copyright by berkay kocababuc 2011

Copyright

by

Berkay Kocababuc

2011

The Thesis Committee for Berkay Kocababuc

Certifies that this is the approved version of the following thesis:

Finite Element Analysis of Wellbore Strengthening

APPROVED BY

SUPERVISING COMMITTEE:

Kenneth E. Gray

Evgeny Podnos

Supervisor:

Co-Supervisor:


by

Berkay Kocababuc, B.S.

Thesis

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

Master of Science in Engineering

The University of Texas at Austin

December 2011

Dedication

To my father.

v

Acknowledgements

First and foremost, I would like to express my deepest gratitude to my supervisor,

Dr. Kenneth Gray for his guidance and encouragement on the project. His endless

patience and support resulted in the final product of this study.

I would also like to thank Dr. Eric Becker and Dr. Evgeny Podnos for taking the

time to provide valuable feedback and knowledge. This thesis would not have been

possible without their help and suggestions.

I want to offer my deepest gratitude to Dr. Paul Bommer for giving me the

opportunity to work as a teaching assistant in Blowout Prevention and Control course. It

was a valuable experience.

I’m also very thankful to Turkish Petroleum Corporation for their financial

support.

Finally, I’m very grateful to my family for their love, encouragement and support.

I would not have been able to accomplish any of this without them.

December 2, 2011

vi

Abstract


Berkay Kocababuc, M.S.E

The University of Texas at Austin, 2011

Supervisor: Kenneth E. Gray

Co-Supervisor: Evgeny Podnos

As the world energy demand increases, drilling deeper wells is inevitable. Deeper

wells have abnormal pressure zones where the difference between pore pressure and

fracture pressure gradient, is very small. Smaller drilling margins make it harder to drill

the well and result in high operation costs due to the increase of non-productive time.

One of the major factors influence non-productive time in drilling operations is lost

circulation due to drilling induced fractures. The most common approach is still plugging

the fractures by using various loss circulation materials and there are several wellbore

strengthening techniques present in the literature to explain the physics behind this

treatment. This thesis focuses on development of a rock mechanics/hydraulic model for

quantifying the stress distribution around the wellbore and fracture geometry after

fracture initiation, propagation and plugging the fracture with loss circulation materials.

In addition, fracture behavior is investigated in different stress states, for different

vii

permeability values and in the presence of multiple fractures. The following chapters

contain detailed description of this model, and analysis results.

viii

Table of Contents

List of Tables ......................................................................................................... ix

List of Figures ..........................................................................................................x

1. Introduction .....................................................................................................1

1.1. Lost Circulation .....................................................................................2

1.2. Drilling Induced Fractures .....................................................................3

1.3. Wellbore Strengthening .........................................................................6

2. Finite Element Analysis of Wellbore Strengthening ......................................7

2.1. Two-Dimensional Model Description ...................................................7

2.1.1. Geometry of the Model ........................................................7

2.1.2. Accuracy of the Model and Mesh ........................................9

2.1.3. Fracture Modeling ..............................................................13

2.1.4. Multiple Fractures ..............................................................15

2.2. Three-Dimensional Model Description ...............................................16

2.2.1. Geometry of the Model ......................................................16

2.2.2. Initial and Boundary Conditions ........................................17

2.2.3. Accuracy of the Model and Mesh ......................................18

2.2.4. Fracture Modeling ..............................................................21

2.2.4.1. Mechanical Behavior of the Fracture Zone .................21

2.2.4.2. Pore Fluid Flow Properties at the Fracture Zone ........23

3. Results ...........................................................................................................25

3.1. Two-Dimensional Analysis of Wellbore Strengthening ......................25

3.2. Three-Dimensional Analysis of Wellbore Strengthening ....................39

4. Conclusions ...................................................................................................48

References ..............................................................................................................50

ix

List of Tables

Table-1 Input data for two-dimensional model verification ....................................8

Table-2 Input data for three-dimensional model verification ................................19

Table-3 Input data for two-dimensional analysis ...................................................25

Table-4 Input data for three-dimensional analysis .................................................39

x

List of Figures

Figure-1.1. Problem incidents, Gulf of Mexico shelf gas wells (Grayson 2009) ....2

Figure-1.2. Stress states superposition for equal horizontal stresses (Hubert, M.K. and

Willis, D.G. 1957) ...............................................................................3

Figure-1.3. Stress fields for different horizontal stress ratios (Hubert, M.K. and

Willis, D.G. 1957) ...............................................................................4

Figure-1.4. Stress states superposition with wellbore pressure (Hubert, M.K. and

Willis, D.G. 1957) ...............................................................................5

Figure-2.1. Two-dimensional model geometry .......................................................7

Figure-2.2. Boundary conditions of the two-dimensional model ............................8

Figure-2.3. Quad shape elements with different geometric orders ..........................9

Figure-2.4. Mesh with different element densities ................................................10

Figure-2.5. Hoop stress distribution around the wellbore for different number of

elements ............................................................................................10

Figure-2.6. Hoop stress for different geometric order ...........................................11

Figure-2.7. Hoop Stress Error for Four Node Quad Elements ..............................12

Figure-2.8. Hoop stress error for different geometric order ..................................12

Figure-2.9. Hoop stress distribution with a fracture ..............................................13

Figure-2.10. Hoop stress distribution with a bridge at the fracture mouth ............14

Figure-2.11. Multiple fractures around the wellbore .............................................15

Figure-2.12. Three-dimensional model loads ........................................................16

Figure-2.13. Boundary conditions of the three-dimensional model ......................17

Figure-2.14. Hex shape elements with different geometric orders (Abaqus 6.10-EF

Documentation, 2010).......................................................................19

xi

Figure-2.15. Three-dimensional model mesh ........................................................20

Figure-2.16. Hoop stress distribution for different geometric order and mesh .....20

