (bottom) for

PKN geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

xvii

2.18 Pressure derivative during falloff for PKN geometry . . . . . . . . . . 43

2.19 Wellbore pressure history for KGD geometry. . . . . . . . . . . . . . . 44

2.20 Fracture length history for KGD geometry. . . . . . . . . . . . . . . . 45

2.21 Average width profiles during fracturing (top) and falloff (bottom) for

KGD geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.22 Pressure derivative during falloff for KGD geometry. . . . . . . . . . . 46

2.23 Pressure derivative during falloff for different closure parameter γ. . . 47

2.24 Comparison between the pressure derivative during falloff for fracture

closure and no closure. . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.25 Pressure derivative for falloff period after fracturing injection in a

50 md reservoir. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.26 Temperature response measurement for different sensor locations: in-

jection followed by shut-in. . . . . . . . . . . . . . . . . . . . . . . . . 49

2.27 Temperature response measurement for different sensor locations: in-

jection followed by flowback. . . . . . . . . . . . . . . . . . . . . . . . 50

3.1 Sensitivity analysis geometry. . . . . . . . . . . . . . . . . . . . . . . 53

3.2 Pressure derivative with respect to Agarwal equivalent time during

falloff, sensitivity to permeability. . . . . . . . . . . . . . . . . . . . . 57

3.3 Fracture volume evolution over time during falloff, sensitivity to per-

meability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.4 van den Hoek (2002) semianalytical pressure solution for closing frac-

ture during falloff. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.5 Width profile at the end of injection, sensitivity to permeability. . . . 59

3.6 Temperature inside the well at the bottom-hole for injection and falloff,

sensitivity to permeability. . . . . . . . . . . . . . . . . . . . . . . . . 60

3.7 Well temperature logarithmic derivative during warmback, sensitivity

to permeability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.8 Temperature behind the casing, sensitivity to permeability. . . . . . . 61

3.9 Well temperature during flowback, sensitivity to permeability. . . . . 62

xviii

3.10 Well temperature logarithmic derivative during flowback, sensitivity to

permeability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.11 Pressure derivative during falloff, sensitivity to injection rate. . . . . . 63

3.12 Well temperature history for injection and warmback periods, sensitiv-

ity to injection rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.13 Well temperature logarithmic derivative for warmback period, sensi-

tivity to injection rate. . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.14 Well temperature first derivative with respect to shut-in time, sensi-

tivity to injection rate. . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.15 Well temperature history for constant injection temperature (Tinj) at

the reservoir depth for injection and warmback periods, sensitivity to

injection rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.16 Well temperature logarithmic derivative for constant injection temper-

ature (Tinj) during warmback, sensitivity to injection rate. . . . . . . 66

3.17 Behind the casing temperature history for warmback, sensitivity to

injection rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.18 Behind the casing temperature logarithmic derivative for warmback,

sensitivity to injection rate. . . . . . . . . . . . . . . . . . . . . . . . 67

3.19 Well temperature history for flowback, sensitivity to injection rate. . . 68

3.20 Well temperature logarithmic derivative for flowback, sensitivity to

injection rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.21 Pressure derivative during falloff, sensitivity to reservoir porosity. . . 69

3.22 Temperature inside the well at the bottom-hole, sensitivity to porosity. 70

3.23 Well temperature logarithmic derivative during warmback, sensitivity

to porosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.24 Temperature behind the casing, sensitivity to porosity. . . . . . . . . 71

3.25 Well temperature during flowback, sensitivity to porosity. . . . . . . . 72

3.26 Well temperature logarithmic derivative during flowback, sensitivity to

porosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.27 Width profile at the end of injection, sensitivity to closure pressure. . 74

3.28 Pressure derivative during falloff, sensitivity to closure pressure. . . . 74

xix

3.29 Temperature inside the well at the bottom-hole, sensitivity to closure

pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.30 Logarithmic derivative during warmback of temperature inside the

well, sensitivity to closure pressure. . . . . . . . . . . . . . . . . . . . 75

3.31 Temperature behind the casing, sensitivity to closure pressure. . . . . 76

3.32 Well temperature for flowback case, sensitivity to closure pressure. . . 76

3.33 Logarithmic derivative during flowback of temperature inside the well,

sensitivity to closure pressure. . . . . . . . . . . . . . . . . . . . . . . 77

3.34 Width profile at the end of injection, sensitivity to Young’s Modulus. 78

3.35 Pressure derivative during falloff, sensitivity to Young’s Modulus. . . 78

3.36 Temperature derivative during flowback, sensitivity to Young’s Modulus. 79

3.37 Pressure derivative during falloff, sensitivity to asperity’s minimum

width (wfmin). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.38 Reservoir heterogeneity configurations. . . . . . . . . . . . . . . . . . 82

3.39 Reservoir heterogeneity: pressure logarithmic derivative during falloff. 84

3.40 Reservoir heterogeneity: temperature during warmback for cases 1 and

2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.41 Reservoir heterogeneity: temperature during warmback for cases 3 and

4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.42 Reservoir heterogeneity: temperature logarithmic derivative during

warmback for cases 1 and 2. . . . . . . . . . . . . . . . . . . . . . . . 88


warmback for cases 3 and 4. . . . . . . . . . . . . . . . . . . . . . . . 89


flowback for cases 1 and 2. . . . . . . . . . . . . . . . . . . . . . . . . 90


flowback for cases 3 and 4. . . . . . . . . . . . . . . . . . . . . . . . . 91

4.1 Minifrac rate schedule schematic. . . . . . . . . . . . . . . . . . . . . 94

4.2 Bourdet derivative of minifrac falloff pressure for 1md reservoir. . . . 95

xx

4.3 Bourdet derivative and finite conductivity model match for integral

transformed minifrac pressure falloff (psi.h) for 1 md reservoir. . . . . 95

4.4 Comparison between closing fracture, fixed fracture and vertical well

falloff type curves, and equivalent integral transformed analysis. . . . 97

4.5 Fracture volume derivative with respect to pressure and fracture stora-

tivity during the falloff for 1md reservoir. . . . . . . . . . . . . . . . . 99

4.6 Effect of initial pressure on integral transformed data. . . . . . . . . . 101

4.7 Minifrac performed in 100 nd reservoir: falloff Bourdet derivative with

respect to Agarwal equivalent time and of integral transformed data. 103

4.8 Field data of minifrac in an ultralow permeability gas reservoir: Falloff

Bourdet pressure derivative with respect to Agarwal equivalent time

and integral transformation. . . . . . . . . . . . . . . . . . . . . . . . 104

5.1 Horizontal well with multiple fractures schematic. . . . . . . . . . . . 106

5.2 Horizontal wellbore discretization. . . . . . . . . . . . . . . . . . . . . 108

5.3 Multifractured horizontal well grid representation. . . . . . . . . . . . 108

5.4 Sequential multifrac along a horizontal well. . . . . . . . . . . . . . . 111

5.5 Real DTS temperature map along the horizontal well during multistage

hydraulic fracturing (extracted from Sierra et al. (2008)). . . . . . . . 112

5.6 Simulated wellbore temperature profiles over time for sequential hy-

draulic fracturing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.7 Sequential hydraulic fracturing history per stage: fracture half length,

pressure and temperature. . . . . . . . . . . . . . . . . . . . . . . . . 115

5.8 Comparison between wellbore temperature profiles at end of injection

of stage 1 and stage 2. . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.9 Warmback temperature profile for sequential fracturing scenario. . . . 116

5.10 Heterogeneity along the reservoir for sequential hydraulic fracturing

scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.11 Sensitivity analysis to permeability heterogeneity along the reservoir

for sequential hydraulic fracturing scenario: temperature profile. . . . 118

xxi


for sequential hydraulic fracturing scenario: warmback temperature at

stage 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120


for sequential hydraulic fracturing scenario: temperature logarithmic

derivative during warmback at stage 3. . . . . . . . . . . . . . . . . . 120

5.14 Simultaneous fracture growth. . . . . . . . . . . . . . . . . . . . . . . 121

5.15 Pressure map after 30 minutes of injection for 1, 2 and 3 fractures

growing simultaneously. . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.16 Temperature profiles along a horizontal well after 5 minutes injection.

Comparison between one, two and three fractures growing at the same

time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.17 Temperature map along a horizontal well during warmback when one

(top), two (middle), and three (bottom) fractures grow simultaneously. 125

5.18 Temperature first derivative with respect to position along the hori-

zontal well during warmback when one (top), two (middle), and three

(bottom) fractures grow simultaneously. . . . . . . . . . . . . . . . . 126

5.19 Crossflow rates during falloff for three fractures growing simultaneously.127

5.20 Pressure derivative comparison between 1, 2 and 3 fractures growing

at the same time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.21 Schematic cases for 2 fractures growing simultaneously. . . . . . . . . 129

5.22 Flow-rate distribution for heterogeneous cases of simultaneous fracture

growth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.23 Fracture half length for heterogeneous cases of simultaneous fracture

growth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.24 Temperature profile after 5 minutes of injection for heterogeneous cases

of simultaneous fracture growth. . . . . . . . . . . . . . . . . . . . . . 132

5.25 Map of temperature first derivative with respect to distance along the

wellbore during injection period. . . . . . . . . . . . . . . . . . . . . . 133

5.26 Pressure maps at the end of injection for heterogeneous cases of simul-

taneous fracture growth. . . . . . . . . . . . . . . . . . . . . . . . . . 135

xxii

5.27 Warmback temperature maps at the end of injection for heterogeneous

cases of simultaneous fracture growth. . . . . . . . . . . . . . . . . . . 136


wellbore during warmback period. . . . . . . . . . . . . . . . . . . . . 137

5.29 Warmback temperature derivative for heterogeneous cases of simulta-

neous fracture growth. . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.30 Falloff pressure derivative for heterogeneous cases of simultaneous frac-

ture growth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.31 Flowback temperature maps at the end of injection for heterogeneous

cases of simultaneous fracture growth. . . . . . . . . . . . . . . . . . . 141


wellbore during flowback period. . . . . . . . . . . . . . . . . . . . . . 142

5.33 Flowback pressure for heterogeneous cases of simultaneous fracture

growth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.1 Microseismic events showing activity at 800 ft above the well that was

submitted to hydraulic fracturing, extracted from Yang et al. (2013). 146

6.2 Illustration of hydraulic fracture interconnecting two isolated zones. . 147

6.3 Initial temperature map for the base case of fracture interconnecting

different zones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.4 Base case reservoir pressure map (psi) for end of injection period, where

fracture 1 is interconnecting different zones. . . . . . . . . . . . . . . 151

6.5 Base case pressure map (psi) after 60 days of production, where fracture

1 is interconnecting different zones. . . . . . . . . . . . . . . . . . . . 152

6.6 Base case temperature map (oC) after 60 days of production, where


6.7 Base case wellbore temperature map for injection period, where frac-

ture 1 is interconnecting different zones. . . . . . . . . . . . . . . . . 153

6.8 Base case wellbore temperature map for the first 60 days of production,

where fracture 1 is interconnecting different zones. . . . . . . . . . . . 153

xxiii

6.9 Flow rate history per fracture. Fracture 1 is connecting the main reser-

voir with a zone 500 ft above it. . . . . . . . . . . . . . . . . . . . . . 154

6.10 Base case derivative of wellbore temperature with respect to position

along the well (∂T/∂y in oC/ft). . . . . . . . . . . . . . . . . . . . . 154

6.11 Base case derivative of wellbore temperature with respect to time

(∂T/∂t in oC/h) during production. . . . . . . . . . . . . . . . . . . . 155

6.12 Wellbore temperature map for injection period, where fracture 2 is

interconnecting different zones. . . . . . . . . . . . . . . . . . . . . . 155

6.13 Wellbore temperature map for the first 60 days of production, where


6.14 Derivative of wellbore temperature with respect to time (∂T/∂t inoC/h). Fracture 2 is connecting the two zones. . . . . . . . . . . . . . 156

6.15 Pressure change and its logarithmic derivative for production period:

comparison between connection with zone 2 through fracture 1 and 2. 157

6.16 Temperature map at after 60 days of production for zone 2 located

200 ft and 500 ft above the reservoir and 500 ft below it. . . . . . . 159

6.17 Temperature history at fracture 1 and 2 positions, sensitivity analysis

to zone 2 depth when fracture 1 is connection the two zones. . . . . . 160

6.18 Pressure derivative, sensitivity analysis to zone 2 depth when fracture

1 is connection the two zones. . . . . . . . . . . . . . . . . . . . . . . 161

7.1 Formation of secondary fractures within cooled region. . . . . . . . . 164

7.2 Crossing fractures model representation. . . . . . . . . . . . . . . . . 165

7.3 Crossing fracture model for uniform flux. . . . . . . . . . . . . . . . . 167

7.4 Pressure response for vertical well producing from uniform flux main

hydraulic fracture crossed by 20 perpendicular secondary fractures. . 169

7.5 Uniform flux crossing fractures: sensitivity analysis to secondary frac-

tures half length (xfDi). . . . . . . . . . . . . . . . . . . . . . . . . . 170

7.6 Uniform flux crossing fractures: sensitivity analysis to number of cross-

ing fractures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

xxiv

7.7 Uniform flux crossing fractures: sensitivity analysis to crossing fracture

orientation relatively to the main hydraulic fracture. . . . . . . . . . . 172

7.8 Crossing fractures model for infinite-conductivity fractures. . . . . . . 173

7.9 Infinite conductivity crossing fractures: sensitivity analysis to sec-

ondary fractures half length (xfDi). . . . . . . . . . . . . . . . . . . . 176

7.10 Infinite conductivity: sensitivity analysis to number of crossing fractures.177

7.11 Wellbore storage effect for drawdown in vertical well with multiple

crossed fractures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

7.12 Multiple fracture closure schematics. . . . . . . . . . . . . . . . . . . 180

7.13 Storage coefficient for multiple fracture closure, Example 1. . . . . . . 185

7.14 Dimensionless falloff pressure for closing fracture, Example 1. . . . . . 185

7.15 Transformed falloff pressure for closing fracture, Example 1. . . . . . 186

7.16 Storage coefficient for multiple fracture closure, Example 2. . . . . . . 186

7.17 Transformed falloff pressure for closing fracture, Example 2. . . . . . 187

B.1 Mass Balance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

B.2 Numerical problem structure: Jacobian, unknowns and residual. . . . 212

D.1 Numerical model verification against analytical solution for infinite act-

ing radial flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

D.2 Numerical model verification against analytical solution for wellbore

pressure in presence of fixed vertical fracture. . . . . . . . . . . . . . 225

D.3 PKN geometry verification against analytical solution for uniform dis-

tributed stress load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

D.4 KGD geometry verification against analytical solution for uniform dis-

tributed stress load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

D.5 Forced convection verification. . . . . . . . . . . . . . . . . . . . . . . 231

D.6 Comparison between numerical and analytic fracture temperature. . . 233

D.7 Reservoir thermal model verification. . . . . . . . . . . . . . . . . . . 235

E.1 Simplified representation of wellbore during warmback. . . . . . . . . 237

E.2 Logarithm of well temperature versus time during warmback. . . . . . 240

xxv

E.3 Simplified representation of wellbore during flowback. . . . . . . . . . 242

xxvi

Chapter 1

Introduction

The process of fracture initiation and propagation in a rock mass by injection of a

pressurized fluid from a borehole is called hydraulic fracturing. Hydraulic fracturing

is a widely applied well stimulation technique. Oil and gas operators around the

world have used hydraulic fractures successfully to increase production and reserves.

Recently the application of multiple hydraulic fractures along horizontal wells has

made possible the exploitation of unconventional reservoirs, such as shale gas and

shale oil. Hydraulic fracture operations allow us to bypass near-wellbore damage and

return a well to its “natural” productivity, and extend a conductive path deep into

a formation and thus increase productivity beyond the natural level (Economides

and Nolte, 2000). Although operational practices have shown considerable evolution

over a few decades, hydraulic fracturing characterization still has a lot of room for

improvement.

Hydraulic fracturing is a complex mathematical problem that involves the me-

chanical interaction of the propagating fracture with the fluid dynamics of the in-

jected fluid (Gidley et al., 1989). The modeling of the hydraulic fracturing process

involves the coupling of at least three processes: the mechanical deformation induced

by the fluid pressure on the fracture surfaces; the flow of fluid within the fracture;

and the fracture propagation. Usually, the solid (rock) deformation is modeled using

the theory of linear elasticity, which is represented by an integral equation that de-

termines the nonlocal relationship between the fracture width and the fluid pressure

1

2 CHAPTER 1. INTRODUCTION

(Economides and Nolte, 2000; Howard and Fast , 1970). An accurate modeling and

placement of created fractures allows better reservoir management.

The oil and gas industry has made considerable progress in fracture modeling.

Numerous computer simulators exist for design and evaluation of hydraulic fracture

treatments. Despite recent advances, there is still much room to improve the charac-

terization, design and execution of fracturing treatments.

Pressure is by far the most commonly used data for fracture analysis, but there are

others tools to investigate fracture geometry and effectiveness, such as near-wellbore

radioactive tracing or microseismic fracture imaging (Barree et al., 2002). Recent

developments in bottom-hole data acquisition techniques have allowed real-time mon-

itoring of hydraulic fracturing using fiber-optic distributed temperature sensing sys-

tems (DTS) to estimate the fracture initiation depth, vertical coverage, number of gen-

erated fractures, effects of diverting agents and undesired flow behind casing (Sierra

et al., 2008). One of the biggest advantages of DTS systems is that it enables us

to observe a dynamic temperature profile along the wellbore during treatments as

opposed to conventional temperature logs which can only provide us with a snapshot

of the temperature profile during a shut-in period. DTS systems generally do not

interfere with flow and can be used for short-term as well as permanent monitoring.

Originally, the temperature profiles, usually obtained by production logging tools

(PLT), have been a tool to estimate qualitatively where the flow took place. However,

temperature data are a rich source of information that has been collected for years,

together with the pressure, but not fully used. Recently, Duru and Horne (2008)

built a comprehensive reservoir thermal model considering conductive and convective

mechanisms and also other thermal phenomena, like viscous dissipation and the adi-

abatic expansion heating/cooling effect. Following their previous development, Duru

and Horne (2010) introduced the potential of using temperature data during history

matching. The use of temperature information improved the accuracy of estimation

of the porosity field.

Sui et al. (2008) applied temperature transient analysis to commingled reservoirs.

They presented a method to determine multilayer formation properties from pressure

and distributed temperature data, and concluded that layer permeability, damage

3

permeability and damage radius can be determined uniquely using single-point tran-

sient pressure and transient distributed temperature data. App and Yoshioka (2013)

have performed a dimensionless analysis of steady-state temperature in multilayer

reservoirs, which was used to demonstrate that the layer permeability has a strong

impact on layer sand-face flowing temperatures.

An example of the particularities of temperature measurements along a horizontal

wellbore in comparison with pressure is presented in Figure 1.1. This figure shows a

very important difference between the pressure and the temperature records: while

the pressure history from all the six sensors overlay each other, each temperature

curve is different from the others. Some of the temperature differences can be related

with difference in depth, and consequently in geothermal gradient, but others show a

distinct behavior, probably a consequence of upstream flowing zones.

When the focus turns to temperature modeling of the hydraulic fracturing process,

many temperature models presented in the literature are focused on better prediction

of fracturing fluid behavior to optimize proppant placement and/or acid reactions in

acid treatments (Kamphuis et al., 1990; Settari , 1980; Tabatabaei , 2011). However,

the use of temperature analysis applied to quantitative fracture and reservoir charac-

terization is not a common practice. As was pointed out by Silva et al. (2012), the

temperature transient analysis potential was not explored fully yet.

With the development of new measurement technologies, such as DTS, the char-

acterization practices of temperature analysis are likely to be improved. Seth et al.

(2010) presented a numerical solution for temperature response during a hydraulic

fracture job using a simplified fracture growth model by approximating the inter-

action between fracture and reservoir by an overall leak-off coefficient, without any

pressure consideration. They showed that the temperature data can give useful infor-

mation about the rock thermal properties, such as conductivity, and also about the

leak-off coefficient. Tabatabaei and Zhu (2011) have presented an inversion procedure

for interpretation of fracture fluid distribution from temperature data. This proce-

dure was applied to hypothetical cases, and the fracture representation was actually

equivalent to specifying a high permeability value to certain grid-blocks and allowing

fluid entrance only at those locations. There was no consideration of fluid-induced


Figure 1.1: Real data of distributed pressure and temperature surveys along horizon-tal well, extracted from Valiullin et al. (2009). The top figure shows well trajectoryand the location of the six gauges, the lower two figures show temperature and pres-sure histories of those six gauges.

1.1. LITERATURE REVIEW 5

fracture growth.

The development of thermal models linked with the pressure response dedicated

to hydraulic fracturing is necessary for a more accurate investigation of potential

application of temperature transient analysis in such scenario. Specially, continuous

temperature profiles provided by DTS should be considered.

A comprehensive pressure and temperature model can clarify the advantages and

limitations of temperature data analysis, and how it can complement pressure data

interpretation. Temperature may be the only source of information available for

wells equipped with DTS, for example. The need to understand better the informa-

tion that are contained by temperature, and its complementary use in addition to

the traditional pressure rate analysis during hydraulic fracturing has motivated the

developments presented in this thesis.

1.1 Literature Review

The literature review presented in this chapter is divided in three main parts: hy-

draulic fracturing modeling and characterization, classical well testing analysis in

fractured wells, and temperature modeling and analysis.

1.1.1 Hydraulic fracturing modeling and characterization

The classic hydraulic fracturing studies look for a better characterization of fluid and

rock properties in order to optimize the fracturing job. To achieve this goal, tests

prior to the main fracture are performed. These tests are made inside the lab and

also at the actual well, such as a step-rate test, minifrac or calibration test, pump-

in/flowback, etc.

Nolte and Smith (1981) presented the basis for the interpretation of pressure be-

havior during hydraulic fracturing based on Carter’s leakoff model (Carter , 1957).

They demonstrated that a log-log plot of fracturing pressure above the closure stress


versus treatment time can be used to identify periods of unrestricted extension, con-

fined height, excessive height growth and restricted penetration. Nolte (1979) pre-

sented a pressure decline analysis theory (pressure after injection and before fracture

closure). He suggested procedures to quantify the fluid loss coefficient, fracture length

and width, fluid efficiency and time of fracture closure. The practice of performing

minifracs was introduced by his development.

In 1997, Nolte (Nolte, 1997; Nolte et al., 1997) extended the attention to the

after-closure period, to determine not only fracture geometry characteristics, but also

reservoir properties such as transmissibility from calibration tests. This kind of test

is performed before the main fracture job, usually injecting fluid without proppant.

The duration of the injection is small, of order a few minutes. The theory was based

on considering the injection as if it was instantaneous (Gu et al., 1993; Ayoub et al.,

1988). The assumption is reasonable, because the injection period is much shorter

than the shut-in (falloff). This characteristic allows the pressure response during the

shut-in to be independent of the injection history, and consequently not dependent

on how the fracture originated. The impulse solution is given by the derivative with

respect to time of the equivalent solution for constant rate injection (Soliman et al.,

2005).

Craig and Blasingame (2005) developed a more general interpretation procedure

treating the data in a similar way to that suggested by Peres et al. (1993) for slug tests.

Integrating the impulse response with respect to time, it is possible to interpret the

modified data using the traditional well testing solutions. Usually the prefracturing

tests are performed without addition of proppant, therefore during the shut-in the

fracture recedes and closes.

A frequent assumption among the hydraulic fracturing models is that fluid loss

from fracture to reservoir rock is described by Carter’s leakoff model (Carter , 1957),

where leak-off velocity is a function of time and the leak-off coefficient. Different from

the previous assumptions, Plahn (1996) presented a numerical model that uses the

reservoir pressure as a control on the flux from the fracture to the reservoir. Plahn

et al. (1995) investigated the pressure response during pump-in/flowback tests by

numerical simulation. Their results showed that the characteristic flowback pressure


signature is caused by near-wellbore fracture “pinching”, not global closure of the

entire fracture.

The geometry of the induced fracture is dominated by the rock mechanical prop-

erties, in-situ stresses, the rheological properties of the fracturing fluid, and local

heterogeneities such as natural fractures and weak bedding planes (Dahi , 2009). It

is well known that hydraulic fractures are indeed more complex than the idealized

planar model. As was mentioned by Fisher and Warpinski (2012), the mineback

works performed in the 1970s and 1980s showed clearly that fractures are much more

complex than envisioned by conventional models of the process. On the other hand, a

large amount of real data have shown a very limited hydraulic fracture height growth,

where most of the fractures stay in the nearby vicinity of the target reservoir.

In order to include the full calculation of stress field and geomechanical aspects

of rock failure it is necessary apply finite elements, as stated in Dean and Schmidt

(2008). They developed a three-dimensional simulator coupling geomechanics, multi-

phase/multicomponent fluid flow and heat conduction and convection during fracture

growth. However, the paper only showed examples of single-phase and isothermal

problems.

Suri et al. (2011) considered the pressure response for an injector well injecting

above the fracture pressure and causing fracture growth. Their model took into

account not only the pressure change in the reservoir and the fracture during injection,

but also considered additional pressure drops due to permeability reduction by solids

deposition near the well and the buildup of an external filter cake inside the well and

the fracture. In their model the fracture growth follows the PKN model (Nordgren,

1972).

1.1.2 Classical well testing fractured well pressure solutions

The pressure response of a vertical fractured well producing at constant rate was

presented by Gringarten et al. (1974) . Their work considers two different models of

the flow inside the fracture: the uniform flow (each point along the fracture length

produces the same rate) and the infinite conductivity (there is no pressure difference


along the fracture). Although the two models are distinct, their responses differ only

modestly in the transition between the linear and the pseudoradial regimes.

The fractured well characteristic flow regimes are the linear or bilinear flow at

early times, for infinite and finite fracture conductivity, respectively. The linear flow

is identified on the log-log plot by pressure and its logarithmic derivative presenting

straight lines of 1/2 slope behavior. The bilinear signature is the 1/4 slope (Cinco-Ley

and Samaniego-V , 1981).

Horne and Temeng (1995) showed the analytical solution for transient pressure

and prediction of performance of a horizontal well with multiple transverse fractures.

The model considers a series of identical (same dimensions and properties) fully pen-

etrating fractures. It was shown that the fractures will ultimately interact with each

other, which will reduce the effectiveness at later times. Usually multiple fractures in

horizontal wells are implemented in low permeability formations.

A vertical fracture crossing a multilayer reservoir was considered by Bennett et al.

(1985). They made a series of simplifications in order to obtain an approximate

analytical solution. Earlier papers have solved the problem numerically, but the

analytical solution increases the physical understanding. It was stated that if the

layers are communicating through a vertical fracture the flow rates measured at the

wellbore at early and intermediate times (before pseudoradial flow) depend on the

properties of the fracture system, rather than those of the reservoir. This statement

means that an attempt to define layer properties based on wellbore measurements

will be unsuccessful. The analytical pressure solution for a fractured well in a double-

porosity reservoir was developed by Houze et al. (1988).

Different from the hydraulic fracturing literature, the traditional well testing ap-

proaches for fractured well characterization deal with a static fracture. That is, the

solutions and pressure interpretation techniques assume the fracture exists with the

same geometry and dimensions from the beginning and remains the same during the

full rate history. Some exceptions are found in studies of pressure behavior in injector

wells performing injection above the rock fracture pressure. Koning and Niko (1995)

and van den Hoek (2002) considered the fracture closure during the falloff period

(well shut-in after an injection period). The assumption of long-time injection allows


the solution be independent of the way the fracture growth happened. The solution

is basically the step-change wellbore storage, where after fracture closure the storage

coefficient decreases.

1.1.3 Temperature modeling and analysis

Several authors have built models to study the heat transfer problem between wellbore

and formation in the context of the oil and gas industry. One of the earliest works

on temperature prediction was by Ramey (1962). Ramey’s method approximates the

pressure gradient of vertical wellbores by the hydrostatic difference, neglecting fric-

tional pressure drop, and assumes steady-state heat transfer inside the wellbore and

transient conduction from the reservoir. The solution was obtained semianalytically

under these assumptions. Ramey’s temperature prediction model works for either a

single-phase incompressible liquid or a single-phase ideal gas in vertical injection and

production wells. Recently, Wang and Horne (2011) have presented a thermal well

model focused on the DTS scenario. The study considered multiphase multicompo-

nent flow.

With the development of new measurement technologies, such as DTS, the char-

acterization practices of temperature analysis are likely to be improved. The current

DTS systems provide continuous and complete wellbore temperature profiles over the

duration of the monitoring period. Discrete wellbore temperatures can be obtained

as frequently as every 30 seconds with a measurement point for every meter along the

wellbore. These systems use a fiber-optic cable assembly that can be deployed in sev-

eral configurations in the well, and more importantly, across the perforated interval

(Glasbergen et al., 2009).

Ouyang and Belanger (2006) built a numerical wellbore model, specially for DTS

data interpretation. Using their wellbore model, Ouyang and Belanger (2006) suc-

ceeded in estimating the flowrate profile by solving an inverse problem. DTS systems

generally do not interfere with flow and can be used for short-term as well as perma-

nent monitoring.

Distributed temperature surveillance has been used conventionally to monitor


the performance of water injectors with the warm-back technique, where the well is

shut in for a period of time and the temperature response is recorded while the well

warms back toward the geothermal gradient (Brown et al., 2003). Holley et al. (2010)

have pointed out the benefits that DTS can bring when applied simultaneously with

microseismic interpretation during hydraulic fracturing, decreasing the uncertainties

such as fracture initiation point, interval isolation, among others.

Huckabee (2009) presented the application of DTS measurements to the scenario

of hydraulic fracturing for both vertical and horizontal wells. Examples of multistage

hydraulic fracturing, zones isolation and warmback differences between injection in-

tervals demonstrate the qualitative application of distributed temperature surveys.

Still in the stimulation context, the DTS was used to determine fluid distribution

during matrix treatment in Glasbergen et al. (2009).

Maubeuge et al. (1994) described an energy equation that takes into account the

temperature effects due to the decompression of the fluid and the frictional heating

that occurs in the formation, and how this equation is coupled with the pressure

equations in a finite-element numerical well model. The application to temperature

log interpretation was demonstrated on real data sets. They have emphasized that

it is very difficult to estimate virgin geothermal gradient, it is even more difficult to

calculate thermal gradient after production or injection operations.

Recently, Duru and Horne (2008) built a comprehensive reservoir thermal model

considering conductive, convective mechanisms and also other thermal phenomena,

like viscous dissipation and adiabatic expansion heating/cooling effect. Following the

previous development, Duru (2011) has introduced the potential of using temperature

data during history matching and has performed laboratory experiments to prove

it. The use of temperature information improved the accuracy of estimation of the

porosity field.

Sui et al. (2008) applied temperature transient analysis to commingled reservoirs.

They presented a method to determine multilayer formation properties from multiple-

point temperature data and single-point pressure, and concluded that layer perme-

ability, damage permeability and damage radius can be determined uniquely using

single-point transient pressure data and distributed transient temperature data. The


study showed that temperature transient analysis is able to determine the kind of

damage that exists around the well. While the pressure analysis only gives the total

skin value, the temperature was able to indicate the damage radius and the perme-

ability of the area. The temperature characteristic that allows this information is its

slower diffusive response when compared with pressure.

App and Yoshioka (2013) have performed a dimensionless analysis of sandface

flowing temperature in a multilayer reservoir. They considered a steady-state model,

which was used to demonstrate the dependency of the flowing temperature on both

the Peclet number and a dimensionless Joule-Thomson expansion coefficient. As the

Peclet number and the pressure drop are associated with the layer permeability, the

layer permeability has a strong impact on layer sandface temperatures during flow.

