ALI_TAGHICHIAN.pdf

UNIVERSITY OF OKLAHOMA

GRADUATE COLLEGE

ON THE GEOMECHANICAL OPTIMIZATION OF HYDRAULIC FRACTURING

IN UNCONVENTIONAL SHALES

A THESIS

SUBMITTED TO THE GRADUATE FACULTY

in partial fulfillment of the requirements for the

Degree of

MASTER OF SCIENCE

By

ALI TAGHICHIAN Norman, Oklahoma

2013

ON THE GEOMECHANICAL OPTIMIZATION OF HYDRAULIC FRACTURING IN UNCONVENTIONAL SHALES

A THESIS APPROVED FOR THE MEWBOURNE SCHOOL OF PETROLEUM AND GEOLOGICAL ENGINEERING

BY

______________________________ Dr. Musharraf Zaman, Chair

______________________________ Dr. Ahmad Ghassemi

______________________________ Dr. Deepak Devegowda

© Copyright by ALI TAGHICHIAN 2013 All Rights Reserved.

iv

Acknowledgements

First of all, I would like to express my deep gratitude and sincere thanks to my helpful and

kind advisor, Dr. Musharraf Zaman, because of all his help and support, key advices, and

his special style of encouragement; without these I would never be able to complete this

research work.

I am deeply grateful of my parents and my dear siblings for all their help, support, and

inspirations regarding my education at each level.

I also want to express my appreciation to Mr. Timothy L. Beard, Manager-ETG

operations, at Chesapeake Energy Corporation, Dr. Ali Daneshy, Head of Daneshy

Consultant International, for allocating their valuable time to me, advising me regarding

completion of unconventional resources, and being so patient and helpful.

I am indeed thankful of Dr. Arul Britto, Emeritus Faculty at University of Cambridge, Dr.

Jean-Claude Roegiers, Emeritus Faculty at University of Oklahoma, Dr. Ahmad Ghassemi

and Dr. Deepak Devegowda, faculty members of Mewbourne School of Petroleum and

Geological Engineering, University of Oklahoma for their kindness and sympathetic helps

and advices.

My gratitude also goes to Mr. Amin Mousavi, PhD candidate of Petroleum Geomechanics,

Tarbiat Modares University and Dr. Nicolas Roussel, Reservoir Engineer at ConocoPhillips

for their useful discussions.

Final thanks are also given to Oklahoma Department of Transportation (ODOT) and

Oklahoma Transportation Center because of their financial support during the course of

this study.

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Table of Contents

Acknowledgements .......................................................................................................................... iv Table of Contents .............................................................................................................................. v List of Tables ................................................................................................................................... viii List of Figures ................................................................................................................................... ix Abstract .............................................................................................................................................. xi Chapter 1: Introduction .................................................................................................................... 1

1.1. Why unconventional resources ................................................................................... 1 1.2. Challenges and solutions in unconventional resources ........................................... 1 1.3. Critical questions in optimization of unconventional reservoirs ........................... 2 1.4. Approaches of fracing optimization in unconventional resources ........................ 3

1.4.1. Production approach ............................................................................................. 3 1.4.2. Geomechanics approach ....................................................................................... 3

1.5. Methods in geomechanical optimization of fracturing design ............................... 4 1.6. Strategies to control stress shadow and stress intensity factor ............................... 5 1.7. Contents of this thesis .................................................................................................. 6

Chapter 2: Literature Survey ............................................................................................................ 8 2.1. General ........................................................................................................................... 8 2.2. History of fracture mechanics ..................................................................................... 8 2.3. History of stress shadow around hydraulic fractures ............................................. 10 2.4. Methods of calculating stress field around cracks .................................................. 11

2.4.1. Stress field in the vicinity of crack tip ............................................................... 12 2.4.2. Stress field for the entire medium ...................................................................... 15

2.5. Calculation of stress shadow around hydraulic fractures ...................................... 17 2.5.1. Analytical determination of stress shadow size ............................................... 17 2.5.2. Numerical determination of stress shadow size .............................................. 18

2.6. Analytical calculation of crakc aperture ................................................................... 19 2.7. Need for a comprehensive study for optimization of fracing design .................. 20

Chapter 3: Methodology and Verification ................................................................................... 21 3.1. General ......................................................................................................................... 21 3.2. Significance of numerical methods for this problem ............................................. 21 3.3. Different scenarios for calculation of stress shadow size ..................................... 22 3.4. Major assumptions and data range in stress shadow analysis ............................... 24

3.4.1. Hydraulic fracture geometry ............................................................................... 25 3.4.2. Different boundary conditions for shadow analysis ....................................... 26 3.4.3. Assumptions ......................................................................................................... 26 3.4.4. Data range ............................................................................................................. 27

3.5. Shadow mechanisms ................................................................................................... 28 3.6. Method of studying stress intensity factor .............................................................. 29

3.6.1. Parameters related to SIF change ...................................................................... 29 3.6.2. Monitoring of SIF change along the fracture edge by a single value ............ 31

3.7. Different scenarios for calculation of SIF change.................................................. 33 3.8. Data range for the SIF change analysis .................................................................... 35 3.9. Verification of stress field for simple problems ..................................................... 35

Chapter 4: Prediction of shadow and aperture of hydraulic fractures ..................................... 38 4.1. General ......................................................................................................................... 38

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4.2. Prediction of aperture for a contained hydraulic fracture ..................................... 38 4.2.1. Effect of in-situ stresses on aperture ................................................................. 38 4.2.2. Mathematical equation for prediction of aperture of a single contained

fracture ................................................................................................................... 39 4.2.3. Reliability of the proposed function for aperture prediction ......................... 40 4.2.4. Visualization of aperture prediction function .................................................. 41

4.3. Prediction of shadow around a contained hydraulic fracture (θT 5°) ............ 42 4.3.1. Numerical results for stress shadow analysis ................................................... 42 4.3.2. Mathematical equation predicting stress shadow of a single contained

fracture ................................................................................................................... 43 4.3.3. Coefficients of the proposed function for stress shadow prediction ........... 44 4.3.4. Reliability of the proposed function for stress shadow prediction ............... 46 4.3.5. Visualization of stress shadow prediction function ........................................ 47 4.3.6. Discussion on the relationship between aperture and stress shadow ........... 48

4.4. Prediction of shadow around a contained hydraulic fracture (θT 5°) ............ 48 4.4.1. Mathematical equation for prediction of stress shadow with different

threshold angles .................................................................................................... 49 4.4.2. Reliability of the equation predicting stress shadow with different threshold

angles ...................................................................................................................... 51 4.5. Effect of uncontainment of the fracture (standalone well fracturing) ................ 52

4.5.1. Change of stress shadow by uncontainment of the fracture ......................... 53 4.5.2. Change of aperture by uncontainment of the fracture ................................... 54 4.5.3. Mathematical equations predicting aperture and shadow size in uncontained

single fractures ...................................................................................................... 54 4.5.4. Reliability of the proposed functions for uncontainment multiplier

determination ........................................................................................................ 56 4.5.5. Use of uncontainment multipliers (standalone fractures) .............................. 57

4.6. Effect of simultaneous fracturing on shadow size and aperture .......................... 57 4.6.1. Change of stress shadow size by simultaneous fracturing ( 5°) .......... 57 4.6.2. Mathematical equation for prediction of simultaneous multiplier ................ 59 4.6.3. Reliability of the proposed equation for simultaneous multiplier prediction

60 4.6.4. Change of stress shadow size by simultaneous fracturing ( 80°) ........ 61

4.7. Effective distance between simultaneous fractures of two parallel wells ........... 61 4.8. Prediction of shadow change for uncontained simultaneous fractures .............. 62

4.8.1. Mathematical equations for uncontainment multiplier in simultaneous fracturing ............................................................................................................... 63

4.8.2. Reliability of uncontainment multiplier for simultaneous fracturing ............ 64 4.8.3. Use of uncontainment multipliers (simultaneous fractures) .......................... 64

Chapter 5: Prediction of Propagation potential in hydraulic fractures .................................... 66 5.1. General ......................................................................................................................... 66 5.2. Effect of aspect ratio of the fractures on the SIF (single fracture) ...................... 66 5.3. Effect of multistage fracturing on the SIF and aperture in standalone wells ..... 67

5.3.1. Qualitative description of multistage fracturing influence on SIF/aperture of each fracture ..................................................................................................... 68

5.3.2. Mathematical equation for prediction of SIF and aperture change in multistage fracturing ............................................................................................ 69

vii

5.3.3. Reliability of the proposed equation for SIF and aperture change in multistage fracturing ............................................................................................ 70

5.3.4. Visualization of the proposed equation ............................................................ 71 5.4. Effect of simultaneous fracing on the SIF of single-stage fractures ................... 72

5.4.1. Qualitative description of SIF change via simultaneous fracturing .............. 72 5.4.2. Mathematical equation for prediction of SIF change in simultaneous

fracturing (single stage) ........................................................................................ 73 5.4.3. Reliability of the proposed function .................................................................. 74

5.5. Effect of simultaneous multistage fracturing on the SIF of the fractures .......... 75 5.5.1. Qualitative description of the SIF change by simultaneous multistage

hydraulic fracturing .............................................................................................. 75 5.5.2. Mathematical equation for prediction of SIF change in multistage

simultaneous fracturing ....................................................................................... 77 5.5.3. Reliability of the proposed equation .................................................................. 78

5.6. Effect of fracture offset on the SIF change in parallel wells ................................ 79 5.7. Use of the SIF change prediction functions .................................................................. 82

Chapter 6: Work flow for optimization of hydrofracing ........................................................... 83 6.1. General ......................................................................................................................... 83 6.2. Successive procedure of hydraulic fracturing optimization .................................. 83

6.2.1. Step one ................................................................................................................. 83 6.2.2. Step two ................................................................................................................. 84 6.2.3. Step three ............................................................................................................... 85 6.2.4. Step four ................................................................................................................ 85

Chapter 7: Concluding remarks and recommendations ............................................................ 86 7.1. Aperture analysis ......................................................................................................... 86 7.2. Stress shadow analysis ................................................................................................ 87 7.3. Analysis of propagation potential ............................................................................. 87 7.4. Optimized fracture network ...................................................................................... 88 7.5. Four steps in optimization of hydraulic fracturing ................................................ 89

References ......................................................................................................................................... 91 Appendix: Variable definition ........................................................................................................ 96

viii

List of Tables

Table 3. 1. Input variable range for the numerical simulation .................................................. 28

Table 3. 2. Input variable range for the numerical simulation .................................................. 35

Table 4. 1. Coefficients of the function in Eq. (4.7) ................................................................... 45

Table 4. 2. Shadow decrease by threshold angle ......................................................................... 50


Table 4. 4. Coefficients for Eqs. (4.12) and (4.13) for standalone fracturing ......................... 56

Table 4. 5. Coefficient of the function in Eq. (4.14) .................................................................. 60

Table 4. 6. Stress shadow change by simultaneous fracturing ( 80°) ............................ 61

Table 4. 7. Coefficients for Eqs. (4.12) and (4.13) for simultaneous fracturing ..................... 63




ix

List of Figures

Figure 2. 1. Configuration of stress field around a 2-D fracture ................................................ 9

Figure 2. 2. Stress field around a center crack with internal pressure in a 2-D medium....... 16

Figure 3. 1 Different scenarios for studying of stress shadow and aperture .......................... 22

Figure 3. 2. Geometry of a contained hydraulic fracture and stress shadow plane ............... 25

Figure 3. 3. Different stress shadow mechanisms by different hydraulic pressures .............. 29

Figure 3. 4. Fracture geometry together with stress change along its edges ........................... 31

Figure 3. 5. Stress variation along the fracture edges ................................................................. 33

Figure 3. 6. Different fracturing techniques/patterns in this study ......................................... 34

Figure 3. 7. Stress validation in the direction vertical away from the fracture center ........... 36

Figure 4. 1. Half-aperture prediction using Eqs. (4.1-4.3) ......................................................... 40

Figure 4. 2. Visualization of aperture determination function .................................................. 41

Figure 4. 3. Shadow change with Poisson’s ratio, numerical values and the predictions...... 43

Figure 4. 4. Shadow function coefficient change by Poisson’s ratio ........................................ 44

Figure 4. 5. Behavior of function coefficients with stress anisotropy ( ) ............................ 46

Figure 4. 6. Shadow size prediction using Eqs. (4.4-4.7) ........................................................... 47

Figure 4. 7. Shadow around a hydraulic fracture ........................................................................ 47

Figure 4. 8. Distribution of SH_S80 over SH_S05 ratio (for all aspect ratios) .......................... 49

Figure 4. 9. Shadow difference with respect to varied threshold angles ................................. 50

Figure 4. 10. Shadow ratio prediction (threshold angle of 80°) ................................................ 52

Figure 4. 11. Stress shadow change by fracture uncontainment ............................................... 53

Figure 4. 12. Aperture change as a result fracture uncontainment ........................................... 54

Figure 4. 13. Prediction of aperture and shadow size change by fracture uncontainment ... 56

x

Figure 4. 14. Shadow reduction by simultaneous fracturing ..................................................... 58

Figure 4. 15. Prediction of shadow size change by simultaneous hydraulic fracturing ......... 60

Figure 4. 16. Effective normalized distance between fracture tips .......................................... 62

Figure 4. 17. Prediction of shadow and aperture change by fracture uncontainment .......... 64

Figure 5. 1. SIF change along fracture edges by different aspect ratios .................................. 67

Figure 5. 2. SIF change along the edges of the fracture by multistage fracturing .................. 68

Figure 5. 3. Prediction of SIF and aperture change by multistage fractures ........................... 71

Figure 5. 4. Prediction of SIF and aperture change by multistage fracturing ......................... 72

Figure 5. 5. SIF change by simultaneous fracturing ................................................................... 73

Figure 5. 6. Precision of the predicted SIF ratios using the proposed Eq. (5.2) .................... 74

Figure 5. 7. Effect of multistage simultaneous fracturing on the SIF of fractures ................ 76

Figure 5. 8. Prediction of SIF ratio of simultaneous multistage fracturing ............................. 79

Figure 5. 9. Effect of distance and offset on the SIF of the fractures ..................................... 80

Figure 5. 10. Effect of fracture offset on SIF change ................................................................ 80

Figure 5. 11. SIF maximum line for different aspect ratios....................................................... 81

xi

Abstract

Hydraulic fracturing of unconventional resources is a way to effectively connect these

reservoirs to the wellbore and to increase their permeability. In order to have an efficient

hydraulic fracturing job, optimization of fracing design should be performed to get the best

production with the least possible cost. Since multistage simultaneous fracing is the method

by which unconventional resources are hydraulically fractured, a systematic strategy in

obtaining the optimal spacing/distance between the multistage/simultaneous fractures is of

paramount importance. From a production perspective, using a fluid flow simulator,

fracture geometry, spacing between fractures, and fracturing pattern can be determined

based on monitoring of the production. In this method, no significant change in

production by reducing the fracture spacing means that fractures with a lower spacing

would not be economical. Porosity, permeability, and fracture conductivity play key roles in

the determination of fracturing geometry/pattern. From geomechanics perspective, on the

other hand, stress shadow analysis and stress intensity factor are used to ensure straight

fractures with no deviation or collapse, and with the highest propagation potential in the

target zone. In this thesis, the two geomechanical aspects are studied in detail. The most

influencing variables on each of these strategies are included in the analysis and a

comprehensive set of equations are proposed for determination of stress shadow size and

propagation potential of the fractures. In addition, since aperture plays a decisive role in

determination of proppant type/size and conductivity of the fractures, this important

parameter is also quantified considering its own influencing variables. Finally, some

strategies are proposed for optimization of hydraulic fracturing for unconventional shales.

1

Chapter 1: Introduction

1.1. Why unconventional resources

Unconventional reservoirs have received more attention in recent years because of the

declining productivity of conventional reservoirs. Deployment of multilateral drilling

technologies together with simultaneous multistage hydraulic fracturing of parallel laterals

have also been much influencing in this regard. In 2009, almost 50% of domestic U.S. gas

production was attributed to unconventional resources and it is estimated to increase to

almost 75% by 2035 (EIA, 2011).

1.2. Challenges and solutions in unconventional resources

This growth in the development of unconventional reservoirs, primarily in shale gas/oil

reservoirs, has, however, encountered several challenges. The permeability of these shale

reservoirs typically ranges from a few to about one hundred nano-darcies. Consequently,

productivity from traditional well architecture and completion schemes was limited and

uneconomical. In early 2000’s though, several key technological developments enabled the

unlocking of this vast resource potential (Andrew et al., 2009).

