Friction Angle, Biot Coefficient and Cohesion

Summary

Mohr-Coulomb strength needs the Internal friction angle, from which come the friction coefficient and cohesion; effective stress needs the Biot coefficient. Friction angle is estimated from velocity or porosity, cohesion follows exactly from UCS and the friction angle, and Biot's coefficient comes from the ratio of frame to grain stiffness or from an empirical porosity relation. All of these are poorly constrained without core.

Inputs and outputs

Equations

Mohr-Coulomb. The failure envelope is a straight line in shear and normal stress, with cohesion \(\gmCoh\) and friction angle \(\gmIfa\). In terms of principal effective stresses the same criterion reads:

Setting \(\sigma_3' = 0\) identifies UCS, and solving for cohesion gives:

Friction angle from logs. Two empirical forms, with the velocity in km/s and porosity as a fraction. A relation for shale (Lal, 1999):

and a linear relation for sandstone (Weingarten and Perkins, 1995):

Biot coefficient. From the drained bulk modulus of the rock frame and the bulk modulus of the grains:

and from porosity with the Krief frame-modulus relation, \(K_{dry}/K_g = (1 - \phi)^{3/(1-\phi)}\):

The effective stress is then \(\sigma' = \sigma - \alpha P_p\).

Single-value calculator

Behavior

Cohesion is proportional to UCS and falls as the friction angle rises, because a stronger frictional contribution at confinement means less of the strength has to come from cohesion. At a UCS of 40 MPa cohesion goes from 15.3 MPa at a friction angle of 15 degrees to 7.3 MPa at 50 degrees; at 80 MPa it goes from 30.7 to 14.6 MPa. At 35 degrees the friction coefficient is 0.70 and q is 3.69: at failure the rock carries 3.7 times as much effective stress axially as laterally, plus its UCS. At the default inputs the shale relation gives 36.9 degrees at a velocity of 4 km/s and the sandstone relation gives 45.2 degrees at a porosity of 0.12. The two differ by 8.3 degrees, which is enough to change the predicted collapse pressure materially. The Krief Biot coefficient rises with porosity as the frame softens: 0.15 at a porosity of 0.05, 0.30 at 0.10 and 0.78 at 0.30. From frame and grain moduli of 20 and 38 GPa it is 0.47.

Parameter guidance

Friction angle. Triaxial tests at several confining stresses give the angle directly and are the only good source. Log relations are a fallback and they depend strongly on lithology: the velocity relation was published for shales and the porosity relation for sandstones, and neither should be used outside its lithology. Typical values are 15 to 30 degrees for shale and 25 to 45 degrees for sandstone and carbonate. A friction angle in the stability model is a strength property of intact rock, and is not the same thing as the friction coefficient of a pre-existing fault, which is taken as about 0.6 in stress-limit calculations.

Cohesion. Take it from the same test that gave UCS and the friction angle, or compute it from them. Do not pair a UCS from one source with a friction angle from another without checking that the resulting cohesion is reasonable. A cohesion that falls below zero or above UCS/2 means that the angle is outside 0 to 90 degrees or that UCS is inconsistent.

Biot coefficient. The upper limit is 1. Use 1 for unconsolidated and very soft rock. For stiff, tight rock it can be 0.3 to 0.6. A fixed value of 0.7 to 1.0 is common in practice when there are no measurements. The grain modulus comes from the minerals (about 37 GPa for quartz and 70 GPa for calcite, mixed by volume), and the drained frame modulus should be a drained value, not the saturated dynamic bulk modulus from the sonic, which is stiffer; using the dynamic bulk modulus underestimates Biot's coefficient.

Worked example

A UCS of 40 MPa and a friction angle of 35 degrees, with the friction angle estimates from a velocity of 4 km/s and a porosity of 0.12, and a check that UCS is recovered from cohesion:

import math
ucs, ifa_deg = 40.0, 35.0
phi = math.radians(ifa_deg)
S0 = ucs * (1 - math.sin(phi)) / (2 * math.cos(phi))
q = (1 + math.sin(phi)) / (1 - math.sin(phi))
mu = math.tan(phi)
print(f"cohesion = {S0:.2f} MPa, mu = {mu:.3f}, q = {q:.3f}")
print(f"check: UCS = 2 S0 cos(phi)/(1 - sin(phi)) = {2 * S0 * math.cos(phi) / (1 - math.sin(phi)):.2f} MPa")
print(f"check: q = (sqrt(1+mu^2)+mu)^2 = {(math.sqrt(1 + mu**2) + mu) ** 2:.3f}")
vp, phit = 4.0, 0.12
print(f"friction angle from Vp = {vp} km/s (shale form):    {math.degrees(math.asin((vp - 1) / (vp + 1))):.1f} deg")
print(f"friction angle from porosity {phit} (sandstone form): {57.8 - 105 * phit:.1f} deg")
kd, kg = 20.0, 38.0
print(f"Biot from moduli = 1 - {kd:g}/{kg:g} = {1 - kd / kg:.3f}")
print(f"Biot from porosity (Krief) = {1 - (1 - phit) ** (3 / (1 - phit)):.3f}")