Figure-2.17. Three-dimensional cohesive element (Abaqus 6.10 Documentation,

2010) .................................................................................................21

Figure-2.18. Flow patterns of pore fluid within cohesive elements (Abaqus 6.10-EF

Documentation, 2010).......................................................................23

Figure-2.19. Permeable layers of a cohesive element ............................................23

Figure-3.1. Hoop stress distribution with zero wellbore pressure .........................26

Figure-3.2. Hoop stress distribution with initiation pressure applied to the wellbore

...........................................................................................................26

Figure-3.3. Radial stress distribution with zero wellbore pressure ........................27

Figure-3.4. Radial stress distribution with initiation pressure applied to the wellbore

...........................................................................................................27

Figure-3.5. Hoop stress change along the wellbore ...............................................28

Figure-3.6. Radial stress change along the wellbore .............................................28

Figure-3.7. Hoop stress distribution after fracture propagation (deformations

magnified 20 times) ..........................................................................29

Figure-3.8. Hoop stress distribution after plugging (deformations magnified 20 times)

...........................................................................................................30

Figure-3.9. Change in fracture opening after plugging..........................................31


...........................................................................................................31

Figure-3.11. Hoop stress distribution with a lower wellbore pressure ..................31

Figure-3.12. Fracture opening for different wellbore pressures ............................32

Figure-3.13. Effects of stress anisotropy on fracture opening ...............................33

xii

Figure-3.14. Effects of stress anisotropy on hoop stress .......................................33

Figure-3.15. Hoop stress distribution after three fractures propagated (deformations

magnified 20 times) ..........................................................................34

Figure-3.16. Fracture lengths in multi fracture model ...........................................35

Figure-3.17. Fracture width comparison between single and multi fracture models35

Figure-3.18. Hoop stress distribution after plugging the primary fracture in multi

fractured model .................................................................................36

Figure-3.19. Change in primary fracture opening in multi fracture model ...........37

Figure-3.20. Change in secondary fracture opening in multi fracture model ........37

Figure-3.21. Hoop stress distribution for multi fracture model .............................38

Figure-3.22. Comparison of hoop stress increase between two models ................38

Figure-3.23. Hoop stress distribution at the end of first step in three-dimensional

model.................................................................................................40

Figure-3.24. Hoop stress distribution after fracture propagation in three-dimensional

model (deformation magnified 50 times) .........................................40

Figure-3.25. Hoop stress distribution after plugging the fracture in three-dimensional

model (deformation magnified 50 times) .........................................41

Figure-3.26. Fracture geometry in three-dimensional model ................................41

Figure-3.27. Hoop stress distributions during wellbore strengthening in three-

dimensional model ............................................................................42

Figure-3.28. Fracture geometry for different loss rates .........................................43

Figure-3.29. Hoop stress distribution after fracture propagation for different loss rates

...........................................................................................................44

Figure-3.30. Hoop stress distribution after plugging for different loss rates .........44

Figure-3.31. Effects of permeability on fracture growth .......................................45

xiii

Figure-3.32. Effects of permeability on hoop stress ..............................................45

Figure-3.33. Effects of mud cake permeability on fracture growth .......................46

Figure-3.34. Pore pressure distribution after fracture propagation ........................47

1

1. Introduction

In a drilling process, it is crucial to keep hydrostatic pressure gradient of the

drilling fluid in between pore pressure and fracture pressure gradients. When the

hydrostatic pressure of the drilling fluid is lower than the formation pore pressure at any

zone, formation fluid begins to flow into the wellbore. This influx can lead to a blowout

if it is not controlled. On the other hand, when the hydrostatic pressure of the drilling

fluid is higher than the fracture pressure, drilling fluid begins to flow into the formation

through drilling induced fractures. The flow of the drilling fluid into the formation

through drilling induced fractures can cause significant mud losses. To prevent influx and

mud losses, upper and lower limits of the operational hydrostatic pressure gradient of the

drilling fluid are defined by fracture pressure and pore pressure gradients. This margin,

between pore pressure and fracture pressure gradients, is known as mud-weight window.

Abnormalities in pressure zones result in smaller drilling margins. When the drilling

margin is small, it is harder to maintain drilling fluid gradient at desired levels which

increases the operation costs due to the increase of non-productive time. In addition, non-

productive time influences total drilling costs a lot more in deeper wells because daily

operating costs of deeper wells are much higher than shallower ones. According to the

data from James K. Dodson Company, average non-productive time for more than 1700

wells drilled between 1993 and 2002 in the Gulf of Mexico is 24% of the total drilling

time and 13% of the non-productive time caused by lost circulation (Grayson 2009).

2

Figure-1.1. Problem incidents, Gulf of Mexico shelf gas wells (Grayson 2009)

1.1. LOST CIRCULATION

Lost circulation can be defined by leakage of drilling fluid into the formation.

Drilling fluid losses may range from a few barrels to hundreds of barrels. It may

acceptable to keep on drilling for small amounts of mud losses. However, uncontrolled

increase in loss rate decreases the hydrostatic pressure of the drilling fluid, which

increases the risk of getting a kick and having an underground blowout. Also, failing to

maintain enough wellbore pressure to support the wellbore walls can cause stuck pipe

due to borehole collapse. Furthermore, lost circulation can damage the formation in

productive zones. Four common types of lost circulation zones are highly permeable

formations, natural fractures, vugular or cavernous formations and drilling induced

3

fractures. Seepage losses mostly occur at highly permeable formations located at shallow

depths. The fluid loss property of the mud cake has significant importance during

penetration of these highly permeable zones. Deeper formation matrices do not allow

fluid flow easily because they have lower permeability values compared to shallow zones

due to the overburden. However, severe lost circulation usually occurs through fractures

in deeper formations and the most common type of lost circulation is caused by induced

vertical fractures (Messenger, J. U. et al. 1981). In addition, 90% of the loss circulation

incidents are caused by fracture propagation (Dupriest 2005).

