Abstract
In this paper, we study the use of virtual element method (VEM) for geomechanics. Our emphasis is on applications to reservoir simulations. The physical processes behind the formation of the reservoirs, such as sedimentation, erosion, and faulting, lead to complex geometrical structures. A minimal representation, with respect to the physical parameters of the system, then naturally leads to general polyhedral grids. Numerical methods which can directly handle this representation will be highly favorable, in particular in the setting of advanced work-flows. The virtual element method is a promising candidate to solve the linear elasticity equations on such models. In this paper, we investigate some of the limits of the VEM method when used on reservoir models. First, we demonstrate that care must be taken to make the method robust for highly elongated cells, which is common in these applications, and show the importance of calculating forces in terms of traction on the boundary of the elements for elongated distorted cells. Second, we study the effect of triangulations on the surfaces of curved faces, which also naturally occur in subsurface models. We also demonstrate how a more stable stabilization term for reservoir application can be derived.
Similar content being viewed by others
References
Ahmad, B., Alsaedi, A., Brezzi, F., Donatella Marini, L., Russo, A.: Equivalent projectors for virtual element methods. Comput. Math. Appl. 66(3), 376–391 (2013)
Andersen, O., Nilsen, H.M., Gasda, S.: Modelling geomechanical impact of co2 injection using precomputed response functions ECMOR XV – 15th European Conference on the Mathematics of Oil Recovery, pp 29–1. EAGE, Amsterdam, Netherlands (2016)
Da Veiga, L.B., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23(1), 199–214 (2013)
Da Veiga, L.B., Brezzi, F., Marini, L.D., Russo, A.: The hitchhiker’s guide to the virtual element method. Math. Models Methods Appl. Sci 24(08), 1541–1573 (2014)
Da Veiga, L.B., Brezzi, F., Donatella Marini, L.: Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51(2), 794–812 (2013)
Da Veiga, L.B., Lipnikov, K., Manzini, G.: Mimetic finite difference method for elliptic problems, vol. 11. Springer (2014)
Bergan, P.G., Nygård, M.K.: Finite elements with increased freedom in choosing shape functions. Int. J. Numer. Meth. Engng. 20(4), 643–663 (1984)
Ding, Z.: A proof of the trace theorem of sobolev spaces on lipschitz domains. Proc. Amer. Math. Soc. 124 (2), 591–600 (1996)
Gain, A.L., Talischi, C., Paulino, G.H.: On the virtual element method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes. Comput. Methods Appl. Mech. Eng. 282, 132–160 (2014)
Gringarten, EJ., Arpat, G.B., Haouesse, M.A., Dutranois, A., Deny, L., Jayr, S., Tertois, A.-L., Mallet, J.-L., Bernal, A., Nghiem, L.X.: New grids for robust reservoir modeling. SPE Annual Technical Conference and Exhibition (2008)
Lie, K.–A., Krogstad, S., Ligaarden, I.S., Natvig, J.R., Nilsen, H., Skaflestad, B.: Open-source MATLAB implementation of consistent discretisations on complex grids. Comput. Geosci. 16, 297–322 (2012)
Mallison, B., Sword, C., Viard, T., Milliken, W., Cheng, A.: Unstructured cut-cell grids for modeling complex reservoirs. SPE J. 19(2), 340–352 (2014)
The MATLAB Reservoir Simulation Toolbox, version 2016a, 7 (2016)
Open Porous Media initiative. Open datasets, 2015. http://wwww.opm-project.org
Ponting, D.K.: Corner point geometry in reservoir simulation ECMOR I-1st European Conference on the Mathematics of Oil Recovery (1989)
Acknowledgments
This work has been partially funded by the Research Council of Norway through grants no. 215641 from the CLIMIT programme.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Andersen, O., Nilsen, H.M. & Raynaud, X. Virtual element method for geomechanical simulations of reservoir models. Comput Geosci 21, 877–893 (2017). https://doi.org/10.1007/s10596-017-9636-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-017-9636-1