- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- BIRS Workshop Lecture Videos /
- A variational phase field model of hydraulic fracturing
Open Collections
BIRS Workshop Lecture Videos
BIRS Workshop Lecture Videos
A variational phase field model of hydraulic fracturing Tanne, Erwan
Description
Since their inception in the mid-90's, variational phase-field models of fracture [1] have steadily gained popularity. One of their strengths, the ability to handle complex topologies with unknown crack paths and the interaction between multiple cracks, which is a fundamental requirement for the numerical simulation of hydraulic fracturing in complex situations. Following the technique of [2], crack propagation subject to given pressure $p$ acting along the fracture surfaces of a brittle material occupying a domain $\Omega$ is computed as the minimizer of the energy functional $$ \mathcal{E}_\ell (u,\alpha)= \int_{\Omega} \frac{1}{2} (1-\alpha)^2 \mathtt{A} e(u):e(u) \mathrm{d}x - p \int_{\Omega} \alpha \div(u) \mathrm{d}x + \frac{3G_c}{8} \int_{\Omega} \frac{\alpha}{\ell} + \ell | \nabla \alpha |^2 \mathrm{d}x $$ where $u$ denotes the displacement field, $\alpha$ the phase field representing the fracture geometry and $\mathtt{A}$ its Hooke's law. As in [2], in the case of a single pre-existing line or penny-shaped crack in an infinite medium the pressure and volume of fluid recovered from minimizers of (1) can be compared with solutions from the literature [3]. This formalism can also be used to address the issue of hydraulic stimulation of multiple cracks. Symmetry arguments are routinely used to suggest that the propagation of an infinite array of cracks of equal length in an infinite reservoir is possible. Yet a simple stability analysis reveals that this is not the case and that loss of symmetry is always energetically favored [4]. References [1] Bourdin, B., Francfort, G., and Marigo, J.-J., Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids, (2000), 48(4) 797-826 [2] Bourdin, B., Chukwudozie, C., and Yoshioka, K. (2012). A variational approach to the numerical simulation of hydraulic fracturing. In Proceedings of the 2012 SPE Annual Technical Conference and Exhibition, volume SPE 159154. [3] Sneddon, I. and Lowengrub, M. (1969). Crack problems in the classical theory of elasticity. The SIAM series in Applied Mathematics. John Wiley & Sons. [4] Tanné, E. (2017). Variational phase-field models from brittle to ductile fracture: nucleation and propagation. PhD thesis, Université Paris-Saclay, Ecole Polytechnique.
Item Metadata
| Title |
A variational phase field model of hydraulic fracturing
|
| Creator | |
| Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
|
| Date Issued |
2018-06-06T10:35
|
| Description |
Since their inception in the mid-90's, variational phase-field models of fracture [1] have steadily gained popularity. One of their strengths, the ability to handle complex topologies with unknown crack paths and the interaction between multiple cracks, which is a fundamental requirement for the numerical simulation of hydraulic fracturing in complex situations.
Following the technique of [2], crack propagation subject to given pressure $p$ acting along the fracture surfaces of a brittle material occupying a domain $\Omega$ is computed as the minimizer of the energy functional
$$
\mathcal{E}_\ell (u,\alpha)= \int_{\Omega} \frac{1}{2} (1-\alpha)^2 \mathtt{A} e(u):e(u) \mathrm{d}x - p \int_{\Omega} \alpha \div(u) \mathrm{d}x + \frac{3G_c}{8} \int_{\Omega} \frac{\alpha}{\ell} + \ell | \nabla \alpha |^2 \mathrm{d}x
$$
where $u$ denotes the displacement field, $\alpha$ the phase field representing the fracture geometry and $\mathtt{A}$ its Hooke's law.
As in [2], in the case of a single pre-existing line or penny-shaped crack in an infinite medium the pressure and volume of fluid recovered from minimizers of (1) can be compared with solutions from the literature [3].
This formalism can also be used to address the issue of hydraulic stimulation of multiple cracks. Symmetry arguments are routinely used to suggest that the propagation of an infinite array of cracks of equal length in an infinite reservoir is possible. Yet a simple stability analysis reveals that this is not the case and that loss of symmetry is always energetically favored [4].
References
[1] Bourdin, B., Francfort, G., and Marigo, J.-J., Numerical experiments in revisited
brittle fracture. J. Mech. Phys. Solids, (2000), 48(4) 797-826
[2] Bourdin, B., Chukwudozie, C., and Yoshioka, K. (2012). A variational approach
to the numerical simulation of hydraulic fracturing. In Proceedings of the 2012
SPE Annual Technical Conference and Exhibition, volume SPE 159154.
[3] Sneddon, I. and Lowengrub, M. (1969). Crack problems in the classical theory
of elasticity. The SIAM series in Applied Mathematics. John Wiley & Sons.
[4] Tanné, E. (2017). Variational phase-field models from brittle to ductile fracture:
nucleation and propagation. PhD thesis, Université Paris-Saclay, Ecole Polytechnique.
|
| Extent |
25.0
|
| Subject | |
| Type | |
| File Format |
video/mp4
|
| Language |
eng
|
| Notes |
Author affiliation: University of British Columbia
|
| Series | |
| Date Available |
2019-03-19
|
| Provider |
Vancouver : University of British Columbia Library
|
| Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
|
| DOI |
10.14288/1.0377134
|
| URI | |
| Affiliation | |
| Peer Review Status |
Unreviewed
|
| Scholarly Level |
Postdoctoral
|
| Rights URI | |
| Aggregated Source Repository |
DSpace
|
Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International