Many temperature models presented in the literature related with fracturing are

focused on better prediction of fracturing fluid behavior to optimize proppant place-

ment and/or acid reactions in acid treatments (Kamphuis et al., 1990; Settari , 1980).

The acid fracturing can be represented by adding a source term to the energy balance

equation referring to the chemical reaction (Tabatabaei , 2011). One of the earliest

works dedicated to the use of temperature surveys to identify the fracture zone was

presented by Agnew (1966). The paper pointed out the need of perform temperature

surveys immediately after the treatment, but the analysis was limited to the fracture

height estimate based on purely qualitative temperature log analysis. However, the

use of temperature to quantitative fracture and reservoir characterization is not a

common practice.

Seth et al. (2010) presented a numerical solution for temperature response during

a hydraulic fracture job using a simplified fracture growth model by approximating

the interaction between fracture and reservoir by an overall leak-off coefficient, with-

out any pressure consideration. They showed that the temperature data can give

useful information about the rock thermal properties, such as conductivity, and also

about the leak-off coefficient. Tabatabaei and Zhu (2011) have presented an inver-

sion procedure for interpretation of fracture fluid distribution from temperature data.

This procedure was applied to hypothetical cases, and the fracture representation was

actually equivalent to specifying a high permeability value to certain grid-blocks and


allowing fluid entrance only at those locations. There was no consideration of fluid-

induced fracture growth. Similar work for vertical wells was presented by Hoang et al.

(2011).

One of the critical aspects of temperature data is the location of the gauge. If

the measurements were done inside the tubing, during fluid injection the registered

temperature will be basically correspondent to that of the injected fluid. However,

during shut-in the temperature will tend to warm-back and this behavior will be

influenced by the rock and fracture properties. Sierra et al. (2008) showed that DTS

can be deployed not only inside the tubing (Figure 1.2-left), but also behind the

casing (Figure 1.2-right), which is less sensitive to effects of flow inside the casing.

Figure 1.2: DTS deployment schemes, extracted from Sierra et al. (2008). The leftfigure shows an example of fiber optic placed inside the tubing, while the right sidefigure shows the fiber permanently installed and cemented behind the casing.

The correct heat transfer coefficients are critical for an accurate temperature pre-

diction. As soon as the fluid is injected at the wellhead it exchanges heat with

the surrounding medium changing its temperature and also the temperature of the

area around the well. Willhite (1967) considered the different interfaces between the

tubing and the formation and presented expressions for the equivalent heat transfer

coefficient.

Wellbore effects influence the temperature measurements even more prominently

than pressure measurements. Temperature wellbore storage (TWBS) is addressed

in Ramazanov et al. (2010), where an analytical solution is presented for vertical

1.2. STATEMENT OF THE PROBLEM 13

well case. The solution depends on flow temperature from the formation and well

characteristics.

1.2 Statement of the Problem

As mentioned in the beginning of this chapter, temperature data have a huge poten-

tial that has not been explored fully yet. Specially with the development of sensing

technologies, the distributed measurements are now reliable and have been used in-

creasingly to monitor hydrocarbon reservoirs. The need to understand better the

information that is carried by temperature and its complementary use in addition

to the traditional pressure rate analysis motivated the developments presented in

this thesis. The key research objective for this work has been the investigation of

temperature and pressure analysis applied to hydraulic fracturing scenarios.

We developed a more comprehensive model, which accounts for the pressure ef-

fect on the temperature response, as well as a dynamic fracture that grows during

injection and eventually is allowed to close during falloff. This study focused on ob-

taining a better understanding of fracture, reservoir and well interaction, and which

are the properties that affect pressure and temperature responses the most. Also,

the developed model can be used to study tests prior to the main fracturing job,

such as minifrac tests. Minifracs appear as a reliable alternative for tight formation

characterization. We explored the fracture creation during injection, as well as the

type curves and data transformation technique for pressure decline analysis during

the falloff.

The horizontal well subjected to multiple hydraulic fracturing stages was also

addressed. This type of well and completion technique has become a key factor for

success in development of unconventional resources. The simultaneous growth of

multiple fractures and the interaction between them are important aspects that we

investigated. Additionally, how heterogeneities can affect the results was considered.

Our ultimate goal was to identify what are the advantages of adding temperature

analysis when compared to the single-point pressure interpretation.


1.3 Dissertation Outline

This dissertation is organized in eight chapters, including the Introduction and Con-

clusions.

Chapter 2 describes the mathematical modeling of hydraulic fracture growth and

closure, flow inside the fracture and through porous media, and the energy balance

for well, fracture and reservoir during the hydraulic fracturing process. This chapter

addresses the growth of a single vertical fracture from a vertical well in a homogeneous

and isotropic reservoir.

Chapter 3 presents the sensitivity analyses performed on the model developed in

Chapter 2. Both pressure and temperature are analyzed. Important features and

behavior of the temperature measurement are also presented.

Chapter 4 describes the applications of the developed model to a minifrac test

scenario. The program output was subject to data transformation and it is demon-

strated that the process estimates the input reservoir permeability and final fracture

length from the traditional well testing interpretation techniques. It was also noticed

that the transformed data can give information about the moment of fracture closure

or stabilization. Application to field data is also presented.

Chapter 5 addresses the creation of multiple fractures along a horizontal well. Both

sequential and simultaneous fracture growth was studied. The presence of reservoir

permeability heterogeneity was investigated and the capability of temperature data

to identify the existence of such structure was explored. Capabilities and limitations

of temperature analysis are presented through different geometry analyses.

Chapter 6 is also related to horizontal multifractured wells, but in this chapter it

is considered that one of the fractures interconnects different zones vertically. This

chapter is motivated by microseismic evidence of activity captured out of the target

reservoir zone. The temperature data analysis is the main focus of this chapter.

Chapter 7 is a particular component of this dissertation, composed of analytic

solutions for the pressure response of a single vertical hydraulic fracture crossing

multiple natural fractures. This chapter considers a constant fracture and reservoir

characteristics exposed to a constant rate drawdown test.

1.3. DISSERTATION OUTLINE 15

Chapter 8 summarizes the work, presents conclusions and outlines recommenda-

tions for further work.

Chapter 2

Hydraulic Fracturing Forward

Model

This chapter presents the mathematical model of hydraulic fracture growth and clo-

sure in a vertical well scenario (Figure 2.1). The modeling is composed of three main

parts: the geomechanics of fracture growth, mass balance and energy balance.

The fracture is able to grow and eventually to decrease in volume (to close) due

to fluid loss to the reservoir. The model described in this chapter couples reservoir

and fracture flow to the energy balance. No prior assumption about fluid loss from

fracture to reservoir is required, and so Carter’s model is not assumed (Carter , 1957).

The coupling between mechanics and mass balance follows an approach similar to the

one described by Plahn et al. (1995). This approach generates the pressure response

and predicts the flow velocities throughout the reservoir during fracture creation by

coupling the mass balance inside the reservoir and within the created fracture with

well-known two-dimensional fracture geometries, named PKN (Nordgren, 1972) and

KGD (Geertsma and Klerk , 1969). The choice of simplified geometries is related to

the fact that the goal of this first analysis is not the full mechanical description, but

to understand pressure and temperature behavior when fracture properties are not

constant.

In addition to accounting for variable fracture length and width, the model con-

siders also the pressure effects on the energy balance. The solution is obtained by

17

18 CHAPTER 2. HYDRAULIC FRACTURING FORWARD MODEL

Figure 2.1: Hydraulic fracture in vertical well.

numerical methods, where the mass balance coupled with geomechanics is solved first

and then applied to the energy balance to obtain the equivalent temperature response,

not only during the injection period, but also in the subsequent falloff or flowback.

The schematic of the solution path is presented in Figure 2.2.

Figure 2.2: Solution path.

The following sections address the detailed pressure and temperature models in-

dividually. The derivation of the governing equations for heat and mass transfer in a

porous medium begins with the consideration of the heat and mass transport in the

fluid continuum saturating the void space of the porous medium (Nield and Bejan,

2006).

2.1. MASS BALANCE AND FRACTURE GROWTH 19

2.1 Mass Balance and Fracture Growth

Even in its most basic form, hydraulic fracturing is a complicated process to model,

as it involves the coupling of at least three processes: (i) the mechanical deformation

induced by the fluid pressure on the fracture surfaces; (ii) the flow of fluid within

the fracture; and (iii) the fracture propagation. Usually, the solid (rock) deformation

is modeled using the theory of linear elasticity, which is represented by an integral

equation that determines the non local relationship between the fracture width and

the fluid pressure.

Fracture growth and closure modeling requires coupling two-dimensional fracture

geometries, such as in the PKN and KGD models (Figure 2.3), to the reservoir model.

Both reservoir and fracture are discretized and solved numerically by the finite dif-

ference technique. Finite difference reservoir simulators are commonly used to model

fluid flow in reservoirs. In this research, the Plahn (1996) approach was followed to

generate the pressure response and predict the flow velocities through the reservoir

during fracture creation, which are necessary for the temperature prediction. The

important characteristic of this model is that it does not limit the flow to be per-

pendicular to the fracture plane, allowing flow in both x and y directions (Figure

2.4).

The basic assumptions adopted in this model are:

1. There is no filter cake present or being built on the fracture faces.

2. The friction loss due to fluid passing through the perforations is not considered.

3. The fluid behavior is Newtonian.

4. Fracture is vertical and grows along constant direction.

5. Fracture is contained in the reservoir layer.

2.1.1 Mass conservation

The derivation of the mass conservation equation starts by considering a representa-

tive elemental volume (REV) of reservoir. In two dimensions (Figure 2.5), the mass


Figure 2.3: PKN (Nordgren, 1972) and KGD (Geertsma and Klerk, 1969) fracturegeometries.

Figure 2.4: Single fracture in a vertical well model representation. The fracture heightis the same as the reservoir thickness (h), it stays contained in the reservoir layer.


balance is represented for a block (i, j) as:

(mass flowrate into block i,j through the west face) - (mass flowrate out of block i,j

through the east face) + (mass flowrate into block i,j through the south face) - (mass

flowrate out of block i,j through the north face) - (mass flowrate removed from block

i,j via sinks) = (rate of change of mass in block i,j)

Figure 2.5: Mass balance.

Translating the mass balance to symbolic form:

(qρ)i−1/2,j − (qρ)i+1/2,j + (qρ)i,j−1/2 − (qρ)i,j+1/2 − Si,j =∂(φρV )i,j

∂t(2.1)

where (qρ) denotes the mass flow rate normal to a particular face of the grid-block,

S is the rate at which mass is removed from the grid block via sinks (or added via

sources), φ is the porosity of the grid-block, and V is the bulk volume of (i, j) grid-

block. The analysis assumes a two-dimensional system with uniform thickness, h, and

the flow velocities inside the reservoir are given by Darcy’s law (Equation 2.2). The

Darcy model shows the relationship between area-averaged fluid velocity, pressure


gradient and fluid viscosity.

qn = −Anknµ

∂p

∂n(2.2)

qn is the volumetric flowrate in the direction specified by n (e.g., n = x, y), kn is

the effective permeability in direction n, An is the cross sectional area of the surface

normal to n through which fluid is transported, and µ is the viscosity of the flowing

fluid. Substituting Darcy’s Law (Equation 2.2) into the mass balance (Equation 2.1)

with appropriate definitions for each An term, yields:

(

−ρk∆yh

µ

∂p

∂x

)

i−1/2,j

−(

−ρk∆yh

µ

∂p

∂x

)

i+1/2,j

+

(

−ρk∆xh

µ

∂p

∂y

)

i,j−1/2

−(

−ρk∆xh

µ

∂p

∂y

)

i,j+1/2

− Si,j = ∆x∆yh∂

∂t(φρ)i,j (2.3)

Equation 2.3 is solved numerically and the detailed discretization is presented in

Appendix B.

The initial and boundary conditions describe a closed reservoir in equilibrium

prior to the injection:

p(x, y, t = 0) = pi (2.4)

S(x = xw, y = yw, t) = (qρ)inj (2.5)

(qρ)(x = xe, y, t) = (qρ)(x, y = ye, t) = 0 (2.6)

After the discretization of the reservoir, and stating the conservation laws, a sys-

tem of equations can be written honoring the boundary and initial conditions. One

grid-block is dedicated exclusively to represent the well. The well grid-block is in

direct contact with fracture and reservoir, and it can account for open completion or

perforated cemented casing. The porosity and permeability are set to represent the

actual well volume and the very high conductivity. This grid-block has a source term


to account for the injected (or produced) volume.

2.1.2 Fracture growth and closure

The fracture growth is controlled by predefined propagation criteria. In this work,

we applied the two-dimensional geometry models: PKN (Nordgren, 1972) and KGD

(Geertsma and Klerk , 1969). The propagation criterion for a PKN fracture is based

on fluid velocity at the tip. While the fracture is growing, pressure at the tip is equal

to the closure pressure pc. For KGD fractures the propagation criterion is based on

the magnitude of the mode I stress intensity factor, KI :

KI(t) = 2

√

12xf (t)

π

∫ xf (t)

0

σf (x, t)− pc√

x2f − x2

dx (2.7)

When the KI overcomes the rock critical value (KIC) the fracture is allowed to

grow, otherwise the fracture stays at the same position.

The modeled fracture is vertical with fixed height equal to the reservoir thickness

(h), as shown in Figure 2.4. Fracture deformation (i.e., fracture aperture) is modeled

using England and Green’s well-known solution for a pressurized plane-strain crack,

in an infinite, linearly elastic medium (England and Green, 1963):

w(ξ, t) =8(1− ν2)

πE

∫ d(t)

ξ

u√

u2 − ξ2

∫ u

0

σf (s, t)− pc√u2 − s2

dsdu (2.8)

where d is the characteristic length of the fracture, ξ is the position along d, w(x) is the

fracture width distribution, σf (x) is total internal fracture stress, and pc is the pres-

sure of fracture closure. E and ν are the mechanical properties of the rock, Young’s

modulus and Poisson’s ratio. During the injection period σf (x) can be considered

equal to pf (x), where pf (x) is the fluid pressure distribution in the fracture. During

fracture closure, the existence of asperities is considered (Figure 2.6), which intro-

duces an additional contact stress (σm) that acts against complete fracture closure.

In this case the total fracture stress is written as:


σf = pf + σm (2.9)

Figure 2.6: Asperities on fracture walls (modified from Danko (2013)). The asperitieson the fracture walls touch during the closure, retaining a high conductivity path alongthe fracture.

The inclusion of asperities in the model ensures that an enhanced conductivity

path along the fracture will be preserved, even though the fracture conductivity de-

creases until a stable fracture volume is reached during shut-in (or flowback). The

minimum fracture aperture at which the walls will first touch (asperities), wf,min, is

an input in this approach, which considers that the asperities contact one another

along the center line of the fracture. The magnitude of minimum aperture is of the

order of 10−4 ft. The contact stress equations are computed using the nonlinear

Barton-Bandis discontinuity closure model (Brady and Brown, 2004):

σm(x, t) =∆wf (x, t)

A− B∆wf (x,t)

wf,min

(2.10)

where A and B are constants, and:

∆wf (x, t) = wf,min − wf (x, t) (2.11)

The effect of A and B on the contact stress is presented in Figure 2.7. The reference

value of A equal to 9.375 × 10−7 ft/psi is obtained from experiments presented in


Bandis et al. (1983). The value of B [ft/psi] can be obtained by the maximum closure

aperture where σm →∞.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10−4

0

200

400

600

800

1000

1200

1400

1600

1800

2000

∆wf (ft)

σm

(∆w

f)

A=9.375e−7, B=9.375e−7

A=1e−6, B=1e−6

A= 1e−5, B= 1e−5

A=9.375e−7, B=4.6875e−7

A=9.375e−7, B=0

Figure 2.7: Contact stress model: effect of parameters A and B [ft/psi]. The lowerthe A value the stronger the contact stress as the fracture is closing against theasperities. The ratio B/A influences the nonlinearity of the contact stress with theclosure width.

The vertical plane-strain assumption simplifies the elasticity problem because frac-

ture width for each vertical cross section depends only on the local pressure (i.e., there

is no lateral coupling), which is the basis for the PKN fracture model. The PKNmodel

is indicated as the appropriate geometry model when fracture length, 2xf , exceeds

fracture height, h (Economides and Nolte, 2000). The vertical plane-strain assump-

tion requires that each vertical cross section deforms independently of all others. This

results from the implicit assumption that pressure gradients along the fracture are

relatively small. Since pressure is uniform over each vertical cross section, Equation

2.8 reduces to a simple form:

wf =2(1− ν2)(σf − pc)

E

√

h2f − 4z2 (2.12)

Equation 2.12 gives an elliptical width profile with maximum width, wmax, at the

center-line of the fracture. The average width that gives the same area as the elliptical


cross section is:

wf =π

4wmax =

πhf (1− ν2)

2E(σf − pc) (2.13)

For the KGD geometry Equation 2.8 has to be solved by numerical integration.

2.1.3 Coupling between fracture, well and reservoir

The coupling between fracture and reservoir starts by assuming a direction of growth

in advance, defining the potential fracture as special grid-blocks (Figure 2.8). The

fracture grids have variable properties, which relate pressure, stress, fracture width

and length. For a homogeneous reservoir, lines of symmetry can be traced from

the wellbore, so the reservoir can be represented by only one fourth of the total,

as is shown in Figure 2.9. Dynamic fracture behavior (growth and shrinkage) is

incorporated into the numerical model by changing fracture grid-block properties

such as porosity and permeability. As pressure increases due to the injection of

fluid, the fracture grows and the grid-block properties are not constant anymore. As

the fracture grows, the fracture grid-block has to advance and the permeability and

porosity have to change due to the increase in fracture width and length. When the

injection stops, the fluid pressure decreases and so does fracture aperture. To honor

fracture volume and conductivity, grid-block permeability and porosity also decrease.

The flow of a Newtonian fluid between parallel plates separated by a distance w

is given by the equation (Lamb, 2000):

q = −w3hf

12µ

∂p

∂x(2.14)

In this way the fracture grid properties need to be modified at each time-step to

honor the fracture geometry. The fracture permeability is determined by the fracture

aperture stated by the following equation:

kf =wf

2

12(2.15)


Figure 2.8: Two-dimensional reservoir, well and fracture volume discretization. Themodel couples well, fracture and reservoir. The well at the reservoir depth is repre-sented by one grid-block, in green in this figure. The active fracture grid-blocks arerepresented in dark blue and the potential grid-blocks are represented in light blue.Reservoir grid-blocks are in white. For homogeneous and isotropic medium lines ofsymmetry can be traced (red traced lines).

Figure 2.9: Isotropic and homogeneous reservoir, well and fracture simplified gridrepresentation (a quarter of the total model). The well at the reservoir depth isrepresented by one grid-block, in green in this figure. The active fracture grid-blocksare represented in dark blue and the potential grid-blocks are represented in lightblue. Reservoir grid-blocks are in white.


Porosity is also changed in order to honor variable fracture volume in a fixed-

dimension grid-block representation, as described in Equation 2.16. The system of

equations is solved by a fully implicit numerical scheme.

φfi =wf

∆yi(2.16)

As permeability and porosity along the fracture grid-blocks are not constant any-

more, their changes and derivatives have to be incorporated into the Jacobian of the

Newton-Raphson scheme. In addition to that, contact stress terms also have to be

considered during shut-in or flowback, when asperities on the opposite fracture walls

touch. In this case, the Jacobian increases in size (Figure 2.10), as for each frac-

ture grid-block there will be a contact stress σm as unknown in addition to pressure.

The additional equations added to the system are the contact stress equations for

each fracture grid-block (Equation 2.10). The Jacobian is computed analytically and

updated at the end of each iteration.

Figure 2.10: Numerical problem structure: Jacobian, unknowns and residual.

In this way the residual equations are formed by the mass balance for every grid-

block in the model and the stress balance for each fracture grid-block, as follows:


Rp(i, j) = −Υn+1xi−1/2,j

(pn+1i,j − pn+1

i−1,j) + Υn+1xi+1/2,j

(pn+1i+1,j − pn+1

i,j )

−Υn+1yi,j−1/2

(pn+1i,j − pn+1

i,j−1) + Υn+1yi,j+1/2

(pn+1i,j+1 − pn+1

i,j ) + Si,j/h (2.17)

−∆xi∆yj∆t

[(φρ)n+1 − (φρ)n]; i = [1, Nx] and j = [2 : Ny]

Rσm = σn+1mi

−Ω(∆wn+1

fi)∆wn+1

fi

[A− B∆wn+1

fi

wf,min]

(2.18)

Here Ω is a smoothing function used to eliminate the discontinuity in the contact

stress (Equation 2.19). Figure 2.11 shows the behavior of Ω with decrease in aperture

for different γ coefficients. The higher the value of γ, the faster the contact stress

starts to influence the solution after the aperture goes below wf,min. The Jacobian

terms derived from Equations 2.17 and 2.18 are presented in Appendix B.

Ω(∆wf ) =

0, if ∆wf ≤ 0;

1− eγ∆w2f , if ∆wf > 0;

(2.19)

During the injection period, for each time the growth criterion is reached and the

fracture advances one grid-block it is necessary to ensure that negative aperture is not

created. If after the fracture tip position change the fracture geometry calculations

give negative width the tip returns to the previous iteration position, which will be

considered the stable tip position for that time-step. In this way the time-step is

completed and the next starts. For the shut-in period, if a negative aperture happens

at early time the problem is solved by cutting the time-step size, which will allow

the contact stress to commence smoothly and avoid such nonphysical behavior. The

fracture growth algorithm within one time-step is presented in Figure 2.12. More

details about fracture growth coupling can be found in Plahn (1996).


0 1 2 3 4

x 10−4

0

0.2

0.4

0.6

0.8

1

∆wf (ft)

Ω(∆

wf)

γ=1e4

γ=1e6

γ=1e8

γ=1e10

Figure 2.11: Smoothing function.

2.2 Energy Balance

During hydraulic fracturing, cold fluid is injected into a warm reservoir. The heat

transfer starts inside the well when the fluid is traveling down towards the injection

zone. When arriving at the reservoir depth, part of the injected fluid creates the

fracture and part of it is lost to the formation. The difference between injected fluid

and reservoir temperatures creates an altered temperature zone not only inside the

well and fracture, but also inside the reservoir, in the fracture neighborhood. In

the energy balance equation we consider not only heat exchange by convection and

conduction, but also heat changes due to pressure effects. The pressure influence is

due to the Joule-Thomson effect and the adiabatic expansion. The Joule-Thomson

effect is a change in the temperature of a fluid during expansion in a steady flow

process involving no heat transfer or work at constant enthalpy (Steffensen and Smith,

1973).

The temperature change per unit pressure change at constant enthalpy is defined

by the Joule-Thomson coefficient,

µJT =∂T

∂p=

βT − 1

ρC(2.20)

2.2. ENERGY BALANCE 31

Figure 2.12: Fracture growth algorithm. This figure describes the fracture tip move-ment check within one time-step, where itip is the grid-block position where the frac-ture tip is located.


where T is temperature, p is pressure, β is the coefficient of thermal expansion, ρ

fluid density, and C the specific heat capacity.

This effect means that as liquid or gas flows in the reservoir toward the well,

depending on its Joule-Thomson coefficient, it heats up or cools down because of the

pressure drop at the wellbore. A liquid or gas also heats up or cools down as it flows

up (or down) the wellbore because of the friction and pressure drop along the way.

In general, a pressure drop causes slight heating of flowing oil and water but a large

Joule-Thomson cooling for flowing gas (Brown et al., 2003).

In the same way as it was done for the mass conservation, the energy balance

equation accounts for the reservoir, well and fracture. The reservoir energy balance

equation is:

(λeffTr)− ρlCl−→v Tr + φβTr

∂p

∂t+ (βTr − 1)−→v p+ Seff = Ceff

∂Tr

∂t(2.21)

where:

λeff = φλl + (1− φ)λr (2.22)

and:

Ceff = φCl + (1− φ)Cr (2.23)

The right hand side of Equation 2.21 represents the transient temperature vari-

ation. The left hand side terms are heat conduction, heat convection, temperature

change caused by temporal fluid expansion, temperature change caused by spatial

fluid expansion and viscous dissipation, and finally the heat source term. A detailed

derivation of Equation 2.21 can be found in Sui et al. (2008).

In addition to that, the growing fracture problem accounts for the work done by

the fluid to increase the size of the fracture by breaking the reservoir rock:

W = pdAf

dt(2.24)

2.2. ENERGY BALANCE 33

where Af = wfh = f(x, t). For the fracture the equation is:

−ρlCl−→vfrTfr + βTfr

∂p

∂t+ (βTfr − 1)−→vfrp+

2hl

wf

(Tr − Tfr) +W = Ceff∂Tfr

∂t(2.25)

For the fracture equation (Equation 2.25) in addition to the previously described

terms there is the work done by the fluid to deform the rock and open the fracture

(W ) and heat conduction through the fracture wall between fluid inside the fracture

and the reservoir.

The models that describe distributed temperature behavior in a reservoir appear

as advection-diffusion equations with forcing terms that include compressibility effects

due to Joule-Thomson and adiabatic expansion. In contrast to diffusion, advection

does not destroy information. Advection is also nonsmoothing and preserves sharp

boundaries. Pressure propagation in a reservoir is strictly a diffusive process and as

a result, the temporal measurements give little indication about what is happening

spatially. On the other hand, temperature propagation has both advective and diffu-

sive components. The strength of each is determined by the Peclet number of the flow

process. The Peclet number is defined as a product of the Reynolds number and the

Prandtl number and measures the relative strength of advection over diffusion (Equa-

tion 2.26). In relatively high Peclet number processes (reservoir flow systems fall into

this category), the advective transport part becomes predominant. The mechanism of

advective transport and the general model for temperature distribution in a reservoir

make it useful to use temperature measurements to estimate the spatial location of

structures in the reservoir. This characteristic was explored in Duru (2011) when

using long-term temperature data in history-match problems.

Pe = Re× Pr =ρvL

µ

cfµ

λ(2.26)

The third part of the model is the wellbore. Ramey (1962) presented the classic

analytical solution for injection fluid temperature change from the well-head to the

reservoir based on the heat transfer between well and reservoir, and the convection


of fluid downhole. This analytic solution is incorporated in our model to calculate

the fluid temperature that arrives at the reservoir depth. The analytic solution states

that:

Tinj(z, t) = Tsurf + az − aA+ (Tinj(z = 0, t)− Tsurf + aA)ezA (2.27)

where:

A =ρqClf(t)

2πκ(2.28)

z is the distance down from the well head, a is the geothermal gradient, Cl is fluid

thermal heat capacity and f is a dimensionless time function representing the tran-

sient of the heat transfer coefficient between the well and the formation. More details

can be found in Ramey (1962). This temperature will be part of the source term for

the well grid-block energy balance equation (Sinj in Equation 2.29).

The well provides fluid to the fracture and to the reservoir. The overall heat

transfer coefficient depends on the kind of completion, for a cased and cemented

well, for example, both materials have to be considered while calculating U (Willhite,

1967). The contact area is given by 2πrwh. So, the well grid-block equation can be

written as:

−ρlCl−→qwTw + πr2wβTw

∂p

∂t+ (βTw − 1)qp+ 2πrwU(Tr − Tw) + Sinj = ρlClπr

2w

∂Tw

∂t(2.29)

The equations presented here are solved numerically by the finite difference method.

A fine discretization is necessary for the finite difference scheme in order to increase

the accuracy and capture the thermal transients. More details about the thermal

model discretization and the numerical solution are given in Appendix C.

2.3. COUPLINGMASS BALANCE, FRACTUREGROWTHAND ENERGY BALANCE35

2.3 Coupling Mass Balance, Fracture Growth and

Energy Balance

The wellbore, fracture and reservoir are always coupled in practice, as they provide

pressure and temperature boundary conditions for each other. The pressure equations

of wellbore, fracture and reservoir can be combined into a single coefficient matrix,

and the pressure in the wellbore, fracture and reservoir can be solved simultaneously.

At each time-step the Newton-Raphson algorithm is run and the fracture growth

criterion is verified. If the criterion is reached the fracture advances one grid-block

and the Newton-Raphson runs until convergence, when the fracture growth criterion

is tested again. If the fracture reaches a stable position the time-step is finished, the

new fracture dimensions and pressure are accepted, and a new time-step is initiated.

The process goes on until the injection stops and the fracture tip reaches the final

stable position. The algorithm allows the fracture to keep growing after the end of

injection, but during shut-in its length does not decrease. This assumption is based

on the fact that the rock was broken and deformed, so the closure happens by the

decrease of the crack aperture.

After a time-step is finished and the pressure, fracture characteristics and fluid

velocities are known, they are input to the energy balance equation, which is solved

numerically also. Once again, the equations of wellbore, fracture and reservoir can

be combined into a single coefficient matrix.

Figure 2.13 shows the general algorithm for the problem solution during injection

and falloff or flowback cases. Each time-step starts by assuming that fracture length

is fixed, then the Newton-Raphson algorithm is used to find p and σm. The starting

guesses for p and σm are the solution vectors from the last time-step (or iteration, if

iterating at the current time-step). Once the converged p and σm are found using the

fixed value of fracture length, xf , they are used to evaluate the fracture propagation

criteria. If the fracture tip is stable (i.e., does not need to move), the time-step is

completed. If the fracture tip is unstable, it is moved exactly one full grid-block in

the direction dictated by the propagation criteria (either forward or backward). The

Newton-Raphson scheme is used again to find p and σm with the new fracture length.


This process of moving the tip and solving for p and σm is repeated until the fracture

tip occupies a stable position. With the fracture geometry and pressure field known,

the thermal problem is solved and the time-step is completed.

2.4 Results

The finite difference model described in the previous sections was used to study pres-

sure and temperature transients during hydraulic fracturing. The base case consists

of a vertical fracture propagating in a homogeneous reservoir. Reservoir dimensions

were set in a way that boundary effects are not felt during the entire simulation

time. The injected and reservoir fluids are assumed to have similar properties, and

the single-phase solution represents the scenario adequately. The base case input

parameters are presented in Tables 2.1 and 2.2 .

The case considered 30 minutes injection of 12 ft3/min of fluid at 20 oC at the

surface. The injected fluid exchanges heat with the wellbore wall and the surrounding

formation during its way down to the reservoir, which makes the injected temperature

at the sand face different from the specified Tinj at the surface. After 30 minutes of

injection the well is shut in and the falloff period starts. Figure 2.14 presents the

reservoir pressure maps at the end of injection (Figure 2.14 - top) and during falloff

(Figure 2.14 - bottom). During the injection period the fracture propagates away

from the wellbore and the injected fluid creates a pressure disturbance around the

fracture. During shut-in, the pressure alteration diffuses further into the reservoir,

reducing its magnitude.

The following subsections show the pressure and temperature signatures for the

two fracture geometries considered in this work. For the simulated cases a fine dis-

cretization was used, where the grid-block dimension in x -direction (∆x) is 1 ft closer

to the wellbore and the dimension in along y (∆y) increases exponentially starting

at 0.2 ft in the fracture neighborhood. The model contains 14250 grid-blocks, 150

in x-direction and 95 in y-direction. The time-step is also small to capture the tran-

sients. During the injection ∆t starts at 0.1 minutes and increases by 5% at each

subsequent time-step to equilibrate the fracture growth with the higher leak-off area

2.4. RESULTS 37

Figure 2.13: Solution algorithm overview.


Table 2.1: Reservoir and fluid properties.

