Chapter 01
The porous medium
Before any of the physics can be written down, a decision has to be made: we will stop looking at grains and pores, and start pretending the material is a smooth continuum in which solid and fluid coexist at every point. That pretence is called homogenisation, and it is only legitimate under conditions worth stating carefully.
The representative elementary volume
Porosity at a point is a meaningless quantity: a point is either inside a grain or inside a pore, so $\phi$ is either $0$ or $1$. What we actually mean by “the porosity here” is the void fraction of a small volume centred here. That makes porosity a function of the averaging volume, and the whole continuum description hinges on there being a range of volumes over which the answer stops caring which volume you chose.
Drag the sampling window across the pack and widen it. Three regimes appear, and the middle one is the only place a continuum model may live.
Porosity as a function of the averaging volume
Left: a synthetic pack whose grain size drifts across the sample, so the medium is genuinely heterogeneous at the large scale. Right: the measured void fraction of the square window, evaluated exactly from a summed-area table over the rasterised pack. Below roughly one grain diameter the measurement is noise; above the sample’s correlation length it starts tracking a real trend. Between them lies the plateau — the REV.
Homogenisation is valid only when $\ell \ll L_{\text{REV}} \ll L$, where $\ell$ is the pore or grain size and $L$ the scale over which the macroscopic fields vary. Fractured rock is the classic offender: a single metre-scale fracture in an otherwise tight matrix has no REV at all, and the honest response is a discrete-fracture or dual-continuum model rather than a larger averaging volume.
Volume fractions and saturation
With an REV in hand, every phase gets a volume fraction. For a medium of porosity $\phi$ holding a wetting fluid at saturation $S_w$ and a gas at $S_g = 1 - S_w$:
Three families of quantity follow, and confusing them is a standard source of factor-of-two errors. Volumetric quantities such as bulk density $\rho = (1-\phi)\rho_s + \phi S_w \rho_w$ average over everything. Intrinsic quantities such as the fluid pressure belong to one phase and are averaged only over that phase’s volume. Ratios such as the void ratio $e = \phi/(1-\phi)$ and the gravimetric water content $w = m_w/m_s$ are what soil laboratories actually report.
Phase composition of a unit volume
Every derived index is computed from the three volume fractions on the left. Note how quickly bulk density moves: the difference between a dry and a saturated sand is several hundred kilograms per cubic metre, which is why saturation matters for slope stability long before any pore pressure is considered.
From pore scale to continuum
The continuum equations are not postulated — they are derived by averaging the pore-scale equations over the REV. Two averages are needed for each phase $\beta$: the superficial (or phase) average, taken over the whole REV, and the intrinsic average, taken over the volume that phase actually occupies.
Averaging a divergence is where the physics enters. The spatial averaging theorem says that the average of a gradient is not the gradient of the average — a surface integral over the solid–fluid interface survives, and it is exactly that integral that becomes the drag term in Darcy’s law and the interfacial heat-transfer term in a two-temperature energy balance:
Hover or tap a highlighted group.
- $\langle\nabla\psi\rangle$ Average of the pore-scale gradient What the microscopic physics actually contains.
- $\nabla\langle\psi\rangle$ Gradient of the averaged field What a continuum model can compute — the smooth field we want to solve for.
- $\int_{A}$ Interfacial term The mismatch between the two. It carries all the pore-geometry information and must be closed by a constitutive law: close it for momentum and you get Darcy’s law with a permeability; close it for energy and you get an effective conductivity, or a heat-exchange coefficient if the phases are not in equilibrium.
This is the closure problem, and it is the reason permeability is measured rather than derived. Every macroscopic coefficient in a THM model — $k$, $\lambda_m$, $\alpha$, the retention curve — is a stand-in for an interfacial integral nobody wants to evaluate.
Effective properties and their bounds
Some effective properties are exact averages. The volumetric heat capacity is one: heat is stored additively, so $(\rho c)_m = (1-\phi)(\rho c)_s + \phi (\rho c)_f$ with no geometry dependence at all. Others depend on how the phases are arranged, and can only be bounded. Thermal conductivity is the canonical example.
Layers parallel to the flux conduct in parallel — arithmetic mean, the best any arrangement can do. Layers perpendicular conduct in series — harmonic mean, the worst. These are the Wiener bounds. If the medium is additionally known to be isotropic, the tighter Hashin–Shtrikman bounds apply. Real granular materials sit inside the HS envelope, and the geometric mean $\lambda_s^{1-\phi}\lambda_f^{\phi}$ is a serviceable engineering estimate.
Effective thermal conductivity and its bounds
The spread between the bounds is small when the two phases have similar conductivities (mineral and water differ by about a factor of five) and enormous when they do not — which is exactly the situation in a dry or partially saturated soil, where air at 0.026 W m⁻¹ K⁻¹ sits beside mineral at 3. In that regime the geometry genuinely controls the answer and no mixing rule is trustworthy.
For saturated soil or rock, take $\lambda_m$ as the geometric mean and accept a few per cent of error. For unsaturated soil, measure it, or use a saturation-dependent model — the conductivity of a silt can triple between air-dry and saturated, and that alone changes a ground-source heat-pump design.
Tortuosity: geometry that survives averaging
Not all pore-geometry information is lost. Two numbers escape the averaging almost intact. Connectivity decides whether porosity is even available for flow — a pumice with 60 % porosity can be effectively impermeable if the vesicles are isolated, so we distinguish total from effective porosity. Tortuosity $\tau = (L_e/L)^2$ measures how much further a fluid particle must travel than the straight-line distance, and it appears in every transport coefficient as a divisor.
Permeability is where connectivity and tortuosity combine most consequentially, and it deserves its own treatment — which is where the next chapter starts.