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Stresses Around a Borehole

Prof. Dr. Eissa Mohamed Shokir

Common Borehole Stability Symbols

s1,s2,s3: Major, intermediate, minor stress

Sv, Sh, SH: Total earth stresses, or Sv, Shmin, SHMAX, or sv, shmin, sHMAX

sr, sq: Radial, tangential, borehole stresses

sr, sq, sv, shmin, sHMAX, etc: Effective stresses

r, ri: Radial direction, borehole diameter

po, p(r): Initial pressure, p in radial direction

MW, pw: Mudweight, pressure in borehole

E, n: Youngs modulus, Poissons ratio

f, r, g: Porosity, density, unit weight

k: Permeability

These are the most common symbols we use

Terminology and Symbols Problems

Often, the terminology and symbols used are confusing and irritating

This complexity arises because: The area of stresses and rock mechanics is

somewhat complex by nature

The terminology came from a discipline other than classical petroleum engineering

There is still some inconsistency in symbology, such as Sh, Sh, Shmin, sh, all for shmin

We will try to be consistent

Please spend the time to understand

Physical principles are the most important

Other Conundrums How do we express stresses?

As absolute stresses? As stress gradients? As equivalent density of the overburden? As equivalent mud weights?

e.g. PF = 18 ppg means 18 pounds per US Gallon is the fracture pressure at some (unspecified) depth (fracture gradient = (s3/z).

e.g. shmin gradient is 21 kPa/m (or 21 MPa/km)

e.g. The minimum stress is 2.16 density units

e.g. shmin is 66 MPa (at z = 3.14 km depth)

All of these are the same! (or could be)

Which method is used usually depends who you are talking to! (Drillers like MW)

The Basic Symbols, 2-D Borehole

Far-field stresses are natural earth stresses and pressures, genera-ted by gravity, tectonics

Borehole stresses are generated by creation of an opening in a natural stress field

Far-field stresses: scale: 100s of metres

Borehole stresses scale: 20-30 ri (i.e. local- to small-scale)

Far-field stress

r q

sr

sq

ri

pw

shmin

sHMAX

po

Borehole stress

Important to Remember

sq is the tangential stress, also called the hoop stress, you will see it repeatedly referred to in these terms

sq lies parallel (tangential) to the wall trace The magnitude of sq is affected by:

In situ stresses MW and cake efficiency Temperature and rock behavior

It is the most critical aspect of the stress condition around a borehole High sq values lead to rock failure Lower sq values usually imply stability

Borehole Stability Analysis Concept

First, we need stresses around the borehole In situ stresses are vital

p, T, chemistry affect these stresses

Mud cake efficiency

In some cases, rock properties are also needed

Then, we must compare the maximum shear stress with the rock strength We need to know the rock strength

We need to know if the rock has been weakened by poor mud chemistry and behavior

If matrix stress exceeds strength, we say the rock has yielded (or failed)

Plotting Stresses Around a Borehole

Usually, we plot sq, sr values along one or the other of the principal stress directions

Vertical

borehole

sr

sq

radius

s

pw = 0

smin

smax

Far-field stresses

Vertical borehole

smax smin

Stresses Around a Borehole One Dimensional Case:

A borehole induces a stress concentration

Two- and three-dimensional cases are more complicated (discussion deferred)

Stress lost must be redistributed to the borehole flanks (i.e.: s concentration)

F (F/A =

stress) F F

Initial stress

High sq near

the borehole,

but low sr!