Parameter Value

Porosity (φ) 0.15Permeability (k) 0.5 mdReservoir thickness (h) 50 ftWell radius (rw) 0.3 ftReservoir depth 2200 ftInitial pressure (pi) 2500 psiFluid compressibility (cl) 5.04 ×10−6 psi−1

Fluid density at standard conditions (ρsc) 49 lb/ft3

Reservoir density (ρr) 125 lb/ft3

Fluid viscosity (µ) 1 cpMinimal horizontal stress (σhmin or pc) 3500 psiPoisson ration (ν) 0.2Youngs Modulus (E ) 2.0× 106 psiAsperities size (wf,min) 7× 10−4 ft

Table 2.2: Rock and fluid thermal properties.

Parameter Value

Fluid heat capacity (Cl) 4186.8 J/kg.KRock heat capacity (Cr) 921.1 J/kg.KRock thermal conductivity (λr) 1.44 W/m.KFluid thermal conductivity (λl) 0.52 W/m.KInjection temperature at surface (z = 0ft) 20 oCInitial Temperature (Ti) 80 oC

2.4. RESULTS 39

Figure 2.14: Half reservoir pressure maps (psi) at the end of injection (top) andfalloff (bottom) periods for PKN fracture geometry. The pressure in the reservoiris reflecting the perturbation caused by the fracture growth and fluid leak-off. Thefracture plane is coincident with the x -axis.


as the fracture dimensions increase. The temporal and spatial discretization were de-

fined based on grid refinement studies and comparison between the numerical result

and the analytic solution for simplified cases. This numerical solution verification is

presented in Appendix D.

2.4.1 PKN fracture geometry

The net pressure and fracture half-length histories for PKN fracture geometry are

shown in Figures 2.15 and 2.16. Net pressure (pnet) is defined as the wellbore

pressure (pwf ) minus the closure pressure, pc. The well pressure increases moderately

during injection, while the fracture is growing. The injection causes not only fracture

length growth but also width growth related with the increase in pressure along the

fracture, as shown in Figure 2.17 (top). During shut-in the pressure declines and

the fracture volume decreases. The fracture closure happens through a decrease in

fracture aperture, while the fracture length generated during injection stays constant

(Figure 2.16). Figure 2.17 (bottom) presents width profiles along the fracture during

shut-in. As pressure decreases inside the fracture, it is not able to stay open, so

it gradually gets narrower. When the width reaches the size of the asperities, the

opposite walls start to touch and contact stress is generated against the complete

closure. In part, this is responsible for the behavior of pressure change during falloff

presented in Figure 2.15. The pressure reflects the behavior of dynamic geometry,

where the fracture grows and closes, changing volume and conductivity.

The log-log plot of falloff pressure and its derivative with respect to the shut-in

time (∆t = t− tinj) for PKN fracture is presented in Figure 2.18. The linear flow is

observed in early times, followed by a steep bump in the derivative, which decreases

towards a slope of -1 after that. The derivative spike is coincident with the moment at

which the fracture width around the well becomes as small as the specified minimum

width of the asperities. The derivative tends to zero because during the shut-in the

pressure tends to return to the initial pressure of the reservoir, remembering that the

injection period is small compared with the total falloff time.

Using the Agarwal equivalent time (Agarwal , 1980) to calculate the derivative we

2.4. RESULTS 41

0 10 20 30 40 50 60−500

−400

−300

−200

−100

0

100

t (min)

Pnet (

psi)

Figure 2.15: Wellbore pressure history for PKN geometry.

Figure 2.16: Fracture length history for PKN geometry.


−200 −100 0 100 2000

2

4x 10

−3

xf (ft)

wf (

ft)

−200 −100 0 100 2000

2

4x 10

−3

xf (ft)

wf (

ft)

Time

Time

Figure 2.17: Average width profiles during fracturing (top) and falloff (bottom) forPKN geometry. Each curve color represents a different moment in time. Duringinjection (top figure) the fracture expands in length and aperture. During falloff(bottom figure) the fracture closes by decreasing the aperture, but keeping the length(xf ).

2.4. RESULTS 43

obtain the green curve in Figure 2.18. Even though the definition of the Agarwal

time is taken based on the radial flow regime, which is not happening at the early to

medium time in this example, it is worth to analyze it because most of the commercial

well testing software use it as a default. One of the differences in this case is the fact

that the derivative rises higher than the pressure change (∆p), and at the late time

it tends to radial flow in the same flat behavior as it does in the drawdown test.

102

100

102

100

101

102

103

104

t[h]

p[p

si] &

dp/d

ln(

t)

Figure 2.18: Pressure difference (blue) and its logarithmic derivatives with respect to∆t (red) and Agarwal equivalent time (green) during falloff for PKN geometry.

2.4.2 KGD fracture geometry

The pressure behavior of the KGD fracture during its propagation differs from the one

observed for the PKN fracture. For the KGD model the pressure decreases slightly as

the fracture propagates (Figure 2.19). The width profiles during injection show how

the fracture grows and during shut-in they indicate how fracture volume decreases

and the effect of asperities on the residual fracture aperture (Figure 2.21). Another

difference between KGD and PKN is that the KGD geometry produces a shorter and

wider fracture (Figures 2.20 and 2.21).


The equivalent log-log plot for KGD fracture geometry is presented in Figure 2.22.

0 20 40 60 80−600

−500

−400

−300

−200

−100

0

100

t [min]

Pnet[p

si]

Figure 2.19: Wellbore pressure history for KGD geometry.

2.4.3 Effect of contact stress on the fracture closure

The way that the fracture closes will be reflected by the pressure derivative. The

contact stress model has a very important role in this process. Depending on how

fast and strong this effect is, the change in pressure characteristics can be more or

less prominent. Figure 2.23 presents pressure derivative for the falloff period of two

different stress parameter (Equation 2.19): one considers a smaller value of γ, and

so the contact stress start to influence the pressure behavior more smoothly; and the

other represents a high value of γ (red curves). As can be seen, the jump of fracture

closure is steeper for the high value of γ.

Considering the possibility that the fracture can stay with constant properties

during the whole shut-in (or flowback) the derivative is going to change smoothly

from the early time behavior to radial flow, but slower than if the fracture were

allowed to close (Figure 2.24).

2.4. RESULTS 45

Figure 2.20: Fracture length history for KGD geometry.

−200 −100 0 100 2000

2

4

6

8x 10

−3

xf (ft)

wf (

ft)

−200 −100 0 100 2000

2

4

6

8x 10

−3

xf (ft)

wf (

ft)

Time

Time

Figure 2.21: Average width profiles during fracturing (top) and falloff (bottom) forKGD geometry. Each curve color represents a different moment in time. Duringinjection (top figure) the fracture expands in length and aperture. During falloff(bottom figure) the fracture closes by decreasing the aperture, but keeping the length(xf ).


102

100

102

101

100

101

102

103

104

t[h]

p[p

si] &

dp/d

ln(

t)

Figure 2.22: Pressure difference (blue) and its logarithmic derivatives with respect to∆t (red) and Agarwal equivalent time (green) during falloff for KGD geometry.

If the reservoir permeability is high the fracture will close very quickly and the

spike in the derivative might not be seen. An example of falloff analysis where the

closure is not identified by the pressure derivative is presented in Figure 2.25. This

example represents the pressure falloff after 20 minutes of fluid injection at 6bpm on

a 50md reservoir. The created fracture wing length is 11.3ft. The small fracture

together with high leak-off are the main reasons for the fast closure, not able to be

captured by the pressure analysis.

2.4.4 Temperature response

The program developed in this work also provides the temperature data inside the

well, fracture and reservoir. The first important observation about the temperature

characteristics is related with the location of temperature sensor: the temperature

data are very sensitive to the location, as can be seen in Figures 2.26 and 2.27.

In Figure 2.26 each curve represents a temperature recorded by a different sensor

location: at the reservoir wall (green line, Tr) and at the wellbore (blue line, Tw),

2.4. RESULTS 47

10−4

10−2

100

102

104

101

102

103

104

105

∆t (h)

∆p (

psi)

∆p’

γ=1e10

γ=1e6

Figure 2.23: Pressure derivative during falloff for different closure parameter γ. Thered curve represents a case where the contact stress start to be applied fully almostinstantaneously, and the blue curve is a example where the contact stress smoothlystart to influence the fracture properties.

10−4

10−2

100

102

104

101

102

103

104

∆ t

∆ p

No−closure

Closure

Figure 2.24: Comparison between the pressure derivative during falloff for fractureclosure (red) and no closure (blue).


10−4

10−2

100

102

101

102

103

104

∆t (h)

∆P

(psi)

d∆

P/lnt

Figure 2.25: Pressure derivative for falloff period after fracturing injection in a 50 mdreservoir.

in addition to the fluid temperature inside the fracture (red, Tfr). The same color

legend is describing the temperature response during flowback in Figure 2.27. When

the well is flowed back the temperature warms faster and the temperature tends to

be the same for the different locations after a couple of hours.

For practical applications there are two main installation configurations in which

DTS can be deployed in a well: the optical fiber can be installed inside the tubing,

in direct contact with the fluid inside the well; or it can be installed permanently

behind the casing and cemented (Figure 1.2), which will lead to higher sensitivity to

the reservoir temperature (Sierra et al., 2008). If installed behind the casing, several

redundant fibers are placed around the well to ensure that after the perforating job

at least one of the fibers will not be damaged by the guns.

When the temperature sensor is placed inside the well it measures the average fluid

temperature at the reservoir depth. At the end of injection the temperature inside

the well is close or equal to the temperature of the injection fluid at the surface, and

the surrounding reservoir is at higher temperature.

When the temperature is recorded behind the casing (Tr), close to the reservoir

rock, the measurement might behave as represented by the green line in Figure 2.26.

2.4. RESULTS 49

When the injection stops, the temperature still decreases due to the heat transfer

through the wall of the well. This period will be shorter for smaller well radius

and also when the flowback is performed. Inside the fracture the temperature (Tfr)

changes very quickly when the fracture is closing. The temperature transits from the

well temperature to the formation temperature as the fluid film gets thinner. Inside

the well the temperature (Tw) will reflect the injected temperature during injection.

During the shut-in the well temperature will rise moved by the interaction between

the volume of fluid at lower temperature inside the well and the heat transfer through

the casing. In case of flowback the temperature increases faster mainly because of

the entrance of warmer fluid that is produced from the fracture and the reservoir.

0 50 100 150 20020

30

40

50

60

70

80

90

t(min)

T (

oC

)

Tw

Tfr

Tr

Figure 2.26: Temperature response measurement for different sensor locations: in-jection followed by shut-in. The blue curve represents the average fluid temperatureinside the wellbore at the reservoir depth, the green curve is the sandface temperature,and the red curve is the average temperature inside the fracture.


0 50 100 150 200 250 300 350 40020

30

40

50

60

70

80

90

t(min)

T (

oC

)

Tw

Tfr

Tr

Figure 2.27: Temperature response measurement for different sensor locations: injec-tion followed by flowback. The blue curve represents the average fluid temperatureinside the wellbore at the reservoir depth, the green curve is the sandface temperature,and the red curve is the average temperature inside the fracture.

2.5 Chapter Summary

This chapter describes the numerical model for hydraulic fracture growth during

injection in a hydrocarbon reservoir, also accounting for fracture closure during shut-

in or flowback. The model couples fracture, well and reservoir accounting for mass

conservation and simplified geomechanics given two well-known fracture geometries:

PKN and KGD. The fluid loss from fracture to reservoir is governed by reservoir

properties and fracture characteristics. No preassumption about leak-off coefficient is

made. In addition to mass and stress balances, the energy balance is also considered.

This describes a comprehensive model that not only accounts for conduction and

convection, but also the pressure effect, like Joule-Thomson and adiabatic expansion.

Even though those pressure effects have small magnitude when compared with the

temperature change caused by cold fluid injection, the effect can still influence some

early transient behavior.

It is important to notice that the same spike in the pressure derivative is present,

2.5. CHAPTER SUMMARY 51

regardless the assumption of fracture geometry. The spike is related to the fracture

closure at the wellbore, when the wall asperities start to touch. If there is no frac-

ture volume change during the falloff (i.e., no fracture closure) the transition from

early time behavior to radial flow happens smoothly, with no jump in the derivative.

Depending on the geometry of the asperities, the jump can be more or less steep.

In particular, the smoothing function Ω(∆wf ) is going to play a very important role

when the fracture apertures reach the specified asperities width. For high permeabil-

ity cases the fracture closure happens very fast and the spike in the pressure derivative

might not be identified.

With regard to the temperature results, the model shows how sensitive this mea-

surement is in relation to the sensor location (inside the well or behind the casing).

The flowback brings information from deep inside the fracture and the reservoir to

the wellbore and makes the temperature recover faster than in the warmback case.

Chapter 3

Results: Single Vertical Fracture

The previous chapter gave the details about the modeling of fracture growth, mass

and energy balances. This chapter describes how the developed model was used for a

series of studies which form the bases of understanding of pressure and temperature

responses during and after hydraulic fracturing. All the scenarios considered a vertical

well and the fracture stays contained to the reservoir thickness, growing along the x

direction (Figure 3.1).

Figure 3.1: Sensitivity analysis geometry: vertical well with fracture growing con-tained to the reservoir thickness.

Beyond the traditional pressure analysis, the analysis of the temperature signa-

tures for early, intermediate and late times was explored. The comparison between

53

54 CHAPTER 3. RESULTS: SINGLE VERTICAL FRACTURE

shut-in and flowback highlights the physical information that can be carried by a

temperature signal measured not only inside the well but also at the sandface.

3.1 Sensitivity Analysis

Sensitivity analysis is a key tool to understand the effect of each model parameter

on the final fracture dimensions, pressure and temperature behaviors. This exercise

allows us to determine what can actually be estimated by the pressure and temper-

ature data. This section also explores the temperature response at different sensor

locations.

Table 3.1: Reservoir and fluid base case model parameters for sensitivity analyses.

Parameter Value

Porosity (φ) 0.2Permeability (k) 1 mdReservoir thickness (h) 50 ftWell radius (rw) 0.3 ftReservoir depth 2200 ftInitial pressure (pi) 2500 psiFluid compressibility (cl) 5.0× 10−6 psi−1



Fluid viscosity (µ) 1 cpMinimal horizontal stress (σhmin or pc) 3500 psiPoisson ration (ν) 0.2Youngs Modulus (E) 3.0× 106 psiAsperities size (wf,min) 7× 10−4 ft

The following sections describe the effect of reservoir permeability, injection rate,

reservoir porosity, closure pressure, Young’s Modulus, and asperity width on pressure

and temperature. The base case input parameters are presented in Tables 3.1 and

3.2. For all the simulations, the injection time was equal to 30 minutes and injection

rate is 22 bpm. After injection the well is shut-in for 200 hours. In case of flowback

the production rate is 1.6 ft3/min.

3.1. SENSITIVITY ANALYSIS 55

Table 3.2: Base case thermal properties for sensitivity analyses.

Parameter Value

Fluid heat capacity (Cl) 4186.8 J/kg.KRock heat capacity (Cr) 921.1 J/kg.KRock thermal conductivity (λr) 1.44 W/m.KFluid thermal conductivity (λr) 0.52 W/m.KInjection temperature at surface (z = 0ft) 20 oCInitial Temperature (Ti) 80 oCµJT 5×10−7η 1×10−8

3.1.1 Sensitivity to reservoir permeability

This section describes our investigation of the effect of reservoir permeability on

the hydraulic fracturing process. Besides the reservoir permeability, all the other

parameters were kept at the same value as the base case (Tables 3.1 and 3.2).

The final fracture length for each reservoir permeability is presented in Table 3.3.

The results show that the higher the permeability the shorter is the fracture length.

The relation permeability/fracture length is explained by the fluid loss: the higher the

permeability the higher the fluid loss. The higher the fluid loss the lower the hydraulic

fracturing efficiency and the shorter the created fracture is. Figure 3.5 presents the

quarter of fracture aperture profile for each considered reservoir permeability.

Table 3.3: Fracture half length (xf ) for different reservoir permeability.

Permeability Fracture Half Length (xf)

0.01 md 3084 ft0.1 md 2119 ft1 md 1009 ft10 md 379.3 ft

The pressure transient analysis for the different reservoir permeabilities is shown

in Figure 3.2. This figure presents the falloff pressure difference and its logarithmic

derivative with respect to Agarwal equivalent time. The commonly observed signature


contains the linear flow at early times, when the fracture is still open, followed by

a steep jump in the logarithmic derivative. For all the cases analyzed, this jump

occurs at the moment when the asperities touch around the wellbore. The lower the

permeability, the later the spike in the logarithmic derivative happens.

It is important to remember that our model does not account for proppant, which

will change how the fracture closes and the final fracture aperture. The length and

volume of the fracture, together with the minimum horizontal stress and rock perme-

ability will dictate the time of closure. The bigger the fracture volume and the lower

the permeability the longer it will take for the fracture to close. This is confirmed by

the fracture volume evolution during falloff, which is presented in Figure 3.3. This

time difference influences the location of the spike in the pressure derivative plot.

If the fracture volume is large and the permeability is low enough it is possible to

observe the fracture storage effect, as happens for the 10 µd permeability (blue curve

in Figure 3.2).

For the moderate permeability reservoir (10 md), the transition to the radial flow

is seen during the simulated shut-in time. The radial flow happens earlier due to the

high diffusivity coefficient and shorter fracture (faster closure and shorter linear flow

period).

Koning and Niko (1995) and van den Hoek (2002) presented semianalytical solu-

tions for fracture closure during falloff applied to the water flooding scenario (long

time of injection when compared with the falloff), where a similar spike is present

when fracture closes. Those models described the fracture closure by changing the

wellbore storage, keeping the fracture conductivity constant and infinite. The main

difference between our numerical model result and their semianalytical solutions is

the early time behavior. The van den Hoek (2002) problem presents a unit slope

of pure storage (Figure 3.4), while our numerical model has the transition between

linear flow and, depending on the permeability and fracture characteristics, the effect

of fracture storage. The first difference between the assumptions of the two problems

is the fact that for our case the injection time is relatively short, while for the water

flooding scenario the injection time is longer than the shut-in. The other point is that

the semianalytical models do not consider the change in fracture conductivity and


describe the fracture closure by changing the total storage, where an open fracture

has high storage values and a closed fracture has low.

10−3

10−2

10−1

100

101

102

103

100

101

102

103

104

105

∆t (h)

∆P

d∆

P(p

si)

k=0.01md

k=0.1md

k=1md

k=10md

Figure 3.2: Pressure difference and its logarithmic derivatives with respect to Agarwalequivalent time during falloff, sensitivity to permeability.

For the same permeability cases described above, the average temperature inside

the wellbore, at reservoir depth, is presented in Figure 3.6. As can be seen, the

average temperature inside the wellbore shows a very small sensitivity to small per-

meabilities, while for the 10 md case the impact of permeability is more pronounced.

For low permeability formations, the impact of convection is small compared with

conduction. Because of the dominant conduction heat transfer mechanism and the

fact that the conduction coefficient is the same for all cases, the difference between

the small permeability reservoirs is not identified clearly.

It can be observed that the lower the permeability, the faster will be the temper-

ature recovery over time. Low permeability reservoirs will have small perturbation

in the temperature inside the reservoir during cold fluid injection. For these cases

the fracture temperature will warm up faster towards the equilibrium with the initial

reservoir temperature (Tres). For all the cases, after the fracture closure, the heat

transfer during shut-in is governed by conduction (diffusive mechanism).

The logarithmic derivative of warmback temperature with respect to shut-in time


10−2

10−1

100

101

102

0

500

1000

1500

2000

2500

∆t (h)

Vfr(f

t3)

k=0.01md

k=0.1md

k=1md

k=10md

Figure 3.3: Fracture volume evolution over time during falloff, sensitivity to perme-ability.

10−3

10−2

10−1

100

101

102

10−4

10−3

10−2

10−1

100

101

tD

pw

fD,

dp

wfD

/dln

t D

delpat=0.01

delpat=0.1

delpat=0.5

delpat=1

delpat=5

Figure 3.4: van den Hoek (2002) semianalytical pressure solution for closing fractureduring falloff. Each collor represents a different closure speed, the higher the delpatinput, the later the fracture closes.


0 500 1000 1500 2000 2500 3000 3500 4000 45000

1

2

3

4

5x 10

−3

x (ft)

wf(f

t)

k=0.01md

k=0.1md

k=1md

k=10md

Figure 3.5: Width profile at the end of injection, sensitivity to permeability.

is presented in Figure 3.7. The difference between the curves is prominent at inter-

mediate times.

As we discussed in the previous chapter, the location of the gauge has a high

impact on the recorded temperature behavior. If the optical fiber is placed behind

the casing and cemented, the temperature will be more sensitive to the formation

temperature. Figure 3.8 shows the sensitivity analysis for reservoir temperature when

the measurement is taken behind the casing. As can be seen, the distinction between

the curves is more pronounced, being less sensitive to the volume of fluid inside the

wellbore and more to the phenomena that have happened inside the fracture and

reservoir (specially the temperature disturbance created by convection). Another

characteristic of behind the casing measurements is the fact that temperature keeps

decreasing after the end of injection. This decrease in temperature is related to the

heat transfer with the fluid inside the wellbore, which is at low temperature. At early

times the heat that is lost to the wellbore is greater than the heat that comes from far

inside the reservoir. At intermediate to later times the temperature starts to increase

towards the equilibrium with the original reservoir temperature (Tres).

During shut-in it is observed that the highest distinction in terms of temperature

occurs inside the fracture and reservoir. If instead of shutting the well we flow it back,

the temperature disturbances that are clear inside the fracture and its neighborhood


will be recovered at the wellbore. Figure 3.9 presents the temperature response mea-

sured inside the well during injection and flowback. As can be seen, the difference

between permeabilities is reflected in the temperature measurement, and this differ-

ence is able to be observed inside the wellbore. The logarithmic derivative highlights

this difference (Figure 3.10).

0 5 10 15 20 2520

30

40

50

60

70

80

t(h)

T (

oC

)

k=0.01md

k=0.1md

k=1md

k=10md

Figure 3.6: Temperature inside the well at the bottom-hole for injection and falloff,sensitivity to permeability.

3.1.2 Sensitivity to injection rate

This section describes the effect of different injection rates during the hydraulic frac-

turing process, specially on pressure and temperature data. The rate is assumed to

be constant during the total injection time.

Table 3.4 presents the final fracture length for the different injection rates. The

higher the injection rate the longer will be the fracture length, and consequently the

longer it takes for the fracture to close, as can be seen in the pressure derivative

signature that are presented in Figure 3.11. The later the fracture closure, the later

the spike in the pressure derivative occurs.

From the thermal part of the analysis, the higher the injection rate the faster the

temperature decreases inside the wellbore during injection period (Figure 3.12). In

this way, the high injection rate will cause a greater volume of colder fluid to enter


10−3

10−2

10−1

100

101

102

103

0

2

4

6

8

10

12

14

∆t (h)

dT

/dln

t (o

C)

k=0.01md

k=0.1md

k=1md

k=10md

Figure 3.7: Well temperature logarithmic derivative during warmback, sensitivity topermeability.

0 5 10 15 20 2555

60

65

70

75

80

85

t (h)

T (

oC

)

k=0.01md

k=0.1md

k=1md

k=10md

Figure 3.8: Temperature behind the casing, sensitivity to permeability.

Table 3.4: Fracture half length (xf ) for different injection rates.

Injection Rate (qinj) Fracture Half Length (xf)

4 bpm 250.4 ft12 bpm 701.9 ft20 bpm 1106 ft28 bpm 1460 ft


0 0.5 1 1.5 2 2.5 3 3.5 4 4.520

30

40

50

60

70

80

t(h)

T (

oC

)

k=0.01md

k=0.1md

k=1md

k=10md

Figure 3.9: Well temperature during flowback, sensitivity to permeability.

10−2

10−1

100

0

5

10

15

20

25

30

∆t (h)

dT

/dln

t (o

C)

k=0.01md

k=0.1md

k=1md

k=10md

Figure 3.10: Well temperature logarithmic derivative during flowback, sensitivity topermeability.


10−3

10−2

10−1

100

101

102

103

101

102

103

104

∆t (h)

∆P

d∆

P(p

si)

qinj

=4bpm

qinj

=12bpm

qinj

=20bpm

qinj

=28bpm

Figure 3.11: Pressure derivative during falloff, sensitivity to injection rate.

the sandface and fracture, increasing the temperature disturbance. The temperature

logarithmic derivative also presents a distinction between the cases (Figure 3.13).

The temperature derivative with respect to the warmback time presents not only

a distinction between the injection rates, but also the disturbance at the fracture

closure point (Figure 3.14). Even in small magnitude the pressure effect is present

in the temperature signal. During the fracture closure the change in pressure is very

pronounced, which enables it to be seen in the first derivative.

To understand what part of the temperature differences is caused by the well effect

and what is related with the larger amount of fluid injected and the size of the fracture,

we removed the well component by specifying a constant injection temperature at the

reservoir depth and performed the same sensitivity analysis described at the beginning

of this section. As the well effect on temperature we refer to the temperature solution

accounting for the heat transfer between fluid inside the well and the earth on its way

down from the wellhead to the sandface.

Figures 3.15 and 3.16 show the temperature history and temperature logarithmic

derivatives, respectively. Without the well effect, the difference between the curves

for the four injection rates decreases drastically (comparing with Figures 3.12 and

3.13). The bigger volume injected (and longer fracture created) does not influence

the characteristics of the temperature recovery measured at the wellbore. For flowback


some difference can be identified, but not as pronounced as when the heat transfer is

tracked from the wellhead.

Considering the well completion where the fiber is placed behind the casing, the

temperature for injection and warmback period is presented in Figure 3.17. It is

observed that even after the end of the injection, the temperature keeps decreasing.

The explanation for this behavior is the heat transfer with the cold fluid inside the

wellbore. Through the well wall, the sandface loses heat faster to the well than it

receives from deeper in the reservoir. The negative sign in the temperature derivative

marks this period, and when it becomes positive the temperature starts to increase

(Figure 3.18).

For the flowback case the effect of injection rate is preserved (Figures 3.19 and

3.20). The temperature derivative highlights the differences, which are related with

the length of the fracture and the amount of cold fluid that was injected inside the

reservoir and stayed inside the fracture at the flowback moment.

0 5 10 15 20 2520

30

40

50

60

70

80

t(h)

T (

oC

)

qinj

=4bpm

qinj

=12bpm

qinj

=20bpm

qinj

=28bpm

Figure 3.12: Well temperature history for injection and warmback periods, sensitivityto injection rate.


10−3

10−2

10−1

100

101

102

103

0

2

4

6

8

10

12

14

∆t (h)

dT

/dln

t (o

C)

qinj

=4bpm

qinj

=12bpm

qinj

=20bpm

qinj

=28bpm

Figure 3.13: Well temperature logarithmic derivative for warmback period, sensitivityto injection rate.

10−3

10−2

10−1

100

101

102

103

0

5

10

15

20

∆t (h)

dT

/dt

(oC

/h)

qinj

=4bpm

qinj

=12bpm

qinj

=20bpm

qinj

=28bpm

Figure 3.14: Well temperature first derivative with respect to shut-in time, sensitivityto injection rate.


0 5 10 15 20 2520

30

40

50

60

70

80

t(h)

T (

oC

)

qinj

=4bpm

qinj

=12bpm

qinj

=20bpm

qinj

=28bpm

Figure 3.15: Well temperature history for constant injection temperature (Tinj) atthe reservoir depth for injection and warmback periods, sensitivity to injection rate.

10−3

10−2

10−1

100

101

102

103

0

2

4

6

8

10

12

14

∆t (h)

dT

/dln

t (o

C)

qinj

=4bpm

qinj

=12bpm

qinj

=20bpm

qinj

=28bpm

Figure 3.16: Well temperature logarithmic derivative for constant injection tempera-ture (Tinj) during warmback, sensitivity to injection rate.


0 5 10 15 20 2555

60

65

70

75

80

85

t (h)

T (

oC

)

qinj

=4bpm

qinj

=12bpm

qinj

=20bpm

qinj

=28bpm

Figure 3.17: Behind the casing temperature history for warmback, sensitivity toinjection rate.

10−3

10−2

10−1

100

101

102

103

−6

−4

−2

0

2

4

6

8

10

∆t (h)

dT

/dln

t (o

C)

qinj

=4bpm

qinj

=12bpm

qinj

=20bpm

qinj

=28bpm

Figure 3.18: Behind the casing temperature logarithmic derivative for warmback,sensitivity to injection rate.


0 0.5 1 1.5 2 2.5 3 3.5 4 4.520

30

40

50

60

70

80

t(h)

T (

oC

)

qinj

=4bpm

qinj

=12bpm

qinj

=20bpm

qinj

=28bpm

Figure 3.19: Well temperature history for flowback, sensitivity to injection rate.

10−3

10−2

10−1

100

101

0

5

10

15

20

25

∆t (h)

dT

/dln

t (o

C)

qinj

=4bpm

qinj

=12bpm

qinj

=20bpm

qinj

=28bpm

Figure 3.20: Well temperature logarithmic derivative for flowback, sensitivity to in-jection rate.


3.1.3 Sensitivity to reservoir porosity

The sensitivity analysis to reservoir porosity shows that it has a modest effect on the

fracture length (Table 3.5). The pressure falloff analysis presented in Figure 3.21

shows that the influence of the porosity is more related with the different fracture

length, volume and consequently its closure time. When compared with the previously

discussed effects of reservoir permeability and injection rate, the influence of porosity

is small.

Table 3.5: Fracture half length (xf ) for different reservoir porosities.

Porosity (φ) Fracture Half Length (xf)

0.1 1331 ft0.15 1106 ft0.20 1009 ft0.25 921.3 ft

10−3

10−2

10−1

100

101

102

103

100

101

102

103

104

105

∆t (h)

∆P

(psi)

d∆

P/d

lnt

(psi)

φ=10%

φ=15%

φ=20%

φ=25%

Figure 3.21: Pressure derivative during falloff, sensitivity to reservoir porosity.

The temperature inside the well only starts to show distinction for the different

porosity values at intermediate to late times of warmback (Figure 3.22). This is


related to the effective reservoir thermal conductivity, which is the driven for temper-

ature warmback.

The temperature logarithmic derivative during warmback period is able to show

the impact of porosity (Figure 3.23). The curves have a different maximum value,

which is related with the different effective thermal conductivity. The higher the

porosity the lower the effective thermal conductivity inside the reservoir (Equation

2.22), and the smaller the maximum temperature logarithmic derivative value.

When the data are recorded behind the casing the sensitivity is higher, as shown

in Figure 3.24.

0 5 10 15 20 2520

30

40

50

60

70

80

t(h)

T (

oC

)

φ=10%

φ=15%

φ=20%

φ=25%

Figure 3.22: Temperature inside the well at the bottom-hole, sensitivity to porosity.

For the flowback case (Figures 3.25 and 3.26), the difference between the tempera-

ture curves is smaller than was experienced by the warmback. The difference from the

warmback behavior is explained by the fact that during the flowback the convection

is dominant compared to conduction, and so the different thermal conductivity does

not alter the final temperature result significantly.


10−3

10−2

10−1

100

101

102

103

0

2

4

6

8

10

12

14

∆t (h)

dT

/dln

t (o

C)

φ=10%

φ=15%

φ=20%

φ=25%

Figure 3.23: Well temperature logarithmic derivative during warmback, sensitivity toporosity.

0 5 10 15 20 2555

60

65

70

75

80

85

t (h)

T (

oC

)

φ=10%

φ=15%

φ=20%

φ=25%

Figure 3.24: Temperature behind the casing, sensitivity to porosity.


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 520

30

40

50

60

70

80

t(h)

T (

oC

)

φ=10%

φ=15%

φ=20%

φ=25%

Figure 3.25: Well temperature during flowback, sensitivity to porosity.

