Copyright by Yongcun Feng 2016

Copyright

by

Yongcun Feng

2016

The Dissertation Committee for Yongcun Feng Certifies that this is the approved

version of the following dissertation:

Fracture Analysis for Lost Circulation and Wellbore Strengthening

Committee:

Kenneth E. Gray, Supervisor

Hugh C. Daigle

John T. Foster

John F. Jones

Mark W. McClure


by

Yongcun Feng, B.E.; M.E.

Dissertation

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

The University of Texas at Austin

August 2016

Dedication

To my parents, Jiatang Feng and Xiuhong Zhang,

To my wife, Xiaorong Li,

for their endless love, support and encouragement.

v

Acknowledgements

It is impossible to complete this dissertation without the help from a number of

people. First of all, I would like to express my sincerest gratitude to my supervisor, Dr.

Kenneth E. Gray, for his inspiration, encouragement, and guidance throughout my study at

The University of Texas at Austin. His friendly character and professional supervision have

made my study at UT-Austin fruitful and enjoyable. I would also like to thank my

dissertation committee members, Dr. Hugh C. Daigle, Dr. John T. Foster, Mr. John F.

Jones, and Dr. Mark W. McClure, for taking time to review my dissertation and providing

me with valuable feedback and comments. Thanks are also extended to Dr. Evgeny G.

Podnos for helping me get started with Abaqus®.

I wish to thank the Wider Windows Industrial Affiliate Program, the University of

Texas at Austin, for financial support of this dissertation and my graduate program. Project

support and technical discussions with industrial colleagues from Wider Windows

sponsors BHP Billiton, British Petroleum, Chevron, ConocoPhillips, Halliburton,

Marathon, National Oilwell Varco, Occidental Oil and Gas, and Shell are gratefully

acknowledged.

I would like to thank my colleagues in the Wider Windows Industrial Affiliate

Program for their support in my study and life at UT-Austin: Yao Fu, Anthony Ho, Zhi Ye,

Cesar Soares, Arjang Gandomkar, Chiranth Hegde, Peidong Zhao, Yangyang Chen,

vi

Hamza Jaffal, Lucas Barros, Scott Wallace, Tyler Adams, Xuyue Chen, Chao Gao, and

Bishwas Ghimire.

I also would like to express my gratitude to the following friends for their friendship

and help, which gave me a wonderful life with numerous good memories during my Ph.D.

study at UT-Austin: Bo Ren, Xiaoning Tan, Guang Yang, Yun Wu, Weiwei Wu, Huizi

Han, Haotian Wang, Yu Liang, Chunbi Jiang, Tianbo Liang, Hao Pang, Kai Zhang, and

Nineng Xu.

Finally, thank you to my parents, Jiatang Feng and Xiuhong Zhang, for their endless

love, encouragement, and support. My special gratitude is to my wife Xiaorong Li for her

love, understanding and encouragement all the time in my life.

vii


Yongcun Feng, Ph.D.

The University of Texas at Austin, 2016

Supervisor: Kenneth E. Gray

Lost circulation is the partial or complete loss of drilling fluid into a formation. It

is among the major non-productive time events in drilling operations. Most of the lost

circulation events are fracture initiation and propagation problems, occurring when fluid

pressure in a wellbore is high enough to create fractures in a formation. Wellbore

strengthening is a common method to prevent or remedy lost circulation problems.

Although a number of successful field applications have been reported, the fundamental

mechanisms of wellbore strengthening are still not fully understood. There is still a lack of

functional models in the drilling industry that can sufficiently describe fracture behavior in

lost circulation events and wellbore strengthening.

A finite-element framework was first developed to simulate lost circulation while

drilling. Fluid circulation in the well and fracture propagation in the formation were

coupled to predict dynamic fluid loss and fracture geometry evolution in lost circulation

events. The model provides a novel way to simulate fluid loss during drilling when the

boundary condition at the fracture mouth is neither a constant flowrate nor a constant

pressure, but rather a dynamic wellbore pressure.

There are two common wellbore strengthening treatments, namely, preventive

treatments based on plastering wellbore wall with mudcake before fractures occur and

remedial treatments based on bridging/plugging lost circulation fractures. For preventive

viii

treatments, an analytical solution and a numerical finite-element model were developed to

investigate the role of mudcake. Transient effects of mudcake buildup and permeability

change on wellbore stress were analyzed. For remedial treatments, an analytical solution

and a finite-element model were also proposed to model fracture bridging. The analytical

solution directly predicts fracture pressure change before and after fracture bridging; while

the finite-element model provides detailed local stress and displacement information in

remedial wellbore strengthening treatments.

In this dissertation, a systematic study on lost circulation and wellbore

strengthening was performed. The models developed and analyses conducted in this

dissertation present a useful step towards understanding of the fundamentals of lost

circulation and wellbore strengthening, and provide improved guidance for lost circulation

prevention and remediation.

ix

Table of Contents

List of Tables ......................................................................................................... xi

List of Figures ....................................................................................................... xii

CHAPTER 1: Introduction ......................................................................................1

1.1 Statement of the Problem .......................................................................2

1.2 Research Objectives ...............................................................................3

1.3 Literature Review...................................................................................6

1.4 Outlines of This Dissertation ...............................................................13

CHAPTER 2: Understanding Fracture Initiation and Propagation Pressures .......16

2.1 Introduction ..........................................................................................17

2.2 Lost Circulation “Thresholds” .............................................................21

2.3 Fracture Initiation Pressure (FIP).........................................................22

2.4 Fracture Propagation Pressure (FPP) ...................................................30

2.5 Preventive and Remedial Wellbore Strengthening ..............................46

2.6 Summary ..............................................................................................50

CHAPTER 3: Developing a Framework for Lost Circulation Simulation ............53

3.1 Introduction ..........................................................................................54

3.2 Numerical Method and Governing Equations .....................................55

3.3 Lost Circulation Model ........................................................................64

3.4 Fluid Loss Simulation Results .............................................................70

3.5 Summary ..............................................................................................88

CHAPTER 4: Modeling Study of Preventive Wellbore Strengthening Treatments:

The Role of Mudcake....................................................................................90

4.1 Introduction ..........................................................................................91

4.2 An Analytical Mudcake Model ............................................................92

4.3 A Numerical Model for Time-dependent Mudcake ...........................112

4.4 Summary ............................................................................................140

x

CHAPTER 5: Modeling Study of Remedial Wellbore Strengthening Treatments142

5.1 Introduction ........................................................................................143

5.2 A Fracture-Mechanics-Based Analytical Model for Remedial Wellbore

Strengthening Applications ................................................................147

5.3 A Finite-element Model for Remedial Wellbore Strengthening

Applications .......................................................................................164

5.4 Summary ............................................................................................192

CHAPTER 6: Cement Interface Fracturing .........................................................193

6.1 Introduction ........................................................................................194

6.2 Development of Cement Interface Fracture Model ...........................197

6.3 Results and Discussion ......................................................................205

6.4 Summary ............................................................................................213

CHAPTER 7: Role of Field Injectivity Tests on Combating Lost Circulation ...214

7.1 Introduction ........................................................................................215

7.2 A Review of Filed Injectivity Tests ...................................................216

7.3 Test Signatures ...................................................................................223

7.4 Test Interpretation ..............................................................................228

7.5 Field Examples...................................................................................235

7.6 Developing a Simulation Framework for Injectivity Tests................239

7.7 Lost Circulation as a Function of Formation Lithology ....................255

7.8 Summary ............................................................................................259

CHAPTER 8: Conclusions and Future Work ......................................................261

8.1 Conclusions ........................................................................................262

8.2 Future Work .......................................................................................268

References ............................................................................................................270

xi

List of Tables

Table 3.1: Material properties of the static fluid loss model. ................................67

Table 3.2: Input parameters for the dynamic fluid loss model. .............................70

Table 4.1: Summary of the parameters used in the example cases. .....................104

Table 4.2: Input boundary-condition values for the mudcake model. .................117

Table 5.1: Input parameters for model validation. ...............................................156

Table 5.2: Base input parameters used in the proposed model. ...........................157

Table 5.3: Input parameters for the finite-element model. ..................................169

Table 6.1: Cement properties. ..............................................................................202

Table 6.2: Formation properties. ..........................................................................202

Table 6.3: Interface bond properties. ...................................................................202

Table 6.4: In-situ stresses, pore pressure and wellbore pressure applied to the model.

.........................................................................................................203

Table 6.5: Geometry of the initial cracks at the casing shoe. ..............................204

Table 7.1: Material properties for the simulations. ..............................................245

xii

List of Figures

Figure 2.1: Left: Pore pressure and fracture gradient plot in depleted zone. Pore

pressure decrease leads to a decrease in fracture gradient. Right: Pore

pressure and fracture gradient plot in deep-water formation with

abnormally high pressure. There is a reduced mud-weight window.18

Figure 2.2: Fracture-initiation pressure of a vertical well. Left: with different

horizontal stress anisotropies and pore pressure; Right: with different

𝛈 and pore pressure. .........................................................................25

Figure 2.3: FIP (micro-fracture-propagation pressure) decreases dramatically with an

increase in horizontal stress anisotropy, and increases moderately with

an increase in fracture toughness; it can be much smaller than the

minimum horizontal stress with high stress anisotropy and low fracture

toughness...........................................................................................27

Figure 2.4: Schematic pressure-volume/time curves in leak-off tests. Left: no visible

leak-off response at fracture initiation, the leak-off pressure is very close

to formation breakdown pressure; Middle: a clear leak-off point before

formation breakdown; Right: multiple leak-off points before formation

breakdown. ........................................................................................30

Figure 2.5: A small fracture on the wellbore wall before formation breakdown: Left:

no filter cake plugging with clean fluid; Middle: high solids

concentration or filter cake inside the fracture; Right: fracture is plugged

at the inlet on wellbore wall. .............................................................33

Figure 2.6: A large hydraulic fracture (wellbore at the fracture center is neglected).

...........................................................................................................35

xiii

Figure 2.7: Pressure response during fracture propagation. Left: theoretical result;

Right: DEA-13 lab test result. ...........................................................36

Figure 2.8: A fracture plugged by solid particles. .................................................38

Figure 2.9: FIP with non-penetration zone length, pore pressure and minimum

horizontal stress. ...............................................................................39

Figure 2.10: A stationary fracture model. ...........................................................40

Figure 2.11: Fracture volume (left) and stress intensity factor (right) changes with

applying pore pressure to the fracture face only, while keeping the

traction on the fracture face constant. ...............................................41

Figure 2.12: Hydraulic fractures in impermeable (left) and permeable (right)

formations with high solids content fluid. ........................................42

Figure 2.13: Fluid leak-off and filter plug development controlled by capillary

pressure for water-wet sandstone with larger pore size and shale with

smaller pore size. Formation fluid is water while fracture fluid is

oil/synthetic based mud.....................................................................44

Figure 2.14: Fluid leak-off and filter cake development controlled by fluid

immiscibility and capillary pressure. ................................................45

Figure 2.15: An example of a preventive wellbore strengthening test on a sandstone

block (after Guo et al., 2014). ...........................................................48

Figure 2.16: The sandstone test block for pressure build-up curve in Figure 12. The

block was fractured to the edges without obvious fluid leak-off due to

LCM sealing effect (after Guo et al., 2014). .....................................49

xiv

Figure 2.17: A repeated hydraulic fracturing test with LCM. First injection cycle

(preventive treatment - intact wellbore, with LCM): high leak-off

pressure and high propagation pressure. Second injection cycle

(fractured wellbore, without LCM): low leak-off pressure and low

propagation pressure. Third injection cycle (fractured wellbore, with

LCM): low leak-off pressure, high (increased) propagation pressure.

(after Black et al., 1988). ..................................................................49

Figure 2.18: Preventive wellbore strengthening treatment enhances both leak-off

pressure and FPP, while remedial wellbore strengthening treatment only

enhances FPP. ...................................................................................50

Figure 3.1: Illustration of lost circulation system with well, formation and fracture.

...........................................................................................................56

Figure 3.2: Schematic of fluid flow in pipe. ..........................................................58

Figure 3.3: Friction factor determined from Blasius model and Churchill model:

Blasius model shows discontinuous transition from laminar to turbulent

flow at Re=2500, while Churchill model shows smooth transition. .59

Figure 3.4: A typical traction-separation law ........................................................61

Figure 3.5: Schematic of fluid flow in the cohesive fracture (Modified after Zielonka

et al., 2014) .......................................................................................62

Figure 3.6: Schematic configuration of well and formation. The formation is in a

plane-strain condition........................................................................65

Figure 3.7: Static lost circulation model. ...............................................................66

Figure 3.8: Dynamic fluid loss model....................................................................68

Figure 3.9: Fluid loss rate with different mud densities in the static fluid loss model.

...........................................................................................................72

xv

Figure 3.10: BHP with different mud densities in the static fluid loss model. ......72

Figure 3.11: Lost circulation fracture geometry with different mud densities in the

static fluid loss model. ......................................................................73

Figure 3.12: Return circulation rate with different mud densities in the dynamic fluid

loss model. ........................................................................................75

Figure 3.13: Fluid loss rate with different mud densities in the dynamic fluid loss

model.................................................................................................76

Figure 3.14: BHP with different mud densities in the dynamic fluid loss model. .76

Figure 3.15: Lost circulation fracture geometry with different mud densities in the

dynamic fluid loss model. .................................................................77

Figure 3.16: Comparison of fracture geometry between the static and dynamic fluid

loss models. .......................................................................................78

Figure 3.17: Return circulation rate with different mud viscosities in the dynamic

fluid loss model. ................................................................................79

Figure 3.18: Fluid loss rate with different mud viscosities in the dynamic fluid loss

model.................................................................................................80

Figure 3.19: BHP with different mud viscosities in the dynamic fluid loss model.80

Figure 3.20: Lost circulation fracture geometry with different mud viscosities in the


Figure 3.21: Return circulation rate with different pump rates in the dynamic fluid

loss model. ........................................................................................83

Figure 3.22: Fluid loss rate with different pump rates in the dynamic fluid loss model.

...........................................................................................................83

Figure 3.23: BHP with different pump rates in the dynamic fluid loss model. .....84

xvi

Figure 3.24: Lost circulation fracture geometry with different pump rates in the


Figure 3.25: Return circulation rate with different annulus clearances in the dynamic

fluid loss model. ................................................................................86

Figure 3.26: Fluid loss rate with different annulus clearances in the dynamic fluid loss

model.................................................................................................87

Figure 3.27: BHP with different annulus clearances in the dynamic fluid loss model.

...........................................................................................................87

Figure 3.28: Lost circulation fracture geometry with different annulus clearances in

the dynamic fluid loss model. ...........................................................88

Figure 4.1: Schematic of the cross section of wellbore, mudcake, and formation.93

Figure 4.2: Pore Pressure distribution around wellbore with different mudcake

thickness 𝑤. ....................................................................................105

Figure 4.3: Total tangential stress induced by fluid flow with different mudcake

thickness 𝑤. ....................................................................................106

Figure 4.4: Effective tangential stress induced by fluid flow (compared with no flow

case) with different mudcake thickness 𝑤. ....................................106

Figure 4.5 Fracture pressure with different mudcake thickness 𝑤. ....................107


permeability 𝐾1. ............................................................................109


permeability 𝐾1. ............................................................................109

Figure 4.8: Effective tangential stress induced by fluid flow (compared with no flow

case) with different mudcake permeability 𝐾1. .............................110

Figure 4.9: Fracture pressure with different mudcake permeability 𝐾1. ............110

xvii

Figure 4.10: Effective tangential stress around wellbore with different mudcake yield

strength 𝑌. ......................................................................................111

Figure 4.11: Fracture pressure with different mudcake yield strength 𝑌. ...........112

Figure 4.12: Illustration of mudcake model with constant mudcake thickness 𝑤𝑜 and

equivalent mudcake permeability 𝑘𝑒𝑡. ..........................................114

Figure 4.13: Geometry of the finite-element mudcake model. ............................115

Figure 4.14: Boundary conditions of the finite-element mudcake model. ..........116

Figure 4.15: Variation of mudcake thickness with time. .....................................119

Figure 4.16: Variation of mudcake permeability with time. ................................119

Figure 4.17: Variation of equivalent mudcake permeability with time. ..............120

Figure 4.18: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 3 hours for

wellbore with extremely permeable mudcake and permeable wellbore

without mudcake. ............................................................................122

Figure 4.19: Effective tangential stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 3

hours for wellbore with extremely permeable mudcake and permeable

wellbore without mudcake. .............................................................122

Figure 4.20: Effective radial stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 3 hours for

wellbore with extremely permeable mudcake and permeable wellbore

without mudcake. ............................................................................123

Figure 4.21: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 3 hours for

wellbore with impermeable mudcake and impermeable wellbore without

mudcake. .........................................................................................123


hours for wellbore with impermeable mudcake and impermeable

wellbore without mudcake. .............................................................124

xviii



mudcake. .........................................................................................124

Figure 4.24: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30 minutes. 126

Figure 4.25: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 4 hours. ......126

Figure 4.26: Effective radial stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30

minutes. ...........................................................................................127

Figure 4.27: Effective radial stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 4 hours.

.........................................................................................................127


minutes. ...........................................................................................128


hours. ...............................................................................................128




minutes. ...........................................................................................131


.........................................................................................................131


minutes. ...........................................................................................132

Figure 4.35: Effective tangential stress profiles along SH direction at t = 4 hours.132



xix


minutes. ...........................................................................................135


.........................................................................................................136


minutes. ...........................................................................................136


hours. ...............................................................................................137



minutes. ...........................................................................................139


minutes. ...........................................................................................140

Figure 5.1: Comparison of fracture pressure with and without remedial wellbore

strengthening treatment in DEA 13 experimental study (reproduced from

Onyia, 1994). ..................................................................................144

Figure 5.2: Results of a laboratory wellbore strengthening test (reproduced from Guo

et al., 2014). ....................................................................................145

Figure 5.3: Results of LOT field tests before and after taking remedial wellbore

strengthening treatment (reproduced from Aston et al., 2004). ......146

Figure 5.4: Schematic of remedial wellbore strengthening problem. ..................149

Figure 5.5: Problem decomposition. Left: an intact wellbore subject to wellbore

pressure and anisotropic far-field horizontal stresses; Right: wellbore

with two symmetric fractures subject to fluid pressure on fracture

surfaces. ..........................................................................................150

xx

Figure 5.6: Fracture in an elastic solid subject to a couple of point loads. ..........151

Figure 5.7: Wellbore with two symmetric fractures subject to uniform pressure on the

wellbore and fracture surfaces and anisotropic far-field horizontal

stresses. ...........................................................................................155

Figure 5.8: Comparison of fracture tip stress intensity factors between the proposed

model and Tada’s model (Tada et al., 1985). .................................156

Figure 5.9: Fracture-tip stress intensity factor with different horizontal stress

anisotropies and bridge locations. ...................................................159

Figure 5.10: Fracture pressure with different horizontal stress anisotropies and bridge

locations. .........................................................................................159

Figure 5.11: Fracture-tip stress intensity factor with different pore pressure and bridge

locations. .........................................................................................161

Figure 5.12: Fracture pressure with different pore pressure and bridge locations.161

Figure 5.13: Fracture pressure with different fracture toughness of the formation rock.

.........................................................................................................162

Figure 5.14: The remedial wellbore strengthening model. Left: geometry and

boundary conditions of the model; Rigth: detailed fracture process zone.

.........................................................................................................167

Figure 5.15: Comparison of stress intensity factor for unbridged fracture. .........171

Figure 5.16: Comparison of stress intensity factor for bridged fracture with various

bridging locations............................................................................172

Figure 5.17: Hoop stress distribution before (left) and after (right) bridging the

fracture in remedial wellbore strengthening treatment. ..................173

xxi

Figure 5.18: Hoop stress on wellbore wall for different horizontal stress contrasts: (a)

without fracture, (b) with unbridged fracture, (c) with bridged fracture,

and (d) comparison of hoop stresses before and after bridging for

horizontal stress contrast equal to 1.3. ............................................175

Figure 5.19: Hoop stress on wellbore wall for different bridging locations. .......176

Figure 5.20: Hoop stress on wellbore wall for different leak-off rates: (a) without

fracture, (b) with unbridged fracture, (c) with bridged fracture, and (d)

comparison of hoop stresses before and after bridging the fracture for

leak-off rate equal to 4 inches/s. .....................................................177

Figure 5.21: Hoop stress along fracture face for different horizontal stress contrasts:

(a) without fracture, (b) with unbridged fracture, (c) with bridged

fracture, and (d) comparison of hoop stresses before and after bridging

the fracture for horizontal stress contrast equal to 1.3. ...................179

Figure 5.22: Hoop stress along fracture face for different bridge locations. .......180

Figure 5.23: Hoop stress along fracture face for different leak-off rates: (a) without


comparison of hoop stresses before and after bridging the fracture for

leak-off rate equal to 4 inches/s. .....................................................182

Figure 5.24: Hoop stress along fracture face for different pressures behind LCM

bridge, varying from formation pressure of 1800 psi to wellbore pressure

of 4000 psi.......................................................................................183

Figure 5.25: Vertical displacement distribution in the model before and after bridging

the fracture in wellbore strengthening. ...........................................185

xxii

Figure 5.26: Fracture half-width distribution for different horizontal stress contrasts:

a) before bridging the fracture, b) after bridging the fracture, and c)

comparison of fracture half-widths before and after bridging the fracture

for horizontal stress contrast equal to 1.3. ......................................186

Figure 5.27: Fracture half-width distribution for different LCM bridge locations.186

Figure 5.28: Fracture half-width distribution for different leak-off rates: (a) before

bridging the fracture, (b) after bridging the fracture. ......................187

Figure 5.29: Fracture half-width distribution for different Young’s Modulus before

and after bridging the fracture.........................................................188

Figure 5.30: Fracture half-width distribution for different Poisson’s ratios before and

after bridging the fracture. ..............................................................189

Figure 5.31: Fracture half-width distribution for different pressure behind LCM

bridge, varying from formation pressure 1800 psi to wellbore pressure

4000 psi. ..........................................................................................190

Figure 6.1: Cement sheath model. (a) the one-quarter geometry; (b) top view of the

casing/cement/formation system; (c) the interface between casing and

cement. ............................................................................................200

Figure 6.2: Interface fracture opening of two cases with uniform horizontal stress

(SR1) and non-uniform horizontal stress (SR1.2). The pictures are top

views of the cut sections of the casing/cement/interface system at 0.5 m

above the casing shoe......................................................................206

Figure 6.3: Interface fracture width around wellbore for the two cases in Figure 6.2.

0-degree and 90-degree correspond to the directions of the maximum

and minimum horizontal stress, respectively. .................................207

xxiii

Figure 6.4: Developments of fracture geometry, fracture pressure and cement plastic

strain with different initial crack sizes. The pictures are front views of

the one-quarter model, with maximum horizontal stress in the X-axis

direction and minimum horizontal stress in the Y-axis direction. The

circumferential extents of the initial cracks for cases Q1 through Q4 are

30𝑜, 45𝑜, 60𝑜 and 90𝑜, respectively, as given in Table 5.5. ....209

Figure 6.5: Developments of fracture geometry and fracture pressure with different

cement properties. The pictures are front views of the one-quarter

model, with maximum horizontal stress in the X-axis direction and

minimum horizontal stress in the Y-axis direction. The properties for

cement types C1 through C3 are given in Table 6.1. ......................211

Figure 6.6: Development of fracture geometry and fracture pressure with different

formation properties. The pictures are front views of the one-quarter

model, with maximum horizontal stress in the X-axis direction and

minimum horizontal stress in the Y-axis direction. The properties for

formation types F1 through F3 are given in Table 2. .....................212

Figure 7.1: Schematic illustration of pressure-time/volume plot in a leak-off test.218

Figure 7.2: Near wellbore hoop stress concentration with uniform far-field stresses

𝑆ℎ𝑚𝑖𝑛 = 𝑆𝐻𝑚𝑎𝑥 (blue color = more compression, red color = less

compression). ..................................................................................218

Figure 7.3: Typical XLOT plot (Modified after Gaarenstroom et al., 1993)....221

Figure 7.4: An example of pressure-volume plot in a pump-in and flow-back test

(Modified after Gederaas and Raaen, 2009). ..................................222

Figure 7.5: Pressure-time plot of a typical PIFB test. .......................................225

xxiv

Figure 7.6: “Saw-tooth” pressure response during fracture propagation (Reproduced

from Okland et al., 2002). ...............................................................225

Figure 7.7: Pressure-volume plot of a typical PIFB test (Modified after Fjar et al.,

2008). ..............................................................................................227

Figure 7.8: Total volume pumped into the well before fracture creation (Modified

after Altun, 1999 and Fu, 2014). .....................................................227

Figure 7.9: FIP=LOP=FBP in idealized condition. ..........................................228

Figure 7.10: FIP, LOP and FBP are not identical with drilling mud as injection

fluid. ................................................................................................230

Figure 7.11: Permeable formation has a large pressure decline during shut-in due to

sufficient leak-off from fracture, while a relative flat pressure response is

usually observed in an impermeable formation. .............................234

Figure 7.12: An example of predicting FCP using flow-back data (Reproduced from

Gederaas and Raaen, 2009). ............................................................235

Figure 7.13: Test in Well 11-2 in formation with relatively high permeability

(Reproduced from Okland et al., 2002). .........................................236

Figure 7.14: Test in Well 10-7 in formation with relatively low permeability

(Reproduced from Okland et al., 2002). .........................................236

Figure 7.15: General PIFB test system with well, formation and fracture. .........242

Figure 7.16: PIFB test model. Top: geometry and boundary conditions of the model;

Bottom: refined mesh around the wellbore. ....................................244

Figure 7.17: Bottom hole pressure versus time plots. Top: PIFB test in permeable

formation; Bottom: PIFB test in impermeable formation. ..............249

Figure 7.18: Pressure versus square root of time plot during shut-in. Top: Permeable

formation; Bottom: Impermeable formation. ..................................251

xxv

Figure 7.19: BHP vs time during flowback phases in impermeable formation.254

Figure 7.20: BHP versus added fluid volume in the system during shut-in and

flowback phases in the impermeable test. .......................................254

Figure 7.21: BHP versus added fluid volume in the system during shut-in and

flowback phases in the permeable test. ...........................................255

Figure 7.22: Two lost circulation and wellbore breathing events occurred in silty

shale formations, rather than in clean shale or clean sand formations.259

1

CHAPTER 1: Introduction

In this dissertation, a systematic study on the lost circulation and wellbore

strengthening is performed. This chapter introduces the background of the problem, the

objectives of this dissertation, a literature review on lost circulation and wellbore

strengthening, and the organization of this dissertation.

2

1.1 STATEMENT OF THE PROBLEM

Lost circulation is the loss of partial or whole drilling fluid into the rock formation

during drilling a well. It is a major contribution to non-productive time (NPT) in drilling

industry (Cook et al., 2011). Lost circulation can lead to issues such as differential sticking

and well control events which can further increase NPT and drilling cost (Shahri et al.,

2014).

More than 12% of NPT has been reported for Gulf of Mexico area shelf drilling

due to lost circulation alone (Wang et al., 2007a). The US Department of Energy reported

that on average 10% to 20% of the cost of drilling high-pressure and high-temperature

(HTHP) wells is expended on mud losses (Growcock et al., 2009). The impact of lost

circulation on well construction is significant, representing an estimate 2 to 4 billion dollars

annually in lost time, lost drilling fluid and materials used to stem the losses (Cook et al.,

2011).

Most of the lost circulations events are fracture initiation and propagation problems,

occurring when the fluid pressure in the wellbore is high enough to create fractures on the

wellbore wall. Lost circulation usually happens in formations with narrow drilling mud

weight window between pore pressure/collapse pressure gradient and fracture gradient.

Three typical scenarios are fluid losses in depleted zones, deepwater formations and

deviated wellbores. In depleted zones, pore pressure reduction usually leads to a significant

decrease in fracture gradient; so it’s much easier to fracture the wellbore during drilling. In

deepwater formations, the fracture gradient is relative low and the mud weight window is

relative narrow since sea water cannot provide as much as overburden loading as rock does;

therefore it’s a challenge to maintain wellbore pressure in this narrow window considering

of friction pressure and swab/surge pressure during drilling. In deviated wellbores, with

the increase of borehole inclination, the mud weight window can diminish very quickly,

3

even resulting in a zero drilling margin; so wellbore pressure is very likely to overcome

fracture gradient, leading to a lost circulation event.

In order to drilling through problematic zones with high risk of lost circulation,

drilling engineers use some approaches to artificially increase the fracture pressure

(maximum pressure a wellbore can sustain without significant fluid loss) and hence widen

drilling mud weight window by bridging, plugging or sealing the fractures. These

approaches are called “Wellbore Strengthening”. Currently, there are mainly two types of

wellbore strengthening methods in drilling industry. They are hoop stress enhancement

method, e.g. Stress Cage (Alberty and McLean, 2004) and Fracture Closure Stress

(Dupriest, 2005), and fracture resistance enhancement method, e.g. Fracture Propagation

Resistance (van Oort et al., 2011) and Tip Screen-out (Fuh et al., 1992; Morita et al., 1990).

Although a few successful applications of wellbore strengthening have been

reported, there is still a lack of a clear understanding of its fundamentals and a lot of

disagreements still exist. Therefore, a compressive investigation of fracture behaviors

during lost circulation and wellbore strengthening is necessary and crucial for

understanding their fundamentals and designing proper wellbore strengthening treatments.

In this dissertation, an integrated research on lost circulation and wellbore strengthening is

conducted based on both analytical and numerical studies.

1.2 RESEARCH OBJECTIVES

The overall objective of this dissertation is to study fracture behaviors during lost

circulation and wellbore strengthening and provide useful implication for lost circulation

prevention/remediation. The research objective will be achieved by carrying out the

following tasks.

4

1.2.1 Understanding Fracture Initiation and Propagation Pressures

The first objective of this dissertation is to perform a detailed investigation on

fracture initiation and propagation pressures for understanding lost circulation and

wellbore strengthening. For significant fluid loss to occur, fractures must initiate on an

intact wellbore or reopen on a wellbore with preexisting fractures, and then propagate into

far-field region. Wellbore strengthening operations are designed to increase one or both of

these two pressures in order to inhibit fracture growth. Some theoretical models used in

drilling industry assume fracture initiation and propagation pressures are only functions of

in-situ stress and rock mechanical properties. However, as demonstrated by numerous field

and laboratory observations, they are also highly related to drilling fluid properties, and

interactions between drilling fluid and formation rock. A detailed discussion on factors that

can affect fracture initiation and propagation pressures will be first presented in this

dissertation. These factors include: micro-fractures on the wellbore wall, in-situ stress

anisotropy, pore pressure, fracture toughness, filter cake development, fracture

bridging/plugging, bridge location, fluid leak-off, rock permeability, pore size of rock, mud

type, mud solid concentration, and critical capillary pressure.

1.2.2 Numerical Investigation of Lost Circulation

The second objective of this dissertation is to develop a coupled fluid flow and

geomechanics model to simulate lost circulation while drilling. The numerical model

should take into account the main elements during fluid loss, including drilling fluid

circulation, fracture growth, fracture fluid flow, pore fluid flow, and deformation of

formation rock. The model should be able to not only capture the essential signatures of

lost circulation known from field observations such as the reduction in return circulation

rate, but also predict those we cannot measure during drilling such as the development of

fracture geometry with fluid loss.

5

1.2.3 Understanding the Role of Mudcake in Preventive Wellbore Strengthening

Treatments

The third objective of this research is to develop models to investigate the

preventive wellbore strengthening method based on plastering wellbore surface with

mudcake. First an analytical model assuming steady-state fluid flow will be developed to

investigate the effects of mudcake thickness, permeability and strength on stress

distribution and fracture pressure of a wellbore. Second, a numerical model will be

developed to exam the transient effects of dynamic mudcake thickness buildup coupled

with dynamic mudcake permeability reduction on the evolution of pore pressure and stress

around wellbore during drilling.

1.2.4 Modeling Studies of Remedial Wellbore Strengthening Treatments

The fourth objective of this research is to develop models to investigate the

remedial wellbore strengthening method based on plugging the fractures using lost

circulation materials (LCMs). First, an analytical solution based on linear elastic fracture

mechanics will be proposed for investigating the geomechanical aspects of wellbore

strengthening operations. The proposed solution should be able to take into account the

effects of wellbore-fracture geometry, in-situ stress anisotropy, and LCM bridge location

on wellbore strengthening, and provide a fast procedure to predict fracture pressure change

before and after bridging a fracture. Second, a finite-element model will be developed for

obtaining the detailed stress and deformation information in remedial wellbore

strengthening treatment, with a focusing attention on the evolutions of near-wellbore hoop

stress and fracture width.

1.2.5 Investigating Fluid Loss through Casing Shoe

The fifth objection is to develop a coupled fluid flow and geomechanics model to

simulate fluid leakage through casing shoe and cement interface. A three-dimensional

6

finite-element model will be developed to simulate cement interface debonding and fluid

flow along the debonding fracture due to pressure buildup at the casing shoe.

1.2.6 Revealing the Importance of Pump-in and Flow-back Tests for Combatting

Lost Circulation

The sixth objective of this dissertation is to reveal the importance of pump-in and

flow-back tests for combatting lost circulation. Striking similarities exist between the

pump-in and flow-back test and lost circulation, since both of them are issues of fracture

growth from wellbore wall to far-field region. A detailed phase by phase interpretation of

pump-in and flow-back tests will be conducted for helping understand fracture behaviors

(initiation, propagation and closure) during lost circulation. A finite-element framework

for modeling field injectivity tests with fluid injection, well shut-in and fluid flowback will

also be developed. Several methods for estimating minimum principal stress using shut-in

data and flow-back data will be investigated.

1.3 LITERATURE REVIEW

1.3.1 Experimental Studies of Lost Circulation

Even though lost circulation is a major NPT event in drilling industry, very few

experimental studies have been conducted on it. The joint-industry DEA-13 experimental

study conducted in the middle 1980’s to early 1990’s (Fuh et al., 1992; Morita et al., 1996a,

1990, 1990; Onyia, 1994) is an early experimental investigation into lost circulation. The

aim of this study is to determine why lost circulation occurs less frequently while drilling

with water based mud (WBM) than with oil based mud (OBM), or why OBM apparently

causes a lower fracture gradient than WBM. Most of the DEA-13 experiments were

conducted using 30×30×30 inch sandstone blocks with 1.5-inch boreholes. Two major

observations of DEA-13 project are: (1) fracture initiation pressure (FIP) and fracture

7

reopening pressure (FRP) are almost independent of mud type and use of LCM additives,

and (2) fracture propagation pressure (FPP) was found to be strongly related to mud type

and significantly increased by use of LCM additives. The experimental results were

explained by a physical model called “tip screen-out” (Fuh et al., 1992; Morita et al., 1996a,

1996b, 1990; Morita and Fuh, 2012), which indicates that fracture gradient (i.e. FPP)

increases due to a sealing plug near fracture tip which isolates fracture tip from wellbore

pressure.

Another experimental effort to study lost circulation is the GPRI 2000 joint-

industry project conducted in the late 1990’s to early 2000’s ( Dudley et al., 2001; van Oort

et al., 2011). Different from DEA-13 experiments conducted on relative large rock blocks,

thus allowing the researchers to observe fracture propagation, GPRI 2000 experiments

were mostly performed with 4-inch diameter cores with 5/8-inch boreholes. Only FIP and

FRP can be evaluated, while FPP cannot be observed with such small samples. The purpose

of GPRI 2000 study was to evaluate the capabilities of different LCMs on increasing

fracture pressure, with a focus on increasing FRP. It was found that when LCMs were used

in the mud, FRP can be significantly increased, which was contrary to the observation in

DEA-13 study; and LCM was more effective in increasing FRP in WBM than in OBM or

synthetic based mud (SBM).

A recent experimental study project on lost circulation was conducted in M-I

SWACO from late 2000’s to early 2010’s (Guo et al., 2014), which was called Lost

Circulation and Wellbore Strengthening Research Cooperative Agreement (RCA) project.

RCA project has been conducted to investigate wellbore strengthening mechanism and

effectiveness of various wellbore strengthening methods, including preventive

strengthening method and remedial strengthening method. Both sandstone and shale blocks

of 6×6×6 inch with borehole of 1 inch were used in the tests. The results show that: (1) a

8

preventive wellbore strengthening treatment is more effective than remedial treatment; (2)

particle size distribution (PSD) and concentration of LCM are critical in sealing/bridging

the fractures; and (3) fracture pressure achieved with wellbore strengthening is always

higher than the breakdown pressure, which means wellbore strengthening can not only

repair weak formation with induced/natural fractures, but also strengthen the formation.

1.3.2 Physical Models for Wellbore Strengthening

Different from its literal meaning, “Wellbore Strengthening” does not target on

increasing the strength of wellbore or formation. It is an operation to alter the stress

distribution in the vicinity of wellbore and fracture and/or fluid pressure distribution inside

the fracture to increase the fracture pressure (i.e. the maximum sustainable pressure of a

wellbore without significant fluid loss). Many field applications and laboratory tests have

shown that fracture pressure can be significantly increased by wellbore strengthening

operations (Alberty and McLean, 2004; Aston et al., 2007, 2004a; Duffadar et al., 2013;

Dupriest, 2005; Dupriest et al., 2008; Guo et al., 2014; van Oort et al., 2011; van Oort and

Razavi, 2014; Wang et al., 2009, 2007a, 2007a, 2007b). Currently, there are three major

physical models in the drilling industry for explaining why wellbore strengthening

treatments can “strengthen” a wellbore.

Stress Cage (SC) model (Alberty and McLean, 2004). When a fracture is created

on wellbore wall, LCM particles are forced in to the fracture. The largest particles first

wedge the fracture mouth on the wellbore wall. Then, smaller LCM particles come in and

plug the spaces among larger particles and between particles and fracture surfaces, and

then seal the fracture mouth together with filtration control agents. Next, trapped fluid in

the fracture filters into the formation through fracture surfaces and compressive forces are

transferred to LCM bridge at fracture mouth. Finally, the fracture is bridged at fracture

9

mouth resulting in an increased hoop stress near the bridge location, which makes the

fracture more difficult to open.

Fracture Closure Stress (FCS) model (Dupriest, 2005). Fractures on wellbore

wall are first created and widened to increase the compressive stress, i.e. fracture closure

stress, in the adjacent rock. The greater the fracture width, the greater the fracture closure

stress. Next, LCM particles within mud slurry are forced into the fracture. Liquid leaks off

from the slurry to the formation rock. LCM particles consolidate and finally form a bridge

inside the fracture that keeps fracture open and isolates fracture tip from wellbore pressure.

The increased fracture closure stress and isolation of fracture tip make the fracture more

difficult to open and extend.

Fracture Propagation Resistance (FPR) model (Morita et al., 1996b, 1990; van

Oort et al., 2011). FPR model does not aim to alter the near wellbore stress to increase hoop

stress or fracture closure stress, instead it attempts to increase the formation’s resistance

against fracture propagation. FPR model supposes that a filter cake can form inside the

fracture as fracture propagation. The filter cake can seal fracture tip and prevent pressure

communication between fracture tip and wellbore, therefore the resistance for fracture

propagation can be increased. FPR model argues that fracture initiation and reopen pressure

cannot be increased by wellbore strengthening treatments, but rather fracture propagation

pressure can be significantly increased.

1.3.3 Analytical Studies of Lost Circulation and Wellbore Strengthening

Compared to other drilling problems, e.g. wellbore instability, there are very few

analytical studies for lost circulation and wellbore strengthening. Several parameters that

researchers want to know from analytical studies are fracture pressure, fracture width and

fracture-tip stress intensity factor before and after bridging/sealing the fracture. For

10

calculating these parameters, it is important to consider the near wellbore stress

concentration for short fracture and pressure drop along fracture for large fracture.

Based on a dimensional analysis and superposition principle of linear elastic

fracture mechanics, Guo et al. (2011) gave an approximate, closed-form solution for the

crack mouth opening displacement (CMOD) of two fractures symmetrically located at

wellbore wall with the fracture surface subjected to either uniform wellbore pressure or

pore pressure. Their model is only valid for a relative short fracture with a length less than

4 wellbore radii and for the maximum to minimum horizontal stress ratio less than 2.

Shahri et al. (2014) proposed a semi-analytical solution for wellbore strengthening

analysis based on singular integral formulation of stress field and solved using Gauss-

Chebyshev polynomials. Their model considers far field stress anisotropy and near

wellbore stress perturbation, but doesn’t take into account pressure drop along fracture and

can’t give closed-form solutions for fracture width and fracture pressure.

Ito et al. (2001) applied the penny-shaped hydraulic fracture model developed by

Abé et al. (1976) to analyze fracture pressure increase after plugging the fracture. However,

there model does not take into account near wellbore stress concentration (i.e. the existence

of wellbore is neglected) and therefore can only be used for a fracture with a size much

larger than wellbore size. What’s more, this model assumes the pressure is uniform inside

the fracture from wellbore to LCM plug and does not consider pressure drop, which is not

realistic for a large fracture.

Using linear elastic fracture mechanics and superposition principle, Morita and Fuh

(2012) proposed two sets of closed-form solutions for stress intensity factor and fracture

pressure after bridging the fracture at fracture mouth and away from fracture mouth. They

assumed the pressure inside the fracture from fracture mouth to bridge location was

uniform and equal to wellbore pressure. Their model for bridging fracture away from

11

fracture mouth neglects the effect of wellbore on stress intensity factor induced by pressure

from LCM bridge to fracture tip, therefore it is only valid for large fracture for which it is

reasonable to ignore the effect of wellbore.

van Oort and Razavi (2014) extended the KGD hydraulic fracture model (Geertsma

and De Klerk, 1969; Linkov, 2013; Zheltov, 1955) to analyze wellbore strengthening.

Fracture pressure and fracture width after sealing the fracture is derived, but this model

neglects both wellbore effect (i.e. near wellbore stress concentration) and pressure drop in

the fracture.

1.3.4 Numerical Studies of Lost Circulation and Wellbore Strengthening

Since, analytical solutions, that describe the fracture geometry and

stress/displacement field near wellbore and fracture and satisfy our needs for wellbore

strengthening analysis, are still not available in the literature (Wang et al., 2009, 2007a),

numerical methods can be applied to simulate the problem and provide valuable

implications for field application. But, so far, only few numerical studies have been

conducted for lost circulation and wellbore strengthening problems, even though there have

been numerous numerical studies for wellbore stability and hydraulic fracturing problems

which are closely related to lost circulation/wellbore strengthening.

Wang et al. (2009, 2007a) proposed a 2D boundary element model to simulate two

symmetric fractures on wellbore wall under anisotropic in-situ stresses and get the stress

and fracture width distribution before and after bridging the fracture in wellbore

strengthening. Guo et al. (2011) used a 2D finite element model to investigate the fracture

width distribution for two pre-existing fractures symmetrically located at the wellbore wall

under various in-situ stress and fracture length; but the simulation did not give any

information about fracture behavior after applying wellbore strengthening treatments.

