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Title: Geoscience and Rock Mechanics

1
Module C -1 Stresses Around a Borehole - I
Argentina SPE 2005 Course on Earth Stresses and
Drilling Rock Mechanics Maurice B.
Dusseault University of Waterloo and Geomec a.s
2
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
s?r, s?q, s?v, s?hmin, s?HMAX, 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

3
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

4
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 (?3/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)

5
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)

sHMAX
Far-field stress
shmin
po
s?q
s?r
ri
q
r
pw
Borehole stress
6
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

7
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)

8
Plotting Stresses Around a Borehole

Usually, we plot ??, ?r values along one or the
other of the principal stress directions

9
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)

Initial stress
(F/A)
(F/A stress)
F
F
F
F
(2F/A)
High sq near the borehole, but low sr!
F force, A Area, F/A stress
10
Stress Redistribution

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

Everyone carries an equal load (theoretical
socialism) In reality, some carry more load
than others (higher ??? near the borehole
wall) Far away (5D) no effect
elastic rocks have rigidity (stiffness)
lost s
elastic rocks resistribute the lost stress
D
These guys may yield if they are overstressed
11
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

s?hmin
circular opening, pw
s?HMAX
12
Arching of Stresses
load
arches
lintels
stress arching
13
Shear Stresses

Shear stress is the cause of shear failure
The maximum shear stress at a point is half the
difference of ?1 and ?3
?max (??1 - ??3)/2, or (??? - ??r)/2 in the
figure

s
s?q
Vertical borehole
s?max
s?min
s?r
pw 0
radius
Vertical borehole
14
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

Symbols used
Far-field stress
s?max
s?q
s?min
s?r
q
ri
r
Known as the Kirsch Equations
pw 0
15
Comments

Note that the equations are written in terms of
effective stresses (s?q, s?r, ??min), with no
pore pressure in the hole
Far-field effective stresses are the earth
stresses, and they have fixed directions
s?q, s?r 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

16
Calculations with In Situ Stresses

For a vertical borehole, the least critical
condition is when ??hmin ??HMAX ??h
???max in this case 2 ??h 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

17
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

s?a s?1
s?r s?3
s?a
18
Lessons from the Elastic Model - I

Even in an isotropic stress field (e.g. shmin ?
sHMAX for a vertical hole in the GoM), 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

19
Lessons from the Elastic Model - II

Fracture breakdown pressure is calculated to be
Pbreakdown 3shmin - sHMAX po
In practice, this is not used for design
Fracture propagation is Ppropagation shmin,
also taken to be PF (fracture pressure) for
planning of MW programs
This is often taken to be MWmax
MW is usually maintained to be less than shmin
In practice, it is often possible to use some
methods to strengthen the borehole
This allows drilling somewhat overbalanced,
when pw gt shmin, (this must be done carefully!)

20
Borehole Stresses if shmin ? sHMAX

Here, we plot the tangential stress, s?q
Higher stress difference is serious! It gives
rise to higher s?q values. Rupture??

s?HMAX
s?HMAX
Calculated from Kirsch equations, along principal
stress directions
s?hmin
3.2s?
2s?hmin
s?hmin
hmin
pw
pw
2s?hmin
1.6s?
hmin
1.4)
(
1.0)
(
Far-field stresses, s?hmin, s?HMAX, are shmin
po, sHMAX po wellbore pressure pw assumed to be
equal to po
Sing06.021
21
High sHMAX - shmin Cases (Tectonic)

It gets worse in tectonic cases!
When shmin - sHMAX is large, the borehole wall in
the sHMAX direction is in tension! Induced
fractures can be generated during pw surges

s?
s?
HMAX
HMAX
s?
s?q 8s?
s?
hmin
pw
s?q 5s?
hmin
hmin
pw
hmin
s?
hmin
Sing06.022
Note here, borehole pressure, pw, is assumed
po
22
Plot of the Tangential Stresses

Here, s? stresses at the wall (ri) are plotted as
a function of ?
Note the symmetry

90
s?(ri)
sHMAX
rw
?
0
sHMAX
-90
Refer to paper by Grandi for details
23
Borehole Wall Stresses (_at_r ri)

Now, introduce effective stresses e.g. symbols ?
for total, s? for effective
Maximum stress at the borehole wall
sqmax 3sHMAX - shmin po (total stresses)
s?qmax 3sHMAX - shmin (effective stresses)
Minimum stress at the borehole wall
sqmin 3shmin - sHMAX - po (total stresses)
sqmin 3shmin - sHMAX (effective
stresses)
For a general 3-D solution for inclined
wellbores use a software solution (big
equations!)