1.2.DRILLING INDUCED FRACTURES

Drilling induced fractures are tensile fractures which initiate when the tension due

to wellbore pressure exceeds the compressive stresses around the wellbore wall. Based on

the Kirsch solution, for an isotropic stress state in an infinite plate with a circular opening

tangential stress, which is often called hoop stress, is equal to two times to the horizontal

stress (Hubert, M.K. and Willis, D.G. 1957).

Figure-1.2. Stress states superposition for equal horizontal stresses (Hubert, M.K. and

Willis, D.G. 1957)

4

However, for an anisotropic stress state, hoop stress changes as the degree of

anisotropy increases. Figure-1.3 shows the stress fields for different horizontal stress

ratios.

Figure-1.3. Stress fields for different horizontal stress ratios (Hubert, M.K. and Willis,

D.G. 1957)

5

In addition, a wellbore pressure, greater than the formation pressure, changes the

stress state around the wellbore and can be estimated by using the Lame equations for a

non-penetrating fluid (Hubert, M.K. and Willis, D.G. 1957). Figure-1.4 shows a stress

state superposition where the horizontal stress ratio is 1.4 and wellbore pressure is equal

to 1.6 times of the minimum horizontal stress.

Figure-1.4. Stress states superposition with wellbore pressure (Hubert, M.K. and Willis,

D.G. 1957)

The tensile strength of the rock is ignored, tensile fractures start to initiate as soon

as the wellbore wall goes into tension. For a homogeneous, anisotropic, linearly elastic

and vertical wellbore, minimum and maximum hoop stress values defined by Kirsch

solutions are given below in the following equations (Peng, S. and Zhang, J. 2006).

6

1.3.WELLBORE STRENGTHENING

Wellbore Strengthening can be defined as increasing the resistance of the

formation to initiation or propagation of a fracture to obtain a wider mud window. There

are different techniques in the literature to strengthen a wellbore. Generally, wellbore

strengthening methods are either based on near wellbore region stress field alteration or

enhancing fracture propagation pressure by isolating the fracture tip (Van Oort et al.,

2009). The concepts “Stress Caging” and “Fracture Closure Stress” are two examples for

the wellbore strengthening methods based on near wellbore region stress field alteration

(Alberty and Mclean, 2004, Dupriest, 2005). Stress caging aims to plug the fracture

mouth with certain size materials to increase the tangential stress around the borehole

(Alberty and Mclean, 2004). In addition, the fracture closure stress method focuses on

fracture opening maximization and maintaining the width by filling the fractures with

wellbore strengthening materials where size of the particles is not important (Dupriest,

2005). On the other hand, “Fracture Propagation Resistance” is built on the pressure felt

at the fracture tip, which is the driving mechanism of fracture propagation, and it is

defined as extension of drilling margin by isolating fracture tip (Van Oort et al., 2009).

However, there will be no increase in fracture gradient; it can only stop propagation of an

existing fracture. In spite of successful field results reported for different approaches,

industry hasn’t found a common ground on factors effecting wellbore strengthening.

In the following chapters, a finite element model will be explained and simulation

results will be interpreted based on these approaches to identify which factors have the

most significant effect on wellbore strengthening. These factors can lead to design of the

best treatment for strengthening a wellbore.

7

2. Finite Element Analysis of Wellbore Strengthening

2.1.TWO-DIMENSIONAL MODEL DESCRIPTION

2.1.1. Geometry of the Model

The model is constructed in two dimensions with plane strain elements by using

ABAQUS software. Only half of the wellbore is modeled due to symmetry. Length and

width of the model is shown in Figure-2.1.

Figure-2.1. Two-dimensional model geometry

Rock is assumed to be linearly elastic and far field stresses are applied as in figure

2.1. Symmetry boundary condition is applied to y-axis on the left and to make sure

nothing moves in the model, one node in the middle of the right face is fixed for zero

displacement in the y direction, as shown in figure 2.2. The hoop stress values around the

wellbore are calculated for a pre-fractured state with zero wellbore pressure. Input data

used in the model are given in table 1.

85 inches

85

inches

8.5 inches

8

Figure-2.2. Boundary conditions of the two-dimensional model

Table-1 Input data for two-dimensional model verification

Wellbore diameter 8.5 inch

Max Horizontal Stress 2200 psi

Min Horizontal Stress 1800 psi

Young Modulus 1740000 psi

Poisson’s ratio 0.25

Displacement

of this node is

zero in y

direction

Symmetry

boundary

condition

9

2.1.2. Accuracy of the Model and Mesh

Hoop stress distributions calculated by the model are compared with the Kirsch

solution for different number of elements around the borehole. Equations used to

calculate the analytical solution are given below.

A structured mesh has been generated with quad elements. Denser mesh improves

the accuracy; however it increases the computation time required. Geometric order of the

elements is another factor that affects accuracy. Geometric order of quad shape elements

can either be linear with four nodes or quadratic with eight nodes. First, four different

mesh densities are used for the model with four node elements. Figure 2.5 shows the

hoop stress distribution around the wellbore for different number of elements.

Figure-2.3. Quad shape elements with different geometric orders

10

Figure-2.4. Mesh with different element densities

Figure-2.5. Hoop stress distribution around the wellbore for different number of elements

2000

2500

3000

3500

4000

4500

5000

0 20 40 60 80 100

Ho

op

Str

ess

(p

si)

Angle (deg)

2D model with 4 node quad elements

Kirsch solution

160 elements around thehalf wellbore


80 elements around the halfwellbore

40 elements around the halfwellbore

11

Secondly, effects of geometric order have been studied for two different element

densities and the results are given in Figure 2.6.