10−3

10−2

10−1

100

101

0

5

10

15

20

25

∆t (h)

dT

/dln

t (o

C)

φ=10%

φ=15%

φ=20%

φ=25%

Figure 3.26: Well temperature logarithmic derivative during flowback, sensitivity toporosity.


3.1.4 Sensitivity to closure pressure

The sensitivity to closure pressure, or minimum horizontal stress, shows that the

higher the closure pressure the shorter will be the fracture (Table 3.6 and Figure

3.27). The size of the fracture affects the pressure signature, specially the closure time

and the pressure derivative spike associated with it. Figure 3.28 presents the pressure

derivative during the falloff with respect to Agarwal equivalent time (Agarwal , 1980)

for three different closure pressures. As can be seen, the higher the closure pressure

the earlier the fracture closure happens.

When it comes to temperature there is no clear influence of closure pressure on the

temperature recorded inside the well for injection and warmback (Figures 3.29 and

3.30). The temperature behind the casing shows minor influence of closure pressure

(Figure 3.31).

As opposed to the warmback, the temperature result for the flowback period

allows us to differentiate among the different closure pressures (Figures 3.32), specially

the temperature derivative, which is presented in Figure 3.33. This difference is a

consequence of the different fracture length that causes two main influences on the

flowback: the amount of fluid inside a fracture at shut-in and the pressure changes

are different for each minimum stress value.

Table 3.6: Fracture half length (xf ) for different closure pressures.

Closure Pressure (pc) Fracture Half Length (xf)

3000 psi 1460 ft3500 psi 1009 ft4000 psi 768.2 ft

3.1.5 Sensitivity to Young’s Modulus

This section addresses the sensitivity analysis to Young’s Modulus (E). Young’s

Modulus is defined as the ratio of stress to strain for uniaxial stress (Gidley et al.,

1989).


0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.5

1

1.5

2

2.5

3

3.5

4x 10

−3

x (ft)

wf(f

t)

Pc=3000

Pc=3500

Pc=4000

Figure 3.27: Width profile at the end of injection for different closure pressures: 3000psi (blue), 3500 psi (green), and 4000 psi (red).

10−4

10−2

100

102

104

106

100

101

102

103

104

105

∆t (h)

∆P

(psi)

d∆

P/d

lnt

(psi)

Pc=3000

Pc=3500

Pc=4000

Figure 3.28: Pressure difference and its logarithmic derivatives with respect to Agar-wal equivalent time during falloff for different closure pressures: 3000 psi (blue), 3500psi (green), and 4000 psi (red).


0 5 10 15 20 2520

30

40

50

60

70

80

t(h)

T (

oC

)

Pc=3000

Pc=3500

Pc=4000

Figure 3.29: Temperature inside the well at the bottom-hole for injection and fallofffor different closure pressures: 3000 psi (blue), 3500 psi (green), and 4000 psi (red).

10−3

10−2

10−1

100

101

102

103

0

2

4

6

8

10

12

14

∆t (h)

dT

/dln

t (o

C)

Pc=3000

Pc=3500

Pc=4000

Figure 3.30: Logarithmic derivative during warmback of temperature inside the well,sensitivity to closure pressure.


0 5 10 15 20 2555

60

65

70

75

80

85

t (h)

T (

oC

)

Pc=3000

Pc=3500

Pc=4000

Figure 3.31: Temperature behind the casing for injection and falloff for differentclosure pressures: 3000 psi (blue), 3500 psi (green), and 4000 psi (red).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.520

30

40

50

60

70

80

t(h)

T (

oC

)

Pc=3000

Pc=3500

Pc=4000

Figure 3.32: Well temperature for flowback case, sensitivity to closure pressure.


10−3

10−2

10−1

100

101

0

5

10

15

20

25

∆t (h)

dT

/dln

t (o

C)

Pc=3000

Pc=3500

Pc=4000

Figure 3.33: Logarithmic derivative during flowback of temperature inside the well,sensitivity to closure pressure.

The analysis of sensitivity to Young’s Modulus shows that this parameter influ-

ences the fracture aperture (Figure 3.34), but not the final fracture length. All the

cases have the same final length of 1009 ft. As the aperture changes the fracture

volume also varies and affects the closure time. The pressure response reflects the

influence of time of fracture closure by the spike on its derivative, as it can be seen

in Figure 3.35.

From the temperature side of the analysis, different Young’s Modulus values do not

change the temperature response during injection and warmback. The only signature

that can be differentiated is the small perturbation in the temperature first derivative

during falloff, which characterizes the moment of fracture closure. This effect is

related to the adiabatic expansion, due to the sudden pressure change at the closure

time. In the presence of noise this perturbation would be masked.

For the flowback the temperature history does not show a pronounced difference

between the simulations with different Young’s Modulus values, but the logarithmic

derivative can reflect distinct behavior in an environment where both the fractures

have different volumes (Figure 3.36).


0 200 400 600 800 1000 1200 14000

1

2

3

4

5x 10

−3

x

wf(f

t)

E=1000000

E=3000000

E=4000000

Figure 3.34: Width profile at the end of injection for different Young’s Modulus.

10−2

10−1

100

101

102

100

101

102

103

104

∆t (h)

∆P

(psi)

E=500000

E=1000000

E=3000000

Figure 3.35: Pressure difference and its logarithmic derivatives with respect to Agar-wal equivalent time during falloff, sensitivity to Young’s Modulus.


10−3

10−2

10−1

100

101

0

5

10

15

20

25

∆t

dT

/dln

(t)

E=1000000

E=3000000

E=4000000

Figure 3.36: Temperature derivative during flowback, sensitivity to Young’s Modulus.

3.1.6 Sensitivity to asperity width

This section addresses the effect of minimum asperity width (wfmin) on the modeled

hydraulic fracturing process. This parameter only plays a role after the injection

period is over. The parameter changes the moment when the constant stress starts

to help to keep a residual aperture and also its magnitude.

Figure 3.37 shows the pressure derivative during falloff for three different minimum

widths. The results show that asperity width changes the closure time, because the

narrower asperity width the longer it takes to the fracture walls to touch and start

the contact stress influence. This conclusion can be expanded for the case of propped

fractures, where the contact will happen much earlier due to the gravel filling the void

volume inside the fracture.

No significant influence of minimum asperity width on temperature was observed

in our simulations.


10−4

10−3

10−2

10−1

100

101

102

103

100

101

102

103

104

∆t (h)

∆P

(p

si)

d∆

P/d

lnt

(psi)

Wfmin

=0.0003

Wfmin

=0.0006

Wfmin

=0.001

Figure 3.37: Pressure difference and its logarithmic derivatives with respect to Agar-wal equivalent time during falloff, sensitivity to asperity’s minimum width (wfmin

).

3.2 Temperature Transient Derivative

The semilog plot of temperature logarithmic derivative during both flowback (Figure

3.10) and falloff (Figure 3.7) presents a maximum point for all the analyzed cases.

For the flowback cases the maximum is located at the same time for most of the

cases described in this chapter, as long as the fluid properties, well characteristics

and flowback rate remain the same.

This observation can be explained by solving a simplified equation for falloff and

flowback. The analytic solutions for well temperature for both shut-in and flowback

are presented in Appendix E.

From the analytic solution for flowback presented in the Appendix E.2, the log-

arithmic derivative of wellbore temperature has a maximum, which is given by the

ratio of well volume by the flowback rate:

∆tmax =Vw

qfb(3.1)

For the simulated flowback cases the ∆tmax is equal to 10.5 min. Multiplying

3.2. TEMPERATURE TRANSIENT DERIVATIVE 81

∆tmax by qfb (1.6 ft3/min) we recover the input volume of the well in front of the

porous medium (16.95 ft3). This effect can be seen as an analog of wellbore storage

in pressure transient analysis. The temperature wellbore storage (TWBS) was also

acknowledged by Ramazanov et al. (2010). This relationship between the temperature

maximum and the wellbore volume and rate was true for many different well radii and

flowback rates tested, but for low permeability where the fracturing was performed

at high rates the maximum position is shifted a little later in time (see 0.01 md and

0.1 md curves in Figure 3.10). A possible explanation is that for large fractures, and

consequently large volume of liquid that remains inside longer after the at shut-in, the

fluid that flows first to the well is at low temperature and it decreases the accuracy

of this simplification.

For the falloff scenario, a possible simplification is to describe the heat transfer

between the well and reservoir as a fluid-filled cylinder at low initial temperature (Ti)

surrounded by a medium at high temperature (Tres), where the heat is transferred

by conduction through the cylinder wall (Figure E.1). The higher the temperature

variation in the reservoir in the vicinity of the well the worse will be this approxima-

tion compared with the full physics solution. The governing equation for simplified

warmback is presented in Appendix E, section E.2.

The logarithmic derivative of the analytical solution for well temperature during

falloff has a maximum which happens at a shut-in time given by:

∆tmax = ηT =ρlClrw2U

(3.2)

From Equation 3.2 we can obtain the effective heat transfer coefficient between

well and reservoir (U) by taking the time of maximum dT/dlnt semilog plot, when

the fluid density and heat capacity are known. The slope of the straight line formed

by the semilog plot ln(dT/dt) vs. t can also be used to calculate U , because it is

given by −1/ηT . As described in Appendix E.1, the calculated value of U obtained

by the semilog analysis is smaller than the one used as an input to the simulator.

By comparing the derivative of the numerical solution with the ones for simplified

analytic equations, it is concluded that the value of U obtained by Equation 3.2 can


be seen as a lower bound for the effective heat transfer coefficient. This would be

useful information in that the heat transfer coefficient is often unknown.

For the falloff case the analytic approximations are very rough and become less

accurate as the wellbore radius increases, because the decrease in the wall temper-

ature and heat conduction inside the reservoir are not accounted for, which reflects

a smaller heat transfer coefficient estimate than the actual one. High injection rates

and permeabilities also contribute to decrease the accuracy of this representation.

3.3 Impact of Reservoir Heterogeneity

The impact of heterogeneity in reservoir permeability was investigated through four

different example configurations, as presented in Figure 3.38. The colored zone in the

figures represents the part of the reservoir where permeability was varied from 0.01

md to 10 md, and the white part remained at constant permeability equal to 1 md.

Figure 3.38: Reservoir heterogeneity configurations. The white part represents theconstant permeability equal to 1md, and the colored one is the part of the reservoirwhere different permeabilities are specified.

For all the cases the width of the inner zone is 60 ft. The final fracture half

length is presented in Table 3.7. The main influence on the final fracture dimensions

comes from the permeability of the zone into which the fracture is growing. Among

the analyzed heterogeneity configurations, only case 1 and 4 configurations had a

pronounced effect on the final fracture volume. In cases 2 and 3 the fracture grows

most of the time in 1 md medium, which led to no significant difference between the

3.3. IMPACT OF RESERVOIR HETEROGENEITY 83

fracture lengths.

Table 3.7: Fracture half length (xf ) for different heterogeneous cases.

Permeability Case 1 Case 2 Case 3 Case 4

0.01 md 3084 ft 1009 ft 1009 ft 3084 ft0.1 md 2119 ft 1009 ft 1009 ft 2119 ft1 md 1009 ft 1009 ft 1009 ft 1009 ft10 md 413.3 ft 1009 ft 1009 ft 450.7 ft

The pressure signature of case 1 is similar to the homogeneous reservoir for the

respective permeability of the colored zone (0.01, 0.1, 1 and 10 md), as shown in

Figure 3.39. Different from pressure, the temperature did not show any difference for

case 1 configuration during injection and warmback. The temperature history and its

logarithmic derivative for warmback are presented in Figures 3.40-top and 3.42-top,

the curves have the same behavior as the 1 md homogeneous reservoir.

The pressure curves for case 2 have small difference from each other and from

the 1 md homogeneous reservoir. On the other hand the temperature shows the

characteristic of the inner part permeability (Figures 3.40-bottom and 3.42-bottom),

showing the same behavior as for a homogeneous reservoir (sensitivity to permeability

shown in Section 3.1.1).

For case 3 the warmback temperature has no sensitivity to the heterogeneity,

confirming that warmback temperature can only see the properties that are close to

the wellbore (Figures 3.41-top and 3.43-top).

In case 4 the temperature follows the influence of the inner permeability part

as presented in Figures 3.41-bottom and 3.43-bottom. All the warmback cases in

the presence of heterogeneity showed that the temperature is most affected by the

permeability properties close to the wellbore.

The flowback scenario for the four hetereogeneous configurations (Figure 3.38)

shows an interesting characteristic of the influence of fracture length and permeability

of the near-wellbore region. From the logarithmic derivative for the flowback period

we see the differences between early and late times. Case 1 has differences in the

derivative behavior at early times, but at later times all the curves converge to the


10−3

10−2

10−1

100

101

102

103

100

105

∆t (h)

∆P

(psi)

d∆

P/d

lnt

Case 1

10−3

10−2

10−1

100

101

102

103

100

102

104

∆t (h)

∆P

(psi)

d∆

P/d

lnt

Case 2

10−3

10−2

10−1

100

101

102

103

100

105

∆t (h)

∆P

(psi)

d∆

P/d

lnt

Case 3

10−3

10−2

10−1

100

101

102

103

100

105

∆t (h)

∆P

(psi)

d∆

P/d

lnt

Case 4

k=0.01md k=0.1md k=1md k=10md

Figure 3.39: Reservoir heterogeneity: pressure logarithmic derivative during falloff.The legend is the same for all the figures and the permeability values are referent tothe colored area presented in Figure 3.38, while the permeability in the white area isequal to 1md for all the curves.


same behavior. Case 1 is characterized by different fracture dimensions for each

considered permeability and the same permeability as the inner part.

Case 2, as opposed to case 1, presents a similar behavior at early times and a clear

difference between the temperature derivatives at later times. In case 2 configuration

all the fractures have the same length, but different inner permeability values. Com-

paring the temperature signature during flowback from cases 1 and 2 (Figure 3.44) we

can suspect that at early time the response is influenced by the fracture dimensions

and the later time by the permeability.

The same comparison can be made for cases 3 and 4 (Figure 3.45). For case 3,

all the curves have the same final fracture length and same inner permeability. In

this case the temperature logarithmic derivatives are the same, completely overlaying

each other (Figure 3.45-top). Case 4 has both different fracture lengths and different

inner permeability values. For this case, all temperature logarithm derivative curves

are distinctive at early and later times (Figure 3.45-bottom).

3.4 Chapter Summary

Throughout this chapter a series of sensitivity analyses were performed and from those

it can be concluded that reservoir permeability and injection rate are the parameters

that most affected the temperature behavior during injection and shut-in/flowback

periods.

For permeability values lower than 1 md it was observed that heat transport is

dominated by diffusion, which makes the temperature independent of the transport

properties, like permeability. For this reason there is no clear distinction among

the curves for shut-in that describe values below 1 md presented in Figure 3.6. If

instead of shutting the well the flowback is performed, the heat transfer starts to

be dominated by convection, which makes the problem again a function of transport

parameters. In this case even the low permeability curves show a definitive distinction

between them. It is important to point out that flowback immediately after injection

has some practical complications from the operational point of view that are not

considered here.


0 5 10 15 20 2520

30

40

50

60

70

80

90

t(h)

T (

oC

)

Case 1

0 5 10 15 20 2520

30

40

50

60

70

80

90

t(h)

T (

oC

)

Case 2


Figure 3.40: Reservoir heterogeneity: temperature during warmback for cases 1 and2. The legend is the same for all the figures and the permeability values are referentto the colored area presented in Figure 3.38, while the permeability in the white areais equal to 1 md for all the curves.


0 5 10 15 20 2520

30

40

50

60

70

80

90

t(h)

T (

oC

)

Case 3

0 5 10 15 20 2520

30

40

50

60

70

80

90

t(h)

T (

oC

)

Case 4


Figure 3.41: Reservoir heterogeneity: temperature during warmback for cases 3 and4. The legend is the same for all the figures and the permeability values are referentto the colored area presented in Figure 3.38, while the permeability in the white areais equal to 1 md for all the curves.


10−3

10−2

10−1

100

101

102

103

0

5

10

15

∆t (h)

dT

/dln

t (o

C)

Case 1


10−3

10−2

10−1

100

101

102

103

0

5

10

15

∆t (h)

dT

/dln

t (o

C)

Case 2

Figure 3.42: Reservoir heterogeneity: temperature logarithmic derivative duringwarmback for cases 1 and 2. The legend is the same for all the figures and thepermeability values are referent to the colored area presented in Figure 3.38, whilethe permeability in the white area is equal to 1 md for all the curves.


10−3

10−2

10−1

100

101

102

103

0

5

10

15

∆t (h)

dT

/dln

t (o

C)

Case 3

10−3

10−2

10−1

100

101

102

103

0

5

10

15

∆t (h)

dT

/dln

t (o

C)

Case 4


Figure 3.43: Reservoir heterogeneity: temperature logarithmic derivative duringwarmback for cases 3 and 4. The legend is the same for all the figures and thepermeability values are referent to the colored area presented in Figure 3.38, whilethe permeability in the white area is equal to 1 md for all the curves.


10−3

10−2

10−1

100

101

0

5

10

15

20

25

30

∆t (h)

dT

/dln

t (o

C)

Case 1

10−3

10−2

10−1

100

101

0

5

10

15

20

25

30

∆t (h)

dT

/dln

t (o

C)

Case 2


Figure 3.44: Reservoir heterogeneity: temperature logarithmic derivative during flow-back for cases 1 and 2. The legend is the same for all the figures and the permeabilityvalues are referent to the colored area presented in Figure 3.38, while the permeabilityin the white area is equal to 1 md for all the curves.


10−3

10−2

10−1

100

101

0

5

10

15

20

25

30

∆t (h)

dT

/dln

t (o

C)

Case 3

10−3

10−2

10−1

100

101

0

5

10

15

20

25

30

∆t (h)

dT

/dln

t (o

C)

Case 4


Figure 3.45: Reservoir heterogeneity: temperature logarithmic derivative during flow-back for cases 3 and 4. The legend is the same for all the figures and the permeabilityvalues are referent to the colored area presented in Figure 3.38, while the permeabilityin the white area is equal to 1 md for all the curves.


The influence of injection rate on temperature is more related with the well ef-

fect than with the change in the total volume of fluid injected and final fracture

dimensions.

Due to the pressure effects on the energy balance equation, the fracture closure

time can be captured by the temperature first derivative when the pressure change

has a high magnitude. The indication of fracture closure from the temperature data

can be used to confirm the estimate from traditional pressure analysis, or as the only

estimate when pressure is not available.

The local characteristics of temperature makes the sensor location very important

for the response measured. The placement of the fiber behind the casing increases

the sensitivity to the reservoir behavior, having less influence from well effects.

The heterogeneous reservoir investigation during warmback showed that temper-

ature can only see the properties that are close to the wellbore. For the flowback the

early time is influenced by the fracture dimensions and late times by the permeability

near wellbore.

All the analyses were performed considering a single zone where the created frac-

ture stays contained perfectly. No vertical growth and interaction between zones was

accounted for.

Chapter 4

Minifrac

A straightforward application of the model presented in Chapter 2 is the study of

minifrac tests. Minifrac is characterized by a very short injection time, usually a few

minutes, at high rate capable of breaking the rock and creating a small fracture. The

test is followed by a long falloff, where the pressure decline is recorded during the

shut-in period (Figure 4.1). This test is normally performed in order to estimate the

fracture closure pressure, and when the falloff is long enough to reach the radial flow

it also can provide a valuable estimate of permeability, which will be used for the

main fracturing job. Minifracs are not conventional well tests, because a fracture is

created during injection and closes during falloff.

As the time of injection is short compared to the falloff, the minifrac could be

approximated as an instantaneous injection. As described by Gu et al. (1993) and

Abousleiman et al. (1994), the instantaneous injection pressure solution for the falloff

period is given by the derivative of corresponding constant rate solution multiplied

by the volume injected:

∆pwv(t) = Vinjd∆pw(t, CsT )

dt(4.1)

Noticing this fact, Craig (2006) realized the slug-test analysis methods (Peres

et al., 1993) can be applied to the falloff data as though the created fracture was

preexisting. From Equation 4.1 the equivalent constant and unitary rate injection

93

94 CHAPTER 4. MINIFRAC

Figure 4.1: Minifrac rate schedule schematic.

pressure solution (∆pw) can be obtained by integrating the recorded falloff pressure

(∆pwv). Equation 4.2 describes the data transformation, and using the same principle

the logarithmic derivative is calculated directly (Equation 4.3).

∆pw(t, CsT ) = I(∆pwv) =1

Vinj

∫ t

0

∆pwv(τ)dτ (4.2)

d∆pw(t, CsT )

dln(t)=

tpwv(t)

Vinj

(4.3)

We used our numerical model to simulate a minifrac performed in a 25 ft thick

reservoir with a permeability of 1 md, where the injection time was 5 minutes at

12 ft3/min. The pressure result is presented in Figure 4.2 in terms of logarithmic

derivative during falloff. At the early times the log-log plot presents the effect of

changing size fracture, with the equivalent time derivative assuming higher values

than the pressure difference for intermediate and later times.

95

Figure 4.2: Bourdet derivative of minifrac falloff pressure for 1md reservoir.

Figure 4.3: Bourdet derivative and finite conductivity model match for integral trans-formed minifrac pressure falloff (psi.h) for 1 md reservoir.


Subsequently, we applied the integral transformation suggested by Craig (2006)

to our simulated falloff data when the fracture is closing (Equation 4.2). Figure 4.3

shows the effect of such transformation. The Bourdet derivative of the transformed

data presents a more familiar characteristic: wellbore storage followed by fracture

influence, and subsequent radial flow.

Using the integral transformation of closing fracture pressure during the falloff

period as input data into a commercial well test analysis software allows the data to

be interpreted as a regular constant-injection test, with unit rate (-1). The model is

able to recover the reservoir permeability and also the fracture dimensions (Table 4.1).

Figure 4.3 shows the model (black lines) fitting the transformed data with parameters

given by nonlinear regression. The initial closing fracture behavior is transformed into

the characteristic unit slope of wellbore storage.

Table 4.1: Comparison between input and interpreted parameters for minifrac test.

Parameter Simulated Case Data Interpretation

Permeability (k (md)) 1 1.03Fracture half length (xf (ft)) 117 118

To understand the signatures displayed by the falloff pressure in a closing fracture

scenario we compared the dynamic fracture simulation with the respective response

for a fixed fracture that has the same characteristics as the stable final stage of the

closing fracture (same reservoir properties, fracture length and conductivity). We also

compared with the response of a simple vertical well. The falloff response comparisons

are shown in Figure 4.4 (top). As can be seen, after the closing fracture reaches a

more stable geometry its derivative overlays the derivative for the traditional fixed

fracture scenario, showing not only the fracture effect but also transitioning together

to the radial flow. The fact that for later time the falloff response for fixed fracture

overlays the derivative for growing/closing fracture case implies that after sufficient

time the reservoir behavior is not influenced by the fracture propagation, but by

the final fracture shape. After stable fracture geometry is reached, the influence of

bilinear flow is observed. For even later time the radial flow is obtained, which is in

97

agreement with Gu et al. (1993) and Abousleiman et al. (1994). When the integral

transformation is applied the log-log plot shows a more familiar characteristic, as

shown in the lower plot in Figure 4.4.

10−3

10−2

10−1

100

101

102

103

101

102

103

104

105

∆t (h)

∆P

; d

∆P

/dln

(t)

[psi]

Closing Frac

Fractured Well

Vertical Well

10−3

10−2

10−1

100

101

102

103

100

101

102

103

104

I(∆

P)

[

psi.h]

∆t (h)

Closing Frac

Fractured Well

Vertical Well

Figure 4.4: Comparison between closing fracture, fixed fracture and vertical wellfalloff type curves (top), and equivalent integral transformed analysis (bottom).

The integral operation transforms the early behavior of falloff data for the closing

fracture into pure wellbore storage (Figures 4.3 and 4.4-bottom). For the growing

fracture problem the total storage coefficient can be expressed by:


CsT (t) = clVw + clVfr(t) +dVfr(t)

dp(4.4)

A comparison between the terms on the right hand side of Equation 4.4 (Figure

4.5) shows that the magnitude of the derivative of fracture volume with respect to

pressure is higher than the storativities of the fracture (clVfr) and wellbore. More

than that, the derivative has a high value for the majority of time when the integral

transforne data (I(∆p)) exhibits the unit slope. Then dVfr/dp drops and the trans-

formed pressure behavior starts to transition to linear flow. The value of the wellbore

storage coefficient interpreted from the transformed data is 0.009 bbl/psi.

For the PKN fracture the derivative of fracture volume with respect to pressure

during the falloff for ideal cases depends on fracture stiffness. Substituting the input

values used for our simulation into the fracture volume derivative expression, the

derivative of fracture volume with respect to pressure can be calculated as follows:

dVfr(p)

dp= 2hxf

dwf (p)

dp≈ 2hxf

Sf

= 0.0095 bbl/psi (4.5)

where Sfr is the fracture stiffness and the expressions for PKN and KGD fractures

are presented in Table 4.2.

Table 4.2: Fracture stiffness (Sf ) for PKN and KGD geometries.

Fracture Model PKN KGD

Stiffness (Sfr)2E

π(1− ν2)h

E

π(1− ν2)xf

For the case of constant wellbore storage in static reservoir geometry, the wellbore

pressure is given in the Laplace space by:

∆pwv(s) = Vinjs∆pwc(s)

1 + CsT s2∆pwc(s)(4.6)

The integral of falloff pressure (Equation 4.6) will be equivalent to the constant

rate at the wellhead with storage coefficient CsT :

99

L∫

∆pwv(t)dt

=1

s∆pwv(s) = Vinjs

∆pwc(s)

1 + CsT s2∆pwc(s)= Vinj∆pw(s) (4.7)

The fact that storage is high when the fracture is closing can explain why the unit

slope is seen at the early times of the transformed data, and also the storage behavior

is covering the change in fracture conductivity with decrease in aperture. The data

integration brings the wellbore and fracture storages to influence the data in the same

way the regular wellbore storage does in a traditional drill stem test (DST).

10−2

10−1

100

101

102

0

0.01

0.02

0.03

0.04

0.05

0.06

∆t (h)

dV

fr/d

P(f

t3/p

si)

10−3

10−2

10−1

100

101

102

103

4

5

6

7

8x 10

−5

∆t (h)

cfV

fr +

cfV

w(f

t3/p

si)

Figure 4.5: Fracture volume derivative with respect to pressure (top) and fracturestorativity (bottom) during the falloff for 1md reservoir.


A useful conclusion that was possible by analyzing the fracture behavior during

closure and the data transformation by integral is the fact that the moment in time

when the pressure and its derivative start to first separate at the end of pure storage is

equivalent to the decrease in dVfr/dp. This statement can be visualized by comparing

Figures 4.4 (bottom) and 4.5 (top). The same was observed in all the simulated cases

so far, with permeability ranging from 100 nd to 10 md.

One problem of the integral transformation approach is that it requires knowledge

of the correct initial pressure (pi). Even a small mistake can cause a large modification

in the pressure change (∆p) and its logarithmic derivative such that they do not show

the radial flow clearly, as exemplified in Figure 4.6. 5 psi deviation from the correct

initial pressure makes the integral data deform and the interpretation is compromised

severely. The plots in Figure 4.6 are based on simulated results, but the same behavior

is seen in real data.

4.1 Low Permeability Reservoirs

Hydraulic fracturing is commonly applied to low permeability reservoirs, with per-

meability ranging from micro- to nanodarcies. In order to analyze the reservoir and

fracture behavior in those scenarios we simulated a minifrac performed in a 100 nd

reservoir.

As we can see in Figure 4.7, the Bourdet derivative with respect to Agarwal

equivalent time (teq) for the falloff period represents the common characteristics of

many real data from unconventional reservoirs published in the literature (Hawkes

et al., 2013, for example). The 3/2 slope appears in cases in which the permeability

is very low and the fracture remains open for a considerable amount of time. This

is in agreement with Nolte’s solution (Nolte, 1979) for high efficiency fracturing jobs

(100% efficiency), because there is very low leakoff due to the formation being tight.

This behavior has been explored by recent papers in order to estimate the closure

pressure, which would be the point where the pressure derivative deviates from 3/2

slope (Marongiu-Porcu et al., 2011; Mohamed et al., 2011; Bachman et al., 2012).

For this specific example the fracture does not stop growing after 5 minutes of

4.1. LOW PERMEABILITY RESERVOIRS 101

10−3

10−2

10−1

100

101

102

103

100

101

102

103

104

∆t (h)

I(∆

p),

dI(

∆p)/

dln

∆t

[psi.h]

I(∆p)

dI(∆p)/dln∆t

10−3

10−2

10−1

100

101

102

103

100

101

102

103

104

∆t (h)

I(∆

p)

dI(

∆p)/

dln

∆t

[psi.h]

I(∆p)

dI(∆p)/dln∆t

10−3

10−2

10−1

100

101

102

103

100

101

102

103

104

∆t (h)

I(∆

p)

dI(

∆p)/

dln

∆t

[psi.h]

I(∆p)

dI(∆p)/dln∆t

Figure 4.6: Effect of initial pressure on integral transformed data. The actual pi usedto create the simulated data is 2500psi. The figures from the top to the bottom weregenerated from ∆p calculated pi equal 2500, 2505 and 2495 psi, respectively.


injection. The fracture grows for 2 minutes more at the beginning of falloff. In this

case, also due to a very small leakoff, the fracture stays open for several hours, which

influences the pressure derivative directly. If we perform the integral transformation

on the falloff part of the data we can see that unit slope remains for 21 hours (Figure

4.7- bottom). There is a correspondence between the end of unit slope and a more

stable fracture volume. For this case, after the transformation we can see that radial

flow is not seen during the 200 hours of falloff. As the reservoir permeability is very

low even the short injection was able to create a considerable fracture length (400

ft), and the transition to fracture flow regime (linear or bilinear) is what we see after

the unit slope.

Figure 4.8 (upper) shows pressure derivative for the falloff period of a field test

in which a minifrac was performed in a high-pressure ultralow permeability reservoir.

The injection time was 3 minutes, followed by 48 hours of falloff. As can be seen, the

pressure derivative shows a similar behavior to our simulated examples. The integral

transformation led to the permeability estimate of 130 nd. In this case the estimated

fracture length from interpretation of integral transformation in a commercial well

testing software was small (6 ft) and it is in agreement to the observation that unit

slope starts to separate after 1 minute of falloff.

4.2 Chapter Summary

The program described in Chapter 2 was applied to minifrac analysis and used to

investigate the behavior of a data transformation based on the integral operation

technique. For all the examples analyzed so far, the end of unit slope of the trans-

formed data happens at the same time as when the value of dVfr/dp drops, and can

be taken as the fracture closure time.

It was possible to recover final fracture properties and reservoir permeability from

traditional well testing technique, given that the duration of falloff was long enough

to develop the linear/bilinear and radial flow regimes after the fracture walls have

touched. The advantage of minifrac tests is the fact that short injection time allows

those regimes to happen sooner than would be seen in a traditional DST.


10−3

10−2

10−1

100

101

102

103

101

102

103

104

105

106

∆t

∆p

[p

si]

d∆

p/d

ln(∆

t)

d∆p/dln(∆t)

∆p

3/2 slope

10−3

10−2

10−1

100

101

102

103

100

101

102

103

104

105

106

∆t

I(∆

p)

[

psi.h]

dI(

∆p)/

dln

(∆t)

I(∆p)

dI(∆p)/dln(∆t)

Figure 4.7: Minifrac performed in 100 nd reservoir: falloff Bourdet derivative withrespect to Agarwal equivalent time (top) and of integral transformed data (bottom).