The issues surrounding connectivity to the productive regions of the reservoir was largely

mitigated by developments leading to maturation in horizontal and multilateral well drilling

and completion. Of course, the real growth in the development occurred as a result of a

combination of unconventional well architecture and massive hydraulic fracture treatments,

thereby connecting the horizontal laterals to increasingly larger volumes of the reservoir

(Warpinski et al., 1997). Eventually, single-stage fracture treatments evolved to multistage

stimulation treatments and fracing of standalone well was progressed to simultaneous

2

fracing of multilateral wells in order to increase reserves per well, enhance well

productivity, and improve project economics (King et al., 2008). Currently, industry is

practicing for horizontal drilling orientation along the direction of the minimum horizontal

stress. The aim of this practice is to create fractures oriented perpendicular to the axis of

the wellbore (Dasseault, 2011). This is because transverse hydraulic fractures provide better

coverage of the reservoir than do longitudinal fractures, and are more efficient in

producing gas/oil formations (Soliman et al., 1996). It was proposed that “as the formation

permeability gets smaller, it would expectedly take a longer time to deplete the reservoir,

and it may be necessary to create more fractures to quickly deplete the reservoir” (Soliman

et al., 1997).

1.3. Critical questions in optimization of unconventional reservoirs

According to the above-mentioned progresses in the field of hydraulic fracturing in

unconventional reservoirs, the critical questions would be “How to design the fracing job

in such reservoirs? What is the optimized distance between hydraulic fractures? What

strategies should be used in order to determine fracturing pattern? What are the most

influencing variables in this regard? What is the share of each variable in defining fracturing

pattern in unconventional reservoirs? Finally and most importantly, is it possible to

quantify the effect of each variable to be used when designing fracturing treatment for such

reservoirs?”

Any acceptable answer to any of these questions can be really helpful in defining fracture

treatment in unconventional reservoirs. In fact, addressing satisfactory responses for all the

above questions, one can do the fracturing design in an optimized way with minimum cost

and maximum production.

3

1.4. Approaches of fracing optimization in unconventional resources

There are two different approaches, by which fracing optimization is conducted; the first

one is production perspective, and the other is geomechanics perspective.

1.4.1. Production approach

Some researchers have studied the fracturing pattern only considering the distance between

fractures by which the highest production is obtained by the least cost. This means that

using reservoir simulators, they were able to start from a base fracture spacing, reduce that

spacing between fractures, and monitor production change. Negligible change in

production in this method means that placing closer fractures is not economical and not

contributing to production (see e.g., Yu and Sepehrnoori, 2013).

Many researchers, however, mentioned that accurate geomechanical information about the

rock and its variation through the shale is also important since stresses predominantly

control fracture initiation and development (Abousleiman, 2007; Barree, 2009; Britt, 2009;

Xu, 2009).

1.4.2. Geomechanics approach

The beginning step for this purpose is defining the term “optimized fracture pattern” by

which we can find solutions to each of the critical questions in Section 1.3. An optimized

fracture pattern should connect the most productive areas of the reservoir to the wellbore.

As multistage fractures perpendicular to the wellbore in horizontal laterals have been

proven to be the best general pattern in this field, the key point here is finding the

optimized spacing between fractures to have maximum production, with minimum cost. In

addition, propagation potential of hydraulic fractures should be considered which means by

4

having different fracturing patterns, how fracturing pressure of the reservoir is influenced

and whether it is required to apply more/less pressure to frac the rock. This means that the

energy required for fracturing of the reservoir is also an important matter which should be

considered as well.

According to these explanations, we can define the optimized fracture pattern as “parallel

fractures, perpendicular to the wellbore axis, having the highest propagation potential in

the target formation, creating the highest production, with an optimized distance to

prevent any deviation/collapse of them.”

Therefore, from a geomechanical perspective, controlling of fracture direction and its

propagation potential are of paramount importance in optimization of fracing design in

unconventional shales. This is because by determining these two factors, questions of

fracing design in unconventional shales can be addressed in a proper and systematic way.

Therefore, it is intended in this thesis to investigate designation of fracture pattern from a

geomechanical point of view.

1.5. Methods in geomechanical optimization of fracturing design

Controlling of fracture path is dominated by a phenomenon called stress shadow effect which

is a disturbed zone around any hydraulic fracture in which direction of principal stresses is

changed. Any other fracture in this region or fractures with overlapping stress shadow

zones will result deviation or collapse1 (Fisher et al., 2004). Hence, this concept is used in

this study for determination of optimized distances by which fracture direction is

controlled in unconventional reservoirs.

1 Stress shadow causes one fracture to collapse on the adjacent fracture and further growth is observed only for one fracture.

5

Propagation potential of hydraulic fractures in the target zone, on the other hand, is

dominantly controlled by stress intensity factor and of course, toughness of the fractures.

Stress intensity factor (SIF) is an indicative of the stress field around the crack tip and its

corresponding strength variable in this regard is known as fracture toughness. As soon as

stress intensity factor reaches the fracture toughness, propagation occurs (Yew, 1997).

Thus, any increase/decrease in stress intensity factor directly results propagation to happen

with lower/higher hydraulic pressures, respectively. As a result, any change in stress

intensity factor is directly related to the amount of energy required for fracturing of the

formation and should be considered in different fracturing patterns as the concept to

examine fracturing potential change.

1.6. Strategies to control stress shadow and stress intensity factor

According to the fact that reducing stress shadow around hydraulic fractures without or at

least with the minimum decrease in propagation potential in the target zone brings the

opportunity to have more closely-spaced, non-deviating fractures, some technological

considerations have been reported in the literature in this regard. As stated previously,

simultaneous hydraulic fracturing is one of these methods performed in multi-lateral

horizontal wells. It is believed that the spacing between hydraulic fractures can be reduced

by utilizing this strategy. Mutalik and Gibson (2008) showed that simultaneously fractured

wells have 21-100% enhancement of initial production rates over the standalone horizontal

wells. King (2010) mentioned simultaneous fracturing as one effective strategy which

significantly raises the reservoir face-contact fractures. Waters et al. (2009) indicated

simultaneous fracturing as a powerful tool to reduce the spacing between fractures in

horizontal wells. Rafiee et al. (2012) studied simultaneous hydraulic fracturing of parallel

6

horizontal wells and modified the conventional approach of zipper frac to a more efficient

pattern called modified zipper frac. In this modified method, an alternating approach is

used for fracturing of parallel wells where far field path of hydraulic fracture is controlled

in a more acceptable way. Roussel and Sharma (2011b) suggested alternating fracturing

strategy to minimize the fracture spacing. In this method, stress shadow around outer

hydraulic fractures is used for making the middle fracture not to deviate. Of course, change

of propagation potential was not considered in their study.

Considering the most influencing variables in the size of stress shadow zone and stress

intensity factor of hydraulic fractures, aperture can also be predicted which is really useful

in term of proppant type/size and fracture conductivity determination.

Therefore, having the optimized distance between hydraulic fractures with no deviation or

collapse, having the propagation potential change in the acceptable range/direction, and

aperture of the fractures at hand, fracing crew can have an acceptable, optimized, and

desired fracturing design for an unconventional reservoir.

1.7. Contents of this thesis

In this thesis, it is tried to propose a work flow for optimization of hydraulic fracturing in

unconventional shales from a geomechanical point of view. Straight fractures without any

deviation/collapse, with the highest propagation potential in the target zone are the aim in

geomechanical optimization. This is done by utilizing two stress concepts called “stress

shadow” and “stress intensity factor”. In Chapter 2, the theoretical bases for these two

concepts are presented and the most important studies in these two fields regarding

optimization of hydraulic fracturing are also reported. In Chapter 3, the methodology of

7

determining stress shadow and stress intensity factor is presented. The most influencing

variables, the range of these variables, different boundary conditions, and different

mechanisms affecting the study are presented. Moreover, the need for investigating the

problem using numerical simulation is also reported and the used numerical technique is

verified by means of simulating simple crack problems and comparing the numerical with

analytical results. In Chapter 4, stress shadow and aperture of hydraulic fractures is studied

and two comprehensive sets of equations for estimation of stress shadow and aperture of

hydraulic fractures is proposed. In Chapter 5, stress intensity factor is investigated in a

comprehensive way and a complete set of equations for estimation of propagation

potential of hydraulic fractures having different geometries/patterns is proposed. In

Chapter 6, summing up the ideas regarding optimization of hydraulic fracturing in shales,

from production and geomechanics perspectives, a workflow is proposed for optimization

of hydraulic fracturing in shales. In Chapter 7, the summary and concluding remarks of

each step of the work is discussed and propositions are made in term of a systematic

method for optimization of hydraulic fracturing for this type of reservoirs.

8

Chapter 2: Literature Survey

2.1. General

In Chapter 1, the major factors in determining an optimized fracturing job in

unconventional reservoirs were introduced from a geomechanical point of view as stress

intensity factor and stress shadow effect. According to the fact that these two factors are

both indicative of stress field around hydraulic fractures, in this chapter, stress field around

hydraulic fractures is discussed in detail.

2.2. History of fracture mechanics

The birth point of obtaining stress field around cavities goes back to the work done by

Kirsch (1989), who succeeded to obtain stress field around a circular cavity in an infinite

two-dimensional (2-D) field. His closed form solution is still used for borehole stability

problems in elastic region. In this procedure, theory of elasticity together with complex

variable functions was utilized to obtain the stress function satisfying all the boundary

conditions. Then, Inglis (1913) obtained stress field around an elliptical cavity and proved

that by decreasing the curvature radius of the two ends of the ellipse, the increase in stress

concentration can be expressed by the following equation:

1 2 (2.1)

in which is the major axis of the ellipse and is the radius of curvature at the ellipse

sharp corners. As it is seen from Eq. (2.1), by tending the curvature of the corners to zero,

9

stress concentration approaches to infinity. This point was an eye-opening conclusion

which made engineers aware of the stress concentration at crack tips. Following this work,

a new field of mechanics was borne called “fracture mechanics”, in which stress field

around different fractures with different patterns and geometries were to be studied and

their propagation was to be pondered. Figure 2.1 shows an open fracture in a 2-D medium

with stress field around it.

Figure 2. 1. Configuration of stress field around a 2-D fracture

As it can be seen from Figure 2.1, a crack possessing two edges, have two half-lengths

called . The plate having this fracture inside can be applied by far-field tensile stress

( ;perpendicular to the crack plane) or internal pressure ( ). By having any of these two

stress/pressure boundary conditions; a stress field is created around the crack which can be

obtained.

r2

r1

r

X

Y

yy

xx

xy

yy

xx

P0

c c

10

2.3. History of stress shadow around hydraulic fractures

Many researchers mentioned the importance of stress shadow around hydraulic fractures.

Fisher et al. (2004) demonstrated that creation of a hydraulic fracture generates a zone of

altered local stresses that may impact the orientation of subsequent fractures in a

phenomenon known as the stress shadowing effect. Cheng (2009) utilized boundary

element method for 2-D geomechanical modeling of hydraulic fractures and indicated that

the number and spacing of the fractures need to be carefully selected considering stress

change in order to create effective fractures with appropriate geometries. Wong et al.

(2013) studied the interaction between adjacent hydraulic fractures using analytical and

numerical methods in two dimensions. They observed the divergence of hydraulic fractures

outward or even collapse of inside fractures on the outside ones as a result of stress

shadow effect. Singh and Miskimins (2010) indicated that an increase in spacing between

the fractures induced less interference, and hence requires less breakdown pressure to

initiate a fracture.

Waters et al. (2009) also stated that shadow following a hydraulic fracture treatment

involves the creation of a localized region of high compressive stresses perpendicular to the

fracture face in the vicinity of the fracture center. This causes the direction of maximum

stress to be reoriented in the region of the stress shadow. By locating the next treatment in

this region, fracture growth is likely to deviate or even occur parallel to the borehole axis

and consequently, necessitates optimizing fracture spacing to obtain the maximum number

of fractures oriented perpendicular to the wellbore (see also Roussel and Sharma, 2011a;

Morrill and Miskimins, 2012).

11

In general, closely spaced hydraulic fractures lead to an increase in the stress in the

minimum stress direction with values higher than the original maximum horizontal stress.

Succeeding fractures, therefore, may tend to propagate in undesirable directions parallel to

the wellbore axis, thereby reducing the efficacy of the fracture treatments and

compromising well productivity.

Stress shadow effect is a useful concept not only for determining the spacing between

transverse hydraulic fractures but also for fracture mapping and distance between

tectonically created fractures. Daneshy et al. (2012) were able to observe the shadow effect

from readings of pressure gauges installed in observation wells and used that information

for determination of fracture orientation and extent, spacing between wells, and optimum

number and spacing between fracture stages. Fischer et al. (1995) utilized the concept of

stress shadow for prediction of distance between pressurized fractures in sedimentary

rocks.

It is evident from the above-mentioned reviews that it is important to study stress shadow

of hydraulic fractures for optimization purpose.

2.4. Methods of calculating stress field around cracks

There are two different ways of determining the stress field around hydraulic fractures. In

the first method, it is assumed that radius of stress calculation point in polar system is

much smaller than the half-length of the fracture ( ≪ ; see Figure 2.1). Therefore, some

terms are neglected in the stress function, and a simple form of a complex variable

function is solved. In this method, stress field is obtained for only a small region around

the crack tip. According to the fact that propagation of a fracture is the main focus in

12

fracture mechanics and it mainly depends on stress field at the crack tip, this method

suffices for propagation potential of hydraulic fractures. In the second method, on the

other hand, no term is neglected from stress function and it is directly/numerically solved

for the entire field. This method is useful for determination of stress shadow around

hydraulic fractures.

2.4.1. Stress field in the vicinity of crack tip

Several researchers proposed the first solution for such problem as a 2-D crack embedded

in an infinite medium under far-field tensile stress or internal pressure. Among them,

works of Muskhelishvili (1933), Westergaard (1939), Sneddon and Elliot (1945), Paris

(1965), and Eshelby (1968) can be considered pioneering. They proposed that stress field

just around the crack tip can be obtained using Eq. (2.2), the notation of which

corresponds to the right tip ( , ) in Figure 2.1.

2cos

2

1 sin2sin

32

1 sin2sin

32

sin2cos

32

(2.2)

in which is the stress intensity factor (SIF) of the crack for mode one fracturing

(opening). It is worth indicating that both of our loading boundary conditions (far-field

tensile stress or internal pressure) belong to mode one loading. In addition, the SIF of a

crack is the same for the cases of far-field tensile stress and internal pressure with the same

13

magnitude (superposition principle; see Janssen et al., 2009). The generalized form of stress

field around the crack tip can be written as:

√2 (2.3)

It is important to notice that the trigonometric function, , is always the same for one

stress component. It is evident from Eqs. (2.2) and (2.3) that having different geometries of

cracks under mode one loading, only the SIF is changing and all the remaining terms are

kept unchanged. Therefore, having the SIF of each crack geometry, one can have stress

field around the crack tip. As an example, the SIF for this particular problem for far-field

stress is given by Eq. (2.4a) and for internal pressure it is calculated using Eq. (2.4b):

√2 (2.4a)

√2 (2.4b)

It is evident from Eqs. (2.4) that in case of equal magnitudes for far-field tensile stress and

internal pressure, the resultant SIF is the same. Moreover, it is seen that in the elastic

region, stress field is proportional to stress/pressure, square root of crack size, and

inversely related to the root distance from the tip.

Many 2-D crack geometries have been analytically solved and their SIFs have been

reported in the literature (e.g., Tada et al., 1973). However, there have also been some

problems in which the geometry of cracks were challenging and stress field for these

fractures were cumbersome to be analytically solved. SIF of such problems were defined by

14

utilizing Boundary Element (BE) and Finite Element (FE) methods (see e.g., Sih, 1973;

Tada et al., 1973; Murakami, 1987).

Regarding three-dimensional (3-D) crack problems, determination of stress distribution

near a flat crack, embedded in an infinitely extended homogeneous, isotropic solid, opened

up due to prescribed internal pressure, analytical solutions have been proposed by a

number of investigators for circular and elliptical geometries. Some of these solutions are

discussed by Keer (1964), Willis (1968), Sneddon and Lowengrub (1969), Shah and

Kobayashi (1971), and Guidera and Lardner (1975). Using the 2-D Fourier transform

method, Kassir (1981) succeeded in solving the SIFs around a rectangular crack. Moreover,

Mastrojannis et al. (1978) developed a method for determination of SIF of a general-

shaped crack with internal pressure in an infinite medium utilizing numerical integration.

