2. Benchmark solution

2.1. Calculation method

The reference solution is one-dimensional because it only depends on the vertical coordinate.

The water mass conservation equation \({m}_{l}\) is given by the following expression:

\[\]

: label: EQ-None

frac {d {m} _ {l}}} {mathrm {dt}} +D {M} _ {l} =0

with \({M}_{l}\) the flow of water such that, if we neglect gravity:

\({M}_{l}={\rho }_{l}{\lambda }_{l}(-\nabla {p}_{l})\) with \({p}_{l}\) the liquid pressure, \({\rho }_{l}\) its density, and \({\lambda }_{l}\) its hydraulic conductivity such as \({\lambda }_{l}=\frac{{K}_{i}}{\mu }\). \({K}_{i}\) is the intrinsic permeability and \(\mu\) the viscosity of the liquid.

Considering an undeformable solid, we can write that

\(\frac{d{m}_{l}}{\mathrm{dt}}={\rho }_{l}\frac{\phi }{{K}_{l}}\frac{d{p}_{l}}{\mathrm{dt}}\) with \(\phi\) the porosity and \({K}_{l}\) the compressibility of water. We note \(N=\frac{\phi }{{K}_{l}}\).

Equation (1) then becomes:

\({\rho }_{l}N\frac{d{p}_{l}}{\mathrm{dt}}+D({\rho }_{l}{\lambda }_{l}(-\nabla {p}_{l}))=0\)

whose variational formulation is as follows:

\[\]

: label: EQ-None

{int} _ {omega} Nfrac {d {p} _ {l}} {mathrm {dt}}mathrm {.} stackrel {} {{p} _ {l}} + {int}} + {int}} _ {omega} {lambda} _ {l}nablastackrel {}} {{p}} {{p}}} {{p}} _ {l}} =- {int}} =- {int}} _ {partialomega}}frac {{M}} ^ {mathrm {ext}}} {{rho} _ {l}}stackrel {} {{p} _ {l}}

with \({M}^{\mathrm{ext}}\) the exterior load.

To establish the solution, we use the one-dimensional case and we adopt a discretization corresponding to a single element of degree 1 since in modeling THM the hydraulic part is treated on linear elements. It is also assumed that the nonlinearities are low in this case and that the coefficients \(N\) and \({\rho }_{l}\) are constant, which implies a relatively small variation in pressure.

Let it be a linear element:

We then write the pressure based on form functions such as:

\({p}_{l}(z,t)=\sum _{i=1}^{i=2}{p}^{i}(t){\lambda }_{i}(z)\)

with

\({\lambda }_{1}(z)=\mathrm{0,5}(1+\mathrm{2z})\)

\({\lambda }_{2}(z)=\mathrm{0,5}(1-\mathrm{2z})\)

The following matrices are then introduced:

\(\left[A\right]=\left[{A}_{\mathrm{ij}}\right];{A}_{\mathrm{ij}}={\int }_{-\mathrm{0,5}}^{\mathrm{0,5}}{\lambda }_{i}{\lambda }_{j}\mathrm{dz}\)

\(\left[B\right]=\left[{B}_{\mathrm{ij}}\right];{B}_{\mathrm{ij}}={\int }_{-\mathrm{0,5}}^{\mathrm{0,5}}\frac{\partial {\lambda }_{i}}{\partial z}\frac{\partial {\lambda }_{j}}{\partial \mathrm{dz}}\mathrm{dz}\)

Which leads to

\(\left[A\right]=\frac{1}{6}\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right]\) and \(\left[B\right]=\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]\)

We then note classically:

\(\{{p}_{l}\}=\left\{\begin{array}{c}{P}_{l}^{1}\\ {P}_{l}^{2}\end{array}\right\}\) \(\{{M}^{\mathrm{ext}}\}=\left\{\begin{array}{c}{{M}^{\mathrm{ext}}}^{1}\\ {{M}^{\mathrm{ext}}}^{2}\end{array}\right\}\)

Knowing that we are injecting a flow of water Q on the upper face, we therefore have

\(\{{M}^{\mathrm{ext}}\}=\left\{\begin{array}{c}0\\ -Q\end{array}\right\}\)

Equation (2) becomes \(\frac{N}{{\lambda }_{l}}\left[A\right]\{\frac{{\mathrm{dp}}_{l}}{\mathrm{dt}}\}+\left[B\right]\{{p}_{l}\}=\frac{-1}{{\lambda }_{l}\rho }\{{M}^{\mathrm{ext}}\}\)

Considering that for short time variations Dt, the evolution of p is almost linear, we can write that

\(\{\frac{{\mathrm{dp}}_{l}}{\mathrm{dt}}\}=\frac{1}{\Delta t}\left\{\begin{array}{c}{P}_{l}^{1}-{p}_{0}^{1}\\ {P}_{l}^{2}-{p}_{0}^{2}\end{array}\right\}\)

and since we start here from an initial state of zero pressure, we will have

\(\{\frac{{\mathrm{dp}}_{l}}{\mathrm{dt}}\}=\frac{1}{\Delta t}\left\{\begin{array}{c}{P}_{l}^{1}\\ {P}_{l}^{2}\end{array}\right\}\)

Finally, we obtain the system of two equations with two unknowns:

\({p}_{l}^{1}(\frac{N}{{\lambda }_{l}\Delta t}+6)-{\mathrm{6p}}_{l}^{2}=\frac{-\mathrm{2Q}}{{\lambda }_{l}{\rho }_{l}}\)

\({p}_{l}^{2}(\frac{N}{{\lambda }_{l}\Delta t}+6)-{\mathrm{6p}}_{l}^{1}=\frac{-\mathrm{4Q}}{{\lambda }_{l}{\rho }_{l}}\)

We then have the following result:

\(\left\{\begin{array}{c}{P}_{l}^{1}\\ {P}_{l}^{2}\end{array}\right\}={\left[\begin{array}{cc}\frac{N}{{\lambda }_{l}\Delta t}+6& -6\\ -6& \frac{N}{{\lambda }_{l}\Delta t}+6\end{array}\right]}^{-1}\mathrm{.}\left\{\begin{array}{c}\frac{-\mathrm{2Q}}{{\lambda }_{l}{\rho }_{l}}\\ \frac{-\mathrm{4Q}}{{\lambda }_{l}{\rho }_{l}}\end{array}\right\}\)

2.2. Reference quantities and results

The value of the pore pressure on the lower and upper faces of the mass is tested.

2.3. Uncertainties about the solution

No uncertainty about the solution. The solution is analytical.

2.4. Bibliographical references

1. Mechanics of porous environments. O. Coussy, Editions Technip, 2000.