24
Preliminary Comments

Creation of a borehole ? high tangential
stresses (sq), low radial stresses (sr)
The larger sHMAX - shmin, the higher s?q is (in
the direction of shmin), the lower s?q is (in the
direction of sHMAX)
Radial effective stress (s?r) is low near the
borehole wall, zero right at the wall

s?q
pw 0
s?r
25
More Preliminary Comments

If both stresses are equal (s?h) and MW po at
borehole wall s?q 2s?h, and s?r 0
If sHMAX shmin is large, s?q is increased, and
s?r doesnt change too much
This greatly increases the shear stresses
These shear stresses are responsible for failure
of the rock, breakouts, sloughing
How do we control this?
High effective mud weights reduce this
Mud cooling shrinks rock, reduces stresses
Avoid shale swelling, promote shale shrinkage

26
Mud Weight Effect (equal s case)
Here, we assume for simplicity that we have
perfect mud cake, and that the pore pressure in
the rock is zero
s?q
s?r
pw 0.3s
s?q
s?r
pw 0.8s
27
Lets Include Pore Pressures
s
Mud pressure - pw
s?q
Assume sHMAX shmin s
s?r
Pore pressure - po
pw 0.6s
perfect cake
radius
Positive support force pw po is applied in
the case of a perfect mud cake this is a strong
stabilizing force because it increases confining
stress, this will be discussed later, when we
introduce rock strength
Much of what we do in mud chemistry and MW
management is to try and keep a positive support
force right at the wall. This acts like a liner
in a tunnel, keeping the rock from deteriorating
and reducing the shear stresses. If it is lost
by poor cake, deterioration can be expected,
especially in shale.
28
Filter Cake Efficiency

The better the filter cake, the better the
support pressure on the borehole wall
Support pressure pw - pi
If there is poor filter cake, support pressure on
a shale may be almost zero!
This support pressure is a true effective stress
that is acting in a radial outward direction,
holding rock in place!
In WBM in shales, the support pressure tends to
decay with time!
Soon after increase in MW good stability
After some time (days, weeks), sloughing can
start again because support p decays

29
Horizontal vs. Vertical Wellbore?

sv 0.9 psi/ft, sh 0.6 psi/ft, p 0.4 psi/ft

Vertical Hole
In non-tectonic systems (shmin sHMAX) vertical
holes are subjected to lower shear stresses they
are generally more stable than horizontal holes
s?q 0.4 psi/ft
0.2
0.2
Stress State
Horizontal Hole
s?v 0.5 psi/ft

0.5
s?q 0.1 psi/ft, top, bottom
0.2
s?h 0.2 psi/ft
s?q 1.3 psi/ft, sides
s?h 0.2 psi/ft
30
Tectonic Stress Conditions
Vertical well
0.1
2.7
2.7
0.1
This orientation is the best one for this case,
showing the importance of knowing the in situ
stresses
s?v 0.5 psi/ft

Horizontal well aligned with minimum stress, ?hmin
2.5
s?hmin 0.3 psi/ft
0.5
0.5
Horizontal well aligned with minimum stress, ?HMAX
1.2
2.5
s?HMAX 1.0 psi/ft
0.4
0.4
Vertical effective stress 0.5 psi/ft
Min. horizontal effective stress 0.3 psi/ft
1.2
Max. horizontal effective stress 1.0 psi/ft
31
TABLE 1
Stress at borehole wall (s?) in a tectonically
active area (Compressive stresses are ve
Tensile stresses are -ve) Depth of investigation
is 5,000 ft
Maximum Stress
Minimum Stress
(s?min)
(s?MAX)
Hole
No.
Configuration
Gradient
Magnitude
Gradient
Magnitude
(
psi/ft)
(
psi)
(
psi/ft)
(
psi)
1
Vertical
2.7
13,500
-0.1
-500
Parallel to
2
minimum
2.5
12,500
0.5
2,500
horizontal stress
Parallel to
3
maximum
1.2
6,000
0.45
2,000
horizontal stress
32
3-Dimensional Borehole Stresses
Borehole radial, axial tangential stresses, sr,
sa, sq
F
Y
F, Y are dip and dip direction (wrt x) of
the borehole axis x, y, z are coordinates
oriented parallel to s1, s2, s3 s1, s2,
s3 are the principal total stress
magnitudes po is the pore pressure
x
Effective stresses s?1 s1 - po s?2 s2 -
po s?3 s3 - po
s2
y
s1
po
s3
Almost always, principle stresses can be taken as
? and ?? to the earths surface
z
33
What About the Axial Stress??

