1. Reference problem#
1.1. Geometry#
This one-dimensional test case comes from the literature and has an analytical solution [1]. It is inspired by the problem of consolidation of a porous soil saturated with water at room temperature (Terzaghi problem). Water is assumed to be incompressible (\(1\mathrm{/}{K}_{\mathit{lq}}\mathrm{=}\mathrm{0,0}\)). The temperature is uniform. In this case, the action of gravity is neglected. The coupled saturated poro-mechanical problem is described by the displacement variables \({u}_{y}\) (soil compaction) and fluid pressure \({P}_{\mathrm{lq}}\) (the hydraulic load being \({P}_{\mathrm{lq}}/{\rho }_{\mathrm{eau}}\)) variables. In order to obtain a purely one-dimensional solution, the Poisson’s ratio is chosen to be equal to \({\nu }_{0}=\mathrm{0,0}\). The dimensions are as follows: \(L=\mathrm{1,00}m\), \(H=\mathrm{10,00}m\).
Y
F0 (t)
C
D
X
H
L A B
Drained Young’s Modulus: \({E}_{0}=10\mathrm{MPa}\) |
Intrinsic Permeability: \({K}_{\mathrm{intr}}={\mathrm{10x10}}^{-8}\) |
Poisson’s ratio: \({v}_{0}=\mathrm{0,0}\) |
Fluid density: \({\rho }_{\mathrm{lq}}=1000\mathrm{kg}/{m}^{3}\) |
Density: \({r}_{0}=2800\mathrm{kg}/{m}^{3}\) |
Porosity: \({\phi }^{0}=\mathrm{0,5}\) |
Biot coefficient: \(b=\mathrm{1,0}\) |
Dynamic viscosity of water: \({\mu }_{\mathrm{lq}}(T)=1\) |
Saturation \({S}_{\mathrm{lq}}({p}_{c})=\mathrm{1,0}\) |
Relative fluid permeability: \({k}_{\mathrm{lq}}^{\mathrm{rel}}({S}_{\mathrm{lq}})=1\) |
Behavioral and thermal coupling characteristics are not significant.
The hydraulic permeability of the medium to water is then: \({\lambda }_{\mathrm{lq}}^{H}=\frac{{K}_{\mathrm{intr}}(\varphi )\mathrm{.}{k}_{\mathrm{lq}}^{\mathrm{rel}}({S}_{\mathrm{lq}})}{{\mu }_{\mathrm{lq}}(T)}\) in (\({m}^{3}s/\mathrm{kg}\)).
In soil mechanics, permeability is often denoted by \(k={\lambda }_{\mathrm{lq}}^{H}{\rho }_{\mathrm{lq}}g\), or here: \(k\approx {10}^{-14}m/s\).
The consolidation coefficient \({c}_{\nu }={\lambda }_{\mathrm{lq}}^{H}{E}_{0}/{b}^{2}\) applies here: \({c}_{\nu }=\mathrm{0,1}{m}^{2}/s\).
1.2. Boundary conditions and loads#
1.2.1. Boundary conditions#
The interstitial fluid pressure remains zero on the entire upper side \(\mathit{CD}\): \({P}_{\mathit{lq}}=0\). The lateral faces have movements blocked in \(x\). The underside \(\mathit{AB}\) has movements blocked in \(x\) and \(y\), and it is waterproof: \({P}_{\mathrm{lq},y}=0\)
1.2.2. Initial conditions#
The column is initially at rest in a blank state: \({P}_{\mathrm{lq}}=0\), \({\sigma }_{\mathrm{yy}}=0\).
1.2.3. Loading#
A pressure step \({F}_{0}=-\mathrm{1,0}\mathrm{Pa}\) is exerted on the upper side CD to \(t=\mathrm{0s}\).
Gravity is neglected here.