Thermo·Hydro·Mechanical
Processes in Porous Media

One material, three fields

Continuum theories of porous media don't track individual grains. They average over a representative elementary volume (REV) — a patch large enough to contain many pores, small enough to act like a "point" of the large-scale problem. The key averaged quantity is the porosity

$$ n \;=\; \frac{V_{\text{pores}}}{V_{\text{total}}}, \qquad e \;=\; \frac{n}{1-n} \;\; \text{(void ratio)} $$

Typical values: dense sand \(n \approx 0.3\), soft clay \(n \approx 0.5{-}0.7\), granite \(n \approx 0.01\). If water fills only part of the pore space we also track the degree of saturation \(S_w\); this primer mostly assumes full saturation (\(S_w = 1\)).

Conduction: Fourier's law

Heat flows down the temperature gradient,

$$ \mathbf{q}_T \;=\; -\,\lambda_{\mathrm{eff}}\,\nabla T $$

where \(\lambda_{\mathrm{eff}}\) is an effective conductivity of the grain–water composite (quartz ≈ 7, water ≈ 0.6, dry air ≈ 0.025 W·m⁻¹·K⁻¹ — which is why moisture content matters so much). Energy conservation gives the heat equation, extended with an advective term when pore water moves with Darcy flux \(\mathbf{q}\):

$$ (\rho c)_{\mathrm{eff}}\,\frac{\partial T}{\partial t} \;=\; \nabla\!\cdot\!\big(\lambda_{\mathrm{eff}}\nabla T\big) \;-\; \rho_f c_f\,\mathbf{q}\cdot\nabla T \;+\; Q_T $$

with the volume-averaged heat capacity \((\rho c)_{\mathrm{eff}} = n\,\rho_f c_f + (1-n)\,\rho_s c_s\).

Who wins — conduction or advection?

The Péclet number compares the two over a length \(L\):

$$ \mathrm{Pe} \;=\; \frac{\rho_f c_f\, q\, L}{\lambda_{\mathrm{eff}}} $$

\(\mathrm{Pe}\ll 1\): temperature spreads as a smooth diffusive halo. \(\mathrm{Pe}\gg 1\): a sharp thermal front marches with the flow. Rock thermal diffusivity is tiny (\(\kappa \sim 10^{-6}\,\mathrm{m^2/s}\)): the annual surface temperature wave dies out a few metres down, and a heat pulse needs decades to conduct 30 m — but flowing groundwater can carry it there in months.

Darcy's law

$$ \mathbf{q} \;=\; -\,\frac{k}{\mu}\,\big(\nabla p \;-\; \rho_f\,\mathbf{g}\big) $$

\(\mathbf{q}\) is the Darcy flux (discharge per unit cross-section, m/s); the actual water particles move faster, at the seepage velocity \(v_s = q/n\). The material property is the intrinsic permeability \(k\) (m²), a pure geometry of the pore network; dividing by viscosity \(\mu\) brings in the fluid. Hydrogeologists often lump them into the hydraulic conductivity \(K = k\rho_f g/\mu\) (m/s) and write \(q = -K\nabla h\) with head \(h\).

Permeability is the most variable material property in all of geoscience — 13 orders of magnitude separate clean gravel (\(K\sim10^{-1}\) m/s) from intact clay (\(K\sim10^{-13}\) m/s). That contrast is exactly why clay is used as a barrier around nuclear waste and beneath landfills.

Mass balance & storage

Conservation of fluid mass turns Darcy's law into a diffusion equation for pressure:

$$ S_s\,\frac{\partial h}{\partial t} \;=\; \nabla\!\cdot\!\big(K\,\nabla h\big) + Q $$

The specific storage \(S_s\) measures how much water a unit volume releases per unit head drop — by decompressing the water and by squeezing the skeleton. Storage is where hydraulics first shakes hands with mechanics.

The effective stress principle

$$ \boldsymbol{\sigma}' \;=\; \boldsymbol{\sigma} \;-\; \alpha\,p\,\mathbf{I} $$

Deformation and strength are governed by the effective stress \(\boldsymbol{\sigma}'\) carried by the grain skeleton — not by the total stress \(\boldsymbol{\sigma}\). The Biot coefficient \(\alpha\) (= 1 for soils, 0.5–0.9 for rocks) measures how efficiently pore pressure unloads the skeleton. Shear strength follows Mohr–Coulomb,

$$ \tau_f \;=\; c' \;+\; \sigma'_n \tan\varphi' $$

so raising pore pressure weakens the ground without touching the load. That one sentence explains rainfall-triggered landslides, injection-induced earthquakes and quicksand.

Equilibrium and constitutive law

Quasi-static equilibrium and small-strain kinematics close the mechanical problem:

$$ \nabla\!\cdot\!\boldsymbol{\sigma} + \rho\,\mathbf{g} = \mathbf{0}, \qquad \boldsymbol{\varepsilon} = \tfrac{1}{2}\big(\nabla\mathbf{u} + \nabla\mathbf{u}^{\mathsf T}\big), \qquad \boldsymbol{\sigma}' = \mathbb{C} : \boldsymbol{\varepsilon} $$

\(\mathbb{C}\) may be linear elasticity for small perturbations, or an elasto-plastic / critical-state model (Cam-Clay, Mohr–Coulomb) when soils yield. Density here is the bulk value \(\rho = n\rho_f + (1-n)\rho_s\).