2. Benchmark solution

2.1. Calculation method

It is an analytical solution. This test makes it possible to validate the discontinuity of the pressure of the massif, we will focus only on the theoretical resolution of the mass conservation equation:

\(\frac{{b}^{2}}{{E}_{0}}\frac{\partial {p}_{\mathit{lq}}(y,t)}{\partial t}-\left(\frac{{K}^{\text{int}}.{k}_{\text{lq}}^{\text{rel}}}{{\mu }_{\text{lq}}^{\text{}}}\frac{{\partial }^{2}{p}_{\text{lq}}(y,t)}{\partial {y}^{2}}\right)=0\)

Since the above differential equation is homogeneous, with constant coefficients, the method of resolution by separable variables is used (see annex 1 for the resolution of this equation).

Taking into account the initial conditions and the limits considered in paragraph 1.4, the expression of pore pressure for the left column is expressed by:

\({P}^{G}(y,t)=\frac{-{\mathrm{4F}}_{G}}{\pi b}\sum _{m=1}^{+\infty }\frac{{\left(-1\right)}^{m-1}}{\mathrm{2m}-1}{e}^{-\lambda E{\pi }^{2}{(\mathrm{2m}-1)}^{2}\frac{t}{{\mathrm{4b}}^{2}{H}^{2}}}\mathrm{cos}\left(\frac{\pi y(\mathrm{2m}-1)}{\mathrm{2H}}\right)\)

and the expression for pore pressure for the right column is expressed as:

\({P}^{D}(y,t)=\frac{-{\mathrm{4F}}_{D}}{\pi b}\sum _{m=1}^{+\infty }\frac{{\left(-1\right)}^{m-1}}{\mathrm{2m}-1}{e}^{-\lambda E{\pi }^{2}{(\mathrm{2m}-1)}^{2}\frac{t}{{\mathrm{4b}}^{2}{H}^{2}}}\mathrm{cos}\left(\frac{\pi y(\mathrm{2m}-1)}{\mathrm{2H}}\right)\)

2.2. Reference quantities and results

Pore pressure PRE1 and stress SIYY are tested at different heights in the column and at different times.

2.3. Uncertainties about the solution

None the solution is analytical.