The coupled poroelastic problem

What an FE code has to solve once the material returns two fluxes instead of one, and what is integrated in time. The homogenized coefficients themselves — , — are a property of the microstructure, not of the coupling: see Poromechanics.

The two balances

Unknowns are the skeleton displacement and the pore pressure . Momentum balance, fluid mass balance and Darcy's law:

closed by the material, which returns both fluxes and the permeability:

What drives the microstructure

Not , and not the Biot effective stress, but the Terzaghi one. The loading splits into a dry problem under    , during which fractures open and close, plus a uniform field carrying no strain singularity — which cannot move a flat crack ([71] § 1.1). In terms of what the FE code hands over,

so the pressure term disappears for an incompressible solid ( ), as it must. Each family then follows

with the family's own contribution to the macroscopic compliance (crack_family_compliances) and the second relation the cubic (Poiseuille) law carrying the aperture into the fracture conductivity.

Discretized in time and space

Backward Euler on , unknowns at , everything dualized by a test pair :

Multiplying the mass balance by is what keeps the two equations of comparable magnitude, and makes the steady limit   the plain Darcy problem.

Newton on that pair uses exactly the four tangent blocks the material declares, plus the Darcy term:

which is mfh_poro_element!, line for line.

The four time-integration choices

	choice	why
Implicit in	backward Euler	unconditionally stable; a well test spans four decades in time and an explicit scheme would be unusable at the small end
Explicit in	the mobility is evaluated once per step from the converged state	follows the apertures through a self-consistent solve; differentiating it would couple the flow block to the mechanics through that solve, for a term of order
Sub-stepped at events	the step is split at every closure and reopening	between two events the law is exactly linear, so the tangent is exact and the answer does not depend on how the loading was subdivided
Logarithmic steps	geometric in each phase	the pressure diffuses as ; uniform steps resolve nothing early and waste everything late

Equal-order interpolation

is used for . The inf-sup condition would bite for an incompressible undrained limit; here the storage term   — the fractures are compressible even though the fluid is not — keeps the pressure block regular. A vanishing would need a Taylor–Hood pair instead.

The worked model is the ARMA 2011 well test.