An interactive primer
Soil and rock are a three-body problem.
Squeeze a saturated clay and the water carries the load. Heat a granite and the water expands faster than the mineral cage holding it. Push water through either — through any porous medium, geological or engineered — and the heat goes along for the ride, but slower. Thermo–hydro–mechanical modelling is the arithmetic of those arguments, and this is a guided tour of it, with every figure computed live in your browser.
Live. Pressure is solved on a masked Darcy grid (grains are near-impermeable), tracers follow the resulting velocity field, and temperature is advected and diffused on the same grid — so the thermal front you see genuinely lags the fluid front. The skeleton response is illustrative; the quantitative version starts in chapter 04.
Why it refuses to be three problems
A porous medium is two materials pretending to be one. There is a skeleton — a connected cage of grains or a cemented mineral frame — and there is fluid filling the space between. Every question you can ask about such a material is a question about how those two share something: share the load, share the volume, share the heat.
That sharing is the coupling. It is not a modelling nicety bolted on at the end; it is what the material is. Push on a saturated clay and, for a while, the water carries the load and the clay does not deform at all. Only as water escapes does the skeleton take up the stress — which is why a building settles for decades after it is finished. Inject cold water into hot granite and the rock contracts, opening fractures and shifting stress far outside the cooled volume, sometimes far enough to unclamp a fault kilometres away.
THM modelling solves three balance laws — momentum for the skeleton, mass for the fluid, energy for the mixture — on the same body at the same time, because each one’s coefficients and source terms are functions of the other two’s unknowns.
Three balance laws, one body
Every THM formulation, however elaborate, is a dressing of the same three statements applied to a representative volume of the medium. The primary unknowns are the skeleton displacement $\vec{u}$, the fluid pressure $p$, and the temperature $T$.
The skeleton must hold itself up
$\nabla\!\cdot\!\boldsymbol{\sigma} + \rho\vec{g} = \vec{0}$ — quasi-static, because inertia matters only for waves and earthquakes. Unknown: displacement $\vec{u}$.
Water is neither created nor lost
What accumulates in the pores equals what flows in. Flux from Darcy’s law. Unknown: pressure $p$.
Heat is conducted, carried, and stored
Conduction through everything, advection by the moving fluid, storage in both phases. Unknown: temperature $T$.
Written out for a fully saturated medium under small strains, and stripped to the terms that matter most, the system looks like this. Hover any coloured group to see what it does.
Hover or tap a highlighted group — or an entry in the list.
- $\ten{C}\!:\!\boldsymbol{\varepsilon}$ Skeleton stiffness The elastic response of the grain frame to strain. Everything mechanical starts here.
- $\alpha\nabla p$ H → M · effective stress Pore pressure carries part of the load. Raise $p$ and the frame is unloaded — the single most consequential coupling in the set.
- $3K\alpha_T\nabla T$ T → M · thermal stress A skeleton that cannot expand answers a temperature change with stress instead — of order 0.5 MPa per kelvin in granite, an order of magnitude less in soft soil.
- $S_\varepsilon\pd{p}{t}$ Storage Fluid and pore space are compressible, so a pressure change stores or releases mass without any flow.
- $\alpha\,\dot\varepsilon_v$ M → H · pore-volume change Squeezing the skeleton squeezes fluid out. This term is what makes consolidation a diffusion process rather than an instant settlement.
- $\beta_\phi\pd{T}{t}$ T → H · thermal pressurisation Water expands roughly an order of magnitude more than the mineral frame. Heat a sealed sample and the pressure climbs — hard.
- $\tfrac{k}{\mu}\nabla p$ Darcy flux Mobility times pressure gradient. Both $k$ (via porosity, via strain) and $\mu$ (via temperature) are coupling channels in disguise.
- $(\rho c)_m$ Mixture heat capacity Volume-weighted over solid and fluid. The solid dominates, which is exactly why thermal fronts lag.
- $\vec v\!\cdot\!\nabla T$ H → T · advection Heat rides along with the water. The Péclet number measures whether this beats conduction.
- $\lambda_m\nabla T$ Conduction Through solid and fluid together; in saturated ground the solid path carries most of it.
Two couplings are missing from the display above because they are almost always negligible: mechanical dissipation heating the medium, and the thermoelastic cooling of a solid as it is pulled apart. They reappear — with numbers — in the map below.
The coupling map
Six arrows connect three physics. Some are strong enough to dominate an engineering answer, others are the kind you can defensibly drop. Knowing which is which is the modelling skill. Select an arrow.
Directed couplings between the three physics
Which coupling wins here?
Whether a coupling matters is never a property of the physics alone — it is a property of the physics at your length scale, at your time scale, in your material. Four dimensionless groups settle most arguments. Set a scenario and watch them move.
Regime calculator
How to read this
The chapters build the system one physics at a time and then put it back together. Every figure is a live computation: sliders change parameters, and the solution is re-solved, not re-scaled from a stored picture. Where a closed-form solution exists it is used; where it does not, a small solver runs in your browser and the scheme is stated in the caption.
Continuum mechanics sign convention: tension positive for stress, so a compressive vertical stress is negative and effective stress reads $\sigma' = \sigma + \alpha p$. Where geotechnical practice is quoted (compression positive) it is flagged explicitly. Strains are small, $\boldsymbol\varepsilon = \tfrac12(\nabla\vec u + \nabla\vec u^{\mathsf T})$, and the fluid is a single wetting phase unless stated otherwise.