1. Reference problem

1.1. Geometry

The field studied is composed of two \(\mathrm{0,5}m\) environments each.

Material \(\mathrm{M1}\) will be called engineered barrier (\(\mathit{BO}\)) and material \(\mathrm{M2}\) geological barrier (\(\mathit{BG}\))

1.2. Material properties

Only the properties on which the solution depends are given here, knowing that the command file contains other material data that plays no role in solving the problem at hand.

BO
Liquid water	Density (\({\mathrm{kg.m}}^{-3}\)) Viscosity	\(1000\) \({10}^{-3}\)
Homogenized parameters	Permeability \(K\) Porosity	\({10}^{-20}{m}^{2}\) \(0.3\)
Van-Genuchten Settings	\(N\) \(\mathrm{Pr}\) \(\mathrm{Sr}\) \(\mathrm{Smax}\)	\(\mathrm{1,064}\) \(\mathrm{1,5}\mathrm{Mpa}\) \(0\) \(\mathrm{0,999}\)
Initial state	Pressure	\({P}_{c}^{0}=89\mathrm{MPa}(S=0.77)\) \({P}_{\mathrm{gz}}=1\mathrm{atm}\)

BG
Liquid water	Density (\({\mathrm{kg.m}}^{-3}\)) Viscosity	\(1000\) \({10}^{-3}\)
Homogenized parameters	Permeability \(K\) Porosity	\({10}^{-19}{m}^{2}\) \(0.05\)
Van-Genuchten Settings	\(N\) \(\mathrm{Pr}\) \(\mathrm{Sr}\) \(\mathrm{Smax}\)	\(\mathrm{1,7}\) \(10\mathrm{Mpa}\) \(0\) \(\mathrm{0,999}\)
Initial state	Pressure	\({P}_{c}^{0}=0(S=1.)\) \({P}_{\mathrm{gz}}=1\mathrm{atm}\)

The saturation and permeability curves follow the Mualem-Van-Genuchten model (HYDR_VGM). It is therefore necessary to define in the materials the parameters \(N\), \(\mathrm{Pr}\),, \(\mathrm{Sr}\), \(\mathrm{Smax}\).

It should be noted that these models are:

\({S}_{\mathrm{we}}=\frac{S-{S}_{\mathrm{wr}}}{1-{S}_{\mathrm{wr}}}\) and \(m=1-\frac{1}{n}\)

\({S}_{\mathrm{we}}=\frac{1}{{\left[1+{\left[\frac{{P}_{c}}{{P}_{r}}\right]}^{n}\right]}^{m}}\)

Relative permeability to water is expressed by integrating the prediction model proposed by Mualem (1976) into Van Genuchten’s capillarity model.

\({k}_{r}^{w}=\sqrt{{S}_{\mathrm{we}}}{(1-{(1-{S}_{\mathrm{we}}^{1/m})}^{m})}^{2}\)

The permeability to gas is formulated in a similar way:

\({k}_{r}^{w}=\sqrt{(1-{S}_{\mathrm{we}})}{(1-{S}_{\mathrm{we}}^{1/m})}^{\mathrm{2m}}\)

We recall that for \(S>\mathrm{Smax}\), these curves are interpolated by a polynomial of degree 2 \(\mathrm{C1}\) in \(\mathrm{Smax}\).

1.3. Boundary and initial conditions

We are in boundary conditions: zero flow everywhere (default).

\(\mathit{BG}\) is saturated (\(S=1\)) and \(\mathit{BO}\) is partially desaturated (\(S=\mathrm{0,77}\)). In terms of capillary pressure, this results in \(\mathrm{Pc}=0\) in \(\mathrm{BG}\) and \(\mathrm{Pc}=89\mathrm{Mpa}\) in the engineered barrier.

1.4. Bibliographical references

1. Granet, S. (2006). Thermo-hydraulic test case on a bi-material: Comparison of various numerical diagrams. Note HT-64-06-012.