By increasing of the internal pressure, SIF increases and reaches a critical value, called

fracture toughness, in which fracture tends to propagate and crack is elongated. It is worth

mentioning that different modes of fracturing have different fracture toughnesses.

For Mode I fracture toughness there exist ISRM (International Society of Rock Mechanics)

suggested methods to determine the fracture toughness (Ouchterlony, 1988; Fowell, 1995).

Several other methods have also been proposed in the literature (e.g., Evans, 1972; Barker,

1977; Atkinson et al., 1982; Chong and Kuruppu, 1984; Sun and Ouchterlony, 1986; Guo

et al., 1993; Chang et al., 2002, and Whittaker et al., 1992).

Based on the fact that SIF of fractures in three-dimensions is very challenging to be

analytically determined and in most of cases it is required to apply numerical integration,

15

therefore, researchers are using numerical techniques (e.g., FE and BE). As an example, in

hydraulic fracturing of unconventional shales with multistage simultaneous fracturing

strategy, it is important to consider the interaction of adjacent and meeting fractures on the

SIF. Numerical modeling is indeed a useful device in this regard because multi-driven

fractures in a medium may influence the SIF of each other and may change the

propagation potential. It is important to mention that propagation potential of a hydraulic

fracture is a relative term defined by the ratio of SIFs of multiple fractures over the SIF of

a single fracture. Thus, it gives a dimensionless number larger/smaller than unity because

of interacting crack tips effect.

2.4.2. Stress field for the entire medium

As explained in the previous section, Eqs. (2.2-2.4) only give stress field around the tip not

for the whole medium. This can be explained by considering a plate with a crack in the

center with far-field stress; if we put → ∞ in Eqs. (2.2), it is evident that all the stresses

tend to zero far away from the tip, which is not the case. Therefore, in order to judge the

disturbed stress zone or stress shadow size, a complete formulation for the whole field is

required to have all the components of stress at every point in the medium. In the world of

linear elastic fracture mechanics (LEFM), this is not an easy task to determine the stress

field for the whole medium, specifically for 3-D crack problems. Sneddon and Elliot (1945)

proposed Eq. (2.5) that describes stress field for the whole medium of a 2-D plate with an

internally pressurized crack in the center.

16

12

cos12

12

1 (2.5a)

12

sinsin

32

(2.5b)

sincos

32

(2.5c)

It is worth mentioning that the notation in Eqs. (2.5) corresponds to the notation in Figure

2.1. It is seen from Eqs. (2.5) that the stress field determined by six coordinate variables,

namely ( , ), ( , ), ( , ), is very challenging when such determination involves. The

stress field ( , , ) for this 2-D medium (e.g., E=30 GPa; 0.3) with an

imbedded center crack which is internally pressurized (e.g. 100 ) is presented as

Figure 2.2.

Figure 2. 2. Stress field around a center crack with internal pressure in a 2-D medium

It is seen from Figure 2.2 that stress concentration exists for all the components of stress at

crack tips. Shear stress is zero along the crack lips and on a line perpendicular to the crack

17

faces, passing through the crack center. This phenomenon happens because of the

symmetry in the model, which does not let the shear stress to exist on these symmetry

lines. The final point about Figure 2.2 is the contour shape of vertical stress ( ) around

the crack tips, which is a peanut-shaped inclined away from the crack tip. The plotted

contour shapes in Figure 2.2 exist for all the fracture geometries and can be used for

checking of the contour shapes in numerical simulations.

2.5. Calculation of stress shadow around hydraulic fractures

There are two different ways for determining the stress shadow size around hydraulic

fractures. In the first method, which is limited to some special cases of fracture geometries,

analytical stress field is used for determination of shadow size. In the second method,

numerical simulation is used for determination of stress field around hydraulic fractures

with any geometry and boundary conditions. Then, stress shadow is calculated from the

stress field. In the following sections both of these methods are presented.

2.5.1. Analytical determination of stress shadow size

Jo (2012) proposed an analytical method for prediction of stress shadow based on

analytical stress distribution around plane strain and penny-shaped fractures (see Green

and Sneddon, 1950). This method can only be used for plane strain and penny-shaped

cracks. Equations (2.6) show the proposed equations for predicting shadow size around

these two limiting cases of hydraulic fractures.

2 3 2

for plane strain fractures infinite length

14 5

for penny shaped fractures AR unity

(2.6)

18

in which, is the distance from the center of a hydraulic fracture to a point where

maximum stress contrast between and is observed, is the Poisson’s ratio, and

is height of the hydraulic fracture. From Eq. (2.6) it is seen that the only influencing

variables in the shadow size are the Poisson’s ratio and fracture height. In addition, Eq.

(2.6) accounts for the aspect ratio but it only addresses stress shadow size for two limiting

cases of plane strain and penny-shaped cracks. Effect of internal pressure, stress

anisotropy, and aspect ratios of the hydraulic fracture, however, have also been found to

influence stress shadow size (Roussel and Sharma, 2011a,b; Morrill and Miskimins, 2012).

2.5.2. Numerical determination of stress shadow size

In this method, having the total stress field from numerical simulation, the change in

magnitude and direction of the principal stresses can be obtained from Eq. (2.7) and Eq.

(2.8), respectively.

, 2 2

(2.7)

tan 22

(2.8)

in which , , are induced stresses in the shadow region, and are changed

principal stresses in this region, and is the direction of principal stresses. Deviation

angle of principal stresses ( ) can also be obtained based on the following criteria:

19

9045

(2.9)

It is worth mentioning that maximum principal stress deviation, depending on the

magnitude of internal pressure, can be near the crack tip or crack center and it vanishes

away from the crack. Therefore, by defining a threshold angle ( ) the area around the

crack with stress deviations more than this threshold angle can be determined. Considering

Eq.(2.8), it is also evident that maximizing the stress contrast means lowering the deviation

angle. Using this numerical method, there is no limitation on the fracture geometry and

boundary condition. Thus, a wide range of problems can be solved satisfactorily using this

approach.

2.6. Analytical calculation of crakc aperture

Sneddon and Elliot (1946) also documented the analytical solution for crack opening

displacement field. This analytic expression for half-aperture of a 2-D hydraulic fracture

was derived as:

1 ;2 1

(2.10)

in which is the Poisson’s ratio, is the Young’s modulus of net play, is the

hydraulic pressure applied inside the fracture, is the half length of the fracture, is

the half aperture, is the maximum half-aperture (located at the center of the fracture

( 0), and is defined on the z axis originated at crack center in the direction of fracture

20

height. From Eq. (2.10), we observe that the displacement of crack tips (at ) is zero

and displacement of the edges increases from the tips to the center of the fracture where

maximum displacement occurs. The equation also highlights the inverse relationship of

fracture aperture with Young’s modulus and Poisson’s ratio of the rock. Furthermore, the

equation describes an elliptical shape for the aperture of the fracture. It is important to

note that for a 3-D hydraulic fracture, maximum aperture is located at half-height and half-

length of it. So, an equation similar to Eq. (2.9) is written in direction, along the length of

the hydraulic fracture. Therefore, having the maximum aperture of the fracture, one can

use Eq. (2.9) and a similar one for direction for determination of aperture distribution on

the entire crack face.

2.7. Need for a comprehensive study for optimization of fracing design

According to the above-mentioned studies in the field of stress shadow and propagation

potential of hydraulic fractures, it is realized that most of these studies have been

performed in two dimensions or are more or less descriptive. Therefore, it can be seen that

there is a lack of comprehensive 3-D studies of stress shadow with propagation potential.

In this thesis, we plan to comprehensively study the stress shadow size and propagation

potential of hydraulic fractures together. In the following chapter, the methodology of

solving the present problem together with verification of the method is presented.

21

Chapter 3: Methodology and Verification

3.1. General

In Chapter 2, basic concepts of stress distribution around hydraulic fractures with

particular attention to propagation potential and stress shadow effect in the target medium

were explained using a 2-D infinite medium with an internally pressurized fracture in the

middle. In addition, a clear perspective from which optimization of hydraulic fracture

should be seen, was introduced by explaining the two concepts of stress intensity factor

(SIF) and stress shadow effect. In this chapter, the method by which we are going to

investigate the problem of optimization of hydraulic fractures in unconventional shale

reservoirs is introduced. The presented method is verified by solving a number of single

fractures in 2-D/3-D media with simple geometries and making a comparison between the

obtained results and analytical solutions reported in the literature for these problems.

3.2. Significance of numerical methods for this problem

Obtaining the stress field around a fracture with its own geometry and boundary

conditions, in a 3-D medium is a challenging matter. Although there are some analytical

solutions available in the literature for some cracks with simple geometries, but finally, a

numerical integral should be solved to obtain the stress field for these problems. As a

result, because of the limited flexibility in analytical solutions, a numerical method is

utilized in this thesis to study the problem of optimization of hydraulic fractures. Finite

Element Method (FEM) which is a powerful tool in solving governing equations of

problems in solid/fluid or coupled solid-fluid mechanics, is chosen for this purpose.

22

ABAQUS CAE 6.12, which is a general purpose FEM-based software package, is selected

for the modeling purposes. The procedure of modeling is designed in such a way to first

give us the opportunity to obtain stress shadow size which is indicative of hydraulic

fracture distance, based on different influencing variables. Then, propagation potential of

these hydraulic fractures is studied in detail to examine which geometry/pattern has what

effect on the propagation potential of the fractures. Finally, according to the constructed

numerical models, aperture of the hydraulic fractures which is highly important in the

selection of proppant type and size and determination of fracture conductivity is estimated

and its change with the defined variables is studied.

3.3. Different scenarios for calculation of stress shadow size

Four different scenarios are defined in this section, based on which stress shadow and

aperture analysis of hydraulic fractures are performed (Figure 3.1).

Figure 3. 1 Different scenarios for studying of stress shadow and aperture (pictures show fracture plane)

SingleContainedHFstandalonewell

SimultaneousContainedHFParallelwells

SingleUn‐containedHFstandalonewell

SimultaneousUn‐containedHFParallelwells3

1 2

4

23

In the first scenario (No. 1 in Figure 3.1), a 3-D hydraulic fracture, with different aspect

ratios is contained in a medium with different stress regimes, moduli, and net pressures. In

Chapter 2, theoretical background of stress shadow was presented and it was shown that

assuming different threshold angles, different shadow sizes are obtained (see Chapter 2,

Section 2.5.1). According to the fact that in the literature (see Roussel and Sharma, 2011),

region of 90° reorientation of maximum horizontal stress is assumed as the region of

minimum shadow effect, hence, it is tried to predict the shadow with all the threshold

angles.

In the second scenario (No. 2 in Figure 3.1), it is assumed that the hydraulic fracture under

consideration is not contained. Since shale net plays are located in between bounding layers

with different moduli from the net play, it is important to consider any fracture penetration

inside the bounding layers. This is done by assuming different Young’s modulus between

the net play and bounding layers, and different penetration extent to the bounding layers.

In the third and fourth scenarios (Nos. 3 and 4 in Figure 3.1), stress shadow change as a

result of simultaneous hydraulic fracturing of laterals is studied in detail. Effect of distance

between hydraulic fracture tips in parallel wells is investigated on the shadow size. In this

part of the work, effective length between fracture tips is also obtained for fractures with

different aspect ratios. It is worth mentioning that effective distance is a distance between

fracture tips beyond which no shadow size change is observed. In addition, effect of

penetration of the simultaneous hydraulic fractures inside the bounding layers (uncontained

fractures) is studied and the shadow change is investigated.

24

Using the above mentioned scenarios, stress shadow and aperture of a hydraulic fracture

can be predicted satisfactorily. First, a set of equations is proposed for a single contained

hydraulic fracture (Scenario 1). After that, in Scenario 2, effect of bounding layers for the

hydraulic fracture is presented as a ratio to be multiplied with the result of the first scenario

to result the value for an uncontained hydraulic fracture. Likewise, in the third scenario, a

ratio is obtained which is the effect of simultaneous fracturing for contained fractures. This

ratio should also be multiplied with the result of the first scenario to provide the result for

the case of simultaneous contained fractures. Finally, for the fourth scenario, effect of

uncontainment of simultaneous fractures is obtained as a ratio to be multiplied with the

result of the third scenario to give the results for simultaneous uncontained fractures.

As an example, in case one is interested in obtaining the shadow size of uncontained

simultaneous fractures between two parallel wells, it is first required to have the shadow for

the basic case; contained hydraulic fracture in a standalone well (Scenario 1). Then, effect

of simultaneous fracturing of parallel laterals is obtained for a contained fracture (Scenario

3) as a ratio to be multiplied with the result of the basic case. Finally, effect of

uncontainment of simultaneous fractures (Scenario 4) is obtained as a ratio to be multiplied

with the resultant value in the previous step.

3.4. Major assumptions and data range in stress shadow analysis

Geometry of the hydraulic fracture with respect to the net play and bounding layers,

different boundary conditions, the most influencing variables and their varying range, and

assumptions of the analyses are defined in this section.

25

3.4.1. Hydraulic fracture geometry

Geometry of the hydraulic fracture with respect to the wellbore, net play, and bounding

layers is shown in Figure 3.2a. As it can be seen from this figure, a rectangular hydraulic

fracture located in plane is made perpendicular to the wellbore. According to the fact

that maximum stress change is observed in the center of the hydraulic fracture (see Waters

et al., 2009), a plane perpendicular to the height axis ( direction) passing through the half-

height of the fracture is used for stress shadow, and maximum aperture analysis. Q plane in

Figure 3.2b shows the plane of aperture and stress shadow calculation. Length, height, and

width (aperture) of the hydraulic fracture are also shown in Figure 3.2b. It is necessary to

indicate that in this analysis, (half of the maximum aperture; see Eq. (2.10)) is

numerically calculated and can be predicted using the proposed equations in the following

sections. Therefore, aperture of the hydraulic fracture can simply be obtained by doubling

the value of .

Figure 3. 2. Geometry of a contained hydraulic fracture and stress shadow plane

Boundinglayer

wellbore

c

z

yx

H

width

Length:2c

Height:H

Aperture:2Wmax

Boundinglayer

NetPlay

Q

(a) (b)

26

3.4.2. Different boundary conditions for shadow analysis

In general, modeling approaches for evaluating the stress shadow effect adopt either one of

two boundary conditions for their case studies. These are:

1. Choice of a fixed hydraulic pressure inside the fracture ( ): This implies that

the fracture aperture is changed by varying values of the moduli of the rock and

hydraulic pressure inside the fracture.

2. Choice of a fixed maximum fracture aperture ( ): With a fixed fracture

aperture, the influence of other parameters on the stress shadowing may be

investigated.

The assumption of a constant pressure within the fracture mentioned as the first boundary

condition is a more reasonable approximation of reality and therefore, in this study, we

employ this boundary condition.

3.4.3. Assumptions

The key assumptions employed in the numerical simulation are:

The 3-D domain is assumed to be a completely elastic medium without any

plasticity-based constitutive laws;

Propagation of the fracture is not considered in this work; instead the hydraulic

fracture is assumed to have been created;

27

The aspect ratio defined as the ratio of height to the length of the fracture is

equal/less than unity;

The threshold angle ( ) is varied from 5°-80°;

The aperture of the fracture is 2 and therefore is twice the ;

In case shadow analyses are carried out for a threshold angle of 80° ( _

and _ ), only analyses with hydraulic pressures larger than maximum

horizontal stress ( ) are considered (see shadow mechanism part;

Section 3.5).

3.4.4. Data range

In this section, the adopted data range for the massive numerical simulation is presented.

Since stress shadowing is most challenging when we have lower stress anisotropies (see

Morrill and Miskimins, 2012), the ratio of minimum horizontal stress over maximum

horizontal stress is assumed to be in the range of 0.95-0.99. Moreover, previous

studies in shale have indicated that Young’s modulus of shale changes from 7 to 77 GPa

with an average value of 26.9 GPa (Hay and Sondergeld, 2011). Therefore, Young’s

modulus of the net play is assumed in the range of 10-70 GPa. For studying of the

effect of bounding layers, modulus of bounding layers is assumed as 0.25-4.0 times of that

of net play and 0.1-0.3 of the fracture height is assumed to penetrate inside the bounding

layers. For the case of simultaneous hydraulic fracturing of horizontal wells, the distance

between parallel wells is varied with similar fracture length and aspect ratios for both of the

wells. The input variables and their ranges are shown in Table 3.1.

