Characterizing the full in situ stress tensor and its applications for petroleum activities

NOMENCLATURE OF SYMBOLS C: compressive strength C0: uniaxial compressive strength Cb: biaxial compressive strength g: acceleration due to gravity P: stress tensor due to pore pressure Pc

Doctoral Dissertation

Characterizing the Full In-Situ Stress Tensor and Its Applications for

Petroleum Activities

Department of Energy and Resources Engineering

Graduate School, Chonnam National University

Do Quang Khanh

August 2013

Characterizing the Full In-Situ

Department of Energy and Resources Engineering

Graduate School Chonnam National Universitv

Do Quang Khanh

Supervised by Professor YANG, Hyung-Sik

A dissertation submitted in partial fulfillment of the requirements for the Doctor of

Philosophy in Energy and Resources Engineering

CONTENTS Characterizing the Full In-Situ Stress Tensor

and Its Applications for Petroleum Activities

1.4 Outline of thesis 7

CHAPTER 2: IN-SITU STRESS TENSOR

2.2 In-situ stress tensor 9 2.3 State of in-situ stress 13 2.4 Pore pressure and effective stress 14 2.5 Frictional limits to stress 15 2.6 Stresses and rock failure 17

CHAPTER 3: STRESS AND FAILURE ANALYSIS FOR WELLBORES 22

3.2 Stress and failure analysis for a vertical cylindrical wellbore 22 3.2.1 Stresses around a vertical cylindrical wellbore 22 3.2.2 Failure analysis for a vertical wellbore 26 3.3 Stress and failure analysis for an arbitrarily deviated wellbore 28 3.3.1 Stresses around an arbitrarily deviated wellbore 28 3.3.2 Failure analysis for an arbitrary deviated wellbore 33

CHAPTER 4: METHODS FOR DETERMINING IN-SITU STRESS 35

4.1 In-situ stress measurements in drilling boreholes 35 4.1.1 Hydraulic fracturing methods 35 4.1.2 Overcoring methods 37 4.1.3 Breakout methods 39 4.1.4 Drilling induced tensile fractures methods 43 4.1.5 Earth focal mechanism (FMS) 45 4.2 New integrated method for determining ISS

using petroleum exploration data 47 4.2.1 Introduction 47 4.2.2 Determining the orientations of horizontal stresses 49 4.2.3 Determining the vertical stress 51 4.2.4 Determining the minimum horizontal stress magnitude 54 4.2.5 Constraining the maximum horizontal stress magnitude 57 4.2.6 Determining pore pressure 62

CHAPTER 5: MODEL DEVELOPMENT FOR

FAILURE ANALYSIS OF WELLBORE (FAOWB) 65

5.2 Structures of the FAoWB software packages 65 5.3 Validation of the results of the packages FAoWB 71 5.3.1 Case 1: Cross-checking Barton’s study (1998) on compressive failure

and breakout width analysis at the KTB wellbore, Germany 71 5.3.2 Case 2: Cross-checking Meyer’s study (2002) on the well stability

at the Swan Lake field, South Australia 78

CHAPTER 6: CASE STUDIES AND IMPLICATIONS 89

6.2 Geological framework of the main studied area 89 6.3 The White Tiger (Bach Ho) field, Centre of the Cuu Long basin, Vietnam 93 6.3.1 Statement of problem 93 6.3.2 In-situ stress determination techniques 96 6.3.3 In-situ stress tensor at the White Tiger field 105 6.3.4 Implications 106 6.3.5 Summary of results 120

6.4 The X field, Northern of the Cuu Long basin, Vietnam 121 6.4.1 Statement of problem 121 6.4.2 In-situ stress determination techniques 123 6.4.3 In-situ stress tensor at the X field 129 6.4.4 Implications 130 6.4.5 Summary of results 135

CHAPTER 7: CONCLUSIONS AND RECOMMENDATIONS 137

Acknowledgements 150

LIST OF FIGURES AND TABLES

Figure 2.1: Components of stresses acting on a plane 10 Figure 2.2: Components of stresses acting on the faces of a cube 11 Figure 2.3: The three states of stress and associated types of faulting 13 Figure 2.4: Frictional limits to stress based on the frictional strength

of favourably oriented fault planes for μ = 0.6 and 1.0 17 Figure 2.5: Two-dimensional Mohr circle 18 Figure 2.6: Three-dimensional Mohr circle 19 Figure 2.7: Mohr diagram with a failure envelope that fits closely to

laboratory rock testing data 20 Figure 2.8: Three-dimensional Mohr diagram and Coulomb failure criterions

for pre-existing planes of weakness and for intact rock 21

Figure 3.1: Vertical cylindrical wellbore with the orientations of the

circumferential stress σϴϴ, axial stress σzz and radial stress σrr

.

23 Figure 3.2: Stress concentration around a vertical in a bi-axial stress field

based on the Kirsch equations 24 Figure 3.3: The stress concentration around a circular borehole subject to

only uniaxial compression 25 Figure 3.4: An arbitrarily deviated wellbore with the orientations of

the cirumferential (σϴϴ), axial (σzz), radial (σrr),

minimum (tmin) and maximum (tmax) stresses 29 Figure 3.5: Three coordinate systems used to transform

for an arbitrarily deviated wellbore 30 Figure 3.6: Lower hemisphere projection used to display relatively stability

of wellbores with different azimuths and deviations 34

Figure 4.1: A schematic diagram with the equipment set-up and

the propagation direction of the induced fracture during

a hydraulic fracturing test 36 Figure 4.2: Typical procedure used in the overcoring technique 38 Figure 4.3: Circumferential stress around a vertical wellbore

with respect to the orientation of the maximum horizontal stress

for formation of BOs and DITFs 40 Figure 4.4: Section of four-arm dipmeter log data showing consistent

breakouts in a north-south direction 41 Figure 4.5: An imaging log data with borehole breakouts 42 Figure 4.6: Hollow cylinder laboratory test 42 Figure 4.7: An imaging log data with drilling induced tensile fractures 44 Figure 4.8: The three main fault regimes and their corresponding

fault plane solutions 46 Figure 4.9: Integration of density logs to estimate overburden stress at depths 52 Figure 4.10: Resistivity image, density log (RHOB),

density correction log (DRHO) and caliper log (CALI) 53 Figure 4.11: Pressure vs time record showing LOP, breakdown, Pc and Pr 55 Figure 4.12: Pressure versus root time plot showing Pc 57