Figure-2.6. Hoop stress for different geometric order

Even though denser mesh and second order elements improve the accuracy, they

increase the computation time significantly. In order to decrease the computation time

most basic mesh within acceptable error limits should selected. Error percentage of the

hoop stress values compared to the analytical solution has been calculated to select the

most appropriate mesh. In this case, eight node quad elements, with eighty elements

around the half wellbore are used for two-dimensional model.

2000

2500

3000

3500

4000

4500

5000

0 20 40 60 80 100

Ho

op

Str

ess

(p

si)

Angle (deg)

Effects of Geometric Order

Kirsch solution

8 node elements with 80 elements around the half wellbore




12

Figure-2.7. Hoop Stress Error for Four Node Quad Elements

Figure-2.8. Hoop stress error for different geometric order

0.0%

2.0%

4.0%

6.0%

8.0%

10.0%

12.0%

14.0%

16.0%

18.0%

20.0%

0 20 40 60 80 100

Ho

op

Str

ess

Erro

r C

om

par

ed t

o

An

alyt

ical

So

luti

on

Angle (deg)





0.0%

2.0%

4.0%

6.0%

8.0%

10.0%

12.0%

14.0%

16.0%

18.0%

20.0%

0 20 40 60 80 100

Ho

op

Str

ess

Erro

r C

om

par

ed t

o

An

alyt

ical

So

luti

on

Angle (deg)

4 node quad elements with40 elements around the halfwellbore




13

2.1.3. Fracture Modeling

In stress caging, the pressure required for fracture reopening is considered

essentially to same as the pressure required to propagate a fracture, because it is assumed

that the drilling fluid can seep into the previous fracture even if the near wellbore region

is in compression (Alberty and Mclean, 2004). A fixed-length crack has been defined to

simulate this behavior in a two-dimensional, plane strain, linearly elastic model. To do

this, two quarters of the wellbore are sketched individually and then tied together by

using tie constraint except along the fracture zone. Not using the tie constraint along the

fracture would allow fracture faces to move freely. At the first step, wellbore pressure is

applied to the crack faces as fracture pressure. Figure 2.9 shows forces applied to the

model and the hoop stress distribution while there is a crack present. The dark blue region

is the region with maximum compression, while the red region is the region with

maximum tension.

Figure-2.9. Hoop stress distribution with a fracture

Pw

Pfrac

Pfrac

14

For the second step, a certain number of nodes at the fracture is fixed in terms of

displacement by using velocity boundary condition and a load which is equal to

formation pressure is applied to the fracture faces behind these nodes to simulate

bridging. By doing this, it is assumed that the drilling fluid inside the fracture fully

dissipated into the formation and there will be no increase in local formation pressure. In

addition, the bridge is considered incompressible and it provides an effective seal

between the wellbore pressure and fracture pressure. Location of the bridge and length of

the bridge is variable and can be changed by the user. If there is space between the bridge

and wellbore, wellbore pressure is applied to the fracture faces in front of the bridge. To

simulate closing of the fracture and prevent overlapping of the fracture faces after

bleeding off the pressure inside the fracture, hard contact interaction is defined along the

fracture.

Figure-2.10. Hoop stress distribution with a bridge at the fracture mouth

Pw

Po

Po

15

2.1.4. Multiple Fractures

If the near wellbore region is still in tension after the first fracture propagation,

more fractures might initiate and grow around the wellbore. A multiple fractured model

has been designed to investigate effects of multiple fractures. This model has two

symmetric fractures in addition to the fracture perpendicular to the minimum horizontal

stress. To create this model four parts are sketched individually and tied together by using

tie constrain as shown in Figure 2.11. Instead of using tie constrains along the fractures

hard contact used, so fracture faces can move freely. All three fractures have equal

lengths and the same pressure is applied to all fracture faces for the first step. For the

second step, only the middle fracture is plugged with loss circulation materials, while

wellbore pressure is kept the same inside of the other two fractures. The motivation

behind this method is previous research which pointed out that a primary fracture is more

likely to control stress alteration around the borehole where multiple fractures are present

(Wang, H. et al. 2009).

Figure-2.11. Multiple fractures around the wellbore

16

2.2.THREE-DIMENSIONAL MODEL DESCRIPTION

2.2.1. Geometry of the Model

The model is constructed in three dimensions with pore pressure elements by

using ABAQUS software. Only half of the wellbore is modeled due to symmetry. Length,

height, and width of all models are equal to ten times the wellbore diameter. The solid

section is assumed to be linearly elastic and far field stresses are applied as in Figure

2.12. According to the aforementioned theory, vertical fractures propagate perpendicular

to the minimum horizontal stress. For this reason, a fracture zone is defined in the middle

of the model as shown in Figure 2.12.

Figure-2.12. Three-dimensional model loads

SH

Sh

Sh

σv

Fracture zone

(perpendicular to σhmin)

17

2.2.2. Initial and Boundary Conditions

Symmetry boundary condition is applied in the y direction, bottom nodes are

fixed for zero displacement in the z axis, and to make sure nothing moves in the model,

one node in the middle is fixed for zero displacement in the y direction, as shown in

Figure 2.13. Pore pressure, and void ratio is assigned to all nodes as initial conditions.

Void ratio is defined as the ratio of the volume of voids to the volume of solid material.

Single phase flow is assumed and saturation is taken as unity for all simulations.

Permeability is defined in material properties as a function of void ratio and assumed

homogeneous and isotropic.

Figure-2.13. Boundary conditions of the three-dimensional model

Displacement of

this node is zero

in y direction

Displacements of

bottom nodes are

zero in z direction

Symmetry boundary

condition

18

2.2.3. Accuracy of the Model and Mesh

Hoop stress distributions are calculated by the three-dimensional model and

compared with the Kirsch solution for different number of elements around the borehole.

Effective stress components considering pore pressure are given below.

