Thermo-Hydro-Mechanical Processes in Porous Media

01 — Foundations

What is a porous medium?

Zoom in on a grain of sandstone and you see solid mineral, void space, and fluid in a tangled microstructure. We cannot track every pore. Instead we average over a small volume large enough to be statistically meaningful but small compared to the structure — the Representative Elementary Volume (REV). Properties like porosity become smooth, well-defined fields.

Finding the REV Grow the averaging window and watch porosity settle

Window size small

—measured porosity φ

Porosity

The fraction of the bulk volume that is void:

$$ \phi = \frac{V_\text{void}}{V_\text{total}}, \qquad 0 < \phi < 1 $$

At tiny scales the window sits inside a single grain (φ → 0) or a single pore (φ → 1). The reading swings wildly. Enlarge the window past the REV and it converges to a stable bulk value — the scale at which continuum equations become valid.

The three phases & three physics

solid skeleton bears stress, deforms, conducts heat
pore fluid stores & transmits pressure, flows, advects heat
thermal field shared by both, drives expansion & flow

Each phase obeys a conservation law; the couplings between them are what make THM rich — and the subject of everything below.

02 — The big picture

The coupling triangle

Three processes, six one-way couplings. Hover or tap an arrow to see how one field drives another — and the exact term it adds to the governing equations.

03 — Thermal

Heat in a porous solid

Heat moves by conduction through the solid–fluid mixture and is carried bodily by moving fluid (advection). Energy is conserved.

Fourier's law

Heat flux flows down the temperature gradient:

$$ \mathbf{q}_h = -\,\lambda_\text{eff}\,\nabla T $$

$\lambda_\text{eff}$ is the effective conductivity of the saturated mixture — a blend of solid and fluid conductivities weighted by porosity.

Energy conservation

Storage = conduction in − advected out + sources:

$$ (\rho c)_\text{eff}\,\frac{\partial T}{\partial t} \;+\; (\rho c)_f\,\mathbf{q}\!\cdot\!\nabla T \;=\; \nabla\!\cdot\!\big(\lambda_\text{eff}\nabla T\big) \;+\; Q $$

$(\rho c)_\text{eff}\,\partial_t T$ heat stored in skeleton + fluid
$(\rho c)_f\,\mathbf{q}\!\cdot\!\nabla T$ heat carried by Darcy flux → H→T coupling
$\nabla\!\cdot(\lambda_\text{eff}\nabla T)$ conduction
$Q$ internal sources (reactions, decay heat…)

Transient heat diffusion Click & drag on the field to inject heat

Thermal diffusivity

A simple explicit finite-difference solution of $\partial_t T = D\,\nabla^2 T$.

04 — Hydraulic

Fluid flow through pores

A pressure difference drives fluid through the tortuous pore network. At the continuum scale the messy microflow collapses into one elegant statement: Darcy's law.

Darcy flow around a low-permeability lens Tracers follow the seepage field; colour = pressure

Lens permeability Pressure gradient

Pressure solves $\nabla\!\cdot(k\,\nabla p)=0$; velocity is $\mathbf{q}=-\tfrac{k}{\mu}\nabla p$.

Darcy's law

The volumetric flux (specific discharge) is proportional to the pressure gradient:

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

$k$ intrinsic permeability (geometry of the pores)
$\mu$ fluid viscosity (falls as $T$ rises → T→H)
$\rho_f\,\mathbf{g}$ buoyancy / gravity drive

Mass conservation & storage

Combine fluid mass balance with Darcy's law to get a pressure-diffusion equation:

$$ S\,\frac{\partial p}{\partial t} \;+\; \alpha\,\frac{\partial \varepsilon_v}{\partial t} \;=\; \nabla\!\cdot\!\Big(\tfrac{k}{\mu}\nabla p\Big) $$

$S$ storage coefficient (fluid + pore compressibility)
$\alpha\,\partial_t \varepsilon_v$ skeleton volume change feeds pressure → M→H coupling

05 — Mechanical

Deformation & effective stress

The skeleton carries load and deforms. The decisive idea of poromechanics: the grains feel only the effective stress — total stress minus the pressure the pore fluid already supports.

