Chapter 04 · M
Poromechanics
A saturated soil or rock has two ways to carry a load, and only one of them can fail. Working out how the load is shared — and what happens when the sharing changes — is the whole of poromechanics, and it is the reason a change in water pressure can move a mountainside.
Effective stress
Terzaghi’s insight, in 1923, was that the deformation and strength of a saturated soil depend not on the total stress but on the part carried by the grain skeleton. Water fills the pores and pushes outward equally in all directions; it can carry pressure but not shear, so it supports the grains against the applied load without contributing any strength of its own.
Terzaghi used $\alpha = 1$, which is right for soils: the grains are thousands of times stiffer than the skeleton, so compressing the pore fluid does nothing to the grains themselves. Biot generalised it. In a cemented rock the frame stiffness $K$ is an appreciable fraction of the grain stiffness $K_s$, and the pore pressure also compresses the solid material; only the difference does mechanical work on the frame. Hence $\alpha = 1 - K/K_s$, which runs from about 1 in soil and weak sandstone down to 0.2–0.4 in dense granite.
The Biot coefficient that appears in the deformation law is $1-K/K_s$. The coefficient that appears in a failure criterion need not be the same number — for frictional strength it is close to 1 even in stiff rock, because failure is governed by grain-contact forces rather than bulk strain. Many codes use one $\alpha$ for both. It is usually a small error, and occasionally not.
The Biot constants and how they connect
Linear poroelasticity needs four independent constants; everything else is derived. A convenient set is the drained bulk modulus $K$, the shear modulus $G$, the Biot coefficient $\alpha$, and the Biot modulus $M$. The constitutive pair is then
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- $\ten{\sigma}$ Total stress What a load cell measures. Elastic response of the frame, minus the part the fluid is carrying.
- $\alpha p\ten{I}$ Pressure contribution Isotropic, because a fluid cannot sustain shear. This is why pore pressure moves a Mohr circle sideways but never changes its radius.
- $\zeta$ Increment of fluid content Volume of fluid added per unit bulk volume — the variable whose divergence appears in the mass balance.
- $\alpha\varepsilon_v$ Squeezed out by straining The M→H coupling: compress the frame and the fluid must go somewhere.
- $p/M$ Stored by compression $1/M = \phi/K_f + (\alpha-\phi)/K_s$ — the compressibility of the fluid plus that of the pore space at fixed frame volume.
Two further constants do most of the intuitive work. The undrained bulk modulus $K_u = K + \alpha^2 M$ is the stiffness you feel when the fluid has no time to escape — always larger than $K$, because the trapped fluid resists. And Skempton’s coefficient $B = \alpha M/K_u$ is the fraction of an applied confining stress that immediately appears as pore pressure. In a soft saturated clay $B \to 1$: load it and every pascal goes straight into the water.
Poroelastic constants explorer
All derived constants are recomputed from $K$, $G$, $K_s$, $K_f$ and $\phi$ using the standard Biot relations. The curves show how $\alpha$ and $B$ move as the frame stiffens relative to the grains — the single most useful chart for deciding whether a material will behave like Terzaghi’s soil or like a stiff, weakly coupled solid.
Drained, undrained, and the time in between
Poroelastic materials have two elastic responses and a diffusion process connecting them. Load quickly and the fluid is trapped: the response is undrained, stiff, and generates pore pressure. Wait, and the pressure dissipates by Darcy flow; the material relaxes onto its drained stiffness while the deformation grows. Nothing is viscous, yet the material creeps — the time dependence is entirely borrowed from the fluid.
The same material, two stiffnesses
Isotropic loading of a saturated sample. The undrained path is steeper by exactly $\alpha^2 M$, and the pressure generated along it is $B\,\Delta\sigma$. Release the drainage and the sample walks horizontally from the undrained line to the drained one at constant total stress — that horizontal walk is consolidation.
Failure, and the arithmetic of triggering it
Soil and rock alike fail in shear when the stress on some plane exceeds the frictional resistance, which is proportional to the effective normal stress on that plane. In Mohr–Coulomb form, $\tau = c' + \sigma'_n\tan\varphi'$. Because pore pressure enters only through $\sigma'_n = \sigma_n - \alpha p$, raising it translates the entire Mohr circle towards the failure envelope without changing its size. The deviatoric loading stays the same; only the resistance falls.
Mohr circle, pore pressure and fault reactivation
The circle is drawn in effective stress, compression positive. Raise the pore pressure and it slides left until it touches the envelope; the readout gives the remaining margin in megapascals. The inset shows the critically oriented plane at $\theta = 45^\circ + \varphi'/2$ to the minor principal stress — and, if a fault is present at a different angle, whether that fault reaches failure earlier or later.
Induced-seismicity work uses the change form directly: $\Delta\mathrm{CFS} = \Delta\tau - \mu\,(\Delta\sigma_n - \Delta p)$, positive meaning “closer to failure”. Note that all three terms are live in a THM problem: injection raises $\Delta p$, poroelastic expansion changes $\Delta\sigma_n$ even where the pressure has not arrived, and cooling adds a thermal contribution that can dominate near an injector after a few years.
What linear poroelasticity leaves out
Everything above is linear and reversible. Real geomaterials are neither. Three departures matter most in practice.
Plasticity and dilatancy
Shearing a dense sand or a well-cemented rock makes it try to expand. In undrained conditions that generates negative pore pressure and apparent strengthening; in a loose material the opposite happens and it can liquefy. The volumetric plastic strain feeds straight back into the fluid mass balance.
Damage and permeability
Microcracking increases permeability by orders of magnitude, and it is directional. A damage variable that only softens the stiffness, without updating $k$, misses the M→H feedback that matters most around excavations and along faults.
Fractures
An open fracture has its own normal and shear stiffness, its own aperture-dependent permeability ($k \propto b^2$, transmissivity $\propto b^3$), and it slips before the matrix yields. Below a critical spacing, no continuum description is defensible.