28

Table 3. 1. Input variable range for the numerical simulation Parameter Range Unit Fracture aspect ratio 1.00-0.20 - In-situ stress anisotropy 0.95-0.99 - Poisson’s ratio 0.00-0.40 - Young’s Modulus 10-70 GPa Excess pressure 0.0-3.0 MPa Young’s modulus ratio1 0.25-4.0 - Penetration2 0.1H-0.3H m Distance between tips 3c -

3.5. Shadow mechanisms

In order to study the stress shadow, different mechanisms of shadow around a hydraulic

fracture should be recognized. The mechanisms of stress shadowing generally belong to

two categories which will be investigated in this study:

1. The pressure applied lies between the minimum and maximum horizontal

stress ( ). This results in only a marginal deviation of

the in-situ stresses around the crack tips ( 45°) and consequently,

does not lead to a principal stress reversal (90° rotation). Figure 3.3a

illustrates a marginal deviation of 40° for this scenario.

2. In cases where the applied pressure is larger than the maximum horizontal

stress ( ), a reversal of stress occurs ( 45°) and this

phenomenon is illustrated in Figure 3.3b.

It should be noted that principal stress deviations plotted in Figs. 3.3 are from the

maximum deviation plane (plane Q defined in Figure 3.3b). The pink lines show the

fracture plane.

1 Young’s modulus ratio is the ratio of Young’s modulus of the bounding layer over that of the net play 2 Penetration is the part of the fracture height penetrating the bounding layers

Figure 3.

Consequentl

fracture and

mechanism

shadow. It i

only be ana

zero for thre

3.6. Met

In this secti

edge. In thi

studied alon

3.6.1. Param

According t

knowing the

3. Different

ly, stress sha

may cause a

difference s

is important

lyzed for th

eshold angles

hod of stud

on, the tech

is thesis, as

g length and

meters related t

o Irwin (195

e SIF of the c

t stress shad

adow is seen

a diverse ran

should be k

to notice th

e second me

s of larger tha

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a result of

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to SIF change

57), stress fie

crack. Consid

(a)

29

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ept in mind

hat stress sh

echanism be

an 45°.

intensity fac

vestigating o

assuming re

e fractures.

eld around th

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Dev

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nction of the

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d when inter

hadow for th

ecause shado

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of SIF chang

ectangular fr

he crack tip

2.2), we have:

fferent hydra

e hydraulic p

principal stres

rpreting the

hreshold angl

ow for the fi

ge is present

ractures, SIF

can be dete

:

aulic pressu

pressure insid

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results of

les of 80° sh

irst mechani

ed for each

F change is t

rmined simp

(b)

ures

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. This

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hould

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30

∝ (3.1)

in which is linearly related to the internal pressure ( ), directly related to a function of

the geometry of the fracture ( ), and a function incorporating the effect of boundaries on

the fracture ( ). Effect of boundary represents itself as the effect of having cracked media

with finite dimensions, or having multiple adjacent fractures. Therefore, we can write:

(3.2)

As an example, for a 2-D fracture in an infinite medium under internal pressure (see

Chapter 2), √ and 1. Determining , for different crack geometries and

different boundary conditions is of paramount importance. This is because by having these

functions, one can determine the SIF of a specific crack geometry, thereby having the

stress field around the crack tip and consequently, its potential for propagation.

Combining Eqs. (3.1) and (3.2), we can write:

∝ (3.3)

Therefore, ratio of a stress component over the internal pressure can be assumed as

indicative of the effect of crack geometry and its boundary conditions on the SIF.

According to the fact that in this thesis, we intend to change the geometry of the hydraulic

fracture (aspect ratio) and consider different fracture patterns (simultaneous and multistage

fracturing), all the changes in the SIF can be considered as a result of changes in these two

functions. Therefore, by the above-mentioned changes, the ratio of stress over the internal

31

pressure will result the SIF change as a result of geometry and boundary condition

functions ( .

3.6.2. Monitoring of SIF change along the fracture edge by a single value

Crack tip of a 3-D hydraulic fracture is a closed line in space which may be an irregular

shape depending on many parameters including in-situ stress, hydraulic pressure change,

moduli variation, containment, heterogeneity of the medium, etc. In this study, this closed

loop has been simplified to a rectangle along which the SIF is changing by position (Figure

3.4).

Figure 3. 4. Fracture geometry together with stress change along its edges

Therefore, according to the prescribed mesh size of the FE model, a large number of data

(stress components) are associated with each case of study. In order to make a comparison

much simpler between different geometries/fracturing patterns, a function is fit to the data

and the resultant coefficients of the function are compared instead. The proposed function

which gives satisfactory fit on the numerical stress values is as given in Eq. (3.4).

Length,2c

Height,2b

x

y

zz

zz

32

1 (3.4a)

1 (3.4b)

in which is the applied hydraulic pressure, is the normal stress at the corner, is

normal stress along the length or height of the fracture, axis is in the direction of length

originated from the corner, is the fracture half-length, is the axis in the direction of

fracture height originated from the corner, is the half-height of the fracture, and are

coefficients of the function ( 1,2). Investigating the proposed function, it is revealed

that stress change is highly controlled by rather than . This is because the negative

sign of in an exponential format makes a small value to be much smaller. Since this

exponential value is subtracted from unity, no noticeable change is observed for stress in

the coefficient range we are dealing with in this regard.

Figure 3.5 shows a typical example of normal stress change along the crack edge. As shown

in the figure, behavior of stress change along crack length or height has satisfactorily been

predicted using Eqs. (3.4). It is also evident from the figure that normal stress is maximum

in the middle point of the crack length and it is reduced to the minimum value at the crack

corner. Since normal stress is normalized by being subtracted by stress at the corner and

divided by the internal pressure, the values at the corners are zero. In addition, stress

change with height or length of fractures with aspect ratio of unity is similar as expected

but for cracks with aspect ratio lower than unity, stress decreases with both length and

height of the fracture, however, the magnitude of decrease is higher in fracture height

33

compared to that of the length. The conspicuous point is that despite the fact that length

of the fracture is not changing and aspect ratio is being reduced by decreasing the fracture

height, still stress along the crack length is reduced.

Figure 3. 5. Stress variation along the fracture edges

Considering the dominant coefficient of the function proposed in Eqs. (3.4), one can have

a good comparison between SIF of different crack geometries and multi-fracture

configurations. As it can be seen from Eqs. (3.4), this value is multiplied with the

exponential function and as a result, any difference in this coefficient means the same

change in normal stress and consequently, the SIF of the crack.

3.7. Different scenarios for calculation of SIF change

There are four different scenarios that are considered for SIF change analysis in this thesis.

Figure 3.6 shows the adopted scenarios for the hydraulic fracturing pattern/technique.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Norm

alized norm

al stress, ( z

z‐ z

z 0)/PH

Normalized tip position, (x/c,y/b)

Szz_Height (AR=1.0)

Szz_Length (AR=1.0)

Szz_Height (AR=0.2)

Szz_Length (AR=0.2)

1.00 0.0 1.00

34

Figure 3. 6. Different fracturing techniques/patterns in this study

It is evident from Figure 3.6 that Scenario 1 is the basic scenario in which we have a single

stage fracture from a standalone well. Scenario 2 shows multistage hydraulic fracturing in a

standalone well. In order to compare propagation potential of the second scenario

(multistage fracturing) with the basic case (Scenario 1), the numerically obtained SIF of the

second scenario is divided by the SIF of the first scenario. Any value lower/higher than

unity means that propagation potential for the scenario under study is smaller/larger than

that in the basic case. Scenario 3 considers the effect of simultaneous single stage fracturing

of the medium between two parallel wells. The obtained SIF for this scenario should also

be compared with that in scenario one in order to quantify the effect of simultaneous

fracturing in a single stage mode. Finally, Scenario 4 is studied in which the effect of

simultaneous multistage hydraulic fracturing is considered between two parallel wells.

Scenario 4 has two influencing variables acting simultaneously on the SIF of the fractures.

The resultant SIF from this scenario should be compared with results of Scenario 1 in

order to have the effect of simultaneous multistage fracturing pattern on propagation

potential of the fractures.

1 2

3 4Ls

2Lp

35

Therefore, changing the spacing between fractures in multistage fracturing and distance

between the meeting fractures in simultaneous fracturing brings the opportunity of

quantifying the SIF change for each scenario. Of course, the effect of fracture geometry

(aspect ratio) on the SIF is also considered.

The above-mentioned scenarios are established to quantify the SIF change according to the

geometry and pattern of the fractures. One qualitative analysis is also done on the effect of

offset between meeting fractures on the SIF change of the fractures. The term offset in this

regard means the distance between two planes on which hydraulic fractures are located in a

simultaneous fracturing mode.

3.8. Data range for the SIF change analysis

In this study, an effective range has been assumed for all the studies. The input variables

and their range are defined in Table 3.2.

Table 3. 2. Input variable range for the numerical simulation No. Parameter Range1 Fracture aspect ratio 0.2-1.02 Fracture spacing1 0.1 -7.0 3 Fracture distance2 0.1 -3.0 4 Fracture offset 0.1 -1.0

Table 3.2 shows that an extensive number of numerical models are required to be built and

analyzed to predict the SIF change via simultaneous multistage fracing job.

3.9. Verification of stress field for simple problems

In this section, the modeling strategy used in ABAQUS is examined by numerically solving

simple crack geometries, the analytical solutions of which are available in the literature. 1 The term “spacing” in this thesis is only used as a measure of the interval between the adjacent fractures in multistage fracturing mode. 2 The term “distance” in this thesis only refers to the distance between meeting fracture tips in simultaneous fracturing mode.

36

Semi-infinite and penny-shaped cracks with internal pressure are chosen for this purpose

(Sneddon, 1945). In this step of the work, appropriate size of the model is defined in order

to minimize the boundary effects to the model response.

Cracks in a material are associated with three geometrical parameters; length, width (called

aperture), and height. The ratio of any two of these can be considered as the aspect ratio.

In this paper, the ratio of height over length of the fracture is called aspect ratio (

/ ). We numerically simulate two cracks with different geometries, one with

an infinite height (plane strain crack; ∞), and the other with aspect ratio of unity

(penny-shaped crack; 1.0), both with an internal pressure. The results of these

numerical simulations are compared with the analytical solutions.

The validation approach described herein compares the horizontal and vertical stresses

around the simulated crack with the analytical results presented by Sneddon and Elliot

(1946). These are shown in Figure 3.7 for horizontal and vertical stresses along a line

perpendicular to the face of the hydraulic fracture as shown in the inset.

Figure 3. 7. Stress validation in the direction vertical away from the fracture center

‐1.20

‐1.00

‐0.80

‐0.60

‐0.40

‐0.20

0.00

0.20

0 2 4 6 8 10

Norm

alized

Stress, S/P

H

Normalized distance, x/c

plane strain crack

SXX/PH (NUMERICAL)

SYY/PH (NUMERICAL)

SXX/PH (ANALYTICAL)

SYY/PH (ANALYTICAL)

xx

xx

yyyy

‐1.2

‐1

‐0.8

‐0.6

‐0.4

‐0.2

0

0.2

0 1 2 3 4 5

Norm

alized Stress, S /PH

Normalized distance, x/c

Penny‐shaped crack

Syy/P (ANALYTICAL)

Sxx/P (ANALYTICAL)

Syy/PH (NUMERICAL)

Sxx/PH (NUMERICAL)

xx

xx

yyyy

37

The figure demonstrates excellent agreement between the analytic and numeric simulation

results for plane-strain and penny-shaped cracks and, therefore, the numerical modeling

strategy adopted here can be considered valid for modeling of hydraulic fractures and stress

shadowing effects. As a result, according to the used numerical method and all the

conditions/assumptions we can run numerical simulations to study the influence of the

considered variables on stress shadow size and stress intensity factor. In Chapters 4 and 5,

all the results of numerical simulations are presented.

38

Chapter 4: Prediction of shadow and aperture of hydraulic fractures

4.1. General

In Chapter 3, the methodology of investigating stress shadow and stress intensity factor

(SIF) of a hydraulic fracture was proposed. All the applied boundary conditions and major

assumptions were also presented. In this chapter, based on explanations in Chapter 3, the

results of all the analyses conduced for stress shadow of hydraulic fractures are reported

and a detailed discussion on the stress shadow is also presented.

4.2. Prediction of aperture for a contained hydraulic fracture

First, aperture of hydraulic fractures is measured according to the numerical results and it is

predicted by proposing an equation. This is done first because aperture of hydraulic

fractures is of crucial significance in proppant size/type and estimating fracture

conductivity.

4.2.1. Effect of in-situ stresses on aperture

As mentioned in Chapter 2, Eq. (2.10) describes half-aperture of a hydraulic fracture in a 2-

D medium as a function of the applied hydraulic pressure, relative distance from fracture

center, and rock moduli. The formulation of Eq. (2.10) ignores the impact of any far-field

stress. However, for the case of aperture determination, when in-situ stresses are present,

the excess hydraulic pressure ( ) should be considered instead. This is

because according to the numerical results, maximum horizontal stress (applied parallel to

the fracture face) has negligible effect on the aperture of the internally pressurized fracture

(see also Jeffrey, 1989) and therefore, a simple subtraction of minimum horizontal stress

from the applied pressure and neglecting far-field stresses suffices to estimate the aperture

39

similar to the case mentioned in Eq. (2.10). Of course, this elimination is only valid for the

case of aperture, where stress anisotropy is not an influencing variable.

4.2.2. Mathematical equation for prediction of aperture of a single contained fracture

In this section, aperture of a contained hydraulic fracture in a standalone well is predicted

considering different mechanical properties for the shale, variant stress anisotropies,

different internal pressures, and fracture aspect ratios (see Scenario 1, Figure 3.1; Chapter

3).

First, it was observed that there is similar effect of Young’s modulus in aperture

determination of 2-D and 3-D fractures. Influence of Poisson’s ratio, on the other hand,

was observed to be different from the 2-D fractures in such a way that aperture of a 2-D

fracture shows a quadratic dependence on Poisson’s ratio (see Eq. (2.10)), while for a 3-D

hydraulic fracture, there is a third order polynomial dependence between aperture and

Poisson’s ratio. In addition, aperture of 3-D hydraulic fractures shows a quadratic

dependence on the fracture aspect ratio as well. Following the above explanations, aperture

of a contained, 3-D hydraulic fracture can be determined using Eq. (4.1).

, , (4.1)

in which is a third order polynomial of the Poisson’s ratio, multiplier is a

quadratic function of aspect ratio, is maximum half-aperture, and , , are

excess pressure, net play Young’s modulus, and half length of the hydraulic fracture,

respectively. These functions are defined in Eqs. (4.2) and (4.3):

40

;1.1286 0.89190.1494 2.0286 (4.2)

0.2757 1.3092 0.0316 (4.3)

This equation underscores the independence between the stress anisotropy and the half-

aperture of hydraulic fractures, implying that aperture only depends on the minimum

horizontal stress ( ).

4.2.3. Reliability of the proposed function for aperture prediction

A comparison of the numerical results and the predictions for Eq. (4.1) are shown in

Figure 4.1 which demonstrates excellent agreement between numerically obtained values

and the predicted ones.

Figure 4. 1. Half-aperture prediction using Eqs. (4.1-4.3)

Therefore, employing Eqs. (4.1-4.3), one can have a satisfactory estimate of the aperture of

a hydraulic fracture by giving the aspect ratio, Poisson’s ratio, excess pressure, and Young’s

modulus. Excess pressure of a hydraulic fracture is determined via subtraction of minimum

horizontal stress from hydraulic pressure inside the fracture. Therefore, minimum

w'max = 0.9991 wmax

R² = 0.9999

0.00E+00

1.00E‐04

2.00E‐04

3.00E‐04

4.00E‐04

0.00E+00 1.00E‐04 2.00E‐04 3.00E‐04 4.00E‐04

Predicted half‐ap

erture ratio, w

' max/ c

Numerical half‐aperture ratio, wmax / c

41

horizontal stress should also be considered as an influencing variable in determination of

aperture of hydraulic fractures.

4.2.4. Visualization of aperture prediction function

The proposed equation for prediction of aperture in 3-D fractures has four inputs

(Poisson’s ratio, aspect ratio, Young’s modulus, and excess pressure) and aperture as the

function of these inputs. Visualization of this function is only possible when plotting is

done by the ratios, and having different plots for different aspect ratios.

The expression for the fracture half-aperture in Eq. (4.1) is re-drawn in Figure 4.2 for

different Poisson’s ratios, excess hydraulic pressures, two aspect ratios, and two different

values of Young’s moduli.

