Chapter 08
Reference
Symbols and units, the property ranges you need before a model can be built, the dimensionless groups that tell you which physics to keep, where all of this gets used, and where to read the real thing.
Nomenclature
| Symbol | Meaning | SI unit | Typical range |
|---|---|---|---|
| Solid displacement | m | 10⁻⁶–10⁰ | |
| Pore fluid pressure | Pa | 10⁴–10⁸ | |
| Temperature | K or °C | 0–350 °C | |
| Total stress (tension positive) | Pa | — | |
| Effective stress, | Pa | — | |
| Volumetric strain, | — | 10⁻⁶–10⁻² | |
| Porosity | — | 0.001–0.6 | |
| Intrinsic permeability | m² | 10⁻²³–10⁻⁷ | |
| Hydraulic conductivity, | m s⁻¹ | 10⁻¹³–10⁰ | |
| Fluid dynamic viscosity | Pa s | 1.8×10⁻⁴–1.8×10⁻³ | |
| Biot coefficient, | — | 0.2–1.0 | |
| Biot modulus | Pa | 10⁸–10¹¹ | |
| Skempton coefficient, | — | 0.3–1.0 | |
| Storage at constant strain | Pa⁻¹ | 10⁻¹¹–10⁻⁸ | |
| Drained bulk and shear moduli | Pa | 10⁷–10¹¹ | |
| Undrained bulk modulus, | Pa | — | |
| Linear thermal expansion of the solid | K⁻¹ | 5×10⁻⁶–4×10⁻⁵ | |
| Volumetric thermal expansion of the fluid | K⁻¹ | 2×10⁻⁴–1.4×10⁻³ | |
| Effective thermal conductivity | W m⁻¹ K⁻¹ | 0.3–6 | |
| Volumetric heat capacity of the mixture | J m⁻³ K⁻¹ | 1.2×10⁶–3.2×10⁶ | |
| Hydraulic diffusivity, | m² s⁻¹ | 10⁻⁹–10² | |
| Thermal diffusivity, | m² s⁻¹ | 2×10⁻⁷–3×10⁻⁶ | |
| Thermal pressurisation source, | K⁻¹ | 10⁻⁵–3×10⁻⁴ |
Property ranges by material
| Material | φ | k (m²) | E (GPa) | ν | α | λ (W/mK) | αT (10⁻⁶/K) |
|---|---|---|---|---|---|---|---|
| Soft clay | 0.45–0.60 | 10⁻¹⁸–10⁻¹⁶ | 0.002–0.02 | 0.35 | 1.00 | 1.2–1.8 | 10 |
| Stiff clay / shale | 0.10–0.30 | 10⁻²¹–10⁻¹⁸ | 0.5–20 | 0.25 | 0.9–1.0 | 1.1–2.1 | 10 |
| Loose sand | 0.35–0.45 | 10⁻¹²–10⁻¹⁰ | 0.01–0.08 | 0.30 | 1.00 | 1.5–2.5 | 10 |
| Sandstone | 0.05–0.30 | 10⁻¹⁷–10⁻¹² | 5–40 | 0.15–0.25 | 0.6–0.9 | 2.0–3.5 | 10 |
| Limestone | 0.02–0.25 | 10⁻¹⁸–10⁻⁹ | 20–70 | 0.20–0.30 | 0.5–0.8 | 2.2–3.4 | 8 |
| Granite (intact) | 0.003–0.02 | 10⁻²¹–10⁻¹⁸ | 40–80 | 0.20–0.30 | 0.2–0.5 | 2.5–3.4 | 8 |
| Granite (fractured) | 0.005–0.05 | 10⁻¹⁶–10⁻¹² | 15–50 | 0.25 | 0.3–0.7 | 2.5–3.2 | 8 |
| Rock salt | <0.01 | 10⁻²³–10⁻²⁰ | 25–35 | 0.25 | <0.1 | 5.0–6.0 | 40 |
| Bentonite buffer | 0.35–0.45 | 10⁻²¹–10⁻¹⁹ | 0.02–0.3 | 0.35 | 1.00 | 0.8–1.5 | 10 |
Dimensionless groups
| Group | Definition | Compares | Decision it drives |
|---|---|---|---|
| Péclet | advection vs conduction | Whether the flow field matters for heat; whether the grid must resolve a front | |
| Rayleigh (porous) | buoyancy vs diffusion | Whether free convection occurs at all () | |
| Time factor | elapsed vs consolidation time | Degree of consolidation; whether the response is drained or undrained | |
| Diffusivity ratio | thermal vs hydraulic diffusion | Whether thermal pressurisation can build up before it drains away | |
| Coupling strength | fluid stiffness vs frame stiffness | Whether a sequential solver converges; how strong Mandel–Cryer will be | |
| Thermal retardation | total vs fluid heat storage | How far the thermal front lags the tracer front | |
| Reynolds (pore) | inertia vs viscosity | Whether Darcy’s law still applies (below ≈ 1–10) | |
| Cell Péclet | advection vs diffusion per cell | Whether the advection term must be upwinded (above 2) |
Where this gets used
Deep geothermal and EGS
Cold injection into hot rock. Thermal breakthrough sets the resource life; thermal contraction opens fractures and can reactivate faults years after injection starts. All three couplings are first order, which is why geothermal drove much of the modern THM literature.
CO₂ storage
Pressure build-up limits injectivity and is bounded by the fracture pressure of the caprock; effective-stress changes on nearby faults govern the seismic hazard. Two-phase flow and a buoyant, compressible fluid make the H part harder than in water problems.