12

Alberty and McLean (2004) employed a 2D finite-element model to study fracture width

distribution and hoop stress field after bridging the fracture near its mouth under nearly

isotropic in-situ stresses. All the above numerical models assume the rock is linearly elastic

and do not take into account the porous features of the rock, therefore the effect of fluid

flow inside the rock and fluid leak-off through wellbore and fracture surfaces are not

considered.

With the goal to perform a comprehensive parametric study for wellbore

strengthening, Arlanoglu et al. (2014), Feng et al. (2015a) and Feng and Gray (2016a)

developed a 2D finite element model assuming the rock is poroelastic material and

including fluid flow inside the rock and across fracture and wellbore surfaces. Based on

their model, stress and pore pressure fields and fracture width distribution before and after

bridging a fracture were investigated, and a comprehensive parametric sensitivity studies

for wellbore strengthening were performed.

With the aim to investigate the hypothesis of wellbore hoop stress

increases when fractures are wedged and/or sealed as presented in Stress Cage

theory, Salehi (2012) and Salehi and Nygaard (2011) used cohesive element method to

study fracture propagation and sealing during lost circulation and wellbore strengthening.

According to their simulation results, they argued that fracture sealing/bridging is not able

to increase wellbore hoop stress more than its ideal state when no fracture exists. But in

their model, a constant injection rate (fluid loss rate) boundary condition was defined at

the fracture inlet which is not consistent with the actual drilling situation where the

downhole condition at fracture inlet is neither a constant flow rate nor a constant pressure.

13

1.4 OUTLINES OF THIS DISSERTATION

This dissertation consists of eight chapters. This chapter (Chapter 1) provides a

brief statement of the problems of lost circulation and wellbore strengthening, an

introduction to the objectives of this dissertation, and a literature review on lost circulation

and wellbore strengthening.

Chapter 2 discusses the role of fracture initiation and propagation pressures on lost

circulation and wellbore strengthening. In view of the existing disagreements on the

fundamentals of lost circulation and wellbore strengthening, a critical and detailed analysis

of fracture initiation and propagation pressures is conducted. Factors that may affect these

two pressures are investigated, which include micro-fractures on the wellbore wall, in-situ

stress anisotropy, pore pressure, fracture toughness, filter cake development, fracture

bridging/plugging, bridge location, fluid leak-off, rock permeability, pore size of rock, mud

type, mud solid concentration, and critical capillary pressure.

Chapter 3 introduces a finite-element model to simulate lost circulation during

drilling with circulation of drilling fluid. Circulation flow of drilling fluid in the well “U-

Tube” consisting of drilling pipe and annulus is simulated based on Bernoulli’s theory.

Fracture propagation into the porous rock is modeled using coupled pore pressure cohesive

zone method. The two parts are coupled together to predict the dynamic fluid loss and

fracture geometry evolution in drilling process. The numerical model provides a unique

new way to model lost circulation while drilling when the boundary condition at the

fracture mouth is neither a constant flowrate nor a constant pressure, but rather a dynamic

bottom-hole pressure.

In Chapter 4, an analytical solution and a numerical model are developed to

investigate the role of mudcake on preventive wellbore strengthening treatments based on

plastering wellbore wall with mudcake. The analytical solution is derived based on steady-

14

state flow assumption and superposition principle. It incorporates the effects of thickness,

permeability and strength of mudcake on near-wellbore pore pressure and stress

distribution, hence the effectiveness of preventive wellbore strengthening treatment. In

order to describe the effects of dynamic mudcake buildup and time-dependent mudcake

property (permeability) on wellbore strengthening. A finite-element framework based on

poroealstic theory is developed to investigate the transient effects of mudcake thickness

buildup and mudcake permeability change on the near-wellbore stress and pore pressure,

and thus the strengthening of wellbore.

In Chapter 5, an analytical solution and a finite-element model are proposed for

modeling remedial wellbore strengthening treatment based on plugging/bridging the

fractures using LCMs. The analytical model, based on linear elastic fracture mechanics

theory, provides a fast procedure to predict fracture pressure change before and after

fracture bridging. The numerical model takes into account poromechanical effects and

provides a more accurate prediction of local stress distribution and fracture width with

wellbore strengthening operations. Sensitivity analyses are then performed using both of

the models to quantify the effects of rock properties, in-situ stresses, bridge locations and

fluid flow on remedial wellbore strengthening.

In Chapter 6, a three-dimensional finite-element framework is developed to exam

the possibility of fluid leakage through casing shoe and along the weak cement interface

when there is pressure buildup in the wellbore due to change of drilling or completion fluid,

conduction of injectivity tests, and etc.

Chapter 7 highlights the importance of field injectivity tests for understanding the

fundamentals of lost circulation and wellbore strengthening, with a review of different

kinds of field tests and a discussion of their advantage and limitations. A coupled fluid flow

and geomechanics injectivity model is also developed which can capture the key elements

15

of injectivity tests known from field observations and aid the interpretation and design of

field tests.

Chapter 8 summarizes the major work conducted in this dissertation, presents the

main conclusions, and provides some recommendations for future work related to this

research.

16

CHAPTER 2: Understanding Fracture Initiation and Propagation

Pressures1

Fracture-initiation pressure (FIP) and fracture-propagation pressure (FPP) are both

important considerations for preventing and mitigating lost circulation. For significant

fluid loss to occur, a fracture must initiate on an intact wellbore or reopen on a wellbore

with preexisting fractures, and then propagate into the far-field region. Wellbore

strengthening operations are designed to increase one or both of these two pressures in

order to combat lost circulation. Most of the existing theoretical models assume fracture

initiation and propagation pressures are only functions of in-situ stress and rock mechanical

properties. However, as demonstrated by numerous field and laboratory observations, they

are also highly related to drilling fluid properties, and interactions between drilling fluid

and formation rock.

This chapter discusses the mechanisms of lost circulation and wellbore

strengthening, with an emphasis on factors that can affect FIP and FPP. These factors

include: micro-fractures on the wellbore wall, in-situ stress anisotropy, pore pressure,

fracture toughness, filter cake development, fracture bridging/plugging, bridge location,

fluid leak-off, rock permeability, pore size of rock, mud type, mud solid concentration, and

critical capillary pressure. The conclusions of this chapter include information seldom

considered in lost circulation studies, such as the effect of micro-fractures on FIP and the

effect of capillary forces on FPP. Research results described in this chapter may be useful

for lost circulation mitigation and wellbore strengthening design, as well as leak-off test

interpretation.

1 Parts of this chapter have been published in: Feng, Y., Jones, J.F. and Gray, K.E., 2016. A Review on

Fracture-Initiation and-Propagation Pressures for Lost Circulation and Wellbore Strengthening. SPE Drilling

& Completion, 31(02), pp.134-144. This paper was supervised by K. E. Gray.

17

2.1 INTRODUCTION

Most lost circulation events occur when the hydraulic pressure in the wellbore

exceeds FIP and FPP of the formation rock. Lost circulation is common in wellbores with

a narrow drilling mud-weight window, which is the difference between the maximum mud

weight before the occurrence of lost circulation and the minimum mud weight to balance

formation pore pressures or avoid excessive wellbore failure. Typical scenarios include

drilling within depleted reservoirs, drilling highly inclined wellbores where increased fluid

densities are required for hole stability, and drilling highly over-pressured formations,

where the margin between formation pore pressure and the overburden pressure is reduced.

Commonly encountered pressure ramps and pressure regressions may also lead to

significant reductions in the drilling mud-weight window. It is well known that carbonate

formations (limestone/dolomite) are usually characterized by presence of natural fractures,

vugs, and cavities, and consequently lost circulation occurs frequently (Masi et al., 2011;

Wang et al., 2010). However, lost circulation in carbonate formations is outside the scope

of this dissertation and the discussion in this chapter is mainly for clastic formations such

as sandstones and shales.

The reduction in pore pressure in depleted reservoirs results in a corresponding,

albeit smaller, reduction in fracture gradient (Hubbert and Willis, 1957; Matthews and

Kelley 1967). Conversely, bounding and inter-bedded shale layers, as well as any isolated

and un-drained sands, will maintain their original pore pressure and fracture gradient.

Therefore, as shown in the left plot of Figure 2.1, it may be difficult or impossible to reduce

the drilling fluid density sufficiently to maintain equivalent circulating densities (ECD)

below the depleted zone fracture gradient. ECD is defined as the effective density of the

circulating fluid in the wellbore, resulting from the sum of the hydrostatic pressure imposed

by the static fluid column and the friction pressure (API STD 2010). In deep-water

18

formations, the total vertical stress is relatively low since sea water does not provide as

much overburden loading as sediment and rock. A reduction in total vertical stress also

results in a lower lateral stress and fracture gradient. If abnormal pressures are also present,

the mud-weight window may be very narrow, as shown in the right plot of Figure 2.1.

Under these circumstances, it may be challenging to avoid hydraulic fracturing both while

tripping due to surge/swab effects, and while circulating due to high annular friction losses

and ECDs.

Figure 2.1: Left: Pore pressure and fracture gradient plot in depleted zone. Pore pressure

decrease leads to a decrease in fracture gradient. Right: Pore pressure and

fracture gradient plot in deep-water formation with abnormally high pressure.

There is a reduced mud-weight window.

FIP and FPP are two important considerations for preventing and mitigating lost

circulation. Only after a fracture initiates on an intact wellbore or reopens on a wellbore

with preexisting fractures, and then propagates into the far-field region, can significant

fluid loss occur. Therefore, accurate pre-drill estimates of these two pressure values are

critical for reducing lost circulation events. A common theoretical method to estimate FIP

19

for a vertical well, compares a simple tensile failure criterion to the hoop stress defined by

the Kirsch equation. FIP predicted by this approach is related to formation rock strength,

in-situ stresses and the formation fluid pressure, and it is assumed that the fracture initiates

at the wellbore wall. However, the Kirsch equation assumes zero leak-off, i.e. impermeable

rock or perfect mudcake.

FPP can be determined from injectivity tests, analysis of fluid losses while drilling,

or from fracture mechanics modeling. In field practice, FPP is often estimated from leak-

off tests (LOTs), performed at casing or liner shoes. However, these tests are generally

insufficient for this analysis which may lead to significant error (Ziegler and Jones, 2014).

It is worth noting that FIP and FPP are commonly taken as parameters of the

formation rock, dependent on the in-situ stresses, mechanical properties of the rock, and

inclination and orientation of deviated wells. However, field experience suggests they may

also be influenced by other parameters related to the drilling fluid (e.g., mud type, fluid

leak-off, solid particles within the fluid, temperature etc.), as well as other properties of the

rock (e.g., lithology, permeability, wettability and capillary effect). A detailed study of

these factors’ effects on them is therefore needed for better understanding of lost

circulation.

In order to drill through problematic zones with a high risk of lost circulation,

various drilling technologies may be useful, including managed pressure drilling (MPD),

dual gradient drilling (DGD) and casing/liner drilling. Alternatively, “Wellbore

Strengthening” is a different approach that seeks to artificially increase the pressure the

wellbore can sustain and hence widen the mud-weight window. Rather than actually

increase the strength of the wellbore rock, as its name implies, this methodology is believed

to work by plastering wellbore surface and/or bridging/plugging lost circulation fractures.

20

As mentioned in the literature review in Section 1.3, there are two main types of

wellbore strengthening methods currently used in the petroleum industry: the hoop stress

enhancement method, e.g., Stress Cage (Alberty and McLean, 2004), and the fracture

resistance enhancement method, e.g., Fracture-Propagation Resistance (Fuh et al., 1992;

Morita et al., 1990; van Oort et al., 2011). The first method is based on inducing and

plugging a fracture to increase the local hoop stress, thus raising fracture reopening

resistance. Feng et al. (2015a) have conducted detailed numerical studies and found that

theoretically at least, hoop stress can be increased significantly if the fracture can be

plugged effectively. Although theoretical studies show there is large potential in hoop

stress increase (Alberty and McLean, 2004; Wang et al., 2009, 2007b, 2008) and numerous

successes are reported for the Stress Cage method (Aston et al., 2007, 2004a; Song and

Rojas, 2006; Whitfill et al., 2006), lost circulation problems are still commonly

encountered with an ECD much lower than the hoop stress around the wellbore. Therefore

a number of doubts still persist, including (1) whether hoop stress is a good indicator of

lost circulation and the evaluation of wellbore strengthening success and (2) when wellbore

strengthening works, is it actually due to an increase in hoop stress? This chapter will

discuss these questions in detail.

Fracture-Propagation Resistance theory is based jointly on experimental and field

observations, including the DEA 13 (Fuh et al., 1992; Morita et al., 1996a, 1996b, 1990;

Onyia, 1994) and GPRI 2000 (van Oort et al., 2011) laboratory studies. Both theory and

experience indicate fracture-propagation resistance can be effectively enhanced using

appropriate wellbore strengthening methods. Although several models (Fuh et al., 2007;

van Oort et al., 2011; van Oort and Razavi, 2014) have been introduced to explain how

fracture-propagation resistance may be increased, there remains a lack of understanding of

the precise role a list of influencing factors may play. These factors include in-situ stresses,

21

wellbore pressure, fracture geometry and size, mud type and properties, rock lithology and

properties, lost circulation material locations and properties, fluid leak-off, mudcake, and

capillary force. Therefore, significant disagreement about the fundamental physics of

wellbore strengthening still exists in the industry.

The purpose of this chapter is to analyze the mechanisms of lost circulation and

wellbore strengthening, by investigating the factors that may affect both FIP and FPP. In

view of the existing disagreement about the fundamentals of lost circulation and wellbore

strengthening, a critical and detailed analysis of these two pressure thresholds is conducted.

It should be noted that wellbore strengthening discussed in this dissertation is physical or

mechanical strengthening of the wellbore by development of mudcake on wellbore wall

and/or bridging plug in the fracture in relatively permeable formations. In impermeable

shales with very low leak-off, chemical strategies are commonly used to strengthen the

wellbore, either by changing chemical composition of the formation (Growcock et al.,

2009) or by forming chemical sealants in the fracture (Aston et al., 2007). The chemical

wellbore strengthening technique is outside the scope of this dissertation. It should also be

noted that most of the discussions in this chapter are based on the case of a vertical well,

but the principles and perspectives are also applicable to deviated and horizontal drilling.

2.2 LOST CIRCULATION “THRESHOLDS”

For significant fluid loss to occur through either a drilling induced or closed pre-

existing natural fracture, the wellbore pressure must overcome both FIP and FPP. These

two pressure limits may be regarded as “thresholds” to lost circulation, which are critical

for well construction and drilling fluid design.

In theory, FIP is usually greater than FPP, if the wellbore is an intact cylinder.

However, when the stress anisotropy is relatively high and/or there are pre-existing

22

fractures, FPP may be equal to or greater than the calculated FIP. In general, this condition

should not cause significant concern.

There are four general conditions related to lost circulation, depending on the

relative magnitudes of ECD, FIP, and FPP:

(1) When ECD is lower than both FIP and FPP, fluid loss will not occur.

(2) When ECD is higher than FIP but lower than FPP, only very small fractures will

generate near the wellbore wall and no significant fluid loss will occur.

(3) When ECD is larger than FPP but lower than FIP, the situation is less stable. No

fluid loss will occur as long as the wellbore remains intact, and the far field stress

region of each formation is isolated from the pressure in the wellbore. However,

lack of wellbore isolation may result from inadequate filter cake development in

permeable formations or where pre-existing natural or mechanically induced

fractures are present in any type of formation.

(4) When ECD is above both FIP and FPP, fluid loss is expected to occur. In this case,

remedial actions must include some form of ECD reduction and/or wellbore

strengthening operations

2.3 FRACTURE INITIATION PRESSURE (FIP)

Conventional interpretation theories for FIP generally assume a perfectly intact

wellbore. Fracture initiation is predicted when the tangential stress (also called hoop stress)

at the wellbore wall equals the tensile strength of the rock. It is widely accepted that FIP

depends much more on in-situ stresses, which determine the hoop stress around the

wellbore, than the tensile strength of the rock, which is comparatively very small. In reality,

the assumption of a perfectly intact wellbore is rarely true. The most likely imperfect

wellbore condition is a wellbore with micro-fractures (Morita et al., 1990). Micro-fractures

23

may develop naturally from tectonic movement, rapid sediment compaction and/or thermal

fluid expansion, as well as from destructive drilling operations. In the case of pre-existing,

hydraulically conductive micro-fractures at the wellbore wall, the above method to predict

FIP is no longer valid. In this case, the wellbore pressure that begins to fail the formation

rock is the propagation pressure for the micro-fractures, rather than the initiation pressure

for any new fractures. However, for the purposes of this discussion, micro-fracture-

propagation pressure is considered as FIP, since the fracture size is very small and the

assumption of a perfect wellbore is seldom satisfied.

2.3.1 FIP of a Perfect Wellbore

FIP for an intact cylindrical wellbore, may be easily determined from continuum

mechanics (Kirsch equations). However, FIP may be very different for permeable and

impermeable formations. For an impermeable formation with negligible tensile strength,

FIP of a vertical wellbore can be estimated by the Hubbert-Willis equation (Hubbert and

Willis, 1957; Jin et al., 2013):

𝑝𝑖𝑛𝑖 = 3𝑆ℎ𝑚𝑖𝑛 − 𝑆𝐻𝑚𝑎𝑥 − 𝑝𝑝 (2.1)

where, 𝑝𝑖𝑛𝑖 is FIP; 𝑆ℎ𝑚𝑖𝑛 and 𝑆𝐻𝑚𝑎𝑥 are the minimum and maximum horizontal

stresses, respectively; 𝑝𝑝 is the pore pressure.

However, FIP for a permeable rock may be significantly affected by an additional

induced stress term, related to fluid penetration from the wellbore to the formation. For a

permeable rock, FIP can be estimated by the Haimson-Fairhurst equation (Haimson and

Fairhurst, 1967):

𝑝𝑖𝑛𝑖 =3𝑆ℎ𝑚𝑖𝑛−𝑆𝐻𝑚𝑎𝑥−𝜂𝑝𝑝

2−𝜂 (2.2)

𝜂 = 𝛼𝑝 (1−2𝑣

1−𝑣) (2.3)

24

where, 𝜂 is a poroelastic parameter of the rock, which determines the magnitude of the

stress induced by fluid penetration, and various in the range [0, 1], from zero fluid

penetration to unimpeded fluid penetration, respectively; 𝛼𝑝 is Biot’s coefficient; and 𝑣

is Poisson’s ratio.

The left plot of Figure 2.2 shows the relationship between FIP, horizontal stress

anisotropy and pore pressure for a vertical well with a constant poroelastic parameter, 𝜂 =

0.5. It is clear that FIP decreases with an increase in stress anisotropy. It is also clear that

for a given 𝑆ℎ𝑚𝑖𝑛 and 𝑆𝐻𝑚𝑎𝑥 , FIP also decreases with an increase in pore pressure.

However, this observation must be viewed in proper context, since 𝑆ℎ𝑚𝑖𝑛 and 𝑆𝐻𝑚𝑎𝑥 are

generally a function of pore pressure and overburden stress (Hubbert and Willis, 1957;

Matthews and Kelley 1967) and increase with increasing pore pressure, if the overburden

is held constant or increases. With horizontal stress ratio 𝑆𝐻𝑚𝑎𝑥 𝑆ℎ𝑚𝑖𝑛 = 1.3⁄ , the right

plot of Figure 2.2 shows a very interesting observation for the effect of 𝜂 on FIP of a

vertical well. That is, FIP increases with the increase of 𝜂 when the pore pressure is

lower than a certain value but decreases when pore pressure is higher than that value. In

this case, the crossover point is 0.85 ∙ 𝑆ℎ𝑚𝑖𝑛. However, with the decrease of horizontal

stress ratio, the crossover point will move to the right. The crossover point will no longer

exist on the X-axis scale when the horizontal stress ratio is smaller than 1.2.

25

Figure 2.2: Fracture-initiation pressure of a vertical well. Left: with different horizontal

stress anisotropies and pore pressure; Right: with different 𝛈 and pore

pressure.

2.3.2 FIP of a Wellbore with Micro-fractures

As mentioned previously, when hydraulically conductive drilling induced or pre-

existing natural micro-fractures exist on the wellbore, the wellbore pressure that begins to

fail the formation rock is the propagation pressure for the micro-fractures rather than the

initiation pressure for any new fractures. Therefore, the continuum mechanics method

using the Kirsch equation to determine FIP is no longer valid. Instead, a fracture mechanics

approach should be used to determine FIP (or micro-fracture-propagation pressure).

Seeking to interpret leak-off tests for estimating horizontal stress, Lee et al. (2004)

analytically studied the propagation pressure of a fracture extending from a wellbore in the

direction of maximum horizontal stress. This analysis is based on the Barenblatt condition,

which dictates a balance between the tensile stress intensity factor produced by fluid

pressure in the fracture and the negative stress intensity factor caused by the compressive

in-situ stress (Lee et al., 2004; Yew and Weng, 2014). According to their study, the FIP of

a wellbore with micro-fractures (or micro-fracture propagation pressure) should be:

𝑝𝑖𝑛𝑖 =3𝑆ℎ𝑚𝑖𝑛−𝑆𝐻𝑚𝑎𝑥

2+

𝐾𝐼𝐶

𝜋√2𝐿 (2.4)

26

where, 𝐾𝐼𝐶 and 𝐿 are the fracture toughness of the formation and the length of the micro-

fracture, respectively. Based on this equation, FIP is not only related to horizontal stress

but is also a function of fracture toughness 𝐾𝐼𝐶 of the rock, and micro-fracture length 𝐿.

For Eq. 2.4, dimensionally normalizing the pressure and stress terms with minimum

horizontal stress 𝑆ℎ𝑚𝑖𝑛, fracture length with wellbore radius 𝑎, and fracture toughness with

the product of minimum horizontal stress and the square root of wellbore radius 𝑆ℎ𝑚𝑖𝑛√𝑎,

it can be transformed to:

𝑝𝑖𝑛𝑖′ =

3−𝑅

2+

𝐾𝐼𝐶′

𝜋√2𝐿′ (2.5)

where, 𝑝𝑖𝑛𝑖′ , 𝐾𝐼𝐶

′ , 𝐿′ and 𝑅 are dimensionless FIP, dimensionless fracture toughness,

dimensionless fracture length and horizontal stress anisotropy, respectively. Note that Eq.

2.5 has a mathematic singularity signature, as the normalized FIP goes to infinitely high

with a normalized fracture length approaching zero. Dimensional analysis shows that with

reasonable values for R and 𝐾𝐼𝐶′ , Eq. 2.5 is not suitable for a fracture length less than 0.01

inches. In fact, the wellbore can be considered intact, with a fracture as short as 0.01 inches.

The fracture toughness of sedimentary rocks varies approximately in the range of

500 to 2000 psi-in0.5 (Senseny and Pfeifle, 1984; Wang, 2007), and horizontal stress

anisotropy under most geologic settings, ranges from 1 to 2 based on the author’s

experience. With the following assumptions: 𝑆ℎ𝑚𝑖𝑛 = 3000 𝑝𝑠𝑖 , wellbore radius 𝑎 =

4.25 𝑖𝑛, and micro-fracture length 𝐿 = 0.5 𝑖𝑛, Figure 2.3 shows the FIP of a vertical well

under various sets of horizontal stress anisotropy and fracture toughness conditions. It

indicates that FIP (1) is very sensitive to and decreases dramatically with an increase in

horizontal stress anisotropy, (2) increases moderately with an increase in fracture

toughness, and (3) can be much smaller than the minimum horizontal stress with a

relatively high stress anisotropy and low fracture toughness.

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27

From this analysis, it is critical to highlight the influence of micro-fractures on FIP.

For instance, in impermeable rocks, the continuum mechanics (Kirsch) equation predicts a

FIP equal to the minimum horizontal stress when stress anisotropy is 2.0. However, with

the same stress anisotropy, Figure 5.3 shows FIP will be far below the minimum horizontal

stress, with a fracture length of only about 10% of the wellbore radius.

Figure 2.3: FIP (micro-fracture-propagation pressure) decreases dramatically with an

increase in horizontal stress anisotropy, and increases moderately with an

increase in fracture toughness; it can be much smaller than the minimum

horizontal stress with high stress anisotropy and low fracture toughness.

2.3.3 FIP versus Leak-off Pressure (LOP)

In conventional field practice, leak-off tests are often used to estimate FIP, which

is taken as the pressure value at the first inflection point where the pressure ramp-up curve

deviates from linearity before formation breakdown. A typical pressure-volume/time

response of a leak-off test is shown in the middle figure of Figure 2.4.

Although it is commonly accepted that the leak-off pressure (LOP) indicates the

start of a fracture and should be identical to the FIP, a careful analysis indicates they are

28

not necessarily the same, especially when “dirty” mud (drilling fluid with high solids

content) is used for a leak-off test in a permeable formation.

For an intact wellbore with solids-free fluid or clean mud, fracture initiation is

largely dominated by in-situ stresses. For a wellbore with micro-fractures and clean mud,

fracture initiation is controlled by the fluid pressure distribution inside the fracture. The

LOP in these two cases should be approximately equal to FIP. However, for a drilling fluid

with high solids content, e.g. LCMs, the mud properties may affect the observed leak-off

behavior and lead to a LOP very different from FIP. This can be explained as follows.

When a short hydraulically conductive micro-fracture is created during a leak-off

test, in theory it should be easily extended with sufficient wellbore pressure. In reality, the

micro-fracture may be quickly sealed by mud solids, forming a filter cake within the

fracture. This “internal” filter-cake can then isolate the fracture from the wellbore, and not

enough fluid pressure will reach the fracture face in order to extend it. This “opening and

healing” or “fracturing and packing” behavior within the fracture, can theoretically restore

the pressure containment capability of the wellbore, and perhaps increase it to a higher

value than the ideal case where no fracture exists.

This phenomenon is similar to wellbore strengthening. However, the “opening and

healing” of such small fractures is not likely detectable in a field leak-off test or even in a

lab test (Guo et al., 2014). In many field leak-off tests, it is difficult to identify a clear leak-

off response at fracture-initiation point, and the FIP can be very close to the formation

breakdown pressure (FBP) as shown in the left figure of Figure 2.4. Therefore, the lack of

a visible leak-off response reasonably below the FBP doesn’t necessarily mean a small

fracture has not been generated.

Numerous elements may influence the signature of a leak-off test, including the

compressibility and elasticity of the mud, casing, cement and formation rock, fluid seepage

29

from the wellbore wall, and fluid leak-off into fractures. Among these factors, only the

effect of leak-off into fractures is observably nonlinear (Fu, 2014). Therefore, when there

is a clear leak-off response as shown in the middle figure of Figure 2.4, a relatively large

fracture is likely to have been created, and the leak-off point, commonly considered to be

fracture initiation, is actually micro-fracture propagation. Undetectable micro-fracture

generation has already occurred before this leak-off point, so LOP is somewhat higher than

FIP.

It is also possible to observe multiple leak-off points on the pressure-volume/time

curve. The right figure of Figure 2.4 shows a case where there are two inflection points.

This signature is more common for leak-off tests conducted in permeable formations, with

a low pump rate and high solids content fluid. These observations may be explained by a

filter cake break within the fracture, where wellbore pressure breaks the filter cake, leading

to additional fracture extension. The fracture will be quickly sealed again by solids in the

mud, and the wellbore pressure will continue to build. If the subsequent wellbore pressure

increases enough, the filter cake may fail again, and the process is repeated. This repeated

fracturing and healing behavior might continue until formation breakdown. It should be

emphasized that a clear slope-change in the pressure-volume/time response during a leak-

off test is usually subsequent to fracture initiation. This response is most likely a filter cake

break in a fracture larger than a micro-fracture, but still in the vicinity and under the

influence of the near wellbore stress concentration. Lab tests show that fractures can grow

significantly without any clear leak-off signature (Guo et al., 2014).

It may not be possible to accurately predict FIP from a leak-off test using a high

solids content fluid. A slope-change point may be undetectable prior to formation

breakdown, or if detected, it may indicate filter cake breakdown rather than fracture

initiation.

30

Figure 2.4: Schematic pressure-volume/time curves in leak-off tests. Left: no visible leak-

off response at fracture initiation, the leak-off pressure is very close to

formation breakdown pressure; Middle: a clear leak-off point before

formation breakdown; Right: multiple leak-off points before formation

breakdown.

2.4 FRACTURE PROPAGATION PRESSURE (FPP)

After initiation, a fracture will tend to propagate from the wellbore wall to the far-

field, under sufficient wellbore fluid pressure. Typically, this fracture propagation consists

of both a stable and an unstable stage. During a leak-off test, the stable fracture-propagation

stage begins at fracture initiation or leak-off and ends roughly at formation breakdown.

Initially the fracture grows very slowly and its volume increases at a rate lower than the

pump rate. Therefore, the wellbore pressure continues to rise prior to formation breakdown,

which is the upper pressure limit for stable fracture growth.

The unstable fracture-propagation stage begins immediately following formation

breakdown. Over a very short time period, the fracture volume expands at a much greater

rate than the pump rate and the wellbore experiences a sudden pressure drop. Ultimately,

the wellbore pressure stabilizes as fracture propagation continues, with a rate of fracture

volume increase roughly equal to the pump rate. From a theoretical viewpoint, the FPP

with clean injection fluid will gradually decrease with the continued increase in fracture

31

length, as will be shown later in the chapter. However, from a practical viewpoint, the FPP

can either increase or decrease with fracture growth, likely due to the high friction pressure

in a relatively large fracture and the complex nature of formation rock.

FPP is a very important parameter for well construction and drilling fluid design,

especially for lost circulation prevention. In challenging areas with severe lost circulation

problems, extended leak-off tests (XLOTs) are recommended in order to obtain reliable

estimates of FPP.

2.4.1 Formation Breakdown Pressure (FBP)

With a clean fluid, the pressure required to initiate a fracture on the wellbore wall

is usually greater than that required to propagate the fracture into the formation.

Furthermore, formation breakdown is often assumed to occur when the hoop stress at the

wellbore wall equals the tensile strength of the rock (Hubbert and Willis, 1957).

Using a fluid with a high solids content, numerous laboratory and field tests

(Aadnøy and Belayneh, 2004; Guo et al., 2014; Liberman, 2012; Morita et al., 1990) have

shown that FBP is often significantly higher than that predicted by conventional continuum

mechanics theories. This phenomenon may be elegantly explained by the filter cake sealing

effect.

Prior to formation breakdown, the fracture size (length) remains small, and fracture

propagation is determined by fracture toughness. When the fracture length is small, the

toughness term in Eq. 2.5 can be much larger than the stress term. According to linear

elastic fracture mechanics, a tensile fracture will start to extend when the stress intensity

factor 𝐾𝐼 reaches fracture toughness 𝐾𝐼𝐶, i.e.

𝐾𝐼 = 𝐾𝐼𝐶 (2.6)

32

The stress intensity factor 𝐾𝐼 is a function of fracture size and geometry, as well as

load condition. Fracture toughness 𝐾𝐼𝐶 is a material constant representing the strength of

the material. For a short fracture on the wellbore wall as shown in Figure 2.5, based on

linear elastic fracture mechanics theory, the stress intensity factor can be estimated by:

𝐾𝐼 = 1.12(𝑃𝑓 − 𝑆𝜃𝜃̅̅ ̅̅̅)√𝜋𝐿 (2.7)

where, 𝑃𝑓 is the pressure inside the fracture; 𝑆𝜃𝜃̅̅ ̅̅̅ is the average normal stress (closure

stress) acting on the fracture face and can be roughly calculated by the Kirsch equation

(neglecting the presence of the fracture). Hence, for a given fracture, 𝑆𝜃𝜃̅̅ ̅̅̅ is only a function

of the wellbore pressure and the far-field stresses. In most geologic settings, 𝑆𝜃𝜃̅̅ ̅̅̅ is a

compressive stress, unless a very high horizontal stress anisotropy (larger than 3) exists. In

order for 𝐾𝐼 to reach 𝐾𝐼𝐶 to propagate the fracture, 𝑃𝑓 must be large enough to overcome

the closure stress 𝑆𝜃𝜃̅̅ ̅̅̅. Therefore, for a given fracture, wellbore pressure and horizontal

stresses, 𝑃𝑓 acting on the fracture face should dominantly control fracture propagation.

When the fracturing fluid is clean, wellbore fluid can easily flow into the fracture,

and apply pressure to the fracture face, approximately the same magnitude as wellbore

pressure. Thus, a stress intensity factor higher than the fracture toughness is more easily

achieved and the fracture will propagate. However, when the fluid contains solids, the

following mechanisms will significantly reduce or eliminate the pressure acting on the

fracture face, preventing fracture propagation:

Solids are transported with fluid flow into the fracture, resulting in a high solids

density and fluid viscosity in the fracture. The fracture may also be plugged/sealed

by a filter cake, as shown in the middle figure of Figure 2.5. A high solids density

and/or fluid viscosity will significantly increase the fracture pressure drop from

fracture inlet to tip, leading to a much lower 𝑃𝑓 and smaller 𝐾𝐼. Filter cake sealing

inside the fracture can further decrease 𝑃𝑓 as well as 𝐾𝐼.

33

Due to the small aperture of the fracture, it is very likely to be bridged and sealed

quickly by the filter cake, before solids can enter the fracture, as shown in the right

figure of Figure 2.5. The low permeability of the filter cake will further restrict fluid

flow into the fracture, and finally lead to 𝑃𝑓 in the fracture equal to pore pressure,

due to pressure bleed-off into porous rock. The excess pressure (𝑃𝑓 − 𝑆𝜃𝜃̅̅ ̅̅̅) will

then decrease or become negative under most conditions. Therefore, the stress

intensity factor will not reach the fracture toughness magnitude, unless the wellbore

pressure builds high enough to break the filter cake at the fracture mouth. As

mentioned previously, the “fracturing and healing” process can be repeated several

times prior to formation breakdown, and therefore the FBP may be significantly

higher than the theoretically predicted FIP.

Figure 2.5: A small fracture on the wellbore wall before formation breakdown: Left: no

filter cake plugging with clean fluid; Middle: high solids concentration or

filter cake inside the fracture; Right: fracture is plugged at the inlet on

wellbore wall.

34

2.4.2 Fracture Propagation Pressure (FPP)

2.4.2.1 Theoretical Prediction

At formation breakdown, the filter cake in the micro-fracture breaks completely,

allowing fluid to enter the fracture. The fracture then grows quickly in both width and

length, extending to the far-field region. The wellbore pressure drops to the FPP. A great

number of field and lab hydraulic fracturing tests have indicated that FPP decreases with

the increase in fracture length. This phenomenon might be partly due to the minimal excess

pressure required to maintain fracture propagation with a large fracture face, and partly due

to the high accessibility of a weak point, with a large fracture circumference (Okland et al.,

2002). However, this phenomenon can be interpreted more elegantly with a coupled fluid

mechanics and solid (fracture) mechanics approach.

After the fracture has propagated a significant distance, the influence of the

wellbore on fracture propagation behavior is greatly diminished (Zheltov, 1955). Figure

2.6 shows a hydraulic fracture with a neglected wellbore in the fracture center. Consider a

fracture with a length of 2𝐿, perpendicular to the minimum horizontal stress 𝑆ℎ𝑚𝑖𝑛, as

shown in Figure 2.6. The formation rock is considered isotropic, homogeneous, linearly

elastic, and impermeable. The fluid is assumed to be incompressible, non-viscous

Newtonian fluid. It is injected through the well at fracture center at a constant rate 𝑄. The

pressure everywhere inside the fracture is the same as wellbore pressure. Using a coupled

fluid and solid mechanics method, similar to that of Detournay (2004), both the fracture

half-length and pressure during fracture propagation can be determined as functions of

time:

𝑎(𝑡) = (𝐸′𝑄(𝑡−𝑡0)

2𝜋1 2⁄ 𝐾𝐼𝐶)

2 3⁄

∝ 𝑡2 3⁄ (2.8)

𝑝(𝑡) =𝐸′𝑄(𝑡−𝑡0)

2𝜋𝑎2∝ 𝑡−1 3⁄ (2.9)

35

where 𝑡 is injection time; 𝑎(𝑡) is the fracture half-length at time 𝑡; 𝑝(𝑡)is the pressure

inside the fracture at time 𝑡; 𝐸′ =𝐸

1−𝑣2 is the plane strain modulus, which is a function of

Young’s modulus 𝐸 and Poisson’s ratio 𝑣 ; and 𝑡0 is the start time of fracture

propagation.

The pressure behavior theoretically predicted by Eq. 2.9 is schematically shown in

the left plot of Figure 2.7. Before 𝑡0, the pressure builds up linearly inside the fracture

without fracture propagation. At 𝑡0 , the stress intensity factor of the fracture reaches

fracture toughness, triggering sudden fracture propagation. Following 𝑡0 , the pressure

drops nonlinearly and proportional to 𝑡−1/3. Figure 2.7 is the FPP when water based mud

was used as the fracturing fluid in a lab test of DEA-13 project (Fuh et al., 1992; Morita et

al., 1990). Apart from the fluctuating signature, a fitted curve shows the pressure decreases

proportionally to 𝑡−0.305, which is reasonably close to the predicted result.

Figure 2.6: A large hydraulic fracture (wellbore at the fracture center is neglected).

36

Figure 2.7: Pressure response during fracture propagation. Left: theoretical result; Right:

DEA-13 lab test result.

The above model is established under the assumptions of an ideal condition: the

rock is impermeable and the fluid is clean with zero viscosity. Eq. 2.9 shows FPP for the

model depends only on injection rate, injection time, and fracture toughness. In reality,

FPP also depends on a list of other factors including in-situ stress, pore pressure, solids

plugging, base fluid leak-off, lithology, permeability, aqueous/non-aqueous fluid, rock

wettability, capillary force and others. Most of these factors’ effects are not independent,

but related to others. Several of these factors’ effects on FPP are discussed as follows.

2.4.2.2 In-situ Stress, Pore Pressure and Solids Plugging

Consider a fracture similar to that in Figure 2.6, which is perpendicular to the

minimum horizontal stress, but now the fracture is in a formation with pore pressure 𝑝𝑝,

and is effectively plugged by solid particles in the fracturing fluid at some location inside

the fracture as shown in Figure 2.8. Assume the plug is perfect without permeability, thus

it completely stops fluid penetration. The fracture domain ahead of the plug, from the

wellbore to the plug, is wetted by fluid and its pressure is the same as wellbore pressure.

The pressure in the fracture section behind the plug (non-penetrated zone) is equal to pore

37

pressure, due to fracture pressure bleed-off into porous rock. This problem was first solved

analytically by Abé et al. (1976). On the basis of their work, the FPP for a large fracture

can be given roughly by the following equation:

𝑝𝑝𝑟𝑜𝑝 =1

1−√1−(1−𝐿𝑛𝑤

𝐿)

2[𝑆ℎ𝑚𝑖𝑛 − 𝑝𝑝√1 − (1 −

𝐿𝑛𝑤

𝐿)

2

] (2.10)

where, 𝑝𝑝𝑟𝑜𝑝 is the FPP; 𝐿𝑛𝑤 is the fracture length of the non-penetrated zone. Note that

Eq. 2.10 is only valid for a fracture with a length much larger than the wellbore radius.

Therefore, the effect of the wellbore can be ignored. Another limitation of this equation is

that it should not be used when the plug location is close to the wellbore, because the

detailed stress-concentration in the wellbore vicinity is neglected. Eq. 2.10 also neglects

the effect of fracture toughness due to its unimportant role when the fracture is large (this

is one of the major differences between large fracture and micro-fracture propagation: the

influence of fracture toughness might be the dominate factor for micro-fractures, but trivial

for large fractures).

It is indicated by Eq. 2.10 (also see Figure 2.9) that FPP after plugging is primarily

determined by the minimum horizontal stress, pore pressure and the non-penetrated zone

length or the location of the plug. As shown in Figure 2.9, for a given minimum horizontal

stress 𝑆ℎ𝑚𝑖𝑛 , with an increase in pore pressure 𝑝𝑝 , FPP decreases. Another important

observation from Figure 2.9 is that for low values of 𝑝𝑝 𝑆ℎ𝑚𝑖𝑛⁄ , corresponding to

formations with hydrostatic or abnormally low pressure, the FPP is very sensitive to non-

penetrated zone size, the larger the non-penetrated zone size, the higher the FPP. This

confirms the statement in the Stress Cage concept (Alberty and McLean, 2004; Feng et al.,

2015a) that the best place to plug a fracture for wellbore strengthening is the fracture inlet

or mouth, and also the statement in the Fracture-Propagation Resistance concept (van Oort

38

et al., 2011) that plugging the fracture to isolate its tip from wellbore pressure can

significantly enhance the fracture propagation resistance (pressure). FPP in low pressure

formations, as shown in Figure 2.9, can be increased to several times higher than the

minimum horizontal stress. However, the influence of plugging is smaller for high values

of 𝑝𝑝 𝑆ℎ𝑚𝑖𝑛⁄ , e.g. formations with abnormally high pressure. Therefore, from the pore

pressure point of view only, wellbore strengthening methods based on plugging the fracture

might be more effective for depleted reservoirs with larger differences between 𝑝𝑝 and

𝑆ℎ𝑚𝑖𝑛than for high-pressure formations with relatively small differences between 𝑝𝑝 and

𝑆ℎ𝑚𝑖𝑛.

Figure 2.8: A fracture plugged by solid particles.

39

Figure 2.9: FIP with non-penetration zone length, pore pressure and minimum horizontal

stress.

2.4.2.3 Fluid Leak-off through Fracture Faces

In order to investigate the effect of fluid leak-off on fracture propagation behavior,

a stationary fracture model as shown in Figure 2.10 is used. In a hypothetical fracture

extending perpendicular to the minimum horizontal stress direction, in a poroelastic rock

with initial pore pressure 𝑝𝑝 and time 𝑡 = 0, a fluid pressure 𝑃𝑓 (greater than 𝑝𝑝) is

applied inside the fracture. The fracture fluid and pore fluid have identical properties.

Therefore, after applying fluid pressure, the normal traction on the fracture face changes

from 𝑆ℎ𝑚𝑖𝑛 to −𝑃𝑓 (here tension is positive) while the pore pressure on the fracture face

changes from 𝑝𝑝 to 𝑃𝑓. Detournay and Cheng (1991) indicated that this problem can be

examined by decomposing it into two separate problems (1) applying normal traction (fluid

pressure) to the fracture face while keeping pore pressure unchanged, and (2) applying pore

pressure while keeping the traction constant. The solutions of each problem are then

40

superposed to obtain the full solution of the original problem. In this case, only problem 2

is of interest (the effect of pore pressure increase on fracture behavior, due to fluid leak-off

thorough the fracture face). According to the analysis results reported by Detournay and

Cheng (1991), a pore pressure increase in this case will lead to a negative change in both

fracture volume and stress intensity factor. As schematically shown in Figure 2.11, the

instantaneous fracture volume 𝑉𝐶0 at 𝑡 = 0 decreases to the long-term volume 𝑉𝐶∞ ,

when pore pressure reaches 𝑃𝑓. The instantaneous stress intensity factor 𝐾𝐼0 also drops to

the long-term value 𝐾𝐼∞ . The decrease of fracture volume and stress intensity factor

reveals the fact that a pore pressure increase as a result of fluid leak-off tends to close the

fracture and inhibit fracture growth.