Figure 4. 2. Visualization of aperture determination function

The key observation from Figure 4.2 is the relationship between Young’s modulus and

fracture half-aperture. More compliant rocks characterized by a lower Young’s modulus are

associated with a larger fracture aperture. Increases in excess pressure also lead to a

widening of the fracture and so do decreases in the value of Poisson’s ratio. In order to

42

justify the behavior of aperture with respect to moduli of the rock, consider elastic moduli

relationship. Bulk compressibility of rock is obtained as 3 1 2 / . This means

by decreasing of Poisson’s ratio and Young’s modulus, compressibility of rock increases,

thereby raising the aperture of the hydraulic fracture.

4.3. Prediction of shadow around a contained hydraulic fracture ( °)

This section focuses on the description of the stress shadowing around a single contained

hydraulic fracture (see Figure 3.1; Scenario 1 in Chapter 3). The input parameters were

varied according to Table 3.1 to quantify the shadow size around a hydraulic fracture. It is

worth mentioning that because of using pressure boundary condition, Young’s modulus of

the medium does not show any effect on stress shadow.

4.3.1. Numerical results for stress shadow analysis

In order to study the stress shadow, excess pressure ( ) is normalized with ten times of

the atmospheric pressure (10 ) and shadow length is normalized with the fracture half

length ( ). The normalized shadow lengths predicted as a function of the normalized

excess pressures are typically shown in Figure 4.3 for an aspect ratio of 1.0, horizontal

stress anisotropy of 0.95 and varying Poisson’s ratios.

It can be seen that there is a nonlinear direct relationship between the shadow size and

internal pressure. This means that by raising the internal pressure, shadow size increases

but with a decreasing gradient. In Figure 4.3, we also describe the variations in the stress

shadow effect with Poisson’s ratio which shows that shadow size increases with Poisson’s

ratio.

43

Figure 4. 3. Shadow change with Poisson’s ratio, numerical values and the predictions

4.3.2. Mathematical equation predicting stress shadow of a single contained fracture

We use all the numerical results of a single contained hydraulic fracture, like the ones

typically shown in Figs. 4.3, to develop an expression for predicting the stress shadow size,

as shown in Eq. (4.4):

_ , 10

1 , 10

(4.4)

The proposed Eq. (4.4) contains two coefficients as , and , which are

functions of Poisson’s ratio and stress anisotropy. It also contains multiplier , a

quadratic function of aspect ratio, which applies the effect of hydraulic fracture geometry

using Eq. (4.5).

0.2596 1.2888 0.0283 (4.5)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3

Norm

alized Shad

ow, SH_S 05 / c

Normalized Excess Pressure, PN/(10 P0)

Poisson's Ratio=0.00

Poisson's Ratio=0.40

Stress ratio=0.95Aspect ratio=1.00

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3

Norm

alized Shad

ow, SH_S 05 / c

Normalized Excess Pressure, PN/(10 P0)

Poisson's Ratio=0.00Poisson's Ratio=0.05Poisson's Ratio=0.10Poisson's Ratio=0.15Poisson's Ratio=0.20Poisson's Ratio=0.25Poisson's Ratio=0.30Poisson's Ratio=0.35Poisson's Ratio=0.40

Stress ratio=0.95Aspect ratio=1.00

44

Any aspect ratio, between 1.0 and 0.0 results a value for multiplier to be used in

Eq. (4.4) for stress shadow size prediction. It is seen that aspect ratio of the fractures are

really important in determination of stress shadow size. In addition, effect of Poisson’s

ratio and stress anisotropy is present in the coefficients of Eq. (4.4).

4.3.3. Coefficients of the proposed function for stress shadow prediction

Coefficients of the function, Eq. (4.4), can be obtained from mathematically relating stress

shadow size with Poisson’s ratio and stress anisotropy. The first observation was that the

coefficients of the proposed function are linear functions of Poisson’s ratio. Having

obtained the values of the linear regression between function coefficients and Poisson’s

ratio, one can ponder the dependency on the stress anisotropy afterwards. The linear

relationship between the coefficients of the function and Poisson’s ratio is typically shown

for two different stress anisotropies ( = 0.95, 0.99) in Figure 4.4.

Figure 4. 4. Shadow function coefficient change by Poisson’s ratio

The coefficients of , and , are then expressed as:

A = ‐0.302v + 3.6115

B = ‐0.736v + 1.2408

0

1

2

3

4

0 0.1 0.2 0.3 0.4

Shad

ow function coefficients, A

,B

Poisson's ratio, v

STRESS ANISOTROPY=0.95

AB

A = 1.7331v + 10.698

B = ‐0.9431v + 2.7923

0

2

4

6

8

10

12

0 0.1 0.2 0.3 0.4

Shad

ow function coefficien

ts, A

,B

Poisson's ratio, v

STRESS ANISOTROPY=0.99

AB

45

,

(4.6a)

, (4.6b)

As it is shown in Eqs. (4.6), there is a linear relationship between coefficients of the

proposed function and Poisson’s ratio. Additionally, effect of stress anisotropy shows itself

in the linear regression coefficients , in Eqs. (4.6). According to the numerical results,

there is a nonlinear relationship between and coefficients and stress anisotropy ( )

which are given as:

1;

1

1 ;

1

(4.7a)

(4.7b)

in which the values of the defined coefficients are listed in Table 4.1:

Table 4. 1. Coefficients of the function in Eq. (4.7) coefficient value coefficient value

-0.06415 0.81730 0.06631 -0.52469 -1.97737 -0.44572 0.97792 -0.54166 -25.97845 0.61366 25.92616 -0.50759 39.27341 -0.25782 -40.35336 -0.71923

Substituting values of Table 4.1 inside Eqs. (4.7), one can draw and coefficients with

respect to . The resulting curves are shown in Figure 4.5. It is seen from Figure 4.5a that

there is a nonlinear relationship between the coefficients , and . From Eq. (4.6),

46

one can have Figure 4.5b for coefficients , assuming a value for the Poisson’s ratio

( 0.2 for this case). As it can be seen from Figure 4.5b, gradient of the curve ( )

is higher than that of ( ). Based on the fact that coefficient is divided by

coefficient in stress shadow determination (see Eq. (4.4)), their ratio can be indicative of

the behavior of shadow versus stress anisotropy. Figure 4.5c shows the ratio of / for

the special case of 0.2. As it can be seen from Figure 4.5c, there is a nonlinear

relationship with increasing gradient between shadow size and stress anisotropy.

Figure 4. 5. Behavior of function coefficients with stress anisotropy ( )

4.3.4. Reliability of the proposed function for stress shadow prediction

Equations (4.4-4.7) now can be employed to describe the shadow size ( _ ) around a

contained hydraulic fracture for any given aspect ratio. The comparison between the

predicted values and numerical results is shown in Figure 4.6 which demonstrates excellent

agreement between the predicted and numerically obtained stress shadow sizes for

threshold angles of 5° and any specified aspect ratio.

‐2

0

2

4

6

8

10

12

0.94 0.96 0.98 1

Coefficient value (M,N)

Horizontal stress anisotropy , Kh

MA

NA

MB

NB

(a)

0

2

4

6

8

10

12

0.94 0.96 0.98 1

Coefficient value (A,B)

Horizontal stress anisotropy, Kh

A

B

(b)

v=0.2

3.00

3.25

3.50

3.75

4.00

4.25

4.50

0.94 0.96 0.98 1

Coefficient ratio (A/B)

Horizontal stress anisotropy, Kh

A/B

(c)

47

Figure 4. 6. Shadow size prediction using Eqs. (4.4-4.7)

4.3.5. Visualization of stress shadow prediction function

As in the case of aperture predicting function, there are four input variables, namely as

stress anisotropy, Poisson’s ratio, excess pressure, and aspect ratio and one output function

in case of stress shadow size. Equations (4.4-4.7) may be replotted as shown in Figure 4.7

which expresses that shadow size has a direct relationship with excess pressure, stress

anisotropy, and Poisson’s ratio.

Figure 4. 7. Shadow around a hydraulic fracture

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Predicted shad

ow, SH_S

05/c

Numerical shadow, SH_S05 / c

AR=1.0AR=0.8AR=0.6AR=0.4AR=0.2

(SH05/c)Pre = 0.9866 (SH05/c)NumR² = 0.9966

48

4.3.6. Discussion on the relationship between aperture and stress shadow

As indicated in Section 4.2, stress anisotropy is not an influencing variable on the aperture

(despite of its noticeable effect on shadow size) and Young’s modulus does not have any

significant effect on the shadow size (even though it inversely controls the aperture). In

addition, Poisson’s ratio inversely controls the aperture of the fracture, while it directly

influences the shadow size. As a result, although there is some direct relationship between

aperture and stress shadow size because of the effect of excess pressure, stress shadowing

of a hydraulic fracture cannot be assumed entirely dependent on its aperture.

Based on the numerical results, more stress contrast is observed for lower Poisson’s ratios

(see also Roussel and Sharma, 2011). Larger stress contrast ( ) means that we are

getting closer to the original state of stress. This concept was used by Jo (2012) to calculate

the shadow size around hydraulic fractures in an analytical approach. Considering Eq. (2.8),

it is also induced that larger stress contrast means smaller deviation angles which confirms

more closeness to the original state of stress. This means that lower Poisson’s ratios will

result the original state of stress to be at a smaller distance to the fracture face compared to

the case of larger Poisson’s ratios. Furthermore, from analytical Eqs. (2.6), it is implied that

there is a direct relationship between shadow size and Poisson’s ratio which confirms our

numerical results.

4.4. Prediction of shadow around a contained hydraulic fracture ( °)

This section focuses on the variations of the shadow size around a contained hydraulic

fracture with different threshold angles. Again considering Figure 3.3, it is evident that by

adopting different threshold angles, the size of stress shadow will change. In the literature,

49

sometimes complete reorientation of in-situ stresses (90°) meaning _ has been called

half of minimum fracture spacing (see Roussel and Sharma, 2011). More importantly,

having a decreasing trend of stress shadow size with respect to the threshold angle, one can

have a better insight of shadow size gradient with threshold angle. A detailed study of the

numerical results showed that _ cannot be satisfactorily predicted using an equation

like Eq. (4.4). Figure 4.8 also shows that the ratio of shadow with threshold angle of 80°

over that of 5° ( _ / _ ) for all the analyses varies from 0.13 to 0.80. The mean

value using a normal distribution is0.5662 0.1152. Because of the high variation of the

values, a more acceptable method is applied to have a satisfactory prediction of shadow for

higher threshold angles.

Figure 4. 8. Distribution of SH_S80 over SH_S05 ratio (for all aspect ratios)

4.4.1. Mathematical equation for prediction of stress shadow with different threshold angles

A precise investigation of the numerical results showed that the difference between

_ and _ is approximately a fixed value changing with threshold angle

and Poisson’s ratio. This means that the curve of shadow decrease with threshold angle

(assuming a fixed Poisson’s ratio) is almost a fixed function which is only shifted. The

0.80.70.60.50.40.30.2

160

140

120

100

80

60

40

20

0

SH_S 80/SH_S 05

Freq

uen

cy

Mean 0.5662StDev 0.1152N 1855

50

mean differences ( _ ) are shown in Table 4.2 for different aspect ratios. It is worth

indicating that these values are obtained in an iterative approach which results the least

difference between the calculated and numerical shadow results ( _

_ _ ).

In order to observe the behavior of shadow difference with respect to the threshold angle,

the values in Table 4.2 are also plotted in Figure 4.9. As it is evident from Figure 4.9, there

is a large difference between _ and other shadows with threshold angles up to 30°

and the difference for the rest of the threshold angles is gently sloped.

Table 4. 2. Shadow decrease by threshold angle Threshold angle,

1_ .

1_ .

1_ .

1_ .

1_ .

5 0.000 0.000 0.000 0.000 0.00010 0.426 0.352 0.279 0.202 0.09615 0.632 0.527 0.424 0.303 0.14620 0.758 0.637 0.510 0.366 0.17530 0.903 0.759 0.610 0.432 0.21140 0.963 0.817 0.658 0.468 0.23050 0.981 0.831 0.674 0.484 0.23960 1.002 0.840 0.681 0.490 0.24470 1.019 0.859 0.690 0.495 0.24780 1.064 0.893 0.710 0.502 0.250

Figure 4. 9. Shadow difference with respect to varied threshold angles

0.000

0.200

0.400

0.600

0.800

1.000

1.200

0 10 20 30 40 50 60 70 80 90

Shad

ow difference, (SH

_S05‐SH_S

)/c

Threshold angle, (Degree)

AR=1.0AR=0.8AR=0.6AR=0.4AR=0.2

51

Using these values (Table 4.2), one can predict the values of shadow in different threshold

angles. However, the results showed some scattering when using these values. This

observation shows that Poisson’s ratio also contributes in shadow change versus threshold

angle. As a result, the formulation for shadow for any threshold angle can be written as Eq.

(4.8) for different aspect ratios.

1_

1_

1_ (4.8)

The values of , , are given for different aspect ratios in Table 4.3.

Table 4. 3. Coefficients of the function in Eq. (4.8) Aspect ratio (AR)

1.0 0.7825 0.4927 0.1560 0.8 2.2038 0.0415 0.0636 0.6 0.9264 0.5535 0.0270 0.4 3.1194 0.0119 0.0774 0.2 0.8824 0.1480 0.0284

Therefore, by having Poisson’s ratio of rock and coefficients mentioned in Table 4.3, the

shadow difference (see Table 4.2) and shadow for threshold angle of 5° (see Eqs. (4.4-4.7)),

one can have stress shadow for any threshold angle ( _ ).

4.4.2. Reliability of the equation predicting stress shadow with different threshold angles

In this section, shadow size with complete reorientation ( _ ) is predicted using Eq.

(4.8) and the resultant values are compared with the numerical results. Figure 4.10 shows

the reliability of predicted data for this analysis. It is important to notice that this

formulation is only valid for the cases that excess pressure is more than the maximum

horizontal stress (see Figure 3.3; second shadow mechanism).

52

Figure 4. 10. Shadow ratio prediction (threshold angle of 80°)

From Figure 4.10, one can see that minimum shadow is predicted using Tables 4.2, 4.3 and

Poisson’s ratio in a fairly accurate manner. As a result, stress shadow around a contained

hydraulic fracture can be predicted for different aspect ratios and for different threshold

angles. It is worth mentioning that in case aspect ratio is in-between any of the mentioned

aspect ratios (See Eq. (4.8); Table 4.3), stress shadow for the bounding aspect ratios should

be determined and then, by a linear interpolation, stress shadow of the hydraulic fracture is

determined for the desired aspect ratio.

4.5. Effect of uncontainment of the fracture (standalone well fracturing)

All the calculations of stress shadow and aperture of a single hydraulic fracture in the

previous sections were done for a contained hydraulic fracture. This means that hydraulic

fracture is entirely located inside the net play without any penetration into the bounding

layers. In this section, we are intended to investigate the effect uncontainment of the

hydraulic fracture. As it is shown in Table 3.1, Young’s modulus of the bounding layers is

assumed 0.25-4.0 times of that in the net play to investigate the shadow size and aperture

of the uncontained hydraulic fracture. Moreover, penetration extent of the hydraulic

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3Predicted shad

ow (SH

_S 8

0/c) N

um

Numerical shadow (SH_S80/c)pre

(SH80/c)Num= 1.0031(SH80/c)pre ‐ 0.0584R² = 0.9832

53

fracture into the bounding layers has been assumed to be between 0.1-0.3 of the hydraulic

fracture height. It is important to mention that for modeling of hydraulic fracture

penetration into the bounding layers, fracture geometry was not changed, but the net play

thickness was assumed to be smaller (see Figure 3.1). Therefore, bounding layers were

becoming effective on the upper and lower tips of the hydraulic fracture.

4.5.1. Change of stress shadow by uncontainment of the fracture

The shadow sizes of this study (uncontained fracture) were compared with those of the

contained hydraulic fracture by an investigation on the shadow size ratio of uncontained

over those of the contained hydraulic fracture. The generated graphs for this comparison

are shown in Figure 4.11 for two selected threshold angles of 5° and 80°.

Figure 4. 11. Stress shadow change by fracture uncontainment

As it can be seen from Figure 4.11, penetration of a hydraulic fracture inside the bounding

layers may result significant shadow size change depending on the ratio of Young’s moduli

of the bounding layers and the net play, and the penetration extent. For the case of lower

Young’s modulus for bounding layers, shadow size increases and the increase is more

significant for higher penetrations. For the circumstance of higher Young’s modulus of

54

bounding layers, on the other hand, shadow size decreases and the decrease is likewise

more significant for higher penetrations. It is important to note that both of the shadow

sizes obtained based on different threshold angles ( 5°, 80°) show approximately the

same change. It is worth mentioning that shadow multipliers mentioned in Figure 4.11

work for all the cases of single fracture mentioned in Table 3.1.

