2. Benchmark solution

2.1. Calculation method used for the reference solution

Given the symmetry of the boundary conditions, the solution is independent of \(y\). For the mechanical part, the mechanical balance of the skeleton is written as a projection on the \((\mathrm{0x})\) axis:

\(({\lambda }_{1}^{M}+2{\lambda }_{2}^{M})\frac{{\partial }^{2}{u}_{x}}{\partial {x}^{2}}-b\frac{\partial p}{\partial x}-\mathrm{rg}=0\)

where \({\lambda }_{1}^{M}\) and \({\lambda }_{2}^{M}\) designate the Lamé coefficients of the material. For the hydraulic part, the conservation of the mass of water is written

\(\frac{\partial {M}_{x}}{\partial x}=0\)

with

\(\frac{{M}_{x}}{\rho }={\lambda }^{H}(\frac{\partial p}{\partial x}-\rho g)\)

where \({\lambda }^{H}\) refers to hydraulic conductivity. The pressure is then given by the formula

\(p={P}_{0}+\rho g(L-x)\)

Horizontal displacements \({u}_{x}\) are given by

\({u}_{x}=\frac{1}{2}\frac{r-b\rho }{{\lambda }_{1}^{M}+2{\lambda }_{2}^{M}}\mathrm{gx}(x-2L)+\frac{b{P}_{0}}{{\lambda }_{1}^{M}+2{\lambda }_{2}^{M}}x\)

2.2. Benchmark results

We recall the formulas giving the Lamé coefficients as a function of the Young’s modulus and the Poisson’s ratio.

\({\lambda }_{1}^{M}=\frac{E\nu }{(1+\nu )(1-2\nu )}\) and \({\lambda }_{2}^{M}=\frac{E\nu }{2(1+\nu )}\)

The reference result is the value of the movements and the pressure at point \(P\).

2.3. Uncertainty about the solution

Analytical solution.