Figure 2.10: A stationary fracture model.

41

Figure 2.11: Fracture volume (left) and stress intensity factor (right) changes with

applying pore pressure to the fracture face only, while keeping the traction

on the fracture face constant.

2.4.2.4 Permeability

The above analyses of fracture propagation indicate that both solids plugging and

fluid leak-off induced pore pressure increases can contribute to preventing fracture

propagation or enhancing fracture-propagation resistance. Fluid leak-off, however, is well

recognized as a critical prerequisite for creating an effective filter plug (Aston et al., 2007).

Therefore, any factors affecting filter plug formation and/or fluid leak-off can influence

fracture propagation, hence lost circulation and wellbore strengthening.

Permeability attracts much attention in lost circulation and wellbore strengthening

analysis, since it is generally believed that only in permeable formations (i.e. sandstone)

can an effective filter plug be formed. Conversely, in impermeable rocks (i.e. shale) it is

generally believed that wellbore strengthening is not likely to be successful, although

several successful cases in shale are reported using specific pre-engineered drilling fluids

and LCMs (Aston et al., 2007). When permeability is low as depicted in the left figure of

Figure 2.12, the base fluid (or filtrate) leak-off rate is too low to allow mud solids or LCMs

to aggregate in the fracture and therefore an effective filter plug is not formed. Low leak-

42

off rates also mean very limited fracture pressure/energy is released into the formation.

Therefore, pressure is trapped inside the fracture, facilitating fracture growth. On the

contrary, in permeable formations as depicted in the right figure of Figure 2.12, filtrate

leak-off rate is high enough to form an effective filter plug, and the pore pressure increase

due to fluid leak-off inhibits fracture growth as previously discussed. In addition, fracture

pressure/energy is easily released into the formation, hence less pressure/energy acts

toward extending the fracture.

Figure 2.12: Hydraulic fractures in impermeable (left) and permeable (right) formations

with high solids content fluid.

2.4.2.5 Capillary Entry Pressure

Additionally, if the wellbore fluid (filtrate) and pore fluid are immiscible, capillary

entry pressure 𝑃𝑐𝑒 , also known as threshold capillary pressure, is an important

consideration for analyzing fluid leak-off behavior, especially if pore throat openings or

capillaries are relatively small (Nelson, 2009). Unfortunately, this parameter is often

neglected for lost circulation mitigation and wellbore strengthening design. High 𝑃𝑐𝑒 can

significantly inhibit fluid leak-off, filter-cake/plug development and pore pressure

increase. 𝑃𝑐𝑒, usually estimated by the Young-Laplace equation, depends highly on the

43

largest pore opening (throat) size, wettability (contact angle) and miscibility (interfacial

tension) of the drilling and pore fluids.

𝑃𝑐𝑒 = 2𝛾𝑓,𝑚1

𝑟𝑐𝑜𝑠𝜃 (2.11)

where, 𝑃𝑐𝑒 is the capillary entry pressure, 𝛾𝑓,𝑚 is the interfacial tension between the

wellbore fluid and pore fluid, 𝑟 is the largest pore opening radius, and 𝜃 is the wetting

(contact) angle.

It is clear from Eq. 2.11 that 𝑃𝑐𝑒 increases as pore opening size decreases. When

the difference between wellbore pressure and pore pressure exceeds 𝑃𝑐𝑒, wellbore fluid

(filtrate) will be pushed (leak-off) into the formation and displace the pore fluid. The typical

pore opening size for sandstone is from several to dozens of microns, but much smaller for

shale, in the range of several to dozens of nanometers (Nelson, 2009). In sandstone, 𝑃𝑐𝑒

for hydrocarbons and brine is roughly 10 to 50 psi, and for shale it is roughly 200 to 800

psi in deepwater Gulf of Mexico according to the study by Dawson (2004) and Dawson

and Almon (2005). Since sandstone has significantly lower 𝑃𝑐𝑒 than shale, it is easier for

the wellbore fluid to leak-off into sandstone, facilitating filter plug development and pore

pressure elevation. In contrast, this is less likely to happen in shale due to its high 𝑃𝑐𝑒.

Figure 2.13 schematically shows the fluid leak-off and the corresponding filter-cake/plug

development controlled by capillary pressure for water-wet sandstone and shale when the

fracture and pore fluids are oil/synthetic based mud (OBM/SBM) and water (brine),

respectively.

44

Figure 2.13: Fluid leak-off and filter plug development controlled by capillary pressure

for water-wet sandstone with larger pore size and shale with smaller pore

size. Formation fluid is water while fracture fluid is oil/synthetic based mud.

In addition to pore opening size, rock wettability and fluid immiscibility also

control 𝑃𝑐𝑒, and therefore fluid leak-off and filter plug formation. It is also important to

note that for extremely low permeability shale, fluid leak-off may be very restricted

regardless of fluid type. For brine saturated, water-wet rocks with relatively small pore

opening size: (1) If the fluid is OBM/SBM, it cannot easily enter the pore openings due to

high interfacial tension between immiscible fluids, and therefore there is little if any fluid

leak-off or filter-cake/plug development (Figure 2.14); (2) in contrast, if the fluid is WBM,

the water in the mud may readily invade the pore openings, leaving the solid particles

behind and thereby forming a filter-cake/plug.

45

Figure 2.14: Fluid leak-off and filter cake development controlled by fluid immiscibility

and capillary pressure.

The above capillary entry pressure analysis may further explain the following field

observations:

(1) Lost circulation in fractured and silty shale formations occurs much more

frequently with OBM/SBM than with WBM, and it is often more difficult to cure

fluid losses with OBM/SBM. Due to high capillary entry pressures, OBM/SBM

cannot easily invade the pores of water-wet shale and silty shale (most shale is

water-wet), and therefore, all the fluid pressure acts toward propagating the fracture

tip. No effective filter plug is developed to isolate wellbore pressure and increase

fracture propagation resistance.

(2) Wellbore “breathing” is a phenomenon that occurs when formations take drilling

fluid when the pumps are on and give the fluid back when the pumps are off, due

to the opening and closing of drilling induced fractures. This phenomenon is usually

observed in water-wet shale (especially silty-shale) formations, while drilling with

OBM/SBM. One plausible explanation is that OBM/SBM will often flow back to

the wellbore rather than leak-off into the formation, due to very high capillary entry

http://www.drilling-mud.org/

http://www.drilling-mud.org/

46

pressures (Ziegler and Jones, 2014). Conversely, WBM will leak-off readily into

these same formations, rather than flow back to the wellbore.

2.5 PREVENTIVE AND REMEDIAL WELLBORE STRENGTHENING

There are two kinds of wellbore strengthening treatments in the drilling industry:

preventive wellbore strengthening and remedial wellbore strengthening. Simply put,

preventive wellbore strengthening methods attempt to “strengthen” the wellbore to prevent

fluid loss due to hydraulic fracturing. Remedial wellbore strengthening methods attempt to

“strengthen” the wellbore after fluid loss due to hydraulic fracturing has already occurred.

Preventive methods focus on increasing both FIP and FPP, while remedial methods focus

primarily on FPP, since a fracture has already been created.

In a preventive wellbore strengthening treatment, LCM has a dual purpose for

preventing fracture initiation. First, LCM helps develop a mud filter cake with low

permeability and high ductility. As discussed previously, this filter cake will help maintain

a high FIP, by effectively isolating the formation from pressure in the wellbore, thus

inhibiting any pore pressure increase in the vicinity of the wellbore wall.

Second, LCM particles can immediately plug any generated micro-fractures,

preventing both fluid flow into the fracture and pressure communication between the

wellbore and fracture tip. In theory, this process should restore (maintain) a leak-off

pressure higher than FIP. This claim can be confirmed by the experimental wellbore

strengthening study by Guo et al. (2014).

Figure 2.15 shows the pressure build-up curve, when using a drilling fluid with 20

ppb graphitic LCMs, during a preventive treatment test (Guo et al., 2014). The pressure

was increased to 2500 psi (test device limit) without apparent leak-off. Since no leak-off

response was observed, it might be concluded that the formation had not been fractured. In

47

fact, the test block was fractured completely to its edges (see Figure 2.16). A reasonable

explanation for this observation is that fractures initiated on the wellbore wall were

immediately sealed by LCMs in the mud. Therefore, no significant mud loss or noticeable

pressure response occurred. This test demonstrates that observed leak-off pressure is not

necessarily equal to FIP, especially when LCM is used. Leak-off pressure can be somewhat

higher than FIP and fracture initiation can occur without fluid loss. This test also confirms

that leak-off pressure can be significantly increased by using LCMs in a preventive

wellbore strengthening treatment.

After fracture initiation and fluid loss have occurred, preventive wellbore

strengthening treatments work in the same manner as remedial treatments. With fluid loss,

LCM particles are forced into the fracture to form a solid bridge or filter cake within the

fracture, and thereby increase fracture propagation resistance. However, fracture initiation

and leak-off pressures are not generally restored to their original values by remedial

treatments.

This assertion is confirmed by the DEA-13 experimental study (Black et al., 1988;

Onyia, 1994) (see Figure 2.17). This repeated test included 3 cycles. In the first cycle the

intact wellbore was fractured with a fluid containing LCMs, and a relatively high leak-off

pressure was observed. The second cycle was a repeat of the first cycle, but without LCMs.

In this case, leak-off pressure was much lower than in the first cycle, because the mud

barrier on the wellbore wall was previously destroyed. FPP was also much lower since

there was no effective bridge or filter cake development without LCM. In the third cycle,

LCM was added back to the fluid, simulating a remedial wellbore strengthening treatment.

In this case, leak-off pressure did not change compared to the second cycle. However, FPP

did significantly increase.

48

From this analysis, it can be claimed that both FIP and FPP can be increased by

wellbore strengthening treatments. Preventive treatments can increase both pressures,

while remedial treatments only alter FPP. However, significant mud loss in a drilling well

will only occur if FPP is exceeded. It is also worth repeating, that these treatments work

much better in more permeable formations with low capillary entry pressures. An

illustration of both methods is shown in Figure 2.18.

Figure 2.15: An example of a preventive wellbore strengthening test on a sandstone block

(after Guo et al., 2014).

49

Figure 2.16: The sandstone test block for pressure build-up curve in Figure 12. The block

was fractured to the edges without obvious fluid leak-off due to LCM

sealing effect (after Guo et al., 2014).

Figure 2.17: A repeated hydraulic fracturing test with LCM. First injection cycle

(preventive treatment - intact wellbore, with LCM): high leak-off pressure

and high propagation pressure. Second injection cycle (fractured wellbore,

without LCM): low leak-off pressure and low propagation pressure. Third

injection cycle (fractured wellbore, with LCM): low leak-off pressure, high

(increased) propagation pressure. (after Black et al., 1988).

50

Figure 2.18: Preventive wellbore strengthening treatment enhances both leak-off pressure

and FPP, while remedial wellbore strengthening treatment only enhances

FPP.

2.6 SUMMARY

FIP of a perfectly cylindrical wellbore can be determined by continuum mechanics

methods (Kirsch equations). However, for a wellbore with micro-fractures, fracture

mechanics methods should be used to predict FIP.

FIP of a wellbore with micro-fractures is controlled not only by pore pressure and

in-situ stresses, but also by fracture length and fracture toughness of the formation

rock. It can be much lower than that of a perfect wellbore.

Leak-off pressure from a leak-off test may not be equivalent to the FIP when a high

solids content “dirty” mud is used. Due to the continuous sealing effect of dirty

mud, the observable “leak-off” pressure may instead be the filter cake breakdown

pressure (i.e. propagation pressure) of a relatively larger sealed fracture, rather than

FIP of an intact wellbore wall.

51

Formation-breakdown pressure is the upper pressure limit for the stable fracture

propagation stage. During this stage, the fracture size remains small, and a fracture

mechanics method can be used to determine formation-breakdown pressure, which

is controlled to a very large extent by fracture toughness. A high solids

concentration in the drilling fluid, a filter-cake inside the fracture and/or a filter

plug at the fracture mouth can significantly increase formation-breakdown

pressure.

FPP is the fracture pressure during the unstable propagation stage. A coupled fluids

and solids mechanics method predicts a decrease in FPP with an increase in fracture

length.

Plugging a fracture can significantly increase its propagation pressure, especially

in formations with large differences between pore pressure 𝑃𝑝 and minimum

horizontal stress 𝑆ℎ𝑚𝑖𝑛 . Therefore, wellbore strengthening methods based on

plugging the fracture should be more effective in depleted reservoirs with large

differences between 𝑃𝑝 and 𝑆ℎ𝑚𝑖𝑛 than in deepwater over-pressured formations

with relatively small differences between 𝑃𝑝 and 𝑆ℎ𝑚𝑖𝑛.

Fluid leak-off through the fracture face hinders fracture growth by facilitating filter

cake development and reducing the fluid energy available to propagate the fracture.

Capillary entry pressure 𝑃𝑐𝑒 is an important and often neglected consideration for

lost circulation mitigation and wellbore strengthening. High capillary entry

pressures, associated with small pore openings and immiscible fluids, can

significantly restrict fluid leak-off and filter-cake/plug development. Field

observations indicate lost circulation in fractured and silty shale formations occurs

more frequently with OBM/SBM than with WBM. Additionally, the observation

52

that wellbore breathing typically occurs in water-wet formations drilled with

OBM/SBM may be elegantly explained by capillary theory.

A preventive wellbore strengthening treatment can increase both FIP and FPP,

while a remedial treatment can only increase FPP.

53

CHAPTER 3: Developing a Framework for Lost Circulation Simulation

Understanding the growth of drilling induced fracture is critical for lost circulation

prevention and remediation. It can help provide useful information for optimization of

drilling fluid rheology, well configuration, pump schedule, and LCM particle size

distribution. In this chapter, a lost circulation simulation framework is developed based on

finite-element method using Abaqus®. The framework can be used to simulate static fluid

loss without mud circulation in the wellbore and dynamic fluid loss with mud circulation.

The framework consists of two components: wellbore and formation rock. It

successfully couples the fluid circulation in the wellbore, fluid seepage on wellbore wall,

propagation of induced fractures, fluid flow in induced fractures, pore fluid flow and

deformation of formation rock. The fluid circulation in the wellbore is modeled based on

the Bernoulli’s equation taking into account gravitational and viscous pressure losses of

fluid flow. Fracture propagation and fluid flow in the fracture are modeled based on a pore

pressure cohesive zone method. A traction-separation constitutive law for describing

fracture propagation and a fluid flow constitutive law for describing fluid flow in the

fracture are incorporated into the cohesive zone model. Fluid seepage on wellbore wall,

pore fluid flow and porous rock deformation are modeled using poroelastic theory. The

numerical model provides a novel way to simulate fluid losses during drilling when the

boundary condition at the fracture mouth is neither a constant flowrate nor a constant

pressure, but rather a dynamic wellbore pressure.

54

3.1 INTRODUCTION

Understanding drilling induced fracture behavior can provide an effective tool for

guiding equivalent circulation density (ECD) optimization and LCM selection for lost

circulation prevention. Traditionally, analytical models are used to predict the wellbore

pressure or ECD at which fractures begin to initiate on wellbore wall (Kostov et al., 2015).

As mentioned in Section 2.3, there are two commonly used analytical models. The first one

is the Hubbert-Willis model (Hubbert and Willis, 1957) which does not consider the effect

of fluid diffusion from wellbore to surrounding formation and usually predicts an upper

limit for fracture initiation pressure (FIP). The other one is the Haimson-Fairhurst model

(Haimson and Fairhurst, 1967) that considers the effect of fluid diffusion and usually

predicts a lower limit for FIP. While these analytical models can provide reasonable

prediction for FIP, they cannot give any information about the fracture pressure and

fracture dimensions once a fracture has been induced while drilling.

Different from injectivity tests or hydraulic fracturing stimulation treatments in

which fluid driven fractures are intentionally created, most of the drilling induced fractures

are unintentional. Even though there are a lot of numerical simulation studies on hydraulic

fracturing in the literature, very few studies are performed on modeling drilling induced

fractures. While a fracture model with pre-designed constant or time-dependent injection

rate can capture the fracture behaviors in an injectivity test or hydraulic fracturing

operation, it cannot describe the fracture induced during drilling. Drilling induced fractures

are “dynamic-pressure-driven”, rather than “constant-rate-driven” or “constant-pressure-

driven”. In other words, there is neither a constant flow rate nor a constant pressure at the

fracture mouth, but rather a dynamic pressure or ECD in the bottom hole which drives

fracture propagation.

55

The dynamic bottom hole pressure (BHP) or ECD while drilling is influenced by

the gravity pressure of the fluid column and viscous pressure loss of fluid flow in the

annulus. For a well with a certain depth, gravity pressure is dominated by drilling mud

density, while viscous loss is controlled by a number of factors including mud density, mud

viscosity, flow (pump) rate, annulus clearance, and roughness of annulus surfaces. These

factors, together with fluid flow and fracture propagation in the formation, are successfully

coupled in the lost circulation simulation framework developed in this chapter.

The lost circulation model development is described in the below sections. Section

3.2 provides the numerical method and governing equations for wellbore fluid flow,

fracture propagation, fracture fluid flow, rock deformation and pore fluid flow involved in

the model development. Section 3.3 describes the geometries, boundary conditions and

materials of the finite-element framework for simulating both static and dynamic fluid

losses without and with drilling mud circulation in the well. Finally, in Section 3.4, the

simulation results are presented, and factors that influence static and/or dynamic fluid loss

are investigated and discussed based on the results.

3.2 NUMERICAL METHOD AND GOVERNING EQUATIONS

Lost circulation while drilling is simulated using a coupled fluid flow and

mechanics numerical model in this study based on the finite-element method. A lost

circulation system generally consists of three components: the well, the fracture, and the

formation. Figure 3.1 shows a typical configuration of the lost circulation system. The

following physical processes that happen in a lost circulation event are included and

simulated simultaneously in the proposed numerical model:

(1) Fluid circulation in the well.

(2) Lost circulation fracture propagation and fluid flow in the fracture.

56

(3) Formation rock deformation and pore fluid flow.

Figure 3.1: Illustration of lost circulation system with well, formation and fracture.

In this study, Abaqus®, a general purpose finite-element-method code for solving

linear and non-linear stress-analysis problems, is used for simulating the above physical

processes in the lost circulation problem.

3.2.1 Fluid Circulation in the Well

Fluid circulation in the well is modeled based on Bernoulli’s equation considering

gravity and viscous pressure losses. Flow between two points in a flow pipe, as shown in

Figure 3.2, is modeled based on Bernoulli’s equation as:

57

∆𝑃 − 𝜌𝑔∆𝑍 = 𝐶𝐿𝜌𝑣2

2 (3.1)

𝐶𝐿 =𝑓𝐿

𝐷ℎ (3.2)

where, ∆𝑃 is the pressure difference between the two points; ∆𝑍 is the elevation

difference between the two points; 𝑣 is the fluid velocity in the pipe; 𝜌 is the fluid

density; 𝑔 is the gravity acceleration factor; 𝐶𝐿 is the loss coefficient; 𝐿 is the pipe

length; 𝑓 is the friction factor; 𝐷ℎ is the hydraulic diameter of the pipe, which is a

function of the cross-section area 𝐴 and wetted perimeter 𝑆 of the pipe and expressed as

𝐷ℎ =4𝐴

𝑃.

The friction factor 𝑓 in Eq. 3.2 is an important parameter controlling the friction

loss of fluid flow. In the simulation, the friction factor can be determined by two methods.

The first method is using the Blasius friction loss formula which uses an empirical relation

based on Reynold’s number 𝑅𝑒 to determine friction factor (Hager, 2003; SIMULIA,

2016). This method distinguishes two different flow regimes according to Reynold’s

number, i.e. laminar flow when 𝑅𝑒 < 2500 and turbulent flow when 𝑅𝑒 > 2500. The

friction factors for the two different flow regimes are expressed as

{𝑓 =

64

𝑅𝑒 (𝑅𝑒 < 2500)

𝑓 =0.3164

𝑅𝑒0.25 (𝑅𝑒 ≥ 2500) (3.3)

The second method to determine the friction factor 𝑓 is based on the Churchill’s

formula which takes into account both the flow regimes (Reynold’s number) and the

roughness of the pipe (Churchill, 1977; SIMULIA, 2016). The friction factor is expressed

as

𝑓 = 8 [(8

𝑅𝑒)

12

+1

(𝐴+𝐵)1.5]

1

12

(3.4)

where,

𝐴 = [−2.457𝑙𝑛 ((7

𝑅𝑒)

0.9

+ 0.27𝐾𝑠

𝐷ℎ)]

16

;

58

𝐵 = (37350

𝑅𝑒)

16

;

and 𝐾𝑠 is pipe roughness.

It should be noted that there is a discontinuous jump in the friction factor when the

flow transitions from laminar to turbulent regime at 𝑅𝑒 = 2500 in the Blasius expression

(Eq. 3.3). This discontinuity may cause convergence issues in the simulation. However, the

Churchill’s formula (Eq. 3.4) transitions smoothly from laminar to turbulent flow. Figure

3.3 shows the transition behaviors of the two models for 𝐾𝑠 𝐷ℎ⁄ = 10−5. In the simulation

studies in this research, Churchill’s formula is used to model the friction loss behavior of

fluid flow in well. Pipe element in Abaqus® is used to represent the well in the model.

Figure 3.2: Schematic of fluid flow in pipe.

59

Figure 3.3: Friction factor determined from Blasius model and Churchill model: Blasius

model shows discontinuous transition from laminar to turbulent flow at

Re=2500, while Churchill model shows smooth transition.

3.2.2 Fracture Propagation and Fluid Flow in the Fracture

Fracture propagation and fluid flow in the fracture are modeled based on a cohesive

zone method using coupled pore pressure and deformation cohesive elements in Abaqus®.

A traction-separation constitutive law for describing fracture propagation and a fluid flow

constitutive law for describing fracture fluid flow are incorporated into the cohesive zone

model.

Fracture opening and propagation are modeled as the damage evolution between

two initially bonded interfaces with zero interfacial thickness. The traction-separation

constitutive law consists of three components: initial (before damage) loading behavior,

damage initiation, and damage evolution of the cohesive interface. Figure 3.4 shows the

traction-separation constitutive law used in the study. The initial loading process is

assumed to follow linear elastic behavior, determined by the stiffness of the interface which

relates stress and strain across the interface. Before damage initiation, the stiffness of the

interface remains constant. Damage begins when the stress/traction applied on the interface

60

satisfies certain damage initiation criteria. In this study, a maximum nominal stress

criterion is used to model damage initiation, which assumes that damage begins when the

traction on the interface reaches the tensile or shear strength (𝑇𝑜 in Figure 3.4) of the

interface.

Beyond damage initiation, the stiffness of the interface drops and damage evolution

occurs. Damage evaluation basically describes the rate at which the stiffness is degraded

once damage initiation is reached. There are two methods to define damage evaluation

either based on the relationship between the displacement at final failure 𝛿𝑓𝑖𝑛𝑎𝑙 (interface

stiffness drops to zero) and the displacement at damage initiation 𝛿𝑖𝑛𝑖 (interface stiffness

begins to drop), or based on the energy 𝐺𝐶 dissipated due to failure (SIMULIA, 2016). In

this study, damage evolution is defined based on an energy criterion called the

Benzeggagh-Kenane fracture criterion (Benzeggagh and Kenane, 1996):

𝐺𝑛𝐶 + (𝐺𝑠

𝐶 − 𝐺𝑛𝐶) (

𝐺𝑆

𝐺𝑇)

𝛽

= 𝐺𝐶 (3.5)

where, 𝐺𝑆 = 𝐺𝑠 + 𝐺𝑡 is the total energy dissipated due to deformations in the first and

second shear directions; 𝐺𝑇 = 𝐺𝑛 + 𝐺𝑠 + 𝐺𝑡 is the total energy dissipated due to

deformations in the normal, the first shear and the second shear directions; 𝐺𝐶 = 𝐺𝑛𝐶 +

𝐺𝑠𝐶 + 𝐺𝑡

𝐶 is the total critical fracture energy in the normal, the first shear and the second

shear directions; 𝐺𝑛, 𝐺𝑠 and 𝐺𝑡 are the energies dissipated due to deformations in the

normal, the first shear, and the second shear directions, respectively; 𝐺𝑛𝐶, 𝐺𝑠

𝐶 and 𝐺𝑡𝐶 are

the critical energies required to case failure in the normal, the first shear, and the second

shear directions, respectively. This fracture criterion assumes that when the ratio of energy

dissipated due to shear deformations to the total energy dissipated in the damage process

reaches a critical value determined by the critical fracture energies of the material, the

fracture will begin to propagate. This criterion is more appropriate for situations where the

61

critical fracture energies purely along the first and second shear direction are similar

(Wang, 2015; Yao et al., 2010). It is very suitable for modeling failure in geomaterials.

Figure 3.4: A typical traction-separation law

For modeling hydraulic driven fracture using the cohesive element method, upon

complete failure of the pore pressure cohesive element, i.e. the separation of the interface

reaching the critical value 𝛿𝑓𝑖𝑛𝑎𝑙 and stiffness of the element reducing to zero, fluid will

flow into the element. Fluid flow in the fracture includes two components as shown in

Figure 3.5: longitudinal flow along the fracture and normal fluid flow (leak-off) from

fracture faces to the surrounding porous medium. The mass conservation of the fluid inside

the fracture is governed by Reynold’s lubrication theory and can be expressed by the

continuity equation (Zielonka et al., 2014):

𝜕𝑤

𝜕𝑡+

𝜕𝑞𝑓

𝜕𝑠+ 𝑣𝑡 + 𝑣𝑏 = 0 (3.6)

where 𝑤 is the fracture aperture; 𝑞𝑓 is the longitudinal fluid flow rate in the fracture; 𝑣𝑡

and 𝑣𝑏 are the normal flow velocities through the top and bottom faces of the fracture,

which can be interpreted as fluid leak-off rate from the fracture to the surrounding porous

medium (Wang, 2015; Zielonka et al., 2014).

62

Incompressible and Newtonian fracturing fluid is assumed in this study. The

momentum equation of the tangential flow in the fracture can be expressed as that of a

Newtonian fluid flow between narrow parallel plates:

𝑞𝑓 = −𝑤3

12𝜇𝑓

𝜕𝑝𝑓

𝜕𝑠 (3.7)

where 𝜇𝑓 is the fluid viscosity, 𝑝𝑓 is the fluid pressure inside the fracture.

Fluid leak-off rates or normal fluid velocities are computed as:

𝑣𝑡 = 𝑐𝑡(𝑝𝑓 − 𝑝𝑡) (3.8)

𝑣𝑏 = 𝑐𝑏(𝑝𝑓 − 𝑝𝑏) (3.9)

where 𝑝𝑡 and 𝑝𝑏 are the pore fluid pressure in the porous medium adjacent to the top and

bottom faces of the fracture; 𝑐𝑡 and 𝑐𝑏 are the parameters control the fluid flow across

the top and bottom fracture faces, which is usually referred as “leak-off coefficients”. This

normal flow model can be interpreted as a thin layer of filter cake on the fracture faces,

which increases or reduces effective permeability of the fracture faces (Yao et al., 2010;

SIMULIA, 2016).

Figure 3.5: Schematic of fluid flow in the cohesive fracture (Modified after Zielonka et al.,

2014)

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3.2.3 Formation Rock Deformation and Pore Fluid Flow

The deformation of the porous formation and pore fluid flow in it are modeled using

coupled pore pressure and deformation continuum finite elements. The formation rock is

assumed to be an isotropic and poroelastic medium. With the sign convention used in the

finite-element analysis throughout this dissertation that tension is positive and compression

is negative, the relationship between total stress 𝝈, effective stress 𝝈′ , and pore pressure

𝑝𝑝, can be expressed as (Biot, 1941):

𝝈 = 𝝈′ − 𝛼𝑝𝑝𝑰 (3.10)

where 𝑰 is unit matrix; 𝛼 is the Biot coefficient, which is a material property of the porous

medium.

Stress equilibrium for the solid phase of the porous material is expressed using the

principle of virtual work for the volume under its current configuration (SIMULIA, 2016;

Wang, 2015; Yao et al., 2010):

∫ 𝝈 .. 𝛿𝜺 𝑑𝑉 = ∫ 𝒕 ∙ 𝛿𝒗 𝑑𝑆 + ∫ 𝒇 ∙ 𝛿𝒗 𝑑𝑉

𝑉

𝑆

𝑉 (3.11)

where 𝑉 is the control volume; 𝑆𝜎 is the surface area under surface traction; 𝝈 is the

total stress matrix, 𝛿�̇� is the virtual strain rate matrix; 𝒕 is the surface traction vector; 𝒇

is the body force vector; and 𝛿𝒗 is the virtual velocity vector. This equation is discretized

using a Lagrangian formulation for the solid phase, with displacements as the nodal

variables. The porous medium is thus modeled by attaching the finite element mesh to the

solid phase. Fluid is allowed to flow through these meshes.

Fluid flow should satisfy the continuity equation, which equates the rate of increase

in fluid volume stored at a point to the rate of volume of fluid flowing into the point within

the time increment:

𝑑

𝑑𝑡(∫ 𝜌𝑓

𝑉𝜑𝑑𝑉) = − ∫ 𝜌𝑓𝒏 ∙ 𝒗𝒇𝒑𝑑𝑆

𝑆 (3.12)

64

where 𝜌𝑓 is the density of the pore fluid; 𝜑 is the porosity of the medium; 𝒗𝒇𝒑 is the

average velocity of the pore fluid relative to the solid phase; 𝒏 is the outward normal to

surface 𝑆 . The continuity equation is integrated in time using the backward Euler

approximation and discretized with finite elements using pore pressure as the variable.

The pore fluid flow behavior in the formation in this study is assumed to be

governed by Darcy's law. It can be describe as:

𝒗𝒇𝒑 = −1

𝜑𝑔𝜌𝑓𝒌 ∙ (

𝜕𝑝𝑝

𝜕𝑿− 𝜌𝑓𝒈) (3.13)

where 𝒈 is the gravity acceleration vector; 𝑔 is the gravity acceleration magnitude; 𝒌

is the hydraulic conductivity of the porous medium; 𝑝𝑝 is pore pressure; 𝑿 is a spatial

coordinate vector. As can be seen from Eqs. 3.10 through 3.13, the stress and pore fluid

pressure in the porous medium are nonlinearly coupled with each other to form a control

equation. When it is converted into a weak form of the equivalent integral, it can be solved

by the finite element discretization method.

3.3 LOST CIRCULATION MODEL

This section describes the assumptions, geometries, boundary conditions, and

materials of two finite-element models developed for simulation of static and dynamic fluid

losses without and with circulation of drilling fluid.

3.3.1 Static Fluid Loss Model

Lost circulation in vertical wellbore is simulated in this study as illustrated in Figure

3.1. While drilling, the drilling fluid is pumped into the well through drilling pipe in the

center of the wellbore and returns to the surface through the annulus between the drilling

pipe and wellbore wall. In static state, pump is stopped. Therefore, there is no fluid flow in

the wellbore. The bottom hole pressure (BHP) is solely dependent on the gravity of the

fluid column in the wellbore.

65

3.3.1.1 Model Geometry

The well and formation are assumed to be in a plane-strain condition at a certain

depth, therefore a 2D geometry is used for the model, as shown in Figure 3.6. Owing to

symmetry, only one half of the formation is considered as shown in Figure 3.7.

The size of the modeled formation is 60×20 m. The well depth is 1000 m. The

overburden stress is in the Z-direction perpendicular to the horizontal plane, and the

maximum and minimum horizontal stresses are in the X- and Y- direction respectively in

the horizontal plane. A predefined fracture path is assigned in the middle of the model

perpendicular to the direction of the minimum horizontal stress. The formation is modeled

as an isotropic, poroelastic material, using coupled pore pressure and deformation

continuum finite elements; the fracture is modeled using a layer of pore pressure cohesive

elements; and the well is model with pipe elements. Since significant stress/displacement

gradients are expected in the wellbore vicinity, the mesh is refined around the wellbore.

Figure 3.6: Schematic configuration of well and formation. The formation is in a plane-

strain condition.

66

Figure 3.7: Static lost circulation model.

3.3.1.2 Boundary Conditions

A symmetric boundary condition is defined on the left edge of the model as shown

in Figure 3.7. The formation is assumed to be at a depth of 1000 m with a normal pore

pressure of 10 MPa, a minimum horizontal stress of 13 MPa and a maximum horizontal

stress of 15 MPa. The minimum and maximum horizontal stresses and undisturbed pore

pressure (10 MPa) are applied on the outer boundaries of the model, as shown in Figure

3.7. Initial pore pressure of 10 MPa is applied to the whole formation.

It is also assumed that the well is always filled up with drilling fluid even after fluid

loss occurs. Gravity force is applied to the fluid in the wellbore. The pressure at the

wellhead is equal to atmospheric pressure (assumed to be zero in this study since its small

value compared with fluid pressure in the wellbore).

The pipe element at the end of the wellbore is tied to the formation and fracture

elements on the wellbore wall to make sure the fluid pressure in the bottom hole is equal

67

to the pore pressure on the wellbore wall. A dynamic pressure that equals the bottom-hole

fluid pressure is imposed onto the inner wellbore wall to model the pushing pressure

applied by fluid column. Besides, the tie constraint enforces the fluid conservation and

equilibrium between the wellbore and formation before facture initiation. When fracture

initiates, the tie constraint handles the fluid conservation and equilibrium between the

wellbore, the formation, and the fracture.

3.3.1.3 Material Properties

The formation rock is modeled as a poroelastic material. It is assumed to be a

sandstone formation with a high permeability of 100 mD. Drilling fluids with different

densities are used to investigate its effect on static fluid loss. Table 3.1 summarizes all the

material parameters used for the simulations.

Table 3.1: Material properties of the static fluid loss model.

Parameters Values Units

Young’s modulus 7000 MPa

Poisson’s ratio 0.2

Fluid density 1.0, 1.2, 1.4, 1.5, 1.6 g/cm3

Fluid viscosity 1 cp

Porosity 0.25

Permeability 100 mD

Tensile strength 0.4 MPa

Critical fracture energy 28 J/m2

3.3.2 Dynamic Fluid Loss Model

The objective of the dynamic fluid loss model is to model lost circulation in vertical

wellbores with drilling fluid circulation. While drilling, the drilling fluid is pumped into

the well through drilling pipe in the center of the wellbore and returns to the ground through

the annulus between the drilling pipe and well wall. The BHP or ECD during this process

68

is dependent not only on the gravity of the fluid column but also on the viscous loss in the

annulus.


The geometry of the dynamic fluid loss model is very similar to the static fluid loss

model as describe in Section 3.3.1.1. The only difference is that the well is modeled as a

“U-tube” configuration, rather than as a single tube as defined in the static model. The

drilling pipe is modeled using fluid pipe connector element in Abaqus® in order to simulate

fluid pumping into and flow through it. The wellbore annulus is modeled using fluid pipe

element in Abaqus®. And they are connected to each other at the bottom of the wellbore as

shown in Figure 3.8.

Figure 3.8: Dynamic fluid loss model.

69


It is assumed that the drilling pipe and annulus are filled up with drilling fluid

during circulation. The pressure at the annulus head is equal to atmospheric pressure

(assumed to be zero in this study due to its small magnitude compared with fluid pressure

in the wellbore and formation). A fluid pumping rate is applied at the top of drilling pipe.

The drilling pipe is connected to the annulus at the bottom of the wellbore. Since the

drilling fluid is assumed to be incompressible, the fluid flow rate into the annulus at the

wellbore bottom is equal to the pump rate. Because BHP is only controlled by the fluid in

the annulus, therefore fluid gravity and viscous pressure loss are only considered for this

part of fluid. The other boundary conditions are the same as those defined in the static fluid

loss model in Section 3.3.1.2.

3.3.2.3 Input Parameters

As aforementioned, BHP or ECD is affected by the gravity pressure of the fluid as

well as the viscous pressure loss due to fluid flow in the annulus. At a certain depth, the

gravity pressure of the fluid is only determined by the density of drilling mud, while the

viscous pressure loss is influenced by several factors, including mud density, mud

viscosity, flow (pump) rate, and the size/clearance of the annulus. In order to investigate

the effects of these factors on lost circulation, different fluid properties, pump rates and

annulus clearances are used in this study as summarized in Table 3.2. The other parameters

used in the simulations are the same as those given in Section 3.3.1.3.

70

Table 3.2: Input parameters for the dynamic fluid loss model.

Parameters Values Units

Mud Density 1.0, 1.1*, 1.2, 1.3, 1.35, 1.4, 1.5 g/cm3

Mud Viscosity 1*, 10, 50, 100, 500 cp

Pump Rate 0.36*, 0.42, 0.48, 0.54, 0.60 m3/min

Wel

l C

onfi

gura

tion Drilling Pipe

Radius 5* 5 5 5 cm

Wellbore Radius 10* 9 8 7 cm

Annulus Clearance 5* 4 3 2 cm

Cross-section Area 235.6* 175.9 122.5 75.3 cm2

Hydraulic Diameter 10* 8 6 4 cm

Note: Values with * are the inputs for the base case of the simulations discussed in the flowing

section.

3.4 FLUID LOSS SIMULATION RESULTS

Using the lost circulation simulation framework described above, fluid losses in

static and dynamic states without and with well circulation are simulated and analyzed

under various conditions. This section summarizes the analysis results.

3.4.1 Static Fluid Loss

For a formation at a certain depth, BHP and thus lost circulation are only dependent

on the drilling mud density under static state with mud circulation. Therefore, different

mud densities are used in the simulations to investigate its effect on static fluid loss.

Figures 3.9 and 3.10 show the fluid loss rates and BHP with different drilling mud

densities. The following conclusions are made based on the simulation results.

With a mud density of 1.0 𝑔 𝑐𝑚3⁄ , the BHP is equal to the pore pressure (10 MPa)

in the formation (see Figure 3.10), therefore no fluid loss occurs, i.e. the fluid loss

rate is zero as shown in Figure 3.9.

71

When the mud density increases to 1.2 or 1.4 𝑔 𝑐𝑚3⁄ , the BHP becomes larger

than pore pressure and there is a very small fluid loss rate. The simulation results

do not show occurrence of fractures for this two cases, therefore this small amount

of fluid loss is due to fluid seepage from wellbore wall to the formation, rather than

fluid flow into fractures.

When the mud density is equal or higher than 1.5 𝑔 𝑐𝑚3⁄ , fracture occurs. It can

be seen that there is a rapid increase in the loss rate at the early time due to the

initiation and propagation of the fractures. But finally it approaches a relatively

constant value. As expected, it is observed from Figure 3.9 that the larger the mud

density, the higher the fluid loss rate. Figure 3.10 shows that the BHP declines

rapidly with the creation of the fractures and finally approaches a constant fracture

propagation pressure which is even smaller than the BHP with 1.4 𝑔 𝑐𝑚3⁄ mud.

It is also observed that the larger the mud density, the faster the BHP drops,

meaning the faster of the fracture creation at the early time.

72

Figure 3.9: Fluid loss rate with different mud densities in the static fluid loss model.

Figure 3.10: BHP with different mud densities in the static fluid loss model.

73

Figure 3.11 displays the lost circulation fracture geometry with different mud

densities at the end of the simulation (100s). With a mud density of 1.4 𝑔 𝑐𝑚3⁄ , no fracture

happens. And then, increasing the mud density by 0.1 𝑔 𝑐𝑚3⁄ results in significant

fracture growth. With the continuous increase of mud density, both of the fracture width

and fracture length increase.

Figure 3.11: Lost circulation fracture geometry with different mud densities in the static

fluid loss model.

3.4.2 Dynamic Fluid Loss

The numerical model framework developed in this chapter allows for a unique new

way of modeling fractures while drilling fluid circulation as “dynamic-pressure-driven”

fractures. Given the conditions of formation depth, rock properties, wellbore

configurations, mud properties and pump schedules, the model can capture the dynamic

loss rate, reduction of return circulation, BHP, and fracture geometry with dynamic fracture

growth. The analysis of the fluid loss and fracture propagation using the simulation

74

framework can help us understand how to prevent lost circulation, optimize mud rheology,

and select LCMs.

3.4.2.1 Effect of Mud Density

Figures 3.12, 3.13 and 3.14 show return circulation rate, fluid loss rate and BHP

while circulating drilling fluids with different densities. The following observations can be

reached from the simulation results.

With a mud density of 1.0 𝑔 𝑐𝑚3⁄ , while the BHP is equal to formation pressure

(10 MPa) without fluid circulation in the static state as shown in Figure 3.10, it is

not equal to formation pressure anymore in the dynamic model with fluid

circulation as shown in Figure 3.14. The BHP with fluid circulation is about 2.2

MPa higher than the hydrostatic formation pressure. This increase of BHP is caused

by the viscous loss due to fluid flow in the annulus. No fracture occurs in this case.

However, because of the pressure difference between BHP and formation pore

pressure, a slow fluid loss occurs as evidenced by the slight reduction in the return

circulation rate and the small fluid loss rate in Figures 3.12 and 3.13 respectively.

When the mud density increases to 1.2 𝑔 𝑐𝑚3⁄ , the BHP becomes larger. But no

fracture occurrence is observed. There is a small amount of fluid loss due to seepage

through wellbore wall.

With a mud density of 1.3 𝑔 𝑐𝑚3⁄ , fracture occurs with a dramatic reduction in the

return circulation rate and increase in the fluid loss rate. The BHP also drops with

fracture creation. Partial loss occurs in this case since the fluid loss rate is smaller

than the pump rate. Recall that for the static model described in the above section,

fracture does not occur until the mud density is increased to 1.5 𝑔 𝑐𝑚3⁄ .

75

With continuous increase of mud density to 1.35, 1.4 and 1.5 𝑔 𝑐𝑚3⁄ , total losses

are observed. It can be seen from Figures 3.12 and 3.13 that the return circulation

rates drop to negative values and the loss rates become larger than the pump rate,

which means no fluid returns to the ground surface through the annulus, rather the

fluid in the annulus will flow backward into the fractures. Since the annulus is

assumed always filled with fluid, the decline of fluid level in the annulus is not

considered. As expected, the larger the fluid density, the larger the fluid loss rate is

observed. Figure 3.14 shows that the BHP declines rapidly with the creation of the

fractures and finally approaches a constant fracture propagation pressure of 13.5

MPa.

Figure 3.12: Return circulation rate with different mud densities in the dynamic fluid loss

model.

76

Figure 3.13: Fluid loss rate with different mud densities in the dynamic fluid loss model.

Figure 3.14: BHP with different mud densities in the dynamic fluid loss model.

77

Figure 3.15 displays the fracture geometry with different mud densities after a

circulation period of 100s. With a mud density of 1.2 𝑔 𝑐𝑚3⁄ , no fracturing happens.

