Heat, Water, Stress.

A soil or rock is two things at once: a skeleton of grains or crystals that carries load, and a connected network of voids that stores and transmits fluid. Continuum theories do not follow individual pores. They average over a representative elementary volume: a window large enough that the averaged property stops fluctuating with window size, yet small compared with the distance over which pressure, temperature and stress vary.

The first number that comes out of the averaging is the porosity, the void fraction of the volume. Almost everything else is built on it: the void ratio favoured in soil mechanics, and the mixture rules that give the medium a single heat capacity and thermal conductivity.

\[ n = \frac{V_v}{V},\qquad e = \frac{n}{1-n},\qquad (\rho c)_{\mathrm{eff}} = n\,\rho_w c_w + (1-n)\,\rho_s c_s,\qquad \lambda_{\mathrm{eff}} \approx \lambda_w^{\,n}\,\lambda_s^{\,1-n} \]

Each process has a conserved quantity, a state variable that measures it, and a flux law that says how it moves. Water mass is measured by pore pressure and moves by Darcy's law. Heat is measured by temperature and moves by Fourier conduction plus whatever the water carries. Momentum is measured by displacement and strain, and is balanced by stress. Taken alone, each is a textbook problem; the couplings in §3 are what make the ground interesting.

Each field enters the other two balance equations somewhere. That gives six directed couplings, of very unequal strength. Two are so strong that no geotechnical calculation can ignore them (pore pressure on effective stress, and volume change on pore pressure). Two depend on the material and the time scale. One is almost always dropped.

Select an arrow to see the mechanism, the term it adds to the equations, and a feel for its size.

Here is the full system for a saturated, linearly thermo-poroelastic medium at small strain, in the form used by Biot, McTigue and Coussy. Sign convention in this section: stress positive in tension and pore pressure positive in compression, the continuum-mechanics habit, so the total stress is σ = σ′ − α p I. §2 and §5 use the geotechnical convention, compression positive, where the same law reads σ′ = σ − α p. Hover or tap any term to see what it does and which coupling it carries.

Three diffusivities and one pressurization coefficient summarise how the system behaves. Their ratios, not their absolute values, decide which couplings matter.

\[ \kappa = \frac{\lambda_{\mathrm{eff}}}{(\rho c)_{\mathrm{eff}}},\qquad c = \frac{k}{\mu\,S},\quad S = \frac{\alpha^2}{K_d} + \frac{\alpha - n}{K_s} + \frac{n}{K_w},\qquad \Lambda = \frac{n\,(\beta_w - \beta_s)}{S},\qquad \mathrm{Pe} = \frac{\rho_w c_w\, q\, L}{\lambda_{\mathrm{eff}}} \]

The oldest coupled problem in geotechnics. Load a saturated clay layer and the water cannot leave at once, so at first it carries the load as excess pore pressure and the skeleton feels nothing. As water drains toward the free surface the load transfers to the skeleton, which compresses, and the surface settles. Terzaghi wrote this in 1923 as a diffusion equation for the excess pressure u; it is the M → H and H → M couplings acting together, with the mechanics eliminated.

\[ \frac{\partial u}{\partial t} = c_v\,\frac{\partial^2 u}{\partial z^2},\qquad c_v = \frac{k\,E_{\mathrm{oed}}}{\mu},\qquad T_v = \frac{c_v\,t}{H^2} \]

The solution for a layer of thickness H drained at the top and sealed at the bottom is a sine series in depth that decays exponentially in the dimensionless time T_v. The average degree of consolidation U is the fraction of the final settlement reached.

\[ \frac{u}{u_0} = \sum_{m=0}^{\infty} \frac{2}{M_m}\,\sin\!\left(M_m \frac{z}{H}\right) e^{-M_m^2 T_v},\qquad U = 1 - \sum_{m=0}^{\infty}\frac{2}{M_m^2}\,e^{-M_m^2 T_v},\qquad M_m = \tfrac{\pi}{2}(2m+1) \]

Heat a saturated element of clay and forbid it to drain. The water wants to expand by β_wΔT, but the pore space that holds it only grows by about β_sΔT, an order of magnitude less. The mismatch must be taken up by compressing the water and pushing the skeleton apart, and both cost pressure. Solving the mass balance of §4 for an undrained element at constant total stress gives the undrained pressurization coefficient Λ, the pressure rise per kelvin.

\[ \Lambda = \left.\frac{\partial p}{\partial T}\right|_{\text{undrained},\,\sigma} = \frac{n\,(\beta_w - \beta_s)}{\dfrac{\alpha^2}{K_d} + \dfrac{\alpha-n}{K_s} + \dfrac{n}{K_w}} \]

Stiff, low-porosity rock has the largest Λ because nothing yields to relieve the pressure. β_w itself more than doubles between 20 and 60 °C, which is why heating tests pressurize faster as they go.

Heating a point in the ground

In the ground, drainage competes with heating. Heat spreads at the thermal diffusivity κ; excess pressure dissipates at the hydraulic diffusivity c. Booker and Savvidou solved the case of a constant point heat source of power Q in an infinite saturated medium in closed form. The temperature is the familiar continuous-source solution; the pressure is the difference of two such solutions, one spreading at κ and one at c.

\[ \Delta T(r,t) = \frac{Q}{4\pi\lambda r}\,\operatorname{erfc}\!\frac{r}{2\sqrt{\kappa t}},\qquad p(r,t) = \frac{\Lambda\,Q}{4\pi\lambda r}\;\frac{\kappa}{\kappa - c}\left[\operatorname{erfc}\frac{r}{2\sqrt{\kappa t}} - \operatorname{erfc}\frac{r}{2\sqrt{c t}}\right] \]

If c ≪ κ the pressure front lags the heat front: pressure rises almost undrained, peaks, then bleeds away over the hydraulic time scale. If c ≫ κ the bracket stays tiny and pressure never accumulates. The ratio c/κ is the single most useful number for judging whether a heated ground will pressurize.

A horizontal heater, 4 m by 1 m in section, buried in a saturated ground 24 m wide and 15 m high with groundwater flowing left to right. Temperature obeys advection–diffusion with the heater as a source; pore pressure diffuses with a source Λ ∂_tT. The far boundary stays at ambient temperature and is drained. It is the §4 system with the mechanics eliminated, solved by explicit finite differences on 96 × 60 cells of 0.25 m, in your browser, in real time.

Watch the pressure field: it appears with the heat, but it is governed by the hydraulic diffusivity, so in clay it lingers long after the temperature has settled, and in sand it never shows up at all.

Every diffusive process has one time scale, t ≈ L²/D. Because κ is nearly the same for all geomaterials while c spans twelve orders of magnitude, the thermal line is a fixed reference and the hydraulic lines fan out around it. Where a hydraulic line sits above the thermal one, heat outruns drainage and the ground pressurizes; where it sits far below, the ground stays drained and heat and water decouple.