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
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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

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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
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Arching of Stresses
load
arches
lintels
stress arching
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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
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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
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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

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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

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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
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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

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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!)

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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
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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
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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
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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!)

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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
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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
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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.
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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

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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
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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
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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
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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
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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
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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

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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
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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
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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
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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

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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
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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

Hydraulics
Rock mechanics
43
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

44
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