Increasing the mud density to 1.3 𝑔 𝑐𝑚3⁄ results in significant fracture growth. However,

in the static model without fluid circulation, fracture begins to occur at a much higher mud

density of 1.5 𝑔 𝑐𝑚3⁄ . With the continuous increase of mud density, both of the fracture

width and fracture length increase.

Figure 3.15: Lost circulation fracture geometry with different mud densities in the dynamic

fluid loss model.

Figure 3.16 compares the fracture geometry in the static loss case and dynamic loss

case with the same mud density of 1.5 𝑔 𝑐𝑚3⁄ at the same time of 100s. It is obvious that

considering fluid circulation and the corresponding viscous loss in the annulus can result

in significantly larger fracture size, even though the mud density is the same. Therefore, it

is important to take into account the dynamic circulation effect on lost circulation

prediction and prevention.

78

Figure 3.16: Comparison of fracture geometry between the static and dynamic fluid loss

models.

3.4.2.2 Effect of Mud Viscosity

As described in Section 3.2, drilling mud viscosity is one of the dominating factors

for viscous pressure loss. Therefore, it can influence BHP and ECD, and hence lost

circulation, while drilling. In this section, dynamic fluid loss and fracture geometry for

mud with different viscosities are analyzed. A constant mud density of 1.1 𝑔 𝑐𝑚3⁄ is used

in this section.


while circulating drilling fluids with different viscosities.

With a mud viscosity of 1cp, the dynamic BHP with fluid circulation is about 2.3

MPa higher than the static BHP (11 MPa with a mud density of 1.1 𝑔 𝑐𝑚3⁄ ) as

shown in Figure 3.18. This increase of BHP is caused by the viscous pressure loss

in the annulus. When the mud viscosity is increased to 10cp, the BHP increases to

about 3.2 MPa above the static BHP due to the increased viscous loss. No fracture

79

occurs in this two cases. However, because of the pressure difference between BHP

and formation pore pressure, a slow fluid loss occurs as evidenced by the slight

reduction in the return circulation rate and the small flow loss rate in Figures 3.17

and 3.18 respectively.

For the cases with a mud viscosity of 50, 100 and 500 cp, fracture occurs with a

dramatic reduction in the return circulation rate and increase in the fluid loss rate

as shown in Figures 3.17 and 3.18. Figure 3.19 shows that the BHP declines rapidly

with the creation of the fracture and finally approaches a constant fracture

propagation pressure of 13.5 MPa. Partial losses occur in these cases since the fluid

loss rate is smaller than the pump rate. Fluid loss rate increases with the increase of

mud viscosity due to the larger viscous loss in the annulus and thus higher ECD at

the bottom hole.

Figure 3.17: Return circulation rate with different mud viscosities in the dynamic fluid

loss model.

80

Figure 3.18: Fluid loss rate with different mud viscosities in the dynamic fluid loss

model.

Figure 3.19: BHP with different mud viscosities in the dynamic fluid loss model.

81

Figure 3.20 displays the fracture geometry with different mud viscosities after a

circulation period of 100s. With a mud viscosity of 10 cp, no fracturing occurs. Increasing

the mud viscosity to 50 cp results in clear fracture growth. With the continuous increase of

mud viscosity, both of the fracture width and fracture length increase due to the increased

ECD in the bottom annulus.

Figure 3.20: Lost circulation fracture geometry with different mud viscosities in the

dynamic fluid loss model.

3.4.2.3 Effect of Pump Rate

Pump rate dominates the fluid flow velocity in the annulus which is another control

factor for viscous loss. Increasing pump rate will lead to increased BHP or ECD in the

annulus associated with the increase of viscous loss. In this section, the dynamic fluid loss

and fracture geometry with different pump rates are analyzed.


while circulation with different pump rates.

82

With a pump rate of 0.36 𝑚3 𝑚𝑖𝑛⁄ , the BHP is around 13.3 MPa which is 2.3 MPa

higher than the static BHP as shown in Figure 3.23. This increase of BHP is caused

by the flow viscous loss in the annulus. When the pump rate is increased to 0.42

𝑚3 𝑚𝑖𝑛⁄ , the BHP increases to 14.2 MPa due to the increased viscous loss. No

fracture occurs in this two cases. However, because of the pressure difference

between BHP and formation pressure, a slow fluid loss occurs as evidenced by the

slight reduction in the return circulation rate and the small flow loss rate in Figures

3.21 and 3.22 respectively.

For the cases with pump rates of 0.48, 0.54 and 0.60 𝑚3 𝑚𝑖𝑛⁄ , fracture occurs with

a dramatic reduction in the return circulation rate and increase in the fluid loss rate

as shown in Figures 3.21 and 3.22. Partial fluid losses occur in these cases. The

return circulation rates approach to a similar value as fracture grow for this three

cases with different pump rates. However, the loss rate increases as the pump rate

increases. Figure 3.23 shows that the BHP declines rapidly with the creation of the

fractures and finally approaches a constant fracture propagation pressure of 13.5

MPa for this three cases.

83

Figure 3.21: Return circulation rate with different pump rates in the dynamic fluid loss

model.

Figure 3.22: Fluid loss rate with different pump rates in the dynamic fluid loss model.

84

Figure 3.23: BHP with different pump rates in the dynamic fluid loss model.

Figure 3.24 displays the fracture geometry with different pump rates at the end of

a circulation period of 100s. With a pump rate of 0.42 𝑚3 𝑚𝑖𝑛⁄ , no fracture occurs.

Increasing the pump rate to 0.48 𝑚3 𝑚𝑖𝑛⁄ results in significant fracture growth. With the

continuous increase of pump rate, both of the fracture width and fracture length increase

due to the increase of viscous loss in the annulus and ECD in the bottom hole.

85

Figure 3.24: Lost circulation fracture geometry with different pump rates in the dynamic

fluid loss model.

3.4.2.4 Effect of Annulus Clearance

Changing the annulus clearance between the drilling pipe and wellbore wall lead to

changes in flow velocity and hydraulic diameter of the flow conduit, which can further

influence the viscous loss as described in Section 3.2. Generally, smaller annulus clearance

has larger flow velocity and smaller hydraulic diameter, and therefore results in larger

viscous loss according to Eqs. 3.1 and 3.2. In this section, dynamic fluid loss and fracture

geometry with different annulus clearances as given in Table 3.2 are analyzed based on the

simulation results.


while circulation with different annulus clearances. The other input parameters are the

same as those given in Table 3.2.

With a relatively larger annulus clearance of 5cm between the drilling pipe and

wellbore wall, no fracture occurs. However, because of the pressure difference

86

between BHP and formation pressure, a slow fluid loss occurs as evidenced by the

slight reduction in the return circulation rate and the small flow loss rate in Figures

3.25 and 3.26 respectively.

For the cases with an annulus clearance of 4, 3 and 2 cm, fracture occurs due to the

increased viscous loss or ECD in the annulus. Significant reduction in the return

circulation rate and increase in the fluid loss rate are observed as shown in Figures

3.25 and 3.26. Partial losses occurs in these cases. The smaller the annulus

clearance, the smaller the return circulation rate and the larger the fluid loss rate.

Figure 3.27 shows that the BHP declines rapidly with the creation of the fractures

and finally approaches a constant fracture propagation pressure of 13.5 MPa for

this three cases.

Figure 3.25: Return circulation rate with different annulus clearances in the dynamic fluid

loss model.

87

Figure 3.26: Fluid loss rate with different annulus clearances in the dynamic fluid loss

model.

Figure 3.27: BHP with different annulus clearances in the dynamic fluid loss model.

88

Figure 3.28 displays the fracture geometry with different annulus clearances at the

end of a circulation period of 100s. With a relative larger annulus clearance of 5 cm, no

fracturing occurs. Decreasing the annulus clearance to 4 cm results in significant fracture

growth. With the continuous decrease of annulus clearance, both of the fracture width and

fracture length increase due to the increase of viscous loss in the annulus.

Figure 3.28: Lost circulation fracture geometry with different annulus clearances in the

dynamic fluid loss model.

3.5 SUMMARY

In this chapter, a finite-element framework is developed which allows predicting

the dynamic fluid loss and fracture geometry during lost circulation in drilling process. It

can be used to simulate fluid loss in static and dynamic state without and with drilling fluid

circulation in the wellbore. The model successfully couples the fluid circulation in the

wellbore, fluid seepage on wellbore wall, fracture propagation, fluid flow in fracture, pore

fluid flow and deformation of formation rock during fluid loss through induced fractures.

89

The fluid flow in the wellbore is modeled based on the Bernoulli’s equation taking into

account the viscos loss. Fracture propagation and fluid flow in the fracture are modeled

based on a pore pressure cohesive zone method.

Factors that can affect ECD and thus lost circulation, including mud density, mud

viscosity, pump rate and annulus clearance, are investigated using the proposed model. The

results show that the viscous pressure loss term due to fluid circulation in the annulus can

lead to significant ECD increase and fluid loss. Drilling mud with relatively low density

which does not cause lost circulation in static state without circulation may lead to

significant fluid loss after resuming mud circulation. So it is important to take into account

the dynamic circulation effect on lost circulation prediction and prevention.

The numerical framework provides a unique new way to model lost circulation in

drilling when the boundary condition at the fracture mouth is neither a constant flowrate

into the fracture nor a constant pressure, but rather a dynamic BHP. In drilling operations,

we are interested in preventing fractures from occurring by controlling BHP/ECD, or

plugging the fractures at the early time of their growth using LCMs, so the capability of

capturing the dynamic fluid loss and fracture geometry development of the proposed

framework can help us understand how to prevent lost circulation, optimize mud rheology,

and select LCMs.

90

CHAPTER 4: Modeling Study of Preventive Wellbore Strengthening

Treatments: The Role of Mudcake

Preventive wellbore strengthening treatment is to add additives, such as LCMs, to

drilling mud to build a layer of low-permeability mudcake on wellbore wall which can

inhibit fracture creation and increase the sustainable pressure of the wellbore. Mudcake

buildup on wellbore wall plays an important role on preventing lost circulation. However,

the time-dependent mudcake buildup and properties have been plaguing the drilling

industry for years. In this chapter, an analytical model is first derived to quantify the effects

of mudcake thickness, permeability and strength on wellbore stresses and fracture pressure.

Steady-state fluid flow is assumed for the analytical model, so it does not consider the time-

dependent effects. For a further step to take into account the time-dependent effects, a

finite-element model is developed to simulate the evolution of near-wellbore stresses and

pore pressure with dynamic mudcake buildup and property (permeability) change while

drilling based on poroelastic theory using Abaqus® and FORTRAN® subroutines.

91

4.1 INTRODUCTION

As the bit drills into a new permeable formation, fresh rock surface is exposed to

drilling mud. The wellbore pressure is usually higher than pore pressure in the surrounding

formation. Under the pressure difference, liquid component in the drilling mud may

seepage into the formation, while some solid mud components may be filtered out onto the

wellbore surface in this process. The filtered solid components can form a thin layer of

low-permeability cake on wellbore wall which is usually called mudcake or filtercake. The

buildup and properties of the mudcake depend on a number of factors such as drilling mud

constituent, wellbore pressure and temperature, pore pressure, formation permeability and

porosity, annulus flow regimes, and time.

Extensive experimental work in the literature has revealed the importance of

the mudcake in inhibiting fracture growth and preventing lost circulation (Cook et al.,

2016; Salehi et al., 2016; Salehi and Kiran, 2016). Researchers have found that adding

additives such as LCMs to drilling mud to build a layer of low-permeability mudcake can

enhance the effective strength of the wellbore (Song and Rojas, 2006; Soroush et al., 2006;

Sweatman et al., 2004). While some researchers insisted that wellbore strengthening is

achieved by bridging fractures on wellbore to increase the wellbore hoop stress (Alberty

and McLean, 2004; Aston et al., 2004b; Dupriest, 2005), other researchers have argued that

similar wellbore strengthening results can arise by build a low-permeability mudcake on

wellbore wall to alter the effective stresses on and around wellbore (Abousleiman et al.,

2007; Tran et al., 2011). Therefore, preventive wellbore strengthening technique based on

plastering the wellbore wall with mudcake before lost circulation occurs has been widely

used and proven to be very effective in the drilling industry, especially for preventing lost

circulation problems in depleted sandstone formations with relatively high permeability.

92

However, preventive wellbore strengthening treatment is a time-dependent

process, because not only the fluid flow in the porous rock around the wellbore is dependent

on time, but also the thickness and physical properties of mudcake are functions of time.

The time-dependent mudcake thickness and properties can significantly affect fluid flow

and therefore stress state around wellbore. This complex feature of mudcake has been

remaining a challenging problem in mudcake modeling in the drilling industry.

In this chapter, an analytical model assuming steady-state fluid flow around

wellbore and thus without considering the time-dependent effects is first derived to

quantify the effects of mudcake thickness, permeability, and strength on pore pressure and

effective stresses in the vicinity of the wellbore. Next, in order to take into account the

time-dependent effects, a finite-element model is developed to simulate the evolution of

near-wellbore stresses and pore pressure with time-dependent fluid flow and dynamic

mudcake thickness buildup and permeability reduction, based on poroelasticity theory

using Abaqus® and FORTRAN® subroutines.

4.2 AN ANALYTICAL MUDCAKE MODEL

4.2.1 Modeling Assumptions

In this section, an analytical mudcake model is derived which takes into account

the effects of thickness, permeability and strength of mudcake on near-wellbore pore

pressure and stress states, and thus fracture pressure of wellbore. Before deriving the

model, the following assumptions are introduced:

1. The wellbore, mudcake, and formation rock are in a plane-strain condition as

depicted in Figure 4.1.

2. The outer formation boundary has a constant pore fluid pressure 𝑃𝑒; the wellbore

is under a constant mud pressure 𝑃𝑖; and 𝑃𝑖 is higher than 𝑃𝑒.

93

3. The fluid flow from wellbore to formation is in steady state and obeys Darcy’s flow.

4. The formation rock is isotropic, homogeneous and poroelastic material.

5. The mudcake is very soft/flexible compared to the rock and has a very low yield

strength; therefore it undergoes perfectly plastic yielding under wellbore pressure;

its Poisson’s ratio for post-yield deformation is assumed to be 0.5.

6. Mudcake thickness and properties do not change with time.

7. The wellbore radius (outer mudcake radius) is 𝑅𝑜; the inner mudcake radius is 𝑅𝑖;

the outer formation radius is 𝑅𝑒 ; the mudcake thickness is 𝑤 = 𝑅𝑜 − 𝑅𝑖 ; the

mudcake permeability is 𝐾1; and the formation permeability is 𝐾2.

8. The maximum and minimum far-field total stresses are 𝜎𝐻 and 𝜎ℎ, respectively.

9. The sign convention for stress in this analytical study is that compression is positive

and tension is negative.

Figure 4.1: Schematic of the cross section of wellbore, mudcake, and formation.

94

4.2.2 Wellbore Stress and Fracture Pressure

Since there is fluid flow from the wellbore to formation, the pore pressure varies

with radial distance. In this problem, three components contributes to the stress around

wellbore: (1) wellbore pressure and field stresses; (2) varying pore pressure distribution

induced by fluid flow; and (3) the contribution of the plastic mudcake. For deriving the

solution, we assume these terms are uncoupled, so the final expressions can be obtained by

a superposition of the three problems. The following subsections describe the detailed

derivation process.

4.2.2.1 Total Stress Induced by Varying Pore Pressure

Assuming the pore pressure at wellbore wall (interface between mudcake and

formation) is 𝑃𝑜, then according to Darcy’s law for radial flow, one can have

1 22 2

ln lno i e o

o e

i o

K KP P P P

R R

R R

(4.1)

Solving this equation, one can get the pore pressure at the wellbore wall

2

1 2

ln

ln ln

o

io i i e

e o

o i

RK

RP P P P

R RK K

R R

(4.2)

Introduce

o i eP P P (4.3)

2

1 2

ln

ln ln

o

i

e o

o i

RK

RB

R RK K

R R

(4.4)

Eq. 4.2 can be simplified as

o i oP P B P (4.5)

95

Next, we want to determine the pore pressure in the formation around wellbore.

Define 𝑃𝑟 is the pore pressure in the formation at distance 𝑟 from wellbore center,

therefore 𝑅𝑜 ≤ 𝑟 ≤ 𝑅𝑒. The following equation can be written from Darcy’s law

2 22 2

ln lnr o e r

e

o

K KP P P P

r R

R r

(4.6)

Inserting the pore pressure at wellbore wall, i.e. Eq. 4.2, into Eq. 4.6, pore pressure at

distance 𝑟 can be obtained

2

1 2

ln ln1

ln

ln ln ln

o e

ir i o

e e oo

o o i

R RK

R rrP P P

R R RRK K

R R R

(4.7)

Define

2

1 2

ln ln1

ln

ln ln ln

o e

i

e e oo

o o i

R RK

R rrM

R R RRK K

R R R

(4.8)

Eq. 4.7 can be simplified to

r i oP P M P (4.9)

or

(1 )r e oP P M P (4.10)

Total stress induced by varying pore pressure around wellbore due to fluid flow can

be calculated by (Fjar et al., 2008)

2 2' ' ' ' ' '

, 2 2 2

2 2' ' ' 2 ' ' '

, 2 2 2

' ' '

z, 2 2

2

2

4

e

o o

e

o o

e

o

r Ro

r pR R

e o

r Ro

pR R

e o

R

pR

e o

r Rr P r dr r P r dr

r R R

r Rr P r dr r P r r P r dr

r R R

vr P r dr

R R

(4.11)

96

where, 𝜎𝑟,𝑝 , 𝜎𝜃,𝑝 and 𝜎𝑧,𝑝 are the total radial, tangential and axial stresses around

wellbore induced by varying pore pressure with fluid flow; ∆P(𝑟) = 𝑃𝑟 − 𝑃𝑒 ; 𝜂 is the

poroelastic stress coefficient; and 𝑣 is Poisson’s ratio. Using Eq. 4.7, ∆𝑃(𝑟) can be

expressed as

2

1 2

ln1

1 ln ln

ln ln ln

o

i eo

e e oo

o o i

RK

R RrP r P

R R RR rK K

R R R

(4.12)

Define

1

ln e

o

AR

R

(4.13)

and combine Eq. 4.4, ∆𝑃(𝑟) can be written as

1 ln ln eo

o

RrP r A AB P

R r

(4.14)

Let

' ' '

1o

r

RI r P r dr (4.15)

' ' '

2

e

o

R

RI r P r dr (4.16)

Inserting Eq. 4.14 into Eqs. 4.15 and 4.16, one can get '

' '

1 '1 ln ln

o

re

oR

o

RrI r A AB P dr

R r

(4.17)

'' '

2 '1 ln ln

e

o

Re

oR

o

RrI r A AB P dr

R r

(4.18)

After integration, 𝐼1 and 𝐼2 are obtained as

'2 ''2 '2

1 '

1 12ln 1 2ln 1

2 4 4oo o

rr r

eo

o RR R

Rr rI A r AB r P

R r

(4.19)

97

'2 ''2 '2

2 '

1 12ln 1 2ln 1

2 4 4

ee e

oo o

RR R

eo

o RR R

Rr rI A r AB r P

R r

(4.20)

Inserting Eqs. 4.19 and 4.20 into Eq. 4.11, total stresses around wellbore induced by

varying pore pressure caused by fluid flow can be determined as

2 2

, 1 22 2 2

2 22

, 1 22 2 2

z, 22 2

2

2

4

or p

e o

op

e o

p

e o

r RI I

r R R

r RI r P r I

r R R

vI

R R

(4.21)

4.2.2.2 Total Stress Induced by Wellbore Pressure, Far-field Stresses and Plastic

Mudcake

The total stress around wellbore induced by pressure at wellbore wall and far-field

stresses can be obtained using Kirsch Equation as

2 4 2 2

, 2 4 2 2

2 4 2

, 2 4 2

2

, 2

1 1 3 4 cos 22 2

1 1 3 cos 22 2

2 cos 2

H h o H h o o or s iw

H h o H h o os iw

oz s v H h

R R R RP

r r r r

R R RP

r r r

R

r

(4.22)

where 𝜎𝑟,𝑠 , 𝜎𝜃,𝑠 and 𝜎𝑧,𝑠 are the total radial, tangential and axial stresses around

wellbore induced by far-field stresses and pressure at wellbore wall; 𝜃 is the

circumferential angle to the direction of 𝜎𝐻. Note that in this equation, 𝑃𝑖𝑤 is the pressure

acting on wellbore wall; it is not equal to fluid pressure 𝑃𝑖 acting on the inner surface of

mudcake. 𝑃𝑖𝑤 is influenced by both the fluid flow through the mudcake and the plastic

flow of the mudcake.

From classical mechanics, the mudcake must satisfy the equilibrium condition

98

0rrd

dr r

(4.23)

The stress-strain relationship for the mudcake is

r r z

r z

z z r

E v

E v

E v

(4.24)

Due to its very soft/flexible feature, mudcake can be assumed to be in perfectly

plastic yielding condition with very low yield strength under the wellbore pressure (Tran

et al., 2011). The Poisson’s ratio for mudcake plastic flow is assumed to be 0.5. The

mudcake is considered in plane-strain condition, i.e. 휀𝑧 = 0. Then from the last equation

of Eq. 4.24, one can get

0.5z r (4.25)

According to the von Mises yield theory (also known as maximum distortion

energy theory), when mudcake yields, the following expression should be satisfied

(Aadnøy and Belayneh, 2004)

2 2 21

2r r z zY

(4.26)

where 𝑌 is the yield strength of the mudcake.

Inserting Eq. 4.25 into Eq. 4.26, one can get

3

2rY (4.27)

Inserting Eq. 4.27 into the equilibrium equation (Eq. 4.23), one obtains the radial stress

distribution

2

ln3

r

Yr C (4.28)

where C is an integration constant.

Boundary condition at the inner mudcake surface is

99

r i

i

P

r R

(4.29)

Applying this boundary condition into Eq. 4.28, the integration constant can be determined

as

2

ln3

i i

YC P R (4.30)

Inserting Eq. 4.30 into Eq. 4.28, radial stress distribution in the mudcake can be

determined as

2ln

3r i

i

Y rP

R

(4.31)

where 𝑅𝑖 ≤ 𝑟 ≤ 𝑅𝑜.

On the wellbore wall 𝑟 = 𝑅𝑜, the radial stress is

2( ) ln

3

or o i

i

RYR P

R

(4.32)

Because in most cases the mudcake is extremely soft/flexible compared to the rock

formations, it can be assumed that the mudcake does not exert any shear tractions on the

wellbore wall (Tran et al., 2011), but it can transmit the radial stress exerted by drilling

fluid pressure on the inner surface of mudcake. The radial stress on the inner surface of

mudcake is equal to drilling fluid pressure 𝑃𝑖, while the radial stress on the wellbore wall

is given by Eq. 4.32. By simply replacing 𝑃𝑖𝑤 in the Kirsch solution (Eq. 4.22) with

𝜎𝑟(𝑅𝑜) in Eq. 4.32, the total stress induced by wellbore pressure, far-field stress and

plasticity of mudcake can be determined.

4.2.2.3 Total Stress around Wellbore

Replacing 𝑃𝑖𝑤 in Kirsch solution (Eq. 4.22) with 𝜎𝑟(𝑅𝑜) given by Eq. 4.32 and

adding the stress terms induced by varying pore pressure caused by fluid flow (Eq. 4.21),

one can get the total stress distribution around wellbore with the presence of mudcake

100

2 4 2 2

,2 4 2 2

2 4 2

,2 4 2

2

z,2

21 1 3 4 cos 2 ln

2 2 3

21 1 3 cos 2 ln

2 2 3

2 cos 2

H h o H h o o o or i r p

i

H h o H h o o oi p

i

oz v H h

R R R R RYP

r r r R r

R R R RYP

r r R r

R

r

p

(4.33)

Total stress at the wellbore wall (mud-rock interface) 𝑟 = 𝑅𝑜 is

,

,

z,

2ln

3

22 cos 2 ln

3

2 cos 2

or i r p

i

oH h H h i p

i

z v H h p

RYP

R

RYP

R

(4.34)

The minimum total tangential stress occurs at 𝜃 = 0 𝑜𝑟 𝜋 on wellbore wall, i.e.

,min ,

23 ln

3

oh H i p

i

RYP

R

(4.35)

Rewrite this equation as

,min , ,3 h H i pl pP (4.36)

with

,

2ln

3

opl

i

RY

R

(4.37)

is the total tangential stress induced by plastic deformation of mudcake, and 𝜎𝜃,𝑝 is the

total tangential stress induced by fluid flow (varying pore pressure).

Regarding the typical wellbore radius 𝑅𝑜 in petroleum applications is between 0.1

and 0.13 m (Tran et al., 2011) and the typical mudcake thickness is between 0.002 and

0.006 m (Bezemer and Havenaar, 1966; Chenevert and Dewan, 2001; Griffith and

Osisanya, 1999; Sepehrnoori et al., 2005), i.e. 𝑅𝑖 is between 0.094 and 0.098 m, the value

of the term 𝑙𝑛(𝑅𝑜 𝑅𝑖⁄ ) is very small. Since the mudcake yield strength 𝑌 is also usually

101

very small (Cerasi et al., 2001; Cook et al., 2016), we can conclude from Eq. 4.37 that the

effect of mudcake strength on the wellbore tangential stress should be very small.

4.2.2.4 Effective Stress around Wellbore

Effective stress on wellbore wall can be determined by subtracting pore pressure

(Eq. 4.2) from the total stresses (Eq. 4.34) on wellbore wall and expressed as

'

,

'

,

'

z,

2ln

3

22 cos 2 ln

3

2 cos 2

or i r p o

i

oH h H h p o

i i

z v H h p o

RYP P

R

RYP P

R

P

(4.38)

The minimum effective tangential stress on wellbore wall can be found at 𝜃 =

0 𝑜𝑟 𝜋 as

'

,min ,

23 ln

3

oh H i p o

i

RYP P

R

(4.39)

or '

,min , ,3 h H i pl p oP P (4.40)

4.2.2.5 Fracture Pressure

Fracture pressure is determined based on a tensile failure criterion that fracture

occurs when the minimum effective stress on wellbore wall reaches the tensile strength of

the rock. Usually tensile strength of sedimentary rock is very small, so in this study we

simply ignore the tensile strength of the formation. Therefore, fracture occurs when

𝜎𝜃,𝑚𝑖𝑛′ = 0. 𝜎𝜃,𝑚𝑖𝑛

′ is given by Equation 4.39. Pore pressure 𝑃𝑜 at wellbore wall in Eq.

4.39 is given by Eq. 4.2. However, we still need to determine the fluid flow induced

tangential stress term 𝜎𝜃,𝑝 in Eq. 4.39. This term can be obtained from Eq. 4.21 by setting

𝑟 = 𝑅𝑜 as

102

2

2

, 1 22 2 2

22 op o o o o o

o e o

RR I R R P R I R

R R R

(4.41)

where, 𝐼1(𝑅0), 𝐼2(𝑅0), and ∆𝑃(𝑅0) can be determined from Eqs. 4.19, 4.20, and 4.12

respectively as

1 0oI R (4.42)

2 2

2 2 2 2

2

1 1 1 12ln 1 2ln 1

2 4 4 4 4

e o e eo e o e o o

o o

R R R RI R A R R AB R R P

R R

(4.43)

1 ln eo o

o

RP R AB P

R

(4.44)

Define

2 2' 2 2 2 21 1 1 1

2ln 1 2ln 12 4 4 4 4

e o e ee o e o

o o

R R R RM A R R AB R R

R R

(4.45)

1 ln e

o

RN AB

R

(4.46)

Then Eqs. 4.43 and 4.44 can be shortened as

'

2 o oI R M P (4.47)

o oP R N P (4.48)

Substituting Eqs. 4.42, 4.47 and 4.48 into Eq. 4.41, one can get the tangential stress

induced by fluid penetration at wellbore wall

'

, 2 2

22p o o

e o

MR N P

R R

(4.49)

Define '

2 2

2

e o

MM

R R

(4.50)

Eq. 4.49 is then rewritten as

, 2 2p o o i eR M N P M N P P (4.51)

Substituting Eqs. 4.2 and 4.51 into Eq. 4.39, the minimum effective tangential stress on the

wellbore wall can be determined as

103

'

,min

23 2 2 2 ln

3

oh H e i

i

RYM N B P M N B P

R (4.52)

Applying the tensile failure criterion, the fracture pressure of wellbore with mudcake is

determine as

23 2 ln

3

2 2

oh H e

if

RYM N B P

RP

M N B

(4.53)

This solution takes into account the effects of mudcake permeability, thickness and

strength on fracture pressure of a wellbore plastered by a layer of mudcake. In the following

section, sensitivity studies are performed to investigate the influences of these factors on

wellbore stress and fracture pressure, and the implications on lost circulation prevention

are discussed.

4.2.3 Example Cases

In this section, the effects of the mudcake thickness, permeability and strength on

wellbore stress and fracture pressure are analyzed through several example cases using the

analytical model presented in the above section.

Typical mudcake parameters used in this study are obtained from a literature

review. Final mudcake thickness with dynamic filtration in drilling may range between 2

to 6 mm (Bezemer and Havenaar, 1966; Chenevert and Dewan, 2001; Griffith and

Osisanya, 1999; Sepehrnoori et al., 2005). The permeability of mudcake may range

between 10−4 to 10−2 mD (Bezemer and Havenaar, 1966; Sepehrnoori et al., 2005).

The yield strength of mudcake may range between 0.1 to 0.6 MPa for oil based mud (Cook

et al., 2016). In the following case studies, mudcake parameters are manipulated within

these typical ranges for sensitivity analyses. The parameters are reported in Table 4.1, with

a set of base case and several supplementary cases for sensitivity study purposes.

104

Table 4.1: Summary of the parameters used in the example cases.

General Inputs Mudcake Parameters

Wellbore

Radius, m 0.1

Maximum Horizontal

Stress, MPa 22 Parameters

Base

Case

Supplementary

Cases

Outer

Radius, m 1

Minimum Horizontal

Stress, MPa 18

Mudcake

Thickness,

mm

2 4 6

Formation

Pressure,

MPa

10 Overburden Stress,

MPa 25

Mudcake

Permeability,

mD

0.01 0.001 0.0001

Wellbore

Pressure,

MPa

12 Angle θ, (o) 0 MudcakeYield

Strength, MPa 0.15 0.25 0.5

Rock

Permeability,

mD

1 Parameter η 0.3

4.2.3.1 Effect of Mudcake Thickness

Using the analytical solution derived in the above section, the pore pressure, fluid

flow induced total and effective tangential stresses, and fracture pressure with different

mudcake thickness are calculated. The results are shown in Figures 4.2 through 4.5. The

following conclusions about the effect of mudcake thickness are made based on the results:

Pore pressure around wellbore decreases with the increase of mudcake thickness as

shown in Figure 4.2. Even with a very thin (e.g. 2mm) mudcake, the pore pressure

at or close to the wellbore wall can have a significant decline compared with the

case without mudcake.

In the near wellbore region, total tangential stress induced by fluid flow is positive,

and decreases with the increase of mudcake thickness; adversely, away from the

wellbore, the induced stress is negative and increases with the increase of mudcake

thickness, as shown in Figure 4.3.

105

Figure 4.4 shows that the effective tangential stress induced by fluid flow

(compared with no flow case) is negative (tensile). Its value increases (tension

decreases) with mudcake thickness increase, meaning the thicker the mudcake, the

less likely for tensile fracture and thus lost circulation to occur.

With mudcake thickness increase, fracture pressure of the wellbore has significant

increase as shown in Figure 4.5, which means the wellbore is effectively

strengthened by mudcake. In this particular case, fracture pressure can be increased

by 2 MPa with a 4 mm mudcake on wellbore wall. This confirms the conclusion

from experimental studies that mudcake can effectively increase the pressure that a

wellbore can sustain and reduce the risk of lost circulation (Cook et al., 2016).

Figure 4.2: Pore Pressure distribution around wellbore with different mudcake thickness

𝑤.

106

Figure 4.3: Total tangential stress induced by fluid flow with different mudcake thickness

𝑤.

Figure 4.4: Effective tangential stress induced by fluid flow (compared with no flow case)

with different mudcake thickness 𝑤.

107

Figure 4.5 Fracture pressure with different mudcake thickness 𝑤.

4.2.3.2 Effect of Mudcake Permeability

Pore pressure around wellbore, fluid flow induced total and effective tangential

stresses, and fracture pressure with different mudcake permeability are calculated and

shown in Figures 4.6 through 4.9. The following conclusions about the effect of mudcake

permeability are made based on the results:

Pore pressure around wellbore decreases with the decrease of mudcake

permeability as shown in Figure 4.6. In this specific case, the pore pressure around

wellbore almost does not change (i.e. equal to formation pressure) with an

extremely low mudcake permeability of 10−4 mD. This case can practically be

approximated as the case of an impermeable wellbore.

In the near wellbore region, total tangential stress induced by fluid flow is positive,

and decreases with the decrease of mudcake permeability; but in the region away

108

from wellbore, the induced stress is negative and increases with the decrease of

mudcake permeability, as shown in Figure 4.7.

Figure 4.8 shows the effective tangential stress induced by fluid flow (compared

with no flow case) is negative (tensile). Its value increases (tension decreases) with

the decrease of mudcake permeability, meaning the lower the permeability of the

mudcake, the less likely for tensile fracture and thus lost circulation to occur. In

this specific case, there is almost no tensile tangential stress induced by fluid flow

with a mudcake permeability as low as 10−4 mD.

With mudcake permeability decrease, fracture pressure of the wellbore has

significant increase as shown in Figure 4.9, which means the wellbore can be

effectively strengthened by a low-permeability mudcake. In this particular case,

fracture pressure can be increased by 2 MPa by changing the mudcake permeability

from 10−2 to 10−3 mD. Mudcake permeability depends on mud type, additives

or LCMs in the mud, and bottom hole conditions. It is critical to properly engineer

the mud formulations to build a low-permeability mudcake on wellbore wall in

preventive lost circulation treatments.

109


permeability 𝐾1.


permeability 𝐾1.

110

Figure 4.8: Effective tangential stress induced by fluid flow (compared with no flow case)

with different mudcake permeability 𝐾1.

Figure 4.9: Fracture pressure with different mudcake permeability 𝐾1.

111

4.2.3.3 Effect of Mudcake Strength

Figures 4.10 and 4.11 show the effective tangential stress and fracture pressure of

wellbore with different mudcake (yield) strength. It can be seen from Figure 4.10 that with

the typical values of mudcake yield strength (less than 1 MPa), the effect of mudcake

strength on wellbore stress is extremely small. Figure 4.11 shows that with the increase of

mudcake strength, fracture pressure has a very slight increase, but this increase is negligibly

small for practical considerations. So it can be concluded that the strength of mudcake itself

almost does not contribute to the “strength” of wellbore. Mudcake “strengthens” the

wellbore mainly through preventing/mitigating fluid seepage from wellbore to the

surrounding formation and therefore inhibiting the development of tensile stress around

wellbore.

Figure 4.10: Effective tangential stress around wellbore with different mudcake yield

strength 𝑌.

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Figure 4.11: Fracture pressure with different mudcake yield strength 𝑌.

4.3 A NUMERICAL MODEL FOR TIME-DEPENDENT MUDCAKE

4.3.1 Objective and Challenge

The analytical model proposed in the above section assumes steady-state fluid flow,

so it does not take into account the time-dependent effect on preventive wellbore

strengthening treatments based on plastering the wellbore with mudcake. However, in

realistic situation, mudcake buildup is a time-dependent process. Three time-dependent

processes should be considered in modeling the dynamic mudcake problem:

Time-dependent fluid flow in the porous rock and mudcake

Time-dependent mudcake thickness buildup

Time-dependent mudcake permeability reduction

Therefore, in this section, in order to take into account these time-dependent effects,

a numerical model is developed to simulate the evolutions of near-wellbore stress and pore

pressure with time-dependent fluid flow and mudcake parameters based on finite-element

method using Abaqus® software and FORTRAN® subroutines.

113

The challenge in developing such a finite-element model is that it is difficult to

actually manipulate the mudcake thickness to model mudcake buildup during the

simulation, because with the current finite-element techniques it is difficult to dynamically

and smoothly change the size of the elements or add new elements that represent the

mudcake during the simulation process. However, it is more practical to manipulate the

material parameters (e.g. permeability) as a function of time during the simulation.

Therefore, in this work, an equivalent method is used to describe the effects of both time-

dependent mudcake thickness and dynamic mudcake permeability with one material

parameter – “equivalent mudcake permeability”. By manipulating this single parameter

during the simulation, we can get the same results as simultaneously changing both

mudcake thickness and permeability, but in a more practical and easier way. In the

following subsections, the equivalent method and the finite-element model development

are described in detail.

4.3.2 An Equivalent Method

Since it’s very difficult to dynamically change the mudcake element size, thus the

mudcake thickness during the simulation process of a finite-element model, a constant

mudcake thickness will be defined in our numerical model. However, an equivalent method

will be used to capture the effects of both mudcake thickness buildup and permeability

change with time using a single parameter – equivalent mudcake permeability.

We still consider the mudcake, wellbore and formation are in a plane-strain

condition as shown in Figure 4.12. In real situation while drilling, as mentioned above,

both mudcake thickness 𝑤(𝑡) and permeability 𝑘(𝑡) are functions of time. But in the

numerical model, we assume the mudcake has a constant thickness 𝑤𝑜 and an equivalent

114

permeability 𝑘𝑒(𝑡). The equivalent permeability 𝑘𝑒(𝑡) is a function of both real mudcake

thickness 𝑤(𝑡) and permeability 𝑘(𝑡).

Figure 4.12: Illustration of mudcake model with constant mudcake thickness 𝑤𝑜 and

equivalent mudcake permeability 𝑘𝑒(𝑡).

The equivalent mudcake permeability 𝑘𝑒(𝑡) is determined by achieving the same

fluid flow rate under the same differential pressure across the mudcake between the

numerical model with constant mudcake thickness and the real case with dynamic mudcake

thickness. Therefore, using Darcy’s law for both cases we can have 2𝜋𝐾𝑒(𝑡)

𝜇𝑙𝑛𝑅𝑖+𝑤𝑜

𝑅𝑖

∆𝑝 =2𝜋𝐾(𝑡)

𝜇𝑙𝑛𝑅𝑖+𝑤(𝑡)

𝑅𝑖

∆𝑝 (4.54)

The left side of Eq. 4.54 is the term for the model with constant mudcake thickness 𝑤𝑜

and equivalent mudcake permeability 𝑘𝑒(𝑡); while the right side of Eq. 4.54 is the term

for the real situation with time-dependent mudcake thickness 𝑤𝑜 and permeability 𝑘𝑒(𝑡).

The equivalent mudcake permeability 𝑘𝑒(𝑡) is then determined as

𝐾𝑒(𝑡) = 𝐾(𝑡)𝑙𝑛(1+

𝑤𝑜𝑅𝑖

)

𝑙𝑛(1+𝑤(𝑡)

𝑅𝑖) (4.55)

115

Once the real time-dependent mudcake thickness 𝑤(𝑡) and permeability 𝑘(𝑡)

are known (e.g. from experimental study), a time-dependent equivalent permeability 𝑘𝑒(𝑡)

can be determined using Eq. 4.55 and then assigned as a material parameter to the mudcake

in the finite-element model. It can be seen from Eq. 4.55 that the equivalent mudcake

permeability reflects not only the time-dependent mudcake thickness but also the time-

dependent mudcake permeability.

4.3.3 Model Formulation


Mudcake on a vertical wellbore is considered. The problem is still assumed to be

in 2D plane-strain condition. Due to symmetry, only one quarter of the wellbore is modeled

as shown in Figure 4.13. The wellbore radius is 0.1m. Both of the length and the width of

the quarter model are 2m. Mudcake has a constant thickness of 3mm and is represented

with two layers of elements as shown in Figure 4.13. The mesh is refined in near wellbore

region. Both of the mudcake and formation are molded with coupled pore pressure and

deformation elements in Abaqus®.

Figure 4.13: Geometry of the finite-element mudcake model.

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4.3.3.2 Initial and Boundary Conditions

Symmetric boundary conditions are applied to the left and bottom boundaries of

the model as shown in Figure 4.14. The maximum horizontal stress (𝑆𝐻𝑚𝑎𝑥) and minimum

horizontal stress (𝑆ℎ𝑚𝑖𝑛) are applied to the right and top outer boundaries of the model,

respectively. Constant pore pressure boundary condition 𝑃𝑝𝑜 with its value equal to the

formation pore pressure is also applied to the outer boundaries. Bottom hole pressure 𝑃𝑤

is applied to inner surface of mudcake. Also defined on this surface is a pore pressure

boundary 𝑃𝑝𝑖 with its value equal to the bottom hole pressure 𝑃𝑤. Initial pore pressure

𝑃𝑝𝑜 is applied to the whole domain of the model. A set of boundary condition values is

given in Table 4.2 for the numerical simulation studies in the following sections.

Figure 4.14: Boundary conditions of the finite-element mudcake model.

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Table 4.2: Input boundary-condition values for the mudcake model.

Parameters Values

𝑆𝐻𝑚𝑎𝑥 22 MPa

𝑆ℎ𝑚𝑖𝑛 18 MPa

𝑃𝑝𝑜 10 MPa

𝑃𝑤 12 MPa

𝑃𝑝𝑖 12 MPa

4.3.3.3 Material Properties

Formation. The formation rock is modeled as poroelastic material with Young’s

modulus 𝐸𝑓 = 1.583 × 106𝑘𝑃𝑎 , Poisson’s ratio 𝑣𝑓 = 0.22 , porosity 𝜙𝑓 = 0.14 , and

permeability 𝑘𝑓 = 0.1 𝑚𝐷.

Mudcake. Mudcake is also modeled as poroelastic material. However, the

mudcake is considered to be very soft/flexible compared with the formation rock. For a

very soft mudcake, as shown in the analytical study in Section 4.2 the mechanical

properties or the “strength” of the mudcake has negligibly small effect on the stress and

pore pressure around wellbore. Therefore, proper mechanical properties should be selected

for modeling mudcake to make sure it alters near-wellbore stress only through its porous

properties (permeability and porosity), rather than through its mechanical properties. This

can be achieved by selecting a low Young’s modulus and a very high Poisson’s ratio for

the mudcake in the finite-element model presented in this section. This statement will be

validated later in Section 4.3.4. In this study, Young’s modulus 𝐸𝑚 = 1.583 × 105𝑘𝑃𝑎

and Poisson’s ratio 𝑣𝑚 = 0.49 is selected for mudcake and can satisfy the requirements.

118

Experimental studies show that, with mud circulation in the wellbore, mudcake

thickness increases and finally approaches a maximum equilibrium value, while mudcake

permeability decreases and finally approaches a minimum equilibrium value (Bezemer and

Havenaar, 1966; Osisanya and Griffith, 1997; Tran et al., 2011). For this study, a typical

profile of mudcake thickness buildup with time given by Eq. 4.56 is assumed (Tran et al.,

2011). As shown in Figure 4.15, the mudcake approaches its maximum equilibrium

thickness of 3mm after about 4 hours’ exposure to circulating mud.