(F/A)

(2F/A) F

F = force, A = Area, F/A = stress

Stress Redistribution

Around the borehole, a stress arch is generated to redistribute earth stresses

elastic rocks have rigidity (stiffness)

lost s

elastic rocks resistribute the lost stress

Everyone carries an equal

load (theoretical socialism)

In reality, some carry more

load than others (higher sq near the borehole wall)

Far away (~5D): ~no effect

These guys may yield if they are overstressed

D

Stresses Arch Around Borehole

The pore pressure in the hole is less than the total stresses

Thus, the excess stress must be carried by rock near the hole

If the stresses now exceed strength, the borehole wall can yield

However, yield is not collapse! A borehole with yielded rock can still be stable

shmin

circular

opening,

pw s H

MA

X

Arching of Stresses

arches lintels

load

stress arching

Typical Borehole Instability Issues

Pack-offs

Excessive tripping and reaming time

Excessive mud losses (fracturing losses)

Stuck pipe and stuck or wedged BHAs

Loss of equipment and costly fishing trips

Sidetracks, often several in the same hole

Cannot get casing to bottom

Poor logging conditions, cleaning trips

Poor cementing conditions, large washouts

These are all related in some way to rock failure and sloughing

Yield of Rock Around a Borehole

Borehole pressure

= pw = MW z

sHMAX

shmin

Axial borehole fractures develop

during drilling when MW is higher

than sq (surges, yield). (This is

related to ballooning as well.)

Swelling or other geochemical filtrate

effects (strength deterioration,

cohesion loss) lead to rock yield

High shear stresses cause shear

yield, destroying cohesion

(cementation), weakening the rock

Low sq

High sq

Shear yield

Tensile yield

Shear Stresses

Shear stress is the cause of shear failure

The maximum shear stress at a point is half the difference of s1 and s3

max = (s1 - s3)/2, or (sq - sr)/2 in the figure

Vertical

borehole

sr

sq

radius

s

pw = 0

Vertical borehole

smax smin

Assumptions:

The simplest stress calculation approach is the Linear Elastic rock behavior model

This behavior model is very instructive

It leads to (relatively) simple equations

r

i2

2

i2

2

i4

4

i2

2

i4

4

ri2

2

i4

4

i

= ( + )

2(1-

r

r) +

( - )

2(1-

4 r

r+

3r

r) 2

= ( + )

2(1 +

r

r) -

( - )

2(1 +

3r

r) 2

= -( - )

2(1 +

2 r

r-

3r

r) 2

in all cases, r r , is taken CCW from reference

ss s s s

q

ss s s s

q

s s

q

q

q

q

max min max min

max min max min

max min

cos

cos

sin

.

r

q

sr

sq

ri

Symbols used

smin

smax Far-field stress

pw = 0 Known as the Kirsch Equations

Comments

Note that the equations are written in terms of effective stresses (sq, sr, smin), with no pore pressure in the hole

Far-field effective stresses are the earth stresses, and they have fixed directions

sq, sr can be calculated for any specific point (r, q) around the borehole, for r ri

Later, one may introduce more complexity: T, p(r), non-elastic behavior, and so on

These require software for calculations; various commercial programs are available

Calculations with In Situ Stresses

For a vertical borehole, the least critical condition is when shmin = sHMAX = sh sq]max in this case = 2 sh if pw = po However, we can still get rock yield!

However, in most cases, especially in tectonic regions and near faults The stresses are not the same!

This means that the shear stresses are larger around the borehole after it is drilled

This means that rock yield is more likely!

Borehole stability issues are more severe

Lost circulation more critical

What is a Linear Elastic Model?

The simplest rock behavior model we use Strains are reversible, no yield (failure) occurs

Linear relationship between stress & strain

Rock properties are the same in all directions

a

r = 3

a = 1

a axial strain

str

ess ( 1

3

)

E = Ds/De =

Youngs modulus

Stress-strain plot

From The Elastic Model

Even in an isotropic stress field (e.g. shmin sHMAX for a vertical hole), shear stress concentration exists around the hole This can lead to rock yield. How to counteract?

We can partly counteract with mud weight E.g.: if pw = shmin = sHMAX = sh (i.e.: MW = sh/z)

If the filter cake is perfect (no Dp near hole)

In practice, this is not done: if MW = sh/z, we are at fracture pressure & drilling is slower!

Higher MW reduces the magnitude of the shear stress, which reduces the risk of rock yield, but increases LC risk, slows drlg

From The Elastic Model

Fracture breakdown pressure is calculated to be Pbreakdown = 3hmin - HMAX + po In practice, this is not used for design