So, the effective stress at the wellbore wall is expressed as;

19

As the complexity of the model increases, element type and mesh become more

important. A structured mesh has been generated with hex shape elements. Denser mesh

improves the accuracy as in two-dimension models but increases the computation time

drastically in three-dimensional analysis. Geometric order of the elements is another

concern. Geometric order of hex shape elements can either be linear with eight nodes or

quadratic with twenty nodes.

Figure-2.14. Hex shape elements with different geometric orders (Abaqus 6.10-EF

Documentation, 2010)

The hoop stress values around the wellbore are calculated for a pre-fractured state

with zero wellbore pressure for different geometric order and mesh. The results are given

in Figure 2.16. Input data used in the model is given in Table 2.

Table-2 Input data for three-dimensional model verification

Wellbore diameter 8.5 inch

Overburden Stress 3000 psi

Max Horizontal Stress 2200 psi

Min Horizontal Stress 1800 psi

Formation Pressure 870 psi

Young Modulus 1740000 psi

Poisson’s ratio 0.25

20

Figure-2.15. Three-dimensional model mesh

Figure-2.16. Hoop stress distribution for different geometric order and mesh

2000

2500

3000

3500

4000

4500

5000

5500

6000

0 20 40 60 80 100

Ho

op

Str

ess

(p

si)

Angle (deg)

Kirsch solution

linear hex shape porepressure elements

quad hex shape porepressure elements

linear hex shape porepressure elements with 4times denser mesh

21

2.2.4. Fracture Modeling

The fracture zone is modeled by using pore-pressure cohesive elements based on

traction-separation modeling. Cracks can only initiate and propagate along this zone and

there is no need of a crack to start with. A crack can initiate in cohesive zone as long as

the initiation criteria are satisfied. Similarly, damage evolution criterion must be satisfied

for fracture propagation.

Figure-2.17. Three-dimensional cohesive element (Abaqus 6.10 Documentation, 2010)

2.2.4.1.Mechanical Behavior of the Fracture Zone

The quadratic nominal stress criterion is used as fracture initiation criteria in the

model. This criterion is satisfied when the following quadratic interaction function

reaches one (Abaqus 6.10-EF Documentation, 2010).

Each term represents the square of nominal stress ratios. Each ratio is the nominal

stress value divided by maximum nominal stress value, where , and are the

maximum values of nominal stress.

22

Fracture propagation criteria are defined by using a damage evolution law. After

initiation criterion is satisfied, the damage evolution law represents the magnitude of

propagation. This magnitude is defined by D which is the overall damage in the material.

The value of D lies between 0 and 1, where D=1 means complete damage. Stress

components affected by the damage are given below (Abaqus 6.10-EF Documentation,

2010).

where, , and are the predicted stress components.

With tensile fractures, Benzeggagh-Kenane fracture criterion given below

describes the problem better than any other criteria, because the critical fracture energies

along the first and second shear directions are equal (Benzeggagh and Kenane, 1996,

Abaqus 6.10-EF Documentation, 2010).

where, GC is the fracture energy dissipated.

In this study all the elastic properties of the fracture zone are taken from a

hydraulically induced fracture problem in Abaqus example problems manual (Abaqus

6.10-EF Documentation, 2010).

23

2.2.4.2.Pore Fluid Flow Properties at the Fracture Zone

Two different flow patterns can be modeled by using pore pressure cohesive

elements. These two flow patterns are tangential flow within the gap and normal flow

across the fracture shown in Figure 2.18.

Figure-2.18. Flow patterns of pore fluid within cohesive elements (Abaqus 6.10-EF

Documentation, 2010)

Tangential flow is defined by using gap flow subsection in material properties of

the fracture zone. In this subsection fluid properties must be specified as either power-law

or Newtonian. In this study, Newtonian fluid with a viscosity of 1 × 10-6

kPaS (1 cp) was

selected for all models. The other flow pattern, normal flow, describes fluid leakage into

the formation through the fracture. Leak-off coefficients for top and bottom faces of a

cohesive element can be used to define mud cake permeability inside the fracture as

shown in Figure-2.19.

Figure-2.19. Permeable layers of a cohesive element

24

In the modeled case, it is assumed that leak-off coefficients are equal for both

faces and normal flow is defined as;

( )

where, c is the leak-off coefficient and q is the flow rate into the formation.

Unfortunately, no experimental data have been reported in terms of leak-off

coefficient. Leak-off coefficient values used in this model are also taken from

hydraulically induced fracture problem in ABAQUS example problems manual.

25

3. Results

3.1.TWO-DIMENSIONAL ANALYSIS OF WELLBORE STRENGTHENING

First run of the two-dimensional model for an example application of wellbore

strengthening consists of three steps. First step gives the stress distribution around the

wellbore to identify fracture initiation pressure. In the second step, a fixed length crack

has been added to the model and the initiation pressure found at the first step is applied to

the wellbore and fracture faces. Fracture mouth is fixed by using velocity boundary

condition to simulate plugging at the last step. Input data used in the simulation is given

in Table 3.

Table-3 Input data for two-dimensional analysis

Wellbore diameter 8.5 inch 0.10795 m

Model Dimensions 85x85x85 inch 1.0795x1.0795x1.0795 m

Max Horizontal Stress 2200 psi 15150 kPa

Min Horizontal Stress 1800 psi 12400 kPa

Formation Pressure 870 psi 6000 kPa

Young Modulus 1740000 psi 1.2e7 kPa

Poisson’s ratio 0.25 0.25

Second order quad shape elements are used for all two-dimensional analysis to

yield the most accurate results. For comparison, hoop stress and radial stress distributions

are given in Figure 3.1-3.4. These figures are taken from ABAQUS software. Negative

sign in ABAQUS corresponds to compression, while positive sign corresponds to

tension. Also, units in ABAQUS legend are kPA. ABAQUS stress values multiplied by

“–1” and units converted to field units in Excel graphs. No wellbore pressure is applied

in Figure 3.1 and 3.3. Hoop stress value at zero degree, from Figure 3.1 is used as the

fracture initiation pressure in the next step.

