Oil & Gas Specific Terms

Mohr-Coulomb

The Mohr-Coulomb Criterion: A Cornerstone in Oil & Gas Engineering

The Mohr-Coulomb criterion is a fundamental concept in geotechnical and petroleum engineering, providing a framework for understanding the failure behavior of rocks and soils under stress. This criterion, named after Christian Otto Mohr and Charles Augustin de Coulomb, describes the relationship between shear stress and effective normal stress at which a material will fail.

In essence, the Mohr-Coulomb criterion establishes a failure envelope, a straight line on a plot of shear stress versus normal stress. This line represents the boundary between safe and failure states for a given material. The equation for this line is:

τ = c + σ'tan(φ)

Where:

  • τ is the shear stress
  • c is the cohesion, representing the material's resistance to shear failure when no normal stress is applied.
  • σ' is the effective normal stress, which is the normal stress minus the pore pressure.
  • φ is the angle of internal friction, representing the material's resistance to shearing due to interparticle friction.

Applications in Oil & Gas:

The Mohr-Coulomb criterion finds widespread use in various aspects of Oil & Gas operations:

  • Drilling: Determining the mud weight required to prevent wellbore instability and optimize drilling operations.
  • Wellbore stability: Assessing the risk of borehole collapse or fracturing based on the in-situ stresses and rock properties.
  • Reservoir characterization: Understanding the strength and deformation behavior of reservoir rocks, which impacts hydrocarbon production.
  • Fracturing: Predicting the pressure required to induce fractures in formations, vital for hydraulic fracturing techniques.
  • Geomechanics: Modeling the mechanical behavior of subsurface formations under various conditions, crucial for reservoir simulation and well planning.

The Plot:

The Mohr-Coulomb failure envelope is typically plotted on a graph with shear stress (τ) on the y-axis and normal stress (σ') on the x-axis. The slope of the line is determined by the angle of internal friction (φ), while the intercept on the y-axis represents the cohesion (c).

Mohr-Coulomb Failure Envelope

Key Points:

  • The Mohr-Coulomb criterion provides a simplified representation of material behavior and is more accurate for brittle materials like rocks.
  • It does not account for strain hardening or other complex material behaviors.
  • The values of cohesion (c) and angle of internal friction (φ) are material-specific and can vary significantly depending on factors like rock type, mineralogy, and pore pressure.
  • While a powerful tool, the Mohr-Coulomb criterion requires careful calibration and validation with experimental data for reliable predictions in real-world scenarios.

Conclusion:

The Mohr-Coulomb criterion plays a crucial role in the Oil & Gas industry by providing a framework to understand and predict the behavior of rocks and soils under stress. By understanding the failure envelope defined by this criterion, engineers can optimize drilling, wellbore stability, and reservoir development strategies, ultimately leading to safer and more efficient operations.


Test Your Knowledge

Quiz: The Mohr-Coulomb Criterion

Instructions: Choose the best answer for each question.

1. What does the Mohr-Coulomb criterion describe?

a) The relationship between stress and strain in a material b) The temperature at which a material will melt c) The relationship between shear stress and effective normal stress at failure d) The rate of deformation of a material under load

Answer

c) The relationship between shear stress and effective normal stress at failure

2. Which of the following is NOT a factor influencing the Mohr-Coulomb failure envelope?

a) Cohesion b) Angle of internal friction c) Poisson's ratio d) Effective normal stress

Answer

c) Poisson's ratio

3. In the Mohr-Coulomb equation (τ = c + σ'tan(φ)), what does "c" represent?

a) Shear stress b) Effective normal stress c) Angle of internal friction d) Cohesion

Answer

d) Cohesion

4. How is the Mohr-Coulomb criterion used in drilling operations?

a) To determine the optimal drilling fluid density b) To calculate the rate of penetration c) To predict the formation temperature d) To estimate the drilling cost

Answer

a) To determine the optimal drilling fluid density

5. What is a key limitation of the Mohr-Coulomb criterion?

a) It is only applicable to ductile materials b) It does not account for strain hardening c) It is too complex to apply in practical scenarios d) It is not accurate for predicting fracture initiation

Answer

b) It does not account for strain hardening

Exercise: Applying the Mohr-Coulomb Criterion

Scenario: You are working on a drilling project where the reservoir rock has a cohesion (c) of 20 MPa and an angle of internal friction (φ) of 30 degrees. The pore pressure at the drilling depth is 10 MPa.