The effective stress principle

Pore pressure carries part of the total load; the rest is borne by grain contacts:

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

$\boldsymbol{\sigma}'$ effective stress — controls strength & deformation
$\alpha\,p$ pressure support → H→M coupling
$\alpha = 1 - K/K_s$ Biot coefficient ($0<\alpha\le 1$)

Constitutive & equilibrium

Linear thermo-poro-elasticity links effective stress to strain and temperature:

$$ \boldsymbol{\sigma}' = \mathbb{C}:\boldsymbol{\varepsilon} - 3K\beta_s\,\Delta T\,\mathbf{I} $$

and quasi-static momentum balance closes the system:

$$ \nabla\!\cdot\!\boldsymbol{\sigma} + \rho\,\mathbf{g} = \mathbf{0} $$

The $\Delta T$ term is the T→M coupling: heat the skeleton and it expands.

Load sharing: total = effective + pore pressure Drag the load; toggle drainage

Applied total stress σ

Undrained: a sudden load is taken almost entirely by the trapped fluid (σ′ barely moves). Drainage lets pressure dissipate and transfers load to the skeleton.

06 — Coupling in action

Terzaghi consolidation

The textbook H–M coupling. Load a saturated clay layer: the fluid takes the load instantly as excess pore pressure, then slowly drains away, handing the load to the skeleton, which settles. The pressure obeys a diffusion equation — the same mathematics as heat.

Saturated layer

Excess pore-pressure isochrones

Degree of consolidation

Time factor $T_v=c_v t/H^2$ 0.00

0%consolidation U

Excess pressure: $\;u/u_0 = \sum_{m=0}^{\infty}\tfrac{2}{M}\sin(M Z)\,e^{-M^2 T_v}$, with $M=\tfrac{\pi}{2}(2m{+}1)$, $Z=z/H$. Double drainage (top & bottom).

07 — The full system

The coupled THM equations

Three conservation laws, three unknown fields — displacement $\mathbf{u}$, pressure $p$, temperature $T$ — knitted together by the coupling terms highlighted in colour.

M

Momentum balance

$$ \nabla\!\cdot\!\big(\mathbb{C}:\boldsymbol{\varepsilon} \;{\color{#59e0ff}-\;\alpha p\,\mathbf{I}} \;{\color{#ffb838}-\;3K\beta_s\Delta T\,\mathbf{I}}\big) + \rho\mathbf{g} = \mathbf{0} $$

Deformation driven by loads, pore pressure, and thermal expansion.

H

Fluid mass balance

$$ S\frac{\partial p}{\partial t} \;{\color{#a3e635}+\;\alpha\frac{\partial \varepsilon_v}{\partial t}} \;{\color{#ffb838}-\;3\alpha_m\frac{\partial T}{\partial t}} = \nabla\!\cdot\!\Big(\tfrac{k}{\mu}\nabla p\Big) $$

Pressure fed by storage, skeleton compaction, and thermal pressurization; relaxed by Darcy flow.

T

Energy balance

$$ (\rho c)_\text{eff}\frac{\partial T}{\partial t} \;{\color{#59e0ff}+\;(\rho c)_f\,\mathbf{q}\!\cdot\!\nabla T} = \nabla\!\cdot\!\big(\lambda_\text{eff}\nabla T\big) + Q $$

Heat stored and conducted, plus advection by seepage.

Closing relations

Strain–displacement

$ \boldsymbol{\varepsilon} = \tfrac{1}{2}(\nabla\mathbf{u}+\nabla\mathbf{u}^{\mathsf T}) $

Biot coefficient

$ \alpha = 1 - K/K_s $

Storage

$ S = \dfrac{\phi}{K_f} + \dfrac{\alpha-\phi}{K_s} $

Darcy flux

$ \mathbf{q} = -\tfrac{k}{\mu}(\nabla p - \rho_f\mathbf{g}) $

Solve these together — usually by the finite element method — and you predict how a dam, a geothermal reservoir, a nuclear-waste repository, or a settling foundation responds over time.