26

Figure-3.1. Hoop stress distribution with zero wellbore pressure


27

Figure-3.3. Radial stress distribution with zero wellbore pressure

Figure-3.4. Radial stress distribution with initiation pressure applied to the wellbore

28

Figure-3.5. Hoop stress change along the wellbore

Figure-3.6. Radial stress change along the wellbore

0

500

1000

1500

2000

2500

3000

3500

0 10 20 30 40

Ho

op

Str

ess

(p

si)

Distance from the wellbore (inch)

Hoop Stress Distribution

Pw=3190psi

Pw= 0 psi

0

500

1000

1500

2000

2500

3000

3500

0 10 20 30 40

Rad

ial S

tre

ss (

psi

)


Radial Stress Distribution

Pw=3190psi

Pw= 0 psi

29

Figure-3.7. Hoop stress distribution after fracture propagation (deformations magnified

20 times)

During the second step, a fixed length fracture propagates instantly. Figure 3.7

shows the hoop distribution with a propagated fracture. Compression around the wellbore

increases as the fracture gains width. The increase in compression is much higher in the

region close to the fracture as shown in Figure 3.7. The fracture tip is the region with

high tension. If this tension is higher than rock can withstand, fracture growth is

expected. In this case, fracture is not growing regardless of high tension because the

fracture length is fixed in the two-dimension model. The third step simulates wellbore

strengthening by plugging the fracture with loss circulation materials. For this case, the

location of the bridge is the first element at the fracture mouth. Also, length of the bridge

equals to the length of this element. The particle diameter is assumed to be the same size

as the fracture width and it is not moving or losing volume inside the fracture. After

30

plugging the fracture mouth and dropping the pressure inside the fracture to pore

pressure, the fracture starts to close as shown in Figure 3.8. As the fracture closes,

bridging materials inside the fracture mouth cause more compression at the region close

to the fracture. This increases the hoop stress at low angles, which might prevent

initiation of new fractures.

Figure-3.8. Hoop stress distribution after plugging (deformations magnified 20 times)

0

100

200

300

400

500

600

700

800

0 2 4 6

Frac

ture

op

en

ing

(mic

ron

)


propagation

after plugging

31

Figure-3.9. Change in fracture opening after plugging


For the second run of the simulation, it is assumed that the drilling mud can seep

into the previously opened fracture before the wellbore region becomes tensile. So, this

time a wellbore pressure lower than the initiation pressure is used for the analysis.

Figure-3.11. Hoop stress distribution with a lower wellbore pressure

0

1000

2000

3000

4000

5000

6000

0 20 40 60 80 100

Ho

op

Str

ess

(p

si)

Angle (deg)

initiation

propagation

after bridging

0

500

1000

1500

2000

2500

3000

3500

0 20 40 60 80 100

Ho

op

Str

ess

(p

si)

Angle (deg)

pre fractured state

propagation

after bridging

32

Figures 3.10 and 3.11 have similar trend, the difference in the compressive trend

after fracture propagation is due to the change in fracture width. As the pressure applied

to the fracture faces increase, fracture width increases.

Figure-3.12. Fracture opening for different wellbore pressures

However wellbore pressure is not the only factor affecting the fracture width.

Stress anisotropy can also change the fracture width. For this reason, another simulation

is designed to investigate effects of stress anisotropy. Three different horizontal stress

ratios are used in the simulations by changing minimum horizontal stress. All data

besides minimum horizontal stress are taken constant. Fracture openings for different

stress ratios are given in Figure 3.13. It can be seen that fracture width is directly

proportional to the degree of anisotropy. Hoop stress values calculated for pre-fractured

state, with a fracture present and after plugging show similar trends in Figure 3.14. Even

though the fracture width increases, lower horizontal stress is results in lower

compression. Therefore, they compensate each other and generate the trend in Figure

3.14.

0

100

200

300

400

500

600

700

800

0 1 2 3 4 5 6

Frac

ture

op

en

ing

(mic

ron

)


Pw=3190 psi

Pw=2650 psi

Pw=2100 psi

33

Figure-3.13. Effects of stress anisotropy on fracture opening

Figure-3.14. Effects of stress anisotropy on hoop stress

0

100

200

300

400

500

600

700

800

900

0 1 2 3 4 5 6

Frac

ture

op

en

ing

(mic

ron

)


SH=1.5Sh

SH=2Sh

SH=3Sh

after plugging, SH=1.5Sh

after plugging, SH=2Sh

after plugging, SH=3Sh

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

0 10 20 30 40 50 60 70 80 90 100

Ho

op

Str

ess

(p

si)

Angle (deg)

Pre-fractured state SH=1.5Sh Fractured state SH=1.5Sh After plugging SH=1.5Sh

Pre-fractured state SH=2Sh Fractured state SH=2Sh After plugging SH=2Sh

Pre-fractured state SH=3Sh Fractured state SH=3Sh After plugging SH=3Sh

34

As the degree of anisotropy increases, tension around the borehole increases. This

could cause initiation and growth of additional fractures. Thus, other simulations are

designed to investigate effects of multiple fractures. Two more fractures are included into

the model in addition to the fracture perpendicular to the minimum horizontal stress

direction. New fractures are symmetric and both of them are 22.5º away from the primary

fracture. Fractures have the same length and all fracture pressures are equal to the

wellbore pressure. The hoop stress distribution after fractures are propagated is shown in

Figure 3.15. Similar to the single crack model, the regions around the crack tips are

dominated by tension, and compressive stresses around the borehole increase away from

the fractures. The fracture in the middle has the maximum tension around the fracture tip,

and it is the widest fracture. The other two fractures have smaller widths due to

compression caused by the middle fracture. Fracture lengths are shown in Figure 3.16. In

addition, fracture widths in multi fracture model are compared with the single fracture

model widths in Figure 3.17.