4.5.2. Change of aperture by uncontainment of the fracture

In addition to the shadow size change according to the Young’s modulus ratio and

penetration extent, aperture is also influenced by this uncontainment in the same way.

Figure 4.12 shows the change in aperture as a result of penetration of the hydraulic fracture

in the bounding layers.

Figure 4. 12. Aperture change as a result fracture uncontainment

As it can be seen from Figure 4.12, similar to the shadow size change (see Figure 4.11),

aperture multiplier also shows the same behavior and even approximately the same values.

4.5.3. Mathematical equations predicting aperture and shadow size in uncontained single fractures

In this section, we propose a set of equations for prediction of aperture and stress shadow

multipliers as a result of uncontainment of the fractures. The following set of equations can

55

be used for prediction of aperture and shadow multipliers as a result of hydraulic fracture

penetration into the bounding layers with an extent of / . is the ratio of

penetrated height ( ) of the fracture over its total height ( ) and it is assumed to be in

the range of 0.1-0.3.

, /1 1 /

(4.9)

_ ,_

/1 1 /

(4.10)

_ ,_

/1 1 /

(4.11)

in which is uncontainment aperture multiplier, , , are uncontainment

shadow multipliers for threshold angles of 5° and 80°, and and are functions of

penetration extent ( ) given by:

1 (4.12)

1 (4.13)

The values of coefficients in Eqs. (4.12) and (4.13) are given in Table 4.4 for

aperture, and shadow multipliers with threshold angles of 5° and 80°. These sets of values

56

(Table 4.4) together with Eqs. (4.9-4.13) can be used to predict aperture and shadow

multipliers1 of hydraulic fractures as a result of penetration into the bounding layers.

Table 4. 4. Coefficients for Eqs. (4.12) and (4.13) for standalone fracturing parameter Aperture 0.45553 -3.06420 3.49308 -0.14382 0.72369 0.52965Shadow ( 05°) 0.42801 -0.84289 3.41533 -2.39090 2.72864 0.04526Shadow ( 80°) 3.18372 -2.89633 0.04991 0.08886 0.81315 1.17125

Based on these numerical results, it is implied that shadow and aperture of a single

hydraulic fracture is highly influenced by its containment.

4.5.4. Reliability of the proposed functions for uncontainment multiplier determination

In order to check the precision of the predicted uncontainment multipliers with respect to

the numerical ones, Figs. 4.13 are plotted which show the reliability of predictions using

these equations.

Figure 4. 13. Prediction of aperture and shadow size change by fracture uncontainment

1 The term “multiplier” in this thesis means that it only takes the effect of the extra variable (e.g. uncontainment, or simultaneous/multistage fracing) into account and it should be multiplied to the basic case (prediction without this variable) to have this extra effect incorporated in prediction of the parameter under study (aperture, shadow, or propagation potential).

y = 1.0006x + 0.0011R² = 0.9997

0.8

1

1.2

1.4

1.6

0.8 1 1.2 1.4 1.6

Predicted aperture ratio

Numerical aperture ratio

y = 0.923x + 0.0812R² = 0.9972

0.8

1

1.2

1.4

1.6

0.8 1 1.2 1.4 1.6

Predicted shad

ow ratio

Numerical shadow ratio (T=5°)

y = 1.0015x + 0.0009R² = 0.9992

0.8

1

1.2

1.4

1.6

0.8 1 1.2 1.4 1.6

Predicted shad

ow ratio

Numerical shadow ratio (T=80°)

57

As it can be seen from Figure 4.13, the ratio of aperture and shadow size of an

uncontained hydraulic fracture (penetrating into the bounding layers) over those of the

contained hydraulic fracture can be predicted using Eqs. (4.9-4.13).

4.5.5. Use of uncontainment multipliers (standalone fractures)

The aperture multiplier obtained from section 4.5.3 ( ) can be simply multiplied with

Eq. (4.1) for aperture prediction of a single uncontained hydraulic fracture. Similarly, for

shadow size change, the obtained multipliers of shadow in Section 4.5.3 (

and ) should be multiplied with Eqs. (4.4,4.8) respectively for prediction of Shadow

size for minimum and maximum threshold angles ( 5°, 80°).

4.6. Effect of simultaneous fracturing on shadow size and aperture

Simultaneous hydraulic fracturing of multilateral wells has been proven to be an efficient

way of having a better fracture network for hydrocarbon flow in unconventional shales. In

this part, two contained hydraulic fractures are assumed from two parallel horizontal wells

(see Figure 3.1; Scenario 3). It is assumed that there is no offset between the crack tips,

which means that the two fractures are aligned on a single plane and their distance is only

changed. The distance change has been assumed by changing of the distance between the

wells, not the aspect ratio or length of the fractures. Referring again to the presented

numerical scheme in Table 3.1, one can see that an extensive number of numerical analyses

are required for this purpose.

4.6.1. Change of stress shadow size by simultaneous fracturing ( 5°)

According to the fact that stress shadow has been satisfactorily predicted for a single

contained hydraulic fracture, the best approach was determined to merely compare the

58

shadow size of contained simultaneous fractures with that of a single contained fracture

(comparing Scenario 3 with Scenario 1; see Figure 3.1). Therefore, multiplying a multiplier

to the single fracture equation will result the shadow for simultaneous fractures.

In order to do so, the ratios of _ / _ and _ / _ were investigated

in detail in order to come up with the multiplying factors to be multiplied with the shadow

size around a contained fracture in a standalone well to predict the shadow size around

simultaneous contained fractures. It is worth indicating that _ defines the shadow size

for simultaneous fractures, while _ is the shadow size for a single fracture.

According to the numerical results, it was observed that hydraulic fracture aspect ratio

again plays an important role in the magnitude of multiplying factor for reduction of

shadow size around simultaneous fractures. From a statistical analysis on the ratio of

shadow size of simultaneous over single fractures, it was evident that this ratio is only

changing by the aspect ratio and distance between hydraulic fractures (for threshold angle

of ( 5°)). Figure 4.14 shows the results of this analysis for simultaneous fractures with

crack tip distances ranging from 0.07 to 3.0 .

Figure 4. 14. Shadow reduction by simultaneous fracturing

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3

Mean Shad

ow ratio (SH

_D/SH_S)

Normalized tip distance, (x/c)

(AR=1.0)(AR=0.8)(AR=0.6)(AR=0.4)(AR=0.2)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 0.5 1 1.5 2 2.5 3

SD of Sh

adow ratio (SH

_D/SH_S)

Normalized tip distance, (x/c)

(AR=1.0)(AR=0.8)(AR=0.6)(AR=0.4)(AR=0.2)

59

As it can be seen from Figure 4.14, shadow ratio of _ / _ starts from 0.69 for a

normalized distance of 0.07 (aspect ratio of unity) and it becomes larger by decreasing the

aspect ratio of hydraulic fractures. As an example, shadow ratio starts from 0.83 for

normalized distance of 0.07 (aspect ratio of 0.2). Another important point about this

analysis is that approaching of shadow ratio to unity for hydraulic fractures with lower

aspect ratios is seen in lower distances between the tips. This means that the lower the

aspect ratio is, the closer the fracture tips must be together to have reduction effect of

stress shadow on each other.

Since these values are determined by a statistical analysis, it is beneficial to also report the

associated standard deviations (SD) for the analyses. As it is evident from Figure 14.4b,

SDs of the analyses are in between 0.015-0.060 and show a more or less decreasing trend

by increasing the distance between the hydraulic fractures.

4.6.2. Mathematical equation for prediction of simultaneous multiplier

After these interpretations, it is also required to have a robust equation for shadow

reduction ratio when we are dealing with simultaneous contained hydraulic fractures. It was

feasible to develop a function to predict the simultaneous multiplier with the numerical

results. Eq. (4.14) shows that the multiplier can be determined for all aspect ratios.

__

1 / (4.14)

It can be seen from Eq.(4.14) that there are two coefficients for this function, which can be

obtained for different aspect ratios from Table 4.5.

60

It is worth indicating that the coefficients for any aspect ratio can be determined via a

linear interpolation between those of the boundary aspect ratios.

Table 4. 5. Coefficient of the function in Eq. (4.14) AR 1.0 0.33522 1.057000.8 0.31354 1.402300.6 0.29420 1.997150.4 0.27786 3.412530.2 0.27772 7.38754

4.6.3. Reliability of the proposed equation for simultaneous multiplier prediction

In this section, the numerically obtained multipliers are compared with the mathematically

predicted ones using Eq. (4.14). The calculated simultaneous multipliers with the proposed

function are shown in Figure 4.15.

Figure 4. 15. Prediction of shadow size change by simultaneous hydraulic fracturing

It can be seen from Figure 4.15 that Eq. (4.14) predicts the shadow change satisfactorily. It

is important to notice that all these discussions are stress shadow change for threshold

angle of 5°. In the following section, change of minimum shadow size is investigated.

y = 1.0326x ‐ 0.0265R² = 0.9969

0.6

0.7

0.8

0.9

1

0.6 0.7 0.8 0.9 1

Predicted shad

ow chan

ge

Numerical Shadow change

61

4.6.4. Change of stress shadow size by simultaneous fracturing ( 80°)

According to the numerical results, it was observed that simultaneous hydraulic fracturing

of parallel laterals influences the minimum shadow size ( 80°) not as significantly as

that of the threshold angle of 5°. In fact, shadow size change for threshold angle of 80° is

much smaller compared to that for threshold angle of 5°. The results of this study are

shown in Table 4.6.

It is evident from Table 4.6 that the decrease in shadow is from 0.93 (for aspect ratio of

unity) to almost no change (for aspect ratio of 0.2).

It is worth mentioning that apertures of hydraulic fractures are not influenced by

simultaneous fracturing of parallel wells in case there is no overlap between the hydraulic

fractures. This means that maximum half-aperture of simultaneous fractures can be

determined via using the same equation for hydraulic fractures in standalone wells (see Eq.

(4.1)).

Table 4. 6. Stress shadow change by simultaneous fracturing ( °)

Normalized distance

Simultaneous multipliers (Mean and SD) (AR=1.0) (AR=0.8) (AR=0.6) (AR=0.4) (AR=0.2)

Mean SD Mean SD Mean SD Mean SD Mean SD0.07 0.93 0.10 0.93 0.09 0.93 0.08 0.96 0.05 0.99 0.040.33 0.91 0.07 0.91 0.08 0.94 0.06 0.97 0.05 1.00 0.030.67 0.91 0.05 0.93 0.06 0.94 0.04 0.98 0.04 1.00 0.041.00 0.91 0.07 0.94 0.05 0.94 0.05 0.98 0.04 1.00 0.041.33 0.92 0.05 0.94 0.05 0.95 0.04 0.97 0.08 1.00 0.041.67 0.93 0.04 0.95 0.04 0.97 0.04 0.99 0.04 1.01 0.03 2.00 0.94 0.04 0.96 0.04 0.96 0.04 0.98 0.08 1.00 0.042.33 0.93 0.04 0.95 0.04 0.96 0.04 0.97 0.08 1.00 0.032.67 0.94 0.04 0.95 0.04 0.96 0.04 0.98 0.08 1.00 0.03

4.7. Effective distance between simultaneous fractures of two parallel wells

In addition to the prediction of the effect of simultaneous hydraulic fracturing on shadow

change, it is also required to show the effective distance between hydraulic fracture tips,

62

beyond which there no effect for shadow change in case of simultaneous hydraulic

fracturing. This boundary distance was determined precisely for different aspect ratios,

which can be seen in Figure 4.16.

It can be seen from Figure 4.16 that there is a distance between the tips of two aligned

hydraulic fractures beyond which there is no effect in stress shadow reduction between

parallel wells.

Figure 4. 16. Effective normalized distance between fracture tips

The effective distance is highly dependent on the aspect ratio of the fracture, which means

by having a fracture with lower aspect ratio, the effective distance is smaller compared to

that of fractures with higher aspect ratios.

4.8. Prediction of shadow change for uncontained simultaneous fractures

In this section, shadow change as a result of having two aligned uncontained hydraulic

fractures penetrated in bounding layers is investigated. It is worth mentioning that this

change is similar to the change of shadow for a single hydraulic fracture penetrating inside

the bounding layers. However, the amount of change is not the same as a single hydraulic

(DIST/c) = 1.2964 AR2 + 2.0593 AR ‐ 0.0258R² = 0.9997

0

0.5

1

1.5

2

2.5

3

3.5

0 0.2 0.4 0.6 0.8 1 1.2

Norm

alized effective tip distance, (DIST/c)

Fracture aspect ratio, AR

63

fracture. This is because we have two shadow changing mechanisms, one because of

having aligned hydraulic fractures and the second is because of the uncontainment.

4.8.1. Mathematical equations for uncontainment multiplier in simultaneous fracturing

According to the numerical results, it was observed that the equations for obtaining

shadow multipliers are given by:

_ ,_

/1 1 /

(4.15)

_ ,_

/1 1 /

(4.16)

in which and are uncontainment multipliers of simultaneous fractures for

threshold angles of 5° and 80°. The coefficients and are also related to penetration

extent and can be calculated in the same way as mentioned for uncontainment of single

fractures (see Eq. (4.12) and Eq. (4.13)). It is seen from Eq. (4.12) and Eq. (4.13) that there

are 6 coefficients for , to be obtained. These coefficients for uncontainment of

simultaneous hydraulic fractures are reported in Table 4.7.

Table 4. 7. Coefficients for Eqs. (4.12) and (4.13) for simultaneous fracturing parameter Shadow ( 05°) 1.72063 -1.28616 0.04982 0.04256 1.11211 1.22879Shadow ( 80°) 0.64112 -1.27348 2.01901 -3.39967 3.96728 0.05015

Using the values reported in Table 4.7 in Eqs. (4.12) and (4.13), , coefficients can be

determined and Eqs. (4.15) and (4.16) can be used for determination of uncontainment

multiplier of simultaneous fracturing.

64

4.8.2. Reliability of uncontainment multiplier for simultaneous fracturing

The precision of predicted uncontainment multiplier for simultaneous fracturing using Eq.

(4.15) and Eq. (4.16) can be seen in Figure 4.17. It can be seen from Figs. 4.17 that shadow

multiplier for incorporation of uncontainment of the hydraulic fractures in simultaneous

fracturing is predicted satisfactorily using Eq. (4.15) and Eq. (4.16).

Figure 4. 17. Prediction of shadow and aperture change by fracture uncontainment

4.8.3. Use of uncontainment multipliers (simultaneous fractures)

As a result, for the shadow size around hydraulic fractures in simultaneous fracing in which

hydraulic fractures penetrate inside the bounding layers, it is first required to calculate the

shadow size for the single hydraulic fracture (using Eqs. (4.4-4.7)). Then, multiplier of

simultaneous fracing ( ) should be multiplied with it in order to take the

simultaneous fracing effect into account for calculation of shadow size (Using Eq. (4.14)).

Finally, the multiplier of uncontainment in simultaneous fracturing ( and )

should be multiplied with the resultant value in order to have the effect of bounding layers

(using Eq. (4.15) and Eq. (4.16)).

y = 0.9983x + 0.0032R² = 0.9997

0.6

0.8

1

1.2

1.4

1.6

0.6 0.8 1 1.2 1.4 1.6

Predicted shad

ow chan

ge


y = 1.0007x ‐ 0.005R² = 0.9996

0.6

0.8

1

1.2

1.4

1.6

0.6 0.8 1 1.2 1.4 1.6

Predicted shad

ow chan

ge


65

It is again important to indicate that aperture of simultaneous hydraulic fractures is the

same as proposed for single hydraulic fracture. This is because the effect of simultaneous

fracturing is negligible on the aperture of the fracture. Therefore, Eq. (4.1) and Eq. (4.9)

result the aperture of simultaneous uncontained hydraulic fractures as well.

66

Chapter 5: Prediction of Propagation potential in hydraulic fractures

5.1. General

In Chapter 4, a complete set of equations was presented for calculation of stress shadow

around single/simultaneous, contained/uncontained hydraulic fractures with different

aspect ratios, boundary conditions, moduli, and other geometrical considerations. In

addition, aperture of the fractures which plays a key role in proppant type/size

determination and conductivity of the fractures was also predicted for the same scenarios.

