1. Reference problem#
The test case proposed by Guiducci et al.
1.1. Geometry#
We consider a reservoir separated into two equal parts by a horizontal discontinuity \([\mathrm{EF}]\).
The coordinate \((x,y)\) is defined by the discontinuity and the edge of the structure. We note \(\alpha\) the angle between the \((x,y)\) coordinate system and the global \((X,Y)\) coordinate system.
Coordinates of the points (in meters) in coordinate system \((x,y)\):
\(x\) |
|
\(x\) |
|
||||
\(A\) |
0 |
-150 |
|
\(D\) |
0 |
150 |
|
\(B\) |
2500 |
-150 |
|
\(E\) |
0 |
0 |
|
\(C\) |
2500 |
0 |
|
\(F\) |
2500 |
0 |
1.2. Material properties#
Properties of intersticial fluid (liquid water):
Density |
\(1000{\mathrm{kg.m}}^{-3}\) |
Viscosity |
\({1.10}^{-3}\mathrm{Pa.s}\) |
Compressibility |
\({3.10}^{9}\mathrm{Pa}\) |
Properties of the rock matrix:
The matrix is elastic and has the following properties:
Young’s module |
\(200\mathrm{MPa}\) |
Poisson’s Ratio |
\(\mathrm{0,25}\) |
Porosity |
\(\mathrm{0,4055}\) |
Intrinsic permeability |
\(\mathrm{1,688}{.10}^{-17}{m}^{2}\) |
Properties of the discontinuity:
The laws of discontinuity behavior are detailed in the documentation [R7.02.15].
The mechanical behavior of the discontinuity is given by Bandis’s law.
Note \(\varepsilon\) the crack opening, \({U}_{\text{max}}\) the asymptrotic opening under zero stress, and \(U={U}_{\text{max}}-\varepsilon\) the crack closure.
In the direction normal to the crack, we have
In the tangential direction, we have
Initial normal stiffness \({K}_{\text{ni}}\) |
|
Asymptotic opening \({U}_{\mathrm{max}}\) |
|
Coefficient \(\gamma\) |
|
Tangential stiffness \({K}_{\text{t}}\) |
|
The flow in the crack is given by the cubic law.
1.3. Initial conditions#
The initial conditions are as follows:
initial opening \({\varepsilon }_{0}\): \(\mathrm{3,04}{.10}^{-4}m\)
initial pressure in the massif: \(\mathrm{48,7}\mathrm{MPa}\)
compressive stress in both directions of the plane: \(62\mathrm{MPa}\)
1.4. Boundary conditions#
The mechanical and hydraulic boundary conditions are given by figures to.
On \([\mathrm{AB}]\): movements blocked in \(y\) and zero hydraulic flow
On \([\mathrm{BC}]\): movements blocked in \(x\) and zero hydraulic flow
On \([\mathrm{CD}]\): mechanical pressure of \(62\mathrm{MPa}\) and zero hydraulic flow
On \([\mathrm{DA}]\): mechanical pressure of \(62\mathrm{MPa}\) and imposed pressure \({p}^{\text{*}}\)
Note: mechanical pressure is only applied to the solid (not to the crack)
The evolution of the pressure \({p}^{\text{*}}\) imposed by the well located in \(x=0\) is as follows (see also figure):
Linear decrease from \(\mathrm{48,7}\mathrm{MPa}\) to \(\mathrm{33,7}\mathrm{MPa}\) during the first \(\mathrm{7,5}\) years of operation
Maintains at \(\mathrm{33,7}\mathrm{MPa}\) for the following \(\mathrm{12,5}\) years.
Illustration 1.4.1: Boundary conditions and mechanical initials
Illustration 1.4.2: Boundary conditions and hydraulic initials
Figure 1.4.3: Operating scenario