Figure 5.1: Start screen of the software packages FAoWB 66 Figure 5.2: Main screen of the software packages FAoWB 66 Figure 5.3: Menu File of the software packages FAoWB 66 Figure 5.4: Menu Input Data of the software packages FAoWB 67 Figure 5.5: Tab Description of the software packages FAoWB 67 Figure 5.6: Tab Stress of the software packages FAoWB 68 Figure 5.7: Tab Rock properties of the software packages FAoWB 68 Figure 5.8: Tab Well of the software packages FAoWB 69 Figure 5.9: Menu Failure Criteria of the software packages FAoWB 69 Figure 5.10: Menu Process of the software packages FAoWB 70 Figure 5.11: Menu Output of the software packages FAoWB 70 Figure 5.12: Stress distribution of case 1 (from packages FAoWB) 72 Figure 5.13: Risk diagrams of case 1 (from packages FAoWB) 73 Figure 5.14: The breakout risk diagrams of the KTB wells

for Mohr-Coulomb, Drucker-Prager and Mogi-Coulomb criteria 74 Figure 5.15: The mud weight required of the KTB wells 76 Figure 5.16: Risk diagrams of case 2 (from packages FAoWB) 79 Figure 5.17: The breakout risk diagrams of wells at Swan Lake field

for Mohr-Coulomb, Drucker-Prager and Mogi-Coulomb criteria 81 Figure 5.18: The mud weight required at the Swan Lake field 82

Figure 5.20: Stress polygon and constraints for case 1 and 2 86

Figure 6.1: Location map of the Cuu Long Basin 90 Figure 6.2: Schematic cross-section of the Cuu Long Basin 91 Figure 6.3:Generalized stratigraphy column of the Cuu Long Basin 92 Figure 6.4 Location map of the White Tiger field at the Cuu long basin 93 Figure 6.5: Basement distribution at White Tiger field, Cuu Long basin 94 Figure 6.6: Main fault and fracture system at the White Tiger field 95 Figure 6.7: Generalized stratigraphy column at the White Tiger 95 Figure 6.8: Examples of the occurrence of BOs and DIFTs at

the basement intervals of the wellbores at the White Tiger field 96 Figure 6.9: Histogram and rose diagrams of the orientation of SHmax

from BOs at the basement intervals of the Whiter Tiger field 97 Figure 6.10: Histogram and rose diagrams of of the orientation of SHmax

from DITFs at the basement intervals of the Whiter Tiger field 97 Figure 6.11: Histogram and rose diagrams of the orientation of SHmax from

both BOs and DITFs at the basement intervals of the Whiter Tiger field 98 Figure 6.12: Vertical stress or overburden stress at the White Tiger field 99 Figure 6.13: Plots of treatment pressure in the hydraulic fracturing tests 99 Figure 6.14: Minimum horizontal stress at the White Tiger field 100 Figure 6.15: Pore pressure at the White Tiger field 101 Figure 6.16: Stress Polygon and constraints at depths of the White Tiger field 104 Figure 6.17: Stress distribution at the depth 3900 m of the White Tiger field 106 Figure 6.18: Stress distribution at the depth 4100 m of the White Tiger field 107 Figure 6.19: Stress distribution at the depth 4300 m of the White Tiger field 108 Figure 6.20: Stress distribution at the depth 4500 m of the White Tiger field 109 Figure 6.21: Risk diagrams at the depth 3900 m of the White Tiger field 110 Figure 6.22: Risk diagrams at the depth 4100 m of the White Tiger field 113 Figure 6.23: Risk diagrams at the depth 4300 m of the White Tiger field 116 Figure 6.24: Risk diagrams at the depth 4500 m of the White Tiger field 118 Figure 6.25: Location map of the X field 121 Figure 6.26: The depth structural map at the X field 121 Figure 6.27: The stratigraphy column of the X field 122 Figure 6.28: Example of DITFs of the wellbore X1 at the X field 123 Figure 6.29: Histogram and rose diagrams of DITFs at the wellbore X1 124

Figure 6.30: Vertical stress or overburden stress at the X field 125 Figure 6.31: Plots of surface pressure in the LOTs/FITs at the X field 126 Figure 6.32: Minimum horizontal stress at the X field 126 Figure 6.33: Pore pressure at the X field 127 Figure 6.34: Stress Polygon and constraints at depth 2300 m of the X field 128 Figure 6.35: Stress distribution at the basement depth 2300 m of the X field 130 Figure 6.36: Risk diagrams at the basement depth 2300 m of the X field 131 Figure 6.37: Risk diagrams on evaluation for the applicability of

under-balanced drilling techniques (Pw=22 MPa) 134 Figure 6.38: Stress distribution at two deviated wellbores of the X field 135

