poroelastic

Poroelasticity is the study of porous elastic solids saturated by fluid, where mechanical deformation and fluid flow are intrinsically coupled. In these materials, changes in pore pressure modify the stress state of the solid skeleton, while deformation changes pore space and hence fluid pressure. The framework is most closely associated with Biot's theory of poroelasticity, developed in the 1940s, and is widely used in geotechnical and biomedical contexts.

Key variables include the displacement field of the solid skeleton, the pore pressure, and material properties such as porosity, hydraulic permeability, fluid viscosity, and a Biot coefficient. The effective stress principle states that the stress carried by the solid skeleton is the total stress minus the pore pressure contribution, often written as the effective stress being the total stress minus α p times the identity, where α is the Biot coefficient between 0 and 1. The solid follows a constitutive relation linking the effective stress to strain, typically σ' = C:ε − α p I, with C the elastic stiffness tensor.

Fluid flow follows Darcy's law: the Darcy flux is proportional to the negative gradient of pore pressure, q = −(k/μ)∇p. Mass conservation couples changes in fluid content to volumetric strain through the Biot modulus, yielding a diffusion-like equation for p. The system exhibits both quasi-static behavior, such as consolidation under slow loading, and dynamic behavior, including poroelastic waves, with wave speeds set by permeability, stiffness, and density.

Applications span geotechnical engineering (soil consolidation, subsidence), reservoir and groundwater engineering, hydraulic fracturing, and biomechanics (bone, cartilage, intervertebral discs). Numerical methods, especially finite element analysis, are commonly used to solve poroelastic problems, with Terzaghi’s one-dimensional consolidation as a classical limiting case.