Maximum Horizontal Stress and Wellbore Stability While Drilling Modeling and Case Study

By using Mohr-Coulomb failure criterion, the maximum horizontal stress magnitude can be derived from these equations and from analysis of wellbore breakout obtained from borehole caliper

SPE 139280

Maximum Horizontal Stress and Wellbore Stability While Drilling: Modeling and Case Study

S Li and C Purdy, Halliburton

Copyright 2010, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE Latin American & Caribbean Petroleum Engineering Conference held in Lima, Peru, 1–3 December 2010

This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s) Contents of the paper have not been reviewed

by the Society of Petroleum Engineers and are subject to correction by the author(s) The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied The abstract must contain conspicuous acknowledgment of SPE copyright

Abstract

Maximum horizontal stress is a critical parameter used in drilling optimization and wellbore stability modeling Current maximum horizontal stress prediction from wellbore breakout was based on the maximum tangential stress on the wellbore wall to be equal

to the rock uniaxial compressive strength We assume that the vertical, minimum and maximum horizontal stresses define a specific relationship when the stresses in the formation are in equilibrium Based on a generalized Hooke’s law with coupling the equilibrium of stresses and pore pressure, the maximum horizontal stress can be solved using this relationship This new technique can reduce the uncertainty of in-situ stress prediction by narrowing the area of the conventional polygon of the in-situ stresses We also propose a new method of the maximum horizontal stress determination from analyses of drilling-induced near-wellbore stresses and breakouts The near-wellbore stresses are obtained from poroelastic equations By using Mohr-Coulomb failure criterion, the maximum horizontal stress magnitude can be derived from these equations and from analysis of wellbore breakout obtained from borehole caliper logs The new technique is compared to the existing methods in an example from measured borehole failures in a case study These comparisons demonstrate that new technique provides a much better result than the current available methods, because the wellbore failure is related with all of the near-wellbore stresses

A case study has been conducted in a GOM oil field to predict pre-drill wellbore stability, where borehole instability was the main cause of borehole trouble time in offset wells Using the proposed in-situ stress method, an improved borehole stability model was built to predict the pre-drill mud weight window Applying this prediction, wellbore failures and drilling risks were greatly reduced

Introduction

Wellbore failures represent a significant portion of all drilling related non-productive time Hence, maintaining wellbore stability is

an important and crucial step in the oil and gas industry Wellbore stability has been studied extensively in different contexts (e.g Bradley, 1979; Wiprut, 2001; Zhang et al., 2008) However, challenges still exist such as accurate determination of in situ stresses, which has become increasingly important for the oil and gas industry to reduce drilling trouble time and improve hydraulic fracturing, particularty in unconventional reservoirs and plays

The maximum horizontal stress can be estimated from the extended leak-off test (XLOT) with fracture reopening test (Bredehoeft et al 1976) This method was derived from the Kirsch solution for a circular hole subjected to an internal pressure in

an isotropic, homogenous, and linear elastic medium The assumption in the derivation was that the reopening occurs when the fluid pressure applied on the borehole wall is high enough to cancel out the minimum tangential stress on the wellbore wall Using elasticity theory and Mohr-Coulomb failure criterion for slippage on the faults, Addis, et al (1996) calculated the maximum horizontal stress for normal faulting and thrust faulting regimes Barton, et al (1988) proposed a method for calculating the maximum horizontal stress when rock strength is known utilizing observations of breakout width from vertical wells They assumed when the maximum tangential stress on the wellbore wall is greater than rock uniaxial compressive strength (UCS), then the wellbore would fail Basing on Mohr-Coulomb failure criterion, we extend this method We also derive an additional method

to constrain maximum horizontal stress using Hooke’s law

Maximum horizontal stress from in situ stress configuration

It is commonly accepted that in situ stress of subsurface formations includes three mutually orthogonal principal stresses They are

vertical stress, σv; maximum horizontal stress, σH; and minimum horizontal stress, σh The three principal stresses should

satisfy to Hooke’s law in order to keep the stress-strain equilibrium According to Hooke’s Law, the minimum horizontal strain

can be written as the following form, when the stresses are expressed in effective stress forms:

E

H v h

h

)

σ

ε = − + (1)

where εh is the strain in the minimum horizontal stress direction; E is the Young’s modulus; ',

v

H

σ and σh' are the effective vertical, maximum horizontal and minimum horizontal stresses, respectively ν is the Poisson’s ratio

Solve Eq.1 we have:

'

'

v h h

H

ν

ε σ

σ = − − (2)

Normally the formations extend very long in horizontal directions, therefore, the strain in the minimum horizontal direction is

much smaller than the strains in vertical and maximum horizontal stress directions Particularly, when the formations of interest

are constrained by stiffer formations, the stress state is similar as the condition of uniaxial strain loading In this extreme case, εh

is close to zero Therefore, the upper bound maximum horizontal stress can be expressed as:

'

'

'

v

h

ν

σ

σ ≤ − (3)

In porous media, the effective stress and total stress have the following relationship:

σ' = σ − αBPp (4)

where α B is the Biot’s coefficient; P p is the pore pressure in the formations

Combine Eqs 3 and 4, we have the maximum horizontal stress as follows:

ν

α σ

σ ≤ ( − ) − + 2 (5)

From Eq 5, if α B is equal to one, we can obtain the upper bound maximum horizontal stress:

UB H = ( h − Pp) − σv+ 2 Pp

ν

σ

σ (6)

From Eq.5, the maximum horizontal stress can be estimated when we know the minimum horizontal stress, vertical stress, pore

pressure and Poisson’s ratio The minimum horizontal stress can be obtained from the mini-Frac tests, LOT, and DFIT which are

normally available in exploration and production wells (Peng and Zhang, 2007) Vertical stress can be obtained by integration of

formation bulk density measured from wire-line logs

Z b w w

v

w

gdz z

ρ

σ (7)

where Z w is the water depth; ρw is the density of sea water; ρb is the bulk density as a function of the depth; Z is the true vertical

depth of the well

Case verification for constitutive relationship of in situ stress and pore pressure

The proposed method (Eq 5) is applied to estimate the maximum horizontal stress in an oil field - Field 1 of Visund Field in

Northern North Sea We examine two wells (well 1 and well 3) in this field The vertical stress, minimum horizontal stress, pore

pressure, and mud weight are available (Wiprut, 2001) Using Eq 5, the maximum horizontal stress at different depths in two

wells are calculated by assuming αB=1 and ν=0.25 The calculated maximum horizontal stresses (σH proposed) are listed in Tables

1 and 2 The maximum horizontal stresses obtained by the proposed method are compared to Wiprut’s results The data in Tables 1

and 2 are obtained from Wiprut (2001), except the σH proposed

Depth

(m TVD) σazimuth (°) H σ(MPa) V P(MPa) p σ (MPa) h σ(MPa) H σ(MPa) H proposed

Table 1: In-situ stress and pore pressure (Wiprut, 2001) and proposed σH in Well 1S of Field 1, Visund Field, Northern North Sea

Depth

(m TVD)

σH azimuth (°) σ(MPa) V

Pp (MPa)

σh (MPa)

σH (MPa)

σH proposed (MPa)

Table 2: In-situ stress and pore pressure (Wiprut, 2001) and proposed σH in Well 3 of Field 1, Visund Field, Northern North Sea

Figures 1 and 2 show the comparison between σH magnitudes calculated from the proposed method and the one from Wiprut (2001) in Visund Field of Northern North Sea In Figs.1 and 2 the pore pressure, minimum horizontal stress, vertical stress and maximum horizontal stress were obtained from Wiprut (2001) The proposed values of σH (shown with empty squares) are calculated from the proposed method (Eq 5 with Poisson ratio of 0.25 and Biot coefficient of 1) The results calculated from the proposed method are consistent to Wiprut’s results