Nuclear waste disposal
The near-field problem: decay heat, an initially unsaturated bentonite buffer that swells as it wets, and a host rock whose permeability must stay low for a hundred thousand years. Thermal pressurisation of the host rock is an explicit safety-case question.
Land subsidence
Groundwater or hydrocarbon withdrawal raises effective stress and compacts the reservoir; the surface follows. Jakarta, Mexico City, the Groningen field and the San Joaquin Valley are all the same equation with different coefficients.
Slope stability and liquefaction
Rainfall infiltration raises pore pressure and lowers effective stress on a potential slip surface; cyclic loading of loose saturated sand does the same thing much faster. The unsaturated zone above the water table is where the interesting physics is.
Permafrost and ground freezing
Adds a phase change, latent heat, cryosuction that draws water toward the freezing front, and ice segregation that heaves the ground. A fully coupled THM problem with a strongly nonlinear energy equation.
Further reading
Foundations
- Terzaghi, K. (1923) — Die Berechnung der Durchlässigkeitsziffer des Tones aus dem Verlauf der hydrodynamischen Spannungserscheinungen. The effective-stress principle and the consolidation equation, in one paper.
- Biot, M. A. (1941) — General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164. The generalisation to arbitrary geometry and compressible constituents; everything in chapter 04 is here.
- Rice, J. R. & Cleary, M. P. (1976) — Some basic stress-diffusion solutions for fluid-saturated elastic porous media with compressible constituents. Rev. Geophys. 14, 227–241. The paper that made poroelastic constants intelligible.
- Detournay, E. & Cheng, A. H.-D. (1993) — Fundamentals of poroelasticity. In Comprehensive Rock Engineering, vol. 2. Still the clearest single overview, with the Mandel solution worked through.
Books
- Coussy, O. (2004) — Poromechanics. Wiley. The thermodynamically rigorous treatment; heavier going, and worth it if you need to extend the constitutive theory.
- Lewis, R. W. & Schrefler, B. A. (1998) — The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media. The standard reference for the numerics, including the unsaturated case.
- Cheng, A. H.-D. (2016) — Poroelasticity. Springer. Encyclopaedic, with an unusually complete collection of analytical solutions.
- Nield, D. A. & Bejan, A. (2017) — Convection in Porous Media, 5th ed. Springer. Everything about the thermal side, including the Horton–Rogers–Lapwood analysis behind figure 3.2.
- Zimmerman, R. W. (2000) — Coupling in poroelasticity and thermoelasticity. Int. J. Rock Mech. Min. Sci. 37, 79–87. A short, careful account of which couplings can be dropped and when.
Benchmarks and numerics
- Mandel, J. (1953) — Consolidation des sols (étude mathématique). Géotechnique 3, 287–299; and Abousleiman et al. (1996), Int. J. Solids Struct. 33, 4805–4830, for the form used in figure 5.2.
- Booker, J. R. & Savvidou, C. (1985) — Consolidation around a point heat source. Int. J. Numer. Anal. Methods Geomech. 9, 173–184. The standard THM benchmark.
- Vermeer, P. A. & Verruijt, A. (1981) — An accuracy condition for consolidation by finite elements. Int. J. Numer. Anal. Methods Geomech. 5, 1–14. Where the $\Delta t \ge h^2/(6c)$ limit comes from.
- Kim, J., Tchelepi, H. A. & Juanes, R. (2011) — Stability and convergence of sequential methods for coupled flow and geomechanics. Comput. Methods Appl. Mech. Engrg. 200. The definitive analysis of the four splitting schemes in figure 6.2.
- White, J. A. & Borja, R. I. (2008) — Stabilized low-order finite elements for coupled solid-deformation/fluid-diffusion. Comput. Methods Appl. Mech. Engrg. 197. The stabilisation used in figure 6.1.
- Rutqvist, J. (2011) — Status of the TOUGH-FLAC simulator and recent applications. Comput. Geosci. 37, 739–750. A readable account of how sequential coupling is done in practice, and where it hurts.
Codes worth knowing
- OpenGeoSys — open-source, THM/THMC, strong in repository applications.
- PorePy and PoroTomo/porepy-style DFM tools — open-source Python, mixed-dimensional fracture–matrix coupling.
- TOUGH-FLAC, FEHM, CODE_BRIGHT — established research and industry couplers, each with a different splitting strategy.
- MOOSE / Falcon — a monolithic multiphysics framework with a mature poromechanics module.
Colophon
Every figure on this site is computed in the browser at the moment you look at it. Where a closed-form solution exists — Terzaghi, Mandel, Ogata–Banks, the half-space, Kozeny–Carman, van Genuchten–Mualem, the Hashin–Shtrikman bounds — it is evaluated directly. Where one does not, a small solver runs: a masked Darcy pressure solve on the home page, a summed-area table over a synthetic grain pack in chapter 01, a mass-conservative Richards solver in chapter 02, a Darcy–Boussinesq convection solver in chapter 03, a coupled one-dimensional THM solver in chapter 05, an actual finite-element assembly in chapter 06, and a two-dimensional poro-thermo-elastic solver in chapter 07. The scheme is stated in each caption.
No frameworks, no build step, no network requests: plain HTML, CSS and JavaScript, with KaTeX vendored locally for the mathematics. Colours are drawn from a categorical palette checked for colour-vision deficiency separation and contrast in both light and dark themes; sequential fields use single-hue ramps and semantic heat scales, always with a colour bar. Every chart carries a legend, direct labels where they fit, a hover tooltip, and — where the data is tabular — a table view.
Written as an interactive primer, not a textbook. Any errors are mine; the physics is everyone’s.