𝑤(𝑡) = 0.003(1 − 𝑒−0.0003𝑡) (4.56)

A typical profile of mudcake permeability reduction with time given by Eq. 4.57

(Tran et al., 2011) is used in this study. The mudcake has an initial permeability of 0.002

mD, which decreases to an equilibrium value of 0.001 mD after approximate 4 hours’ mud

circulation as shown in Figure 4.16.

𝑘(𝑡) = 0.001(1 + 𝑒−0.0003𝑡) (4.57)

Therefore, according to Eqs. 4.55, 4.56 and 4.57, the equivalent mudcake

permeability of the mudcake is

𝑘𝑒(𝑡) =0.001(1+𝑒−0.0003𝑡)𝑙𝑛(1+

𝑤𝑜𝑅𝑖

)

𝑙𝑛[1+0.003(1−𝑒−0.0003𝑡)

𝑅𝑖]

(4.58)

Figure 4.17 shows the variation of equivalent mudcake permeability with time. At

the early time, the equivalent permeability is very high because of the small mudcake

thickness (see Figure 4.15) and high mudcake permeability (see Figure 4.16) in real case.

Finally the equivalent permeability approaches 0.001 mD which is equal to the real

equilibrium permeability because in this example the real equilibrium mudcake thickness

is equal to the constant mudcake thickness defined in the model. The variation of equivalent

permeability with time given by Eq. 4.58 is coded into a FORTRAN® subroutine and

119

plugged into the Abaqus® model to simulate the dynamic evolution of mudcake thickness

and permeability.

Figure 4.15: Variation of mudcake thickness with time.

Figure 4.16: Variation of mudcake permeability with time.

120

Figure 4.17: Variation of equivalent mudcake permeability with time.

4.3.3.4 Simulation Procedures

Three steps are defined in the model for the simulating the dynamic mudcake

problem.

Step 1. Geostatic step in Abaqus® is used to obtain initial stress and pore

pressure of the entire simulation domain. In this step, no wellbore is drilled and

the formation in unperturbed condition.

Step 2. A set of elements representing the wellbore is removed and mud

pressure is applied to the wellbore surface to simulate the drilling process and

obtain the stress and pore pressure distribution immediately after drilling.

Step 3. Mudcake with the equivalent permeability defined by Eq. 4.58 is added

onto the wellbore surface and the mud pressure is reassigned onto the mudcake

surface, and then fluid seepage is simulated up to 8 hours.

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4.3.4 Model Calibration

As mentioned in section 4.2, mudcake alters the stress and pore pressure around

wellbore mainly through preventing/mitigating fluid flow through wellbore to the

surrounding formation due to its low permeability. Because mudcake is extremely

soft/flexible compared with formation rock, the mechanical properties or the “strength” of

mudcake should not alter the stress and pore pressure around wellbore very much.

Therefore, when the mudcake is extremely permeable (with very high permeability), the

mud pressure in the wellbore can be totally transmitted onto the wellbore surface, and the

stress state around wellbore should be the same as that of a permeable wellbore without

mudcake. On the other hand, when the mudcake is perfectly impermeable (with zero

permeability), the stress state around wellbore should be the same as that of an

impermeable wellbore wall without mudcake. To calibrate the model, the pore pressure,

tangential stress and radial stress along 𝑆𝐻𝑚𝑎𝑥 direction are compared for the following

two sets of equivalent cases:

Set 1: Mudcake with extremely high permeability ( 107 Darcy) versus

permeable wellbore without mudcake.

Set 2: Impermeable mudcake (zero permeability) versus impermeable wellbore

without mudcake.

Figures 4.18 through 4.23 show the excellent agreements in pore pressure,

tangential stress and radial stress for the two sets of equivalent cases. The results validate

the solution behaviors of the proposed numerical model, and proved the statement that the

mechanical properties of a very soft mudcake with small modulus and high Poisson’s ratio

have negligibly small effect on wellbore stress.

122

Figure 4.18: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 3 hours for wellbore with

extremely permeable mudcake and permeable wellbore without mudcake.

Figure 4.19: Effective tangential stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 3 hours for

wellbore with extremely permeable mudcake and permeable wellbore without

mudcake.

123


wellbore with extremely permeable mudcake and permeable wellbore without

mudcake.

Figure 4.21: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 3 hours for wellbore with

impermeable mudcake and impermeable wellbore without mudcake.

124

Figure 4.22: Effective tangential stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 3 hours for


mudcake.



mudcake.

125

4.3.5 Simulation Results: Time-Dependent Mudcake Effects

4.3.5.1 Time-Dependent Mudcake Thickness Buildup

In this section, the time-dependent effects of mudcake thickness buildup on pore

pressure and effective stresses around wellbore are analyzed using the proposed numerical

model. The mudcake permeability is assumed to be constant at 0.01 mD during the entire

filtration process. Three simplified conditions are also simulated and used as baselines for

comparison, including: (1) no mudcake on wellbore wall, (2) a perfect impermeable

mudcake on wellbore wall, and (3) a permeable mudcake with constant thickness of 3 mm

(equal to the maximum equilibrium thickness of the dynamic mudcake). Pore pressure,

effective radial stress and effective tangential stress along the direction of 𝑆𝐻𝑚𝑎𝑥 after

seepage periods of 30 minutes and 4 hours are shown in Figures 4.24 through 4.29.

Generally, the following effects of time-dependent mudcake thickness buildup are

observed from the simulation results:

At early time (t = 30 minutes), as shown in Figures 4.24, 4.26 and 4.28, the pore

pressure, effective radial stress and effective tangential stress (compression is

negative in this numerical study) with time-dependent mudcake thickness are larger

than those with constant mudcake thickness and closer to those without mudcake,

because the thickness is still small at the early time.

At late time (t= 4 hours), as shown in Figures 4.25, 4.27 and 4.29, as the mudcake

grows and approaches the equilibrium thickness of 3 mm, the pore pressure and

stress profiles approach those of the constant mudcake thickness case. Tangential

stress becomes more compressive compared with that at the early time t=30

minutes, which means mudcake buildup results in positive effect for lost circulation

prevention.

126

Figure 4.24: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30 minutes.

Figure 4.25: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 4 hours.

127

Figure 4.26: Effective radial stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30 minutes.


128

Figure 4.28: Effective tangential stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30 minutes.

Figure 4.29: Effective tangential stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 4 hours.

129

4.3.5.2 Time-Dependent Mudcake Permeability Reduction

In this section, the time-dependent effects of mudcake permeability reduction on

pore pressure and stresses abound wellbore are analyzed using the proposed numerical

model. The mudcake thickness is assumed to be constant at 3 mm during the entire filtration

process. Three simplified conditions are also simulated and used as baselines for

comparison, including: (1) no mudcake on the wellbore wall, (2) a perfect impermeable

mudcake on wellbore wall, and (3) permeable mudcake with constant permeability of 0.001

mD which is equal to the minimum equilibrium permeability of the dynamic mudcake.

Pore pressure, effective radial stress and effective tangential stress along the direction of

𝑆𝐻𝑚𝑎𝑥 after seepage periods of 30 minutes and 4 hours are shown in Figures 4.30 through

4.35.

Generally, the following effects of time-dependent mudcake permeability reduction

are observed from the simulation results:



negative) with time-dependent mudcake thickness are larger than those with

constant mudcake thickness and closer to those without mudcake, because the

permeability is still large at early time.

As the mudcake permeability decreases and approaches the equilibrium

permeability of 0.01 mD at t = 4 hours, as shown in Figures 4.31, 4.33 and 4.35,

the pore pressure and stress profiles approach those of the constant (minimum)

mudcake permeability case. Tangential stress becomes more compressive

compared with that at the early time t=30 minutes, which means mudcake

permeability reduction with time results in positive effect for lost circulation

prevention.

130



131



132


Figure 4.35: Effective tangential stress profiles along SH direction at t = 4 hours.

133

4.3.5.3 Coupled Time-Dependent Mudcake Thickness Buildup and Permeability

Reduction

In this section, the coupled effects of time-dependent mudcake thickness buildup

and permeability reduction on pore pressure and stresses around wellbore are analyzed

using the proposed numerical model. Three simplified conditions are also simulated and

used as baselines for comparison, including: (1) no mudcake on the wellbore wall, (2) a

perfect impermeable mudcake on wellbore wall, and (3) a permeable mudcake with

constant permeability of 0.001 mD (equal to the minimum equilibrium permeability of the

dynamic mudcake) and constant thickness of 3 mm (equal to the maximum equilibrium

thickness of the dynamic mudcake) on wellbore wall. The results are shown in Figures 4.36

through 4.41.

Generally, the following effects of coupled time-dependent mudcake thickness

buildup and permeability reduction are observed from the simulation results:



negative) with time-dependent mudcake thickness and permeability are larger than

those with constant mudcake thickness and permeability and closer to those without

mudcake, because the mudcake thickness is still small and the permeability is still

large at this time. At early time, assuming mudcake with no permeability

(equivalent to impermeable wellbore) or with constant final thickness and

permeability may lead to substantial underestimation of pore pressure and effective

stress around wellbore wall.

At late time (t = 4 hours), as shown in Figures 4.37, 4.39 and 4.41, as the mudcake

approaches the equilibrium thickness of 3 mm and equilibrium permeability of

0.001mD, the pore pressure and stress profiles approach those with constant

134

mudcake thickness and permeability. Tangential stress becomes more compressive

compared with that at the early time, which means the coupled effect of mudcake

thickness buildup and permeability reduction with time results in positive effect for

lost circulation prevention.


135



136



137

Figure 4.41: Effective tangential stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 4 hours.

138

4.3.5.4 Significance of Mudcake Permeability

In this section, the effects of mudcake permeability on near-wellbore pore pressure

and stress distributions are examined. The results with mudcake permeability of 0.01 mD

and 0.001 mD are compared (mudcake thickness is constant at 3 mm). Two simplified

conditions are also simulated and used as baselines for comparison, including: (1) no

mudcake on the wellbore wall, and (2) a perfect impermeable mudcake on wellbore wall.

The comparison results are shown in Figures 4.42 through 4.44.

Generally, the following effects of mudcake permeability on near-wellbore pore

pressure and stresses are observed from the simulation results:

As expected, the pore pressure with smaller mudcake permeability of 0.001 mD

are much smaller than those with larger mudcake permeability of 0.01 mD and

much closer to those with impermeable mudcake.

The compressive tangential stress with smaller mudcake permeability of 0.001

mD are much larger than those with larger mudcake permeability of 0.01 mD

and much closer to those with impermeable mudcake, which implies that

fractures or lost circulation events are more unlikely to occur with low-

permeability mudcake due to the larger compressive tangential stress.

Therefore, it is important to build a low-permeability mudcake on wellbore wall

as soon as possible when taking preventive wellbore strengthening treatments

during drilling.

139



140


4.4 SUMMARY

In this chapter, an analytical solution for the effects of mudcake thickness,

permeability and strength on the wellbore stress and fracture pressure in a steady flow

condition is presented. Sensitivity studies are then performed using this solution. The

results show that both mudcake thickness and permeability have great influence on

wellbore stress and fracture pressure, while the effect of mudcake strength is negligibly

small for practical consideration. In particular, an increasing of fracture pressure with

decreasing mudcake permeability and/or increasing mudcake thickness is observed.

For a further step, a finite-element model taking into account the transient effects

of mudcake thickness buildup coupled with mudcake permeability reduction on near-

wellbore stress state is developed. A one-parameter description of both mudcake thickness

buildup and permeability reduction based on an equivalent flow study is used to facilitate

the model development without compromising the calculation accuracy. The results show

that taking into account the time-dependent mudcake thickness buildup and permeability

141

reduction results in a wellbore stress state between that without considering mudcake effect

and that assuming an impermeable mudcake (or that assuming constant final-equilibrium

mudcake thickness and permeability). The numerical model developed in this chapter

presents a useful tool to analyze the time-dependent stress evolution around wellbore with

dynamic mudcake development for the design and evaluation of preventive wellbore

strengthening treatments based on plastering wellbore surface with mudcake.

142

CHAPTER 5: Modeling Study of Remedial Wellbore Strengthening

Treatments2

Remedial wellbore strengthening methods attempt to “strengthen” the wellbore by

bridging/plugging lost circulation fractures with LCMs after fluid loss already occurs. It is

an effective method to mitigate or stop fluid loss in the drilling industry. Although a

number of successful applications have been reported, remedial wellbore strengthening

operations are still mostly performed on a trial-and-error basis, without a clear

understanding of its fundamentals. The effects of several parameters are still not thoroughly

understood. Thus, for better understanding of the underlying mechanisms of remedial

wellbore strengthening treatments based on bridging lost circulation fractures, this chapter

performs studies on it based on both analytical and numerical modeling. The analytical

model derived based on linear elastic fracture mechanics provides a fast procedure to

predict fracture pressure change before and after bridging the fractures, while the numerical

model developed using finite-element method gives a more detailed description of the

distribution of local stress and fracture width with remedial wellbore strengthening

operations.

2 Parts of this chapter have been published in the following journal papers which were supervised by K. E.

Gray:

Feng, Y., Arlanoglu, C., Podnos, E., Becker, E., Gray, K.E., 2015. Finite-Element Studies of Hoop-

Stress Enhancement for Wellbore Strengthening. SPE Drill. Complet. 30, 38–51. doi:10.2118/168001-

PA.

Feng, Y., Gray, K.E., 2016a. A parametric study for wellbore strengthening. J. Nat. Gas Sci. Eng. 30,

350–363. doi:10.1016/j.jngse.2016.02.045.

Feng, Y., Gray, K.E., 2016b. A fracture-mechanics-based model for wellbore strengthening applications.

J. Nat. Gas Sci. Eng. 29, 392–400. doi:10.1016/j.jngse.2016.01.028.

143

5.1 INTRODUCTION

Remedial wellbore strengthening methods attempt to “strengthen” the wellbore

after fractures has already occurred on wellbore. It is an effective method to mitigate or

stop fluid loss by bridging lost circulation fractures with LCMs. The ultimate objective of

remedial wellbore strengthening treatments is to increase the fracture pressure that a

wellbore can sustain without significant fluid loss and widen the drilling mud weight

window. Various experimental studies on remedial wellbore strengthening have been

carried out in the drilling industry.

In laboratory and field experimental studies, repeated leak-off tests are commonly

used to investigate the effectiveness of remedial wellbore strengthening treatments. By

performing repeated leak-off tests in a wellbore with and without LCMs in the injection

fluid, the fracture pressure for strengthened and un-strengthened wellbore can be

compared. A number of laboratory experiments (Aston et al., 2007, 2004; Guo et al., 2014;

Morita et al., 1996a; Onyia, 1994; Savari et al., 2014) and field applications (Alberty and

McLean, 2004; Dupriest, 2005; van Oort et al., 2011) have shown that fracture pressure

can be effectively increased by bridging small fractures in remedial wellbore strengthening

operations. Figure 5.1 compares the fracture pressure of a wellbore without and with

wellbore strengthening treatment in one experimental study of the DEA 13 project (Onyia,

1994). The wellbore was first fractured by a leak-off test with drilling fluid free of LCMs,

and then a repeated test was conducted with drilling fluid containing LCMs. It is shown

that the wellbore was effectively strengthened with an increase of fracture (breakdown)

pressure about 5000 psi after applying remedial wellbore strengthening treatments.

144

Figure 5.1: Comparison of fracture pressure with and without remedial wellbore

strengthening treatment in DEA 13 experimental study (reproduced from

Onyia, 1994).

Figure 5.2 is the result of an experimental study on remedial wellbore strengthening

performed by Guo et al. (2014). In this test drilling fluid without LCMs was first injected

to fracture the intact wellbore and a fracture (breakdown) pressure of 870 psi was achieved.

Subsequently, two repeated leak-off tests using the same drilling mud without LCMs were

conducted to test the strength of the wellbore with existing fractures created in the first test.

The fracture (breakdown) pressure in these two case was about 400 psi lower than in the

intact wellbore. Subsequently, in the final test the drilling mud was replaced with base

drilling mud plus 30-lb/bbl graphitic LCMs to strengthen the wellbore. It is observed that

an enhanced fracture (breakdown) pressure about 1700 psi was achieved in this case, which

was about 800 and 1200 psi higher than the fracture (breakdown) pressure in the intact

wellbore and fractured wellbore, respectively.

145

Figure 5.2: Results of a laboratory wellbore strengthening test (reproduced from Guo et al.,

2014).

Figure 5.3 shows a field wellbore strengthening test by Aston et al. (2004). A base

mud free of LCM was first pumped to fracture a vertical wellbore, and an original

breakdown pressure of 1200 psi was observed. In a following test, the base mud was

replaced by a designed mud containing 80-lb/bbl LCM solids to investigate the effect of

remedial wellbore strengthening. The solid curve in Figure 5.3 shows the pressure-time

curve using LCMs. A fracture (breakdown) pressure about 2050 psi was reached in this

case, which is about 850 psi higher than the original state. This significant increase in

fracture (breakdown) pressure clearly indicates that the fractures can be successfully

bridged using remedial wellbore strengthening treatment.

146

Figure 5.3: Results of LOT field tests before and after taking remedial wellbore

strengthening treatment (reproduced from Aston et al., 2004).

However, despite successful lab experiments and field applications, the

fundamental physics of remedial wellbore strengthening are not thoroughly understood. A

lot of disagreements still exist in the drilling industry. There is still a lack of proper

mathematic models to quantitatively describe this problem. Thus, for better understanding

of the underlying mechanisms of remedial wellbore strengthening treatments based on

bridging lost circulation fractures, studies using both analytical and numerical modeling

have been carried out in this chapter. The analytical model proposed based on linear elastic

fracture mechanics provides a fast procedure to predict fracture pressure change before and

after bridging the fractures, while the numerical model developed using finite-element

method gives a more detailed description of the distribution of local stress and fracture

width with remedial wellbore strengthening operations.

147

5.2 A FRACTURE-MECHANICS-BASED ANALYTICAL MODEL FOR REMEDIAL

WELLBORE STRENGTHENING APPLICATIONS

Based on linear elastic fracture mechanics, this section illustrates an analytical

solution for investigating geomechanical aspects of remedial wellbore strengthening

operations. The proposed solution (1) provides a fast procedure to predict fracture pressure

before and after bridging a preexisting fracture in remedial wellbore strengthening

treatments, and (2) considers the effects of wellbore-fracture geometry, in-situ stress

anisotropy, and LCM bridge location. The solution is validated by comparison with

available fracture mechanics examples, then used to investigate and quantify the effect of

several parameters on wellbore strengthening with a sensitivity study. Results show that

the wellbore can be effectively strengthened by increasing fracture pressure with remedial

wellbore strengthening operations, and the magnitude of strengthening is affected by LCM

bridge location, in-situ stress anisotropy, and formation pore pressure. The proposed

solution illustrates how remedial wellbore strengthening treatment works, and provides

useful considerations for field operations.

In addition, in order to avoid lost circulation, it is crucial to adjust mud weight

during the drilling in a timely manner, according to fracture pressure. Some essential

parameters controlling fracture pressure, such as geometry of pre-existing fractures on the

wellbore, are usually not known prior to drilling, but they may be measured while drilling.

Therefore, performing real-time analyses is important in order to make prompt adjustments

to drilling operations. The advantages of an analytical model compared to a numerical

model for real-time analyses are obvious, because it is usually very concise and easy to

implement, and building a meshed structure for conducting a large number of iterative

computations is not needed, contrary to numerical analyses, such as finite-element and

boundary-element analyses. The proposed analytical model provides a rapid procedure to

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predict fracture pressure before and after bridging a preexisting fracture. It has the potential

for implementation in near-real time drilling analysis for wellbore strengthening evaluation

and mud weight adjustments.

5.2.1 Analytical Model

5.2.1.1 Model Description

Figure 5.4 illustrates the problem under consideration. It is a plane-strain wellbore

model with two short fractures extending symmetrically from the wellbore wall and

perpendicular to the minimum horizontal stress. Before bridging the fracture (i.e. without

the bridges in Figure 5.4), given the small length of the fractures, the pressure inside the

fracture is assumed to be uniform and equal to the wellbore pressure or the equivalent

circulation pressure during drilling. After bridging the fracture at some location inside the

fracture shown in Figure 5.4, the pressure ahead of the bridge from the fracture mouth to

the bridge is equal to wellbore pressure. Pressure behind the bridge is equal to formation

pore pressure, assuming the bridge perfectly blocks pressure transmission in the fracture;

pressure higher than formation pore pressure behind the bridge will bleed off to pore

pressure due to fluid leak-off into the porous rock. In the following discussion, the fracture

portions ahead of and behind the bridge are referred to as invaded zone and non-invaded

zone, respectively.

In this analytical study, a goal is to develop a fast and easy-to-use analytical solution

to quantify fracture pressure change after bridging the fracture. The solution is derived by

combining Kirsch stress solutions around a circular borehole with a linear-elastic fracture

criterion. The following sections detail the procedures for deriving the solution.

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Figure 5.4: Schematic of remedial wellbore strengthening problem.

5.2.1.2 Net Pressure along Fracture

The system shown in Figure 5.4 can be decomposed into two simper cases

(Carbonell and Detournay, 1995; Shahri et al., 2014) as shown in Figure 5.5: (1) an intact

wellbore subject to wellbore pressure and anisotropic horizontal stresses, and (2) a wellbore

with two symmetric fractures subject solely to pressure loads inside the fracture. The

tangential stress along the fracture direction in the first case can be determined by Kirsch

solution:

𝑆𝜃𝜃 =1

2(𝑆𝐻𝑚𝑎𝑥 + 𝑆ℎ𝑚𝑖𝑛) (1 +

𝑅2

𝑟2) −

1

2(𝑆𝐻𝑚𝑎𝑥 − 𝑆ℎ𝑚𝑖𝑛) (1 + 3

𝑅4

𝑟4) − 𝑃𝑤

𝑅2

𝑟2 (5.1)

where, 𝑆𝜃𝜃 is the tangential stress along the fracture direction; 𝑆𝐻𝑚𝑎𝑥 and 𝑆ℎ𝑚𝑖𝑛 are the

maximum and minimum horizontal stresses, respectively; 𝑅 is the wellbore radius; 𝑟 is

the radial distance from the wellbore center; 𝑃𝑤 is the wellbore pressure.

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Figure 5.5: Problem decomposition. Left: an intact wellbore subject to wellbore pressure

and anisotropic far-field horizontal stresses; Right: wellbore with two

symmetric fractures subject to fluid pressure on fracture surfaces.

The net pressure acting on the fracture surfaces is defined as the difference between

the fluid pressure inside the fracture and the tangential stress in the rock normal to the

fracture surfaces. Assuming short fracture length, the tangential stress solution along the

fracture direction given by Eq. 5.1 is still valid. The net pressure of the fracture can be

approximated as:

𝑃𝑛𝑒𝑡 = 𝑃𝑓𝑙𝑢𝑖𝑑 − 𝑆𝜃𝜃 (5.2)

where 𝑃𝑛𝑒𝑡 is the net pressure along the fracture; 𝑃𝑓𝑙𝑢𝑖𝑑 is the fluid pressure inside the

fracture; it is equal to wellbore pressure and formation pore pressure in the invaded zone

and non-invaded zone, respectively. Therefore, the net pressure in the invaded zone after

bridging the fracture is:

𝑃𝑤𝑒𝑡 = 𝑃𝑤 − 𝑆𝜃𝜃 (5.3)

where, 𝑃𝑤𝑒𝑡 is the net pressure in the invaded zone from fracture mouth to bridge location.

The net pressure in the non-invaded zone is:

𝑃𝑑𝑟𝑦 = 𝑃𝑝 − 𝑆𝜃𝜃 (5.4)

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where 𝑃𝑑𝑟𝑦 is the net pressure in the non-invaded zone from bridge location to fracture

tip; 𝑃𝑝 is formation pore pressure.

5.2.1.3 Solution for Fracture with Point Load

Figure 5.6 shows a fracture in an elastic solid, subject to a couple of point loads

normal to the fracture surfaces. The fracture length is 𝑎, and the distance from the loading

positon to the fracture mouth is 𝑏. Based on linear-elastic fracture mechanics theory, an

approximate solution of fracture-tip stress intensity factor for this problem is (Tada et al.,

1985):

𝐾𝐼0 =2𝑃

√𝜋𝑎∙ 𝐹 (

𝑏

𝑎) (5.5)

where, 𝐾𝐼0 is the fracture-tip stress intensity factor of the fracture subject to a couple of

point loads; 𝑃 is the magnitude of the points loads; 𝐹 (𝑏

𝑎) =

1.3−0.3(𝑏

𝑎)

5 4⁄

√1−(𝑏

𝑎)

2 , is a function

of fracture geometry and loading positon; 𝑎 is the fracture length; and 𝑏 is the distance

from the loading position to the fracture mouth.

Figure 5.6: Fracture in an elastic solid subject to a couple of point loads.

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5.2.1.4 Solution for Fracture with Distributed Load

In this study, it is assumed that the length of the pre-existing fracture emanating

from the wellbore is relatively small compared to the wellbore radius. The fracture-tip

stress intensity factor solution of the wellbore strengthening problem depicted in Figure

5.4 can be found by integrating the point-load solution in Eq. 5.5 from the fracture mouth

to the fracture tip, with the point load replaced by the net pressure at each location. Thus,

the fracture-tip stress intensity factor for the wellbore strengthening problem in Figure 5.4,

with distributed net pressure on fracture surfaces, can be obtained by calculating the

following integrals:

𝐾𝐼 = ∫2𝑃𝑤𝑒𝑡

√𝜋𝑎

𝐷

𝑅𝐹 (

𝑏

𝑎) 𝑑𝑟 + ∫

2𝑃𝑑𝑟𝑦

√𝜋𝑎

𝐿

𝐷𝐹 (

𝑏

𝑎) 𝑑𝑟 (5.6)

𝑎 = 𝐿 − 𝑅 (5.7)

𝑏 = 𝑟 − 𝑅 (5.8)

where 𝐾𝐼 is the fracture-tip stress intensity factor for the fracture with distributed load; 𝐷

is the radial distance from the bridge location to the wellbore center; 𝐿 is the radial

distance from the fracture tip to wellbore center; 𝑟 is the radial distance from a certain

point along the fracture to the wellbore center.

By substituting Eqs. 5.3 and 5.4 into Eq. 6 and computing the integrals, the

following solution for the fracture-tip stress intensity factor can be obtained:

𝐾𝐼 = (𝐹1 + 𝐹2) ∙ [2𝑃𝑤 − (𝑆𝐻𝑚𝑎𝑥 + 𝑆ℎ𝑚𝑖𝑛)] + (𝐹1 + 3𝐹3) ∙ (𝑆𝐻𝑚𝑎𝑥 − 𝑆ℎ𝑚𝑖𝑛) − 2𝐹4 ∙

(𝑃𝑤 − 𝑃𝑝) (5.9)

𝐹1 =1

√𝜋𝑎∫ 𝐺(𝑟)𝑑𝑟

𝐿

𝑅 (5.10)

𝐹2 =1

√𝜋𝑎∫

𝑅2

𝑟2 𝐺(𝑟)𝑑𝑟𝐿

𝑅 (5.11)

𝐹3 =1

√𝜋𝑎∫

𝑅4

𝑟4 𝐺(𝑟)𝑑𝑟𝐿

𝑅 (5.12)

𝐹4 =1

√𝜋𝑎∫ 𝐺(𝑟)𝑑𝑟

𝐿

𝐷 (5.13)

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𝐺(𝑟) =1.3−0.3(

𝑟−𝑅

𝑎)

5 4⁄

√1−(𝑟−𝑅

𝑎)

2 (5.14)

where 𝐹1, 𝐹2, 𝐹3, and 𝐹4 are integral functions, which solely depend on the geometry of

the wellbore-fracture system, i.e. wellbore radius, fracture length, and LCM bridge

location. They are, therefore, referred to as geometry terms in this study.

5.2.1.5 Fracture Pressure

Once the fracture-tip stress intensity factor in Eq. 5.9 is found, the fracture pressure

can be determined based on a linear-elastic fracture mechanics criterion. In this study, an

open mode linear-elastic fracture criterion is used, which assumes that fracture initiation

occurs once the stress intensity factor at the fracture tip reaches the fracture toughness, i.e.

𝐾𝐼 = 𝐾𝐼𝐶 (5.15)

where 𝐾𝐼𝐶 is the fracture toughness of the formation rock, taken to be a material property

which describes the ability of the rock material to resist fracture growth.

Fracture pressure, therefore, can be obtained by combining Eqs. 5.9 and 5.15:

𝑃𝑓 =1

2∙

1

𝐹1+𝐹2−𝐹4∙ 𝐾𝐼𝐶 +

1

2∙

𝐹1+𝐹2

𝐹1+𝐹2−𝐹4∙ (𝑆𝐻𝑚𝑎𝑥 + 𝑆ℎ𝑚𝑖𝑛) −

1

2∙

𝐹1+3𝐹3

𝐹1+𝐹2−𝐹4∙ (𝑆𝐻𝑚𝑎𝑥 − 𝑆ℎ𝑚𝑖𝑛) −

𝐹4

𝐹1+𝐹2−𝐹4∙ 𝑃𝑝 (5.16)

where 𝑃𝑓 is the fracture pressure of the wellbore after bridging the fractures.

Eq. 5.16 illustrates that fracture pressure of the wellbore with bridged fractures is a

function of (1) load conditions, including maximum and minimum horizontal stresses and

pore pressure, (2) rock property, i.e. fracture toughness, and (3) geometry of the wellbore-

fracture system reflected by the geometry terms 𝐹1~ 𝐹4. Recall that the fracture pressure

of an intact wellbore, however, is usually defined as a function of load and rock property

only (Fjar et al., 2008; Jin et al., 2013; Zoback, 2010), by combing a Kirsch stress solution

and a tensile failure criterion.

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5.2.2 Model Validation

Validity of the proposed model is tested against an available example solution, as

depicted in Figure 5.7. An elastic wellbore with symmetric arrangement of two fractures is

subject to uniform pressure on the wellbore and fracture surfaces and anisotropic far-field

horizontal stresses. The fracture-tip stress intensity factor solution for this problem can be

found by superposing the solutions for two simpler problems given by Tada et al. (1985),

i.e. (1) the elastic wellbore with two symmetric fractures subject only to far-field

anisotropic stresses, and (2) the same wellbore-fracture system subject only to uniform

pressure in the wellbore and fractures. Its fracture-tip stress intensity factor solution is

formulated as:

𝐾𝐼 = 𝑆ℎ𝑚𝑖𝑛√𝜋𝑎𝐹𝜆1(𝑆0) + 𝑃𝑤√𝜋𝑎𝐹𝜆2(𝑆0) (5.17)

where,

𝐹𝜆1(𝑆0) = (1 − 𝜆1)𝑀(𝑆0) + 𝜆1𝑁(𝑆0) (5.18)

𝐹𝜆2(𝑆0) = (1 − 𝜆2)𝑃(𝑆0) + 𝜆2𝑁(𝑆0) (5.19)

𝑀(𝑆0) = 0.5(3 − 𝑆0)[1 + 1.243(1 − 𝑆0)3] (5.20)

𝑁(𝑆0) = 1 + (1 − 𝑆0)[0.5 + 0.743(1 − 𝑆0)2] (5.21)

𝑃(𝑆0) = (1 − 𝑆0)[0.637 + 0.485(1 − 𝑆0)2 + 0.4𝑆02(1 − 𝑆0)] (5.22)

𝑆0 =𝑎

𝑅+𝑎 (5.23)

𝜆1 =𝑆𝐻𝑚𝑎𝑥

𝑆ℎ𝑚𝑖𝑛 (5.24)

𝜆2 =𝑃𝑓𝑙𝑢𝑖𝑑

𝑃𝑤(= 1 𝑓𝑜𝑟 𝑡ℎ𝑖𝑠 𝑝𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟 𝑐𝑎𝑠𝑒) (5.25)

𝐹𝜆1(𝑆0) and 𝐹𝜆2(𝑆0) are functions of geometry and load of the wellbore-crack system;

𝑆0 is the ratio of fracture length to the distance between fracture tip and wellbore center;

𝜆1 is the horizontal stress ratio; 𝜆2 is the ratio of pressure in the fracture to the pressure

in wellbore, equal to 1 in this case due to the uniform pressure distribution in the wellbore

and fracture.

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Note that, for Eq. 5.17, the pressure in the fracture is assumed to be uniform and

equal to the wellbore pressure. In order to compare the proposed solution in this study

against the available solution, the LCM bridge location in Figure 5.4 is assumed at the tip

of the fracture (i.e. there is no bridge in the fracture) and the pressure in fracture is equal

to the wellbore pressure due to the small fracture length assumption.

Figure 5.7: Wellbore with two symmetric fractures subject to uniform pressure on the

wellbore and fracture surfaces and anisotropic far-field horizontal stresses.

Input data shown in Table 5.1 were used for model validation. Figure 5.8 shows the

comparison of fracture-tip stress intensity factor between the solution herein and that of

Tada et al. (1985), for different fracture lengths. The two solutions are almost identical

when fracture length is less than or equal to wellbore radius. However, with an increase of

fracture length, the two solutions gradually diverge. But the slight mismatch (less than 10%

relative error) between the two solutions with a fracture less than 3 times that of the

wellbore radius is considered acceptable in this study. It should be noted again that the

proposed analytical solution is limited to predicting fracture-tip stress intensity factor and

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fracture pressure for a wellbore with small pre-existing fractures, a likely situation in

drilling operations.

Table 5.1: Input parameters for model validation.

Parameter Unit Value

Wellbore radius (𝑅) inch 6

Fracture length (𝑎) inch 1~18

Minimum horizontal stress (𝑆ℎ𝑚𝑖𝑛) psi 3000

Maximum horizontal stress (𝑆𝐻𝑚𝑎𝑥) psi 3600

Wellbore pressure (𝑃𝑤) psi 4000

Figure 5.8: Comparison of fracture tip stress intensity factors between the proposed model

and Tada’s model (Tada et al., 1985).

5.2.3 Results of Analytical Study

Results of the proposed analytical solution for remedial wellbore strengthening are

presented in this section. As mentioned earlier, the effects of some wellbore strengthening

parameters are not fully understood and justify further elucidation. This includes in-situ

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stress anisotropy, LCM bridge location, and pore pressure and considerations for field

applications.

Input data for analyses are given in Table 5.2. For illustration purposes, a fracture

length of 6 inches is used here, a value utilized in several previous studies (Alberty and

McLean, 2004; Arlanoglu et al., 2014; Feng et al., 2015a; Wang et al., 2009, 2007b). In

the sensitivity analyses, fracture-tip stress intensity factor and fracture pressure are

calculated for different in-situ stress anisotropies (i.e. 𝑆𝐻𝑚𝑎𝑥 𝑆ℎ𝑚𝑖𝑛⁄ ranging from 1 to 2

with a constant 𝑆ℎ𝑚𝑖𝑛 of 3000psi) and various bridge locations from the fracture mouth

all the way to the fracture tip. The bridge location at the fracture mouth means that no fluid

can penetrate into the fracture, hence the length of invaded zone is zero. As bridge location

moves to the fracture tip, it becomes identical to the situation without facture

bridging/strengthening in which case fluid can penetrate into the full length of the fracture,

all the way to the fracture tip.

Table 5.2: Base input parameters used in the proposed model.

Parameter Unit Value

Wellbore radius (𝑅) inch 6

Fracture length (𝑎) inch 6

Minimum horizontal stress (𝑆ℎ𝑚𝑖𝑛) psi 3000

Maximum horizontal stress (𝑆𝐻𝑚𝑎𝑥) psi 3600

Wellbore pressure (𝑃𝑤) psi 4000

Pore pressure (𝑃𝑝) psi 1800

Fracture toughness (𝐾𝐼𝐶) psi-in0.5 2000

The effects of in-situ stress anisotropy and bridge location on fracture-tip stress

intensity factor and fracture pressure are shown in Figures 5.9 and 5.10, respectively. They

are calculated according to Eqs. 5.9 and 5.16. The results show that, without bridging the

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fracture (i.e. the ratio of invaded zone length to fracture length equals to 1), the fracture

has the largest fracture-tip stress intensity factor and the wellbore has the smallest fracture

pressure, meaning that it is very easy for the fracture to initiate and, therefore, lost

circulation to occur. After bridging the fracture, however, the stress intensity factor

decreases and fracture pressure increases significantly, which demonstrates that the

wellbore is effectively strengthened. In addition, with decrease of the invaded zone length

(i.e. bridge location moving closer to the fracture mouth), fracture pressure increases

quickly, hence better strengthening applications. The best location to bridge the fractures

for strengthening a wellbore is at the fracture mouth, near the wellbore wall. Figures 5.9

and 5.10 also show that, at same bridge location, the larger the in-situ stress anisotropy, the

larger the fracture-tip stress intensity factor and the smaller the fracture pressure, meaning

it’s more likely for fracture initiation and lost circulation to occur. Another observation

from Figure 5.10 is that the fracture pressure, with the same bridge location, experiences a

relatively larger enhancement for a wellbore with smaller in-situ stress anisotropy than that

with a larger stress anisotropy. For example, compared with the situation without fracture

bridging, the fracture pressure has an increase of 3400 psi and 2800 psi when bridging at

the fracture mouth for the in-situ stress anisotropies equal to 1.0 and 2.0, respectively. This

suggests that wellbore strengthening operations should be more effective in formations

with relatively uniform in-situ stresses.

159

Figure 5.9: Fracture-tip stress intensity factor with different horizontal stress anisotropies

and bridge locations.

Figure 5.10: Fracture pressure with different horizontal stress anisotropies and bridge

locations.

160

Figures 5.11 and 5.12 show fracture-tip stress intensity factor and fracture pressure

with different bridge locations and pore pressure. As for previous conclusions, these two

figures also show that the wellbore can be better strengthened while bridging the fracture

near the fracture mouth. Given the same horizontal in-situ stress, the stress intensity factor

or the fracture pressure has the same value without bridging the fracture (i.e. the ratio of

invaded zone length to fracture length equals to 1) for different pore pressure as shown in

Figures 5.11 and 5.12. After bridging the fracture at the same location, the wellbore-

fracture system experiences a larger decrease in fracture-tip stress intensity factor and

increase in fracture pressure for the cases with lower pore pressure to minimum horizontal

stress ratio ( 𝑃𝑝 𝑆ℎ𝑖𝑚𝑛⁄ ) than that with higher 𝑃𝑝 𝑆ℎ𝑖𝑚𝑛⁄ ratio. As mentioned in the

introduction, lost circulation is commonly encountered in pressure depleted reservoirs and

deepwater high pressure formations; both of them usually have a very narrow mud weight

window. The results shown in Figures 5.11 and 5.12 suggests that wellbore strengthening

operations should be more effective in depleted zones as compared to deepwater high

pressure zones, because depleted zones usually have relatively small 𝑃𝑝 𝑆ℎ𝑖𝑚𝑛⁄ values

caused by production-induced pore pressure drop. To the contrary, deepwater high pressure

formations usually have relatively large 𝑃𝑝 𝑆ℎ𝑖𝑚𝑛⁄ values because of high formation

pressure and relatively low in-situ stress.

161

Figure 5.11: Fracture-tip stress intensity factor with different pore pressure and bridge

locations.

Figure 5.12: Fracture pressure with different pore pressure and bridge locations.

162

Figure 5.13 shows the effect of fracture toughness on predicted fracture pressure.

As described above, fracture pressure is calculated with fracture-tip stress intensity factor

equal to fracture toughness. With the same bridge location, a fracture with larger toughness

requires higher pressure to initiate.

Figure 5.13: Fracture pressure with different fracture toughness of the formation rock.

It should be noted that sensitivity analyses presented above in Figures 5.9 through

5.13 are conducted using the base input parameters in Table 5.2. Changing base parameters

only changes the specific values of fracture-tip stress intensity factor and wellbore fracture

pressure, but does not change the trends for effects of horizontal stress anisotropy, bridging

location, pore pressure, and fracture toughness. For example, with a constant minimum

horizontal stress of 3000 psi, it can be observed from Figure 5.10 that a larger in-situ stress

anisotropy leads to a smaller fracture pressure. If the minimum horizontal stress is

increased to 6000 psi, the effect of horizontal stress anisotropy on fracture pressure will

163

have the same trend, but a higher fracture pressure value is expected due to the higher in-

situ stress magnitude.

5.2.4 Conclusions and Implications of the Analytical Study

This section presents a fracture-mechanics-based analytical solution for remedial

wellbore strengthening applications. The proposed solution provides an accurate prediction

of fracture pressure enhancement due to fracture bridging, taking into account near-

wellbore stress concentration, in-situ stress anisotropy, and different LCM bridge

locations. The equations presented in this study can be used to perform quick and

quantitative parameter sensitivity analyses to illustrate the undying mechanisms of

wellbore strengthening. In addition, compared with numerical methods, conciseness of the

analytical approach in this study offers the possibility to conduct fracture pressure

prediction in near-real-time drilling operations. Such capability would lead to timely

wellbore strengthening evaluation and mud weight adjustments. Effects of several essential

parameters on wellbore strengthening are assessed using the proposed solution. Based on

parameter sensitivity analyses, the following conclusions and implications are reached:

Fracture pressure can be significantly increased by bridging the small pre-existing

fractures emanating from the wellbore wall in remedial wellbore strengthening

operations. Compared with other loss control methods, which are usually expensive

and require additional equipment, such as using managed pressure drilling or dual

gradient drilling methods to accurately control downhole pressure, along with

casing drilling technology or setting additional casing strings to isolate loss zones,

wellbore strengthening is usually a more economical way to solve lost circulation

problems.

164

The closer the bridging location to the fracture mouth, the better strengthening can

be achieved. This means for better application of wellbore strengthening

techniques, it is important to accurately predict fracture geometry, especially

fracture mouth opening, for selecting the best LCM size. Fracture geometry on

wellbore wall during drilling processes is usually unknown. To some degree,

numerical simulation or imaging logging methods can be used to predict or measure

fracture geometry. However, improved or new techniques are needed for acquiring

better knowledge of drilling-induced or pre-exiting natural fractures on the

wellbore wall.

Higher in-situ stress anisotropy results in relatively lower fracture pressure for a

wellbore with pre-existing fractures. Therefore, in drilling under such situations,

close attention should be paid to detect lost circulation events. Wellbore

strengthening methods may be applied to prevent or remedy fluid loss.

Remedial wellbore strengthening applications by bridging pre-existing fractures are

more effective for formations with small in-situ stress anisotropy than those with

large stress anisotropy.

Remedial Wellbore strengthening applications are more effective for formations

with low pore pressure and in-situ stress ratio, such as pressure depleted reservoirs,

as compared to formations with large pore pressure and in-situ stress ratio, such as

deepwater high pressure formations.