Table 6.1: The full in-situ stress tensor at the basement depths

of the White Tiger field 105 Table 6.2: The full in-situ stress tensor at the basement depth 2300 m

of the X field 129

NOMENCLATURE OF SYMBOLS

C: compressive strength

C0: uniaxial compressive strength

Cb: biaxial compressive strength

g: acceleration due to gravity

P: stress tensor due to pore pressure

Pc: fracture closure pressure

Pi: fracture initiation pressure

Pp: pore pressure

Pr: fracture reopening pressure

Pw: wellbore fluid pressure

Rb: coordinate transform matrix

RS: coordinate transform matrix

S: applied stress tensor

S’: effective stress tensor

S1, S2, S3: three principal stresses

Sb: stress tensor in the borehole coordinate system

Sg: stress tensor in the geographic coordinate system

SHmax: maximum horizontal stress magnitude

Shmin minimum horizontal stress magnitude

Ss: stress tensor in principal stress coordinate system

Sv: vertical stress magnitude

T: tensile strength

z: depth

α, β, γ: Euler rotation angles

δ: wellbore azimuth

σrr: effective radial stress

σtmax: maximum effective stress tangential to the wellbore wall

σtmin: minimum effective stress tangential to the wellbore wall

σzz: effective axial stress

σϴϴ: effective circumferential stress

φ: wellbore deviation

ω: angle between σtmax and the wellbore axis

: shear stress

Characterizing the Full In-Situ Stress Tensor and Its Applications for Petroleum Activities

Do Quang Khanh

Department of Energy and Resources Engineering Graduate School, Chonnam National University (Supervised by Professor YANG, Hyung-Sik) (Abstract)

Knowledge of the full in-situ stress tensor has an importance for petroleum activities A demand in the determination of in-situ stress using petroleum exploration data available has increased during the last decades over the world The new integrated method for determining the full in-situ stress tensor using the available petroleum data has been accepted as more reliable and widely applicable in many petroleum basins

This thesis developed and applied the new integrated method for determining the full tensor of in-situ stress using the available petroleum data This method involves many aspects in which the constraining related to the magnitude of the maximum horizontal stress is the most challenge It also requires the integration and modification many techniques for studying specific problems using available datasets

The software packages on failure analysis of wellbores (FAoWB) written in the programming language MATLAB were designed and developed from this new integrated method for determining the full stress tensor and the extended theories on stresses and failures around the wellbore They facilitate the determination of the full in-situ stress tensor using the observations of wellbore failures (breakouts BOs and/or drilling-induced tensile fractures DIFTs) in petroleum wellbores The forward calculating of stresses around the wellbores will be constrained with the observations of borehole failures and rock strength, pore pressure or mud pressure depending on available data at a particular petroleum field Moreover, under the full in-situ stress tensor determined they also help to derive easily the implications related to the state of in-situ stress Their accuracy and reliability were confirmed through the cross-checking of two well-known investigations earlier Three different strength criteria including the Mohr-Coulomb, Drucker-Prager and Mogi-Coulomb criteria also were applied to recommend the selection of an appropriate criterion for relatively strong rocks Furthermore, they have been demonstrated to be user-friendly, attractive and easy to develop the codes for other real cases

The software packages FAoWB were used to characterize well the state of the full in-situ stress tensors from the new integrated method with available data of basement reservoirs of the petroleum fields belonging to the Cuu Long basin, Vietnam Those are the White Tiger field located at the centre of

the Cuu Long basin and the X field located at the northern of the Cuu Long basin Results showed that the stress regimes at basement reservoirs of the Cuu Long basin should be the normal faulting (NF) or the strike-slip (SS) with the orientation of the maximum horizontal stress oriented in the direction NW-SE being consistent with the previous studies The change of the stress regimes from NF to SS together with the strength rock measured should affect the risk of the occurrence of BOs and/or DITFs These predictions are suitable to the practical problems at the petroleum fields of this basin as the wellbore collapse (due to BOs) or the lost circulation (due to DITFs) Moreover, with advanced knowledge of the full in-situ stress tensors including both the orientations and magnitudes, we could choose the optimum drilling trajectories oriented in the direction NE-SW, change the suitable mud weight to prevent wellbore instability or evaluate the applicability of under-balanced drilling techniques at the petroleum fields of the Cuu Long basin

Keywords: In-situ stress, wellbore failures, breakouts, drilling-induced tensile fractures, wellbore instability

· plate tectonics and neotectonics;

· earthquake prediction and seal breach by fault reactivation;

· stability of underground excavations (tunnels, mines, caverns, shafts, stopes);

· slope stability;

· drilling borehole stability;

· induced hydraulic fracturing stimulation;

· reservoir drainage and flooding patterns;

· subsurface fluid flow in naturally-fractured reservoirs, and

· storage and extraction of oil and gas from the subsurface

A dramatic increase in the determination of in-situ stress using petroleum exploration data and its applications to problems in petroleum exploration and production has been seen during the last decades over the world One key driver for the increased awareness has been the increasing quality and use of borehole imaging tools, and the geomechanical information yielded by these tools Nowadays, drilling induced failures including breakouts and/or drilling induced tensile fractures from borehole imaging tools are recognized and used to determine in-situ stress (Zoback et al., 1985; Peska and Zoback, 1995) Furthermore, the

increased incidence of deviated drilling has provided both new techniques for constraining the in situ stress tensor and increased demand for solutions to problems related to the state-of-stress such as wellbore stability and fracture stimulation

1.2 Project philosophy and purposes

There have been a number of different methods available to determine the in-situ stress in the Earth’s crust These methods include earthquake focal mechanisms, hydraulic fracturing, overcoring, borehole breakouts, drilling induced tensile fractures and geological indicators Each stress measurement technique has advantages and limitations The relationship between

in situ stress and induced failures in drilling boreholes can have significant implications for in-situ stress determination methods Therefore, the philosophy of this project was to integrate and/or modify techniques as required for studying specific problems using available datasets in the case studies