Fig 1: In-Situ Stress and pore pressure versus depth in Well 1S of Field 1, Visund Field, Northern North Sea

Fig 2: In-Situ Stress and pore pressure versus depth in Well 3 of Field 1, Visund Field, Northern North Sea

In-situ stress constraint from the proposed method in different stress regimes

The magnitudes and orientations of in situ stresses play very important roles in geological engineering In normal faulting stress

regime, the vertical stress is the greatest principal stress which has the following relationship, σν≥σH≥σh In the strike-slip faulting

stress regime, the vertical stress is the intermediate principal stress, i.e σH≥ σν≥σh In the reverse faulting stress regime, the

vertical stress is the least principal stress, σH≥σh≥ σν

Assuming there are critically oriented faults constraining stress magnitudes, the Mohr-Coulomb criterion in faults can be

expressed as follows:

' 3

'

1

sin

1

cos

2

σ ϕ

ϕ

f

f q

C

+

−

≤ (8)

where σ1' is the maximum effective stress; '

3

σ is the minimum effective stress; C is cohesive strength;

) 1 ( sin

1

sin

1

μ μ

ϕ

ϕ

+ +

=

−

+

=

f

f f

q , ϕf is the angle of internal friction of the fault, μ is the friction coefficient of the

fault

For deep formations, the cohesion of the fault can be assumed to be zero Therefore:

' 3

'

σ ≤ qf (9)

From Eq 9, the in-situ stresses can be expressed as follows for different faulting stress regimes:

Normal faulting regime:

p h

p

P

P

≤

−

−

=

σ

σ

σ

σ

'

3

'

1 (10)

Strike-slip faulting regime:

p h

p

P

P

≤

−

−

=

σ

σ

σ

σ

'

3

'

1 (11)

Reverse faulting regime:

f

p v

p

P

P

≤

−

−

=

σ

σ

σ

σ

'

3

'

1 (12)

Hence, from Eqs 10, 11 and 12, the lower bound of minimum horizontal stress is:

f

p f p v

LB

h

q

P q

−

= σ

σ (13)

In the strike-slip regime, it has:

σH ≤ qf( σh − Pp) + Pp (14) The upper bound of maximum horizontal stress is:

p p v f

UB

H = q ( σ − P ) + P

σ (15) The coefficients of friction (μ) of 0.6 -0.7 work very well in most current stress field (Jaeger and Cook, 1979) The stress polygon can be drawn based on the in-situ stress relationship in different stress regimes from Eqs 10-15, as shown in Fig 3 (with μ=0.6) However, the polygon gives a wide range of stress values at depth Therefore, to reduce the uncertainty of the stress polygon, the in-situ stresses need to be constrained The maximum horizontal stress calculated from proposed method in Eq 5 gives reasonable constraints in the stress polygon (the red dash line in Fig 3) In Fig 3, the measured pore pressure, minimum

horizontal stress, vertical stresses at depth of 3850 m are as the following: σ v = 81.5MPa, σ h = 79.2MPa, P p = 62.3MPa (Wiprut, 2001) Poisson’s ratio of 0.25 and Biot coefficient of 1 are applied to Eq 5 to calculate the maximum horizontal stress The proposed maximum horizontal stress (red dash line) states the possible in-situ stress lying only in the stress polygon located in the right hand side of this line Therefore, the proposed maximum horizontal stress method can narrow the stress polygon and constrain the stress state

Fig 3: Stress polygon at depth of 3850 m (TVD) in Well 1S of Field 1, Visund field, Northern North Sea

Maximum horizontal stress from wellbore breakouts using Mohr-Coulomb failure criterion