Output

cohesion = 10.41 MPa, mu = 0.700, q = 3.690
check: UCS = 2 S0 cos(phi)/(1 - sin(phi)) = 40.00 MPa
check: q = (sqrt(1+mu^2)+mu)^2 = 3.690
friction angle from Vp = 4.0 km/s (shale form):    36.9 deg
friction angle from porosity 0.12 (sandstone form): 45.2 deg
Biot from moduli = 1 - 20/38 = 0.474
Biot from porosity (Krief) = 0.353

Assumptions and limitations

• The failure envelope is a straight line. Real rock envelopes are curved at low confinement, so a friction angle and cohesion fitted over a stress range are only valid over that range.
• Cohesion computed from UCS and the friction angle is a convenient parameter, not an independent measurement. Both inputs carry their own uncertainty.
• The log relations for friction angle apply to the lithology they were published for. They carry scatter of several degrees and are not tied to a physical mechanism.
• Biot's coefficient is a scalar, so the rock is isotropic and the pore fluid acts equally in all directions. In stiff low-permeability rock it depends on the process (drained or undrained) and on the stress.
• The grain bulk modulus reflects the real mineralogy, and the frame modulus is drained.

QC checks

• The friction angle is between about 15 and 50 degrees, and is lower in shale than in clean sandstone. A shale-form angle in a clean sandstone interval is a lithology error.
• Cohesion is positive, below UCS/2, and follows UCS in shape. A cohesion curve that goes the other way with the friction angle is a sign of the formula applied in the wrong units.
• The Biot coefficient is between 0 and 1, and is higher in soft, porous rock than in tight rock.
• Where triaxial tests exist, the Mohr-Coulomb parameters from the logs are compared with them, and the test confining stresses should be near the range that matters.

Going Deeper

The Biot coefficient was introduced in the theory of poroelasticity, and its definition as one minus the ratio of frame to grain compressibility comes from Biot and Willis. It is the factor that multiplies pore pressure in the effective stress that controls the deformation of the frame. A separate coefficient, often called Skempton's pore-pressure coefficient, controls the pore pressure response in undrained loading. For failure, the effective stress coefficient can differ from the deformation one, and in low-porosity rock a value of 1 is often used for failure and a lower value for deformation. The Mohr-Coulomb link is the most useful result of the page: UCS, cohesion and friction angle are three parameters of which only two are independent, so a log estimate of UCS and an independent estimate of the friction angle fix the third. Nonlinear criteria (Hoek-Brown, Mogi, Drucker-Prager) have been used in wellbore stability to correct for the over-conservative collapse pressures that a linear criterion gives.

References

1. Chang, C., Zoback, M.D. and Khaksar, A., 2006. Empirical relations between rock strength and physical properties in sedimentary rocks. Journal of Petroleum Science and Engineering, 51(3–4), 223–237.
2. Lal, M., 1999. Shale stability: drilling fluid interaction and shale strength. SPE 54356, SPE Asia Pacific Oil and Gas Conference and Exhibition.
3. Weingarten, J.S. and Perkins, T.K., 1995. Prediction of sand production in gas wells: methods and Gulf of Mexico case studies. Journal of Petroleum Technology, 47(7), 596–600.
4. Krief, M., Garat, J., Stellingwerff, J. and Ventre, J., 1990. A petrophysical interpretation using the velocities of P and S waves (full-waveform sonic). The Log Analyst, 31(6), 355–369.
5. Biot, M.A., 1941. General theory of three-dimensional consolidation. Journal of Applied Physics, 12(2), 155–164.
6. Zoback, M.D., 2007. Reservoir Geomechanics. Cambridge University Press.

Python reference implementation

Related methods

• Dynamic Elastic Moduli
• Brittleness Indices
• Static vs Dynamic Moduli
• UCS Correlations
• Breakout and Breakdown Pressures
• Pore Pressure