In-situ stress determination in any oil field or sedimentary basin involves some aspects, such

as determination of the maximum horizontal stress orientations, the magnitude of the vertical stress, the magnitude of the minimum horizontal stress and the constraining related to the magnitude of the maximum horizontal stress The approach to aspects of stress determination

is dependent upon the dataset available The main purpose of this project is to formulate and apply the new integrated method for determining the full tensor of in-situ stress based on new and existing techniques from available petroleum data Next, the use of these techniques within several case studies at the petroleum fields will be analyzed to examine the wide range of implications of in situ stress data to petroleum exploration and production activities

A significant part of this project has involved designing and developing the software packages on failure analysis of wellbores (FAoWB) written by programming language MATLAB They facilitate the determination of the full in-situ stress tensor using the observations of wellbore failures in petroleum wellbores Moreover, under the full in-situ

stress tensor determined the FAoWB software packages also help to derive easily the implications related to the state of in-situ stress, such as the choice of the optimum drilling trajectories for wellbore planning and the suitable mud weights for well stability

1.3 Review

During the last decades there has been extensive research on the determination of in situ stresses and its applications, particularly in the petroleum industry To provide a contextual framework for the more detailed discussion of the new integrated method for the in-situ stress determination based new and existing techniques, a brief review of existing techniques

is presented here

Generally, in sedimentary basins occurred the petroleum activities, the vertical stress is a principal stress Consequently the full in-situ stress tensor can be reduced to four components These components are the orientation of the maximum horizontal stress, the vertical stress magnitude (Sv), the minimum horizontal stress magnitude (Shmin) and the maximum horizontal stress magnitude (SHmax)

The orientation of the maximum horizontal stress can be determined from observations of breakouts and drilling-induced tensile fractures commonly seen on borehole image logs Borehole breakouts (BOs) were first described by Bell and Gough (1979) as stress-induced compressive failure of the wellbore, and have subsequently been used to determine maximum horizontal stress orientations throughout the world (Zoback and Zoback, 1980; Plumb and Cox, 1987, etc.) The advent of borehole imaging tools has confirmed the nature

of breakouts and has led to the recognition of stress-induced tensile wellbore failure known

as drilling induced tensile fractures (DITFs) DITFs are oriented orthogonal to breakouts and can also be used to determine the orientation of the maximum horizontal stress (Aadnoy, 1990b; Brudy and Zoback, 1993, etc.)

The vertical stress magnitude can be determined from the weight of the overburden (McGarr

and Gay, 1978), which can be calculated using density logs and checkshot velocity surveys Density logs are routinely run during petroleum exploration and conventionally provide a density measurement every 15 cm However, density logs are rarely run to the surface resulting in a lack of shallow data Density in the shallow section can be estimated by transforming sonic velocity from a checkshot velocity survey (Ludwig et al., 1970)

Hydraulic fracture test is an early and reliable method for determining in situ horizontal stress magnitudes and orientations (Haimson and Fairhurst, 1967) Hydraulic fracture tests involve isolating a section of the wellbore and increasing the pressure in the isolated interval

by pumping fluid into it, and thereby creating a fracture in the wellbore wall This fracture forms parallel to the wellbore axis (for a vertical wellbore) and orthogonal to the minimum horizontal stress In general the fracture propagates away from the wellbore in this orientation as fluid continues to be pumped into the interval In a thrust faulting stress regime the fracture may rotate to horizontal, as it propagates away from the wellbore, complicating the analysis However, in general it is the minimum horizontal stress that acts to close the fracture (Hubbert and Willis, 1957), and consequently the pressure at which the fracture closes is a measure of the minimum horizontal stress and can be determined from the pressure versus time record (Haimson and Fairhurst, 1967, etc.)

In petroleum drilling, hydraulic fracture tests are not generally undertaken but the leak-off test (LOT) is somewhat similar in procedure to the initial stages of a hydraulic fracture test and is routinely conducted during petroleum drilling Leak-off tests are conducted to determine the maximum fluid density that can be used in the next drilling section (i.e fracture gradient) and not for stress determination per se During a LOT the pressure is increased until a decrease in the rate of pressurization is observed Consequently the induced fracture is comparatively small compared to that induced during a hydraulic fracture test, resulting in fracture closure not generally being observed However, Breckels and van Eeklen (1982) showed that leak-off test pressures provide an estimate of the Shmin, but not as

accurate an estimate as that yielded by hydraulic fracture tests

Recognizing the similarity between LOTs and hydraulic fracture tests, Kunze and Steiger (1991) proposed the Extended Leak-Off Test (XLOT) This test uses the same equipment as

a LOT, but a procedure more similar to the hydraulic fracture test, with multiple cycles of pressurization and de-pressurization, results in a pressure versus time record that can be used

to determine the Shmin with increased confidence The orientation of the maximum horizontal stress may be determined by observing the orientation of the induced fracture using an impression packer or a borehole imaging tool (Engelder, 1993; Haimson, 1993) The magnitude of the maximum horizontal stress can be determined from XLOTs and hydraulic fracture tests in some circumstances where a re-opening pressure can be interpreted (Haimson and Fairhurst, 1967; Enever et al., 1996, etc.)

