Tight sedimentary covers for CO₂ sequestration

Abstract

CO₂ storage at depth is a promising way to reduce the spread of greenhouse gases in the atmosphere. Obviously the sedimentary cover should ensure the sealing of reservoirs. The latter are periodically fractured to allow injection and we propose a model to predict the maximum bearable gas pressure before reinitiating these fractures in the caprock or incidentally along the interface between the reservoir and the caprock. The method is based on a twofold criterion merging energy and stress conditions. Specific conditions related to the gas pressure acting on the crack faces and the swelling of the reservoir due to the pressure rise require taking into account several terms in addition to the classical singular term that describes the state of stress at the tip of the main crack.

Appendices

Appendix 1

Let us consider $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}$ and $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{g}$ solutions to an elastic problem locally homogeneous to 0 in the vicinity of the main crack tip (Fig. 1), i.e. such that the equilibrium equation has a vanishing right hand side member and such that the boundary conditions are vanishing as well in the vicinity under consideration. Then the following integral is path independent

$$ \uppsi(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{g} ) = \frac{1}{2}\int\limits_{\varGamma } {\left[ {\underline{\underline{\sigma }} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} } \right) .\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{N} .\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{g} - \underline{\underline{\sigma }} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{g} } \right).\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{N} .\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} } \right]{\text{d}}s} $$

Here Γ is any contour starting on one face and finishing on the other face of the main crack (Fig. 1) and N the normal to this contour pointing toward the crack tip.

Then if $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}$ can be expanded in a Williams series

$$ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} \left( {x_{1} ,x_{2} } \right) = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{C} + kr^{\lambda } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} (\theta ) + \cdots $$

The Generalized Stress Intensity Factor (GSIF) k is given by (Leguillon and Sanchez-Palencia 1987; Labossiere and Dunn 1999)

$$ k = \frac{{\uppsi\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} \left( {x_{1} x_{2} } \right),\,r^{ - \lambda } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u}^{ - } (\theta )} \right)}}{{\uppsi\left( {r^{\lambda } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} (\theta ),\,r^{ - \lambda } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u}^{ - } (\theta )} \right)}} $$

where $r^{ - \lambda } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u}^{ - } (\theta )$ is the so-called dual mode to $r^{\lambda } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} (\theta )$, it plays the role of an extraction function. Indeed, in 2D if λ is solution to the eigenvalue problem with and $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} (\theta )$ as eigenmode, then $-\uplambda$ is also a solution with its own eigenvector $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u}^{ - } (\theta )$ (Leguillon and Sanchez-Palencia 1987). The primal mode has a finite energy in the vicinity of the crack tip while the dual mode has not but it plays no role since the contour Γ encompasses the crack tip at a finite distance.

Then, according to (13)

$$ \begin{aligned} k & = \frac{{\uppsi\left( {\underline{{\hat{U}}}^{0} \left( {x_{1} x_{2} } \right),\,r^{ - \lambda } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u}^{ - } (\theta )} \right)}}{{\uppsi\left( {r^{\lambda } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} (\theta ),\,r^{ - \lambda } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u}^{ - } (\theta )} \right)}}; \\ T & = \frac{{\uppsi\left( {\underline{{\hat{U}}}^{0} \left( {x_{1} x_{2} } \right),\,r^{ - 1} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t}^{ - } (\theta )} \right)}}{{\uppsi\left( {r\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} (\theta ),\,r^{ - 1} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t}^{ - } (\theta )} \right)}} \\ \end{aligned} $$

Appendix 2

Let us consider now two solutions $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}^{0}$ and $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}^{{l_{d/p} }}$ to an elastic problem, again locally homogeneous to 0 in the vicinity of the main crack tip, corresponding respectively to the initial state illustrated in Fig. 1 and to a final state embedding a crack extension (deflection or penetration) with length $l_{d/p}$ (Fig. 2). Then the change in potential energy between these two states is given by

$$ \delta W =\uppsi\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}^{{l_{d/p} }} (x_{1} ,x_{2} ),\,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}^{0} (x_{1} ,x_{2} )} \right) $$

The term $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}^{0}$ admits a Williams expansion

$$ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}^{0} \left( {x_{1} ,x_{2} } \right) = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{C} + kr^{\lambda } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} (\theta ) + \cdots $$