5.3 A FINITE-ELEMENT MODEL FOR REMEDIAL WELLBORE STRENGTHENING

APPLICATIONS

While the analytical model proposed in the above section provides a fast procedure

to predict fracture pressure change before and after fracture bridging, it cannot provide

detailed stress and displacement information local to wellbore and fracture in remedial

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wellbore strengthening treatments. Besides, it does not take into account the

poromechanical effect of the formation rock. However, numerical approaches, such as the

finite-element method, can be applied to obtain the detailed information about the evolution

of local stress and deformation in wellbore strengthening operations.

Wang et al. (2009, 2007b) proposed a 2D boundary-element model to simulate two

symmetric fractures on the wellbore wall under anisotropic in-situ stresses, and to obtain

the stress distribution and fracture width before and after bridging the fracture. Guo et al.

(2011) used a 2D finite-element method to investigate fracture width distribution for two

pre-existing fractures, symmetrically located at the wellbore wall under various in-situ

stresses and fracture length. However, the simulation gave no information about fracture

behavior after fracture bridging. Alberty and McLean (2004) employed a 2D finite-element

model to simulate fracture width distribution and hoop stress field after bridging the

fracture, but they only studied cases under nearly isotropic in-situ stresses and bridging

near fracture mouth. What’s more, in all these previous numerical studies, it was assumed

that the rock is linearly elastic. Thus the effects of fluid flow inside the rock and fluid leak-

off through the wellbore wall and fracture surfaces were not considered.

To better simulate remedial wellbore strengthening treatment, a poroelastic finite-

element numerical model, considering fluid flow and fluid leak-off is developed in this

chapter. And then a comprehensive parametric study for remedial wellbore strengthening

based on bridging lost circulation fractures is performed using the proposed numerical

model. The finite-element numerical model discussed in this chapter quantifies near-

wellbore stress and fracture geometry, before and after bridging fractures. Effects of

various parameters are investigated in the parametric study, and the results lead to a number

of useful implications for field applications.

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5.3.1 Numerical Model Development

As mentioned above, linearly elastic rock behavior has been utilized in most

previous studies. The effects of pore pressure and fluid leak-off have not been considered.

The goal of this numerical study is to develop a poroelastic model, taking into account fluid

flow inside the rock and fluid leak-off across fracture faces and the wellbore wall, and to

investigate near-wellbore hoop stress and fracture geometry before and after applying

wellbore strengthening operations. Although the numerical model cannot directly provide

a solution for maximum sustainable pressure of the wellbore in wellbore strengthening

analysis, there is no doubt that, from the points of view of both continuum mechanics and

fracture mechanics, increasing hoop stress – and, hence, stress acting on closing the

fracture – will facilitate the prevention of fracture growth, thus achieving wellbore

strengthening. The link between enhancing hoop stress and increasing maximum

sustainable pressure of the wellbore has also been discussed in detail in a series of papers

(Alberty and McLean, 2004; Aston et al., 2007, 2004a; Dupriest, 2005; Dupriest et al.,

2008; Feng et al., 2015a).


Remedial wellbore strengthening treatments for a vertical wellbore is considered.

The wellbore is assumed to be in a plane-strain condition. Owing to symmetry, only one

quarter of the wellbore is used in the finite-element numerical analysis, as shown in Figure

5.14. Wellbore radius is 4.25 inches. The length and width of the quarter model are 40.25

inches.

A pre-existing fracture is assumed on the wellbore as shown in Figure 5.14, with

its face aligned with the maximum horizontal stress (𝑆𝐻𝑚𝑎𝑥 ) or X-axis direction. The

fracture length of 6 inches is used so as to be consistent with previous work by other

investigators (Alberty and McLean, 2004; Wang et al., 2009, 2007a, 2007b). The fracture

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is plugged with LCMs as shown in Figure 5.14. Through this section, the plug formed

inside the fracture by LCMs across the fracture width is called an LCM bridge. It should

be noted that, although whether LCM bridge material properties (e.g. strength and

permeability) are important or not for wellbore strengthening has been one of the most

debatable questions in the drilling industry, the effect of those material properties is beyond

the study scope of this chapter. The LCM bridge is assumed to be a “perfect” bridge in this

study, which is a rigid body with zero permeability. So there is no fluid flow across the

bridge. To create the effect of bridging the fracture, the velocity in the Y direction at the

bridging location is set equal to zero. As illustrated in Figure 5.14, the quadrant angle is 0

in the X-axis, and increases to 90 degrees around the quarter wellbore.

Figure 5.14: The remedial wellbore strengthening model. Left: geometry and boundary

conditions of the model; Rigth: detailed fracture process zone.

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As shown in Figure 5.14, symmetric boundary conditions are applied to the left and

top boundaries of the model. The maximum horizontal stress (𝑆𝐻𝑚𝑎𝑥) along the X-direction

and minimum horizontal stress (𝑆ℎ𝑚𝑖𝑛) along the Y-direction are applied to the right and

bottom outside boundaries, respectively. Various horizontal stress values are used as shown

in Table 5.3 for investigating the effect horizontal stress contrast.

Wellbore pressure is applied to the inner wall of the wellbore. Pressure in the

fracture is equal to wellbore pressure before bridging the fracture with LCM. After bridging

the fracture, pressure in the fracture ahead of the LCM bridge (from wellbore to the bridge)

is still equal to wellbore pressure. However, pressure behind the LCM bridge (from the

bridge to fracture tip) is set equal to pore pressure, because the pressure in this region will

drop to formation pore pressure with fluid leaking off into the formation and no continuous

fluid supply from wellbore due to the impermeable bridge. Fluid leak-off velocities are

applied on the wellbore wall and fracture faces to simulate fluid leak-off. Various fluid

leak-off velocities and rock properties are also used in the simulations.

5.3.1.3 Input Parameters

Table 5.3 provides the input parameters for the finite-element numerical

simulations.

Total size of the model is about ten times the wellbore size in order to eliminate

boundary effects on near-wellbore stress and strain states.

Formation rock properties are selected for a typical sandstone.

Maximum and minimum horizontal stresses with different stress contrasts, from

1 (𝑆𝐻𝑚𝑎𝑥 𝑆ℎ𝑚𝑖𝑛 = 1⁄ ) to 1.5 (𝑆𝐻𝑚𝑎𝑥 𝑆ℎ𝑚𝑖𝑛 = 1.5⁄ ) are used.

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Different pressures behind the bridge 𝑃𝑓𝑏 are used to simulate the sealing

capacity of the LCM bridge, from complete sealing (𝑃𝑓𝑏 = 𝑃𝑝 = 1800 𝑝𝑠𝑖) to

no sealing (𝑃𝑓𝑏 = 𝑃𝑝 = 40000 𝑝𝑠𝑖).

Various Young’s modulus, Poisson’s ratios, LCM bridge locations and leak-off

rates are also selected for parametric sensitivity studies.

Table 5.3: Input parameters for the finite-element model.

Parameter Values Units

Model length 40.25 inches

Model width 40.25 inches

Wellbore radius (𝑅) 4.25 inches

Young’s modulus (𝐸) 1×106, 2×106 psi

Poisson’s ratio (υ) 0.2, 0.3

Minimum horizontal stress (𝑆ℎ𝑚𝑖𝑛) 3000 psi

Maximum horizontal stress (𝑆𝐻𝑚𝑎𝑥) 1~1.5·Smin psi

Wellbore pressure (𝑃𝑤) 4000 psi

Pressure in fracture before bridging (𝑃𝑓𝑜) 4000 psi

Pressure ahead of bridge after bridging (𝑃𝑓𝑎) 4000 psi

Pressure behind of bridge after bridging

(𝑃𝑓𝑏)

1800 ~4000 psi

Fracture length (a) 6 inches

Initial pore pressure (Pp) 1800 psi

Leak-off rate 1.0, 2.0, 3.0, 4.0 in/min

Permeability 0.0023 in/min

Void ratio 0.3

LCM bridge location away from wellbore 0.5, 2.0, 3.5, 5.0 inches

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5.3.2 Model Validation

In order to verify the accuracy of the proposed numerical model, existing analytical

solutions are used for validation purpose in this section. The main objective of this

numerical study is to analyze the stress distribution around the wellbore in remedial

wellbore strengthening treatments. However, to the author’s knowledge, there are no

analytical models which can be readily implemented for predicting near-wellbore stress

with fractures emanating from the wellbore wall. Fortunately, several analytical solutions

for stress intensity factor for this problem are available in the literature. Therefore, in this

section, the stress intensity factors calculated from numerical model are validated against

the results of analytical solutions. First, stress intensity factors of an unbridged fracture

with constant length are compared; and then results of a bridged fracture with different

bridging locations are calculated and compared.

5.3.2.1 Stress Intensity Factor of Unbridged Fracture

As described in Section 5.2.2, the solution of stress intensity factor at fracture tip

for the problem of two fractures emanating symmetrically from a pressurized borehole in

an elastic media is given by Tada et al. (1985). For the problem with two unbridged

fractures in the direction of maximum horizontal stress as depicted in Figure 5.7, the stress

intensity factor at both fracture tips can be calculated using Eq. 5.17 according to their

solution.

With the same wellbore and fracture dimensions as given in Figure 5.14 and Table

5.3, wellbore pressure of 4000 psi, and the maximum and minimum horizontal stresses of

3600 psi and 3000 psi respectively, the values of stress intensity factor obtained from the

analytical model and the proposal numerical model without considering porous fluid flow

are shown in Figure 5.15. A comparison between them shows that the results are in good

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agreement with a relative error less than 5%, serving to verify reliability of the proposed

model.

Figure 5.15: Comparison of stress intensity factor for unbridged fracture.

5.3.2.2 Stress Intensity Factor of Bridged Fracture

The proposed numerical model is further validated by comparing against the

analytical model developed in Section 5.2 for a wellbore with bridged fractures. The

problem geometry is very similar to the above validation example. The only difference is

that an LCM bridge is assigned in each fracture wing as shown in Figure 5.4. As assumed

for the numerical model, fluid pressure in the fracture ahead of the LCM bridge (from

wellbore to the bridge) and behind the LCM bridge (from the bridge to the fracture tip) are

also set equal to wellbore pressure and pore pressure, respectively. The analytical solution

of stress intensity factor is given by Eq.5.9. Further details about the development of the

solution can be found in Section 5.2 and in Feng and Gray (2016b).

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Stress intensity factors are calculated and compared using the analytical solution

and the numerical model for the problem with various LCM bridge locations. The

maximum and minimum horizontal stresses are equal to 3600 psi and 3000 psi,

respectively; wellbore pressure is 4000 psi; and pore pressure is 1800 psi. It should be

noted that even though pore pressure is included here, fluid leak-off through the wellbore

and fracture faces is not considered in this validation study. Figure 5.16 shows the

comparison between the two approaches. The good match between the results further

demonstrates the validity of the proposed numerical model.

Figure 5.16: Comparison of stress intensity factor for bridged fracture with various

bridging locations.

5.3.3 Simulation Results and Parametric Analysis

In this section, results from finite-element numerical simulations using the input

parameter in Table 5.3 are presented. Hoop stress around the wellbore and along the

fracture, and fracture width are investigated utilizing the list of influential parameters.

These parameters include horizontal stress contrast, LCM bridge location, fluid leak-off

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rate, pressure behind LCM bridge, Young’s modulus and Poisson’s ratio of the rock

formation.

5.3.3.1 Hoop Stress

Using the finite-element model described above, hoop stress states in the vicinity

of the wellbore and the fracture are analyzed for various combinations of influential

parameters. Figure 5.17 shows the hoop stress state before and after bridging the fracture

with LCM. The bridge location is 2 inches away from the fracture mouth. Throughout this

section, negative and positive stress values mean compressive and tensile stresses,

respectively. It is clear that, before bridging, the fracture tip is under tension and near-

wellbore rock is under compression. However, after bridging the fracture, there is a

compressive stress increase area near the bridging location, meaning the fracture is more

difficult to open; whereas the tensile stress near the fracture tip decreases, meaning the

fracture is more difficult to propagate. After bridging the fracture in wellbore

strengthening, lost circulation is less likely to continue.

Figure 5.17: Hoop stress distribution before (left) and after (right) bridging the fracture in

remedial wellbore strengthening treatment.

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Hoop Stress on Wellbore Wall

Horizontal Stress Contrast 𝑺𝑯𝒎𝒂𝒙/𝑺𝒉𝒎𝒊𝒏. Hoop stress on the wellbore wall from

0 to 90 degrees, with different horizontal stress contrasts, for scenarios without fracture,

with unbridged fracture, and with bridged fracture, are computed and shown in Figures

5.18a, 5. 18b and 5. 18c, respectively. For wellbore without fracture, hoop stress on the

wellbore within 30 degrees to X-axis (𝑆𝐻𝑚𝑎𝑥 direction) is tension. With increase in

horizontal stress contrast, the tensile stress increases. When there is a fracture created as

shown in Figure 5.18b, the tensile stress on the wellbore wall close to fracture changes to

compression. Figure 5.18c is the hoop stress after bridging the fracture at 2 inches away

from the wellbore. It’s not easy to see the stress differences between wellbore with

unbridged fracture and wellbore with bridged fracture from Figures 5.18b and 5.18c. To

facilitate observation, hoop stresses in a single case with 𝑆𝐻𝑚𝑎𝑥 𝑆ℎ𝑚𝑖𝑛⁄ = 1.3 before and

after bridging the fracture are compared in Figure 5.18d. It is clearly shown that

compressive hoop stress on the wellbore wall increases in the area near the fracture mouth

from 0 to 45 degrees and decreases in the area beyond 45 degrees. This is because after

bridging the fracture, fluid pressure behind the bridging point will decrease, as a result the

fracture will try to close. The healing of the fracture will stretch the formation around it,

leading to an increased tension (decreased compression). However, a rigid bridge restrains

healing of the fracture portion near the fracture mouth, resulting in a locally increased

compression; in the far area beyond 45 degrees increased tension still occurs due to the

overall healing behavior of the fracture. Increased compression near the fracture means

that the fracture is less likely to be opened after bridging. Decreased compression beyond

45 degrees means bridging the fracture actually weakens this wellbore portion and new

fractures may generate here with increased wellbore pressure. Figure 5.18d only shows a

single case with 𝑆𝐻𝑚𝑎𝑥 𝑆ℎ𝑚𝑖𝑛⁄ = 1.3 for clarity and conciseness. For other stress

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anisotropies (i.e. 𝑆𝐻𝑚𝑎𝑥 𝑆ℎ𝑚𝑖𝑛⁄ = 1.0, 1.1, 1.2, 1.4 𝑎𝑛𝑑 1.5 ), the same trend with

increased compression near and decreased compression far from the fracture mouth will

also be observed.

Figure 5.18: Hoop stress on wellbore wall for different horizontal stress contrasts: (a)

without fracture, (b) with unbridged fracture, (c) with bridged fracture, and

(d) comparison of hoop stresses before and after bridging for horizontal

stress contrast equal to 1.3.

LCM Bridge Location. Figure 5.19 shows hoop stress on the wellbore wall before

and after bridging the fracture at 0.5, 2 and 5 inches away from wellbore. When the LCM

bridge is close to the wellbore wall, e.g. the 0.5-inch case, there is a dramatic compressive

hoop stress increase on the wellbore wall near the fracture. However, with the bridging

location further away from the fracture, e.g. the 2.0-inch case, there is less hoop stress

increase. For a bridge at 5.0 inch, there is no clear hoop stress change – the stress curve in

Figure 5.19 overlaps with the one before bridging the fracture. Figure 5.19 shows that for

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a bridge at 0.5 inch, at the fracture mouth (0 degree) there is a small compression decrease

compared to the case of no bridging. This unexpected result needs further explanation. One

possible reason is that the stress at 0 degree is read from the corner element at the fracture

mouth in the numerical model; and when the bridge is very close to this element for the

case of bridging at 0.5 inch, some unavoidable computational error in the numerical

simulation might lead to this discrepancy, due to the very large displacement in this

particular corner element. However, the overall result illustrates that, from a hoop stress

enhancement point of view, the best place to bridge the fracture is at the fracture mouth.

Figure 5.19: Hoop stress on wellbore wall for different bridging locations.

Leak-off Rate. Depending on permeability of the rock formation and filter cake on

the wellbore wall, fluid can leak off into the formation at different rates. In this study,

different pre-defined leak-off rates on the wellbore wall and fracture face are applied to

investigate the effect of leak-off. Figures 5.20a, 5.20b, and 5.20c illustrate hoop stress on

the wellbore with different leak-off rates, for scenarios without fracture, with unbridged

fracture, and with bridged fracture, respectively. With no fracture on the wellbore wall,

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there is only a small difference in hoop stresses on wellbore for different leak-off rates, as

shown in Figure 5.20a. However, leak-off rate can have an obvious effect on wellbore hoop

stress when there is an unbridged or bridged fracture, especially in the area near the fracture

mouth, as shown in Figures 5.20b and 5.20c. The higher the leak-off rate, the smaller the

compressive hoop stress on the wellbore wall. Figure 5.20d compares the hoop stress

change on the wellbore wall before and after bridging the fracture with a leak-off rate of 4

inches/s. Similar to Figure 5.18d, there is a compressive hoop stress increase area on the

wellbore wall from 0 to 45 degrees after bridging the fracture. The results indicate that

considering fluid leak-off on the wellbore wall and fracture faces can improve wellbore

strengthening design and evaluation.

Figure 5.20: Hoop stress on wellbore wall for different leak-off rates: (a) without


comparison of hoop stresses before and after bridging the fracture for leak-

off rate equal to 4 inches/s.

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Hoop Stress along Fracture Faces

Horizontal Stress Contrast. In remedial wellbore strengthening treatment, it is

desired that hoop stress along the fracture faces can be enhanced, thus increasing the stress

acting to close the fracture. Figures 5.21a, 5.21b and 5.21c show hoop stress along the

facture faces with different horizontal stress contrast for scenarios without fracture, with

unbridged fracture, and with bridged fracture, respectively. The horizontal axis is the

distance away from the wellbore wall along the fracture direction. Before fracture creation,

as shown in Figure 5.21a, hoop stress along the fracture direction is tensile in the near

wellbore region, and becomes compressive with increase of distance away from the

wellbore wall. The higher the horizontal stress contrast, the larger the tensile stress and

tensile area. However, the near-wellbore tensile hoop stress becomes compressive when a

fracture is created, as illustrated in Figure 5.21b; whereas the near-fracture-tip compressive

stress becomes tensile. Figure 5.21b also shows horizontal stress contrast has negligible

influence on hoop stress along fracture faces. Note that in this study, different 𝑆𝐻𝑚𝑎𝑥

values are utilized to change horizontal stress contrast, whereas 𝑆ℎ𝑚𝑖𝑛 is kept as a constant

value. Figure 5.21c is the stress after bridging the fracture. Horizontal stress contrast still

has negligible influence on hoop stress along the fracture. But there is a significant

compression increase near the bridge location at 2 inches away from wellbore wall. Hoop

stresses along fracture, with horizontal stress contrast equal to 1.3, before and after

wellbore strengthening are compared in Figure 5.21d. It is clearly indicated that bridging

the fracture will significantly increase the compressive hoop stress near the LCM bridge

location, which makes the fracture harder to reopen.

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Figure 5.21: Hoop stress along fracture face for different horizontal stress contrasts: (a)

without fracture, (b) with unbridged fracture, (c) with bridged fracture, and

(d) comparison of hoop stresses before and after bridging the fracture for

horizontal stress contrast equal to 1.3.

LCM Bridge Location. Bridge location is a central issue in remedial wellbore

strengthening treatment. Figure 5.22 shows hoop stress along the fracture face with

different bridging locations. When the bridge is very close to the wellbore wall, e.g. the

0.5-inch case Figure 5.22, there is a dramatic compressive hoop stress increase around the

bridge location. However, with the bridge is away from the wellbore wall, the increase of

compressive hoop stress becomes smaller. There is almost no hoop stress change while

bridging the fracture near the fracture tip. For example, for the case with bridge at 5.0

inches away from wellbore wall, the stress after bridging is almost the same as that before

bridging, indicated as two overlapping curves in Figure 5.22. These results also

demonstrate that the best place to bridge the fracture is near the wellbore wall, and it is

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important to determine pre-existing fracture width for designing LCM particle size

distribution and optimizing bridging location.

Figure 5.22: Hoop stress along fracture face for different bridge locations.

Leak-off Rate. Figure 5.23 shows hoop stress along the fracture face with different

fluid leak-off rates from the wellbore wall and fracture faces into the formation. Similarly

to hoop stress around the wellbore (Figures 5.20b and 5.20c), after fracture creation and

bridging, the higher the leak-off rate, the smaller the compressive hoop stress along the

fracture, especially in the region close to the wellbore, as shown in Figures 5.23b and 5.23c.

This result seems completely opposite to the field and experimental observation that

conventional wellbore strengthening techniques are usually more effective in high-

permeable formations (e.g. sandstones) than low-permeable formations (e.g. shales). This

discrepancy can be explained as follows. In field and experimental practices, the

development and quality of the LCM bridge is usually the dominate factor for wellbore

strengthening consequences. In high-permeable formations, larger leak-off rate from the

fracture into the formation can greatly facilitate the consolidation of LCM particles in the

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fracture, so a bridge can develop quickly and usually with high quality. Conversely, in low-

permeable formations, the bridge quality is usually poor due to limited leak-off, resulting

in less strengthening benefit. But in this study, we assume a perfect bridge with infinite

strength and zero permeability exists. From a pure stress analysis, the results in Figures

5.23b and 5.23c are obtained, which only illustrate the stress distribution with a perfect

bridge, but does not reflect the development and quality of the bridge. Investigation of the

effect of LCM properties on wellbore strengthening is beyond the scope of this study.

Figure 5.23d compares hoop stresses along the fracture, before and after bridging the

fracture at 2 inches away from the wellbore, with a leak-off rate of 4 inches/s. Increased

compressive stress and decreased tensile stress are observed around the bridging location

and fracture tip, respectively.

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Figure 5.23: Hoop stress along fracture face for different leak-off rates: (a) without


comparison of hoop stresses before and after bridging the fracture for leak-

off rate equal to 4 inches/s.

Pressure Behind LCM Bridge. For the numerical studies above, the bridge is

assumed to be impermeable and therefore prevents pressure communication across the

bridge. It is assumed that pressure behind the LCM bridge will decline to formation

pressure as the fluid leaks off. However, in real situations, the bridge is likely not

completely impermeable in real situations, and fluid can flow across the bridge due to

pressure differential. Depending on fluid penetration across the bridge or the permeability

of the bridge, pressure behind the LCM bridge can vary from formation pressure (perfectly

impermeable bridge) to wellbore pressure (fully permeable bridge). Figure 5.24 shows

hoop stress along the fracture faces for pressure behind the LCM bridge varying from

formation pressure, 1800 psi, to wellbore pressure, 4000 psi. The higher the pressure

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behind the LCM bridge, the smaller the compression near the bridge location and the higher

the tension at the fracture tip. This lessens the effectiveness of wellbore strengthening. The

results in Figure 5.24 illustrate the importance of forming a low permeability bridge.

Figure 5.24: Hoop stress along fracture face for different pressures behind LCM bridge,

varying from formation pressure of 1800 psi to wellbore pressure of 4000

psi.

It should be noted that all of the above hoop stress calculations are performed for

the fracture with a constant length of 6 inches. This length was selected for two reasons.

First, it is argued that 6 inches is an indicative fracture length from practical experience by

Alberty and McLean (2004); and second this value is consistent with several numerical and

analytical wellbore strengthening studies by other investigators (Alberty and McLean,

2004; Arlanoglu, 2012; Mehrabian et al., 2015; Wang et al., 2007a, 2007b). Fracture length

is a major parameter determining the maximum sustainable pressure of the wellbore.

Changing its value will greatly affect wellbore strengthening results. As indicated by

Morita and Fuh (2012), if the fracture is effectively bridged at the same distance from the

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bridging point to the fracture mouth, the larger the total length of the fracture, the better

strengthening result can be achieved. More details for wellbore strengthening with different

fracture lengths can be found in the analytical studies by Morita and Fuh (2012) and Shahri

et al. (2014).

5.3.3.2 Fracture Width

Knowledge of fracture geometry, especially fracture width distribution, is essential

for selecting LCM sizes in designing a wellbore strengthening treatment. In this section,

the effects of a number of parameters on fracture width distribution before and after

bridging the fracture are investigated.

Figure 5.25 shows vertical displacement perpendicular to the fracture face, before

and after bridging the fracture using LCM. The bridge location is 2 inches away from the

wellbore wall. The displacement magnitude along the fracture face is the half width of the

fracture. The minus sign in Figure 5.25 means the fracture opening displacement is

opposite to the direction of the Y-axis. The blue and red colors indicate larger and smaller

fracture opening displacement (or fracture width), respectively. Fracture width decreases

after bridging the fracture, especially in the region behind the bridge location.

Note that in the following figures for plotting fracture width, a minus sign is still

used. But this sign only means the direction of fracture opening displacement, not a

negative fracture width.

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Figure 5.25: Vertical displacement distribution in the model before and after bridging the

fracture in wellbore strengthening.

Horizontal Stress Contrast. Figures 5.26a and 5.26b show fracture half-width

distribution along the fracture length with different horizontal stress contrasts with

unbridged and bridged fracture, respectively. The bridge location is 2 inches away from

the wellbore. For both cases, with increase in horizontal stress contrast, fracture half-width

increases, especially in the area close to the wellbore. Figure 5.26b shows that horizontal

stress contrast has a very small effect on fracture width behind the LCM bridge after

strengthening. Fracture widths before and after bridging the fracture for a particular case

with horizontal stress contrast equal to 1.3 are compared in Figure 5.26c. The results shows

that the fracture width behind the LCM bridge has a significant decrease after bridging the

fracture, which means the fracture is trying to close after the strengthening operation.

However, fracture width experiences a much smaller decrease ahead of bridge location,

likely due to the relatively higher fluid pressure in this part of the fracture.

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Figure 5.26: Fracture half-width distribution for different horizontal stress contrasts: a)

before bridging the fracture, b) after bridging the fracture, and c)

comparison of fracture half-widths before and after bridging the fracture for

horizontal stress contrast equal to 1.3.

LCM Bridge Location. Fracture half-widths before bridging the fracture and after

bridging it at 0.5, 2.0, 3.5 and 5.0 inches away from the wellbore wall are computed and

shown in Figure 5.27. When the fracture is bridged near the wellbore wall, for example the

0.5-inch case, the fracture experiences a larger decrease in its width, compared with

bridging away from the wellbore. For a bridge at 5.0 inch, there is no clear width change –

the width curve in Figure 5.27 overlaps with that before bridging the fracture. Since the

objective of wellbore strengthening is to prevent fracture opening and propagation, Figure

5.27 also shows that the best place to bridge the fracture is at the fracture mouth.

Figure 5.27: Fracture half-width distribution for different LCM bridge locations.

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Leak-off Rate. Leak-off rate also affects fracture width distribution. Figures 5.28a

and 5.28b are fracture half-widths for different leak-off rates before and after bridging the

fracture at 2 inches away from wellbore. For both cases, higher leak-off rate results in

smaller fracture width. Figure 5.28b illustrates that leak-off rate has a relatively larger

influence on fracture width ahead of the LCM bridge location, with a relatively smaller

influence behind the bridge.

Figure 5.28: Fracture half-width distribution for different leak-off rates: (a) before

bridging the fracture, (b) after bridging the fracture.

Young’s Modulus. Young’s modulus has a significant influence on fracture width.

As shown in Figure 5.29, larger Young’s modulus leads to a smaller width for the fracture,

both before and after bridging at 2 inches away from the wellbore. Additionally, a fracture

with smaller Young’s modulus has a much larger width decrease after bridging than a

fracture with larger Young’s modulus, especially the part behind the LCM bridge. This

result demonstrates the importance of accurate Young’s modulus data for reliable

prediction of fracture width, hence LCM size optimization.

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Figure 5.29: Fracture half-width distribution for different Young’s Modulus before and

after bridging the fracture.

Poisson’s Ratio. Figure 5.30 shows fracture half-width before and after bridging

the fracture at 2 inches away from the wellbore wall with Poisson’s ratios of 0.2 and 0.3.

The results show, compared with Young’s modulus, Poisson’s ration has a much smaller

influence on fracture widths before and after bridging. An apparently larger Poisson’s ratio

only results in a slightly smaller fracture width.

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Figure 5.30: Fracture half-width distribution for different Poisson’s ratios before and after

bridging the fracture.

Pressure behind LCM Bridge. As discussed above, pressure behind the LCM

bridge can vary from formation pressure (perfectly impermeable bridge) to wellbore

pressure (fully permeable bridge) depending on the permeability of the LCM bridge. Figure

5.31 shows the fracture half-width distribution for pressure behind the LCM bridge varying

from formation pressure, 1800 psi, to wellbore pressure, 4000 psi. The lower the pressure

behind the LCM bridge, the smaller the fracture width behind the LCM bridge. This, again,

means the less permeable the LCM bridge, the more effective the strengthening operation.

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Figure 5.31: Fracture half-width distribution for different pressure behind LCM bridge,

varying from formation pressure 1800 psi to wellbore pressure 4000 psi.

5.3.4 Conclusions and Implications of the Numerical Study

For effectively strengthening a wellbore, it is necessary to understand the

fundamentals of lost circulation and wellbore strengthening. Because the fracture

growth/closure behavior and corresponding stress and dimension changes are difficult to

measure in the field, numerical simulation (e.g. finite element modeling) is especially

useful for studying these processes and revealing useful information for field applications.

Regarding hoop stress as an important consideration in remedial wellbore strengthening

treatment, the following conclusions and field implications are indicated from the above

finite-element modeling analysis.

After bridging a fracture, there is a compression increase area near the bridging

location, whereas the tensile stress near the fracture tip decreases. This means that

the fracture becomes more difficult to reopen and propagate, and lost circulation is

less likely to continue.

Hoop stress around wellbore can be substantially increased by bridging the fracture

near the wellbore wall. Bridging the fracture near the fracture tip does little to

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increase hoop stress and strengthen the wellbore. It is important to correctly predict

fracture width and optimize LCM size distribution in wellbore strengthening

design.

Fluid leak-off affects both hoop stress and fracture width distributions. It is

important to consider fluid leak-off on the wellbore wall and fracture faces in

predicting fracture geometry, optimizing LCM size distribution, and evaluating

potential hoop stress enhancement with wellbore strengthening.

For effectively strengthening the wellbore by preventing fluid communication

between the wellbore and fracture tip, a desirable LCM bridge should have low

enough permeability to ensure there is no fluid flow across the bridge and the

fracture portion behind the bridge can close due to fluid leak-off. The permeability

of the bridging plug is likely a more important parameter than its strength.

Young’s modulus has a significant influence on fracture width distribution in

wellbore strengthening. It is critical to obtain reliable Young’s modulus values for

prediction of fracture width, hence optimization of LCM size. However, Poisson’s

ratio only has a very small effect on fracture width distribution.

It is well-known from fracture mechanics that the longer the fracture the less net

pressure (difference between fluid pressure inside the fracture and field stress acting

on fracture faces) is needed to propagate the fracture. However, in a real field

situation, the fracture length is hard to predict. However, from these simulation

results the best place to bridge the fracture, from a point of view of hoop stress

enhancement, is at its mouth or entrance on the wellbore wall. This simplifies the

problem, i.e., bridge the fracture mouth, then fracture length is of much less

consequence.

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5.4 SUMMARY

The goal of this chapter is to develop models to investigate the remedial wellbore

strengthening method based on plugging/bridging lost circulation fractures using LCMs.

An analytical solution and a finite-element model are proposed for modeling this problem.

The analytical solution derived based on liner elastic fracture mechanics and

superposition principle can be used for a fast prediction of fracture pressure enhancement

due to fracture bridging, taking into account near-wellbore stress concentration, in-situ

stress anisotropy, and different LCM bridge locations. The equations presented in the

solution can be used to perform quantitative parameter sensitivity analyses to illustrate the

undying mechanisms of wellbore strengthening. In addition, compared with numerical

methods, conciseness of the analytical approach offers the possibility to conduct fracture

pressure prediction in near-real-time drilling operations. Such capability would lead to

timely wellbore strengthening evaluation and mud weight adjustments.

However, the analytical model cannot provide detailed stress and displacement

information local to wellbore and fracture in remedial wellbore strengthening treatments.

Besides, it does not take into account the porous nature of formation rock. Therefore, a

poroelastic finite-element numerical model, considering fluid flow and fluid leak-off, is

also developed in this chapter. The finite-element model can be used to quantify near-

wellbore stress and fracture geometry, before and after bridging fractures in remedial

wellbore strengthening treatment. Effects of various parameters are investigated through a

comprehensive parametric study using the numerical model, and several useful

implications for field applications are proposed based on the parametric study.

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CHAPTER 6: Cement Interface Fracturing3

With ongoing environmental concerns and increasingly stringent regulations, the

traditional topic of wellbore integrity is more important than ever before. A great deal of

importance is attached to the cement sheath because it is necessary to provide zonal

isolation and well integrity during the life of a well. A quality cement sheath with low

permeability can effectively prevent hydraulic communication of formations to the

borehole and to other formation layers cased off during well construction. However,

leakage from the cement sheath frequently leads to well integrity problems in operations.

These failures may occur at any time throughout the life of a well - during drilling,

completion, production, and even after well abandonment. Debonding at the casing/cement

or cement/formation interfaces, which may result in substantial flow channels and fluid

leakage, is often responsible for the loss of wellbore integrity.

In this chapter, a three-dimensional finite-element framework is developed to exam

the possibility of fluid leakage through casing shoe and along the weak cement interface

when there is pressure buildup in the wellbore due to change of drilling or completion fluid,

conduction of injectivity tests, and etc. Cement interfaces to casing and formation are

represented with zero-thickness pore pressure cohesive element layers, so as to model

debonding fracture initiation and propagation, as well as tangential and normal fluid flow

within the fracture. The model is used to quantify the length, width, and circumferential

coverage of the debonding fracture.

3 Parts of this chapter has been published in the following conference paper which was supervised by K. E.

Gray:

Feng Y., Podnos, E. and Gray K. E. “Well Integrity Analysis: 3D Numerical Modeling of Cement

Interface Debonding" presented at the 50th US Rock Mechanics / Geomechanics Symposium held in

Houston, Texas, USA, 26-29 June 2016.

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6.1 INTRODUCTION

During well construction, fluid may not only lose in the open-hole section through

hydraulically induced fractures, it may also lose through leakage at casing shoe with poor

cement integrity.

Well integrity, always of importance, is becoming more so because of increasing

environmental concerns and regulatory activities. Moreover, increasingly hostile

environments such as HT/HP fields, ultra-deep-water fields, and geothermal fields, along

with more complicated development operations, such as those encountered in gas

producing wells, gas storage wells, water injectors, and cuttings/waste injectors, all present

new challenges for well integrity. Failure of well integrity can lead to costly remedial

operations or total loss of the well, severe environment contamination, and even fatal

accidents to operating personnel.

The cement sheath is the heart of well integrity. It is expected to ensure well

integrity by providing zonal isolation and support for the casing throughout the life of a

well, from well construction, hydrocarbon production, to post-abandonment. A successful

cementing job is expected to result in complete zonal isolation, without leaving any leakage

pathway in the annulus between casing and formation. Unfortunately, this goal is not

always achieved, and failure of cement sheath commonly occurs during the life of a well

(Fourmaintraux et al., 2005).

Failure modes of the cement sheath may include: (1) tensile cracking in the cement,

(2) plastic deformation in the cement, and (3) debonding at the cement/casing and/or

cement/formation interfaces. These modes are thought to result from high stress/pressure

levels encountered in the cement sheath during a well’s life. Cement sheath failure may

also be induced by improper cement placement because of high well inclination, poor hole

calibration, poor centralization, poor selection of chemical agents, or poor mud removal

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(Bois et al., 2011). For this study, attention is focused mainly on debonding at the cement

interfaces when there is pressure buildup in the wellbore due to change of drilling or

completion fluid, conduction of injectivity tests, and etc.

Several researchers have developed models to simulate cement interface

debonding, based on simplifications such as assumptions of linear elastic materials (casing,

cement, rock) and initially intact cement sheath (Bosma et al., 1999; Fleckenstein et al.,

2001; Fourmaintraux et al., 2005; Gray et al., 2009; Pattillo and Kristiansen, 2002; Ravi et

al., 2002; Shahri et al., 2005). However, laboratory tests performed on a Class-G cement

system showed that cement is better characterized as a porous media, and modeled not as

a one-phase linear elastic medium (Bois et al., 2012; Ghabezloo et al., 2008). It is also well

known that the cement and formation may not always behavior elastically, and plastic

deformation commonly occurs, especially in the vicinity of wellbore where high stress

level usually exists due to local stress concentration (Gray et al., 2009). Therefore, selection

of appropriate constitutive laws, which can correctly describe cement and formation

behavior, is a key point for cement sheath modeling (Bois et al., 2011).

Most previous studies (Bosma et al., 1999; Fleckenstein et al., 2001; Fourmaintraux

et al., 2005; Gray et al., 2009; Ravi et al., 2002; Shahri et al., 2005) did not consider initially

existing defects at the cement interfaces due to poor mud removal, cement shrinkage during

hydration, or other operational factors, before simulating interface debonding by applying

stress/pressure boundary conditions. Additionally, most researchers have modeled

interface debonding as a tensile or shear failure due to high local stress induced by the

combined effects of far-field in-situ stress (usually not isotropic), fluid pressure in wellbore

(likely subjected to large changes during the life of a well), temperature changes, and

different mechanical properties of the casing, cement and formation. However, when an

initial crack due to poor mud removal or cement shrinkage, or an induced crack resulting

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from excessive shear/tensile stress, exists at the cement interface, then fluid may invade

into the crack, leading to pressure buildup and, ultimately to a fluid-driven fracture that

propagates along the interface. Consequences of the debonding fracture propagation, such

as leakage of formation fluid to the surface and/or uncontrolled inter-zonal flow, can result

in severe operational troubles and/or substantial environmental pollution. Circumstances

in which fluid-driven debonding fracture may occur include:

Changing mud while drilling

Pressure testing/cycling during drilling, completion, and production

Perforation operations

Hydraulic fracturing

Fluid injection, such as gas injection, produced water re-injection, drilling

cutting re-injection, steam injection, and flooding operations.

Gray et al. (2009) developed numerical models to investigate cement interface

debonding by considering the casing/cement interface as a contact condition, which may

allow zero or some amount of tension transmission across the interface, corresponding to

the cases with no bonding strength and finite bonding strength respectively. Other

researches modeled the cement interface as a layer of interface elements based on a

Coulomb friction model (Bosma et al., 1999; Ravi et al., 2002). While such models may

be satisfactory for analyzing interface debonding when there is no fluid invasion into the

annular cracks after debonding, they cannot simulate the propagation of fluid-driven cracks

along the interface, which requires fully-coupled modeling of the mechanical behavior of

the casing/cement/formation and fluid flow in the cracks.

Great progress has been made in fully-coupled modeling of fluid-driven fracture in

porous media in the past decade, such as the coupled pore pressure cohesive zone method

and coupled pore pressure extended finite element method (Kostov et al., 2015; Searles et

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al., 2016; Wang, 2015; Yao et al., 2010). These methods can successfully account for

several key phenomena of fluid-driven fracture in porous media, including fluid flow

within the fracture, pore fluid flow in the porous media, deformation of porous medium,

and fracture propagation (Zielonka, et al. 2015). However, these techniques have not

been applied to the study of cement sheath integrity until recently (Wang and Taleghani,

2014). To the author’s knowledge, Wang and Taleghani (2015) first applied a coupled pore

pressure cohesive zone method to this problem. They successfully modeled fluid-driven

fracture propagation along the cement interfaces, which improves the understanding of

cement bond failure due to excessive fluid pressure buildup at the interface.

The study presented in this chapter is another implementation of the coupled pore

pressure cohesive zone method to cement sheath integrity modeling. While the overall

approach of this chapter is similar to that of Wang and Taleghani, i.e. both are based on

cohesive zone theory and finite-element analysis, the model assumptions, interpretation

methods, and specific focus are different. Wang and Taleghani (2015) mainly focused on

investigating the effects of interface properties on the propagation of debonding fracture,

while the aim of this study is to simulate non-uniform debonding fracture with various in-

situ stress conditions, cement and formation properties, and pre-existing cracks at the

cement interfaces.

6.2 DEVELOPMENT OF CEMENT INTERFACE FRACTURE MODEL

6.2.1 Modeling Goals

The first goal of this study is to develop a general finite-element framework for

simulating cement sheath debonding in a vertical well due to fluid-driven fracture

propagation along the circular interfaces of cement to casing and formation. The

framework should also take into account other crucial aspects of the

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casing/cement/formation system, including nonlinearity of the material properties,

interaction between different materials, poromechanical nature of cement, plastic

deformation of cement and formation, three dimensionality and anisotropy of in-situ

stresses. The framework should have the flexibility to be readily modified for a non-vertical

well, casing eccentricity, and more complex material models.

The second objective of the paper is to apply this finite-element framework to

investigate the propagation of debonding fracture from a casing shoe, with various initial

micro-annular-fractures around the shoe, different in-situ stress anisotropies, and different

material properties. Also, development of plastic deformation in the cement during

debonding fracture growth will be followed.

To achieve these goals, the SIMULIA Abaqus® Standard finite-element package

was selected for developing the model. It is a general-purpose finite-element code

developed to solve linear and nonlinear stress-analysis problems, with capabilities of

constructing 3D geometry, a variety of nonlinear solvers, and an extended collection of

material models (Gray et al., 2009). The recently developed capabilities of the Abaqus®

package, in modeling fully-coupled, fluid-driven fracture in porous medium, are

particularly useful for studies of hydraulic fracturing related problems in the petroleum

industry (Kostov et al., 2015; Searles et al., 2016; Wang, 2015; Yao et al., 2010). In this

study, fracture propagation and fluid flow in the fluid-driven fracture along the cement

interface are modeled using coupled pore pressure cohesive zone method as described in

Section 3.2. A traction-separation constitutive law and a fluid flow constitutive law are

incorporated into the cohesive zone model to describe the two phenomena respectively.

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6.2.2 Model Geometry and Discretization

Geometry. Since a debonding fracture may develop both circumferentially and

vertically at the cement interface with a non-uniform pattern, the traditional 2D model,

assuming plane stress/strain conditions, cannot correctly predict fracture growth (Wang

and Taleghani, 2014). In addition, accurate modeling of nonlinear effects, such as interface

discontinuity and material plasticity, depends highly on the ability to model 3D stress state

(Gray et al., 2009). Therefore, a three-dimensional model was developed in this study for

the cement sheath integrity problem, as shown in Figure 6.1.

A vertical well is modeled with its axial direction coinciding with the Z-axis of the

Cartesian coordinate system, as depicted in Figure 6.1. The X-axis and Y-axis are assigned

along the direction of the maximum and minimum horizontal stresses, respectively. The

components of the model include casing, cement, formation as shown in the plane view in

Figure 6.1b, and the interface between casing and cement as shown Figure 6.1c. For the

present study, only debonding at the casing/cement interface is considered, while the outer

interface at the cement/formation is assumed to be functionally bonded. However,

debonding at the outer interface can be easily included with minor model modification.

