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
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:
Young’s modulus, Poisson’s 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, generated by gravity,
tectonics…
Borehole stresses are
generated by creation
of an opening in a
natural stress field
Far-field stresses:
scale: 100’s 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
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
 Δp, ΔT, 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
Far-field stresses
smax
smin
s
sq
smax
smin
sr
Vertical
pw = 0
borehole
radius
Vertical borehole
Stresses Around a Borehole

One Dimensional Case:
A
borehole induces a stress concentration
 Two- and three-dimensional cases are more
complicated (discussion deferred)
Initial stress
(F/A)
(F/A =
stress)
F
F
F = force, A = Area, F/A = stress

F
F
(2F/A)
High sq near
the borehole,
but low sr!
Stress “lost” must be redistributed to the
borehole flanks (i.e.: s concentration)
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 s’q
near the borehole wall)
Far away (~5D): ~no effect
D
These guys may “yield”
if they are overstressed
Stresses “Arch” Around Borehole

shmin
circular
opening,
pw
sHMAX



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…
Arching of Stresses
load
arches
stress arching
lintels
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 = (s’1 - s’3)/2, or (s’q - s’r)/2 in the figure

s
sq
smax
smin
sr
Vertical
pw = 0
borehole
radius
Vertical borehole
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min
sq
sr
q
ri
pw = 0
4 r 2i 3 r i4
(s  max + s  min)
(s  max - s  min)
r 2i
(1 - 2 ) +
(1 - 2 + 4 ) cos2q
s r =
2
2
r
r
r
3 r i4
(s  max + s  min)
r 2i (s  max - s  min)
s q =
(1 + 2 ) (1 + 4 ) cos2q
2
2
r
r
2 r 2i 3 r i4
(s max - s min)
(1 + 2 - 4 ) sin2q
 rq = 2
r
r
in all cases, r  r i , q is taken CCW from reference.
r
Known as the “Kirsch” Equations
Comments





Note that the equations are written in
terms of effective stresses (sq, sr,
s’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
Calculations with In Situ Stresses

For a vertical borehole, the least critical
condition is when s’hmin = s’HMAX = s’h
 s’q]max
in this case = 2· s’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
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
σ’a = σ‛1
Stress-strain plot
σ’ – stress (σ‛1 – σ‛3)

σ‛r = σ‛3
E = Ds/De =
Young’s modulus
εa – axial strain
σ‛a
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…
Lessons from the Elastic Model - II

Fracture breakdown pressure is calculated
to be Pbreakdown = 3σ’hmin - σ’HMAX + 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 MW]max
 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 > σhmin, (this must be done carefully!)
Borehole Stresses if shmin  sHMAX

Here, we plot the tangential stress, s’q
σHMAX
σHMAX
Calculated from Kirsch equations,
along principal stress directions
2·σhmin

Sing06.021
σhmin
pw
3.2·σhmin
2σhmin
1.6·σhmin
σ HMAX
(
σ hmin
(
= 1.0)
σhmin
pw
σ HMAX
σ hmin
= 1.4)
Higher stress difference is serious! It
gives rise to higher s’q values. Rupture??
Far-field stresses, shmin, sHMAX, are: shmin – po, sHMAX – po
wellbore pressure pw assumed to be equal to po
High sHMAX - shmin Cases (Tectonic)

It gets worse in tectonic cases!
σ
σ
HMAX
HMAX
sq ~ 5σ
hmin
pw
σ
hmin
sq ~ 8σ
hmin
pw
σ
hmin
σ
hmin
σ
( HMAX = 2.0)
σ hmin

σ HMAX
= 3.0)
(
σ hmin
When shmin - sHMAX is large, the borehole wall
in the sHMAX direction is in tension! Induced
fractures can be generated during pw surges
Sing06.022
*Note: here, borehole pressure, pw, is assumed = po
Plot of the Tangential Stresses

Here, σθ stresses at the
wall (ri) are plotted as a
function of θ
+90°
σHMAX
σθ(ri)
rw
θ
0
σHMAX
-90°

Note the symmetry
Refer to paper by Grandi for details
Borehole Wall Stresses (@r = ri)
Now, introduce effective stresses: e.g.
symbols s for total, s for effective
 Maximum stress at the borehole wall:

σq]max = 3·σHMAX - σhmin – po (total stresses)
sq]max = 3·σ’HMAX - σ’hmin (effective stresses)
Minimum stress at the borehole wall:
σq]min = 3·σhmin - σHMAX - po (total stresses)
s’q]min = 3·σ’hmin - σ’HMAX (effective stresses)
 For a general 3-D solution for inclined
wellbores: use a software solution (big
equations!)

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
sq
sr
pw = 0
radius
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 doesn’t 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
Mud Weight Effect (equal s case)
s
sq
Assume sHMAX = shmin = s
sr
radius
pw = 0
s
sq
sr
pw = 0.3s
Here, we assume for simplicity that we
have “perfect” mud cake, and that the
pore pressure in the rock is zero
radius
s
sq
sr
pw = 0.8s
radius
Let’s Include Pore Pressures…
s
Mud
pressure pw
sq
sr
perfect cake
pw = 0.6s
Assume sHMAX = shmin = s
Pore pressure - po
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.
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
Horizontal vs. Vertical Wellbore?

