1. Reference problem

1.1. Geometry

In 2D, we consider a square with side \(\mathrm{1m}\). In 3D, we consider a cube with side \(\mathrm{1m}\).

1.2. Material properties

Intrinsic permeability: \(\kappa \mathrm{=}0.05{m}^{2}\mathrm{.}{\mathit{Pa}}^{\mathrm{-}1}\mathrm{.}{s}^{\mathrm{-}1}\)

Biot coefficient: \(b\mathrm{=}1.\)

Young’s module: \(E\mathrm{=}2.5\mathit{Pa}\)

Poisson’s ratio: \(\nu \mathrm{=}0.25\)

So we have the following Lamé coefficients: \(\lambda \mathrm{=}\mu \mathrm{=}1.0\text{Pa}\)

1.3. Boundary conditions and loads

The boundary conditions are the Dirichlet conditions corresponding to the available analytical solution.

In 2D, we define \(A\mathrm{=}2\mathrm{\times }{\pi }^{2}\mathrm{\times }\kappa\) and we define the mechanical volume load:

\(f(t,x,y)\mathrm{=}\left[\begin{array}{c}\mathrm{cos}(\pi x)\mathrm{sin}(\pi y)\\ \mathrm{sin}(\pi x)\mathrm{cos}(\pi y)\end{array}\right]\pi {e}^{\mathrm{-}\mathit{At}}\left[\mathrm{-}(\lambda +2\mu )+b\right]\)

In 3D, we define \(A\mathrm{=}3\mathrm{\times }{\pi }^{2}\mathrm{\times }\kappa\) and we define the mechanical volume load:

\(f(t,x,y,z)\mathrm{=}\left[\begin{array}{c}\mathrm{cos}(\pi x)\mathrm{sin}(\pi y)\mathrm{sin}(\pi z)\\ \mathrm{sin}(\pi x)\mathrm{cos}(\pi y)\mathrm{sin}(\pi z)\\ \mathrm{sin}(\pi x)\mathrm{sin}(\pi y)\mathrm{cos}(\pi z)\end{array}\right]\pi {e}^{\mathrm{-}\mathit{At}}\left[\mathrm{-}(\lambda +2\mu )+b\right]\)

1.4. Initial conditions

The initial conditions correspond to the analytical solution that is available.