Because of the symmetry of the problem, only one quarter of the system is modeled.

The casing inner radius is 8.41 cm, and outer radius is 9.69 cm, i.e. the casing

thickness is 1.28 cm. The drilling hole size is 12.07 cm, hence the thickness of the cement

sheath between casing and formation is 2.38 cm. The casing/cement interface is modeled

using a layer of cohesive elements with zero thickness. The total size of the quarter model

is 10×10×40 m. The casing shoe, where leakage occurs and pressure builds up, is at the

bottom of the wellbore.

Discretization. The casing is discretized using 3D linear full-integration elements

without degree of freedom of pore pressure. Cement and formation are discretized using

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3D linear full-integration elements with degree of freedom of pore pressure. The

casing/cement interface is discretized with a layer of coupled pore pressure and

deformation cohesive elements which can model propagation of the debonding fracture

and fluid flow in the fracture, as described in Section 3.2. Figure 6.1 shows the overall

mesh of the model, with 1500, 2000, 20500, and 500 elements being used to mesh casing,

cement, formation, and cohesive interface, respectively.

Since interface failure, large plastic deformation, and high stress levels are expected

close to the wellbore, especially near the casing shoe, the mesh in this region is refined,

while coarser elements are used farther away from the casing shoe, both in lateral and

vertical directions.

Figure 6.1: Cement sheath model. (a) the one-quarter geometry; (b) top view of the

casing/cement/formation system; (c) the interface between casing and

cement.

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6.2.3 Material Variables

Casing. The steel casing is assumed to be a linearly elastic material in this study,

with Young’s modulus, 𝐸, and Poisson’s ratio, 𝑣, equal to 2 × 108 kPa and 0.27 ,

respectively.

Cement and Formation. The cement and formation rock are modeled as

elastic/perfectly plastic porous materials. The Mohr-Coulomb plasticity model is used. The

elastic material parameters include Young’s modulus, 𝐸, and Poisson’s ratio , 𝑣, ; the

Mohr-Coulomb plastic parameters include internal friction angle, 𝜑, and cohesive

strength , 𝑐 ; and the porous properties include porosity, ∅, and permeability, 𝑘 . The

properties of cement and formation used in this study are reported in Tables 6.1 and 6.2,

respectively, with each of them consisting of a base case and several supplementary cases

for sensitivity study purposes.

Interface Bond. As stated above, the interface between casing and cement is

modeled using a layer of zero thickness coupled pore pressure and deformation cohesive

elements, which incorporates a traction-separation law to describe fracture propagation

behavior and a flow rule for fluid flow in the fracture. A major challenge in using a cohesive

model is the determination of cohesive properties of the casing/cement interface. Very little

data are available in the literature. Here the cohesive properties used for the casing/cement

interface, including tensile strength, shear strength, cohesive stiffness, and critical energy,

are based on data reported by Wang and Taleghani (2015), where they estimated cohesive

properties of the cement interface through numerically simulating and matching the pipe

(casing) push-out tests by Carter and Evans (1964). The leakage fluid is assumed to be

water, and a very small leak-off coefficient is used for the fracture surface on the cement

side due to low permeability of the cement, while zero leak-off is defined on the casing

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side due to the impermeable nature of casing. The parameters utilized for the cohesive

interface bond are summarized in Table 6.3.

Table 6.1: Cement properties.

Young’s

Modulus, 𝐸

Poisson’s

Ratio, 𝑣

Friction

Angle, 𝜑

Cohesive

strength, 𝑐

Permeability,

𝑘

Porosity,

∅

Type kPa (o) kPa mD

C1 2.00E+07 0.26 27 8.00E+03 0.05 0.2

C2* 3.00E+07 0.26 27 1.00E+04 0.05 0.2

C3 8.00E+07 0.26 27 1.50E+04 0.05 0.2

Note: C2 is the base case.

Table 6.2: Formation properties.

Young’s

Modulus, 𝐸

Poisson’s

Ratio, 𝑣

Friction

Angle, 𝜑

Cohesive

strength, 𝑐

Permeability,

𝑘

Porosity,

∅

Type kPa (o) kPa mD

F1* 3.30E+06 0.26 30 5.00E+03 1.0 0.2

F2 5.00E+06 0.26 30 7.00E+03 1.0 0.2

F3 4.00E+07 0.26 30 1.20E+04 1.0 0.2

Note: F1 is the base case.

Table 6.3: Interface bond properties.

Cohesive

Stiffness

Shear

Strength

Tensile

Strength

Critical

Energy

Leak-off

Coefficient

Fluid

Viscosity

kPa kPa kPa J/m2 m/s/Pa cp

8.50E+07 2,000 500 100 5.897E-12 1.0

6.2.4 Boundary Conditions and Simulation Steps

Boundary Conditions. A symmetric boundary technique is used to reduce

computation burden of the 3D model, hence only one-quarter of the

casing/cement/formation domain is modeled as shown in Figure 6.1. Symmetry boundary

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conditions are applied to the front and left surfaces of the model in Figure 6.1. The vertical

displacement at the bottom surface is constrained. An overburden pressure of 11,000 kPa

is applied at the top surface of model. Maximum horizontal stress parallel to the X-axis

and minimum horizontal stress parallel to the Y-axis are applied on the right surface and

back surface of Figure 6.1, respectively. Different horizontal stress ratios, as given in Table

6.4, are used to investigate the effect of stress anisotropy on cement interface debonding.

The initial pore pressure is assumed equal to 6,000 kPa. A constant wellbore fluid pressure

of 7,200 kPa is applied on the inner surface of the casing. Because thickness of the

simulation domain is relatively small, no variation in initial stress or pressure is considered

in the vertical direction. Table 6.4 summarizes different types of loads assigned to model.

Table 6.4: In-situ stresses, pore pressure and wellbore pressure applied to the model.

Horizontal

Stress

Ratio*

Minimum

Horizontal

Stress

Maximum

Horizontal

Stress

Overburden

Pressure

Initial

Pore

Pressure

Wellbore

Pressure

Type kPa kPa kPa kPa kPa

SR0* 1 9,600 9,600 11,000 6,000 7,200

SR1 1 8,000 8,000 11,000 6,000 7,200

SR1.1 1.1 8,000 8,800 11,000 6,000 7,200

SR1.2 1.2 8,000 9,600 11,000 6,000 7,200

SR1.25 1.25 8,000 10,000 11,000 6,000 7,200

SR1.3 1.3 8,000 10,400 11,000 6,000 7,200

SR1.35 1.35 8,000 10,800 11,000 6,000 7,200

SR1.4 1.4 8,000 11,200 11,000 6,000 7,200

Note: SR0 is the base case. Horizontal stress ratio is the ratio of maximum horizontal stress to

minimum horizontal stress.

Initially existing cracks, through which fluid leakage occurs and a fracture initiates,

are assigned at the casing/cement interface around the casing shoe. The width and height

of the initial cracks are constant and equal to 2 mm and 20 mm respectively, while various

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circumferential crack extents are used, in an attempt to investigate interface debonding

patterns with different initial crack shapes. The circumferential extent of the initial crack

is quantified using an arc angle. A 0𝑜 arc angle indicates there is no initial crack, and a

90𝑜 angle means a crack extending throughout the circumference of the circular interface.

Table 6.5 provides a detailed description of the initial crack shapes modeled in this study.

Table 6.5: Geometry of the initial cracks at the casing shoe.

Type Q1 Q2 Q3 Q4*

Initial

Crack

Shape

Circumferential

Extent, (o) 30 45 60 90

Thickness, mm 2 2 2 2

Height, mm 20 20 20 20

Note: Q4 is the base case.

Simulation Steps. Two steps were used in each simulation. At the first step, far-

field horizontal stress, overburden pressure, and initial pore pressure are applied to the

mesh. Initial state of stress reaches equilibrium in this step. No debonding occurs during

the equilibrium process except for pre-assigned cracks around the casing shoe. The second

step simulates leakage and pressure buildup at the casing shoe and their consequences,

including debonding fracture propagation along the casing/cement interface and plastic

deformation development in the cement at the same time. Pressure buildup around the

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casing shoe is modeled as a result of fluid charge (leakage) into the cement interface with

a very small charge rate.

6.3 RESULTS AND DISCUSSION

Using the finite-element model described above, the cement interface debonding is

investigated for various combinations of in-situ stresses, pre-exiting cracks around the

casing shoe, and cement and formation properties. The following subsections summarize

the simulation results.

6.3.1 Cement Interface Debonding with Anisotropic Horizontal Stress

In-situ stresses are always compressive and resist opening of cement interface

unless the stress anisotropy is too large. However, with a leakage, fluid pressure may build

up in the interface, ultimately, resulting in sufficient pressure to overcome resistance of the

in-situ compressive stress and, hence, initiation of a hydraulic fracture along the interface.

To investigate the effects of non-uniform (horizontal) in-situ stresses on development of

the debonding fracture, several cases with different horizontal stress ratios as shown in

Table 6.4 are simulated. For comparison purposes, only two cases, SR1 and SR1.2 with

horizontal stress ratios equal to 1.0 (uniform) and 1.2 respectively, are selected for analysis.

Figure 6.2 illustrates creation of the debonding fracture near the casing shoe for the

two selected cases by the end of the second simulation step. A comparison between the two

cases shows that:

For the case (SR1) with uniform horizontal stress, the width of the debonding

fracture, as expected, is uniform around the casing, as shown in Figures 6.2 and 6.3.

This is because everything, i.e. stress and material properties, is uniformly

distributed in this case.

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For the case (SR1.2) with anisotropic horizontal stress, the debonding fracture is

not uniform, with a smaller (or even zero) width in the direction of the maximum

horizontal stress (X-axis) and a larger width in the direction of the minimum

horizontal stress (Y-axis), as shown in Figures 6.2 and 6.3. This is because the

casing/cement interface is under a larger compressive stress in the direction of

maximum horizontal stress, which resists fracture opening, but under a smaller

compression in the direction of minimum horizontal stress. This observation is

consistent with the conclusion of hydraulic fracturing studies that a fracture should

occur in the plane perpendicular to the least principal stress (Gray et al., 2009;

Hubbert and Willis, 1957; Yew and Weng, 2014; Zoback, 2010).

Figure 6.2: Interface fracture opening of two cases with uniform horizontal stress (SR1)

and non-uniform horizontal stress (SR1.2). The pictures are top views of the

cut sections of the casing/cement/interface system at 0.5 m above the casing

shoe.

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Figure 6.3: Interface fracture width around wellbore for the two cases in Figure 6.2. 0-

degree and 90-degree correspond to the directions of the maximum and

minimum horizontal stress, respectively.

6.3.2 Cement Interface Debonding with Different Initial Cracks

Figure 6.4 shows the growth of the fracture, distribution of pressure in the fracture,

and distribution of plastic strain in the cement by the end of the second step for the four

cases with different initially existing cracks around the casing shoe as given in Table 6.5.

The in-situ stress type used is SR1.2 in Table 6.4 and the material properties used are the

base values described in Section 6.2.3. The results show that the patterns of the

developments of fracture, pressure, and plastic strain are consistent for each case, i.e. where

fracture occurs, high pressure in the fracture and large plastic strain in the cement exist.

The large plastic strain induced by fracture propagation may lead to material damage, such

as creation of voids and micro-cracks in the cement (Gray et al., 2009), and consequently

further damage of well integrity.

Another observation is that the fluid-driven debonding fracture tends to develop

vertically, rather than circumferentially around the casing. When a 90𝑜 initial-

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circumferential crack exists, for example, case Q4 in Table 6.5 and Figure 6.4, the fracture

will propagate at the minimum horizontal stress (Y-axis) side because the resistance to

fracture opening is the smallest there as explained above. However, if the initial crack does

not cover the full circumference of the interface, such as cases Q1, Q2, and Q3 in Table

6.5 and Figure 6.4, the fracture is more inclined to propagate in the vertical direction, rather

than in the circumferential direction, and results in a larger fracture height. In the three

cases, the fractures actually had some circumferential growth, but they were not able to

extend completely to the side of the minimum horizontal stress (Y-axis), even though there

is small resistance for fracture propagation.

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Figure 6.4: Developments of fracture geometry, fracture pressure and cement plastic strain

with different initial crack sizes. The pictures are front views of the one-

quarter model, with maximum horizontal stress in the X-axis direction and

minimum horizontal stress in the Y-axis direction. The circumferential

extents of the initial cracks for cases Q1 through Q4 are 30𝑜, 45𝑜, 60𝑜 and

90𝑜, respectively, as given in Table 5.5.

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6.3.3 Cement Interface Debonding with Different Cement and Formation

Properties

To investigate the influence of cement properties on the propagation of debonding

fracture due to excessive pressure buildup around the casing shoe, three types of cement

system with different stiffness, as given in Table 6.1, are considered. Type C3 corresponds

to “stiff-cement” with a high value of Young’s modulus and a high value of cohesive

strength. To the contrary, type C1 is “compliant-cement” with lower values of Young’s

modulus and cohesive strength. Properties of cement type C2 are in between those for stiff-

and compliant-cement. All the other input variables are according to values of the base

cases provided in Section 6.2.

Figure 6.5 shows the fracture growth and pressure distribution at the cement

interface for the three cases with different cement stiffness by the end of the second

simulation step. The debonding fracture with stiff cement has a larger growth in height than

the facture with soft cement. This is because stiffer cement restricts fracture opening in the

radial direction of the wellbore. Thus, with the same amount of fluid leakage into the casing

shoe, the stiffer cement shows a larger fracture height.

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Figure 6.5: Developments of fracture geometry and fracture pressure with different cement

properties. The pictures are front views of the one-quarter model, with

maximum horizontal stress in the X-axis direction and minimum horizontal

stress in the Y-axis direction. The properties for cement types C1 through C3

are given in Table 6.1.

Effects of formation properties on debonding fracture propagation were also

investigated. Similar to the analysis of cement properties, three types of formation system

with different stiffness as given in Table 6.2 are studied. Formation type F3 corresponds to

“stiff-formation” with high values of Young’s modulus and cohesive strength, and

formation type F1 is “compliant- formation” with lower Young’s modulus and cohesive

strength. Properties of formation type F2 is in between of them. All the other input variables

are based on the values of base cases provided in Section 6.2. Figure 6.6 illustrates the

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simulation results of the three cases. It shows a strong dependence of the development of

fracture height on the stiffness of the formation rock. The higher stiffness (i.e. the higher

the Young’s modulus and the higher the cohesive strength) of the formation, the larger the

fracture height that will develop. This result can be explained with the same statement that

was made above for the influence of cement stiffness on fracture growth.

Figure 6.6: Development of fracture geometry and fracture pressure with different

formation properties. The pictures are front views of the one-quarter model,

with maximum horizontal stress in the X-axis direction and minimum

horizontal stress in the Y-axis direction. The properties for formation types

F1 through F3 are given in Table 2.

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6.4 SUMMARY

A three-dimensional finite-element model is developed to simulate the cement

debonding due to the propagation of a fluid-driven fracture along the cement interface. The

model successfully takes into account the main elements for well integrity analysis, such

as interaction between different materials, poromechanical nature of the cement, plastic

deformation of cement and formation, three dimensionality and anisotropy of in-situ

stresses, propagation of debonding fracture, and flow of fracturing fluid. The model was

used to quantify fracture geometry and fracture pressure distribution for various conditions

with different in-situ stresses, pre-exiting cracks at the casing shoe, and cement and

formation properties. The results show that non-uniform debonding fractures occur under

these conditions. With anisotropic horizontal stress, the debonding fracture has a smaller

width in the direction of the maximum horizontal stress and a larger width in the direction

of the minimum horizontal stress. With initial cracks in the cement interface, debonding

fractures tend to develop vertically along the axis direction of a vertical well, rather than

circumferentially around the well. The results also demonstrate that the debonding fracture

propagation is highly influenced by the stiffness of cement and formation. The proposed

model provides a useful tool for simulating the debonding of cement interface caused by

leakage and pressure buildup around the casing shoe. It is useful for evaluating the risk of

cement sheath failure and fluid leakage from cement interface during drilling, pressure

tests, perforation, hydraulic fracturing, and any kinds of fluid or gas injection operations

during the production phase.

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CHAPTER 7: Role of Field Injectivity Tests on Combating Lost

Circulation4

Fracture parameters such as fracture initiation and propagation pressure and in-situ

stresses such as minimum horizontal stress and overburden, are critical inputs for

petroleum exploration design. They are particularly important for lost circulation

evaluations and wellbore strengthening applications. Field injectivity tests during drilling,

including leak-off test (LOT), extended leak-off test (XLOT), and pump-in and flow-back

test (PIFB), are promising and highly reliable methods for determining fracture parameters

and/or underground stress information. Moreover, such tests are extremely helpful for

understanding the fracture growth mechanism from the wellbore wall to the far field region,

which is equally important for study of lost circulation and wellbore strengthening. This

chapter highlights the importance of field injectivity tests for understanding the

fundamentals of lost circulation and wellbore strengthening, with a review of different

kinds of field tests and a discussion of their advantages and limitations. A coupled fluid

flow and geomechanics injectivity test model is also developed which can capture the key

elements of injectivity tests known from field observation and aid the interpretation and

design of field tests.

4 Parts of this chapter has been published in the following conference papers which were supervised by K.

E. Gray:

Feng Y., Jones J. F., Gray K. E. "Pump-in and Flow-back Tests for Determination of Fracture Parameters

and In-situ Stresses" presented at the 2015AADE National Technical Conference and Exhibition, San

Antonio, Texas, April 8-9, 2015.

Feng Y., Gray K. E. “Signatures and Interpretations of Pump-in and Flow-back Tests in High

Permeability and Low Permeability Formations " presented at the 5th GEOProc International Conference

on Coupled Thermo-Hydro-Mechanical-Chemical (THMC) Processes in Geosystems, Salt Lake City,

Utah, February 25-27, 2015.

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7.1 INTRODUCTION

For drilling wells safely and efficiently, precise determination of minimum

horizontal stress and fracture parameters (e.g. fracture initiation and propagation pressure)

is critical for the drilling industry. These data are required for evaluating wellbore stability,

determining drilling mud weights, predicting wellbore breathing and lost circulation,

designing wellbore strengthening treatments, and selecting casing shoe depths.

Indirect methods from logs and geomechanical modeling are often used to predict

fracture gradient and minimum horizontal stress (Zoback, 2010). However, where a high

degree of precision is required, a direct field injectivity test measurement is usually needed

to calibrate the predicted results. Field injectivity tests performed in drilling operation may

include formation integrity tests (FITs), leak-off tests (LOTs), extended leak-off tests

(XLOTs), and pump-in and flow-back tests (PIFBs). But not all these tests can provide

reliable stress information, either due to insufficient injection time/volume (resulting in

insufficient fracture length for stress evaluation), or due to a number of factors distorting

test signatures and leading to interpretation difficulties and uncertainties.

Misinterpretation of minimum horizontal stress and fracture parameters may cause

a number of drilling problems, while increasing non-productive time and well costs. If

fracture pressure is overestimated, unexpected lost circulation and wellbore breathing can

occur. Additional unplanned casing strings required to mitigate these problems, may in

extreme cases jeopardize the success of the well. Conversely, if fracture pressure is

underestimated, planned well costs will be unrealistically high. As a result, other projects

may suffer from a lack of funding or the well may not even be drilled (Postler, 1997).

This chapter first reviews the different kinds of field injectivity tests used to

evaluate minimum horizontal stress and fracture parameters, with a description of their

advantages and limitations. Secondly, test signatures and factors that may lead to

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difficulties to injectivity test interpretation are discussed. Thirdly, a phase by phase

comparison and interpretation of two pump-in and flow-back test examples in different

formations is performed. The fourth objective of this chapter is to develop an injectivity

test simulation framework that can capture the key elements of a field injectivity test.

Finally, due to the similar fracture behaviors in an injectivity test and lost circulation, the

importance of the field tests for understanding the fundamentals of lost circulation and

wellbore strengthening is highlighted.

7.2 A REVIEW OF FILED INJECTIVITY TESTS

With the increasing importance of geomechanics in the petroleum industry,

accurate determination of fracture parameters (e.g. fracture pressure/gradient) and in-situ

stress are increasingly appreciated by petroleum engineers. Therefore, more and more field

hydraulic fracturing tests are designed and performed to obtain field stress information. In

drilling and completions, lost circulation are strongly influenced by fracture pressure and

in-situ stresses. Field injectivity tests, including FITs, LOTs, XLOTs, and PIFBs, are

designed to determine fracture pressure and minimum horizontal stress during the drilling

process. These tests are typically performed after a casing string has been set, after drilling

10 to 20 feet of new formation (van Oort and Vargo, 2008).

7.2.1 Formation Integrity Test (FIT)

A FIT is a test to confirm the cement and formation integrity near the casing shoe,

and to ensure the fracture gradient at the shoe is sufficient to withstand any expected or

potential loads while drilling the subsequent hole section. In a FIT, the bottom-hole

pressure is gradually increased to a pre-determined value, which is lower than predicted

fracture initiation pressure. Typically no fracturing occurs during a FIT, and the wellbore

remains intact. The pressure-time curve remains as a straight line during the test, while the

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slope of the line indicates the stiffness/compliance of the wellbore systems. Accurate

fracture parameters or in-situ stress information cannot be obtained from a FIT.

7.2.2 Leak-off Test (LOT)

A LOT, as schematically shown in Figure 7.1, provides an estimate of fracture

gradient at the casing shoe. This information is used for mud weight design and lost

circulation prevention in the subsequent hole section. It is often assumed that the lowest

fracture gradient in a given wellbore will exist at the casing shoe, however for a variety of

reasons this is not always the case. During the LOT, the well is shut-in and drilling fluid is

gradually pumped into the wellbore at a constant small rate, until a noticeable inflection

point is observed on the pressure-time curve. The wellbore pressure at the inflection point

is known as leak-off pressure, indicating that a fracture has been created. Leak-off pressure

is commonly assumed equal to fracture initiation pressure, which is used to determine the

upper limit of wellbore fluid pressure.

However, it is the author’s contention that leak-off pressure is not necessarily equal

to fracture initiation pressure. It is usually somewhat higher than fracture initiation

pressure, especially when “dirty” fluid (i.e. drilling mud with high solids content) is used

for the test. This idea has been mentioned in Chapter 2 and will be further discussed latter

in this chapter.

It is worth noting that a leak-off test rarely provides far-field stress information,

since the fracture created in a leak-off test remains very short (within several radii from the

wellbore) due to minimal injection volume. This short fracture is still under the influence

of the near wellbore stress concentration. Figure 7.2 shows the hoop stress concentration

in the near wellbore region, for a vertical well in a formation with uniform far-field

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horizontal stresses. To correctly measure the far-field minimum horizontal stress, a fracture

must extend through the near wellbore stress concentration region.

Figure 7.1: Schematic illustration of pressure-time/volume plot in a leak-off test.

Figure 7.2: Near wellbore hoop stress concentration with uniform far-field stresses

𝑆ℎ𝑚𝑖𝑛 = 𝑆𝐻𝑚𝑎𝑥 (blue color = more compression, red color = less

compression).

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7.2.3 Mini-frac Test (MFT)

A mini-frac test (MFT) is a small scale fracturing test commonly performed before

a conventional hydraulic fracturing stimulation operation. Fracture gradient and minimum

horizontal stress can be determined from a MFT, and then used for hydraulic fracturing

design. However, MFTs are usually performed in reservoir formations, and very commonly

in cased and perforated well intervals. So, the prediction results from MFTs cannot

represent the situation for the entire well section, most of which is in non-reservoir

formations, where most lost circulation events occur.

7.2.4 Extended Leak-off Test (XLOT)

A fracture generated during an XLOT will typically have sufficient length to pass

through the near wellbore stress concentration region. Therefore, this test may provide

sufficient data for estimating far-field stress and fracture information. In an XLOT, fluid

injection continues until a relatively steady fracture propagation pressure is reached,

followed by a shut-in phase as shown in Figure 7.3. In addition to leak-off pressure, several

other fracture parameters can be estimated from an XLOT, including formation breakdown

pressure, fracture propagation pressure, instantaneous shut-in pressure and fracture closure

pressure.

Formation breakdown pressure is the divide separating the stable and unstable

fracture propagation stages. During the time between leak-off and formation breakdown,

the fracture experiences stable propagation. The fracture growth during this period is very

slow, with a volume increase rate less than the pumping rate. Therefore, the wellbore

pressure continues to rise prior to formation breakdown, which is the upper pressure limit

for stable fracture growth. The volume increase during this stage is primarily due to the

growth of fracture width rather than length.

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Subsequently, during the time between formation breakdown and fracture

propagation, the fracture extends unstably with a rapid increase in fracture length. The

wellbore experiences a sudden pressure drop immediately after formation breakdown,

since the fracture volume expands at a rate much greater than the pumping rate. With

continued injection, the pressure ultimately levels off to fracture propagation pressure,

indicating the fracture growth rate is roughly equal to the pumping rate. However, with

continued pumping and an increase in fracture length, the fracture propagation pressure

will gradually decrease in a wave-like pattern.

Instantaneous shut-in pressure (ISIP) is the pressure observed immediately after

pumping is stopped. Once pumping ceases, the additional pressure required to overcome

flowing friction along the fracture and tubing (if pressure is recorded at the surface) and

the fracture tip resistance, immediately drop to zero. Therefore, instantaneous shut-in

pressure is always somewhat lower than fracture propagation pressure.

Fracture closure pressure is frequently estimated from shut-in data using

interpretation methods developed for mini-fracturing tests. Fracture closure pressure is

commonly taken as the best estimate of minimum horizontal stress, based on the

assumption that the wellbore pressure is equal to minimum horizontal stress as the

mechanical fracture starts to close. It is worth noting that mini-fracturing test interpretation

methods were initially developed for diagnostic fracture injection tests in permeable

reservoirs with clean fluids. However, XLOTs are usually performed in tight shale with

mud. Therefore, direct use of these methods for XLOT interpretation may give inaccurate

results. In addition, fracture closure during the shut-in phase is the result of fluid leak-off

into the formation, which is highly dependent on permeability and threshold capillary entry

pressure. In low-permeability formations like tight shale, leak-off may be too slow for the

fracture to close within a reasonable amount of time. In this case, minimum horizontal

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stress cannot be properly evaluated using the aforementioned methods. Alternatively, an

XLOT with a flow-back phase may provide a better solution. This test is referred to as

pump-in and flow-back test (PIFB) in this chapter.

Figure 7.3: Typical XLOT plot (Modified after Gaarenstroom et al., 1993).

7.2.5 Pump-in and Flow-back Test (PIFB)

A pump-in and flow-back (PIFB) test is a standard XLOT followed by a flow-back

phase with either a constant choke or constant flow-back rate at the surface. Both XLOTs

and PIFB tests can include multiple cycles for confirming test results. Since fluids flow

back directly to the surface, fracture closure in a PIFB test is almost always assured and

not dependent on fluid leak-off into the formation. These tests provide a superior method

of measuring fracture closure pressure in low permeability formations, especially when

mud is used as the injection fluid. In a PIFB test, wellbore pressure and flow-back

time/volume are recorded. A plot of pressure versus flow-back time/volume will show an

inflection point, which indicates a change in system stiffness/compliance (see Figure 7.4).

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This inflection point is commonly interpreted as fracture closure pressure, equal to

minimum horizontal stress.

Figure 7.4: An example of pressure-volume plot in a pump-in and flow-back test

(Modified after Gederaas and Raaen, 2009).

In addition to minimum horizontal stress, PIFB tests also provide leak-off pressure

(fracture initiation pressure), formation breakdown pressure, fracture propagation pressure,

instantaneous shut-in pressure, and fracture reopening pressure. These values (especially

fracture initiation and propagation pressure) are important considerations for mud density

selection, casing program design and lost circulation prevention, particularly in wells with

narrow drilling margins.

With clean injection fluid, fracture initiation and propagation pressures are

primarily dominated by in-situ stresses and rock strength. Conversely, when drilling mud

is used as the injection fluid, these parameters are also highly dependent on mud properties

(e.g., mud type and solid particle size and concentration), petro-physical properties of the

rock other than strength (e.g., lithology, permeability, and wettability), and interactions

between the drilling mud and formation rock (e.g., fluid leak-off and filter cake

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development on the wellbore wall and fracture faces, and capillary pressure effects). A

detailed understanding of these and other factors is required for a proper interpretation of

PIFB tests, and understanding fracture growth behavior and lost circulation fundamentals.

7.3 TEST SIGNATURES

7.3.1 Pressure-time Signature

The plot of the pressure-time response of a typical PIFB test is shown in Figure 7.5.

With fluid pump-in, wellbore pressure first increases linearly to the LOP point. At the LOP

point, the straight line deviates to the right, and then continues to rise with a smaller slope

until it reaches the FBP. Immediately after FBP, the wellbore pressure experiences a

significant drop. Then, with continuous pumping, the pressure declines to the relative stable

FPP. At the beginning of the shut-in phase after stopping the pump, the wellbore pressure

has another sudden drop from FPP to ISIP. Then, the pressure decline becomes slower and

slower. Finally, a flow-back phase follows the shut-in phase. During the flow-back phase,

wellbore pressure first decreases with a relatively smaller rate, then with a relatively larger

rate.

The deviation at LOP reflects a significant change in the stiffness (or compliance)

of the pressure system. Several factors may influence this change, including the

compressibility of mud, casing, cement and formation rock, fluid penetration from

wellbore wall, and fluid leak-off into fractures. Among these factors, only the effect of

fluid leak-off into fractures is observably nonlinear (Fu, 2014). Therefore, when there is a

clear LOP, it is considered that a fracture must have been created.

After fracture creation, usually, the fracture first grows at a rate smaller than the

pumping rate. So, the wellbore pressure continues to rise until FBP. FBP divides the stable

and unstable fracture propagation stages. From LOP to FBP, the fracture grows stably and

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slowly. After FBP, the fracture experiences an unstable propagation with a fast increase in

fracture length and volume. The fracture volume expands at a rate higher than the pump

rate, so there is a dramatic pressure drop after FBP.

After formation breakdown, the pressure finally declines to the relatively constant

FPP. The fracture growth rate is roughly equal to the pump rate. The wellbore pressure

features a “saw-tooth” shape during fracture propagation in field tests. An example is

shown in Figure 7.6.

With a rapid pressure drop, wellbore pressure declines to ISIP after the pump is

stopped. This pressure drop is caused by the elimination of frictional pressure and pressure

required to overcome fracture tip resistance. ISIP is often referred to as minimum fracture

extension pressure or fracture gradient within the fracture stimulation and well completion

communities. Some researchers argue that ISIP is a reasonable estimate of minimum

horizontal stress (Alberty et al., 1999; Postler, 1997).

FCP is commonly recognized as the best estimate of minimum horizontal stress

(Fjar et al., 2008; Okland et al., 2002). FCP can be obtained either by using shut-in data

(plotting pressure vs. square root of time), or by using flow-back data (plotting pressure vs.

flow-back time or volume). In both methods, a change in the slope of the data line indicates

fracture closure. The pressure at this slope-change point is FCP. However, fracture closure

is not an instantaneous process (Fjar et al., 2008; Hayashi and Haimson, 1991; Raaen et

al., 2001; Raaen and Brudy, 2001). So, the slope change is not an instantaneous process

too. According to Hayashi and Haimson (1991) fracture closure is a two-stage process: in

the first stage fracture width reduces while the fracture length remains constant; in the

second stage fracture length decreases until the complete closure of the fracture. It is argued

that the minimum horizontal stress should be interpreted at the end of the first stage, rather

than at full fracture closure (Raaen et al., 2006, 2001; Raaen and Brudy, 2001).

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Figure 7.5: Pressure-time plot of a typical PIFB test.

Figure 7.6: “Saw-tooth” pressure response during fracture propagation (Reproduced

from Okland et al., 2002).

7.3.2 Pressure-volume Signature

The pressure-volume response of a typical PIFB test is shown in Figure 7.7. 𝑉𝑖𝑛 is

the total volume pumped into the well during the pump-in phase in Figure 7.5. 𝑉𝑤 is

volume pumped into the well before a fracture has been created. It actually represents the

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combined volume change due to mud compression, casing expansion, fluid penetration

through wellbore wall, and open hole expansion, as schematically shown in Figure 7.8.

After LOP, a fracture must have been created, but at first its volume grows at a rate

lower than pump rate, resulting in a continuously increasing pressure until formation

breakdown pressure (FBP). The fracture during this stable propagation period remains

small. 𝑉𝑓𝑠 is equal to fluid volume leaked off into the fracture plus any further volume

change due to mud compression, casing expansion, fluid penetration, and open hole

expansion, with wellbore pressure increase.

The wellbore pressure drops from FBP to FPP during the unstable fracture

propagation stage. Thus, in this period, no further mud compression or wellbore expansion

occurs. 𝑉𝑓𝑝 represents the injection volume to extend the fracture and fluid penetration

loss to the formation through wellbore and fracture surfaces. It is suggested that this volume

should be used to check the length of the fracture to evaluate whether it is sufficient to pass

the near-wellbore stress concentration region and reach the far-field area (Fjar et al., 2008).

From instantaneously shut-in pressure (ISIP) to final shut-in pressure (FSIP), there

is no volume change in the system, because the well is shut in. But the pressure has a

significant drop due to the removal of frictional pressure and pressure required to overcome

fracture tip resistance.

During the flow-back phase, 𝑉𝑜𝑢𝑡 is the total volume flow back to the surface. 𝑉𝑓𝑐

is roughly equal to the volume flow back from the fracture during facture closure. 𝑉𝑟𝑜𝑐𝑘

is the additional volume returned after fracture closure, due to inward fluid flow from the

formation to the wellbore, and wellbore shrinkage with pressure decrease. 𝑉𝑙𝑜𝑠𝑠 is the total

fluid volume lost into the formation.

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Figure 7.7: Pressure-volume plot of a typical PIFB test (Modified after Fjar et al., 2008).

Figure 7.8: Total volume pumped into the well before fracture creation (Modified after

Altun, 1999 and Fu, 2014).

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7.4 TEST INTERPRETATION

7.4.1 FIP, LOP and FBP

In an idealized situation with perfectly impermeable rock, clean fluid and relatively

low horizontal stress anisotropy, FIP, LOP, and FBP of a vertical wellbore should be

identical, as shown in Figure 7.9. They all represent the pressure required to initiate a

fracture on wellbore wall.

Figure 7.9: FIP=LOP=FBP in idealized condition.

However, FIP can be much lower in permeable formations than in impermeable

formations as discussed in Section 2.3. This is because local pore pressure near the

wellbore can experience an increase in a permeable formation due to wellbore fluid

penetration. This elevated pore pressure opposes compression stress, thereby leads to a

decreased compression (or increased tension) in the vicinity of the wellbore, resulting in a

relatively low FIP. Conversely, in impermeable rock, there is no fluid penetration, hence,

no pore pressure increase near the wellbore, resulting in a relatively higher FIP.

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When drilling mud is used as the injection fluid, because it usually has high solids

content, a mudcake commonly develops on the wellbore wall in a permeable formation.

With extremely low permeability, mudcake can effectively isolate wellbore fluid and

formation fluid, and inhibit pore pressure increase near the wellbore as analyzed in Chapter

4. Therefore, it can help maintain a relatively high FIP for a permeable wellbore.

LOP is the clear point where the pressure vs. injection time/volume plot begins to

deviate from linearity, indicating a significant change in the stiffness or compliance of the

system. LOP is commonly assumed equal to FIP by many drilling engineers. But this

assumption is only correct when clean injection fluid without solids is used. For drilling

mud with high solids content, LOP is not necessary identical to FIP.

With clean fluid, when a micro-fracture is created on the wellbore wall, the fluid

can easily enter the fracture. Fluid pressure can immediately act on fracture surfaces and

fracture tip to propagate the fracture, leading to a noticeable LOP on the pressure response.

Conversely, when drilling mud is used as injection fluid, the mud solids can quickly seal a

newly created micro-fracture by forming a filter cake as mentioned in Section 2.3. The

filter cake can arrest further fluid flow into the fracture and isolate the fracture tip from

wellbore pressure. Due to sealing effect of the filter cake, micro-fracture initiation (i.e. FIP)

is usually not detectable in the pressure response in lab tests even with a precise gauge, let

alone in field tests, as discussed by Guo et al. (2014) based on their laboratory experiments.

They have also shown that a fracture can grow to a significant size without a detectable

LOP. With continuous injection, a LOP may be observed, but it can be much higher than

the actual FIP, at which a fracture is initially created.

The above analysis implies that it may not be possible to precisely measure FIP in

permeable rock with drilling mud as the injection fluid. An inflection point may be

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undetectable at FIP; or if detected, it may indicate a higher filter cake breakdown pressure

rather than FIP.

Similarly, FBP can also be significantly increased by filter cake development in the

fracture. This phenomenon is more obvious in high permeability formations than in low

permeability formations, since filter cake is easier to develop in the former case. The

difference between FIP (or LOP) and FBP is relatively large in permeable formations,

compared with less permeable formations. Figure 7.10 shows a more realistic case where

FIP, LOP and FBP are not identical in permeable formations when “dirty” drilling mud is

injected.

Figure 7.10: FIP, LOP and FBP are not identical with drilling mud as injection fluid.

7.4.2 FPP

For drilling engineering, FPP is critical for wellbore stability evaluation, lost

circulation prevention, and casing design, especially for drilling in challenging areas with

a narrow drilling mud weight window. In an idealized case, with clean injection fluid and

impermeable formations, FPP is primarily dominated by minimum horizontal stress, and it

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can be a reliable estimate of the minimum horizontal stress. However, when drilling mud

is used and the formation is permeable, fluid leak-off and filter cake development in the

fracture can significantly influence FPP.

During fracture propagation, FPP is the total pressure needed to overcome 1) the

minimum horizontal stress for keeping the fracture open, 2) frictional losses for fluid flow,

3) fluid loss to the formation due to fluid penetration through fracture surfaces, and 4)

fracture tip resistance for further fracture growth. At least the latter three components can

be affected by fluid leak-off and/or filter cake development during fracture propagation.

During pumping, solids in the mud are transported with fluid flow into the fracture.

While mud filtrate leaks off into the formation, the solid particles in the mud are left in the

fracture. This will result in a higher solid density and fluid viscosity in the fracture, which

will significantly increase the frictional losses when the fracture has a large length,

therefore, lead to a higher FPP.

With fluid leak-off from the fracture surfaces to the formation rock, the fluid

volume (energy) inside the facture acting to extend the fracture is reduced. Therefore, for

creating further fracture volume, a higher pressure (energy) is needed to make up this

energy loss due to fluid leak-off. This phenomenon also increases FPP for a permeable

formation.

Moreover, with fluid leak-off in permeable formations, filter cake can easily build

up at the fracture surfaces and fracture tip. It can effectively isolate the fracture tip from

wellbore pressure, thereby significantly reducing pressure acting to propagate the fracture.

Additional pressure is required to break the filter cake for further fracture propagation,

which leads to a higher FPP. Conversely, in impermeable formations, no effective mud

barrier can build up in the fracture because there is no filtrate leak-off. This means there is

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full pressure communication between wellbore and fracture tip. The fracture can extend at

a relatively lower pressure.

The performance of filter cake is critical for lost circulation prevention and

wellbore strengthening treatment. Since a quality filter cake can only form in permeable

formations, it is generally believed that wellbore strengthening treatments work better in

permeable formations (e.g. sand and sandstone). Conversely, in relatively impermeable

rocks (e.g. shale), wellbore strengthening is not likely to be successful. When permeability

is low, the filtrate leak-off rate is too low to allow mud solids to aggregate to form a quality

filter cake or bridge. Low leak-off rates also mean very limited fracture pressure/energy is

released into the formation. Therefore, pressure is trapped inside the fracture, facilitating

fracture growth and lost circulation. Morita et al., (1990) and van Oort et al. (2011) also

argued that an effective filter cake is more likely to build up in water based mud (WBM)

than in oil based mud (OBM). That’s one reason why lost circulation is more likely to occur

when drilling with OBM.

7.4.3 Estimation of Minimum Horizontal Stress

It is widely accepted that fracture closure pressure (FCP) is the best estimate for

minimum horizontal stress (Fjar et al., 2008; Fu, 2014; Gederaas and Raaen, 2009; Raaen

et al., 2001; Raaen and Brudy, 2001; Ziegler and Jones, 2014). In a PIFB test, FCP can be

evaluated either from shut-in data or from flow-back data. When using the shut-in data, the

leak-off rate from fracture to formation should be sufficiently high for the fracture to close

in a reasonable time. So the prediction results depend highly on rock permeability. For the

latter method using flow-back data, fracture closure is almost guaranteed and less

dependent on fluid leak-off (permeability), because fluid can directly flow back to the

surface, rather than to the formation.

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For formations with high permeability and tested with a low-viscosity and low-

solids-content fluid, shut-in data can give a reasonable estimate of minimum horizontal

stress, because the fracture can close in a relative short time after the pump is stopped, due

to rapid leak-off. A highly sloped pressure-time curve indicating a fast pressure drop during

shut-in is expected (see Figure 7.11). Conversely, in formations with low permeability,

such as shale, the fracture usually does not close in a reasonable time due to the low leak-

off rate; therefore, shut-in data are not sufficient to give a good estimate of minimum

horizontal stress. A relatively flat pressure-time response is usually observed for this case

(see Figure 7.11).

However, even if the formation has sufficient permeability for fluid leak-off,

fracture closure can be significantly inhibited by a tight filter cake on the fracture surfaces.

In other words, the minimum horizontal stress measurement is also highly related to filter

cake quality in the fracture. With a tight mudcake, the leak-off may be too slow to close

the fracture. This implies that FCP in the shut-in curve may not be properly defined, similar

to the impermeable case in Figure 7.11.

It should also be noted that the methods for estimating FCP using shut-in data are

based on interpretation methods developed for MFTs. These methods, such as pressure vs.

square root of time plotting technique, were initially developed for pressure transition

analysis in permeable reservoirs with relatively clean fracturing fluid. However, XLOTs

or PIFB tests are usually performed in tight shale with “dirty” mud. So, the direct use of

these methods for shut-in data interpretation in low permeability formation with drilling

mud may lead to inaccurate results.

For a better estimation of minimum horizontal stress, especially for low permeable

formations or permeable formations with well-developed filter cake, a flow-back test is

preferred.

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In a flow-back test, a plot of pressure versus flow-back time will show an inflection

point, as shown in Figure 7.5, which indicates system stiffness/compliance change due to

fracture closure. The slope before this point reflects the stiffness/compliance of system

with an open fracture, while the slope behind this point is the stiffness/compliance of

system with a closed fracture. The pressure at this inflection point is FCP, which is

commonly recognized as a good estimate of minimum horizontal stress.

