Visualizing Principal Stress Trajectory

Patterns for Geoscientific and

Geotechnical Applications

Name: Ned Thomas

Advisor: Ruud Weijermars

Date: February 4, 2017

1

What Can We Learn from

Principal Stress Patterns?

(Weijermars, 2013, Figs 9,12)

→𝜏1𝜏1 →

Kirsch Equations

Tangential Stress:

• 𝜏𝜃 =1

2𝜏1 + 𝜏3 1 +

𝑎

𝑟

2−

1

2𝜏1 − 𝜏3 1 + 3

𝑎

𝑟

4cos 2𝜃 − 𝑃

𝑎

𝑟

2

Radial Stress:

• 𝜏𝑟 =1

2𝜏1 + 𝜏3 1 −

𝑎

𝑟

2+

1

2𝜏1 − 𝜏3 1 − 4

𝑎

𝑟

2+ 3

𝑎

𝑟

4cos 2𝜃 + 𝑃

𝑎

𝑟

2

Shear Stress:

• 𝜏𝑟𝜃 = −1

2𝜏1 − 𝜏3 1 + 2

𝑎

𝑟

2− 3

𝑎

𝑟

4sin 2𝜃

A Concise Algorithm

The Frac Number, F, controls the net

fluid pressure in the borehole as it

relates to the natural background

stress in the host rock and the

pressure in the penetrated formation.

𝑭 =𝑷𝑵𝑬𝑻

𝝉𝟏

𝝌 = −𝝉𝟑|𝝉𝟏|

The stress solutions are governed by two non-dimensional parameters

The Bi-axial Stress Scalar, χ, specifies

the anisotropy of the far-field stresses

Updated Non-dimensional Equations

Tangential Stress:

• 𝜏𝜃∗ =

1

21 − 𝜒 1 +

𝑎

𝑟

2−

1

21 + 𝜒 1 + 3

𝑎

𝑟

4cos 2𝜃 − 𝐹

𝑎

𝑟

2

Radial Stress:

• 𝜏𝑟∗ =

1

21 − 𝜒 1 −

𝑎

𝑟

2+

1

21 + 𝜒 1 − 4

𝑎

𝑟

2+ 3

𝑎

𝑟

4cos 2𝜃 + 𝐹

𝑎

𝑟

2

Shear Stress:

• 𝜏𝑟𝜃∗ = −

1

21 + 𝜒 1 + 2

𝑎

𝑟

2− 3

𝑎

𝑟

4sin 2𝜃

Principal Stresses:

• 𝜏1,3∗ =

1

2𝜏𝑥∗ + 𝜏𝑦

∗ ± 𝜏∗𝑥𝑦2 +

1

4𝜏𝑥∗ − 𝜏𝑦

∗ 21

2 𝑭 =𝑷𝑵𝑬𝑻

𝝉𝟏

𝝌 = −𝝉𝟑𝝉𝟏

(Weijermars and Schultz-Ela, 2015, Fig. 3)

Overbalanced | Balanced | Underbalanced

𝑭 > 𝟎 ExcessMud Weight

𝑭 < 𝟎 InadequateMud Weight

𝑭 = 𝟎

Faults around Salt Stocks

7

a: Radial fractures in the roof of the Pierce salt stocks, North Sea basin (Carruthers, 2012). b: Salt diapir with interpreted concentric fractures, North Sea basin (Stewart, 2006).

tan 2𝛽 =2𝜏𝑥𝑦

𝜏𝑥 − 𝜏𝑦

𝛽 =1

2tan−1

2𝜏𝑥𝑦

𝜏𝑥 − 𝜏𝑦

Principal Stress Trajectories

Bi-axial plane stress (χ = 1) Balanced borehole pressure (F = 0)

Borehole net-pressure is 2 times far-field principal stress:

𝑷𝑵𝑬𝑻 = 𝟐 𝝉𝟏

𝝉𝟏

𝝉𝟑

Overbalanced holes (F>0)

Excess mud weight

Risk of hole opening and lost circulation

Dike Patterns of the Spanish Peaks, CO

10

Dike patterns of the Spanish Peaks region, Colorado (Muller and Pollard, 1977), with solid blue lines representing mode I dike paths (Mériaux and Lister, 2002).

Field photographs of igneous dike emanating from the West Spanish Peak(Photos taken by Weijermars in 1990).

Stress Trajectories around

a Horizontal Wellbore

11

Extensional Basin Compressional Basin

Underbalanced holes (F<0)

Inadequate mud weight

Risk of shear failure including breakout and/or collapsed hole

Neutral Point Stress-ReversalThe neutral point defines a boundary called

a “stress cage” for overbalanced wellbores

and “fracture cage” for underbalanced

For underbalanced holes (F < 0), all

radial stress inside the fracture cage is

negative (tension) while all tangential

stress is positive (compression)

For overbalanced holes (F > 0), all

radial stress inside the stress cage is

positive (compression) while tangential

stress is negative (tension)

Neutral Point Locations

14

Concentric faulting

near large salt diapir

in Lower Congo Basin

15

(fig. 7.8 in Carruthers, 2012)

Mud Volcano Fault Evolution

16

Apsheron anticline in the South Caspian Basin Courtesy: Virtual Seismic Atlas

Mud volcano systems:

High fluid pressures (typically)

→ hydraulic fractures and shearing

→ open fractures and dilatant faults

σ1 is oriented normal to the image for this extensional basin

Concentric faults around the southernmost mud volcano indicate that underbalanced conditions (F < 0) prevailed at some point

Fracture Cage

Right: Initial crack is too short to extend beyond the fracture cage. As a result, this fracture redirects around the borehole.

(Weijermars et al., 2013, Fig. 10)

Recall: A fracture will redirect along planes parallel to the maximum principal stress.

Left: Initial crack is long enough to extend beyond the fracture cage region. As a result, when the fracture redirects parallel to the maximum principal stress, it propagates radially away from the borehole.

Initial Cracks

(Weijermars, 2016)

Controlling the initiation and propagation direction of hydraulic fractures for stimulation of the reservoir zone

Drilling

Prevention of

wellbore damage

during drilling

toward a target

reservoir

Hydraulic Fracturing

Conclusions

• Predict fracture orientations and planes of shear failure• Characterize some

of the most critical aspects of wellbore stability

• Apply to a number of geotechnical and geoscientific fields

• Deduce boundary conditions such as: (1) stress anisotropy, (2) far-field stress magnitudes, and (3) hole pressures

Principal stress trajectory patterns:

Ned Thomas

Texas A&M University

Thank you!

Questions?

Visualizing Principal Stress Trajectory Patterns

for Geoscientific and Geotechnical Applications