10−3

10−2

10−1

100

101

102

102

103

104

105

106

∆t (h)

∆p

d∆

p/d

ln∆

t (p

si)

∆p

d∆p/dln∆t

10−3

10−2

10−1

100

101

102

10−2

100

102

104

∆t (h)

I(∆

p)

dI(

∆p)/

dln

(∆t)

(psi.h) I(∆p)

dI(∆p)/dln(∆t)

Figure 4.8: Field data of minifrac in an ultralow permeability gas reservoir: FalloffBourdet pressure derivative with respect to Agarwal equivalent time (top) and integraltransformation (bottom).

Chapter 5

Horizontal Multifractured Well

Multifracture treatments have proven to be an effective method for developing oil

and gas especially in unconventional reservoirs (Figure 5.1). However, the process of

horizontal well fracturing is uncertain and it is not fully understood yet.

This chapter explores the use of temperature data to improve the understanding

of well, fracture and reservoir behavior during hydraulic fracturing operations in a

horizontal well.

This study was motivated by the recent improvement in temperature measurement

systems, where continuous wellbore temperature profiles can be obtained with high

precision (Drakeley et al., 2006; Ouyang and Belanger , 2006). Small temperature

changes can be detected by modern temperature-measuring instruments, such as fiber-

optic distributed temperature sensors (DTS) in intelligent completions.

A numerical horizontal well model was developed and coupled with reservoir and

fracture. Different from the recent developments in horizontal well modeling (Yosh-

ioka, 2007; Yoshioka et al., 2005), we have considered a transient reservoir model and

fracture creation and closure. The model can generate n hydraulic fractures at any

moment of the simulation time. The location of a fracture along the horizontal well

is given as an input to the model. Mass, stress and energy balances are accounted

for. The model is solved numerically by finite-difference technique and was used to

study horizontal multifractured wells and to understand the temperature response

throughout the fracturing process.

105

106 CHAPTER 5. HORIZONTAL MULTIFRACTURED WELL

The modeling of the horizontal well is described in the next section, followed by the

results for relevant scenarios such as the sequential stage fracturing and simultaneous

fracture growth (more than one fracture growing at the same time). The existence of

simultaneous fracture growth during the same stage was identified by a DTS survey

during hydraulic fracturing according to Huckabee (2009).

Figure 5.1: Horizontal well with multiple fractures schematic (extracted from Miller(2013)).

5.1 Horizontal Well Model

The wellbore model consists of two main parts: the flow and the thermal models. The

wellbore flow model is composed of mass balance and momentum balance, and the

wellbore thermal model is formulated by the energy balance equation. The wellbore

flow model is used for solving wellbore fluid velocity and pressure profiles, and the

wellbore flowing fluid temperature profiles are solved from the wellbore thermal model.

In a way similar to that presented by Yoshioka (2007), the wellbore flow model is

5.1. HORIZONTAL WELL MODEL 107

treated as a sequential steady-state. On the other hand, the thermal model is treated

as transient. This difference in treatment is justified by the fact that wellbore fluid

flow will become stabilized much faster than the wellbore fluid heat transfer process.

At each time-step, wellbore flowing fluid velocity, pressure, and temperature profiles

are updated by using reservoir information.

Pressure drop along the wellbore affects the reservoir pressure distribution and

thus mass transfer between the reservoir and wellbore. The wellbore flow model

is described by Equations 5.1 and 5.2, which represents the mass and momentum

balances, respectively (Ouyang and Belanger , 2006; Ouyang , 2005).

The mass balance consists of rate of mass flow into the well minus the rate of mass

flow out of the well should be equal to the rate of accumulation of mass:

∂ρ

∂t=

2

rwγρIvI +

∂ (ρv)

∂y(5.1)

The momentum balance in the axial direction is written as:

dp

dy= −ρv2

rwf − d (ρv2)

dy− ρgsinθ (5.2)

where g represents gravity and θ is the angle formed between the well and the hori-

zontal position.

From Ouyang (1998) the friction coefficient for injection well assuming laminar

flow is:

f =16

Re

[

1− 0.0625(−Re)

1.3056

(Re + 4.626)−0.2724

]

(5.3)

The steady-state horizontal well thermal model was presented by Yoshioka (2007).

He has developed a numerical model and asymptotic analytic solution with the goal

of estimate flow rate inflow distribution along the well based on the temperature

response. In Yoshioka’s problem the Joule-Thomson effect is the main driven force

for the temperature alteration. In our case the difference between injected fluid and

geothermal temperature at reservoir depth is the main cause of temperature variation,

although Joule-Thomson effect is also included.


Figure 5.2: Horizontal wellbore discretization. The wellbore is represented by anone-dimensional numerical model.

Figure 5.3: Multifractured horizontal well grid representation. Half of the geometryis presented, where the reservoir and fracture are divided in grid-blocks. The wellboreis also discretized and the nodes represented in green are the ones that are connectedwith the grid-block at the center of the fracture and from where the fluid can leaveor enter the well.

5.1. HORIZONTAL WELL MODEL 109

Energy transport in the wellbore is governed by heat convection, heat conduction,

source/sink and friction heating. By summing up all these terms, the energy balance

in the wellbore is written as:

−ρlCl−→vwTw + λl

∂2Tw

∂y2+ βTw

∂p

∂t+ (βTw − 1)−→vwp+

2U

rw(Tr − Tw) + Sw = ρlCl

∂Tw

∂t(5.4)

Equations 5.1, 5.2 and 5.4 are solved numerically and the discretized versions for

each wellbore element (Figure 5.2) are presented in Appendices B and C.

The connection between wellbore, fracture and reservoir is illustrated in Figure

5.3. The boundary condition is specified flowrate at the well-head. The wellbore open

flow perforations are specified to be at the fracture positions along y-direction. When

only one fracture grows at a time the total rate is injected at a single fracture (green

block in Figure 5.3). When more than one fracture grows simultaneously the rate

is distributed among the fractures guided by the fracture and surrounding reservoir

properties. The flow coming from the wellbore to the fracture appears as a source

term in the equations for the grid-block at the center of the fracture.

5.1.1 Fracture representation

The multiple fracture creation in a horizontal well was the target of this section of the

research. More than one fracture can initiate growth from the wellbore simultaneously

or sequentially (stage by stage), as it is done usually in field operations.

We assumed a two-dimensional model, where the PKN fracture geometry (Nord-

gren, 1972) is used to describe the fracture growth. In this way the fracture parame-

ters and growth criteria are the same as presented in Chapter 2.

The number of fractures and the position from where each of them start to grow

are inputs of our model, as well as the moment in time when the injection is going

to start at each stage. The fractures can interact with each other and influence the

growth behavior from the pore-pressure prospective and the flow distribution. The

effect on the local stress field change is not considered in the analyses presented in

this work.


The next sections address the issues of sequential and simultaneous fracture growth

separately.

5.2 Sequential Fracturing

Hydraulic fracturing field operations in horizontal wells are performed by dividing

the well length in many intervals (or stages) and fracturing them sequentially from

toe towards the heel. In cased and cemented formations the operation starts by

perforating the first stage of interest. Following the perforation of the interval, the

high rate fluid injection and fracture creation starts. When a stage injection period

is over, a plug is set to isolate that interval, and the adjacent one is then perforated

and the fracturing fluid injection starts again. This process goes on from the toe to

the heel until the scheduled sequence is completed (Figure 5.4). Another approach is

the use of open-hole packers with fracture sleeves (Holley et al., 2012).

Figure 5.5 shows a real DTS survey during multistage hydraulic fracturing ex-

tracted from Sierra et al. (2008). The injection periods are well marked by the cold

fluid traveling down to the perforation. After an interval is isolated the warmback

starts, where conduction is the main heat transfer mechanism, and the cold fluid

cannot reach this location any more. When the interval isolation is not effective it

can be easily identified from the DTS data, tracking the cold front reaching further

than the isolation point.

According to the forward model, convection has a significant impact on temper-

ature behavior during treatment. As the cold fluid travels along the well it cools

down the wellbore and the near-wellbore region. Convection is controlled mainly by

volumetric flow rate inside the wellbore which is a function of injection rate and the

fracture initiation point along the well. After the end of injection, the target stage

is isolated and the warmback starts. As the section is isolated, it is not likely to be

affected by the following injection periods. Such behavior is confirmed by the real

data presented by Huckabee (2009).

Because the intervals are isolated by sealing plugs, we can assume that there

is no convection, and therefore, conduction is the main heat transfer mechanism

5.2. SEQUENTIAL FRACTURING 111

Figure 5.4: Sequential multifrac along a horizontal well. Each stage is fractured andisolated by a plug before the next stage starts. Each schematic of the horizontal wellrepresents a moment in time during fracturing, where time increases from the top tothe bottom.


Figure 5.5: Real DTS temperature map along the horizontal well during multistagehydraulic fracturing (extracted from Sierra et al., 2008).

affecting the temperature of the isolated portion of the wellbore. Conduction from

the formation causes the wellbore temperature to warm up. Figure 5.6 shows the

map of temperature along the wellbore over time that results from the forward model

calculation. The input parameters are presented in Tables 5.1 and 5.2. This example

considered a sequence of four stages of half an hour injection followed by an interval

of one hour and a half between them. The injection periods can be followed easily by

observing the cold front moving down the wellbore. When the interval is isolated the

temperature increases towards the initial reservoir temperature. The speed at which

the wellbore temperature cools down is influenced highly by the injection flow-rate.

Figure 5.7 shows the pressure and temperature histories for each of the stages.

From this figure we can see that the stages are isolated and the moment at which

each stage was fractured. In this case the injection rate and duration of fracturing

was the same for all the stages. As the reservoir was homogeneous and one stage did

not interfere in the others, all four generated fractures are identical in dimensions.

Figure 5.8 shows two temperature profiles along the wellbore: one at the end of

injection period of stage one (30 minutes after the beginning of injection) and the

second referent to the end of injection of stage two. The point of fluid injection,


Table 5.1: Base case reservoir and fluid properties for horizontal multifractured well.

Parameter Value

Porosity (φ) 0.15Permeability (k) 0.5 mdReservoir thickness (h) 50 ftWell radius (rw) 0.3 ftHorizontal section lenght (L) 1000 ftReservoir depth (ztop) 2500 ftInitial pressure (pi) 2500 psiFluid compressibility (cl) 5.0× 10−6 psi−1



Fluid viscosity (µ) 1cpMinimal horizontal stress (σhmin or pc) 3500 psiPoisson ration (ν) 0.2Youngs Modulus (E) 3.0× 106 psiAsperities size (wf,min) 7× 10−4 ft

Table 5.2: Base case thermal properties for horizontal multifractured well.

Parameter Value

Fluid heat capacity (Cl) 4186.8 J/kg.KRock heat capacity (Cr) 921.1 J/kg.KRock thermal conductivity (λr) 1.44 W/m.KFluid thermal conductivity (λl) 0.52 W/m.KInjection temperature at surface (z = 0ft) 25 oCInitial Temperature (Tres) 95 oCµJT 5×10−7η 1×10−8


corresponding to the fracture initiation in this case, is well seen. Comparing these

two profiles we can also visualize the warm-up of the first stage. The locations where

the fracture was created and fluid was injected during the injection will warm up

slower compared with the locations that did not take fluid (Figure 5.9).

The temperature profiles are influenced strongly by the injection rate. The higher

the injection rate the faster the whole horizontal section of the well is going to be

cooled down.

Figure 5.6: Simulated wellbore temperature profiles over time for sequential hydraulicfracturing. Four fractures are created (one per stage) in a 0.5 md reservoir. The bluecolor is related with the cold injected fluid traveling along the wellbore. Position Lequal zero is the heel and 1500 is the toe of the well. After a stage is isolated, thatsection of the well starts to warm up towards the reservoir temperature.


0 200 400 600 800 1000 12000

100

200

300

time (min)

Fra

ctu

re h

alf length

− X

f(ft)

0 200 400 600 800 1000 1200

2500

3000

3500

4000

time (min)

Pre

ssure

(psi)

0 200 400 600 800 1000 1200

40

60

80

time (min)

Tem

pera

ture

(k)

stage 1

stage 2

stage 3

stage 4

Figure 5.7: Sequential hydraulic fracturing history per stage: fracture half length(top), pressure (middle) and temperature (bottom).


0 200 400 600 800 1000 1200 1400

30

40

50

60

70

80

90

100

Lf(ft)

T(o

C)

Stage 2 (t=180min)

Stage 1 (t=30min)

Figure 5.8: Comparison between wellbore temperature profiles at end of injection ofstage 1 (blue curve) and stage 2 (red curve). During the injection period a cold fronttravels down along the wellbore. After a stage is isolated the injected fluid cannotreach that part of the well anymore, as the red curve is illustrating.

0 200 400 600 800 1000 1200 140060

70

80

90

100

Lf(ft)

T(o

C)

Figure 5.9: Warmback temperature profile for sequential fracturing scenario. Thepoints where the fluid is injected take longer to warm up than the other sectionsof the well. The stages and isolation points are identified by the difference in thetemperature value.


The temperature recovery of each interval is going to depend on how long the zone

was exposed to the cold fluid and how much fluid has been injected in a specific zone.

The longer fracture is not necessarily located at the cooler spot. The sensitivity to

permeability shows that for higher permeabilities more cold fluid is going to be lost

to the formation in the vicinity of the wellbore, and so the longer it is going to take

to warmback. Those high permeability zones usually have the shorter fractures due

to the high leak-off associated with them.

The next section explores in more detail the permeability heterogeneity along the

horizontal well for the sequential multifracture problem.

5.2.1 Reservoir heterogeneity

This subsection addresses the scenario where the reservoir has a heterogeneous perme-

ability zone crossing the horizontal wellbore. To perform such analysis we considered

the same four-stage horizontal multifractured well, as presented previously. The reser-

voir heterogeneity consists of a stripe that extends along the x-direction at stage 3

location. The heterogeneous layer starts along y halfway between stage 4 and 3 and

extends halfway between stages 3 and 2. Figure 5.10 shows a schematic representation

of the problem.

The sensitivity analysis to heterogeneity permeability was performed considering

permeabilities ranging from 0.05 md to 50 md. The reservoir permeability outside

the heterogeneous zone is 0.5 md. The injection rate was 20 bpm. Each stage consists

of 40 minute injection followed by 2 hours shut-in.

The pressure analysis would be able to reflect the contrasts in permeability when

the shut-in period is long enough to reach the radial flow regime. But in reality a

sequence of operations happens, which disturb the falloff pressure record and its time

is very short. With the availability of permanently installed distributed temperature

sensors, the temperature during the warmback can be an alternative to monitor those

isolated zones.

The warmback profiles for different permeabilities are presented in Figure 5.11.


Figure 5.10: Heterogeneity along the reservoir for sequential hydraulic fracturingscenario. The colored area represents the region with different permeability valuethan the rest of the reservoir.

0 200 400 600 800 1000 120080

85

90

95

100

L(ft)

T (

oC

)

420 430 440 450 460 470 480 490 500 510 520

85

90

95

L(ft)

T (

oC

)

0.05md 0.5md 1md 5md 1md 50md

Stage 4 Stage 2Stage 3

Stage 3

Stage 1

Figure 5.11: Sensitivity analysis to permeability heterogeneity along the reservoir forsequential hydraulic fracturing scenario: temperature profile during warmback (top),and zoom in at the stage 3 location (bottom). Each color line is equivalent to adifferent reservoir permeability and the convention is the same for the two figures.

5.3. SIMULTANEOUS FRACTURING 119

The zoom at the heterogeneous location (Figure 5.11-bottom) allows us to visual-

ize the difference between the permeability cases. Temperature is more sensitive to

high permeability. The difference between the 50 md and the 10 md cases is much

more pronounced than the difference between 0.5 and 0.05 md. This fact was already

observed in Chapter 3 for the case of a vertical well single fracture. Another charac-

teristic of the warmback profile presented in Figure 5.11 is the cooler temperature at

stage 3 for the 50 md case than the temperature at stage 4. As stage 4 was fractured

later in time, it might be expected that this zone be cooler than the previous ones,

what is not true for the high permeability heterogeneous zone in stage 3. When a fact

like this is observed it can be a flag for possible different rock characteristics along

the well.

The change in temperature with time for different permeabilities at stage 3 is

presented in Figure 5.12. This figure highlights the fact that permeabilities below 1

md have almost the same behavior. The temperature logarithmic derivative (Figure

5.13) shows the same trend for all the curves, but the magnitude of the maximum

point changes with permeability. The higher derivative value is related with the lower

permeability.

5.3 Simultaneous Fracturing

For each stage of hydraulic fracturing in a horizontal well, the perforated length can

reach hundreds of feet, which opens the possibility for more than one fracture to

propagate from the wellbore. Depending on the distance between the fractures and

the reservoir characteristics in that area, one fracture can interfere with the others’

growth and closure.

This section addresses the simultaneous growth of multiple fractures at the same

stage (Figure 5.14). The initiation point along the horizontal well is chosen in advance,

in the same way as the number of fractures.

The fracture growth is represented by a simplified two-dimensional model and the

interaction between the fractures is set by the change in the local pore pressure due

to fluid injection, while the stress field and the rock properties are kept constant.


10 15 20 25 30 3520

30

40

50

60

70

80

90

100

t(h)

T (

oC

) 0.05md

0.5md

1md

5md

10md

50md

Figure 5.12: Sensitivity analysis to permeability heterogeneity along the reservoir forsequential hydraulic fracturing scenario: warmback temperature at stage 3.

10−3

10−2

10−1

100

101

102

0

5

10

15

20

25

Stage 3

t(h)

dT

/dln

t(oC

)

0.05md

0.5md

1md

5md

10md

50md

Figure 5.13: Sensitivity analysis to permeability heterogeneity along the reservoir forsequential hydraulic fracturing scenario: temperature logarithmic derivative duringwarmback at stage 3.


Figure 5.14: Simultaneous fracture growth at the same fracturing stage.

The total injection rate is divided between the existing fractures. The proportion

is given by:

qinj =

nf∑

i=1

qi (5.5)

where qinj is the total injection rate, nf is the number of fractures and qi is the fluid

rate at fracture i.

The fracture grid-block permeability is in continuous change to honor the fracture

conductivity, which makes the use of constant well-index inappropriate. The well

equation is not only a function of pressure, but also contact stress when the asperities

of opposite walls of the first grid-block are touching.

Figure 5.15 presents the pressure maps at the end of injection, where up to three

fractures are growing at the same time. This figure shows half of the reservoir, with

the symmetry line traced at the center of the horizontal well, which is located at x = 0.

The reservoir is homogeneous and isotropic. From the top to the bottom, Figure 5.15

shows the case where only one, two and three fractures are growing simultaneously

in the same fracturing stage. The injection period consists of 30 minutes of constant

injection rate at 11 bpm, and it is divided among the fractures that are growing

simultaneously. The higher the number of fractures, the shorter is the final fracture

length. The difference in fracture length can be visualized from the warm colors in

the pressure map (Figure 5.15). The reservoir flow and thermal properties that are


not mentioned here are equal the the ones presented in Tables 5.1 and 5.2.

For the analyzed cases, the fractures that have grown together in the same stage

have identical length. For two-fracture case the injection rate is evenly distributed

among the fractures. However, the same is not true for three fractures. When the

middle fracture (fracture 2) starts to feel the interference of the others located on both

sides, the injection rate at fracture 2 decreases slowly as the injection time increases.

Figure 5.15: Pressure map (psi) after 30 minutes of injection for 1 (top), 2 (middle),and 3 fractures (bottom) growing simultaneously.


The temperature profile along the horizontal wellbore after five minutes of injec-

tion is presented in Figure 5.16. The detailed view of the perforated interval (Figure

5.16 bottom) shows the differences between the case where the injection happens in a

single point (single fracture) and the cases of multiple fractures. When the injection

happens at several points along the interval there is a progressive decrease in flowrate

inside the wellbore from the heel to the toe. This phenomenon is evident in the early

time temperature data, and from the change in slope in the temperature profile it is

possible to identify the number of fractures that are likely to be created.

0 100 200 300 400 500 600 700 800 90060

70

80

90

L(ft)

T (

oC

)

1 frac

2 fracs

3 fracs

750 800 850 900

78

80

82

84

L(ft)

T (

oC

)

1 frac

2 fracs

3 fracs

5 min injection

Zoom − 5 min injection

Figure 5.16: Temperature profiles along a horizontal well after 5 minutes injection.Comparison between one (blue), two (green) and three (red) fractures growing at thesame time.

During the warmback, the locations where fluid was injected are well marked by

the slower temperature recovery rate when compared to other sections of the wellbore

that did not take fluid (Figure 5.17). For the cases where there is one or two fractures


the visualization of fracture locations is not very clear in the color map, and this

identification can be improved by taking the spatial derivative of temperature along

the wellbore. The first derivative of temperature with respect to position along the

wellbore is shown in Figure 5.18, which is able to identify the fracture positions in all

the three cases very clearly.

For the three fracture case, the fracture in the middle position flows back at very

low rate, injecting into its neighbors. This cross-flow happens due to the pressure

diffusion and the fact that the middle fracture suffers from interference of the other

two. The temperature map presented in Figure 5.17-bottom shows the moment where

the warmer fluid leaves the position of fracture 2 and travels towards the fracture on

its right and the other on its left. The average cross-flow rate can be obtained by

tracing the warm front path along the interval between the fractures. For the example

presented in Figure 5.17 the calculated rate from temperature is 0.0575 ft3/min,

while the mean of the crossflow rate obtained from simulation is 0.059 ft3/min (mean

obtained from the time interval correspondent to 15 minutes after shut-in, point where

the warm fluid start to leave fracture 2 position, and 300 minutes). The actual rate

distribution during falloff is presented in Figure 5.19. Fractures 1 and 3 have the

same rate behavior over time, the fluid produced from fracture 2 is evenly distributed

between the two side fractures.

The effect of more than one fracture is also felt by the pressure signal. The falloff

pressure derivative for the cases of two and three fractures reflect the interference

between the fractures by the steep increase in the pressure logarithmic derivative,

behavior that is not present in the single fracture case (blue curve). Figure 5.20

shows the pressure derivatives of the three cases.

5.3.1 Simultaneous fracture growth in presence of hetero-

geneities

For the sake of simplicity, we are going to discuss in more detail the effect of hetero-

geneity on the growth of two fractures at the same time. Falloff and flowback were

analyzed in homogeneous and heterogeneous reservoir permeability fields. Table 5.3


Figure 5.17: Temperature maps (oC) along a horizontal well during warmback whenone (top), two (middle), and three (bottom) fractures grow simultaneously.


Figure 5.18: Temperature first derivative with respect to position along the horizontalwell during warmback when one (top), two (middle), and three (bottom) fracturesgrow simultaneously (∂Tw/∂y in oC/ft).


500 1000 1500 2000 2500 3000 3500 4000 4500 5000−0.1

−0.05

0

0.05

0.1

0.15

0.2

qf(f

t3/m

in)

t(min)

frac1

frac2

frac3

Figure 5.19: Crossflow rates during falloff for three fractures growing simultaneously.As the side fractures are equally spaced in relation to the first one and the reservoiris homogeneous the amount produced from fracture 2 is equally divided betweenfractures 1 and 3.

10−2

10−1

100

101

102

103

101

102

103

104

105

∆t (h)

∆p

[p

si]

d∆

p/d

lnt e

q

1 frac

2 fracs

3 fracs

Figure 5.20: Pressure derivative comparison between 1 (blue), 2 (green) and 3 (red)fractures growing at the same time.


contains the combination of cases that are discussed in this section, and their geome-

tries are illustrated in Figure 5.21. Case 1 consists of homogeneous reservoir; cases 2

and 3 are reservoirs that have a region of high permeability that extends along the

x-direction. In case 2 the heterogeneity is located along the y-direction in such a way

that it contains the fracture 1 location, but not the fracture 2. Case 3 has a similar

geometry to case 2, but the high permeability zone is shifted to be only in fracture

2 position, while fracture 1 grows through the original reservoir permeability media.

Case 4 represents a heterogeneous scenario where the high permeability zone extends

along the y-direction and it does not cross the wellbore. Eventually, the fractures can

advance into the high permeability region.

Table 5.3: Cases analyzed for simultaneous fracture growth.

Falloff Flowback

Homogeneous Case 1a Case 1bHeterogeneous Fracture 1 Case 2a Case 2bHeterogeneous Fracture 2 Case 3a Case 3bHeterogeneous along y Case 4a Case 4b

Table 5.4: Input parameters for simultaneous fracture growth in a heterogeneousmedium.

Parameter Value

Injection rate (qinj) 11 bpmFlowback rate (qfb) 0.54 bpmHorizontal section length 1110 ftFracture 1 position 1106 ftFracture 2 position 1039 ftReservoir permeability (k) 0.5 mdHeterogeneity permeability (khet) 5 mdTres 90 oCTinj 35 oC

The input parameters are shown in Table 5.4, and the information that is not

presented is equal to the base case presented earlier in Tables 5.1 and 5.2. For all the


Figure 5.21: Schematic cases for 2 fractures growing simultaneously in presence ofheterogeneity. The colored areas represent higher oermeability than the rest of thereservoir.

presented cases, the wellbore has the same properties and the two fractures are 67 ft

apart from each other, located at stage 1 (closer to the toe). The injection period

consists of 30 minutes constant injection, followed by 50 hours of shut-in (or falloff).

For the b cases, the well is flowed back at low rate for 150 minutes.

Figure 5.22 presents the rate distribution between the two fractures for all eight

cases. During the injection period the permeability contrast created by the hetero-

geneity and the fracture growth dictates the rate distribution. The higher permeabil-

ity zone is going the take the higher portion of the injection. Both cases 1 and 4,

where the two fractures are exposed to the same media, have identical rate distribu-

tions. Case 4 has a zone of higher leak-off (higher permeability), and it causes the

fracture length to be shorter than the final length in case 1, as can be seen in Figure

5.23. For case 2, fracture 1 has a higher portion of the injection rate, however the

final fracture length is shorter than fracture 2 (Figure 5.23). The opposite occurs in

case 3, fracture 2 takes more fluid and it is shorter due to the higher permeability

in that position. Due to the high leak-off, when one of the fractures grows in higher

permeability zone it tends to have smaller length, as seen in Figure 5.23 for cases 2,

3, and 4. The length history is the same for both falloff and flowback cases.

The relative rate distribution can be inferred from the temperature analysis during

the early time of injection (Figure 5.24). Cases 1 and 4 have the same temperature

behavior, given that the injection rate is distributed between the two fractures evenly.


For those two cases the injection temperature would not be useful to identify the

presence of heterogeneity. For case 2, where the fracture 1 (closer to the toe) is growing

in a high permeability medium, and so it takes the highest amount of incoming

injection fluid. Consequently, there is smaller injection at the fracture 2 position.

This higher rate reaching the fracture 1 location causes the well to be cooler between

fractures 1 and 2. The other way around, case 3 has the higher permeability zone

located at the fracture 2 position, and so most of the cold fluid is injected at this

position, taking longer to cool down the interval between the fractures.

For the homogeneous case (cases 1a and 1b), the fractures have identical properties

and the pressure maps are symmetrical, as shown in Figure 5.26. This figure presents

the pressure maps at the end of injection for the four geometries (Figure 5.21).

The temperature derivative with respect to position along the well allows the

identification of the difference in rate distribution between the fractures, as can be

seen from the color map presented in Figure 5.25. This figure displays the information

for the total injection period, and could be used for real-time identification of injection

zones.

When the injection stops, the fracture that is growing in a lower permeability

medium has a tendency to close first, flowing back into the well part of the liquid

that was inside of the fracture. As a consequence it is possible to have cross-flow to

the other fracture. From Figure 5.22 it can be noticed that this happens for all the

cases where the two fractures have different properties. Even though a small flowback

rate is specified, the fracture that is placed in the lower permeability zone closes fast,

sending a large amount of fluid into the well almost instantaneously. After fracture

closure, the region of high permeability has the highest contribution to the flowback.

The existence of cross-flow during warmback was identified in DTS data from a matrix

stimulation job in Clanton et al. (2006), where due to reservoir heterogeneities one of

the zones received most of the injected fluid.

The falloff (or warmback) temperature maps are presented in Figure 5.27. This

figure would be the one most likely to be captured by a DTS installed inside the

wellbore. Analyzing the homogeneous scenario (case 1a), when the well is shut in

and the falloff starts the heat conduction through the well wall is the main heat


0 20 40 60 80−8

−6

−4

−2

0

2

t(min)

q(b

pm

)

case1a

frac1

frac2

0 20 40 60 80−8

−6

−4

−2

0

2

t(min)

q(b

pm

)

case1b

frac1

frac2

0 20 40 60 80−8

−6

−4

−2

0

2

t(min)

q(b

pm

)

case2a

frac1

frac2

0 20 40 60 80−8

−6

−4

−2

0

2

t(min)

q(b

pm

)

case2b

frac1

frac2

0 20 40 60 80−8

−6

−4

−2

0

2

t(min)

q(b

pm

)

case3a

frac1

frac2

0 20 40 60 80−8

−6

−4

−2

0

2

t(min)

q(b

pm

)

case3b

frac1

frac2

0 20 40 60 80−8

−6

−4

−2

0

2

t(min)

q(b

pm

)

case4a

frac1

frac2

0 20 40 60 80−8

−6

−4

−2

0

2

t(min)

q(b

pm

)

case4b

frac1

frac2

Figure 5.22: Flow-rate distribution for heterogeneous cases of simultaneous fracturegrowth. The cases presented are explained in Figure 5.21 and in Table 5.3.


0 20 40 60 80 1000

200

400

600

t (min)

xf(f

t)

case1

frac1

frac2

0 20 40 60 80 1000

100

200

300

400

t (min)

xf(f

t)

case2

frac1

frac2

0 20 40 60 80 1000

100

200

300

400

t (min)

xf(f

t)

case3

frac1

frac2

0 20 40 60 80 1000

100

200

300

Lf(ft)

xf(f

t)

case4

frac1

frac2

Figure 5.23: Fracture half length for heterogeneous cases of simultaneous fracturegrowth.

200 400 600 800 100040

50

60

70

80

90

L(ft)

T (

oC

)

case 1

case 2

case 3

case 4

1010 1020 1030 1040 1050 1060 1070 1080 1090 1100

65

70

75

80

L(ft)

T (

oC

)

case 1

case 2

case 3

case 4

Figure 5.24: Temperature profile after 5 minutes of injection for heterogeneous casesof simultaneous fracture growth.


Figure 5.25: Map of temperature first derivative with respect to distance along thewellbore during injection period (∂Tw/∂y in oC/ft).


transfer mechanism ruling the warmback. For heterogeneous cases, where the two

fractures are exposed to different permeabilities (cases 2a and 3a), the existence of a

cross-flow was identified at the early times from the low to the high permeability zone

(Figure 5.22, cases 2a and 3a). This fact can be observed clearly from the analysis of

temperature map (Figure 5.27, cases 2a and 3a). The fracture that produces warms

faster and the path of the hot fluid traveling from the producing fracture position to

the injecting ones can be tracked.

The fractures created in case 4 are identical, because they are exposed to the same

change in reservoir permeability. The temperature response during both injection and

falloff does not show any clear distinction between this case and the homogeneous one

(case 1a). Therefore the temperature during injection and shut-in can only see what is

very close to the wellbore, or heterogeneity that is strong enough to change the flow

pattern inside the wellbore. The similarity between temperature responses during

warmback is also presented by their logarithm derivatives: Figure 5.29 shows that

the cases 1a nd 4a overlay each other perfectly.