Task:

  1. Calculate the effective normal stress (σ') at a depth of 2,000 meters, assuming a density of the overburden rock of 2.5 g/cm³.
  2. Use the Mohr-Coulomb equation to determine the shear stress (τ) at failure for this effective normal stress.

Note:
- Assume gravitational acceleration (g) = 9.81 m/s² - Convert units as needed.

Exercice Correction

**1. Calculate the effective normal stress (σ')** - Overburden pressure = density * depth * g = 2.5 g/cm³ * 2,000 m * 9.81 m/s² = 49.05 MPa - Effective normal stress = Overburden pressure - pore pressure = 49.05 MPa - 10 MPa = 39.05 MPa **2. Calculate the shear stress (τ) at failure** - τ = c + σ'tan(φ) = 20 MPa + 39.05 MPa * tan(30°) = 42.71 MPa **Therefore, the shear stress at failure for this effective normal stress is 42.71 MPa.**


Books

  • "Soil Mechanics in Engineering Practice" by Terzaghi, Peck, and Mesri: This classic textbook covers the fundamentals of soil mechanics, including the Mohr-Coulomb criterion, in great detail.
  • "Rock Mechanics and Engineering" by Hoek and Brown: A comprehensive resource on rock mechanics, including sections dedicated to the Mohr-Coulomb criterion and its application in various scenarios.
  • "Fundamentals of Geotechnical Engineering" by Braja M. Das: Another well-respected textbook providing a thorough explanation of the Mohr-Coulomb criterion and its application in geotechnical engineering.
  • "Petroleum Engineering: Drilling and Well Completion" by Adam, et al.: This book covers the application of the Mohr-Coulomb criterion in drilling and well completion operations, specifically in terms of wellbore stability and mud weight selection.
  • "Reservoir Simulation" by Aziz and Settari: Explains the use of the Mohr-Coulomb criterion in reservoir simulation models for accurately representing rock deformation and failure behavior.

Articles

  • "The Mohr-Coulomb Failure Criterion: A Review" by J. D. Byerlee: Provides a historical review of the development and applications of the Mohr-Coulomb criterion.
  • "A Comparison of the Mohr-Coulomb and Drucker-Prager Yield Criteria for Rocks" by L. S. Lee: Examines the limitations of the Mohr-Coulomb criterion and compares it to other yield criteria.
  • "The Influence of Pore Pressure on the Mohr-Coulomb Strength of Rocks" by M. A. M. Behnia: Discusses the effect of pore pressure on rock strength and the importance of considering it in the application of the Mohr-Coulomb criterion.

Online Resources

  • Wikipedia - Mohr-Coulomb Theory: A good starting point for a concise overview of the Mohr-Coulomb criterion and its history.
  • Stanford University - Introduction to Rock Mechanics: A lecture series from Stanford University that includes a detailed explanation of the Mohr-Coulomb criterion and its applications in geomechanics.
  • Purdue University - Soil Mechanics Lectures: These lectures cover the fundamentals of soil mechanics, including the Mohr-Coulomb criterion, and its application in various engineering scenarios.
  • GeoMechanics Tutorial - Mohr-Coulomb Failure Criterion: Provides a simple yet informative tutorial on the Mohr-Coulomb criterion, including its graphical representation and applications.
  • Sciencedirect - "Mohr-Coulomb Model for Soil Mechanics": Offers a comprehensive review of the Mohr-Coulomb criterion and its applications in soil mechanics, including numerical modeling.

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