It is commonly assumed that when a vertical well is drilled, the in-situ stress around the wellbore includes three mutually orthogonal principal stresses That is, the vertical stress, σν; the maximum horizontal stress, σH; and the minimum horizontal stress, σh However, for the inclined borehole the in-situ stress needs to convert to a new coordinate system where one axis is in the borehole axial direction, as shown in Fig 4 Therefore, the in-situ stress (or far-field stress) for an inclined wellbore can be expressed as follows:

i i

i

i i

i i

v h

H xz

H h yz

H h xy

v h

H

z

h H

y

v h

H

x

2 sin ) sin

cos (

2

1

sin 2 sin ) (

2

1

cos 2 sin ) (

2

1

cos sin

) sin cos

(

cos sin

sin cos

) sin cos

(

2 2

0

0

0

2 2

2 2

0

2 2

0

2 2

2 2

0

σ α σ

α σ

σ

α σ

σ

σ

α σ

σ

σ

σ α

σ α σ

σ

α σ

α σ

σ

σ α

σ α σ

σ

− +

=

−

=

−

=

+ +

=

+

=

+ +

=

where i is the inclination; α is the drilling direction of the borehole with respect to σ H, as shown in Fig 4

When a borehole is drilled in the rock situated in the in-situ stress state, the stress redistribution (creating near-field stresses)

occurs near wellbore The near-field stresses depend on in-situ stresses, mud pressure, and wellbore inclination Bradley (1979)

derived the stress distribution around an inclined borehole located in an arbitrary stress field The total normal stresses and shear

stresses near a wellbore can be expressed in the following forms:

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎧

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ + +

−

=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

− +

=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

− +

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

−

=

−

−

−

=

−

⎟⎟

⎠

⎞

⎜⎜

⎝

−

⎟⎟

⎠

⎞

⎜⎜

⎝

−

−

⎟⎟

⎠

⎞

⎜⎜

⎝

+

=

+

⎟⎟

⎠

⎞

⎜⎜

⎝

+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

−

− +

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

+

=

2

2 0

0

2

2 0

0

4

4 2

2 0

0 0

2

2 0 2

2 0 0 0

2

2 4

2 0

4

4 0

0 2

2 0

0

2

2 4

4 2

2 0

4

4 2

2 0

0 2

2 0

0

1 ) cos sin

(

1 ) cos sin

(

3 2

1 2 cos 2

sin 2

2 sin 4

2 cos )

( 2

2 sin

3 1 2

cos

3 1 2

) (

1 2

) (

2 sin 3

4 1

2 cos 3

4 1 2

) (

1 2

) (

r R r R

r

R r

R r

R r

R

r

R p r

R r

R r

R

r

R p r

R r

R

r

R r

R r

R

yz z

xz yz

rz

xy y

x r

xy y

x z

z

m xy

y x y

x

m xy

y x y

x r

σ σ

θ σ

θ σ

σ

θ σ

θ σ

σ σ

θ νσ

θ σ

σ ν σ σ

θ σ

θ σ

σ σ

σ σ

θ σ

θ σ

σ σ

σ σ

θ

θ

θ

The stresses at wellbore wall can be obtained from the above equations Therefore, the total normal stresses and shear stresses

at the wellbore wall for a deviated borehole in polar system (r , , θ z) are defined by the following equations:

σr = Pm

m xy

y x y

σθ 0 0 2 ( 0 0) cos 2 4 0 sin 2

] 2 sin 4 2 cos ) (

2

σ

σz = z − x − y + xy

) cos sin

(

σθz = − xz + yz

0

=

= rz

r σ

σ θ

For a vertical well, the effective principal stresses around wellbore wall due to drilling can be simplified as follows, if the

thermal effect stress is considered:

σθ' = σH + σh − αBPp − Pm− 2 ( σH − σh) cos 2 θ − σΔt (16)

σr' = Pm − αBPp (17)

t h

H p

B v

z = σ − α P − ν σ − σ θ − νσΔ

σ' 2 ( ) cos 2 (18) where θ is defined starting at x axis (the maximum horizontal stress direction); '

θ

σ , '

r

σ and '

z

σ are the effective tangential stress,

radial stress, and axial stress at wellbore wall, respectively P m is the mud pressure; σΔtis the thermal stress

