Biot and the Effective-Stress Coefficient

Part 2, Part 2: Strain, Elasticity, and the Poroelastic Rock

Learning objectives

State Biot's refinement: for volumetric deformation, only the fraction alpha of the pore pressure offsets the total stress
Compute alpha as one minus the dry-frame over mineral bulk modulus ratio, and read its two limits
Evaluate the canon example: a 12 GPa frame on 36.6 GPa quartz carries alpha of 0.672
Choose the right coefficient for the job: alpha for deformation and stress paths, the full Terzaghi subtraction for friction and failure

How Much of the Pressure Counts?

Terzaghi's subtraction treats the pore fluid as carrying its full pressure against every normal stress, and for sliding on surfaces it does. But for the rock's volume change the bookkeeping is subtler, because the mineral grains themselves are compressible. Squeeze a rock and part of the deformation is the frame rearranging, which pore pressure resists, and part is the solid mineral itself compressing, which pore pressure cannot help with at all. Biot's theory meters the split with one number: $\alpha = 1 - K_{dry}/K_{min}$ , the effective-stress coefficient, and the deformation-governing stress becomes $\sigma - \alpha P_p$ p.

The formula reads like a story. A soft frame, $K_{dry}$ dry tiny against the mineral, deforms almost entirely by rearrangement: $\alpha \to 1$ and Terzaghi is exact, which is why soil mechanics never needed the refinement. A frame as stiff as its mineral, porosity squeezed toward nothing, leaves the pressure nothing to push on: $\alpha \to 0$ , and a tight granite barely notices its pore fluid. Every reservoir rock lives between the limits.

Dial the frame stiffness above. At the canon reservoir's $K_{dry} = 12$ dry=12 GPa against quartz's $K_{min} = 36.6$ min=36.6, the same mineral constant the Rock Physics course carries, the dial reads $\alpha = 0.672$ : about two thirds of the pore pressure participates in volumetric bookkeeping. Stiffen the frame toward a tight carbonate at 30 GPa and alpha collapses toward 0.18; soften it toward an unconsolidated sand at 2 GPa and it climbs past 0.94.

Which Coefficient, When

The practical rule the course will apply without further ceremony: friction and failure use Terzaghi's full subtraction; deformation and stress-path arithmetic use alpha. Fault slip and breakout criteria live on surfaces where the fluid presses at full strength, so Parts 3, 6, and 9 keep $\alpha = 1$ . The depletion stress path of section 2.6 and the compaction of Part 10 are volumetric stories, so alpha enters their formulas explicitly, and the canon calculations there will carry $\alpha = 1$ only because our reservoir sand is soft enough to justify it, a choice stated, not smuggled. One more distinction guards against a classic confusion: this alpha governs deformation; the Skempton coefficient of the next section governs how pore pressure responds to loading. Same theory, different questions.

References

Biot, M. A. (1941). General theory of three-dimensional consolidation. Journal of Applied Physics, 12(2), 155-164.
Wang, H. F. (2000). Theory of Linear Poroelasticity. Princeton University Press.
Jaeger, J. C., Cook, N. G. W., & Zimmerman, R. W. (2007). Fundamentals of Rock Mechanics (4th ed.). Blackwell.