Math 2.0

Mechanics of deformable solids Definition

Poroelasticity

Builds on Linear elasticityLamé parametersStrain tensorCauchy stress tensorDarcy's law

Definition

Consider a fluid-saturated porous solid with displacement uu of the solid skeleton and infinitesimal strain ε=12(∇u+∇uT)\varepsilon = \tfrac12 (\nabla u + \nabla u^T). Let σ\sigma be the total stress (tension positive), pp the pore pressure, and ζ\zeta the variation of fluid content per unit volume. Let λ,μ\lambda, \mu be the drained Lamé parameters of the skeleton, α∈(0,1]\alpha \in (0, 1] the Biot coefficient and M>0M > 0 the Biot modulus. Let k>0k > 0 be the permeability and ηf>0\eta_f > 0 the fluid viscosity. Linear isotropic poroelasticity is the system of the following four relations, with body force ff.

  1. Stress law: σ=λ(tr⁡ε)I+2με−αpI\sigma = \lambda (\tr \varepsilon) I + 2\mu \varepsilon - \alpha p I.
  2. Fluid content: ζ=αtr⁡ε+p/M\zeta = \alpha \tr \varepsilon + p / M.
  3. Darcy's law for the fluid flux qq relative to the solid: q=−(k/ηf)∇pq = -(k/\eta_f) \nabla p.
  4. Balance laws: ∇⋅σ+f=0\nabla \cdot \sigma + f = 0 and ∂ζ/∂t+∇⋅q=0\partial \zeta / \partial t + \nabla \cdot q = 0.

Remarks

Example: a uniform state with constant pore pressure p0p_0 and strain ε=αp03λ+2μI\varepsilon = \frac{\alpha p_0}{3\lambda + 2\mu} I has σ=0\sigma = 0 by (1). With q=0q = 0 and constant ζ\zeta, it satisfies (1) to (4) with f=0f = 0: the pore pressure inflates the free skeleton. Non-example: if α=0\alpha = 0, the pore pressure leaves the stress law and the fluid content no longer depends on the strain. Solid and fluid then decouple into plain linear elasticity and pressure diffusion, so there is no poroelastic coupling. The quantity σ+αpI\sigma + \alpha p I is Biot's effective stress, the part of the stress carried by the skeleton. When the solid grains and the fluid are incompressible compared with the skeleton, α=1\alpha = 1 and 1/M=01/M = 0.

Reading path

To understand this page from the ground up, read these 17 pages in order. The foundations come first.

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Cauchy stress tensor, concept, no evidence levelCauchy stress tensorconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptLamé parameters, concept, no evidence levelLamé parametersconceptLinear elasticity, concept, no evidence levelLinear elasticityconceptStokes flow, concept, no evidence levelStokes flowconceptStrain tensor, concept, no evidence levelStrain tensorconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptGradient, concept, no evidence levelGradientconceptDivergence, concept, no evidence levelDivergenceconceptDarcy's law, statement, no evidence levelDarcy's lawstatementPoroelasticity, definition, no evidence levelPoroelasticitydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Cauchy stress tensor concept
  2. Euclidean space object
  3. Function concept
  4. Lamé parameters concept
  5. Linear elasticity concept
  6. Stokes flow concept
  7. Strain tensor concept
  8. Total derivative concept
  9. Function of several variables concept
  10. Vector-valued function concept
  11. Differentiable function of several variables concept
  12. Partial derivative concept
  13. Vector field concept
  14. Gradient concept
  15. Divergence concept
  16. Darcy's law statement
  17. Poroelasticity definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Poroelasticity
About this page
TagY87ATBGP9T
URIm2://Y87ATBGP9T
Kinddefinition
WritingW0 stub
MSC74F10, 76S05, 74L10
DefinesPoroelasticity
UsesLinear elasticity, Lamé parameters, Strain tensor, Cauchy stress tensor, Darcy's law
Changed3 November 2026

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