Figure 5.30 shows the falloff pressure derivative for cases 1a, 2a, 3a and 4a. From

this figure we see that pressure is not sensitive to whether the heterogeneity is located

in fracture 1 or fracture 2, because case 2a overlays case 3a exactly. What the

pressure sees is the average final behavior. On the other hand, the temperature

logarithmic derivative (Figure 5.29) captures the difference between cases 2a and

3a. The fact that there is cross-flow changes the temperature derivative considerably

at the fracture locations. This is an example where the use of temperature can

complement the pressure analysis, providing information that would not be available

only with pressure.

It is observed that a larger amount of fluid is injected in a region of higher perme-

ability, and it is not necessarily the zone where the longer fracture was created. The

fact that the fracture might cross high-conductivity fractures while it is growing also

changes the injection distribution.

Although it is less common from the practical point of view, we also present the

analysis of flowback in the four reservoir permeability cases described previously. The

flowback rate is small (Table 5.4), in accordance with the pump-in/flowback tests.


Figure 5.26: Pressure maps (psi) at the end of injection for heterogeneous cases ofsimultaneous fracture growth. The permeability heterogeneity affects the leak-off andconsequently the fracture dimensions.


Figure 5.27: Warmback temperature maps (in oC) at the end of injection for hetero-geneous cases of simultaneous fracture growth.


Figure 5.28: Map of temperature first derivative with respect to distance along thewellbore during warmback period (∂Tw/∂y in oC/ft).


10−2

10−1

100

101

102

0

50

100

150

t(min)

dT

/dln

t(oC

)

Fracture 1

10−2

10−1

100

101

102

0

50

100

150

Lf(ft)

dT

/dln

t(oC

)

Fracture 2

case 1a

case 2a

case 3a

case 4a

case 1a

case 2a

case 3a

case 4a

Figure 5.29: Warmback temperature derivative for heterogeneous cases of simultane-ous fracture growth. The top plot shows the logarithmic derivative of temperatureinside the well at the position of fracture 1, and the bottom plot is the temperaturederivative at fracture 2 position.


10−2

10−1

100

101

102

101

102

103

104

t(h)

∆p &

d∆

p/d

lnt

(psi)

case 1a

case 2a

case 4a

case 3a

Figure 5.30: Falloff pressure derivative for heterogeneous cases of simultaneous frac-ture growth.

The injection period and the fracture characteristics are the same as already presented

for the falloff cases, and all the conclusions and observations are also valid.

During the flowback the well temperature rises much faster than during the falloff.

The comparison between the time scale of the temperature maps presented in Figure

5.31, and Figure 5.27 endorses this argument. For the flowback in a homogeneous

reservoir (case 1b), the cold fluid starts to be produced from the fractures at early

times, until the warm reservoir fluid reaches the wellbore. From this moment on the

warmer fluid starts to flow up to the well and the difference in fluid temperature acts

like a tracer. From the temperature map, the hot front evolution along the well over

time allows the estimate of flowback rate. This property is even more interesting

for the heterogeneous cases. The slope calculated from the temperature map on

the position-time plane gives the cross-flow rate. The comparison between the rates

calculated from the temperature map and the actual simulation result is presented in

Table 5.5.

Similar to the warmback, at early times of flowback the fracture that has grown

in lower permeability medium closes first, injecting a volume of warm fluid into the


reservoir. After the closure the predominant contribution to the flowback rate comes

from the higher permeability zone (cases 2b and 3b in Figure 5.27).

The evolution of temperature derivative with respect to position along the wellbore

over time highlights the trace of warm fluid and rate distribution clearly (Figure 5.32).

In cases 1b and 4b the two fractures have the same rate contribution (50% each), and

we can observe the decrease in slope of the warm color as a straight trace from

the interval between the two fractures and the one between fracture 2 and the heel.

For case 2b, the fracture two ejects a volume of warmer fluid into the wellbore, but

subsequently it is fracture 1 which contributes to the flowback rate. This fact is well

seen from the derivative map, where the warm trace that comes later from fracture

1 does not have any change in slope after passing the fracture 2 position towards the

heel. Case 3b has the highest contribution to rate coming from fracture 2, and from

this point on the slope of the tracer remains constant. The interval between fracture

1 and 2 has a different rate of warm-up.

Table 5.5: Flowback rates estimated from temperature profiles.

Total rate Fracture 1 Fracture 2From T From T Actual From T Actual

Case 1 0.524 bpm 0.29 bmp 0.27 bpm 0.2324 bpm 0.27 bpmCase 2 0.54 bpm 0.37 bpm 0.36 bpm 0.17 bpm 0.18 bpmCase 3 0.548 bpm 0.22 bpm 0.18 bpm 0.328 bpm 0.36 bpmCase 4 0.548 bpm 0.28 bpm 0.27 bpm 0.268 bpm 0.27 bpm

The pressure history during flowback for the four reservoirs considered in this

section is presented in Figure 5.33. Similarly to the falloff case, the flowback pressure

is not able to differentiate between case 2b and 3b. For these two cases the use of

temperature data would be a complement to pressure data to identify the differences.

5.4 Chapter Summary

This chapter addressed the temperature response during multistage hydraulic fractur-

ing in horizontal wells. Two main configurations were accounted for: the sequential


Figure 5.31: Flowback temperature maps (oC) at the end of injection for heteroge-neous cases of simultaneous fracture growth.


Figure 5.32: Map of temperature first derivative with respect to distance along thewellbore during flowback period (∂Tw/∂y in oC/ft).


0 0.5 1 1.5 2 2.51500

2000

2500

3000

3500

4000

P(p

si)

t(h)

case 1b

case 2b

case 3b

case 4b

Figure 5.33: Flowback pressure for heterogeneous cases of simultaneous fracturegrowth.

multifracturing along the horizontal wellbore and the simultaneous growth of multiple

fractures during the same stage. The analysis demonstrated the localized character-

istics of the temperature data, when compared to the pressure data which reflects

only an average behavior along the well.

The map of temperature derivative with respect to distance along the well over

time appears to be a useful tool to identify fracture position and fluid rate at early

times of both injection and flowback periods.

The interference of more than one fracture on the growth of the others was inves-

tigated, and it was demonstrated that the interference can affect not only the final

fracture geometry, but also the flow pattern inside the well during falloff. It was

demonstrated that the usefulness of temperature analysis is not only to qualitatively

identify the existence of multiple fractures and the existence of cross-flow between

them, but also to quantify the flow-rate.

The existence of heterogeneities was also an issue of focus in this chapter. When

two fractures are growing in different permeability zones the pressure is not sensitive

to where along the well the heterogeneity is present. The temperature on the other

hand can give the spatial information to resolve this problem.


This study has shown that it was not necessarily the coldest zone that was the

one with the longest fracture, but it is the one with the highest local permeability.

Chapter 6

Fracture Interconnecting Different

Zones

This chapter presents the temperature response along a horizontal multifractured well

when one of the hydraulic fractures interconnects different zones vertically. Unlike in

the previous chapters, here we consider not only the injection and the falloff periods,

but also the initial production life of the well.

This chapter was motivated by the evidence from microseismic surveys that hy-

draulic fracturing may stimulate different zones far from the main reservoir by acti-

vating a preexisting fault or natural fractures. Microseismic fracturing imaging uses

an array of geophones or accelerometers in the treatment well or an offset well to

measure the acoustic energy transmitted from the slippage of microfractures and fis-

sures adjacent to a propagating fracture. The slippage creates seismic events that

form an “envelope” around the propagating fracture and by mapping the location of

each seismic event, the fracture azimuth, length and height can be inferred (Barree

et al., 2002).

Holley et al. (2010) have pointed out the benefits that DTS can bring when ap-

plied simultaneously with microseismic interpretation during hydraulic fracturing,

decreasing the uncertainties as to fracture initiation point, interval isolation, among

others.

145

146 CHAPTER 6. FRACTURE INTERCONNECTING DIFFERENT ZONES

Figure 6.1 presents an example of a microseismic survey during a multistage hy-

draulic fracturing job shown in Yang et al. (2013). They showed that the microseismic

events are clustered at two distinct depths, one close to the depth of the well being

pressurized and the other about 800 ft above, in a different formation. They ex-

plained the existence of events at long distance from the target reservoir as a result

of the hydraulic stimulation being dominated by flow channeling along preexisting

fractures and faults.

Based on this example we developed a model that accounts for the communication

between different zones through hydraulic fracture growth and activation of a preex-

isting fault. In the next section we present the base case geometry and the pressure

and temperature responses associated with it. The following sections explore the po-

sition along the well where the connection happened (which fracture has connected

the main reservoir with the zone above it) and the effect of depth difference between

the connected zones.

Figure 6.1: Microseismic events showing activity at 800 ft above the well that wassubmitted to hydraulic fracturing, extracted from Yang et al. (2013).

6.1. GEOMETRY DESCRIPTION 147

Figure 6.2: Illustration of hydraulic fracture interconnecting two isolated zones.

6.1 Geometry Description

The geometry described in this chapter consists of two isolated zones with a preex-

isting impermeable fault or a cluster of natural fractures that extends from one zone

to the other. The main reservoir is subjected to hydraulic fracturing treatment and

one of the fractures activates the interconnection between the reservoir and zone 2.

The geometry is illustrated in Figure 6.2. This geometry can be applied for cases of

more than one fracture growing simultaneously and/or sequential fracturing with one

fracture growing at a time.

The base case consists of two hydraulic fractures growing simultaneously and one

of them (the one closer to the toe) connecting the reservoir with another permeable

zone 500 ft above the main reservoir. The formation in between the two zones is

considered to be impermeable and the only way to transfer fluid and to communicate

pressures is through the fracture. The base case properties are presented in Table

6.1.

The initial temperature map is presented in Figure 6.3. The difference in fluid


temperature due to the geothermal gradient is of 11 oC.

When fracture 1 starts to grow it connects vertically with zone 2, having a higher

fluid loss than fracture 2, which stays contained within the main reservoir thickness

(h). Consequently, fracture 1 is shorter, even though the highest amount of the

injected fluid goes into this position. The pressure map at the end of the injection

period (Figure 6.4) shows clearly that hydraulic fracture in the main reservoir has

perturbed the zone 2, 500 ft above.

Following the fracturing, the well was shut in for 24 hours. After that the well was

put in production for 60 days at rate of 1000 barrels per day (STB/D). The pressure

map after 30 days of production shows that zone 2 is contributing to the production

actively through the decrease in pressure (Figure 6.5). The temperature map shows

the colder fluid traveling down through the fracture from zone 2 to the reservoir.

What can actually be measured from a sensing tool, such as DTS, is the tem-

perature along the wellbore over time. During injection the cold fluid front can be

tracked, which is influenced by the rate distribution (Figure 6.7). During the shut-in

the temperature inside the wellbore increases slowly due to the heat transfer with the

reservoir, basically moved by conduction through the well wall. When the production

starts, warm fluid from the reservoir causes an increase in temperature. Another

factor that plays for the temperature increase is the friction heating due to the fluid

motion, but the magnitude of this factor is small compared with the difference in

fluid temperature from the surface and the reservoir.

The map of temperature profiles over time shows that temperature decreases as the

production proceeds (Figure 6.8). The cold fluid enters the well progressively from

the fracture 1 position, and it has a high impact on the mixing well temperature,

because this fracture has the highest rate contribution.

Figure 6.9 presents the flow-rate for each of the two fractures. The highest contri-

bution to the total flow-rate comes from fracture 1, given that zone 2 has high perme-

ability when compared with the main reservoir. As the time progresses the fracture

2 produces less (Figure 6.9). The influence of warm fluid coming from fracture 2 is

to increase the well temperature slightly. From the temperature map presented in

6.2. EFFECT OF POSITION OF FRACTURE CONNECTION 149

Figure 6.8 the presence of fracture 2 is not very clear, but the map of first deriva-

tive of temperature with respect to position along the wellbore can capture the exact

point from where the warm fluid enters the well: the blue vertical line in Figure 6.10

around position 800 ft.

Temperature history shows that the early production time is characterized by a

faster warm-up of the well due to warm fluid from the reservoir. When the total

production comes from a contained zone it is expected that the temperature will keep

warming up until an equilibrium temperature is reached. For the case described here,

part of the late time production comes from a different zone of lower temperature,

which makes the temperature decrease continuously after a maximum warm-up. This

change in behavior is better visualized by the first derivative of temperature with

respect to time (Figure 6.11). The red zone at early times (positive derivative) is

related with the warmer fluid coming from the reservoir entering the wellbore. But

as the fluid coming from zone 2 reaches the well the temperature starts to decrease

and the derivative with respect to time changes to negative sign.

After 60 days of production the inflow temperature is 90 oC, while the original

temperature in zone 2 is 84 oC. The fluid exchanges heat with the fracture/fault wall

on its way down from zone 2 to the main reservoir, and tends to be a little warmer

than in zone 2.

6.2 Effect of Position of Fracture Connection

This section presents an example where the fracture that is connecting the two zones

is fracture 2, which is closer to the heel. In this way there is a longer section between

the fracture that has connected the zones vertically and the toe of the well.

Figure 6.12 shows the temperature map along the wellbore during the injection

period. The fact that fracture 2 takes the highest amount of the injection is very clear

given the decrease in the cold front speed after passing by this point of the wellbore.

The continuous temperature profile along the wellbore is presented as a color

map in Figure 6.13. The first 60 days of production show clearly from which point a

colder fluid coming from zone 2 is entering the wellbore. From the toe to the fracture 2


Table 6.1: Input parameters for base case geometry.

Parameter Value

Injection rate (qinj) 14.25 bpmFlowback rate (qfb) 1000 STB/DInjection time (tinj) 30 minShut-in time (ts) 24 hoursFlowback time (tfb) 60 daysHorizontal section length 1160 ftFracture 1 position 1120 ftFracture 2 position 800 ftReservoir depth (Dtop) 3200 ftReservoir permeability (k) 0.5 mdZone 2 permeability (k2) 500 mdReservoir thickness (h) 50 ftZone 2 thickness (h2) 10 ftDistance between zones (∆d) 500 ftTres 95 oCT2 84 oCLocal geothermal gradient 2.2oC/100 ft


Figure 6.3: Initial temperature map for the base case of fracture interconnecting dif-ferent zones. The reference zero depth is the surface and the temperature is expressedby the color map in Celsius degrees (oC).

Figure 6.4: Base case reservoir pressure map (psi) for end of injection period, wherefracture 1 is interconnecting different zones.


Figure 6.5: Base case pressure map (psi) after 60 days of production, where fracture1 is interconnecting different zones.

Figure 6.6: Base case temperature map (oC) after 60 days of production, wherefracture 1 is interconnecting different zones.


Figure 6.7: Base case wellbore temperature map (oC) for injection period, wherefracture 1 is interconnecting different zones.

Figure 6.8: Base case wellbore temperature map (oC) for the first 60 days of produc-tion, where fracture 1 is interconnecting different zones.


100 200 300 400 500 6000

200

400

600

800

1000

1200

t(hours)

Rate

(S

TB

/D)

Frac 1

Frac 2

Figure 6.9: Flow rate history per fracture. Fracture 1 is connecting the main reservoirwith a zone 500 ft above it.

Figure 6.10: Base case derivative of wellbore temperature with respect to positionalong the well (∂T/∂y in oC/ft).


Figure 6.11: Base case derivative of wellbore temperature with respect to time (∂T/∂tin oC/h) during production.

Figure 6.12: Wellbore temperature map (oC) for injection period, where fracture 2 isinterconnecting different zones.


Figure 6.13: Wellbore temperature map (oC) for the first 60 days of production,where fracture 2 is interconnecting different zones.

Figure 6.14: Derivative of wellbore temperature with respect to time (∂T/∂t in oC/h).Fracture 2 is connecting the two zones.

6.3. EFFECT OF DISTANCE BETWEEN CONNECTED ZONES 157

position the temperature behaves very differently than it does from fracture 2 towards

the heel, as the temperature derivative with respect to time highlights (Figure 6.14).

The comparison between the pressure derivative of the case where fracture 2 is

connecting the zones with the base case (fracture 1 making the connection) shows

that pressure is not sensitive to the location of the connection (Figure 6.15). The

distributed temperature on the other hand can capture the spatial location along the

wellbore.

10−3

10−2

10−1

100

101

102

103

104

100

101

102

103

∆t (h)

∆P

(psi)

d∆

/dln

tP(p

si)

Frac1

Frac2

Figure 6.15: Pressure change and its logarithmic derivative for production period:comparison between connection with zone 2 through fracture 1 and 2.

6.3 Effect of Distance Between Connected Zones

The depth difference between the zones will influence the final temperature inside the

well. From the geothermal gradient, the higher the difference between the zones the

cooler is going to be the late time temperature inside the well, if the connected zone

is above the reservoir. The longer distance that the fluid will travel from one zone to

another the more heat is going to be added due to friction and heat transfer with the

walls of the fracture (or fault).


We explored the difference in depth between the main reservoir and the connected

zone. In addition to the base case, we considered the cases where the connected zone

is 200 ft above the reservoir and 500 ft below the reservoir. In all the three cases

the fracture 1 is the one that connects the two zones, as it is in the base case.

The comparison among the temperature maps at the end of 60 days of production

is presented in Figure 6.16. In this figure the reservoir where the well is present is in

green color. The temperature maps show the fluid with different temperature from

the reservoir travel to in through the connecting fault.

The history of well temperature at positions of fracture 1 and 2 is presented in

Figure 6.17. As can be seen the distinction between the different depths is very clear.

On the other hand, the pressure derivative does not show any difference between the

two zones and all of them reflect the permeability of zone 2, which is higher than the

main reservoir (Figure 6.18).

6.4 Chapter Summary

This chapter highlights the use of long term temperature profiling to identify and/or

confirm the interconnection between reservoirs due to hydraulic fracturing a fault or

natural fracture activation. The temperature profile can identify the location along

the well where the connection is taking place, as well as if the zone connected is above

or below the reservoir where the well is located.

The temperature profile appears to be a very good complement to seismic surveys

in confirming when the fracturing has occurred out of zone, and to confirm whether

the detected events actually communicate between different zones.

This chapter shows another example of information that can be obtained from the

temperature, which pressure cannot provide. If for example water is being produced

and it comes from zone 2, the temperature can identify from which part of the well

it is entering, not needing an intervention in the well to run a production log. In this

way the diagnostic becomes faster and cheaper, increasing the support for decision

making.


Figure 6.16: Temperature map at after 60 days of production for zone 2 located200 ft (top figure) and 500 ft (middle figure) above the reservoir and 500 ft belowit (bottom figure). The main reservoir is in green color (95oC).


0 500 1000 150030

40

50

60

70

80

90

100

110

t(h)

T(o

C)

Fracture 1

0 500 1000 150030

40

50

60

70

80

90

100

110

t (h)

T(o

C)

Fracture 2

∆d=200ft

∆d=500ft

∆d=−500ft

No connection

∆d=200ft

∆d=500ft

∆d=−500ft

No connection

Figure 6.17: Temperature history at fracture 1 (top) and 2 (bottom) positions, sen-sitivity analysis to zone 2 depth when fracture 1 is connection the two zones.


10−3

10−2

10−1

100

101

102

103

104

100

101

102

103

∆t (h)

∆P

(psi)

d∆

/dln

tP(p

si)

∆d=500ft

∆d=−500ft

∆d=200ft

Figure 6.18: Pressure derivative, sensitivity analysis to zone 2 depth when fracture 1is connection the two zones.

Chapter 7

Fracture Crossing Multiple

Fractures

The existence of a complex fracture network connected with a created hydraulic frac-

ture has been supported by microseismic surveys performed during hydraulic frac-

turing treatments in unconventional reservoirs. In shale gas reservoirs, for example,

microseismic surveys show a cloud around the created hydraulic fracture, which can

indicate the slip of preexisting natural fractures. The slip can cause these preexisting

fractures to be stimulated and hydraulically active.

The existence of other fractures crossing a main hydraulic fracture has also been

mentioned in works that studied the possibility of fractures been generated in tight

formations due to cooling effects, like Ghassemi (2007), for example. He stated that:

“Thermally-induced stresses also cause formation of new secondary cracks that can

generate microseismic events. Regions of enhanced shear stress with higher potential

for shear failure are also observed near the crack ends and off the main fracture plane”

(Ghassemi , 2007).

This chapter presents the constant rate drawdown solutions for a uniform flux

and for an infinite conductivity main hydraulic fracture crossed by multiple vertical

fractures in an infinite reservoir. The angles at which a crossing fracture intersects

the main fracture can vary from one to the other, and in the same way the fracture

half-lengths are independent. In addition to that, the storage effect is analyzed.

163

164 CHAPTER 7. FRACTURE CROSSING MULTIPLE FRACTURES

Figure 7.1: Formation of secondary fractures within cooled region, extracted fromGhassemi (2007).

Craig (2006) has presented a semianalytical solution for a well crossed by n vertical

fractures. His approach assumes all the fractures are crossing the well. Here we have

extended his idea to fractures crossing a main hydraulic fracture.

During hydraulic fracturing not only a main fracture is created, but also natural

fractures are assumed to be stimulated. When we have multiple fractures with differ-

ent orientations in an anisotropic stress medium they are likely to close at different

times. The semianalytic solution for fracture closure during falloff is also developed

in this chapter. We also consider the case of the minifrac the sequence, where the

short time of injection allows the problem to be represented by instantaneous injec-

tion. As we are modeling multiple fracture directions, the closure time of each set

will be determined by the stress orientation. The closure is modeled by the storage

coefficient change over time.

7.1 Drawdown Solution for Multiple Crossing Frac-

tures

In this section the constant rate drawdown solutions for main hydraulic fracture

crossed by multiple fractures are developed. The schematic representation of the

problem geometry is shown in Figure 7.2. The fractures are placed in a homogeneous

and isotropic reservoir. The following subsections present the particular cases of

7.1. DRAWDOWN SOLUTION FOR MULTIPLE CROSSING FRACTURES 165

uniform flux and infinite conductivity fractures.

Figure 7.2: Crossing fractures model representation.

7.1.1 Uniform flux

The uniform flux solution assumes the flow is distributed equally along the connected

fractures in the system. This assumption allows a great simplification of the solution

procedure.

The pressure response for uniform flow along the vertical fracture can be obtained

easily by the Newman product theorem, which states that the pressure response in

a given geometry can be obtained by integration of equivalent line-source solutions

along the fracture length:

∆pδ =

∫ xf

−xf

Sr∆pr(r, t)dr (7.1)

where ∆pδ is the pressure drop caused by an instantaneous removal (or addition) of

fluid volume (q dt), Sr represents the source intensity (Equation 7.2) and ∆pr(r) is

the pressure drop for instantaneous production or injection of fluid at distance r from

the observation point.

In case of a single hydraulic fracture the source intensity is given by:

Sr =q dt

2xfhφct(7.2)

The vertical fracture pressure drop over time is obtained by time integration of

Equation 7.3:


∆p =

∫ t

0

∫ xf

−xf

Sr∆pr(r, t)drdt (7.3)

In a similar way, the drawdown solution of crossing fractures is based on super-

position of line-source solutions in the Laplace space. The pressure solution for a

line-source fully penetrating the reservoir thickness is given in the Laplace space by:

∆pD =qDxfDs

K0(rD√s) (7.4)

where rD represents the distance between the observation point and the line source.

The dimensionless variables are defined as:

xfD =xf

xf1

(7.5)

rD =r

xf1

(7.6)

tD =kt

φµctx2f1

(7.7)

qDi =qDi

qw(7.8)

∆pD =2πhk

qwµ∆p (7.9)

where xf1 is the main hydraulic fracture half length, which is taken as a reference,

and qw the well flow-rate.

Figure 7.3 presents the visual representation of the parameters that describe the

geometry of the problem. The distances to the sources above (r) and below (r∗) the

main fracture are defined as:

ri =√

X2i + α2

i − 2Xiαicos(180o − θi) (7.10)


r∗i =√

X2i + α∗2i − 2Xiα∗i cos(θi) (7.11)

Figure 7.3: Crossing fracture model for uniform flux. Each fracture i that is crossingthe created hydraulic fracture can be represented by the position it crosses the mainfracture (Xi), the angle it forms (θi), and its half length (xfi).

In Equations 7.10 and 7.11, Xi represents the point along the main fracture where

it is crossed by the fracture i. αi and α∗i are the distances from the crossing point along

the fracture i above and below the crossing point, respectively. θi is the angle formed

between the crossing fracture i and the main hydraulic fracture, named fracture 1.

The influence of fracture i upon the pressure felt at the well is given by:

pDi =1

2xfDi

∫ 0

−xfDi

qDi(α∗, s)K0(r

∗D(Xi, α

∗)√s)dα∗

+

∫ xfDi

0

qDi(α, s)K0(rD(Xi, α)√s)dα

(7.12)

where:

xfDi =xfi

xf1

(7.13)


qDi =qDi

qw=

xfi

xf1 +∑nf

j=2 xfjhj

(7.14)

By superposition, the total pressure change at the wellbore is given by the sum

of any individual fracture (Equation 7.12) in the system, as described by Equation

7.15.

pwfD =

nf∑

i=1

pDi (7.15)

The main hydraulic fracture is numbered as 1, and the crossing fractures from 2

to nf . The final constant rate drawdown wellbore pressure solution is writen as:

pwD =

nf∑

i=2

qDi

2xfDis

∫ 0

−xfDi

K0(r∗D(Xi, α

∗)√s)dα∗

+

∫ xfDi

0

K0(rD(Xi, α)√s)dα

+qD1

2s

∫ 1

−1

K0(α√s)dα (7.16)

Equation 7.16 is inverted numerically by the Stehfest algorithm (Stehfest , 1970).

The dimensionless wellbore pressure and its logarithmic derivative for a hydraulic

fracture crossed by 20 perpendicular fractures is presented in Figure 7.4. Each crossing

fracture has relative half length (xfDi) equal to 0.1, and they intersect the main

hydraulic fracture at their midpoint. The fractures are distributed evenly along the

two wings of fracture 1 (ten for each side, or wing).

The pressure signature for this specific case reflects the interaction among the

adjacent fractures happening between linear and radial flows (dimensionless time

interval from 0.001 to 0.03). There is a tendency to unit slope, then the flow evolves

to radial, when the plateau in the pressure logarithmic derivative appears.

Sensitivity analyses to the secondary fracture relative half length and density of


fracture are presented in Figures 7.5 and 7.6, respectively. It can be observed that

the longer the crossing fractures and the higher their density along the main fracture

the stronger the interaction effect will appear in the transition between linear and

radial flows. The angle formed by the fractures and the plane of the main fracture

also influences the response, but the effect is less pronounced than the relative length

of the secondary fractures (Figure 7.7).

10−4

10−2

100

102

10−4

10−3

10−2

10−1

100

101

tD

pw

D

Figure 7.4: Pressure response for constant rate drawdown for main hydraulic frac-ture crossed by 20 perpendicular secondary fractures. The 20 crossing fractures aredistributed evenly along the length of the main fracture. All crossing fractures areassumed to have the same length xfDi = 0.1.

7.1.2 Infinite conductivity

For infinite-conductivity fractures the pressure is transmitted instantaneously from

one point to another inside the fracture. In this way the infinite-conductivity fracture

problem requires that the influence of production (or injection) from each point along

fracture on another to be accounted for. Different from the uniform flux case, the


10

−5

10

−4

10

−3

10

−2

10

−1

10

010

110

210

−4

10

−3

10

−2

10

−1

10

0

10

1

tD

pw D

xfD

i =0.0

5

xfD

i =0.2

5

xfD

i =0.6

5

Figu

re7.5:

Uniform

fluxcrossin

gfractu

res:sen

sitivity

analy

sisto

secondary

fractures

half

length

(xfDi ).

Wellb

orepressu

reresp

onse

forcon

stantrate

draw

dow

nfrom

asystem

contain

ingamain

hydrau

licfractu

recrossed

by10

perp

endicu

larsecon

dary

fractures.


10

−4

10

−2

10

010

210

−4

10

−3

10

−2

10

−1

10

0

10

1

t D

pw D

nf=

2

nf=

4

nf=

10

Figure

7.6:

Uniform

fluxcrossingfractures:

sensitivityan

alysisto

number

ofcrossingfractures.

Wellborepressure

respon

seforconstan

trate

drawdow

nfrom

asystem

containingamainhydraulicfracture

crossedbynfperpendicular

secondaryfracturesof

lengthxfDi=

0.2.


10

−5

10

−4

10

−3

10

−2

10

−1

10

010

110

210

−3

10

−2

10

−1

10

0

10

1

tD

pw D

θ=

π/6

θ=

π/3

θ=

π/2

Figu

re7.7:

Uniform

fluxcrossin

gfractu

res:sen

sitivity

analy

sisto

crossingfractu

reorien

tationrelatively

tothe

main

hydrau

licfractu

re.Wellb

orepressu

reresp

onse

forcon

stantrate

draw

dow

nfrom

asystem

contain

ingamain

hydrau

licfractu

recrossed

by10

secondary

fractures

oflen

gthxfDi=

0.1.


rate along each fracture changes over time.

The solution is obtained by superposition of the pressure change caused by pro-

duction of each fracture upon the other. The problem geometry is defined in Figure

7.8, which shows the definition of the distances between a source on fracture i and a

point in fracture k. Equation 7.17 defines this distance.

ri−k(x∗i , x

∗k) =

√

[Xk − (Xi + x∗i cosθi) + x∗kxosθk]2 + [x∗ksinθk − x∗i sinθi]

2 (7.17)

Figure 7.8: Crossing fractures model for infinite-conductivity fractures.

The general solution can be written as:

pxfD =

nf∑

i=2

1

2xfDi

∫ 0

−xfDi

qDi(α∗, s)K0(r

∗1−iD(X1, α

∗)√s)dα∗

+

∫ xfDi

0

qDi(α, s)K0(r∗1−iD(X1, α

∗)√s)dα

+

∫ xfD1

0

qD1

2xfD1

K0(α√s)dα (7.18)

As presented by Gringarten et al. (1974), to solve the infinite-conductivity fracture

problem each fracture is divided into nfs equal length uniform-flux segments.

Each segment size is defined as:


di =LfDi

nfs

(7.19)

The sum of rates of each segment is equivalent to the fracture rate, and the sum

of all fracture rates is equal to the well rate.

nfsi∑

m=−nfsi

di qi|m = qi (7.20)

nf∑

i=1

qi = qw (7.21)

The pressure drop in each segment of fracture k is given by Equation 7.22. There

are∑nf

i=1 2nfsi equations like this. As the fracture have infinite conductivity, the

pressure inside each segment is equal to that in the other segments. The same for

the fractures, the pressure inside each fracture is the same as the pressure that is

measured in the wellbore. In this way the number of variables can be reduced to only

one pressure and the rate contributions of each segment, adding up to∑nf

i=1 2nfsi+1

variables. In case of symmetrically located fractures in the problem can be described

by half of the reservoir and the number of equations is reduced to∑nf

i=2 nfsi+2nf1i+1.

pxfD|x∗k=

nf∑

i=1

nfsi∑

m=1

qiD|m2xfiD

∫ xiD|m+1

xiD|m

K0(ri−k(α, x∗k)√s)dα

−−nfsi∑

m=−1

qiD|m2xfiD

∫ xiD|m+1

xiD|m

K0(ri−k(α, x∗k)√s)dα

(7.22)

The remaining equation to complete the problem comes from the combination of

Equations 7.20 and 7.21 in dimensionless form:

nf∑

i=1

nfsi∑

m=−nfsi

di qi|m =1

s(7.23)

7.2. WELLBORE STORAGE EFFECT 175

In this way a system of∑nf

i=1 2nfsi + 1 equations and∑nf

i=1 2nfsi + 1 unknowns

needs to be solved in order to obtain the pressure in the wellbore and the flowrate

distribution along the fracture segments. The system is solved in the Laplace domain

and inverted to the time domain to obtain the pressure-transient solution.