However, as mentioned in Chapters 1 and 2, stress shadow and stress intensity factor (SIF)

of hydraulic fractures should be considered together in order to have a satisfactory

judgment about hydraulic fracture optimization. Therefore, in this chapter, effect of

different fracture geometries and their relative positions are considered to study the effect

of fracturing pattern on propagation potential of the fractures. It is important to note that

the same method mentioned in Chapter 3 is used for the study of SIF.

5.2. Effect of aspect ratio of the fractures on the SIF (single fracture)

For the case of a single hydraulic fracture in a standalone well, by changing the aspect ratio,

the SIF along the length and height of the fracture changes. This observation has been

shown in Figure 5.1a. As it is evident from Figure 5.1a, the SIF of the fracture is reduced

by decreasing of aspect ratio in an asymmetric way. This means that the decrease along the

height is not equal to that in length of the fracture (lower change is observed for the

length). This change can also be shown in term of (defined in Chapter 3; Eq. (3.4)) as

shown in Figure 5.1b. As it is seen in Figure 5.1b, is the same for length and height of

the fracture when aspect ratio is unity but it starts to diverge to different values along the

67

length and height of the fracture by decreasing of the aspect ratio. The lower curve of is

indicative of the SIF behavior along the height of the fracture, while the upper curve is

indicative of the SIF along the length of the fracture (see Figure 5.1b).

According to the correspondence between the SIF change and , it is first induced that

the introduced coefficient suffices for showing the SIF change along edges of the hydraulic

fracture. Therefore, using the introduced coefficient, one can study the behavior of the SIF

along the edge of a fracture only by making a comparison between the fitted values.

Figure 5. 1. SIF change along fracture edges by different aspect ratios

5.3. Effect of multistage fracturing on the SIF and aperture in standalone wells

In multistage hydraulic fracturing of shales, perforations are created along an interval of the

horizontal wellbore with a pre-calculated spacing. In this section, we are intended to

investigate the effect of adjacent hydraulic fractures from a standalone well on the SIF and

aperture of each fracture. In order to do so, parallel hydraulic fractures with varying

spacing and aspect ratio are assumed and the SIF change with all the available

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Norm

alized norm

al stress, ( z

z‐ z

z 0)/PH

Normalized tip position, (x/c,y/b)

AR=1.0

AR=0.8

AR=0.6

AR=0.4

AR=0.2

1.00 0.0 1.000.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0.0 0.2 0.4 0.6 0.8 1.0

Cefficient of the fitting function, (a 1)

Fracture aspect ratio, (b/c)

Along Fracture Height

Along Fracture Length

(b)(a)

68

configurations is studied. It is worth indicating that the SIF of each fracture with a certain

aspect ratio with different spacing is compared with the SIF of a single-stage fracture with

the same aspect ratio.

5.3.1. Qualitative description of multistage fracturing influence on SIF/aperture of each fracture

According to the numerical results, it was observed that closely-spaced hydraulic fractures

influence the SIF and aperture in a negative manner, which means that aperture and SIF of

the fractures are lowered compared to the case of efficiently spaced fractures. Using the

defined coefficient in Chapter 3 ( ), we can show the change in the SIF along the height

and length of the fracture as shown in Figure 5.2.

Figure 5. 2. SIF change along the edges of the fracture by multistage fracturing

As it can be seen from Figure 5.2, the decrease in the SIF along the height and length of

the fractures shows the effect of spacing between multi-staged fractures on their

propagation potential. This means that having closely-spaced fractures causes the SIF to

decrease significantly and makes the propagation to occur with larger hydraulic pressures.

Having Figure 5.2 seen, one can see that aspect ratio of the fractures also plays an

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

The SIF chan

ge along the height

Normalized half‐spacing beween fractures, Lp/c

AR=1.0

AR=0.8

AR=0.6

AR=0.4

AR=0.2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

The SIF chan

ge along the length

Normalized half‐spacing beween fractures, Lp/c

AR=1.0

AR=0.8

AR=0.6

AR=0.4

AR=0.2

69

important role in the SIF change of the adjacent fractures. It is observed that going back to

the original state of SIF (standalone fracture) occurs in a shorter distance by having

fractures with lower aspect ratios. In addition, comparing the SIF change along the height

and length of the hydraulic fracture, it is induced that the SIF along the fracture height is

back to the original state in a shorter distance between adjacent fractures, while this is done

for fracture length in a longer distance.

5.3.2. Mathematical equation for prediction of SIF and aperture change in multistage fracturing

In order to have a quantified SIF change by the spacing between adjacent fractures, the

following equation is proposed:

, (5.1)1

in which is the half-spacing between adjacent fractures, is the SIF reduction

multiplier along the height or length of the fracture, is the aperture reduction

multiplier, and are the coefficients of the function defined for SIF change along the

length/height or aperture of the fracture. Table 5.1 shows the values for the coefficients of

the function. Using the values mentioned in Table 5.1 together with Eq. (5.1), according to

the allowed decrease for the SIF or aperture, one can have a good estimate about the

spacing between adjacent hydraulic fractures. As it can be seen from Table 5.1, coefficients

of Eq. (5.1) are also dependent on the aspect ratio of the fracture. It is worth indicating

that in case aspect ratio of the fracture is different from the values mentioned in Table 5.1,

1 Note that in case is used for aperture, it is and in case it is used for SIF change, it becomes

70

spacing for two bounding aspect ratios are determined and, then by a simple interpolation

between the results, spacing for the desired aspect ratio is calculated.

Table 5. 1. Coefficients of the function in Eq. (5.1) Usage coefficients AR=1.0 AR=0.8 AR=0.6 AR=0.4 AR=0.2

SIF change along the height

0.16447 0.16582 0.16785 0.12873 -0.20650 0.42449 0.31392 0.20294 0.10429 0.02090 1.01680 1.01308 1.00820 1.00309 1.00179 1.92373 1.93651 1.94993 1.87369 1.85405

SIF change along the length

0.16447 0.17497 0.18725 0.19689 0.19505 0.42449 0.37417 0.29515 0.17889 0.05834 1.01680 1.01373 1.01024 1.00477 1.00275 1.92373 2.02108 2.12896 2.22962 2.28194

Aperture change

0.09109 0.09643 0.10355 0.10695 0.17979 0.46656 0.35414 0.23938 0.13539 0.03863 1.00792 1.00662 1.00542 1.00338 1.00603 2.41794 2.42928 2.40773 2.29029 2.35485

Referring to the concept of stress shadow, it was understood that there is always a

disturbed zone around hydraulic fractures which prevents any fracture to propagate in this

region. Therefore, really short spacing between adjacent fractures in this section (e.g. 0.17c)

should only be considered as a device for intuition and better understanding of the

concept. Likewise to the SIF, aperture of the adjacent fractures is also influenced by close

spacing of the fractures. The same procedure applied to calculate the SIF change of

hydraulic fractures was also utilized for estimation of aperture change of the fractures as a

result of having multistage fracturing. The last four rows of Table 5.1 together with the

same equation used for the SIF change (see Eq. (5.1)) can be used for prediction of

aperture change.

5.3.3. Reliability of the proposed equation for SIF and aperture change in multistage fracturing

In order to check the validity of the proposed Eq. (5.1), numerical results are compared

with the mathematically predicted values for the SIF and aperture change of the fractures

in multistage fracturing. The results of this comparison are shown in Figs. 5.3.

71

Figure 5. 3. Prediction of SIF and aperture change by multistage fractures

As it is evident from the figure, SIF and aperture change in multistage fracturing is

satisfactorily predicted using Eq. (5.1).

5.3.4. Visualization of the proposed equation

In this section, the proposed function for prediction of SIF change is visualized by using 3-

D plots. Since aspect ratio and spacing between fractures are input variables and SIF

decrease in percentage is the output function, Figure 5.4 is drawn for the SIF change along

height and length of the fractures. As it is seen from Figure 5.4, the highest decrease of SIF

belongs to the case with the highest aspect ratio and the lowest spacing between adjacent

fractures. Moreover, the SIF change along the height of the fracture is more influenced by

aspect ratio than that along the length of the fracture. One similar surface alike to those of

SIF change with aspect ratio and spacing between fractures also exists for aperture change.

y = 1xR² = 0.9992

0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1

Predicted SIF m

ultiplier

Numerical SIF multiplier

y = 1xR² = 0.9997

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.20 0.40 0.60 0.80 1.00

Predicted aperture m

ultiplier

Numerical aperture multiplier

72

Figure 5. 4. Prediction of SIF and aperture change by multistage fracturing

5.4. Effect of simultaneous fracing on the SIF of single-stage fractures

In this section, two wells are assumed to be placed parallel to each other and only one

hydraulic fracture exists for each well. It is also assumed that both of the hydraulic fracture

faces align on a single plane with a certain distance between the tips. The distance between

fracture tips is changed and SIF behavior with respect to the basic case (standalone

fracture) is studied. Likewise to the previous case, behavior is investigated as a

representative of the SIF change along the fracture height or length. Figure 5.5 shows the

results of this comparison. Of course, in this section, the ratio of SIF for simultaneous

fracture over standalone fracture is reported.

5.4.1. Qualitative description of SIF change via simultaneous fracturing

According to the numerical results, the first observed point is that the meeting edges

(heights) are influenced by each other, while other edges (lengths) do not show any change.

73

As it can be seen from Figure 5.5, the SIF along the height of simultaneous fractures is

controlled by two key variables; distance between the tips and aspect ratio.

Figure 5. 5. SIF change by simultaneous fracturing

It is evident that SIF increases as a result of simultaneous fracturing where fracture tips

meet each other and their tip stress field is affected by that of the other fracture. In

addition, it is also observed that aspect ratio also plays an important role in this SIF

increase in such a way that higher aspect ratio results higher SIF change.

5.4.2. Mathematical equation for prediction of SIF change in simultaneous fracturing (single stage)

In order to have a quantifying equation for SIF ratio of simultaneous fractures versus

standalone fractures in a single staged mode, the following relationship is proposed:

exp (5.2)

in which is the simultaneous fracing multiplier for SIF change, is the distance

between fracture tips, is the fracture half-length, is the coefficient varying with aspect

ratio of the fractures. Coefficient can be determined regarding Table 5.2.

74

Table 5. 2. Coefficients of the function in Eq. (5.2)

Aspect ratio Function coefficient, n in

Eq. (5.2) 1.0 0.083370.8 0.071610.6 0.058990.4 0.043370.2 0.03642

Of course, all these values are only valid for single stage fractures without having any

fractures beside each other.

5.4.3. Reliability of the proposed function

The reliability of the proposed Eq. (5.2) for prediction of the SIF change along the meeting

edges is shown in Figure 5.6.

Figure 5. 6. Precision of the predicted SIF ratios using the proposed Eq. (5.2)

As it is seen from Figure 5.6, the predicted SIF changes coincide on the numerical values

satisfactorily. Therefore, the proposed equation can be considered as a reliable equation for

prediction of the SIF change along the meeting edges of rectangular hydraulic fractures.

y = 0.9954x + 0.0146R² = 0.9965

1

1.1

1.2

1.3

1.4

1.5

1 1.1 1.2 1.3 1.4 1.5

Predicted SIF ratio

SIF ratio from numerical results

75

5.5. Effect of simultaneous multistage fracturing on the SIF of the fractures

For the case of multistage hydraulic fracturing, existence of simultaneous fractures

influence the SIF in a similar way but with different magnitude. It is obvious that

multistage fracturing has negative influence on the SIF (decreasing effect), while

simultaneous fracturing has positive influence on the SIF (increasing effect).

5.5.1. Qualitative description of the SIF change by simultaneous multistage hydraulic fracturing

In order to investigate this effect, simultaneous multistage hydraulic fracturing is

performed and the effect of these two techniques is studied together. It is important to

mention that the ratio of the SIF for simultaneous multistage over single-stage hydraulic

fractures in a standalone well is calculated and used for this analysis. Figure 5.7 is drawn

typically for two aspect ratios of 1.0 and 0.4. The depicted surfaces shown in Figs. 5.7 are

for the SIF change along the height (5.7a, 5.7b) and along the length of the fractures (5.7c,

5.7d).

As it is evident from Figs. 5.7a and 5.7b, the SIF change has been drawn for different

distances between fracture tips and different fracture spacings. The SIF change surface

shows the same behavior for all the aspect ratios, however, we have only plotted for aspect

ratios of 1.0 and 0.4. As it is seen from Figs. 5.7a and 5.7b, the highest SIF increase belongs

to the smallest distance between the tips and the largest spacing. It is also observed from

Figs. 5.7a and 5.7b that as we reduce the spacing between the multi-staged fractures, the

SIF of the fractures is reduced by this decreased spacing. In fact, the decreasing mechanism

of fracture spacing (multistage fracturing) and the increasing mechanism of distance

between fracture tips (simultaneous fracturing) are fighting together, one for decreasing

76

and the other for increasing of the SIF. The results (Figs. 5.7a and 5.7b) show that in case

of having too closely-spaced fractures, simultaneous fracturing has no effect on

propagation of fractures from parallel wells. This means that spacing between hydraulic

fractures should be selected carefully in order to have the increasing effect of simultaneous

fracturing incorporated in the fracturing treatment.

Figure 5. 7. Effect of multistage simultaneous fracturing on the SIF of fractures

(a) (b)

(d)(c)

77

The boundary spacing below which the SIF starts to decrease (no matter how much the

distance between the tips of simultaneous fractures is) varies with aspect ratio of the

fractures in such a way that for lower aspect ratios, the boundary spacing region is shorter

compared to that for higher aspect ratios. As the aspect ratio of the fracture increases, this

boundary spacing for the SIF change becomes larger.

It is also seen from Figs. 5.7c, 5.7d that the SIF along the length of the fracture is not

influenced by the simultaneous fracturing. For the case of spacing between multistage

fractures, however, the SIF along the length of the fracture is reduced. Likewise to the

height of the fractures, the SIF decrease/the decreasing boundary along the length is

higher/larger for larger aspect ratios.

5.5.2. Mathematical equation for prediction of SIF change in multistage simultaneous fracturing

Likewise to the previous sections, it is important to quantify the SIF change according to

the distance between fracture tips and different spacing between fractures in simultaneous

multistage fracturing of the reservoir. For any spacing between the fractures, SIF change as

a result of simultaneous fracturing can be predicted using an appropriate equation for tip

distances up to three times of the half-length of the fracture (3 ) as follows:

/

/ (5.3)

in which is the ratio of the SIF for simultaneous multistage fracturing over

standalone fracturing and is estimated using four coefficients as for 1 4. The

values of the coefficients for the proposed function are calculated for different tip

distances and are reported in Table 5.3. It is worth indicating that for any other in-range

78

distance, spacing can be calculated for two boundary distances and a simple linear

interpolation can be applied for the desired distance.

Table 5. 3. Coefficients of the function in Eq. (5.3) Tip distance, ( / ) coefficients AR=1.0 AR=0.8 AR=0.6 AR=0.4 AR=0.2

0.25

0.17201 0.18063 0.18685 0.20748 0.10236 0.90047 0.64194 0.39699 0.19147 0.05582 1.34113 1.27754 1.20242 1.12411 1.03763 1.98234 1.98857 1.96326 1.93938 1.70901

0.50

0.18998 0.19526 0.19393 0.19476 -0.24732 0.80502 0.57514 0.35572 0.16592 0.03463 1.16000 1.12493 1.08375 1.04449 1.00091 1.84573 1.83626 1.79059 1.77249 1.55430

0.75

0.17970 0.18134 0.17360 0.16485 -0.19061 0.69429 0.49660 0.30539 0.13928 0.02931 1.10209 1.07760 1.05131 1.02491 0.99304 1.70569 1.70689 1.69860 1.73495 1.66500

1.0

0.16554 0.16431 0.16126 0.15787 -0.74749 0.61076 0.43819 0.26749 0.12180 0.02059 1.07491 1.05268 1.03472 1.01397 0.99449 1.64114 1.65266 1.68927 1.77491 1.54847

2.0

0.15221 0.16056 0.17045 0.19478 -0.03603 0.46249 0.33927 0.21240 0.09934 0.02634 1.02857 1.02140 1.00862 1.00104 0.99125 1.69518 1.75410 1.83902 1.98255 1.83895

3.0

0.16155 0.17036 0.17857 0.21098 0.01983 0.43019 0.31744 0.20265 0.09400 0.02545 1.02019 1.01411 1.00731 1.00177 0.99930 1.78498 1.83897 1.91251 2.05990 1.91793

Table 5.3 shows the values of the function coefficients for different tip distances and

aspect ratios. One practical use of this Table and the proposed Eq. (3.5) is that for an

allowable decrease in SIF, one can have a satisfactory estimation of the spacing for the

multistage fractures in simultaneous multistage fracturing scenario.