With the improvements of wellbore imaging tools, borehole breakouts BOs and/or DITFs can be more accurately interpreted and their geometry observed Zoback et al (1985) proposed a method for determining the magnitude of the maximum horizontal stress using the angular width of breakouts around the wellbore is proposed This technique was used to obtain SHmax in New Mexico (Barton et al., 1988) However, this technique is controversial because attempts to relate size and shape of breakouts to stress magnitudes requiring consideration of the geometrical effects of breakout development and the failure mechanisms

of the material (Detournay and Roegiers, 1986, etc.) Nonetheless if breakouts are observed and compressive rock strength measurements available, a lower bound for SHmax can be determined (Moos and Zoback, 1990, etc.) Like breakout occurrence, DITF occurrence can

be used to constrain SHmax, in this instance given knowledge of tensile rock strength (Moos and Zoback, 1990, etc.) Tensile rock strength is typically low compared to compressive rock strength and rocks typically contain planes of weakness on which the tensile rock strength is negligible Consequently the tensile rock strength can be assumed to be negligible (Brudy and Zoback, 1999)

Widespread application of deviated drilling led to new techniques being utilized for stress determination Aadnoy (1990) proposed a method for inverting three or more LOTs from

wellbores of different trajectories to determine the complete stress tensor Gjønnes et al

(1998) suggested the original method was inaccurate, because it ignored shear stresses, and proposed an improved method However, the improved inversion also contained large uncertainties, in part due to the inaccuracy of LOTs and suggested the use of multiple techniques to determine the in situ stresses Image logging in deviated wells led to the observation that breakout orientations rotate as deviation increases, depending on the stress regime and borehole azimuth (Mastin, 1988) A technique for inverting the variation in breakout orientations with borehole deviation and azimuth to determine the complete stress tensor is proposed (Qian and Pedersen, 1991) Peska and Zoback (1995) developed a similar technique for using rotation of breakout azimuths with deviation to constrain the stress tensor However, the rotation of DITF azimuths and variations in the occurrence of both breakouts and DITFs are considered to constrain the full in-situ stress tensor Using observations of both DITFs and BOs occurrence and change in orientation, the full in-situ stress tensor can

be determined from a single deviated borehole

Besides the frictional failure provides a theoretical limit to the ratio of the maximum to minimum effective stress beyond which failure of optimally-oriented pre-existing faults occurs (Sibson, 1974) A large number of in situ stress measurements in seismically active regions have shown stresses to be at frictional limit (McGarr, 1980; Zoback and Healy, 1984) Where one or more of the stress magnitudes are known, frictional limits can be used to constrain stress magnitudes in seismically inactive regions and estimate stress magnitudes in seismically active regions Most commonly SV and Shmin are known and the frictional limit is used to provide an upper limit to SHmax

1.4 Outline of thesis

The thesis consists of seven chapters The first chapter states the rationale, philosophy and purposes of project It also reviews studies on the determination of in-situ stresses and its applications, particularly in the petroleum industry

Chapter 2 presents an introduction to the full in-situ stress tensor and its relating concepts, such as the state of in-situ stress, the pore pressure and effective stress, frictional limits to stress and rock failures

Chapter 3 provides the theoretical descriptions on stresses around borehole and wellbore failures Especially, the stress concentration and wellbore failures in arbitrary wellbores They are necessary background for the discussion of the methods for determining the in-situ stress tensor used in the petroleum industry

Chapter 4 presents and discusses the methods for determining in-situ stress It introduces to in-situ stress indicators in drilling boreholes from hydraulic fracturing, overcoring, breakout, drilling-induced tensile fractures and earth focal mechanism Especially, it presents and discusses in detail the new integrated method for determining in-situ stress using petroleum exploration data available, covering the techniques for the determination of the maximum horizontal stress, the magnitudes of Sv, Shmin and pore pressure Pp and constraints of the maximum horizontal stress magnitude SHmin from the occurrence of wellbore failures and the frictional limits to stress

Chapter 5 introduces and presents the software packages FAoWB on failure analysis of wellbores They are designed and developed based on the applying of the theories in chapters

2, 3 and the new integrated method for determining the full in-situ stress tensor presented in chapter 4 Two well-known investigations earlier are considered and compared by the software packages FAoWB The results obtained from these software packages have confirmed their accuracy and reliability and showed their other potentials, such as their user-

Chapter 6 reports the use of the software packages FAoWB to study the real cases at the petroleum fields belonging to the Cuu Long basin, Vietnam General geological information

on the Cuu Long basin is introduced Two petroleum fields at the Cuu Long basins are the White Tiger field, located in the centre and the X field, located in the northern of this basin The full in-situ stress tensors at these petroleum fields are characterized by applying the new integrated method using the available petroleum data from two petroleum fields and using the packages FAoWB Next, under these full in-situ stress tensors the risk diagrams of the occurrence of drilling-induce tensile fractures (DITFs) and/or breakouts (BOs) of the packages FAoWB were established and analyzed With advanced knowledge of the full in-situ stress tensors including both the orientations and magnitudes at these petroleum fields, their implications should be derived on the choice of the optimum drilling trajectories, the predictions and prevention the wellbore instability and the evaluation for the applicability of the drilling under-balance techniques at these petroleum fields at the Cuu Long basin

Finally, conclusions and recommendations were drawn from applying this new integrated method for determining the full in-situ stress tensor and the using of the packages FAoWB with the available petroleum data at the petroleum fields

The in situ stress field is the present-day stress field and is responsible for contemporary failures It is the result of a variety of forces acting at differing scales These forces result in variations in the stress field at the scales of several hundred kilometres to less than a kilometre They can be dividing into first and second order forces (Zoback, 1992) First order forces are the result of plate boundary interactions and are responsible for the continental-scale stress field Second order forces are the result of topography, lithospheric flexure, lateral density and strength variations and geologic structure

Stresses within rocks at a depth in the subsurface in the Earth’s crust are referred as in-situ stresses The quantitative investigation of the above phenomena requires a mathematical representation of in situ stresses within the Earth’s crust The state of in-situ stresses can be mathematically described by using the concepts of the full in-situ stress tensor

2.2 In-situ stress tensor

Stress at a point is defined as a force (F) acting over a unit area (A) It is usually divided into

acting normal to the plane The shear stress () is the stress component acting parallel to a plane, inducing sliding along that plane Alternatively, the components of stresses acting on a plane can be completely described by the normal stress (σn) acting on the plane, the maximum shear stress ( max) acting on the plane and the orientation of that shear stress The maximum shear stress ( max) is also divided into two perpendicular directions ( x and

 y) on the plane (Figure 2.1)