Figures 7.9 and 7.10 presents the sensitivity analyses to the length of the crossing

fractures relative to the main hydraulic fracture and the number of crossing fractures,

respectively.

7.2 Wellbore Storage Effect

The solutions presented in the previous section were developed for constant rate at

the sandface. As is well-known, the existence of a volume of fluid inside the wellbore

causes the phenomenon named wellbore storage, where the fluid decompression (or

compression) delays the equivalence between the flowrate imposed at the wellhead

and the flowrate that is actually transmitted to the reservoir at the sandface. The

bigger the volume of fluid the bigger the wellbore storage effect.

This section discusses the addition of the wellbore storage effect to the multiple

crossing fractures, and two scenarios are considered. The first refers to constant

rate drawdown with constant wellbore storage. The second scenario considers the

falloff pressure response where the fractures are open at initial time of shut-in but are

allowed to close as the pressure falls off. Depending on the fracture orientation and

stress field, different fractures are allowed to close at different times. The closure is

modeled by reduction of total wellbore storage coefficient.

7.2.1 Constant wellbore storage coefficient

The constant wellbore storage coefficient the solution is well known and can be ob-

tained easily in the Laplace space by applying Duhamel’s principle (van Everdingen

and Hurst , 1949; Ramey and Agarwal , 1972).

pwD =pxfD

1 + s2CDpxfD

(7.24)


10

−4

10

−3

10

−2

10

−1

10

010

110

210

310

−4

10

−3

10

−2

10

−1

10

0

10

1

tD

pw D

xfD

=0.0

5

xfD

=0.2

5

xfD

=0.6

5

Figu

re7.9:

Infinite

conductiv

itycrossin

gfractu

res:sen

sitivity

analy

sisto

secondary

fractures

half

length

(xfDi ).

Wellb

orepressu

reresp

onse

forcon

stantrate

draw

dow

nfrom

asystem

contain

ingamain

hydrau

licfractu

recrossed

by10

perp

endicu

larsecon

dary

fractures.


10

−4

10

−2

10

010

210

−4

10

−3

10

−2

10

−1

10

0

10

1

t D

pw D

nf=

2

nf=

4

nf=

10

Figure

7.10:Infiniteconductivitycrossingfractures(n

f):

sensitivityan

alysisto

number

ofcrossingfractures.

Wellborepressure

respon

seforconstan

trate

drawdow

nfrom

asystem

containingamainhydraulicfracture

crossed

bynfperpendicularsecondaryfracturesof

lengthxfDi=

0.2.


where

CD =C

2πφcthx2f1

(7.25)

Equation 7.24 is inverted using the Stehfest algorithm (Stehfest , 1970).

Figure 7.11 shows the solution for different wellbore storage coefficients. The

wellbore storage covers the early time behavior, and depending on the its value the

early time data with the linear flow and the interference between different fractures

can be masked completely.

7.2.2 Variable storage coefficient: multiple fracture closure

The massive stimulation of shale reservoirs has been associated with the idea of

production enhancement not only by the creation of new hydraulic fractures but also

from stimulation of preexisting natural fractures by triggering shear slip (Zoback ,

2007; Vermylen and Zoback., 2011).

There has been a considerable effort to understand the causes of the highly variable

shale reservoir response to stimulation, in particular because a better understanding of

what controls the shape (width, length and height) and effectiveness (improved access

to hydrocarbons and improvement in reservoir flow properties) of the stimulated

volume would help to guide selection of fluids, proppants, flow rates and volumes to

maximize stimulation effectiveness (Moos et al., 2011).

The microseismically detectable shear slip is increasingly being recognized as key

to the permanent enhancement in flow properties, and increasing access to the reser-

voir that results from stimulation. This is founded on the idea that productivity

enhancement due to stimulation results not just from creation of new hydraulic frac-

tures but also from the effect of the stimulation on preexisting fractures.

When the main hydraulic fracture is created the preexisting natural fractures can

also be stimulated. Due to the increase in fluid pressure the fractures in contact with

the created hydraulic fracture can open and be filled with fluid.

At the end of injection the pressure starts to decrease and fluid to leakoff from the

fractures to the reservoir. The newly created vertical hydraulic fracture is oriented


10

−5

10

−4

10

−3

10

−2

10

−1

10

010

110

210

−5

10

−4

10

−3

10

−2

10

−1

10

0

10

1

t D

pw D

CD=

0.0

CD=

0.0

1

CD=

0.0

5

CD=

0.1

Figure

7.11:Wellborestorageeff

ectfordrawdow

nin

vertical

wellwithmultiple

crossedfractures.


perpendicular to the minimum horizontal stress. The crossing fractures with direction

different from perpendicular to the minimum stress tend to close first in time.

If we consider the volume of each crossing fracture in addition to the created

hydraulic fracture a storage coefficient can be attributed to it. In this way, the model

of fracture closure during falloff has to consider the change in wellbore storage. As

fractures start to close the storage coefficient decreases.

(a)

(b)

(c)

Figure 7.12: Multiple fracture closure schematics.

As described by Craig (2006), the wellbore storage coefficient at early times when

all the fracture are still open (Figure 7.12-a) is written as:


C1 = cwVw +2Af1

Sf

+

nf∑

i=2

2Afi

Sf

(7.26)

When the crossing fractures are closed by the created hydraulic fracture is still

open (Figure 7.12-b) the storage coefficient changes to:

C2 = cwVw +2Af1

Sf

+ 2

nf∑

i=2

cfVfi (7.27)

Similarly, the after-closure (Figure 7.12-c) storage coefficient is written as:

C3 = cwVw + cf

(

Vf1 +

nf∑

i=2

Vfi

)

(7.28)

During the falloff, multiple closure times will be observed as the pressure declines

below the closure stress. Equations 7.26, 7.27 and 7.28 can be combined in a single

equation using the unit-step function (U):

CT = C1 + (C2− C1)U∆t1 + (C3− C2)U∆t2 (7.29)

where ∆t1 and ∆t2 are the times of falloff when the crossing fractures and the

main hydraulic fracture close, respectively.

In this section we are going to describe the multiple fracture closure for the scenario

of a minifrac test. As was described in Chapter 4, the minifrac rate schedule can

be simplified by assuming the injection of a volume Vinj in the reservoir happened

instantaneously. For the multiple closure described above the sand-face flowrate can

be written as:

qsf = Vinjδ(t)−[

C1 + (C2− C1)U∆t1 + (C3− C2)U∆t2

]

d∆pwv

dt(7.30)

The wellbore pressure for rate schedule given by Equation 7.30 can be obtained

by applying the superposition principle:


∆pwv =

∫ t

0

qsf (τ)d∆pwc(t− τ)

dtdτ (7.31)

where ∆pwv stands for the pressure at the wellbore with variable sand-face flowrate

and ∆pwc is the equivalent wellbore pressure for constant rate case.

Substituting Equation 7.30 in to Equation 7.31:

∆pwv = Vinjd∆pwc(t)

dt−∫ t

0

qCD(τ)

d∆pwc(t− τ)

dtdτ (7.32)

where qCDis:

qCD(τ) =

[

C1 + (C2− C1)U∆t1 + (C3− C2)U∆t2

]

d∆pwv(τ)

dτ(7.33)

Applying the Laplace Transform to Equation 7.32:

∆pwv = Vinj

[

s∆pwc(s)−∆pwc(t = 0)]

− s qCd

[


(7.34)

The Laplace transform of qC is:

qC(s) = C1

[


+ (C2− C1)L

U∆t1

d∆pwv(t)

dt

+(C3− C2)L

U∆t2

d∆pwv(t)

dt

(7.35)

To obtain the Laplace transform of the terms with the unit-step function it is nec-

essary to apply the property developed by Correa and Ramey (1986) (Equation 7.36).

The transform is presented in Equation 7.37.

L

Ukf′(t)

= s f(s)− f(0)−∫ k

0

e−stf ′(t)dt (7.36)


qC(s) = C1s∆pwc(s) + (C2− C1)

s∆pwc(s)−∫ ∆t1

0

e−std∆pwv(t)

dtdt

+(C3− C2)

s∆pwc(s)−∫ ∆t2

0

e−std∆pwv(t)

dtdt

(7.37)

Inserting the changing storage rate in the Laplace space (Equation 7.37) into the

Equation 7.34 we obtain the pressure at the wellbore for multiple crossing fracture

with different closure times:

∆pwv =

Vinj∆pw(s)|C1 , if t < ∆t1;

Vinj∆pw(s)|C2 − (C2 − C1) s∆pw(s)|C2

∫ ∆t10

e−st d∆pwv(t)dt

dt, if ∆t1 ≤ t ≤ ∆t2;

Vinj∆pw(s)|C3 − (C2 − C1) s∆pw(s)|C3

∫ ∆t10

e−st d∆pwv(t)dt

dt

−(C3 − C2) s∆pw(s)|C3

∫ ∆t20

e−st d∆pwv(t)dt

dt, if t > ∆t3.

(7.38)

Applying the properties of the inverse Laplace Transform to Equation 7.38 we

obtain the real space pressure solution for multiple fracture closure during the falloff:

∆pwv =

Vinj∆pw(t)|C1 , if t < ∆t1;

Vinj∆pw(t)|C2 − (C2 − C1)∫ ∆t10

d∆pw(t−τ)|C2

dtd∆pwv(τ)

dτdτ, if ∆t1 ≤ t ≤ ∆t2;

Vinj∆pw(t)|C3 − (C2 − C1)∫ ∆t10

d∆pw(t−τ)|C3

dtd∆pwv(t)

dtdτ

−(C3 − C2)∫ ∆t20

d∆pw(t−τ)|C3

dtd∆pwv(τ)

dτdτ, if t > ∆t3.

(7.39)

Taking a close look at the integral terms of Equation 7.38, one can notice that the

time interval over which the integral is defined corresponds to previous time stages.


Rewriting Equation 7.38 for the intervals ∆t1 ≤ t ≤ ∆t2 and t > ∆t3 we obtain

Equations 7.40 and 7.41, respectively.

∆pwv = Vinj∆pw(t)|C2 − (C2 − C1)

∫ ∆t1

0

d∆pw(t− τ)|C2

dt

d∆pw(t)|C1(τ)

dτdτ (7.40)

∆pwv = Vinj∆pw(t)|C3 − (C3 − C1)

∫ ∆t1

0

d∆pw(t− τ)|C3

dt

d∆pw|C1(t)

dtdτ

−(C3 − C2)

∫ ∆t2

∆t1

d∆pw(t− τ)|C3

dt

d∆pw|C2(τ)

dτdτ

−(C3 − C2)(C2 − C1)

∫ ∆t2

∆t1

d∆pw(t− τ)|C3

dt

∫ ∆t1

0

d∆pw(τ − ξ)|C2

dτ

d∆pw|C1(ξ)

dξdξ

dτ

(7.41)

To compute the wellbore pressure it is necessary to calculate the integrals pre-

sented in Equations 7.40 and 7.41. In this work they are calculated using the numer-

ical integration technique called Romberg’s Method.

Considering the case where the crossing fracture closes at tD equals to 0.001 and

the main fracture closes at tD equals to 0.01 (equivalent storage coefficient presented

in Figure 7.13), the pressure solution is presented in Figure 7.14. From Chapter 4

we have seen that minifrac falloff data can be understood better by applying the

integral transform. The Bourdet derivative is presented in Figure 7.15. It is assumed

that crossing fractures have a small storage coefficient associated to them, and con-

sequently the transition is very smooth. A second example shows the case where the

change in the storage volume is more abrupt (Figure 7.16). The transformed data

analysis show the spike in the derivative (Figure 7.17).


0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020

0.002

0.004

0.006

0.008

0.01

tD

CD

Figure 7.13: Storage coefficient for multiple fracture closure, Example 1.

0 2 4 6 8 100

10

20

30

40

50

tD

pD

Figure 7.14: Dimensionless falloff pressure for closing fracture, Example 1.


10−6

10−4

10−2

100

102

104

10−4

10−3

10−2

10−1

100

101

tD

I(p

D);

dI(

pD

)/dln

(tD

)

Figure 7.15: Transformed falloff pressure for closing fracture, log-log plot equivalentto Example 1.

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020

0.05

0.1

0.15

0.2

tD

CD

Figure 7.16: Storage coefficient for multiple fracture closure, Example 2.


10−5

10−4

10−3

10−2

10−1

100

101

102

10−5

10−4

10−3

10−2

10−1

100

101

tD

I(p

D);

dI(

pD

)/dln

(tD

)

Figure 7.17: Transformed falloff pressure for closing fracture, log-log plot equivalentto Example 2.

7.3 Chapter Summary

The analytic solution for fracture crossed by multiple fractures was derived for uni-

form flux and infinite conductivity. In all the cases, depending on the length of the

crossing fractures in relation to the main fracture the interaction between the pro-

duction (or injection) of adjacent fractures is felt in the transition from the linear

flow to the infinite radial flow. The longer the crossing fractures and the higher

the number of crossing fractures the stronger the interaction effect. Microseismic

surveys performed in unconventional reservoir indicate the activation of natural frac-

tures around the newly created hydraulic fracture by shear slip. The model developed

here addresses this specific scenario. In the same kind of reservoir, pressure transient

analysis has shown a fast transition from linear flow to unit slope. This behavior

has been attributed to the interaction between different stimulation stages (long hy-

draulic fractures created by massive fluid injection), but this study has opened another

possibility: the interaction between the fractures in the stimulated volume can also

produce a tendency to unit slope before the radial flow.

Chapter 8

Conclusions and Final Remarks

This work investigated the combined use of temperature and pressure data to analyze

hydraulic fracturing. This investigation required the development and implementa-

tion of a numerical model for hydraulic fracture growth in a hydrocarbon reservoir in

both vertical and horizontal well cases. The model allows the assessment of pressure

and temperature responses during injection and falloff or flowback. The temperature

model not only accounts for conduction and convection, but also the pressure effects,

like Joule-Thomson and adiabatic expansion. Even though those pressure effects have

small magnitude when compared with the temperature change caused by cold fluid

injection, the effect can still influence some early transient behavior.

During fracture closure the pressure showed a particular behavior, characterized

by an abrupt change in pressure derivative behavior at the moment when the fracture

walls start to touch and the contact stress starts to control the fracture aperture. If

there is no fracture volume change during the falloff (i.e., no fracture closure) or if

this happens immediately after the shut-in, the transition from early time behavior

to radial flow happens smoothly, with no jump in the derivative.

With regard to the temperature results, the vertical well model showed that the

sensor location (if inside the well or behind the casing) can affect the results sig-

nificantly. The flowback brings information from deep inside the fracture and the

reservoir to the wellbore and makes the temperature recover faster than in the warm-

back case.

189

190 CHAPTER 8. CONCLUSIONS AND FINAL REMARKS

The sensitivity analysis shown in Chapter 3 allowed the conclusion that reservoir

permeability and injection rate are the parameters that most affected the temperature

behavior during injection and shut-in/flowback periods. However, for permeability

values lower than 1 md it was observed that heat transport is dominated by diffusion,

which makes the temperature independent of the transport properties, like perme-

ability. If instead of shutting the well the flowback is performed, the heat transfer

starts to be dominated by convection, which makes the problem again a function of

transport parameters. In this case even the low permeability curves show a definitive

distinction between each other. The influence of injection rate on temperature is more

related with the well effect than with the change in the total volume of fluid injected

and final fracture dimensions.

Due to the pressure effects in the energy balance equation, the fracture closure

time can be captured by the temperature derivative when the pressure change has a

high magnitude. The indication of fracture closure from the temperature data can be

used to confirm the estimate taken from traditional pressure analysis, or as the only

estimate when pressure is not available.

The hydraulic fracturing model was applied to minifrac analysis and used to inves-

tigate the behavior of a data transformation based on the integral operation technique.

For all the examples analyzed, the end of unit slope of the transformed pressure data

happens at the same time as when the value of dVfr/dp drops, and can be taken as

fracture closure time. It was possible to recover final fracture properties and reser-

voir permeability from traditional well testing techniques, given that the duration of

falloff was long enough to develop the linear/bilinear and radial flow regimes. The

advantage of minifrac tests is that short injection time allows those regimes to happen

sooner than would be seen in a traditional DST.

One of the main contributions of this work was to address the temperature re-

sponse during multistage hydraulic fracturing in horizontal wells. Two main con-

figurations were accounted for: the sequential multifracturing along the horizontal

wellbore and the simultaneous growth of multiple fractures during the same stage.

The analysis demonstrated the localized characteristics of the temperature data, when

compared to the pressure data which reflects only an average behavior along the well.

191

The map of temperature derivative with respect to distance along the well over

time appears to be a useful tool to identify fracture position and fluid rate at early

times of both injection and flowback periods. The interference of more than one

fracture on the growth of the others was investigated, and it was demonstrated that

the interference can affect not only the final fracture geometry, but also the flow

pattern inside the well during falloff. The temperature analysis was used to identify

the multiple fracture positions, to diagnose the existence of cross-flow between the

fractures, and also to quantify the flow-rate.

The existence of heterogeneities was another issue of focus in this work. When two

fractures are growing in different permeability zones the pressure is not sensitive to

where along the well the heterogeneity is present. The temperature on the other hand

can give the spatial information to resolve this problem. This study has shown that it

was not necessarily the coldest zone that was the one with the longest fracture, but it

is the one with the highest local permeability. To the best of our knowledge, this was

the first time that falloff and flowback wellbore temperature analyses were investigated

for simultaneous multiple hydraulic fracture growth in presence of heterogeneities.

The usefulness of long-term temperature profiling for identifying and/or confirm-

ing the interconnection between reservoirs due to hydraulic fracturing fault or natural

fracture activation was demonstrated. The temperature profile can identify the lo-

cation along the well where the connection is taking place, as well as if the zone

connected is above or below the reservoir where the well is located. The temperature

profile appears to be a very good complement to seismic surveys in confirming when

the fracturing has occurred out of zone, and to confirm whether the detected events

actually communicate different zones.

A series of analyses demonstrated the advantages of adding temperature analysis

when compared to the single-point pressure interpretation. Those cases are specially

evident for horizontal well scenarios.

In addition to numerical models, an analytic solution for pressure response when

a vertical hydraulic fracture is crossing multiple natural fractures was derived. The

solutions showed that depending on the length of the crossing fractures relative to

the main fracture the interaction between the production (or injection) of adjacent


fractures is felt in the transition from the linear flow to the infinite-acting radial flow.

The longer the crossing fractures and the higher the number of crossing fractures the

stronger the interaction effect.

In summary, the key findings of this work are as follows:

1. Temperature transients after hydraulic fracturing are not only affected by the

injection rate, but also by the permeability of the near-well region;

2. The simultaneous multiple fracture growth shows that temperature can predict

the number of created fractures during falloff. In the presence of heterogeneity

it is possible that cross-flow occurs in the well during the falloff. The direction

of cross-flow may indicate the heterogeneity type;

3. Different from the pressure analysis, distributed temperature data can differen-

tiate between different heterogeneity locations along the wellbore;

4. If the hydraulic fracture has communicated between different zones, the continu-

ous temperature monitoring during production (like that provided by DTS) can

identify from which point along the wellbore the fluid is coming. The difference

in geothermal gradient is the main driver of the distinguishable temperature

signature.

8.1 Future Work

This work consisted of an initial effort to better understand and apply temperature

analysis to complement the current practices of well testing interpretation in fractured

wells. In order to further extend this promising field of research there are important

steps that deserve attention for future works, such as:

1. Real data analysis. The presented work was mainly theoretical from the tem-

perature point of view. The next suggested step is to analyze real data from

DTS in a scenario that includes hydraulic fracturing and long-time monitoring

after it.

8.1. FUTURE WORK 193

2. Detailed description of fracture vertical growth, accounting for three-dimensional

behavior. The extension of the fracture growth algorithm to account for the

possibility of vertical growth in addition to the increase in length and width.

This is going to be useful to investigate the existence of vertical heterogeneity,

exploring the differences in temperature due to the geothermal gradient.

3. Multiphase analysis. Effects like different relative permeability and viscosity,

as well as capillary pressure can affect the results considerably and should be

included in a following stage of this wok.

4. Non-Newtonian fluid consideration. Hydraulic fracturing might involve the in-

jection of complicated fluids, like polymer solutions. The thermal behavior of

those fluids is an important topic to be studied. In addition to that the presence

of proppant should be added.

Nomenclature

Af Vertical cross sectional area of fracture

c Compressibility

Cl Heat capacity of the fluid

Clk Leak-off coefficient

Cs Heat capacity of the solid

CsT Wellbore storage coefficient

E Young’s modulus

h Reservoir thickness / fracture height

hl Fracture wall heat transfer coefficient

k Permeability

kf Fracture permeability

KI Stress intensity

KIc Critical stress intensity factor

xf Fracture half length

p Fluid pressure

pc Fracture closure pressure

pnet Fracture net pressure

Pe Peclet number

Pr Prandtl number

q Flowrate

r Radius

Re Reynolds number

s Laplace variable

195


S Source

Sfr Fracture stiffness

t Time

T Temperature

U Equivalent heat transfer coefficient at well wall

v Fluid velocity

V Volume

wf Fracture aperture

wf Average fracture aperture for a cross section

wfmax Maximum fracture aperture for a cross section

wfminAsperity aperture

x Spatial coordinate in direction parallel to fracture

y Spatial coordinate in direction perpendicular to frac-

ture

Greek Letters

β Coefficient of thermal expansion

λ Thermal conductivity

µ Fluid viscosity

φ Porosity

υ Poissons ratio

ρ Density

Υ Transmissibility

Subscripts

D Dimensionless

e External boundary

eff Effective

8.1. FUTURE WORK 197

f Fluid

fb Flowback

fr Fracture

i Initial

inj Injection

l Liquid

lk Leak-off

r Reservoir/rock

s Solid

sc Standard condition

sf Sandface

v Variable rate

w Well

List of Abbrevations

DST Drill Stream Test

DTS Distributed Temperature Sensing

PTA Pressure Transient Analysis

REV Representative Elemental Volume

TTA Temperature Transient Analysis

TWBS Temperature Wellbore Storage

Appendix A

Summary of Equations

This appendix summarizes the equations and boundary conditions that describe the

fracture growth in vertical and horizontal well scenarios. The mass and energy bal-

ances are specified for each part of the model: reservoir, fracture and well.

A.1 Vertical Well

This section addresses the equations and boundary conditions for a single vertical

hydraulic fracture in a vertical well. The problem is divided in two parts: mass

balance and fracture growth, and energy balance.

A.1.1 Mass balance and fracture growth

The reservoir mass balance is described by the two-dimensional diffusivity equation:

kxµ

∂

∂x

(

ρ∂p

∂x

)

+kyµ

∂

∂y

(

ρ∂p

∂y

)

=∂

∂t(φρ) (A.1)



p(x, y, t = 0) = pi (A.2)

199

200 APPENDIX A. SUMMARY OF EQUATIONS

q(x = xw, y = yw, t) = qinj (A.3)

q(x = xe, y, t) = q(x, y = ye, t) = 0 (A.4)

The fracture mass balance is solved in the same way as the reservoir, but the

properties of such grid-blocks are not constant and follow:

wf (p) =π

4wmax =

πhf (1− ν2)

2E(σf − pc) (A.5)

kf =wf

2

12(A.6)

φfi =wf

∆yi(A.7)

When the fracture walls are touching during closure the extra unknown is the

contact stress and the additional equation is:

σm(x, t) = Ω(∆wf )∆wf (x, t)

A− B∆wf (x,t)

wf,min

(A.8)

where:

Ω(∆wf ) =

0, if ∆wf ≤ 0;

1− eγ∆w2f , if ∆wf > 0;

(A.9)

A.1.2 Energy balance

The energy balance equations for reservoir, fracture and well with respective boundary

conditions are presented in this section. The velocity field and fracture geometry used

in the energy balance problem are given by the previously presented coupled mass

balance and fracture growth.

A.1. VERTICAL WELL 201

Reservoir


∂p


∂Tr

∂t(A.10)

Initial condition:

Tr(x, y, t = 0) = Tres (A.11)

Boundary conditions:

−λeff∂Tr

∂y

∣

∣

∣

∣

y=±wf/2

= hl(Tf − Tr); x = [r+w , xf ] (A.12)

−λeff∂Tr

∂r

∣

∣

∣

∣

(r=rw)

= (1− γ)U(Tw − Tr) (A.13)

Tr(x, y = ye, t) = Tr(x = xe, y, t) = Tres (A.14)

Fracture

−ρlCl−→vfrTfr +βTfr

∂p

∂t+(βTfr− 1)−→vfrp+

2hl

wf

(Tr−Tfr)+W = Ceff∂Tfr

∂t(A.15)

Initial condition:

Tf (x, t = 0) = Tres; x = [0, xf ] (A.16)

Boundary condition:

Qf (x = 0) = vwρClTw(t) (A.17)

where Qf is the heat flux for the fracture.


Well

−ρlCl−→qwTw + πr2wβTw

∂p

∂t+ (βTw − 1)qp+ 2πrwU(Tr − Tw) + Sinj = ρlClπr

2w

∂Tw

∂t(A.18)

Initial condition:

Tw(t = 0) = Tres (A.19)

Boundary condition:

qw(z = ztop, t) = qinj(t) (A.20)

Tw(z = ztop, t) = Tinj(zres, t); t = [0, tinj] (A.21)

Sinj = qinjρClTinj(zres, t) (A.22)

where

Tinj(zres, t) = Tsurf + azres − aA+ (Tinj(z = 0, t)− Tsurf + aA)ezresA (A.23)

A.2 Horizontal Well

This section addresses the equations and boundary conditions for pressure and tem-

perature models in horizontal well scenario.

A.2.1 Mass balance and fracture growth

The reservoir and fracture equations are similar to the one presented for the vertical

well, because the model is also two-dimensional. The difference in the horizontal

well scenario is that more than one fracture is modeled and there is no grid-block

fully dedicated to the wellbore. The well in this model is placed along y-axis and it

A.2. HORIZONTAL WELL 203

communicates directly with the fractures through source terms at the center fracture

grid-block.

Reservoir and fracture

kxµ

∂

∂x

(

ρ∂p

∂x

)

+kyµ

∂

∂y

(

ρ∂p

∂y

)

=∂

∂t(φρ)



p(x, y, t = 0) = pi (A.24)

nf∑

i=1

q(x = 0, y = yfi, t) = qinj (A.25)

q(x = xe, y, t) = q(x, y = ye, t) = 0 (A.26)

Well

Considering a simplified case for the flow inside the wellbore, where the fluid density

changes and pressure gradients along the wellbore are negligible, the mass balance

can be written as:

2

rwγρIvI +

∂ (ρv)

∂x= 0 (A.27)

where vI represents the fluid velocity entering or leaving the wellbore. In this case

the only points where vI is different than zero are at the fracture initiation points.

Boundary condition:

q(y = ywheel, t) = qinj (A.28)


q(y = ywtoe , t) = 0 (A.29)

A.2.2 Energy balance

In the multifractured horizontal well scenario, the energy balance equation for each

fracture is similar to the one presented for the single vertical fracture case. On the

other hand, reservoir and well are different, because of the well geometry and the

existence of more than one fracture.

Reservoir

For the reservoir the boundary conditions are different, because more than one frac-

ture can be created and the horizontal well extends from the heel (yheel) to the (ytoe).


∂p


∂Tr

∂t(A.30)

Initial condition

Tr(x, y, t = 0) = Tres (A.31)

Boundary conditions:

−λeff∂Tr(x, y, t)

∂y

∣

∣

∣

∣

y=yfi±wfi/2

= hl(Tf (x, t)− Tr(x, yfi ± wfi/2, t))

x = [r+w , xfi] and i = [1, nf ] (A.32)

−λeff∂Tr(x, y, t)

∂r

∣

∣

∣

∣

(r=rw)

= U (Tw(y, t)− Tr(x, y, t))

y = [ywheel, ywtoe ] (A.33)

A.2. HORIZONTAL WELL 205

Tr(x, y = ye, t) = Tr(x = xe, y, t) = Tres (A.34)

Well

−ρlCl−→vwTw + κl

∂2Tw

∂y2+ βTw

∂p

∂t+ (βTw − 1)−→vwp+

2U

rw(Tr − Tw) + Sw = ρlCl

∂Tw

∂t(A.35)

Sw(y = ywheel) = qinjρClTinj(zres, t)/(πr

2w) (A.36)

where Tinj(zres, t) is given by Equation A.23.

Initial condition:

Tw(y, t = 0) = Tres (A.37)

Boundary condition:

−κl∂Tw

∂y

∣

∣

∣

∣

y=ywheel

= Uy

(

Tr

∣

∣

y=y−wheel

− Tw

∣

∣

y=y+wheel

)

(A.38)

−κl∂Tw

∂y

∣

∣

∣

∣

y=ywtoe

= Uy

(

Tr

∣

∣

y=y+wtoe

− Tw

∣

∣

y=y−wtoe

)

(A.39)

Appendix B

Numerical Discretization of Flow

Model

The mass balance and fracture growth are solved by a fully implicit finite difference

numerical scheme, which requires that reservoir and fracture be divided in grid-blocks

and the governing equations be discretized and written for each element of the sys-

tem. In this appendix the discretization of mass balance and fracture properties are

presented, as well as the Jacobian elements, which are part of the implicit numerical

scheme solution using the Newton-Raphson algorithm.

B.1 Reservoir Discretized Equation

The block-centered discretization of single-phase two-dimensional equation presented

in Chapter 2 (Equation B.1) can be written as in Equation B.2.

(

−ρk∆yh

µ

∂p

∂x

)

i−1/2,j

−(

−ρk∆yh

µ

∂p

∂x

)

i+1/2,j

+

(

−ρk∆xh

µ

∂p

∂y

)

i,j−1/2

−(

−ρk∆xh

µ

∂p

∂y

)

i,j+1/2

− Si,j = ∆x∆yh∂

∂t(φρ)i,j (B.1)

207

208 APPENDIX B. NUMERICAL DISCRETIZATION OF FLOW MODEL

Figure B.1: Mass Balance.

−Υn+1xi−1/2,j

(pn+1i,j − pn+1

i−1,j) + Υn+1xi+1/2,j

(pn+1i+1,j − pn+1

i,j )

−Υyn+1i,j−1/2

(pn+1i,j − pn+1

i,j−1) + Υn+1yi,j+1/2

(pn+1i,j+1 − pn+1

i,j ) (B.2)

+Si,j/h =∆xi∆yj

∆t[ρn+1(φn+1 − φn) + φn(ρn+1 − ρn)]

where:

Υxi±1/2,j= α

(

∆y

∆xkx

)n+1

i±1/2,j

(

1

Bµ

)n+1

i±1/2,j

(B.3)

and

Υyi,j±1/2= α

(

∆x

∆yky

)n+1

i,j±1/2

(

1

Bµ

)n+1

i,j±1/2

(B.4)

In Equations B.3 and B.4 the first term in between parenthesis is the geometric

part of transmissibility and the second is the fluid part. α is the constant for unit

system consistency. Each one of the parts of transmissibility has a different treatment.