5.5.3. Reliability of the proposed equation

Using the values of the coefficients reported in Table 5.3 in Eq. (5.3), one can have the

prediction of the numerical results for the SIF change as a result of simultaneous

multistage fracturing of rock. The reliability of Eq. (5.3) is shown in Figure 5.8.

79

Figure 5. 8. Prediction of SIF ratio of simultaneous multistage fracturing

Figure 5.8 shows that the SIF change as a result of simultaneous multistage fracturing is

satisfactorily predicted using the proposed function.

5.6. Effect of fracture offset on the SIF change in parallel wells

According to the fact that simultaneous fracturing between parallel wells may generate

propagating fractures in between the wells that their meeting tips may have some offset,

effect of offset between the tips should also be studied. Therefore, in this section, hydraulic

fractures with different aspect ratios in simultaneous fracturing of parallel wells are studied

with different distance and offset between the tips. Then, the SIF change is compared with

respect to single-stage hydraulic fractures. The results of this change is typically shown for

AR= 1.0, 0.4 in Figure 5.9.

As it is evident from Figure 5.9, the highest SIF change is when the distance between the

tips is the lowest and there is zero offset between the tips. By increasing the distance and

offset between the tips, SIF change reduces sharply and goes back to its original value in a

standalone well. The magnitude of change, however, is highly dependent on the aspect

y = 1x ‐ 9E‐06R² = 0.9993

1

1.1

1.2

1.3

1.4

1.5

1 1.1 1.2 1.3 1.4 1.5

Predicted SIF ratio

SIF ratio from numerical results

80

ratio of the meeting fractures in such a way that by increasing the aspect ratio, the amount

of change is raised. Figs. 5.10 show the SIF change typically for aspect ratio of unity.

Figure 5. 9. Effect of distance and offset on the SIF of the fractures

Figure 5. 10. Effect of fracture offset on SIF change

As it is evident from Figure 5.10a, the highest decrease is for the shortest offsets. This

means that at the lowest values of distance between fracture tips, the highest SIF change

belongs to the ones with the shortest offset magnitude. It is also seen from Figure 5.10b

that by increasing the offset, maximum SIF change is no longer for the shortest distance

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ratio of SIF simultan

eous/stan

dalone fractures

Distance between the tips, Ls/c

Lo=0.1 cLo=0.2 cLo=0.3 cLo=0.4 cLo=0.5 cLo=0.6 cLo=0.7 cLo=0.8 cLo=0.9 cLo=1.0 c

1

1.02

1.04

1.06

1.08

1.1

1.12

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

SIF ratio of simultan

eous/stan

dalone fractures


Lo=0.4 c

Lo=0.5 c

Lo=0.6 c

Lo=0.7 c

Lo=0.8 c

Lo=0.9 c

Lo=1.0 c

81

between the tips. This means that as the offset between the tips increases, maximum SIF

change is seen at larger distances between the tips. The red line on Figure 5.10b shows the

maximum SIF change for different offset values. As it is seen, the highest SIF change is

moved to higher distances between the tips as the offset increases. The highlighted redline

in Figure 5.10b, is slightly shifted to smaller distances between the tips for lower aspect

ratios. Figure 5.11 shows this shifting of maximum values to lower distances between the

tips.

As it is evident from Figure 5.11, some data points are missing for lower aspect ratios. This

is because maximum values of SIF change for these missing points are located at distances

smaller than 0.1 .

Figure 5. 11. SIF maximum line for different aspect ratios

In addition, the gradient of the SIF change after its maximum value is higher for shorter

offset magnitudes. As the offset increases, gradient of SIF change decreases and this causes

the SIF change to be more for higher offset values at higher distances.

1

1.02

1.04

1.06

1.08

1.1

1.12

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

SIF ratio of simultan

eous/stan

dalone fractures


AR=1.0

AR=0.4

AR=0.6

AR=0.8

82

5.7. Use of the SIF change prediction functions

In the preceding sections, a complete set of equations were proposed for prediction of any

change in SIF as a result of using different fracturing technique/pattern. Using these sets

of equations, one can determine the SIF behavior corresponding to the selected fracing

strategy/pattern. Therefore, according to the availability of equipment and maximum

applicable pressure, the allowable SIF change is obtained and based on that, spacing

between adjacent hydraulic fractures is determined.

83

Chapter 6: Work flow for optimization of hydrofracing

6.1. General

In Chapters 4 and 5, the results of a comprehensive study for stress shadow, aperture, and

propagation potential of hydraulic fractures were presented. The results were qualitatively

and quantitatively discussed and complete sets of equations were proposed for prediction

of stress shadow, aperture, and the SIF change in hydraulic fractures. The reliability of all

the equations was also verified using the numerical results. In this chapter, the final work

flow for geomechanical optimization of hydraulic fracturing is proposed according to the

equations proposed in the two preceding chapters.

6.2. Successive procedure of hydraulic fracturing optimization

As indicated in Chapter 1, optimization of hydraulic fractures is performed from different

perspectives/approaches. Any of these perspectives are of paramount importance in their

own scale and influence on production and the cost for completion. In fact, they are not

separated from each other but instead they should be investigated together for an

optimized hydraulic fracturing. Four important steps for the optimization of hydraulic

fractures in unconventional shales are given below:

6.2.1. Step one

In the first step, optimization of hydraulic fracturing should be investigated from a

production point of view. In this method, porosity, permeability, fracture conductivity,

reservoir geometry and its fluid boundary conditions are used in a reservoir simulator for

optimization purpose. In this step, horizontal well spacing, fracturing pattern, and fracture

84

geometry are investigated and an optimized fracturing pattern including fracture geometries

are defined.

However, according to the fact that aperture of hydraulic fractures plays a key role in

conductivity of the fractures, it is first required to have an estimate about the fracture

aperture according to its influencing variables. Aperture is influenced by fracture geometry

(length and aspect ratio), moduli of the medium, excess pressure, and the used fracturing

technique (multistage fracing). From all the above-mentioned influencing factors in

aperture determination, except for the geometry of the fracture which should come from

the flow simulator, all the other variables are geomechanical factors defined in Chapter 4

and 5. The general equation for prediction of aperture for a hydraulic fracture is

determined as:

, , (6.1)

Estimation of aperture for a hydraulic fracture is done using the above equation together

with Eqs. (4.2), (4.3), (4.9), (4.12), (4.13), (5.1) and Tables 4.4, 5.1. Then, from the aperture,

proppant size/type, and finally, fracture conductivity is estimated. Therefore, in each try

for production optimization by changing fracture geometry/pattern, hydraulic conductivity

of the fractures should also be updated based on the new aperture.

6.2.2. Step two

In this step, having fracture geometry, stress shadow analysis can be used for determination

of shadow zone around the hydraulic fractures according to the in-situ stress regime, rock

moduli, and excess pressure by using the sets of equations proposed in Chapter 4. The

85

general form of shadow prediction for threshold angle of 5° is written as Eq. (6.2) which is

used for stress shadow size of simultaneous uncontained hydraulic fractures.

_ , 10

1 , 10

(6.2)

In this equation, Eqs. (4.5), (4.6), (4.7), (4.14), (4.15), (4.7), (4.12), (4.13) and Tables 4.1, 4.5,

4.7 are used to give the value for shadow size of a hydraulic fracture under simultaneous

uncontained hydraulic fracturing.

6.2.3. Step three

In this step, using the fracture geometry from step one, the allowable propagation potential

decrease (decrease of the SIF in the target zone; mentioned in Chapter 5), and considering

the fracing technique used for the job (simultaneous, multistage), the spacing between

hydraulic fractures are determined. For the case of simultaneous multistage fracturing of

rock, Eq. (5.3) together with Table 5.3 can be used for investigation of propagation

potential of this fracing technique.

6.2.4. Step four

In this final step, according to the fracture geometry from step one, and proposed

fracturing patterns from steps one, two, and three, a final optimized fracturing pattern is

obtained by making decision between the three fracturing patterns in the three preceding

steps.

86

Chapter 7: Concluding remarks and recommendations

Unconventional resources, having low permeability, are considered as potential

hydrocarbon reserves because of unconventional architecture; new drilling and completion

technology. Hydraulic fracturing plays a decisive role in economy of these reservoirs since

it makes the best productive regions connected to the wellbore, thereby raising the

permeability of the reservoir. In fracturing treatment, however, it is required to optimize

the fracing job in an efficient way to reduce the cost and make the best optimized fracture

network.

7.1. Aperture analysis

In this part of the thesis, a comprehensive equation was proposed for prediction of

aperture of hydraulic fractures with in-situ stress anisotropy, rock moduli, net pressure,

fracture aspect ratio, containment, and using different fracturing patterns (simultaneous

and/or multistage fracturing) as input variables. It was observed that stress anisotropy does

not have any effect on the aperture of hydraulic fractures. It was also observed that

aperture is inversely related to the rock moduli and directly related to the aspect ratio of the

fracture. Uncontainment of the hydraulic fracture highly influences the aperture of the

fracture, while simultaneous fracing does not have significant effect on the aperture size. In

case of lower Young’s modulus for the bounding layers and more penetration extent,

aperture increases. On the other hand, for the case of higher Young’s modulus for the

bounding layers and higher penetration, a lower aperture is observed. Finally, it was also

shown that aperture of the hydraulic fractures is negatively influenced by multistage

fracturing. In Chapter 5, this decrease in aperture was quantified considering different

spacing between the fractures.

87

7.2. Stress shadow analysis

First, it was shown that shadow mechanism of hydraulic fractures is different depending on

the net pressure magnitude. In addition, shadow size around a hydraulic fracture was also

calculated assuming the same variables used in aperture determination. It was observed that

Young’s modulus of the rock (using constant pressure boundary condition) does not

change the shadow size. Stress anisotropy, Poisson’s ratio, and net pressure directly

increase the shadow size. Effect of simultaneous fracturing and uncontainment were also

calculated as multipliers to be multiplied with the case of single/contained fractures for

shadow determination for these scenarios. A comprehensive set of equations for shadow

size was also proposed by which shadow size of hydraulic fractures can be predicted

satisfactorily. These equations are really useful in hydraulic fracturing treatment and design

in term of perforation distance and proppant size/type for the fracture.

7.3. Analysis of propagation potential

Based on the consideration that stress intensity factor (SIF) defines the propagation

potential of a hydraulic fracture, a comprehensive numerical simulation framework was

designed in chapter 5 to investigate the interaction between fracture tips and the influence

on the SIF. Different scenarios were considered specifically, fracture aspect ratio, spacing

between multistage fractures, and distance and offset between the tips in simultaneous

fracturing mode. First, it was shown that multistage fracturing in a standalone well

dominantly reduces the SIF of the propagating fractures. The level of this decrease is lower

for the lower aspect ratios. The effect of spacing on the SIF was quantified using a fitting

equation with its proposed coefficients. Secondly, effect of two hydraulic fractures from

two parallel wells (single stage fracturing) was considered to study the effect of

88

simultaneous fracturing of parallel wells on the SIF change. It was observed that SIF of the

meeting hydraulic fractures increases noticeably as a result of simultaneous fracturing. The

magnitude of this change, however, is higher for higher aspect ratios. This effect was also

quantified proposing an equation and its coefficients. Thirdly, the effect of having

simultaneous multistage fracturing of parallel wells were compared with the case of single

stage fracturing of standalone wells in order to quantify the effect of simultaneous

fracturing when multistage fracturing technique is applied. Finally, in order to show the

effect of offset between the fracture tips, effect of offset/distance between the tips were

studied and the behavior of SIF as a result such pattern was also studied. It was observed

that existence of offset between fracture tips is a retarding factor in the SIF increase in

simultaneous fracturing. In addition, for simultaneous fractures with no offset, the lower

the distance between the tips, the higher the SIF increase. For the case of existing offset,

on the other hand, in case offset is more than one-fourth of the fracture length, the highest

SIF change no longer belongs to the least distance and there will be a certain distance in

which SIF is maximum. It is worth mentioning that the range of SIF change in this range

of offset (offset 0.4 ), is between 1.1 to 1.0 of a single stage fracture in a standalone well.

7.4. Optimized fracture network

An optimized fracture pattern was defined in Chapter 1 as “parallel fractures having the

highest production, the highest propagation potential in the target formation, perpendicular

to the wellbore axis, with an optimized distance to prevent any deviation/collapse”.

89

7.5. Four steps in optimization of hydraulic fracturing

In order to reach the goal of optimized fracturing pattern, three important aspects from

geomechanical and production perspectives should be considered simultaneously to come

up with the final decision on the fracturing pattern.

It has been stated in the literature that low permeability of the reservoir makes the

fracturing of it essential to produce at an economical rate. The lower the permeability, the

higher the number of fractures should be generated to have efficient production (Soliman

et al., 1997). On the other hand, fractures cannot be placed too close to each other because

of the geomechanical aspect of fracturing since propagation potential may substantially

decrease and fractures may not propagate in the direction perpendicular to the wellbore.

Therefore, the following four steps are suggested in this thesis for an efficient optimization

of hydraulic fractures considering the most influencing parameters into account.

1. In the first step, a primary fracturing pattern is assumed based on experience or

data from adjacent wells. Then, calculating the aperture of the fractures from the

equations proposed in Chapter 4, proppant type/size is selected and estimation on

the fracture conductivity is made. After that, fracturing pattern and fracture

geometry is optimized noting the fact that in each pattern new fracture conductivity

should be calculated based on the new aperture. Finally, in this step, an initial

estimate is given on fracture geometry and fracturing pattern.

2. In the second step, according to the geomechanical data and the obtained fracture

geometry from the first step, stress shadow size around hydraulic fractures with

their own fracing pattern is estimated using the set of equations proposed in

90

Chapter 4. In this step, the second estimate about fracturing pattern; distance

between parallel wells and spacing between multistage fractures is made.

3. In the third step, the estimated fracturing pattern in the second step is used for

determination of the change in propagation potential of the hydraulic fractures

compared to the case of a single staged fracture in a standalone well.

4. In this final step, according to the obtained propagation potential change in the

third step and considering the allowable propagation potential decrease, the best

spacing between adjacent fractures in multistage fracturing and the best distance

between parallel wells is estimated based on the two estimations made in the first

and second steps. Basically, maximum distance obtained based on economical

production (first step), no deviation or collapse (second step), and allowable

propagation potential decrease (third step) is selected as the optimum spacing

between the fractures.

In this thesis, a detailed qualitative and quantitative study of aperture of a hydraulic fracture

required for the first step, stress shadow size for the second step, and change in

propagation potential in the third steps were done in order to come up with a complete

framework for optimization of fracture treatment in unconventional shales.

91

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Appendix: Variable definition

variable Definition

Maximum induced stress

Applied far-field stress

Curvature radius

Crack half-length

Hydraulic pressure

, Polar coordinate system

Stress intensity factor for mode one

, , Stress components

Young’s modulus

Poisson’s ratio

Crack height

Principal stress reorientation

Deviation angle

Threshold angle

Maximum half-aperture

Half-aperture

_ Shadow size of a single fracture under threshold angle of

_ Shadow size of simultaneous fractures under threshold angle of

Horizontal stress ratio (stress anisotropy)

Young’s modulus of the net play

97

Variable Definition

Young’s modulus of bounding layers

Excess hydraulic pressure

Penetration extent (penetrated fracture height over total height)

Minimum horizontal stress

Maximum horizontal stress

Maximum deviation angle

Geometric function

Boundary condition function

Multistage fracturing spacing

Distance between fracture tips in simultaneous fracturing

Aspect ratio of the fracture

Aspect ratio multiplier for aperture

Bulk compressibility

Aspect ratio multiplier for shadow size determination

, Stress shadow coefficients

, Stress anisotropy coefficients

_ Stress shadow size difference of threshold angle of 05 from that of

for any aspect ratio

Uncontainment aperture multiplier for a single fracture

Uncontainment stress shadow multiplier for a single fracture

(threshold angle of 5)

98

variable Definition

, Coefficient s of Penetration extent function

Uncontainment stress shadow multiplier for a single fracture

(threshold angle of 80)

Simultaneous shadow multiplier

Uncontainment multiplier for shadow size of simultaneous

fracturing (threshold angle of 5)

Uncontainment multiplier for shadow size of simultaneous

fracturing (threshold angle of 80)

, Multistage fracturing multiplier for aperture and stress intensity

factor change

Simultaneous fracturing multiplier for stress intensity factor

change

Simultaneous multistage fracturing multiplier for stress intensity

factor change

Offset distance

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