Figure 2.1: The components of stress acting on a plane

To be more precise, stress is a tensor which describes the density of forces acting on all surfaces passing through a given point The full stress tensor in a three-dimensional space at depth can be described as a second-rank tensor by normal stress acting on each of three orthogonal planes and the two orthogonal components of shear stress acting on those planes (Figure 2.2)

Figure 2.2: Components of stresses acting on the faces of a cube

These nine components of the full three-dimensional stress tensor can be written as:

÷

÷

÷

÷ ø

ö ç

xz

yz yy yx

xz xy

xx

S

s s

s

s s

s

s s

s

(2.1)

where S is the stress tensor, Sii is the normal stress in the i direction, Sij is the shear stress acting in the j direction in the plane containing the j and k directions and Sij = Sji

The stress tensor can be simplified by choosing a coordinate system such that the planes have

no shear stress acting upon them The normals to these planes are principal stress directions

In this case the complete stress tensor can be completely defined by the magnitudes of the three principal stresses and the orientations of two of the principal stresses The stress tensor

is fully constrained if the magnitudes and orientations of the three principal stresses S1, S2

and S3 with the convention S1 ≥ S2 ≥ S3 are known:

÷

÷

÷

÷ ø

ö ç

1

0

0

0 0

0 0

S S

S

where S1 is the maximum principal stress, S2 is the intermediate principal stress and S3 is the minimum principal stress

The earth’s surface is in contact with a fluid which cannot support shear tractions Consequently, one principal stress is generally normal to the earth’s surface with the other two principal stresses acting in an approximately horizontal plane Alternately, the vertical orientation can be assumed to be one of the principal stress orientations Consequently, the in-situ stress tensor, completely constrained by the orientation of one of the horizontal stresses and the magnitudes of the vertical and two horizontal principal stresses, can be written:

÷

÷

÷

÷ ø

ö ç

H

S S

0 0

Since no shear stress can exist in the plane of the earth’s surface, one of the principal stresses in the upper crust must be oriented perpendicular to the surface In general, one principal stress is usually within 200 of vertical except beneath very rough topography It must be true close to the earth’s surface and it is also generally true to the depth to the brittle-ductile transition in the upper crust at about 15 ~ 20 km depth (Zoback, 1992) Thus, the other two principal stresses are oriented approximately horizontal, parallel to the ground surface Throughout this thesis the assumption that the vertical stress (Sv) is a principal stress and hence the maximum and minimum horizontal stresses (SHmax and Shmin) are also principal stresses has been used Consequently, the state of the stress tensor at depth is fully described by three principal stress magnitudes, including the vertical stress Sv corresponding to the weight of overburden; SHmax, the maximum principal horizontal stress; and Shmin, the minimum principal horizontal stress and one stress orientation, usually taken

to be the azimuth of the maximum horizontal stress (M.D Zoback, 2010)

2.3 States of in-situ stress

According to the relative magnitudes of the vertical stress and two mutually perpendicular horizontal stresses, Anderson (1951) assumed vertical and horizontal stresses are principal stresses and classified three possible states of stress and associated faulting styles (Figure 2.3)

Figure 2.3: The three states of stress and associated types of faulting (after Engelder,1993)

The different states of stress and associated fault styles are the normal faulting stress regime ( Sv > SH m a x > Sh m i n) , the strike-slip faulting stress regime ( SH m a x > SV >

Sh m i n) and the reverse or thrust faulting stress regime ( SH m a x > Sh m i n > SV)

These three states of stress correspond to the three commonly seen modes of faulting in the earth’s crust and are used throughout this thesis to describe relative stress magnitudes in the earth’s crust

When studying stress it is often convenient to consider the reference state of stress that may occur in the absence of tectonic forces and density heterogeneities The simplest

reference state is that of lithostatic stress found in magma:

S1 = S2 = S3 => SHmax = Shmin = Sv = Pmag (2.4) where Pmag is the pressure within the magma The lithostatic state of stress implies that the rocks have no long term shear strength and thus behave as a fluid (Engelder, 1993) However, experiments suggest that most rocks support at least a small differential stress for very long periods The lithostatic state of stress is likely to dominate in the deeper mantle and core, but is likely to be rare in the lithosphere

The uniaxial strain reference state assumes the rocks in the sub-surface are constrained laterally and deform elastically Hence, there is no strain in the horizontal direction and horizontal compression is developed due to the inability of the rocks to expand laterally In the uniaxial strain reference state, horizontal stress increases as a function of the depth of burial, but at a slower rate than the vertical stress:

 =   =  ʋ

ʋ =  ʋ

ʋ  ƍ (2.5) where n is Poisson’s ratio, varying between O and 0.5, g is the acceleration due to gravity and z is the depth When n = 0.5 (i.e a fluid) the uniaxial strain reference state is the same

as the lithostatic reference state The uniaxial reference state is thought to approximate that

of newly deposited sediments in a sedimentary basin However, modification in the elastic properties of the sedimentary rocks during diagenesis and creep relaxation may bring the state of stress in sedimentary basins closer to lithostatic (Engelder, 1993)

2.4 Pore pressure and effective stress

Rocks under natural conditions generally contain pore fluid at some pressure Pore fluid pressure has a critical influence on the physical properties of the porous solids (Terzaghi, 1943) Most physical properties of porous rocks obey a law of effective stress where the effective stress (S’) is the difference between the total applied stress (S) and the pore fluid pressure (Pp):

S’ = S – Pp (2.6) Most porous rocks obey Terzaghi’s effective stress law for rocks, which states that a pressure of Pp in the pore fluid of a rock will cause the same reduction in peak normal stress as caused by a reduction of the confining pressure by an amount equal to pore pressure