Raaen et al. (2001) and Gederaas and Raaen (2009) also suggested recording the

flow-back volume data. A plot of pressure versus flow-back volume helps determine the

FCP, as shown in Figure 7.12. They argued that a flow-back test gives more precise

estimate of minimum horizontal stress than traditional methods using shut-in data, which

usually overestimate the minimum horizontal stress.

Figure 7.11: Permeable formation has a large pressure decline during shut-in due to

sufficient leak-off from fracture, while a relative flat pressure response is

usually observed in an impermeable formation.

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Figure 7.12: An example of predicting FCP using flow-back data (Reproduced from

Gederaas and Raaen, 2009).

7.5 FIELD EXAMPLES

Discussion in this section concerns two PIFB tests obtained from the literature. The

two tests were conducted in two neighboring wells, Well 11-2 and Well 10-7, in the North

Sea (Okland et al., 2002). The test in Well 11-2 was performed in a formation with apparent

higher permeability compared with the test formation in Well 10-7. The two tests were

performed at 4097 ft and 4284 ft true vertical depth (TVD), respectively. Test in Well 11-

2 has two cycles, with pump-in and shut-in phases in the first cycle and pump-in and flow-

back phases in the second cycle (Figure 7.13). Test in Well 10-7 also has two cycles, but

there is an additional flow-back phase in the first cycle (Figure 7.14). From Figures 7.13

and 7.14, it can be seen that signatures of the two tests in high and low permeability

formations are very different, which are discussed and compared in this section.

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Figure 7.13: Test in Well 11-2 in formation with relatively high permeability

(Reproduced from Okland et al., 2002).

Figure 7.14: Test in Well 10-7 in formation with relatively low permeability (Reproduced

from Okland et al., 2002).

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Pump-start to FBP. In the phase from pump-start to FBP, it is obvious that there

is a relatively clear LOP in Well 10-7 compared with Well 11-2. In Well 11-2 the

pressure builds up linearly to FBP without a detectable deviation. A possible

explanation for this is that a leak-off response may not be detectable due to the

continuous sealing effect of filter cake in relatively permeable formation (Well 11-

2), as discussed above. Another observation is that FBP in the high permeability

case is much lower than that in the low permeability case, even though the test

formation depths are not much different. This may be related to the permeability of

the formation. High permeability formations (e.g. sand or sandstone) usually cannot

transmit as much overburden to horizontal stress as low permeability formation

(e.g. shale) do. Therefore, the test formation in Well 11-2 may have a relatively

lower in-situ stress due to its relatively higher permeability, thereby resulting in a

low FBP.

Fracture propagation in first cycle. Comparing the two cases, it can be found

that, after formation breakdown, the pressure drop in the low permeability

formation is much faster than that in the high permeability case. This may be related

to a relative higher fluid leak-off rate on fracture faces in the more permeable

formation. This leak-off reduces the energy acting on creating new fracture volume,

thereby resulting in a slow pressure drop. Another possible reason is that mud solids

form a filter cake with fluid leak-off, which seals the fracture tip and inhibits

fracture propagation. Comparing FBP and FPP, the less permeable formation

experiences a larger pressure drop from FBP to FPP than that in the more permeable

formation. An explanation for this could be: compared with low permeability

formation, high permeability formation has a relatively lower FBP due to its lower

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in-situ stress, but a relatively higher FPP due to filter cake development; therefore,

it has a relatively smaller difference between FBP and FPP.

Shut-in and flow-back in first cycle. Pressure decreases quickly during shut-in in

Well 11-2 due to its high permeability, while remaining essentially constant in the

low permeability Well 10-7. This implies that the fracture probably closed during

shut-in for the high permeability Well 11-2, and a FCP (or minimum horizontal

stress) can be properly predicted using the shut-in data. However, the fracture is

less likely to close in the low permeability well due to the very limited fluid leak-

off. This further confirms that FCP (or minimum horizontal stress) cannot be

reasonably determined using shut-in data in relatively impermeable rocks; it also

confirms that the pressure analysis methods, developed for interpretation of MFTs

in high permeability reservoir where fractures can easily close, may not be suitable

for shut-in data analysis in low permeability formations, where most XLOTs and

PIFB tests are performed.

Fracture reopening and re-propagation in second cycle. In both cases, there is

a fracture reopening pressure (FRP) which is much lower than the FBP in the first

cycle. This is because a fracture is already created on the wellbore wall after the

first injection cycle, therefore no additional pressure is needed to overcome the rock

strength to initiate the fracture. In an idealized situation with clean injection fluid

and impermeable rock, the difference between FBP and FRP can be interpreted as

the tensile strength of the rock. In the low permeability test, FPP is almost equal to

FRP. But in the high permeability test, FPP is somewhat higher than FRP. This may

also be related to the sealing effect of filter cake in high permeability rock. After

fracture re-opening, the fracture has full conductivity in impermeable rock, and

fluid can flow freely from the wellbore to the fracture tip. However, filter cake in a

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permeable fracture may significantly restrain pressure communication between the

wellbore and the fracture tip, leading to an FPP higher than FRP.

Flow-back in second cycle. In both tests, the pressure decreases quickly and

nonlinearly during flow-back. An inflection point can be defined in both cases, as

shown in Figures 7.13 and 7.14. As discussed above, the inflection reflects the

system compliance/stiffness change before and after fracture closure. Thus, the

pressure at this point is interpreted as FCP and commonly used as a prediction of

minimum horizontal stress. In the low permeability case, FCP predicted from flow-

back data in the second cycle is consistent with that predicted from the first cycle.

This confirms the test results.

7.6 DEVELOPING A SIMULATION FRAMEWORK FOR INJECTIVITY TESTS

Knowledge of minimum horizontal stress (𝑆ℎ𝑚𝑖𝑛) is important in many aspects

during the life of oilfield development. It is a key factor for prediction of wellbore stability

and lost circulation in the drilling and completion stages (Raaen and Brudy, 2001). FITs

and LOTs are routinely performed at each casing shoe during drilling. However, the main

purpose of these tests is to verify quality of the cement at the casing shoe and strength of

the next borehole section to be drilled. As discussed above, these tests normally cannot

give precise stress information due to limited fracture distance (Raaen and Brudy, 2001;

Ziegler and Jones, 2014). To overcome these limitations, XLOTs and PIFB tests were

introduced for quantifying 𝑆ℎ𝑚𝑖𝑛 with a higher degree of precision. Since a sufficiently

long fracture can be created and sufficient data can be gathered from a PIFB test, it has

been considered a preferred method of obtaining 𝑆ℎ𝑚𝑖𝑛 (Kunze and Steiger, 1992; Raaen

and Brudy, 2001; Zoback and Haimson, 1982), especially for early stages of oilfield

development. In a PIFB test, fluid injection continues until a relatively

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steady fracture propagation pressure (FPP) is reached, followed by a shut-in

phase and a flowback phase. As mentioned in the above section, the basic idea to predict

𝑆ℎ𝑚𝑖𝑛 using PIFB data is that FCP is equal to 𝑆ℎ𝑚𝑖𝑛 (Raaen et al., 2001). However,

prediction of FCP may be difficult (Guo et al., 1993; Plahn et al., 1997), and this difficulty

contributes significantly to the uncertainty of stress measurements (Raaen et al., 2001).

FCP identification using PIFB data frequently borrows techniques from pressure

analysis of hydraulic fracturing which was initially developed for determining reservoir

limits from diagnostic fracture injection tests in permeable reservoir formation. A typical

method is identifying fracture closure by plotting bottom hole pressure versus the square

root of shut-in time, with the pressure value at the slope change point of the plot being

interpreted as FCP. The traditional methods usually require that the fracture can close with

sufficient fluid leak-off from the fracture to the formation during shut-in. If the formation

is sufficiently permeable, this requirement can usually be satisfied with a reasonably long

shut-in period, i.e. the fracture will eventually close.

However, PIFB tests are mostly performed in tight shale formations with very low

permeability (these formations are loosely referred as “impermeable” formations in the

following discussion), where the casing shoes are usually set. For PIFB tests in such

formations, the fluid leak-off during shut-in may be too slow for the fracture to close within

a reasonable amount of time. Therefore, 𝑆ℎ𝑚𝑖𝑛 cannot be properly evaluated using the

aforementioned methods based on the time development of pressure during shut-in. The

direct use of these traditional methods for interpreting 𝑆ℎ𝑚𝑖𝑛 in impermeable formations

may lead to significant errors, usually an overestimate of 𝑆ℎ𝑚𝑖𝑛 (Raaen et al., 2001).

Therefore, a flow-back phase can be included in the test to aid fracture closure for a better

stress measurement. Since fluid flows back directly to the surface, fracture closure in the

flowback phase is always assured and not dependent on fluid leak-off into the formation.

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Therefore, flowback tests provide a superior method of measuring 𝑆ℎ𝑚𝑖𝑛 in impermeable

formations.

A proper understanding of the differences between PIFB tests in permeable

formations and impermeable formations is important for test interpretation and

determination of 𝑆ℎ𝑚𝑖𝑛 . This section first illustrates fully coupled fluid flow and

geomechanics modeling of PIFB tests to investigate and compare the essential pressure

and fracture behaviors of the tests in permeable and impermeable formations. Then, the

reliability of several in-situ stress prediction methods using shut-in data and flowback data

is tested and discussed with the numerical simulation results.

7.6.1 Model Formulation

Pressure responses and fracture behaviors in PIFB tests are simulated using a

coupled fluid flow and geomechanics numerical model in this study based on the Finite

Element Method (FEM). A PIFB test system generally has three components: the well, the

fracture, and the formation. Figure 7.15 shows a typical configuration of the PIFB test

system. The following physical processes that happen in a PIFB test are included and

simulated simultaneously in the proposed numerical model.

Fluid flow in the drill pipe

Fracture propagation and fluid flow in the fracture during fluid injection

Formation rock deformation and pore fluid flow

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Figure 7.15: General PIFB test system with well, formation and fracture.

The first objective of this section is to develop a general finite-element model that

can take into account the main elements in a PIFB test process, such as fluid injection

through the drill pipe, fracture propagation, fluid flow in the fracture, pore fluid flow, and

rock deformation. The model should be able to capture the essential features of a PIFB test,

such as pressure vs time/volume signatures and fracture closure behaviors during the test.

The second objective is to apply this model to simulate and compare PIFB tests in

permeable and impermeable formations.

Fluid flow in the drilling pipe is modeled based on Bernoulli’s equation as

described in Section 3.2.1 using the pipe element technique in Abaqus®. Fracture

propagation and fluid flow in the fracture during the tests are modeled based on a cohesive

zone method using coupled pressure/deformation cohesive elements in Abaqus®. A

traction-separation constitutive law and a fluid flow constitutive law as described in

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Section 3.2.2 are used in the cohesive zone model. The deformation of the porous formation

and pore fluid flow are modeled based on poroelastic theory and Darcy’s law as described

in Section 3.2.3 using coupled pore pressure and deformation continuum elements in

Abaqus®.

7.6.1.1 Assumptions and Geometry of the Model

PIFB tests performed in vertical wellbores are simulated in this study. The

overburden stress is in the vertical direction and the other two principal stresses, i.e.

minimum horizontal stress and maximum horizontal stress, are in the horizontal plane. The

wellbore and surrounding formation are assumed to be in a plane-strain condition as shown

in Figure 7.16. Owing to symmetry, only one half of the wellbore and formation is used in

the model.

The wellbore radius is 0.1 m. The total size of the modeled domain is 60×20 m. In

formations with anisotropic horizontal stresses, it is well known that the fracture will

propagate in the direction perpendicular to Shmin during the test. Therefore a fracture path

in this direction is defined in the middle of the model, as shown in Figure 7.16. The

formation is modeled as an isotropic, poroelastic material, using coupled pore pressure and

deformation continuum finite elements; the fracture is modeled using layer pore pressure

cohesive elements; and the well is model with pipe elements. Since significant

stress/displacement gradients are expected in the wellbore vicinity, the mesh is refined

around the wellbore, as shown in Figure 7.16, to increase the computational accuracy. The

total numbers of elements and nodes of the model are 1320 and 1476, respectively.

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Figure 7.16: PIFB test model. Top: geometry and boundary conditions of the model;

Bottom: refined mesh around the wellbore.

7.6.1.2 Boundary Conditions of the Model

A symmetric boundary condition is defined on the left edge of the model. The

formation is assumed to be at a depth of 1000 m with a normal pore pressure of 10 MPa, a

minimum horizontal stress of 15 MPa and a maximum horizontal stress of 18 MPa. The

minimum and maximum horizontal stresses and undisturbed pore pressure (10 MPa) are

applied on the outer boundaries of the model, as shown in Figure 7.16. Initial pore pressure

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of 10 MPa is applied to the whole formation. Drilling fluid is injected into the well from

the top of the drill pipe with a rate of 1.6 gallon per minute. Note that since a plane-strain

assumption is made, this rate is the fluid volume injected to a formation of unit thickness.

It is also assumed that the well has been circulated enough to ensure it is filled up

with drilling fluid before the PIFB test. So an initial hydrostatic pressure profile is defined

along the well depth before injection starts. The pipe element at end of the drill pipe is tied

to the formation element on the wellbore wall to make sure the fluid pressure in the bottom

hole is equal to the pore pressure on the wellbore wall. Besides, a dynamic pressure that

equals the bottom-hole fluid pressure is imposed onto the inner wellbore wall to model the

pushing pressure applied by fluid column.

7.6.1.3 Material Properties Used in the Model

All of the material properties of permeable and impermeable formations used in the

simulations are the same, except the formation permeability. The permeability values used

for the permeable and impermeable formations are 5 mD and 0.05 mD respectively, which

are typical values corresponding to sandstones and tight shales. Table 7.1 summarizes all

the other material parameters used for the simulations.

Table 7.1: Material properties for the simulations.

Parameter Values Units

Young’s modulus 7000 MPa

Poisson’s ratio 0.2

Fluid density 1000 Kg/m3

Fluid viscosity 1 cp

Porosity 0.25

Tensile strength 1.8 MPa

Critical fracture energy 30 J/m2

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7.6.2 Results and Discussion

Using the coupled fluid flow and geomechanics model described above, PIFB tests

in permeable and impermeable formations are simulated and analyzed. Several methods

for predicting 𝑆ℎ𝑚𝑖𝑛 using pressure versus time or pressure versus volume data from PIFB

tests are analyzed based on the simulation results. The summary of the analyses is reported

in the following subsections.

7.6.2.1 Simulated Test Signatures

PIFB tests with two cycles are simulated in this study. Each cycle include three

phases: injection, shut-in, and flowback. Since the fracturing fluid leak-off rates in the

permeable and impermeable formations are significantly different, different time schedules

were used for PIFB tests in the two types of formations in order to obtain adequate data for

analyses.

For PIFB tests in the permeable formation: first circle includes 600s injection, 600s

shut-in, and 100s flowback; the second circle includes 800s injection, 600s shut-in,

and 200s flowback.

For PIFB tests in the impermeable formation: first circle includes 600s injection,

600s shut-in, and 400s flowback; the second circle includes 1000s injection, 800s

shut-in, and 900s flowback.

Bottom hole pressure (BHP) versus time curves for PIFB tests in the permeable and

impermeable formations are plotted in Figure 7.17. The plots capture some essential PIFB

test signatures learned from field tests.

The pressure increases linearly to formation breakdown pressure (FBP) in the early

time of the injection phase. Since the numerical model does not take into wellbore storage

effect into account, the pressure builds up very quickly. It can be observed that pressure

buildup in a permeable formation is slower than that in an impermeable formation. This is

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because the relatively higher leak-off rate on the permeable wellbore wall leads to faster

fluid loss to the formation, hence slower pressure buildup in the wellbore.

After formation breakdown, the pressure declines to a relatively constant FPP.

Values of FPP in permeable and impermeable formations are 2.5 MPa and 1.7 MPa higher

than 𝑆ℎ𝑚𝑖𝑛, respectively. These results are in qualitative agreement with the numerical

studies by Lavrov et al. (2015) where FPP was reported about 2-2.7 MPa above 𝑆ℎ𝑚𝑖𝑛,

and the field test observations reported by Raaen et al. (2006) and Okland et al. (2002)

where FPP is around 1-1.5 MPa above 𝑆ℎ𝑚𝑖𝑛. By comparing the two cases, it can be seen

that a permeable formation has higher FPP compared to an impermeable formation with

the same in-situ stress condition. This can be explained as, due to the increased total stress

around the fracture with fluid leak-off, the permeable formation requires a higher BHP to

maintain a sufficient net fracture pressure for fracture propagation.

Figure 7.17 also shows that the BHP features a “saw-tooth” shape during fracture

propagation. This phenomenon again agrees well with field test observations by (Raaen et

al. (2006) and theoretical analysis by Feng et al. (2015b). This feature can be explained

from a fracture mechanics point of view. The fracture will have a growth in length when

the stored fracture energy reaches critical fracture energy. However, immediately after

length growth, the stored fracture energy will release to a value lower than the critical

fracture energy, so the fracture will stop propagating until enough fracture energy

(pressure) is achieved again with continued injection. The fracture growth and suspension

process will repeat during fracture propagation, resulting in a “saw-tooth” pressure pattern.

Another interesting observation is that the “saw-tooth” pressure fluctuation is more severe

in the permeable formation than in the impermeable formation. This again can be

interpreted by the higher fluid leak-off in the permeable formation, which results in an

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increased total stress in the surrounding formation acting on closing the fracture, and hence

a larger pressure buildup for fracture propagation.

During shut-in phase, the BHP in a permeable formation has a remarkable decline,

to a value much lower than the 𝑆ℎ𝑚𝑖𝑛 and approaching the formation pore pressure of 10

MPa; while in the impermeable test, the BHP only exhibits a very small decline and it is

always significantly higher than 𝑆ℎ𝑚𝑖𝑛. It is well known that the pressure drop during shut-

in phase is solely dependent on fluid leak-off. Since the impermeable formation has very

slow leak-off, the BHP keeps nearly constant during the entire shut-in phase. However,

during the flowback phase, BHP drops significantly below 𝑆ℎ𝑚𝑖𝑛 , because fluid flows

back directly to the surface and pressure drop is not dependent on fluid leak-off anymore.

In the shut-in and flowback phases, changes of slope are observed in the pressure vs time

curves, which are commonly used to predict 𝑆ℎ𝑚𝑖𝑛. Methods using shut-in and flowback

data to predict 𝑆ℎ𝑚𝑖𝑛 for permeable and impermeable formations are discussed further in

the following sections.

During the second test cycles, shut-in and flowback phases have very similar

pressure behaviors with those in the first cycles for both formation types. However, no FBP

is observed in the early injection time. Instead, an inflection point very close to 𝑆ℎ𝑚𝑖𝑛 is

shown in both cases. The pressure at this point can be interpreted as the pressure required

to reopen the fracture created in the first test cycle, and is commonly referred to as fracture

reopen pressure (FRP). The simulation results clearly indicate FRP is a good representative

of 𝑆ℎ𝑚𝑖𝑛.

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Figure 7.17: Bottom hole pressure versus time plots. Top: PIFB test in permeable

formation; Bottom: PIFB test in impermeable formation.

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7.6.2.2 Discussion on 𝑺𝒉𝒎𝒊𝒏 Prediction

𝑺𝒉𝒎𝒊𝒏 Prediction Based on Shut-in Data

As mentioned earlier, the basic idea to predict 𝑆ℎ𝑚𝑖𝑛 from PIFB tests is that the

FCP is equal to 𝑆ℎ𝑚𝑖𝑛.

Fracture closure, hence FCP, can be identified based on the time development of

pressure. A common method is using a plot of pressure versus square root of shut-in time.

A change in the slope of the plot is generally interpreted as the FCP. However, this method

is only valid when satisfying the following two prerequisites:

Linear fluid flow along the longitudinal direction of the fracture and Darcy’s Law

governed fluid leak-off into the formation;

Fracture can close due to fluid leak-off into the formation in a reasonable shut-in

time.

For permeable formations these requirements may be easily satisfied, while for

impermeable formations this method is not a desired tool for 𝑆ℎ𝑚𝑖𝑛 prediction due to its

negligible leak-off. Figure 7.18 shows the plots of BHP versus square root of shut-in time

for the two cases. It can be seen that there is a distinct inflection point on the permeable

test curve where the BHP equals to 15.4 MPa, which is a fairly reasonable estimate for

𝑆ℎ𝑚𝑖𝑛 (15 MPa). This pressure is therefore interpreted as FCP. This result is in agreement

with the previous studies on predicting 𝑆ℎ𝑚𝑖𝑛 using shut-in data in permeable formations

(Raaen et al., 2001). However, for the impermeable case, there is no clear inflection point

on the plot, and the pressure is considerably higher than 𝑆ℎ𝑚𝑖𝑛 through the entire shut-in

phase; therefore it is impossible to appropriately predict 𝑆ℎ𝑚𝑖𝑛 based on the time

development of pressure in such cases. This again agrees with previous study conclusions

by Raaen et al. (2006, 2001), Raaen and Brudy (2001) and Kunze and Steiger (1992).

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For the plot of the permeable test, the slope is relatively constant before FCP; and

then increases immediately after FCP; finally decreases to a nearly constant value again.

The instant slope increase can be interpreted as a result of system stiffness increase at

fracture closure. The slope decrease is caused by the reduced fluid leak-off due to the

reduced pressure difference between BHP and pore pressure at the later stage of shut-in.

Figure 7.18: Pressure versus square root of time plot during shut-in. Top: Permeable

formation; Bottom: Impermeable formation.

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𝑺𝒉𝒎𝒊𝒏 Prediction Based on Flowback Data

As mentioned above, PIFB tests are mostly performed in impermeable shale

formations. The fluid leak-off during shut-in may be too slow for the fracture to close

within a reasonable amount of time in such formations. In order to ensure fracture closure,

a flowback phase can be introduced in a PIFB test. In the numerical examples, a flowback

phase is added to each cycle of the test. Figure 7.19 shows the time development of BHP

during flowback in the impermeable formation. Clear infection points (FCPs) at the same

BHP are observed on both cycles. The FCP (15 MPa) accurately measures the minimum

horizontal stress 𝑆ℎ𝑚𝑖𝑛.

The slope of the curve at each side of the fracture closure point is a relatively

constant value, with a smaller slope before fracture closure and a larger slope after fracture

closure. This is consistent with existing observations from field tests (Raaen et al., 2001)

and numerical simulations (Lavrov et al., 2015). This slope change can be explained as:

the system has a relatively smaller stiffness with an open fracture before fracture closure,

resulting in a slower pressure decline with fluid flowback; while after fracture closure, the

system stiffness increases, leading to a faster pressure decline and a larger slope.

In field PIFB tests, it has been demonstrated that FCP can be precisely determined

using the pressure versus volume added to or flowback from the

wellbore/fracture/formation system (Gederaas and Raaen, 2009; Okland et al., 2002; Raaen

et al., 2006, 2001; Raaen and Brudy, 2001). This is also illustrated very well in the PIFB

test simulation in the impermeable formation. Figures 7.20 and 7.21 show the amount of

fracturing fluid stored in the system during shut-in and flowback phases of the impermeable

test and permeable test, respectively. Apparently, during shut-in, the added fluid volume

does not change, because there is no fluid leaving the system. The BHP exhibits a

significant reduction of 6 MPa in the permeable formation due to large leak-off, but only a

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negligibly small decline of 0.3 MPa in the impermeable formation. It is evident from Figure

7.20 that the FCP (15 MPa) obtained from the pressure versus volume curves can give a

precise identification of the 𝑆ℎ𝑚𝑖𝑛. Again, before fracture closure a larger slope is observed

because of the relatively lower stiffness of the system with an open fracture; while after

fracture closure a smaller slope is observed due to the relatively larger stiffness of the

system with a closed fracture. Figure 7.21 shows that after the fracture has closed and the

pressure drops below 𝑆ℎ𝑚𝑖𝑛 in the shut-in phase in a permeable test, adding a flowback

phase does not provide any valuable information for stress determination, if not perturbing

the test interpretation. However, if the fracture is not closed due to limited shut-in time,

adding a flowback phase for tests in permeable formations may yield the same benefits as

for the impermeable formations. Therefore, whether for tests in permeable or impermeable

formations, it is important to properly schedule the shut-in and/or flowback periods to make

sure the fracture will close in the tests. The proposed model can be used to predict the time

required for the fracture to close in a shut-in test or in a flowback test based on the

formation properties and operational parameters.

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Figure 7.19: BHP vs time during flowback phases in impermeable formation.

Figure 7.20: BHP versus added fluid volume in the system during shut-in and flowback

phases in the impermeable test.

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Figure 7.21: BHP versus added fluid volume in the system during shut-in and flowback

phases in the permeable test.

7.7 LOST CIRCULATION AS A FUNCTION OF FORMATION LITHOLOGY

The fracture behavior in lost circulation and wellbore breathing is similar to that in

a field injectivity test. A fracture initiates at the wellbore wall, and is then propagated to

the far field by a fluid driven force. The main difference is that field injectivity tests are

generally conducted at a constant rate into a known formation and depth. On the other hand,

lost circulation generally occurs while circulating at a dynamic wellbore pressure. The

formation and depth where lost circulation is occurring is often unknown (at least initially)

especially in long sections of open hole. It is not always correct to assume lost circulation

is occurring near the previous casing shoe, where fracture gradient is presumed lowest.

Factors that affect field injectivity tests also influence lost circulation. From the

discussions above in this chapter and Chapter 2, permeability, capillary entry pressure, and

mudcake development on the wellbore wall and in the fracture can significantly affect

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fracture initiation and propagation pressures and therefore, lost circulation. These factors

can vary significantly for different formations and fluid types.

Accurate identification of formation types where lost circulation is more likely to

occur is extremely important for successful drilling operations (Ziegler and Jones, 2014).

In the following section, we will discuss the relative likelihood of lost circulation in salt,

clean shale, clean sandstone, and silty-shale. Fractured carbonates are intentionally omitted

from this discussion, since they are better handled as a separate topic.

Massive Salt. Due to its high plasticity, in-situ stresses in massive salt (halite) tend

to be uniform. Simple rock mechanics models show that under the same load condition (i.e.

overburden) uniform stresses lead to higher fracture initiation pressures, compared to

anisotropic stresses. Additionally, massive salt has extremely low permeability and is

therefore not penetrated by the drilling fluid. Lost circulation is rarely a problem in massive

salt, except under certain conditions where inclusions or sutures are present.

Clean Shale. Clean shale (typically high clay content, high Poisson’s ratio, low

Young’s modulus mud rock) also has very low permeability and relatively high plasticity

and fracture initiation pressure. Due to its small pore throat sizes, clean (usually water-wet)

shale has relatively large capillary entry pressures for immiscible oil or synthetic based

drilling fluids. The combination of low permeability and high capillary entry pressure

greatly inhibits fluid invasion into clean shale, preserving its relatively high fracture

initiation pressure. Compared to silty-shale, sandstone, siltstone and fractured carbonates,

lost circulation is usually much less of a problem in clean shale.

Sandstone. In theory, sandstone has relatively low fracture initiation and

propagation pressures compared to shale, silty-shale and salt (Ziegler and Jones, 2014).

However, due to its high permeability, large pore throat sizes and low capillary entry

pressures, a high quality filter cake can easily develop on the wellbore wall or inside a

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fracture. This filter cake effectively maintains fracture initiation pressures and can

significantly increase fracture propagation pressure, if a fracture is created. Therefore, lost

circulation is usually manageable in sandstone if appropriate drilling fluids (including

LCM) are used. Pressure depleted sandstones present an additional challenge, since a

reduction in pore pressure will also reduce fracture initiation and propagation pressures.

Fracture initiation pressures may also be reduced depending on wellbore trajectory and the

relative magnitudes of principal in-situ stresses. The last statement is true for all formation

types, except where in-situ stresses are uniform or nearly uniform.

Silty-shale. Silty-shale is a term commonly used to describe formations with

properties between those of clean shale and sandstone. It therefore encompasses a relatively

wide range of formation types and is not a precise lithological description. Silty-shale is

commonly encountered in transition zones between clean shale and sandstone, but it can

occur anywhere in the wellbore. Many reservoir top seals in deep water are comprised of

shale, with silt contents ranging from 17 to 41 percent (Dawson, 2004). Therefore, much

analysis has been performed on these types of formations, with respect to mineralogy,

capillarity and reservoir fluid sealing capacity.

Silty-shale has intermediate permeability and capillary entry pressures, compared

with clean shale and clean sandstone. However, capillary entry pressures for water-wet

silty-shale are too high to be ignored, and likely hold the key to understanding the

differences in apparent fracturing behavior, when silty-shale is drilled with oil or synthetic

based fluid versus water based fluid.

When water-wet silty-shale is drilled with water based fluid, it generally behaves

much like permeable (albeit tight) sandstone, with an effective filter cake formed on the

wellbore wall or inside a fracture. Therefore, fracture initiation and propagation are no

more likely to occur than in sandstone.

258

However, when water-wet silty-shale is drilled with oil or synthetic based fluid,

capillary entry pressures are high enough to significantly inhibit leak off and filter cake

development, similar to clean shale but with lower fracture initiation and propagation

pressures. Therefore, without a protective filter cake, water-wet silty-shale drilled with oil

or synthetic based fluid, is the most likely formation type to undergo hydraulic fracture

initiation and propagation. Since permeability is higher than for clean shale, it may also be

possible that wellbore pressure can be communicated to both the wellbore wall and

potentially to the far-field, via natural fractures or interconnected pore space. These

conditions also mean that both preventive and remedial wellbore strengthening methods

are less likely to be successful.

Wellbore breathing, which is loosely described as fluid losses while circulating,

followed by fluid flow back when the pumps are stopped, is almost always associated with

oil or synthetic based fluids and water-wet silty-shale. Wellbore breathing is very similar

to a pump-in and flow-back test, where a fracture is initiated and propagated while

circulating. Since capillary entry pressures prevent fluid penetration and leak-off inside the

fracture, fluids lost while circulating return to the wellbore when the pumps are stopped

and the fracture closes.

Figure 7.22 shows logging data from two lost circulation and wellbore breathing

events, in the same wellbore, drilled with synthetic based drilling fluid. Both loss zones are

clearly identified by repeated resistivity measurements, while the gamma ray data clearly

indicates that both fluid loss events occurred in silty-shale, rather than clean shale or

sandstone.

259

Figure 7.22: Two lost circulation and wellbore breathing events occurred in silty shale

formations, rather than in clean shale or clean sand formations.

7.8 SUMMARY

Various types of field injectivity tests, for interpreting fracture parameters and

minimum in-situ stress, are reviewed in this chapter, including FITs, LOTs, XLOTs, and

PIFB tests. Effects of several key factors, such as formation permeability, fluid leak-off

and filter cake development, on field injectivity tests are discussed. It has been found that

FIP, LOP, FBP, FPP, and FCP are all related to formation permeability, fluid leak-off and

filter cake sealing. Ignoring their influences may lead to incorrect interpretation of field

injectivity tests, and consequent drilling problems, e.g. lost circulation, and unnecessary

cost. Disregarding fractured carbonates, lost circulation and wellbore breathing are highly

related to formation and fluid type and are most likely to occur in water-wet silty-shale

drilled with oil or synthetic based fluids.

260

A fully coupled fluid flow and geomechanics model for numerical simulation of

field injectivity tests is presented. The model successfully takes into account the main

elements in a field injectivity test, including fluid flow in the well, fracture propagation,

fluid low in the fracture, pore fluid flow and deformation of formation rock. The essential

signatures of field injectivity tests known from previously reported field tests can be

captured by the model. PIFB tests in permeable and impermeable formations are simulated

and compared using the model. It is demonstrated by the simulation results that an accurate

𝑆ℎ𝑚𝑖𝑛 may be determined in a permeable formation using traditional methods based on the

time development of pressure; while for impermeable formations where most of the field

tests are performed during drilling, a flowback test may be needed for a better prediction

of 𝑆ℎ𝑚𝑖𝑛 due to limited fluid leak-off. It is important to properly design the shut-in and/or

flowback schedules to ensure fracture closure in a field test. The model presented in this

chapter provides a useful tool for optimizing field test design to ensure that sufficient and

high-quality data are obtained for test interpretation.

261

CHAPTER 8: Conclusions and Future Work

This chapter first summarizes the major research work and the key conclusions of

this dissertation. Then, some recommendations are proposed for future work related to this

research.

262

8.1 CONCLUSIONS

I. The roles of fracture initiation and propagation pressures on lost circulation and

wellbore strengthening are first discussed in this dissertation. Factors that may affect

these two pressures are analyzed, which include micro-fractures on the wellbore wall,

in-situ stress anisotropy, pore pressure, fracture toughness, filter cake development,

fracture bridging/plugging, bridge location, fluid leak-off, rock permeability, pore size

of rock, mud type, mud solid concentration, and critical capillary pressure. The key

conclusions of this analysis included:

FIP of a wellbore with micro-fractures is controlled not only by pore pressure and

in-situ stresses, but also by fracture length and fracture toughness of the formation

rock. It can be much lower than that of a perfect wellbore.

Plugging a fracture can significantly increase its propagation pressure, especially

in formations with large differences between pore pressure 𝑃𝑝 and minimum

horizontal stress 𝑆ℎ𝑚𝑖𝑛.

Fluid leak-off through the fracture face hinders fracture growth by facilitating filter

cake development and reducing the fluid energy available to propagate the fracture.

Capillary entry pressure 𝑃𝑐𝑒 is an important and often neglected consideration for

lost circulation mitigation and wellbore strengthening. High capillary entry

pressures, associated with small pore openings and immiscible fluids, can restrict

fluid leak-off and filter-cake/plug development. Field observations indicate lost

circulation in fractured and silty shale formations occurs more frequently with

OBM/SBM than with WBM. Additionally, the observation that wellbore breathing

typically occurs in water-wet formations drilled with OBM/SBM may be elegantly

explained by capillary theory.

263

II. A finite-element framework to simulate lost circulation during drilling with circulation

of drilling fluid is developed. Circulation of drilling fluid in the “U-Tube” wellbore and

fracture propagation in the porous formation are coupled together to predict the

dynamic fluid loss and fracture geometry evolution in drilling process. The fluid flow

in the well is modeled based on the Bernoulli’s equation taking into account the viscos

loss. Fracture propagation and fluid flow in the fracture are modeled based on a pore

pressure cohesive zone method. The numerical model provides a unique new way to

model lost circulation in drilling when the boundary condition at the fracture mouth is

neither a constant flowrate nor a constant pressure, but rather a dynamic bottom-hole

pressure or ECD. Factors that can affect ECD and thus lost circulation, including mud

density, mud viscosity, pump rate and annulus clearance, are investigated using the

proposed model. The key conclusions of this work include:

The viscous pressure loss term due to fluid circulation in the annulus can lead to

significant ECD increase and fluid loss. Drilling mud with relatively low density

which does not cause lost circulation in static state without circulation may lead to

significant fluid loss after resuming mud circulation. So it is important to take into

account the dynamic circulation effect on lost circulation prediction and prevention.

In drilling operations, we are interested in preventing fractures from occurring by

controlling ECD, or plugging the fractures at the early time of their growth using

LCMs, so the capability of capturing the dynamic fluid loss and fracture geometry

development of the proposed framework can help us understand how to prevent

lost circulation, optimize mud rheology, and select LCMs.

III. An analytical solution and a numerical model are developed to investigate the role of

mudcake on preventive wellbore strengthening treatments based on plastering wellbore

with mudcake. The analytical solution derived based on steady-state fluid flow

264

assumption incorporates the effects of mudcake thickness, permeability and strength

on wellbore stress and fracture pressure. In order to describe time-dependent mudcake

effect on preventive wellbore strengthening. A finite-element framework based on

poroealstic theory is developed to investigate the transient effects of dynamic mudcake

thickness buildup and permeability reduction on the near-wellbore stress and pore

pressure, and thus the strengthening of wellbore. The key conclusions of this study

include:

Both mudcake thickness and permeability have great influence on wellbore stress

and fracture pressure. With the decrease of mudcake permeability and/or increase

of mudcake thickness, fracture pressure increases.

Mudcake strength has negligibly small effects on wellbore stress and fracture

pressure, and thus wellbore strengthening.

Taking into account the dynamic mudcake thickness buildup and permeability

reduction results in a time-dependent wellbore stress state between that without

considering mudcake and that assuming an impermeable mudcake.

The time-dependent mudcake model provides a useful tool to analyze the stress

evolution around wellbore with dynamic mudcake development for the design and

evaluation of preventive wellbore strengthening treatments based on plastering

wellbore surface with mudcake.

IV. An analytical solution and a finite-element model are proposed for modeling remedial

wellbore strengthening treatment based on plugging/bridging lost circulation fractures

using LCMs. The analytical model, based on linear elastic fracture mechanics theory,

provides a fast procedure to predict fracture pressure change before and after fracture

bridging. The numerical model, taking into account poromechanical effects, provides

a more accurate prediction of the distribution of local stress with remedial wellbore

265

strengthening operations. Sensitivity analyses are performed using both of the

analytical and numerical models to quantify the effects of rock properties, in-situ

stresses, bridge locations and fluid flow on remedial wellbore strengthening. Key

conclusions of this study include:

Fracture pressure can be significantly increased by bridging the small pre-existing

fractures emanating from the wellbore wall in remedial wellbore strengthening

operations.

The closer the bridging location to the fracture mouth, the better strengthening can

be achieved. This means for better application of wellbore strengthening

techniques, it is important to accurately predict fracture geometry, especially

fracture mouth opening, for selecting the best LCM size.

Remedial wellbore strengthening applications by bridging pre-existing fractures are

more effective for formations with small in-situ stress anisotropy than those with

large stress anisotropy.

Remedial wellbore strengthening applications are more effective for formations

with low pore pressure and in-situ stress ratio, such as pressure depleted reservoirs,

as compared to formations with large pore pressure and in-situ stress ratio, such as

deepwater high pressure formations.

After bridging a fracture, there is a compression increase area near the bridging

location, whereas the tensile stress near the fracture tip decreases. This means that

the fracture becomes more difficult to reopen and propagate, and lost circulation is

less likely to continue.

Fluid leak-off affects both hoop stress and fracture width distributions. It is

important to consider fluid leak-off on the wellbore wall and fracture faces in

266

predicting fracture geometry, optimizing LCM size distribution, and evaluating

potential hoop stress enhancement with wellbore strengthening.

For effectively strengthening the wellbore by preventing fluid communication

between the wellbore and fracture tip, a desirable LCM bridge should have low

enough permeability to ensure there is no fluid flow across the bridge and the

fracture portion behind the bridge can close due to fluid leak-off. The permeability

of the bridging plug is likely a more important parameter than its strength.

V. A three-dimensional finite-element framework is developed to exam the possibility of

fluid leakage through casing shoe and along the weak cement interface when there is

pressure buildup in the wellbore due to change of drilling or completion fluid,

conduction of injectivity tests, and etc. The model is used to quantify the length, width,

and circumferential coverage of the cement interface debonding fractures. The key

conclusions of this work include:

Non-uniform debonding fractures may occur under various conditions with

different in-situ stresses, pre-exiting cracks at the casing shoe, and cement and

formation properties

With anisotropic horizontal stress, the debonding fracture has a smaller width in the

direction of the maximum horizontal stress and a larger width in the direction of

the minimum horizontal stress.

With initial cracks in the cement interface, debonding fractures tend to develop

vertically along the axis direction of a vertical well, rather than circumferentially

around the well.

The debonding fracture propagation is highly influenced by the stiffness of cement

and formation.

267

The proposed model provides a useful tool for simulating the debonding of cement

interface caused by leakage and pressure buildup around the casing shoe. It is useful

for evaluating the risk of cement sheath failure and fluid leakage from cement

interface during drilling, pressure tests, perforation, hydraulic fracturing, and any

kinds of fluid or gas injection operations during the production phase.

VI. Finally, importance of field injectivity tests for understanding the fundamentals of lost

circulation and wellbore strengthening are highlighted, with a review of different kinds

of field tests and a discussion of their advantage and limitations. A coupled fluid flow

and geomechanics is also developed to simulate injectivity tests with pump-in, shut-in

and flowback phases. The key conclusions of this study include:

FIP, LOP, FBP, FPP, and FCP in an injectivity test are all related to formation

permeability, fluid leak-off and filter cake sealing. Ignoring their influences may

lead to incorrect interpretation of injectivity tests, and consequent drilling

problems, e.g. lost circulation, and unnecessary cost.

Accurate identification of formation types where lost circulation is more likely to

occur is extremely important for successful drilling operations. Disregarding

fractured carbonates, lost circulation and wellbore breathing are highly related to

formation and fluid type and are most likely to occur in water-wet silty-shale drilled

with oil or synthetic based fluids.

The injectivity test simulation model can capture the key elements of injectivity

tests known from field observation and aid the interpretation and design of field

tests.

It is demonstrated by the injectivity test simulation results that an accurate 𝑆ℎ𝑚𝑖𝑛

may be determined in a permeable formation using traditional methods based on

the time development of pressure; while for impermeable formations where most

268

of the injectivity tests are performed, a flowback test may be needed for a better

prediction of 𝑆ℎ𝑚𝑖𝑛 due to limited fluid leak-off. It is important to properly design

the shut-in and/or flowback schedules to ensure fracture closure in a field test.

8.2 FUTURE WORK

This dissertation made an attempt to perform a systematic study on lost circulation

and wellbore strengthening. Several analytical and numerical models were developed to

model dynamic fluid loss while drilling, preventive wellbore strengthening based on

plastering wellbore with mudcake, and remedial wellbore strengthening based on

bridging/plugging lost circulation fractures. Some recommendations for future research

related to this dissertation are described as following:

For modeling lost circulation while drilling, it is highly recommended to consider

pre-existing fractures on wellbore wall with which the lost circulation fractures may

not propagate perfectly along the direction of maximum horizontal stress as

assumed in this dissertation.

Thermal effect is an important factor to consider in the modeling studies of lost

circulation and wellbore strengthening.

For the study of preventive wellbore strengthening based on strengthening wellbore

with mudcake, dynamic filtration tests are recommended to investigate the time-

dependent developments of both external and internal mudcake, and then

incorporate the test results with advanced numerical models to simulate time-

dependent stress and failure around wellbore.

For the study of remedial wellbore strengthening, advanced numerical models that

have the capabilities of simulating transportation and deposition of LCMs in the

269

lost circulation fractures will be very useful for modeling the dynamic fracture

bridging/plugging process in wellbore strengthening.

For better application of wellbore strengthening techniques, it is important to

accurately and quickly estimate/measure the geometry of lost circulation fractures

during drilling for selecting/adjusting the size distribution of LCMs in a real-time

manner. Improved or new logging while drilling techniques are needed for

acquiring better knowledge of drilling-induced or pre-exiting natural fractures on

the wellbore wall.

The current wellbore strengthening studies mainly focus on addressing fluid loss

through hydraulically induced fractures in sand/shale formations. Severe losses are

also commonly encountered in carbonate formations with vugs, cavities, and large

fractures. More efforts on addressing lost circulation in carbonates are

recommended for further study.

270

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