2. Reference solution

The reference solution is one-dimensional because it only depends on the vertical coordinate (1D loading). The water-saturated system comes down to solving the problem of mass conservation:

\(\frac{\partial (\phi {\rho }_{l})}{\partial t}-\text{div}({K}_{\text{int}}\frac{{\rho }_{l}}{{\mu }_{l}}\nabla {P}_{l})=q\)

where q is the flow imposed on the edge.

• The liquid is incompressible: \({\rho }_{l}=c\)

• The matrix is incompressible (infinite stiffness): the porosity therefore remains constant \(\phi =\mathit{cst}\).

In this case, we therefore obtain a stationary flow which can be summed up in writing the flow:

\(-\text{div}({K}_{\text{int}}\frac{{\rho }_{l}}{{\mu }_{l}}\nabla {P}_{l})=q\)

If we write \({P}_{L}\) the pressure in \(Y=L\), we therefore obtain after discretization:

\({K}_{\text{int}}\frac{{\rho }_{l}}{{\mu }_{l}}\frac{{P}_{L}-{P}_{0}}{L}=q\)

Which works with the data entered previously: \({P}_{L}=1.1{10}^{6}\mathit{Pa}\)