It has been demonstrated both by laboratory testing and in oil fields by the compaction of sediments from which oil has been drained and pore pressure reduced (Teufel et al., 1991) that rock deformation and failure occurs in response to the effective, not total stress

Pore fluid pressure is isotropic and the stress tensor associated with pore fluid pressure is given by:

÷

÷

÷

÷ ø

ö ç

0 0

ö ç

-=

p V

p h

p H

P S

P S

P

S

S

0 0

0 0

0 0

2.5 Frictional Limits to Stress

The stresses at which rocks in the subsurface fail provide useful theoritical limits to the magnitudes of in-situ stresses Shear failure occurs if the ratio of shear stress to normal stress becomes too large (Byerlee, 1978) Shear failure acts to reduce the ratio of principal stresses to below critical levels Shear failure in the normal fault regime acts to increase Shmin,

which moves the Mohr circle below the failure envelope and away from failure (Zoback and Healy, 1992) Shear failure in the reverse fault regime acts to decrease SHmax, which again moves the Mohr circle below the failure envelope and away from failure

The ratio of the maximum to minimum effective stress that causes slip on pre-existing faults that are optimally oriented with respect to the stress field was determined by Jaeger and Cook (1979):



 = f(μ) =  + (+ 1)/ (2.9) This relation provides frictional limits to the ratio of maximum to minimum effective principal stresses provided there exists optimally oriented faults of no cohesion If the ratio exceeds the above function of μ (usually from 0.6 to 1.0), then slip occurs in order to reduce that ratio to within frictional limits For the case of μ = 0.6 the ratio of the effective maximum principal stress to the effective minimum principal stress equals 3.12 For the case

of μ = 1.0 the ratio of the effective maximum principal stress to the effective minimum principal stress equals 5.83 This requires the value of μ to either be known or assumed A large number of in situ stress measurements in seismically active regions have shown stresses

to be at frictional limit (Zoback and Healy, 1984) Consequently, this ratio can be used to constrain the ratio of the magnitudes of the maximum and minimum stress in seismically active regions Furthermore, this ratio can be used to place upper or lower bounds on the maximum and minimum stress magnitude respectively in seismically inactive regions Shminand SV are commonly known and the relationship can be used to place a limit on the more poorly constrained value of SHmax

Depending on the stress regime of in-situ stress, the above equation becomes:

have been marked out according to Anderson's (1951) theory of faulting The in-situ

stresses are expected to plot within the area bound by the frictional limits (Figure 2.4)

Figure 2.4: Frictional limits to stress based on the frictional strength of favourably oriented fault planes for μ = 0.6 and 1.0 RF : reverse fault regime; SS: strike-slip fault regime; NF:

normal fault regime (After Moos and Zoback, 1990)

2.6 Stresses and rock failure

The rock properties and the in-situ stress tensor will control rock failure Failure occurs on planes as a function of the shear and normal stress acting on a plane, its frictional properties

The normal and shear stress acting on a plane in a two-dimensional stress field can be calculated using:

is the minimum principal stress and ϴ is the angle between S1 and the normal to the plane (Jaeger and Cook, 1979) The shear and normal stress calculated from equations 2.13 and 2.14 can be simply displayed on a Mohr diagram Plotting the shear and normal stresses on a Mohr diagram for ϴ varying between 00 and 900 and a given S1 and S2, forms a circle the centre of which is at a normal stress of (S1 + S2)/2 and shear stress of zero, and the radius

of which is (S1 - S2)/2 All two-dimensional states-of-stress lie on the perimeter of this circle (Figure 2.5)

Figure 2.5: Two-dimensional Mohr circle

A three-dimensional Mohr diagram can also be used to represent three-dimensional stress

fields In this case the diagram contains three Mohr circles with centres at {(S1 + S2)/2,0}, {(S1 + S3)/2,0} and {(S2 + S3)/2,0}, and radii of (S1 - S2)/2, (S1 - S3)/2 and (S2 - S3)/2 respectively All three-dimensional states of stress lie within the shaded area defined by the three Mohr circles (Figure 2.6)

Figure 2.6: Three-dimensional Mohr circle

The rock properties can also be displayed on a Mohr diagram in the form of a failure envelope A failure envelope separates two shear/normal stress regions Normal stress/shear stress combinations within the region below the failure envelope do not result in failure while those above the failure envelope do result in failure

Failure envelopes can either be theoretically or empirically determined An empirical failure envelope is based on laboratory rock tests, in which the maximum stress applied to a rock is increased until failure occurs This results in a shear and normal stress value for failure, which can be plotted as a point on a Mohr diagram (Figure 2.7) A series of points, which form a failure envelope, can be determined by failing the rock under many different stress states

Figure 2.7: Mohr diagram with a failure envelope that fits closely to laboratory rock testing

data (after Meyer, 2002)

A commonly used failure envelope is the Mohr-Coulomb failure criterion For pre-existing planes of weakness with no cohesive strength, the Coulomb failure envelope is represented by a straight line passing through zero shear stress (Figure 2.8):

t = m (sn - Pp) (2.15) where sn and t are, respectively, the components of normal and shear stress acting on the failure plane, Pp is pore pressure, and μ is the coefficient of internal friction Laboratory experiments on a wide variety of rock types have suggested that the coefficient of internal friction, m, lies between 0.6 and 1.0 (Byerlee, 1978) This range suggested by laboratory measurements seems to be applicable to crustal faults (Zoback and Healy, 1984) This failure envelope represents frictional sliding on a pre-existing failure plain with no cohesive strength i.e for a normal stress of zero, any shear stress greater than zero causes sliding