As a consequence of matching conditions the inner expansion of $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}^{{l_{d/p} }}$ is

$$ \begin{gathered} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}^{{l_{d/p} }} (x_{1} ,x_{2} ) = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{C} + kl_{d/p}^{\lambda } \left[ {\rho^{\lambda } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} (\theta ) + \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{V}_{d/p}^{1} (y_{1} ,y_{2} )} \right] \hfill \\ + \cdots \quad {\text{with}}\;{\text{y}}_{i} = x_{i} /l\;{\text{and}}\;\rho = r/l \hfill \\ \end{gathered} $$

where $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{V}_{d/p}^{1}$ is solution to an elastic problem posed on the unbounded domain spanned by the y _i ’s, with an homogeneous equilibrium equation, vanishing forces along the mother crack faces, a decay to 0 at infinity and prescribed forces on the two faces $\gamma_{d/p}$ (with normal $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{n}_{d/p}$) of the crack extension

$$ \underline{\underline{\sigma }} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{V}_{d/p}^{1} (y_{1} ,y_{2} )} \right) \cdot \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{n}_{d/p} = - \underline{\underline{\sigma }} \left( {\rho^{\lambda } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} (\theta )} \right) \cdot \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{n}_{d/p} $$

Equations for $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{V}_{d/p}^{2}$, $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{V}_{d/p}^{3}$ and $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{V}_{d/p}^{4}$ (see (14)) can be derived in the same way.

And then

$$ \begin{aligned} \delta W & = A_{d/p} k^{2} l^{2\lambda } + \cdots \quad {\text{with}}\;A_{d/p} \\ & { = \psi }\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{V}_{d/p}^{1} (y_{1} ,y_{2} ),\,\rho^{\lambda } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} (\theta )} \right) \\ \end{aligned} $$

The difficulty in the present analysis comes from the non homogeneous conditions due to the pressure pco ₂ acting on the main crack walls and on the crack extension and the poro-elastic behaviour in the reservoir triggering a swelling. Nevertheless these non homogeneous conditions are taken into account by the special terms (9) and (11), finally all calculations give

$$ A_{d/p} { = \psi }\left( {\underline{{\hat{V}}}_{d/p}^{1} (y_{1} ,y_{2} ),\,\rho^{\lambda } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} (\theta )} \right) $$

$$ \begin{aligned} B_{d/p} \text{ = } &\uppsi\left( {\underline{{\hat{V}}}_{d/p}^{2} (y_{1} ,y_{2} ),\,\rho^{\lambda } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} (\theta )} \right) \\ & +\uppsi\left( {\underline{{\hat{V}}}_{d/p}^{1} (y_{1} ,y_{2} ),\,\rho \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} (\theta )} \right) \\ \end{aligned} $$

$$ \begin{aligned} C_{d/p} \text{ = } &\uppsi\left( {\underline{{\hat{V}}}_{d/p}^{3} (y_{1} ,y_{2} ),\,\rho^{\lambda } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} (\theta )} \right) \\ & +\uppsi\left( {\underline{{\hat{V}}}_{d/p}^{1} (y_{1} ,y_{2} ),\,\rho \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\upsilon } (\theta )} \right) \\ & + \frac{1}{2}\int\limits_{{\gamma_{d/p} }} {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{n}_{d/p} \cdot \underline{{\hat{V}}}_{d/p}^{1} (y_{1} ,y_{2} ){\text{d}}S} \\ \end{aligned} $$

$$ D_{d/p} { = \psi }\left( {\underline{{\hat{V}}}_{d/p}^{3} (y_{1} ,y_{2} ),\,\rho \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} (\theta )} \right) $$

$$ \begin{aligned} E_{d/p} \text{ = } &\uppsi\left( {\underline{{\hat{V}}}_{d/p}^{3} (y_{1} ,y_{2} ),\,\rho \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\upsilon } (\theta )} \right) \\ & + \frac{1}{2}\int\limits_{{\gamma_{d/p} }} {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{n}_{d/p} \cdot \underline{{\hat{V}}}_{d/p}^{3} (y_{1} ,y_{2} ){\text{d}}S} \\ \end{aligned} $$

$$ \begin{aligned} F_{d/p} \text{ = } &\uppsi\left( {\underline{{\hat{V}}}_{d/p}^{2} (y_{1} ,y_{2} ),\,\rho \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\upsilon } (\theta )} \right) \\ & +\uppsi\left( {\underline{{\hat{V}}}_{d/p}^{3} (y_{1} ,y_{2} ),\,\rho \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} (\theta )} \right) \\ & + \frac{1}{2}\int\limits_{{\gamma_{d/p} }} {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{n}_{d/p} \cdot \underline{{\hat{V}}}_{d/p}^{2} (y_{1} ,y_{2} ){\text{d}}S} \\ \end{aligned} $$

Of course, under the simplifying assumption that pco ₂ does not act on the crack extension faces, the integrals on $\gamma_{d/p}$ disappear.