Abstract
In reservoir simulation, it is important to understand the mechanical behaviour of fractured rocks and the effect of shear and tensile displacements of fractures on their aperture. Tensile opening directly enhances the fracture aperture, whereas shear of a preexisting rough-walled fracture creates aperture changes dependent on the local stress state. Since fracture dilatation increases reservoir permeability, both processes must be included in a realistic and consistent manner into the mechanical reservoir simulation model. Here, we use the extended finite volume method (XFVM) to conduct flow and geomechanics simulations. In XFVM, fractures are embedded in a poroelastic matrix and are modelled with discontinuous basis functions. On each fracture segment the tractions and compressive forces are calculated, and one extra degree of freedom is added for both the shear and tensile displacement. In this particular XFVM implementation we assume that linear elasticity and steady state fluid pressure adequately constrain the effective stress. In this paper, shear dilation is not calculated a posteriori, but it enters the equations such that aperture changes directly affect the stress state. This is accomplished by adding shear dilation to the displacement gradients and therefore ascertains a consistent representation in the stress-strain relations and force balances. We illustrate and discuss the influence of this extra term in two simple test cases and in a realistic layer-restricted two-dimensional fracture network subjected to plausible in situ stress and pore pressure conditions.
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The datasets generated and analysed during the current study are available from the corresponding authors on reasonable request.
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Acknowledgements
The authors acknowledge Rajdeep Deb for providing the legacy XFVM code and thank Michael Liem, ETH Zürich and Ranit Monga, ETH Zürich for constructive discussions.
Funding
This work was supported by the Swiss National Science Foundation (SNSF) through grant SNF 200021_178922/1. The funding sources have no involvement in the study design, the preparation of the article and decisions related to its publication. Open access funding provided by Swiss Federal Institute of Technology Zurich
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Giulia Conti: Software, Formal Analysis, Conceptualization, Methodology, Writing - original draft. Patrick Jenny: Conceptualization, Methodology, Writing - review & editing, Supervision, Funding acquisition. Stephan Matthäi: Conceptualization, Methodology, Writing - review & editing, Supervision.
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Appendices
Appendix A: Convergence
To investigate the influence of the additional shear dilation coupling on the convergence, we set up another test case, where the shear opening effect is very high (\(\phi _{dil}=11.3^{\circ }\)). The test case consists of a single fracture (\(L^f={0.971}\,\textrm{km}\), \(34^{\circ }\) to principle stress) embedded in a \({3}\,\textrm{km}\times {3}\,\textrm{km}\) matrix. The boundary conditions are the same as described in Section 3.1. The rock parameters are \(\lambda ={12}\textrm{GPa}\), \(G={14.7}\textrm{GPa}\). Figure 15 shows the shear slip profiles for meshes of \(N\times N\) with \(N\in \{150,300,600\}\) at \(p^f={9.5}\,\textrm{MPa}\). The relative difference in shear displacement due to shear dilation in the middle of the fracture is \(-8.38\%\), \(-8.24\%\), and \(-7.9\%\), for \(N=150\), \(N=300\), and \(N=600\), respectively. So the refinement slightly decreases the effect of shear dilation on the shear slip.
Figure 16 shows the grid convergence for the case with and without shear opening. We are in the asymptotic limit, if the three grid spacings fulfill
where c is the slope, q the convergence rate, \(s_\tau ^{converged}\) the converged solution, and \(s_{\tau ,max}\) the shear slips in the middle of the fracture for the three grid resolutions. Using Richardson extrapolation we get \(q_{\phi =0^{\circ }}=0.898\) and \(q_{\phi =11.3^{\circ }}=1.05\). So for this specific test cases, convergence is slightly better, if the coupling due to shear opening is included.
Convergence of the simulations shown in Fig. 15. For \(\phi _{dil}=0^{\circ }\): \(q=0.898\), \(c=349.92\), and \(s_\tau ^{converged}=43.1\), whereas for \(\phi _{dil}=11.3^{\circ }\): \(q= 1.05\), \(c=579.56\), and \(s_\tau ^{converged}=39.94\)
Appendix B: Displacement and stress fields
In Fig. 17, the displacement of the matrix nodes in x- and y- directions are shown for a pressure of \(p^f={11}\,\textrm{MPa}\) in the single fracture test case of Section 3.1. Only the case with shear opening is plotted, since the difference in stress and displacement for the cases with and without shear opening is too small to be seen.
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Conti, G., Matthäi, S. & Jenny, P. XFVM modelling of fracture aperture induced by shear and tensile opening. Comput Geosci 28, 227–239 (2024). https://doi.org/10.1007/s10596-023-10214-5
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DOI: https://doi.org/10.1007/s10596-023-10214-5