For intact rocks w h e r e t h e c o h e s i v e s t r e n g t h i s n o n - z e r o , the Coulomb failure

criterion for frictional sliding is defined by:

t = C + m (sn - Po) , (2.16) where C is the cohesive or shear strength of the rock In this case, the Coulomb failure envelope is represented by a straight line passing through the value of non-zero shear stress (Figure 2.8)

Figure 2.8: Three-dimensional Mohr diagram and Coulomb failure criterions for pre-existing

planes of weakness and for intact rock (after Reynolds, 2001)

in an elastic material depends on the far field in-situ stresses as well as the wellbore trajectories This can also result in wellbore failures if induced stresses around wellbores are over the rock strength These wellbore failures may be compressive failures known as borehole breakouts BOs and/or tensile failures as drilling-induced tensile fractures DITFs at the wellbore wall Therefore, observations of wellbore failures have also been proposed to determine both stress orientations and magnitudes of the in-situ stress tensor (Bell and

Gough, 1979; Zoback et al., 1985, etc.)

3.2 Stress and failure analysis for a vertical cylindrical wellbore

3.2.1 Stresses around a vertical cylindrical wellbore

Assuming the vertical stress is a principal stress and the rock behaves elastically, stresses around a vertical cylindrical wellbore are considered in cylindrical coordinates Kirsch (1898) developed a set of equations describing the stress components around a circular borehole subjected to far in-situ stresses in a thick, homogenous, isotropic elastic medium The three principal wellbore stresses of a vertical cylindrical wellbore of radius R are the effective radial stress (σrr), the effective axial stress (σzz) and the effective circumferential stress (σϴϴ)

as shown in figure 3.1 The radial stress σrr acts normal to the wellbore wall The axial stress

σzz acts parallel to the wellbore axis The circumferential stress σϴϴ acts orthogonal to σrr and

σzz (in the horizontal direction in the plane tangential to the wellbore wall)

Figure 3.1: Vertical cylindrical wellbore with the orientations of the circumferential stress

Mathematically, the effective stresses around a vertical cylindrical wellbore of radius R are described in terms of a cylindrical coordinate system by the following equations:

difference between the mud pressure Pm in the wellbore and the pore pressure Pp in the surrounding formation

The stress concentration around the vertical cylindrical wellbore predicted by these equation

is illustrated in figure 3.2 The bunching up of stress trajectories at the azimuth of Shmin

indicates strongly amplified compressive stress In contrast, the spreading out of stress trajectories at the azimuth of SHmax indicates a decrease in compressive, which may be tensile

Figure 3.2: Stress concentration around a vertical in a bi-axial stress field based on the Kirsch equations The principal stress trajectories are parallel and perpendicular to the

wellbore wall (after Zoback, 2002)

At the wellbore wall where R = r, these stress components reduce to:

where ʋ is Poisson’s ratio and SV is the vertical stress

The above equations are rewritten in terms of the far-field principal stresses as follows:

σϴϴ = ( +  ) − 2( − ) cos 2 − −  (3.8)

where Pw is the wellbore fluid (mud) pressure

The variation of effective principle stresses around a vertical wellbore wall is a function of relative bearing (ϴ) around the wellbore from the orientation of SHmax

In the case of a uniaxial stress field (SHmax), the stress concentration should be clearer as shown in figure 3.3 The stress at the wellbore wall subjected to only a uniaxial stress field is reduced to the following equation:

σϴϴ = SHmax - 2 SHmax cos2ϴ (3.11) where σϴϴ is the circumferential or hoop stress and θ is the angle around borehole wall measured from the direction of SHmax

Figure 3.3: The stress concentration around a circular borehole subject to only uniaxial

compression

This figure 3.3 shows the stress concentration around a circular borehole in a plate of elastic rock subject to uniaxial compression The maximum circumferential compression around the borehole occurs at points M and N (θ = 90o) In this uniaxial case, equation (3.11) reduces to 3SHmax at those points The minimum amount of compression occurs at points A and B (θ =

0o) where equation 3.11 reduces to -SHmax Therefore, a total circumferential stress difference

of 4S occurs around the open hole In this example the borehole is subjected only to

far-field compression, however due to the stress perturbation around the borehole points A and B are subjected to tension

3.2.2 Failure analysis for a vertical wellbore

3.2.2.1 Breakouts (BOs) or Compressive failure analysis for a vertical wellbore

Borehole breakouts are stress-induced ovalizations of the cross-sectional shape of the wellbore (Bell and Gough, 1979) Ovalization is caused by compressive shear failure on intersecting conjugate shear planes resulting in pieces of rock spalling off the wellbore wall This occurs when the wellbore stress concentration exceeds that required to cause compressive failure of intact rock

Assuming that the rock surrounding the wellbore is subjected to three principal stresses If these stresses exceed the rock strength, the rock will fail The stress state at the wellbore wall

at the azimuth of Shmin (where the stress concentration is most compressive) is compared to the compressive failure law defining the strength of the rock The common used compressive failure law is the Mohr-Coulomb failure criterion expressed in the space of principal stresses

In general σ1 is the circumferential stress and σ3 is the radial stress (Moos and Zoback, 1990) From equation (3.8) the maximum of the circumferential stress occurs at ϴ = ± 900

and can

be rewritten in the form:

For the simple case where σ3 is zero (i.e σ3 is σrr and the wellbore is in balance, Pw = Pp) and σ1 is the circumferential stress, Equation 3.12 can be substituted into the Mohr-Coulomb criterion in the space of principal stresses resulting in a simple failure criterion:

where C is the appropriated compressive rock strength

Compressive failure occurs when the circumferential stress exceeds the rock strength, for an

in balance wellbore (Pw = Pp) The significance of Equation 3.11 to the situation where the wellbore is not in balance (Pw ≠ Pp) is discussed later