σv = 0.9 psi/ft, σh = 0.6 psi/ft, p = 0.4 psi/ft
Vertical Hole
sq = 0.4 psi/ft
In non-tectonic systems (shmin ~
sHMAX) vertical holes are subjected
to lower shear stresses; they are
generally more stable than
horizontal holes
0.2
0.2
Stress State
sv = 0.5 psi/ft
sh = 0.2 psi/ft
sh = 0.2 psi/ft
Horizontal Hole
sq = 0.1 psi/ft,
top, bottom
0.5
0.2
sq = 1.3 psi/ft, sides
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
Horizontal well aligned with
minimum stress, sHMAX
sv = 0.5 psi/ft
shmin = 0.3 psi/ft
1.2
0.4
sHMAX = 1.0 psi/ft
Horizontal well aligned with
minimum stress, shmin
2.5
0.5
0.5
2.5
0.4
1.2
Vertical effective stress = 0.5 psi/ft
Min. horizontal effective stress = 0.3 psi/ft
Max. horizontal effective stress = 1.0 psi/ft
TABLE 1
Stress at borehole wall (σ’θ) in a tectonically active area
(Compressive stresses are +ve; Tensile stresses are -ve)
Depth of investigation is 5,000 ft
Maximum Stress
No.
Hole
Configuration
1
(σθ]MAX)
Minimum Stress
(σθ]min)
Gradient
(psi/ft)
Magnitude
(psi)
Gradient
(psi/ft)
Magnitude
(psi)
Vertical
2.7
13,500
-0.1
-500
2
Parallel to
minimum
horizontal stress
2.5
12,500
0.5
2,500
3
Parallel to
maximum
horizontal stress
1.2
6,000
0.45
2,000
3-Dimensional Borehole Stresses
Borehole radial,
axial & tangential
stresses, sr, sa, sq
F
Y
x
s2
y
po
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
Effective stresses:
s1
s1 = s1 - po
s2 = s2 - po
s3 = s3 - po
s3
z
Almost always, principle stresses can be
taken as  and  to the earth’s surface
What About the Axial Stress??





Axial stress, sa, acts parallel to the sr, sa, sq
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 req’d
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
The Best Well Orientation
In a relaxed (non-tectonic) basin, sv > shmin ~
sHMAX, vertical wells are the most stable
 In a tectonic basin, an estimate of the
stresses is essential; for example:

 If
sHMAX > sv > 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…
Stresses and Drilling
To increase hole stability, the
best orientation is that which
minimizes the principal stress
difference normal to the axis
Favored hole
orientation
sv
60-90° cone
sHMAX
shmin
sv
Drill within a 60°cone
(±30°) from the most
favored direction
sHMAX
shmin
sv >> sHMAX > shmin
sHMAX ~ sv
>> shmin
sv
sHMAX
shmin
sHMAX >> sv > shmin
“Showing” the Best Trajectory
sv
shmin
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)

Typical Troublesome Hole (GoM)
16.00
Stress, pressure in ppg
17 ½” x 20”
14 ¾” x 17 ½”
16” Liner
13 3/8”
15.00
14.00
PP
Pore
pressure
MWmin Lade
Shhmin
s
Svv
s
13.00
12.00
Planned Casing
Planned Csg
Actual Casing
Actual Csg
Drill MW
Drill
MW
MW to Keep Hole Open
MW to keep
hole open
11.00
10.00
9.00
8.00
3000’
4000’
4960 Stuck Pipe: no
rotation, no circulation
5000’
6000’
7000’
8000’
9000’
Depth in feet
Increase MW to
Losing 300 bbl.hr (ballooning?)
get out of hole Pack-off
Hole tight with pumps off
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
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
Well Design and Cost Optimization

Actual (Likely) Well Costs
High Risk      Low Risk


Well Design Costs
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
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
0.6
sloughing
“Ballooning”
0.8
Lost
circulation
Safe
Shear failure
1.0
Fluid influx
Stress to Strength ratio

Mud Weight
The shape of the
cost curve changes,
depending on the
stresses and where
we are in the hole!
Borehole Stability and Hydraulics
Borehole management is not only stresses,
rock strength, MW and mud properties!
 It is also dependent on hydraulics:

 Pumping
strategy and cleaning capabilities
 Gel strength, viscosity, mud density
 BHA design, ECD, even tripping policy
How do We Predict RM Stability?

We need to know the rock stresses in situ
 Vertical,
horizontal usually, sv, shmin
 Pore pressures (especially overpressure cases)

We need to know the rock strength
 Lab
testing of core
 Correlations to geophysical log data bases
 Testing of drill chips (penetrometers, sonic…)
Then, we make predictions of stability MW
 This is an indicator only!

 Careful
monitoring on the active well
 Improvement of our “calibrations”, ECD…
Lessons Learned
Stress concentrations arise naturally when
a hole is drilled
 The tangential stress sq is critical

 Affected
by stress, tectonics, rock behavior…
Borehole cake and mud support are critical
 We can calculate stresses, but rock
parameters are (E, n, Y, Co, To…) needed
 We can reduce the effects of high sq

 MW,

lower T, better cake, OBM…
We can use log data and correlations to
predict the MW for stability