Figure-3.15. Hoop stress distribution after three fractures propagated (deformations

magnified 20 times)

35

Figure-3.16. Fracture lengths in multi fracture model

Figure-3.17. Fracture width comparison between single and multi fracture models

0

100

200

300

400

500

600

0 1 2 3 4 5 6

Frac

ture

Wid

th (

mic

ron

)

Fracture Length (inch)

primary fracture(SH=3Sh)

secondary fracture(SH=3Sh)

primary fracture(SH=2Sh)

secondary fracture(SH=2Sh)

primary fracture(SH=1.5Sh)

secondary fracture(SH=1.5Sh)

0

100

200

300

400

500

600

700

800

900

0 1 2 3 4 5 6

Frac

ture

Wid

th (

mic

ron

)


single fracture model(SH=3Sh)

single fracture model(SH=2Sh)

single fracture model(SH=1.5Sh)

multi fracture model(SH=3Sh)

multi fracture model(SH=2Sh)

multi fracturemodel(SH=1.5Sh)

36

For the last step, only the primary fracture is plugged with lost circulation

materials. Bridge location is in the fracture mouth and fracture pressure behind the bridge

is set to formation pressure. Fracture pressure is kept constant for the other two fractures.

Hoop stress distribution after plugging the primary fracture is shown in Figure 3.18.

During this step, the primary fracture starts to close and tension around its tip decreases,

while secondary fractures gain width and tension increases at their fracture tips.

Figure-3.18. Hoop stress distribution after plugging the primary fracture in multi

fractured model

Fracture geometry changes for primary and secondary fractures are given in

Figures 3.19 and 3.20. Primary fractures behave similar to these in the single fracture

model. Secondary fracture widths stay almost the same at the fracture mouths. However,

secondary fractures gain some width inside the fracture due to the high fracture pressure

inside them. If this pressure decreases as the fluid dissipates to the formation, the

secondary fractures will close.

37

Figure-3.19. Change in primary fracture opening in multi fracture model

Figure-3.20. Change in secondary fracture opening in multi fracture model

0

100

200

300

400

500

600

0 1 2 3 4 5 6

Frac

ture

Wid

th (

mic

ron

)


primary fracture (SH=3Sh)

primary fracture afterplugging (SH=3Sh)

primary fracture (SH=2Sh)

primary fracture afterplugging (SH=2Sh)

primary fracture (SH=1.5Sh)

primary fracture afterplugging (SH=1.5Sh)

0

50

100

150

200

250

0 2 4 6

Frac

ture

Wid

th (

mic

ron

)


secondary fracture (SH=1.5Sh)

secondary fracture after pluggingthe middle fracture (SH=1.5Sh)

secondary fracture (SH=3Sh)

secondary fracture after pluggingthe middle fracture (SH=3Sh)

secondary fracture (SH=2Sh)

secondary fracture after pluggingthe middle fracture (SH=2Sh)

38

Hoop stress distribution at the end of the model shows that the treatment is still

increasing fracture gradient at low angles even if only the primary fracture has been

plugged. But lower fracture widths result in a lower increase compared to single fracture

model.

Figure-3.21. Hoop stress distribution for multi fracture model

Figure-3.22. Comparison of hoop stress increase between two models

0

1000

2000

3000

4000

5000

0 20 40 60 80 100

Ho

op

Str

ess

(p

si)

Angle (deg) after fracture propagation (SH=3Sh) after plugging (SH=3Sh)

after fracture propagation (SH=2Sh) after plugging (SH=2Sh)

after fracture propagation (SH=1.5Sh) after plugging (SH=1.5Sh)

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 20 40 60 80 100

Ho

op

Str

ess

(p

si)

Angle (deg)

after propagation of a single fracture after plugging (single fracture model)

after propagation of three fractures after plugging (multi fracture model)

39

3.2.THREE-DIMENSIONAL ANALYSIS OF WELLBORE STRENGTHENING

To investigate factors effecting wellbore strengthening more precisely, results

from a three-dimensional model with pore pressure needs to be studied carefully. A tree-

dimensional model for this study consists of three steps. Fracture initiation pressure is

applied to the wellbore for the first step to create tension at the fracture zone. After

initiation criterion is satisfied, fluid flow into the fracture is defined and fracture

propagation is observed in the second step. In the third and final step, the fracture is

plugged with lost circulation materials. These materials create an effective sealing

between wellbore pressure and fracture tip. At the end of the third step, pressure inside

the fracture is dissipated to the formation. Input data used in the following simulations

are given in Table 4.

Table-4 Input data for three-dimensional analysis

Wellbore diameter 8.5 inch 0.10795 m

Model Dimensions 85x85x85 inch 1.0795x1.0795x1.0795 m

Overburden Stress 3000 psi 20700 kPa

Max Horizontal Stress 2200 psi 15150 kPa

Min Horizontal Stress 1465 psi 10100 kPa

Formation Pressure 870 psi 6000 kPa

Porosity 0.32 0.32

Permeability 160 md 16x10-14m2

viscosity 1cp 1x10-6 kPas

Young Modulus 1740000 psi 1.2e7 kPa

Poisson’s ratio 0.25 0.25

Flow rate into the fracture 0.015 bbl/min 4x10-5 m3/s

40

Figure-3.23. Hoop stress distribution at the end of first step in three-dimensional model

Figure-3.24. Hoop stress distribution after fracture propagation in three-dimensional

model (deformation magnified 50 times)

41

Figure-3.25. Hoop stress distribution after plugging the fracture in three-dimensional

model (deformation magnified 50 times)

Figure-3.26. Fracture geometry in three-dimensional model

0

50

100

150

200

250

300

350

400

450

0 10 20 30 40 50

Frac

ture

Wid

th (

mic

ron

)


after fracture propagation

after plugging

42

Due to tension created around the wellbore, a fracture is initiated, propagated and

plugged as shown in Figures 3.23 to 3.25. Differently from previous two-dimensional

models, a cohesive zone allows determination of fracture length and incorporation of

pore pressure into the model as shown in Figure 3.26. At the end of three-dimensional

analysis, results are consistent with the two-dimensional model. As in Figure 3.27,

compression increase around the wellbore after fracture propagation. Opening a fracture

creates higher compression in the region close to the fracture, while it creates a lower

compression increase further away from the fracture. After plugging the fracture, it starts

to close while drilling fluid inside the fracture dissipates into the formation. At the end,

there is significant hoop stress increase observed at low angles.