B.2. FRACTURE DISCRETIZED EQUATION 209

The geometric part is given by the harmonic average of absolute permeability of the

two connected grid-blocks:

(

∆y

∆xkx

)n+1

i±1/2,j

=∆yj

(∆xi±1 +∆xi)/2

∆xi±1 +∆xi

∆xi

kxi,j+ ∆xi±1

kxi±1,j

(B.5)

and

(

∆x

∆yky

)n+1

i,j±1/2

=∆xi

(∆yj±1 +∆yj)/2

∆yj±1 +∆yj∆yjkyi,j

+∆yj±1

kxi,j±1

(B.6)

The treatment of the fluid part of transmissibility is dependent on the direction

of flow, which arises from the hyperbolic characteristics of the governing equations

(Aziz and Settari , 1979). Consequently, the fluid part is defined by the up-winding

rule:

(

1

Bµ

)n+1

i+1/2,j

=

(

1Bµ

)n+1

i+1,j, if pi,j < pi+1,j;

(

1Bµ

)n+1

i,j, if pi,j > pi+1,j;

(B.7)

B.2 Fracture Discretized Equation

Inside the fracture grid-blocks, permeability and porosity are not constant as the

fracture grows and the treatment of those equations is different from the reservoir

equations. The permeability and porosity are changed to honor fracture conductivity

and volume, respectively. Those properties are functions of pressure and contact

stress when it is applied.


B.2.1 Mass balance

−Υn+1xif−1/2,jf

(pn+1if ,jf

)(pn+1if ,jf

− pn+1if−1,jf

) + Υn+1xif+1/2,jf

(pn+1if+1,jf

− pn+1if ,jf

)

−Υn+1yif ,jf−1/2

(pn+1if ,jf

− pn+1if ,jf−1

) + Υn+1yif ,jf+1/2

(pn+1if ,jf+1 − pn+1

if ,jf) (B.8)

+Sif ,jf/h =∆xif∆yjf

∆t[ρn+1(φ(p, σm)

n+1 − φ(p, σm)n) + φn(ρn+1 − ρn)]if ,jf

where if and jf represents the fracture location. As the fracture is assumed to grow

along the x direction, jf is constant and if goes from a block besides the well (iwell±1)

to the fracture tip (itip). If the symmetry lines can be traced at the center line of the

fracture jf and jwell are equal to 1, then Equation B.16 can be rewritten as:

−Υn+1xif−1/2,1

(pn+1if ,1

)(pn+1if ,1

− pn+1if−1,1

) + Υn+1xif+1/2,1

(pn+1if+1,1 − pn+1

if ,1)

+Υn+1yif ,1+1/2

(pn+1if ,2

− pn+1if ,1

) (B.9)

+Sif ,1/h =∆xif∆y1

∆t[ρn+1(φ(pn+1, σm)

n+1 − φ(p, σm)n) + φn(ρn+1 − ρn)]if ,1

The dependence of permeability and porosity upon pressure is given by:

φf (pif ,jf , σmif ,jf) = wf (pif ,jf , σmif ,jf

)/∆yjf (B.10)

kf (pif ,jf , σmif ,jf) = wf (pif ,jf , σmif ,jf

)2/(12∆yjf ) (B.11)

B.2.2 Stress balance equation

During fracture closure the contact between asperities at the fracture walls creates

a contact stress that acts against the fracture closure. The equation was described

in Chapter 2 (Equation 2.10). The discretized form of this equation is written in

Equation B.12.

B.3. JACOBIAN ELEMENTS 211

σn+1mfi

=Ω(∆wn+1

fi )∆wn+1fi

[A− B∆wn+1

fi

wmin]

(B.12)

where Ω is a smooth function used to eliminate the discontinuity in the contact stress.

Ω(∆wfi) =

0, if ∆wfi ≤ 0;

1− eγ∆w2fi , if ∆wfi > 0;

(B.13)

∆wfi = wfmin− wfi (B.14)

This equation only appears in the fracture grid-blocks during closure. σm is a

function of fracture aperture, which is also a function of fluid pressure. In this way

the contact stress is a function of fluid pressure and its dependence has to be accounted

for in the Jacobian.

B.3 Jacobian Elements

For the fracture growth and closure the Jacobian matrix can change in size. When the

fracture is growing the contact stress is not present and a reduced part of the Jacobian

that contains only the pressure equations needs to be solved (Figure B.2- top). The

unknowns are the fluid pressure for all the grid-blocks. When the fracture closes the

contact stress starts to be active and the size of the Jacobian matrix increases. In

addition to the fluid pressure in each grid-block, the contact stresses at the fracture

grid-blocks are extra unknowns (Figure B.2- bottom).

For the sake of simplification, let’s assume the symmetric case, where the reservoir

can be represented by only a quarter of its total size. In this case the well is located in

the bottom left corner and the fracture grows along j = 1. The residual expressions

are written as:


Figure B.2: Numerical problem structure: Jacobian, unknowns and residual. The topfigure represents the structure of Newton-Raphson matrix solution where pressuresare the only unknowns. The bottom figure is the matrix representation when contactstress is present.


Rp(i,j) = −Υn+1xi−1/2,j

(pn+1i,j − pn+1

i−1,j) + Υn+1xi+1/2,j

(pn+1i+1,j − pn+1

i,j )

−Υn+1yi,j−1/2

(pn+1i,j − pn+1

i,j−1) + Υn+1yi,j+1/2

(pn+1i,j+1 − pn+1

i,j ) + Si,j/h (B.15)

−∆xi∆yj∆t

[(φρ)n+1 − (φρ)n]; i = [1, Nx] and j = [2 : Ny]

Rp(if ,1)= −Υn+1

xif−1/2,1(pn+1

if ,1)(pn+1

if ,1− pn+1

if−1,1) + Υn+1

xif+1/2,1(pn+1

if+1,1 − pn+1if ,1

)

+Υn+1yif ,1+1/2

(pn+1if ,2

− pn+1if ,1

) + Sif ,1/h (B.16)

−∆xif∆y1

∆t[(ρφ(pif ,1, σmif

))n+1 − (ρφ(pif ,1, σmif))n]; if = [2 : ifrac]

Rσ = σn+1mi

−Ω(∆wn+1

fi)∆wn+1

fi

[A− B∆wn+1

fi

wf,min]

(B.17)

The Jacobian terms are given by the differentiation of Equations B.15, B.16 and

B.17 with respect to pressure and contact stress.

B.3.1 Reservoir grid-blocks

For the reservoir grid-blocks, the properties are constant and the only parameter that

changes with pressure is the fluid density. In this way, each grid-block is dependent

on its own pressure and the pressure in the neighbor blocks. The derivative with

respect to any other pressure is going to be zero. The expressions for this part are

trivial and can be found in reservoir simulation books, as Aziz and Settari (1979), for

example.

B.3.2 Fracture grid-blocks

The fracture grid-blocks have residual equations representing the mass and the stress

balances. For the fracture grid-blocks, in addition to fluid density, permeability and


porosity change with pressure. In those grid-blocks there is an extra unknown, which

is the contact stress (σm).

Taking those relations into account, the Jacobian elements for the energy balance

residual for fracture blocks are:

∂Rp(if ,1)

∂pn+1if ,1

= −Υxif−1/2,1(pn+1

if ,1)−

∂Υxif−1/2,1(pn+1

if ,1)

∂pn+1if ,1

(pn+1if ,1

− pn+1if−1,1

)

+Υxif+1/2,1(pn+1

if ,1) +

∂Υxif+1/2,1(pn+1

if ,1)

∂pn+1if ,1

(pn+1if+1,1 − pn+1

if ,1) (B.18)

+Υyif ,1+1/2(pn+1

if ,1) +

Υyif ,1+1/2(pn+1

if ,1)

∂(pn+1if ,1

)(pn+1

if ,1− pn+1

if ,1) (B.19)

−∆xif∆y1

∆t[(

∂ρ

∂pn+1if ,1

φ(pif ,1, σmif))n+1 + ρ

∂φ(pif ,1, σmif)n+1

∂pn+1if ,1

]; if = [2 : ifrac]

As permeability and porosity are also function of contact stress, the derivative

with respect to this variable is also needed:

∂Rp(if ,1)

∂σn+1if

= −∂Υxif−1/2,1

∂σn+1if

(pn+1if ,1

− pn+1if−1,1

) +∂Υxif+1/2,1

∂σn+1if

(pn+1if+1,1 − pn+1

if ,1)

+Υyif ,1+1/2

∂σn+1if

(pn+1if ,1

− pn+1if ,1

)−∆xif∆y1

∆tρ∂φ(pif ,1, σmif

)n+1

∂σn+1if

; if = [2 : ifrac] (B.20)

where:

∂φn+1i

∂σn+1mi

=1

2∆y1

∂wn+1f

∂σn+1m

(B.21)

∂φn+1i

∂pn+1i

=1

2∆y1

∂wn+1f

∂pn+1i

(B.22)

∂kfi∂pi

=w2

f

2∆y1

∂wn+1fi

∂pn+1i

(B.23)


The stress balance residual has derivatives with respect to pressure and contact

stress, as shown in Equations B.24 and B.25, respectively.

∂Rσf

∂pn+1i,1

= − ∂

∂wfi

Ω(∆wn+1fi

)∆wn+1fi

[A− B∆wn+1

fi

wf,min]

∂wfi

∂pn+1i,1

(B.24)

∂Rσi

∂σn+1i

= 1− ∂

∂wfi

Ω(∆wn+1fi

)∆wn+1fi

[A− B∆wn+1

fi

wf,min]

∂wfi

∂σn+1mi

(B.25)

The derivative of fracture aperture with respect to pressure and contact stress

is going to be specific for each fracture geometry. The PKN geometry has a very

simple expression (1/Sf ), while the KGD has its aperture derivative taken from the

numerical calculation.

Appendix C

Numerical Discretization of

Thermal Model

The energy balance equation is solved by finite difference method. This appendix

presents the discretized energy balance equations for reservoir and fracture.

C.1 Reservoir

The energy balance equation is an advection-diffusion equation with forcing terms

that include compressibility effects due to Joule-Thomson and adiabatic expansion.

The temperature propagation has both advective and diffusive components, which

requires a distinguished treatment when applying numerical discretization techniques.

Taking a representative element volume the energy balance equation for the reser-

voir (Equation 2.21) is discretized as:

217

218 APPENDIX C. NUMERICAL DISCRETIZATION OF THERMAL MODEL

∆t∆yj

(

−λeff∆T

∆x

∣

∣

∣

∣

n+1

(i−1/2, j)

+ λeff∆T

∆x

∣

∣

∣

∣

n+1

(i+1/2, j)

)

+∆t∆xi

(

−λeff∆T

∆y

∣

∣

∣

∣

(i, j−1/2)

+ λeff∆T

∆y

∣

∣

∣

∣

(i, j+1/2)

)

(C.1)

+∆t∆yjCl

[

− (ρvx)n+1(i−1/2, j) MT n+1

(i−1/2, j) + (ρvx)n+1(i+1/2, j) MT n+1

(i+1/2, j)

]

+∆t∆xiCl

[

− (ρvy)n+1(i, j−1/2) MT n+1

(i, j−1/2) + (ρvy)n+1(i, j+1/2) MT n+1

(i, j+1/2)

]

+∆xi∆yjφβTn+1(i, j)(p

n+1(i, j) − pn(i, j)) + ∆t∆yj(βT

n+1(i, j) − 1)vn+1

x(i, j)

(pn+1(i+1, j) − pn+1

(i−1, j))

2

+∆t∆xi(βTn+1(i, j) − 1)vn+1

y(i, j)

(pn+1(i, j+1) − pn+1

(i, j−1))

2+ Sn+1

(i,j) = ∆yj∆xiCeff

[

T n+1(i,j) − T n

(i,j)

]

The convection terms are discretized following the up-winding rule, which states:

MT n+1(i−1/2, j) =

T n+1(i−1, j), if vn+1

x(i−1/2, j) > 0;

T n+1(i, j), if vn+1

x(i−1/2, j) < 0.(C.2)

MT n+1(i+1/2, j) =

T n+1(i+1, j), if vn+1

x(i+1/2, j) < 0;


x(i+1/2, j) > 0.(C.3)

MT n+1(i, j−1/2) =

T n+1(i, j−1), if vn+1

y(i, j−1/2) > 0;


y(i, j−1/2) < 0.(C.4)

MT n+1(i, j+1/2) =

T n+1(i, j+1), if vn+1

y(i, j+1/2) < 0;


y(i, j+1/2) > 0.(C.5)

C.2 Fracture

The energy balance equation for the fracture element (Equation 2.25) is discretized

as:

C.2. FRACTURE 219

∆twfiCl

[

− (ρvx)n+1(i−1/2,1) MT n+1

(i−1/2, j) + (ρvx)n+1(i+1/2, j) MT n+1

(i+1/2, j)

]

+∆xiwfiβTn+1(i, j)(p

n+1(i, j) − pn(i, j)) + ∆t∆xiCl

[

(ρvy)n+1(i, j+1/2) MT n+1

(i, j+1/2)

]

+∆t∆xihl(Tn+1ri,2

− T n+1fi

) + ∆twfi(βTn+1(i, j) − 1)vn+1

x(i, j)

(pn+1(i+1, j) − pn+1

(i−1, j))

2∆x+W n+1

i + Sn+1(i,j) = ∆xiρCl

[

wn+1fi

T n+1fi

− wnfiT nfi

]

220 APPENDIX C. NUMERICAL DISCRETIZATION OF THERMAL MODEL

Appendix D

Numerical Model Verification

The numerical models presented in this dissertation were verified by comparison with

existing asymptotic and/or simplified analytic solutions. This appendix shows in

detail the analytic solutions that were used and the comparison against the numerical

models for pressure, fracture dimensions and temperature results.

D.1 Reservoir Pressure

The numerical reservoir pressure model was verified for two scenarios: vertical well

(analytic solution for infinite-acting radial flow) and vertical fractured well (analytic

solution for infinite-conductivity fracture in an infinite slab reservoir).

D.1.1 Vertical well

In order to verify the finite difference scheme for single-phase flow through a porous

medium, a slightly compressible fluid case was compared with the analytical solution

of the single-phase diffusion equation for an infinite-acting radial flow in a homoge-

neous reservoir.

The diffusivity equation for single-phase flow of a slightly compressible fluid is

given by Equation D.1.

221

222 APPENDIX D. NUMERICAL MODEL VERIFICATION

1

r

∂

∂r

(

r∂p

∂r

)

=φµcTk

∂p

∂t(D.1)

The initial and boundary conditions are:

p(r, t = 0) = pi (D.2)

limr→∞p(r, t) = pi (D.3)

qsf = qB (D.4)

The solution in terms of dimensionless variables is:

pD(rD, tD) = −1

2Ei

(

r2D4tD

)

(D.5)

where:

pD(rD, tD) =2πkh

qBµ(pi − p) (D.6)

tD =kt

φµcT r2w(D.7)

rD =r

rw(D.8)

Figure D.1 shows the comparison between numerical and analytical solutions,

which are in complete agreement. The example represents a constant rate production

of 1000 STB/D in a 200 md reservoir. The reservoir size was specified large enough

to not feel the influence of the boundaries during the production time (Lx = Ly =

90000 ft with the well placed at the center). For this comparison the ∆x and ∆y

values start at 1 ft around the well and grow exponentially towards the boundaries

at rate of 1.2. The reservoir is discretized in 100 grid-blocks in x and 100 grid-blocks

in y-direction. The small time-step size is also an important component to reproduce

D.1. RESERVOIR PRESSURE 223

the transient analytic problem. For this verification example the time step started

at 10 seconds and also increased exponentially by a factor of 1.2 as simulation time

progressed. Due to the symmetry of the problem, only one fourth of the reservoir can

be used to represent the full problem.

10−3

10−2

10−1

100

101

102

103

101

102

103

t (h)

∆ P

; ∆

p’ (P

a)

implicit numerical solution

analytical solution infinite acting reservoir

Figure D.1: Numerical model verification against analytical solution for infinite actingradial flow.

D.1.2 Vertical fractured well

The fracture solution was verified with the infinite-conductivity fracture crossing a

vertical well in an infinite reservoir (Gringarten et al., 1974) . The infinite-conductivity

fracture pressure solution can be obtained from the uniform-flux solution evaluated

at dimensionless position (xD) equal to 0.732.

The uniform-flux pressure solution for a source at position x along the fracture is:


pD(xD, yD = y′D, tDxf ) =

√π

2

√

tDxf

[

erf

(

1 + xD

2√tDxf

)

+ erf

(

1− xD

2√tDxf

)]

−1

4(1 + xD)Ei

(

−(1 + xD)2

4tDxf

)

− 1

4(1− xD)Ei

(

−(1− xD)2

4tDxf

)

(D.9)

The dimensionless pressure is given by Equation D.5, but the dimensionless posi-

tion and time are specified in terms of fracture half-length (xf ):

xD =x

xf

(D.10)

tDxf =kt

φµcTx2f

(D.11)

The comparison between the analytical solution and the numerical model is pre-

sented in Figure D.2. The numerical model dimensions and spatial and temporal

discretazations are the same as the one used to validate the infinite-acting radial

flow. The difference in the fracture model is that high permeability (1010kres) was set

for the grid-blocks along the x -direction crossing the wellbore such that the fracture

half length (xf ) is equal to 30 ft. A fine discretization around the well and fracture

minimized the numerical storage and allowed the transient response to be well rep-

resented. For the example showed in Figure D.2 the grid-block size started at 1 ft

in both x and y directions, increasing at rate 1.2. There are 70 grid-blocks in along

x-direction and 50 along y-direction. The time-step started at 10 seconds, increasing

at 1.2 growth factor.

D.2 Fracture Creation

The numerical model accounting for fracture creation was verified by comparison to

the available analytic solutions for impermeable formation cases (Economides and

Nolte, 2000). The following subsection describe the process for PKN and KGD ge-

ometries.

D.2. FRACTURE CREATION 225

10−3

10−2

10−1

100

101

102

103

104

100

101

102

103

t(h)

∆ P

; ∆

p’ (P

a)

implicit numerical solution − fractured well

analytical solution − fractured well

Figure D.2: Numerical model verification against analytical solution for wellborepressure in presence of fixed vertical fracture.

D.2.1 PKN fracture geometry

For the PKN geometry the analytical solution can be obtained when the rock is

impermeable (no leak-off). The following equations describe the fracture half-length,

fracture average aperture and pressure inside the fracture:

xfD(tD) = 1.56t4/5D (D.12)

wfD(0, tD) = 1.09t1/5D (D.13)

pD(0, tD) = 0.855t1/5D (D.14)

where

xfD =1

4

[

G

(1− ν)µQ0

]1/3

xf (D.15)


wfD =1

4

[

G

(1− ν)µQ0

]1/3

wf (D.16)

pD =π

16

[

(1− ν)2H3

G2Q0µ

]1/3

p (D.17)

tD =1

16

[

G2Q0

(1− ν)2µ2H3

]1/3

t (D.18)

Figure D.3 shows that our numerical model can reproduce the fracture growth

behavior of an impermeable rock. To generate the numerical result the reservoir

permeability was set to zero. This test specified injection rate of 1.5 bpm for 17

minutes. The time-step started at 0.1 minutes and increased with simulation time

by a factor of 1.05. The discretization around the well was 0.5 ft in both x an y

directions. The reservoir was represented by 300 grid-blocks in x-direction, increasing

exponentially by 1.05. In y-direction only 2 grid-block are enough, given that there

is no flow in this direction (ky = 0).

D.2.2 KGD fracture geometry

The fracture aperture for KGD geometry is obtained by a double integral expression

(Equation 2.8). Assuming simplified stress distributions along the fracture it is pos-

sible to solve the double-integral analytically. To verify the numerical integration for

KGD aperture profile we assumed a uniform distributed effective stress:

σn(x, t) = σ1 (D.19)

With this simplification, Equation 2.8 can be solved analytically. The expression

for the fracture aperture (wfa) becomes:

wfa =4(1− ν2)σ1

E

√

1− (x/xf )2 (D.20)

Figure D.4 shows the comparison between the aperture calculated numerically

D.2. FRACTURE CREATION 227

0 2 4 6 8 10 12 14 16 180

0.5

1

1.5x 10

−3

t (min)

wf(m

)

0 2 4 6 8 10 12 14 16 180

100

200

300

t (min)

xf (

m)

0 2 4 6 8 10 12 14 16 180

100

200

300

t (min)

Pnet (

psi)

Analytical

Numerical

Analytical

Numerical

Analytical

Numerical

Figure D.3: PKN geometry verification against analytical solution for uniform dis-tributed stress load.


(from Equation 2.8) and the analytical solution (D.20). This figure shows very good

agreement between the two and verifies the numerical scheme for the KGD. For the

calculations σi was set equal to 1.

0 5 10 150

0.5

1

1.5

2

2.5

3x 10

−5

x(ft)

wf(f

t)

Analytical

Numerical

Figure D.4: KGD geometry verification against analytical solution for uniform dis-tributed stress load.

D.3 Temperature Response

The temperature response from our numerical model was verified based on different

scenarios. The pressure effect on temperature was verified by comparing the numerical

solution with the analytic expression for a simplified pressure profile as presented

by Ramazanov and Nagimov (2007). The temperature changes due to conduction

and convection inside reservoir and fracture were verified using simplified analytic

solutions for cold fluid injection, as described in Kamphuis et al. (1990).

D.3.1 Pressure effect in vertical well

The temperature effects of Joule-Thompson and adiabatic expansions were verified

with the analytical solution presented by (Ramazanov and Nagimov , 2007). Their

D.3. TEMPERATURE RESPONSE 229

solution consists of a simple analytical model, which allows us to calculate the tem-

perature changes in the saturated porous formation at variable bottomhole pressure.

The assumptions of the analytical solution are:

1. it is assumed that the fluid and skeleton compressibility are infinitesimally small;

2. thermal conductivity is absent;

3. temperature change is not influenced by the fluid and reservoir parameters;

4. the porous reservoir is homogeneous and horizontal.

With these simplifications the energy balance equation in cylindrical coordinates

becomes:

−Clv

[

∂Tr

∂r+ ε

∂p

∂r

]

+ ηφCl∂p

∂t= Ceff

∂Tr

∂t(D.21)

The pressure equation is given by:

1

r

∂

∂

(

r∂p

∂r

)

= 0 (D.22)


p|t=0 = pi (D.23)

p|r=rw = ϕ(t) (D.24)

p|r=re = pi (D.25)

Equation D.22 combined with conditions D.23, D.24 and D.25 can be solved an-

alytically and the resulting expression is given by Equation D.26.

p(r, t) = pi +pi − ϕ(t)

ln(ri/rw)ln(r/re) (D.26)


The solution of Equation D.21 can be obtained along characteristic lines, given

by:

Tr(r, t) = T0(r1) + ε[p(r1, 0)− p(rt, t)] + (ε+ η∗)

∫ t

0

∂p(rτ , τ)

∂τ(D.27)

where T0 is the initial temperature profile, η∗ = φCl

Ceff

η, rt is the characteristic curve

that is written as:

r2t = r21 − 2a(pit− s(t)) (D.28)

where:

s(t) =

∫ t

0

ϕ(τ)dτ (D.29)

and:

a =k

µln(ri/rw)

Cl

Ceff

(D.30)

For the verification the well pressure was assumed to decrease linearly during τ

hours:

ϕ(t) =

pi − pi−p0τ

t, if t < τ ;

p0, if t > τ .(D.31)

The comparison between analytical and numerical solutions is shown in Figure

D.5. For this case the spatial discretization starts at 1 ft and the temporal at 1

minute, both increasing at rate of 1.15. One quarter of the reservoir is used with

40 grid-block along x and 40 along y-direction (1600 grid-blocks). τ is specified as 2

hours.


0 2 4 6 8 10 12 14 160

1

2

3

4

t(h)

∆T

(oC

)

analytical

Numerical

0 2 4 6 8 10 12 14 16100

120

140

160

180

200

t(h)

P(K

gf)

analytical

Numerical

Figure D.5: Forced convection verification.


D.3.2 Fracture temperature

In order to verify the numerical scheme for the fracture equation a simplified case

considering zero leak-off and constant rock temperature was analyzed. The problem

becomes a simple heat conduction and convection inside a slot (Equation D.32), which

has an analytical solution.

−∂Tfr

∂x+

1

ρlufrw/2hl(Tr − Tfr) =

1

ufr

∂Tfr

∂t(D.32)

The conditions that simplify the problem are:

∂Tfr

∂y(x, y = 0, t) = 0 (D.33)

Tfr(x, y, t = 0) = Tres (D.34)

Tfr(x = 0, y, t) = Tinj (D.35)

The analytic solution is given by:

Tfr =

Tres, if x− ufr > 0;

Tres − (Tres − Tinj)e−ηx/ufr , if x− ufr < 0.

(D.36)

where:

η =2hl

ρlClw(D.37)

To verify our numerical thermal model for the fracture, the pressure effects were

set to zero and the temperature inside the rock was kept constant and equal to Tres.

The flow velocity inside the fracture was also assumed to be constant and equal to

qinj/(wh).

Figure D.6 presents the comparison between the numerical and analytical results,

showing a very good agreement.


0 20 40 60 80 100 120300

320

340

360

380

400

X

T (

K)

Analytical

Numerical

Figure D.6: Comparison between numerical and analytic fracture temperature.

D.3.3 Reservoir temperature during fracturing

The verification of the rock temperature numerical solution was made by comparing

with the analytical solution that neglects the heat conduction in x direction and also

considers the fracture temperature constant and equal to the injected fluid. In this

way the two-dimensional problem is simplified to one-dimensional.

The partial differential equation for one-dimensional heat transfer inside the reser-

voir accounting for conduction and convection is written as:

λy,eff∂2Tr

∂y2− ρlvlkCl

∂Tr

∂y= Ceff

∂Tr

∂t(D.38)


Tr(x, y = 0, t) = Tinj (D.39)

∂Tr

∂x(x = 0, y, t) = 0 (D.40)


Tr(x, y, t = 0) = Tres (D.41)

The analytic solution is given by:

Tr(y, t) = Tinj + (Tres − Tinj)erf(ξ) + erf(C/

√D)

1 + erf(C/√D)

(D.42)

where:

C =ClkρlCl

(ρC)eff(D.43)

D =λeff

(ρC)eff(D.44)

ξ =y − C

√teqρlCl

2√

Dteq(D.45)

The comparison between the analytic solution and the numerical simulator as-

suming the same simplifications is presented in Figure D.7, which shows the good

agreement between them.


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35300

320

340

360

380

400

Y

T (

K)

Analytical

Numerical

Figure D.7: Reservoir thermal model verification.

Appendix E

Wellbore Temperature Analytic

Solutions

In this appendix the analytic solutions for the wellbore temperature which are dis-

cussed in Chapter 3 are derived. The goal of the analytic simplifications was to

understand the behavior of the full-physics numerical results.

E.1 Temperature Analytic Solution during Warm-

back

Figure E.1: Simplified representation of wellbore during warmback.

The simple analytic solution of temperature during the warmback period is pre-

sented in this appendix. The following assumptions were used to simplify the energy

237

238 APPENDIX E. WELLBORE TEMPERATURE ANALYTIC SOLUTIONS

balance equation:

• Vertical well;

• During warmback convection and pressure effects are negligible;

• Reservoir temperature at the moment of shut-in is constant and equal to the

original reservoir temperature.

The energy balance equation for the average temperature inside the well can then be

simplified to:

VwρCl∂T

∂t= 2πrwU(Tr|r=rw − T ) (E.1)

In order to define the solution of Equation E.1, boundary conditions at the wall

between the well and the reservoir need to be specified. Two scenarios were consid-

ered: a simpler one where the temperature in the reservoir and at the well’s wall is

constant and equal to the original reservoir temperature (Tres); and a more realistic

second scenario which considers the wall temperature changing due to the heat loss

to the fluid inside the wellbore.

E.1.1 Constant temperature around the well

This case considers that the well outer boundary temperature (Tr) is constant and

equal to Tres. Introducing the dimensionless variable TD (Equation E.2) and the

group ηT (Equation E.3), Equation E.1 is transformed into Equation E.4.

TD =Tres − T

Tres − Tinj

(E.2)

ηT =ρClrw2U

(E.3)

ηT∂TD

∂t= −TD (E.4)

E.1. TEMPERATURE ANALYTIC SOLUTION DURING WARMBACK 239

The solution of the first order ordinary differential equation, given the dimension-

less initial condition TD(t = 0) = 0, is straightforward:

TD = e− t

ηT (E.5)

Equation E.5 suggests that a plot of log(T ) versus t will have a slope of − 1η. To

verify this assumption we simulated an injection of cold fluid through a vertical well,

followed by warm back. Different reservoir permeability values were considered and

the simulator accounted for pressure effects as well as the cooling inside the reservoir

due to convection and conduction during the injection period. Figure E.2 shows the

semilog plot where a straight line appears at intermediate to later times. The slope

of the straight line is −1.754× 10−4. This slope can be used to estimate the effective

heat transfer coefficient through the well wall as indicated by Equation E.6. Using

this equation the estimated heat transfer coefficient (U) is 12.5 W/m2, while the

actual value is 15.4 W/m2. The difference between the estimated and the real value

is related to the extremely simple model, which neglected the effect of cooling of the

zone around the well during injection. This makes the heat flux around the well

smaller than the case where the temperature at r = r+w is Tres.

U =ρClrw2

1

ηT(E.6)

E.1.2 Variable temperature around the well

This case considered the reservoir temperature around the well changing with time

during the warmback. The dimensionless radial diffusivity equation represents the

energy balance inside the reservoir when conduction is the dominant energy transfer

mechanism:

1

rD

∂

∂rD(rD

∂TrD

∂rD) =

∂TrD

∂tD(E.7)

where TrD is given by Equation E.2 and tD is defined in Equation E.8.


0 2000 4000 6000 8000 10000 12000 1400010

−4

10−3

10−2

10−1

100

∆t (min)

dT

/d(t

) (K

/min

)

0.25

0.5

0.75

1

2.5

5

Figure E.2: Logarithm of well temperature versus time during warmback: each curverepresents a different reservoir permeability. Intermediate and later times show astraight line behavior.

tD =keff t

Ceffr2w(E.8)


TrD(tD = 0) = 0 (E.9)

TrD(rD −→ reD) = 0 (E.10)

∂TrD

∂rD|rD=0 = hl(TrD − TwD) (E.11)

Applying the Laplace Transform to Equation E.7 and the initial condition, it

becomes the modified Bessel Equation of order zero:

d2TrD

dr2D+

1

rD

dTrD

drD= sTrD (E.12)

From the general solution of Bessel Equation and the outer boundary condition,

E.2. TEMPERATURE ANALYTIC SOLUTION DURING FLOWBACK 241

the solution of Equation E.12 is:

TrD = BK0(rD√s) (E.13)

Applying the inner boundary condition:

B =hlrwkeff

1√sK1(

√s)(TwD − TrD|rD=1) (E.14)

The wellbore equation (E.1) is solved in the Laplace space and written in terms

of dimensionless variables as:

TwD =β

s+ βTrD|rD=1 +

1

s+ β(E.15)

Combining Equations E.13, E.14 and E.15, the final solution is obtained:

TrD|rD=1 =K0(

√s)

keffhlrw

√sK1(

√s)(s+ β) + sK0(

√s)

(E.16)

TwD =1

s+ β(

βK0(√s)

keffhlrw

√sK1(

√s)(s+ β) + sK0(

√s)

+ 1) (E.17)

where

β =2hlCeffrwρClkeff

(E.18)

The solution was numerically inverted to the real space using Stehfest algorithm

(Stehfest , 1970).

E.2 Temperature Analytic Solution during Flow-

back

For the flowback case, the temperature inside the well can be simplified by assuming

that heat convection is the dominant effect in the heat transfer, and also that reservoir


Figure E.3: Simplified representation of wellbore during flowback.

fluid that enters the wellbore has constant temperature equal to Tres (Figure E.3).

The governing equation is:

VwρCl∂T

∂t= qfbρCl(Tr|r=rw − T ) (E.19)

The solution of the first order ordinary differential equation is straightforward and is

presented in Equation E.20.

TD = e−qfbt

Vw (E.20)

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