1. Reference problem#

1.1. Geometry of the 2D problem (A and B modeling)#

Let’s say a floor column with a length \(L=\mathit{LX}\) and a height \(H=\mathrm{10m}\). In \(X={L}_{d}\), this column has an interface-type discontinuity (a non-meshed interface that is introduced into the model through level-sets thanks to the DEFI_FISS_XFEM operator). The bar is entirely crossed by the discontinuity (in terms of approximating the displacement fields and pore pressure of the massif, only Heaviside enrichment is taken into account).

The temperature within the column is uniform regardless of instant \(t\). The column is also fully saturated by a fluid (water for example) and the effects of gravity forces are not taken into account.

In order to have a one-dimensional solution (according to the \(y\) direction of the space coordinate system) the Poisson’s ratio is taken to be zero.

The geometry of the floor column is shown.

_images/1000020100000332000002CF8B66390430F37F15.png

Figure1.1-1:2D Problem Geometry

1.2. Geometry of the 3D problem (C and D models)#

Let’s say a floor column with a length \(L=\mathit{LX}\), a thickness \(E=\mathrm{1m}\) and a height \(H=\mathrm{10m}\). In \(X={L}_{d}\), this column has an interface-type discontinuity (a non-meshed interface that is introduced into the model through level-sets thanks to the DEFI_FISS_XFEM operator). The bar is entirely crossed by the discontinuity (in terms of approximating the displacement fields and pore pressure of the massif, only Heaviside enrichment is taken into account).

The temperature within the column is uniform regardless of instant \(t\). The column is also fully saturated by a fluid (water for example) and the effects of gravity forces are not taken into account.

In order to have a one-dimensional solution (according to the \(z\) direction of the space coordinate system) the Poisson’s ratio is taken to be zero.

The geometry of the floor column is shown.

_images/1000020100000324000002CE5F93DCBF36023B14.png

Figure1.2-1: Geometry of the 3D problem

1.3. Material properties#

The parameters given in the Table correspond to the parameters used for modeling in the hydromechanical coupled case. The coupling law used is” LIQU_SATU “.

Liquid (water)

Viscosity \({\mu }_{w}(\mathit{en}\mathit{Pa.s})\): Compressibility module \(\frac{1}{{K}_{w}}(\mathit{en}{\mathit{Pa}}^{\text{-1}})\): Liquid density \({\rho }_{w}(\mathit{en}\mathit{kg}\mathrm{/}{m}^{3})\): Relative fluid permeability \({k}_{\text{lq}}^{\text{rel}}\left({S}_{\text{lq}}\right)\):

\({10}^{\text{-3}}\) \(0\) \(1000\) \(1\)

Elastic parameters

Drained Young’s Modulus \(E(\mathit{en}\mathit{MPa})\): Poisson’s Ratio \(\nu\):

\(10\) \(0\)

Coupling parameters

Biot coefficient \(b\): Initial homogenized density \({r}_{0}(\mathit{en}\mathit{kg}\mathrm{/}{m}^{3})\): Intrinsic permeability \({K}^{\text{int}}(\mathit{en}{m}^{2}/s)\):

\(1\) \(2800\) \({10}^{\text{-8}}\)

Table 1.3-1 : Material Properties

The porosity of the material is taken to be equal to \(\varphi =\mathrm{0,5}\).

1.4. Boundary conditions, initial conditions and loads#

1.4.1. Boundary conditions case 2D#

The movements are blocked on the faces [AD] and [BC] in the horizontal direction, and on the lower face [AB] in both the vertical and horizontal directions. On the interface, horizontal movements are blocked.

The pore pressure at the top of the column is zero regardless of the instant in question, that is to say \(p\left(H,t\right)=0.0\).

1.4.2. Boundary conditions: 3D case#

The movements are blocked on the face [ABFE]. The movements according to (0x) are blocked on the [EADH], [FBCG] faces as well as on the interface. The movements according to (Oy) are blocked on the faces [ABCD] and [EFGH].

The pore pressure at the top of the column is zero regardless of the instant in question, that is to say \(p\left(H,t\right)=0.0\).

1.4.3. Loads and initial conditions#

In order to start from a different load on either side of the fracture, the aim is to initially create a discontinuity of the pressure field on the roof of the column in the column. Thus for the part of the column located on the left, the imposed load is \({F}_{G}=-1.0\mathit{Pa}\) and for the part of the column located on the right, the imposed load is \({F}_{D}=-1.54\mathit{Pa}\).

The initial pressure conditions for hydraulic equilibrium are therefore written as:

  • \({p}^{G}(y,0)=-\frac{{F}_{G}}{b}=1.0\mathit{Pa}\)

  • \({p}^{D}(y\mathrm{,0})=-\frac{{F}_{D}}{b}=1.54\mathit{Pa}\)

Boundary conditions are summarized on:

_images/1000000000000239000002D2AC365005E66B39BC.png

Figure 1.4.3-1Boundary and initial conditions for 2D modeling

The fracture is shown in red on the. No boundary conditions are applied on this interface.

1.4.4. Note on the modeling used#

Given the discontinuity of the pressure field at the top of the column [V7.30.100], we note that:

\(\{\begin{array}{c}p(y,0)=-\frac{{F}_{0}}{b}\mathit{si}y<H\\ p(y,0)=0\mathit{si}y=H\end{array}\)

This particularity of the solution confers instability in the numerical resolution (occurrence of oscillations) of the coupled problem at the top of the column. This is linked to the name compliance with condition LBB [V7.30.100].

Indeed, in the classical case, D_ PLAN_HMD modeling is used to achieve this. However, in HM- XFEM only the extension of the D_ PLAN_HM modeling was carried out. The results (for the case HM- XFEM) obtained at the top of the column should therefore be taken with care. It is in fact observed that the results obtained with the HM-XFEM model are less accurate at the top of the column, especially for short periods of time. But these results are similar to those obtained with D_ PLAN_HM modeling in the classical case.