We define that the wellbore breakout angle is 2ß, which occurs first in the minimum horizontal stress direction, therefore,

2θ=180-2ß Hence, the effective principal stresses around wellbore can be written as follows in terms of the breakout angle:

t h

H m

p B h

σr' = Pm− αBPp (20)

σz' = σv− αBPp+ 2 ν ( σH − σh) cos 2 β − νσΔt (21)

We assume that when a borehole fails, it follows Mohr-coulomb failure criterion, i.e

3 '

Fig 4: Stress transformation in Cartesian systems (x, y, z), (x’, y’, z’), and polar system (r, θ, z) in a cross-section of the wellbore

Applying Mohr-Coulomb failure criterion, we can solve the minimum mud weight to maintain wellbore stability with breakout

angle of 2 ß However, firstly we need to determine the maximum and minimum stresses In shear failure condition, σr'is always

the minimum stress (σ3' = σr') Therefore, two different cases, based on which one is larger between σθ' andσz', need to be

considered for maximum horizontal stress calculation through comparingσθ' withσz'

Case I:

When σθ' >σ'z>σr', this is the case with small breakout, i.e

) )(

2 1 ( 2

) 1 ( arccos[

2

h H

t m

v h

σ σ ν

σ ν σ

σ σ β

−

−

−

−

−

− +

≤ Δ (23)

In this case, '

θ

σ is the maximum principal stress ( '

1

σ ), and σr'is the minimum principal stress ( '

3

σ ) Substituting σθ' (Eq

19) and '

r

σ (Eq 20) into the Mohr-Coulomb failure criterion, we can obtain the minimum mud pressure with breakout angle of 2

ß:

) 1 ( )

2 cos 2 1 ( ) 2 cos 2

1

(

+

−

− +

−

− + +

f

p f B

t h H

m

q

UCS P

q

(24)

This equation is the minimum mud pressure or shear failure pressure for the wellbore with a breakout angle of 2 ß If wellbore

breakout angle is zero, the minimum mud pressure to maintain wellbore stability (avoiding shear failure) becomes:

) 1 ( 3

+

−

−

− +

−

f

t p

f B h H

m

q

UCS P

q

(25)

This is consistent with the derivation given in the references (Peng and Zhang, 2007) By knowing mud pressure and breakout

width of the wellbore, the maximum horizontal stress (σH) can be obtained from Eq 24:

σ σ β α

σ

2 cos 2 1

) 2 cos 2 1 ( ) 1 ( )

1 (

+

+

−

−

−

− + +

H

P q P

q UCS

(26)

If we assume that the wellbore is in uniaxial loading condition which is unusual case, then q f =0 In this case, Eq 26 simplifies

to the following:

σ σ β α

σ

2 cos 2 1

) 2 cos 2 1 ( +

+

−

− +

+

H

P P

UCS

(27)

This equation is the same to the one given by Barton, et al (1988) Therefore, the method we propose in Eq 26 is an improved

and extended form obtained by Barton et al (1988)

Case II:

When σr'<σθ' <σz', this is the case that the wellbore has large breakouts, i.e.,

) )(

2 1 ( 2

) 1 ( arccos[

2

h H

t m

v h

σ σ ν

σ ν σ

σ σ β

−

−

−

−

−

− +

≥ Δ (28)

In this case, σ'z is the maximum principal stress, and σr' is the minimum principal stress Substituting σz' and σr'to

Mohr-Coulomb failure criterion, we can solve the minimum mud weight with breakout angle of 2 ß:

p f

t h

H p

B v

m

q

UCS P

q P

≥

σ β σ

σ ν α

(29) This is the wellbore minimum mud pressure with large breakout angle of 2 ß From Eq 29 by knowing mud pressure and

wellbore breakout width (2 ß), we can calculate the maximum horizontal stress:

       ν β

σ β νσ

σ α

σ

2 cos 2

2 cos 2

) 1

h v

p f B m f H

P q P

q

Case study

In this section, a deepwater well drilled in the Gulf of Mexico is examined as one of the offset wells of the proposed well The maximum horizontal stress is analyzed by using the proposed method (from Eq 26) and compared to the one obtained from the available method (Barton et al., 1988) The wellbore shear failures using the maximum horizontal stresses calculated from the two methods are also analyzed and compared

In order to predict the minimum mud weight needed to prevent wellbore shear failure (wellbore collapse), it needs the following data for the wellbore stability (shear failure) analysis (Zhang et al., 2008):

· overburden stress

· minimum and maximum horizontal stresses

· pore pressure

· in-situ stress orientation

· wellbore trajectory

· relevant rock-strength data

Shear failure analysis results can be expressed either as shear failure gradient (SFG) or shear failure pressure (SFP) The SFG

is also the minimum mud weight required to prevent shear failure in the wellbore

Figure 5 shows the basic inputs for the wellbore stability in a deepweter well in the Gulf of Mexico This is a post-well analysis for verifying our maximum horizontal stress method Firstly, the overburden stress, pore pressure, minimum horizontal stress, and rock strength are analyzed for wellbore stability analysis, as shown in Fig 5 Secondly, the maximum horizontal stresses are calculated using different methods - the proposed method from Eq 26 and the one from Barton et al (1988) Then, the shear failure gradients are modeled based on the two different maximum horizontal stresses, as shown in Fig 6

Fig 5: Postdrill analysis in a deepwater well in the Gulf of Mexico Pore pressure, minimum horizontal stress, and rock strength are plotted as the inputs of wellbore stability analysis

In Fig 6 the left track plots caliper logs and bit size; the shaded area between caliper and bit size is the wellbore over-gauge or

breakouts The right track shows the maximum horizontal stress curves calculated by the proposed method from Eq 26 (blue

curve) and by the method presented by Barton et al (1988) (green curve) In the middle track, the shear failure gradients (SFG) are

calculated by using the two different maximum horizontal stresses as inputs The other inputs for the shear failure gradient

calculations remain the same The shear failure gradient calculations use Mohr-Coulomb failure criterion, and the rock strengths

are obtained from sonic transit time based on the industry standard approaches (refer to Fig 5)

Fig 6: The maximum horizontal stress and shear failure gradient calculations in a deepwater well in the Gulf of Mexico Two calculation

methods for the maximum horizontal stress (the proposed method in this paper and the available method in publications) are used and

their results are compared

Figure 6 indicates that our proposed method gives larger maximum horizontal stress magnitudes For a higher maximum

horizontal stress, it should need a higher mud weight to maintain wellbore stability Corresponding to the shear failure gradient,

higher SFG (blue curve in the middle track) in the most sections are given using the input of the maximum horizontal stress

proposed by our method Comparing SFG to mud weight and wellbore breakouts, the SFG calculated by the proposed method

gives a better result to predict wellbore failure Also, the proposed wellbore stability model can better calibrate the failure events in

this wellbore For instance, at the depths from 15,000 to 15,500 ft and from 17,800 to 18,300 ft, the used mud weight was less than

the calculated SFG, where the wellbore had breakouts (refer to Fig 6)

Conclusions

It has long been recognized that the maximum horizontal stress is the most difficult component to determine accurately We

proposed two methods to calibrate and determine maximum horizontal stress The first method is based on a generalized Hooke’s

law with coupling the equilibrium of three in-situ stress components and pore pressure The new technique can reduce the

uncertainty of in-situ stress prediction by narrowing the area of the conventional polygon of the in-situ stresses We also propose a

new method of the maximum horizontal stress determination from analyses of drilling-induced near-wellbore stresses and

breakouts by using Mohr-Coulomb failure criterion A case study demonstrates that our proposed method provides better results in

wellbore stability analysis

Nomenclature

UCS Uniaxial/unconfined compressive strength (psi or MPa)

E Young’s modulus

XLOT Extended LOT