Figure-3.27. Hoop stress distributions during wellbore strengthening in three-dimensional

model

0

500

1000

1500

2000

2500

0 10 20 30 40 50 60 70 80 90 100

Ho

op

str

ess

(p

si)

Angle (deg)

initiation pressure applied to the wellbore after fracture propagation after plugging

43

As mentioned before, rate of fluid loss must be defined in the model to determine

fracture geometry. Figure 3.28 shows fracture geometry for different fluid losses.

Fluctuation in the figure is because of model length. Some fractures are actually longer

than the model but boundary conditions restricted into growth. In this case, a trend-line

can be added to determine actual fracture length instead of using a longer model.

Figure-3.28. Fracture geometry for different loss rates

Compression created around the wellbore due to a fracture is directly related with

the fracture width. Wider fractures cause higher compression around the wellbore as

shown in Figure 3.29. Moreover, plugging a wider fracture creates a higher hoop stress

increase during wellbore strengthening as shown in Figure 3.30.

0

100

200

300

400

500

600

0 10 20 30 40 50

Frac

ture

Wid

th (

mic

ron

)


after fracturepropagation (0.23bbl loss)



44

Figure-3.29. Hoop stress distribution after fracture propagation for different loss rates

Figure-3.30. Hoop stress distribution after plugging for different loss rates

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 20 40 60 80 100

Ho

op

Str

ess

(p

si)

Angle (deg)

initiation pressure appliedto the wellbore

after fracture propagation(0.15 bbl loss)



0

500

1000

1500

2000

2500

3000

0 20 40 60 80 100

Ho

op

Str

ess

(p

si)

Angle (deg)

after plugging (0.15 bblloss)



45

Effects of permeability can be observed in this finite element model. During

propagation of a fracture, more fluid flows into the formation in highly permeable rocks.

This will reduce the fracture pressure and decreases fracture growth. Decrease in fracture

growth then decreases compression around the wellbore.

Figure-3.31. Effects of permeability on fracture growth

Figure-3.32. Effects of permeability on hoop stress

-10

0

10

20

30

40

50

60

70

0 10 20 30 40 50

Frac

ture

Wid

th (

mic

ron

)


160 md

80 md

0

500

1000

1500

2000

0 20 40 60 80 100

Ho

op

str

ess

(p

si)

Angle (deg)

initiation pressure applied to the wellbore (160 md) after fracture propagation (160 md)

after plugging (160 md) initiation pressure applied to the wellbore (80 md)

after fracture propagation (80 md) after plugging (80 md)

46

Another important factor is mud cake permeability inside the fracture. The

aforementioned leak-off option is used to simulate this behavior by defining a leak-off

coefficient. A low leak-off coefficient corresponds to low mud cake permeability.

Similarly, as rock permeability, more fluid flow into the formation reduces fracture

pressure and growth.

Figure-3.33. Effects of mud cake permeability on fracture growth

Furthermore, the leak-off coefficient is a very important factor that determines pore

pressure distribution around the wellbore during fracture propagation as well as the rate

of fluid loss. However, there are no correlations between this value and lab experiments

yet (Salehi and Nygaard, 2010). In the simulation in this thesis, the leak-off coefficient is

an arbitrary number. The pore pressure distributions using this value are shown in figure

3.34. The highest values correspond to color grey is the pressure inside the fracture.

-20

0

20

40

60

80

100

120

140

160

0 10 20 30 40 50

Frac

ture

Wid

th (

mic

ron

)


high mud cakepermeability

low mud cakepermeability

47

Figure-3.34. Pore pressure distribution after fracture propagation

48

4. Conclusions

The objective of this thesis was to develop detailed finite element models and

procedures to investigate factors effecting wellbore strengthening. ABAQUS software

was used to develop finite element models. Development started with a two-dimensional,

plane strain, linearly elastic model and upgraded to a three-dimensional model with

consideration of pore pressure effects. The models developed can simulate initiation and

propagation of a fracture. Furthermore, the models can quantify stress distributions

around the wellbore after plugging the fracture with loss circulation materials. Moreover,

fracture geometry length and width can be observed in three-dimensional model. The

following conclusions can be drawn from the simulation results;

Opening a fracture increases compression around the wellbore. This

increase is highest in regions close to the fracture compared to regions

further away.

Plugging the fracture mouth with loss circulation materials does not

increase the hoop stress everywhere around the wellbore. It increases the

hoop stress at the near wellbore region close to the fracture and it may

prevent opening and growth of additional fractures.

Fracture width is the most significant factor to increase hoop stress, if the

bridging material can keep the fracture open and sealed.

After plugging the fracture mouth, the weakest point around the wellbore

changes. Because of this, if the wellbore pressure is increased to new

initiation pressures after treatment, new fractures grow in new directions.

If there is more than one fracture present, fractures will be smaller than

expected and their size differs based on fracture location.

49

As the formation permeability or mud cake permeability increases,

fracture length and width decreases, because more fluid will flow into the

formation. Thus, pressure felt by the fracture tip decreases.

Finally, for future work, finite element models should be tested with detailed field

data and confirmed by using field results of wellbore strengthening applications.

Moreover, additional experimental studies are required to calibrate the models.

50

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