Axial stress, sa, acts parallel to the hole wall,
? to sr, sq
Usually ignored in borehole stability
However, if sa is very large compared to sr sq,
it can also cause yield
More sophisticated analysis reqd
Almost always, using the hole angle and azimuth,
we do the following
Determine maximum and minimum stresses in the
plane of the hole
Carry out a 2-D stability analysis

sr, sa, sq
34
The Best Well Orientation

In a relaxed (non-tectonic) basin, sv gt shmin
sHMAX, vertical wells are the most stable
In a tectonic basin, an estimate of the stresses
is essential for example
If sHMAX gt sv gt shmin, we still have to know the
specific values to decide the best trajectory
If sHMAX 0.7, sv 0.5, shmin 0.4 psi/ft, a
horizontal well parallel to sHMAX is the best
If sHMAX 0.7, sv 0.6, shmin 0.4 psi/ft, a
well parallel to shmin is likely the best
Careful Rock Mechanics analysis is best
0ther factors fissility, fractures

35
Stresses and Drilling
sv
sHMAX sv gtgt shmin
To increase hole stability, the best orientation
is that which minimizes the principal
stress difference normal to the axis
60-90 cone
sHMAX
shmin
sv
Favored hole orientation
sv
Drill within a 60cone (30) from the
most favored direction
sHMAX
sHMAX
shmin
shmin
sHMAX gtgt sv gt shmin
sv gtgt sHMAX gt shmin
36
Showing the Best Trajectory
sv
sHMAX

This is a polar plot of ease of drilling
Related to magnitude of shear stress on wall
This is based in situ stress knowledge
In this example, a horizontal well, W to E, seems
to be easiest
A horizontal well N to S is the worst (all other
factors being equal)

shmin
37
Typical Troublesome Hole (GoM)
16.00
15.00
14.00
PP
Pore pressure
MWmin Lade
13.00
shmin
Sh
Stress, pressure in ppg
sv
Sv
12.00
Planned Casing
Planned Csg
Actual Casing
Actual Csg
11.00
Drill MW
Drill MW
MW to Keep Hole Open
MW to keep hole open
10.00
9.00
8.00
Depth in feet
3000
4000
5000
6000
7000
8000
9000
4960? Stuck Pipe no rotation, no circulation
Increase MW to get out of hole
Losing 300 bbl.hr (ballooning?)
Pack-off
Hole tight with pumps off
38
The Plan The Reality

Hole planned from offset wells (sv, shmin, log
correlations to strength data, po)
Jagged line is a prediction of MW to sustain
reasonable borehole stability
Brown line chosen MW program from stability
calculation (using Lade criterion)
Red line was the actual mud weight needed to cope
with a series of problems
The casings were set higher than expected and an
extra string was eventually needed

39
How do We Sustain Stability?

MW control (up or down)
Mud properties control (reduce ECD)
Trip and connection policy (speed, surge)
Inhibitive WBM minimize chemical effects
OBM eliminate chemical effects
Air or foam UB drilling (shallow, strong rx)
Use fn-gr LCM, gilsonite in fractured shale
Cool the drilling mud to reduce s?q, reducing the
chances of rock failure
When all else fails, sidetrack, set casing

40
Well Design and Cost Optimization

High risks are mainly related to low MW, rapid
drilling, increased well blowout risks Low cost
if successful.
Low risks are mainly associated with slow
drilling and high MW, but drillings time is long
Generally costly
In between, there is a level of acceptable risks
with a lower cost factor

41
Borehole Cost Optimization

Affected by drilling speed, casing string costs,
cleaning problems, cost of drilling mud, risks,
trip problems
Optimizing this in real time is the challenging
task of the Drilling Engineer

Lost circulation
Safe
1.0
The shape of the cost curve changes, depending on
the stresses and where we are in the hole!
Ballooning
Fluid influx
0.8
Shear failure sloughing
Stress to Strength ratio
0.6
Mud Weight
42
Borehole Stability and Hydraulics