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Fv-Mhmm: Local Adaptation Driven By An A Posteriori Error Estimator
- Publisher: European Association of Geoscientists & Engineers
- Source: Conference Proceedings, ECMOR XVI - 16th European Conference on the Mathematics of Oil Recovery, Sep 2018, Volume 2018, p.1 - 13
Abstract
Multiscale methods for simulating groundwater flow and for predicting production of large scale reservoirs have known significant breakthrough over the last decades. These approaches successively solve coarse scale problems from fine-scale local solutions and map the coarse scale solution to the fine scale. They make it possible to include petrophysical information from the fine scale, while keeping acceptable computation time. However, information exchange between coarse scale and fine scale has to be improved to deal with highly heterogeneous reservoirs.
Recently, FV-MHMM method[1] has been derived as an adaptation to the finite volume formalism of the Mixed Hybrid Multiscale Method developed in [2].
The pressure field is obtained by solving a hybrid form of the parabolic system at a coarse scale. The mathematical formulation relies on Lagrange multipliers, viewed as coarse scale fluxes, to ensure pressure continuity between coarse blocks. This paper proposes different strategies for improving the performance of FV-MHMM on heterogeneous media.
On one hand, basis functions of the global problem can be adapted to account for heterogeneities. Two approaches are proposed: a transmissivity weighted (tw) scheme and a multiscale two point flux approximation (mstpfa) scheme. This last approach uses local simulation to build a weighting scheme based on estimates of the heterogeneous fluxes across coarse faces.
On the other hand, the number of degrees of freedom (that is to say of basis functions at the coarse scale) can be increased in order to improve the solution. Such an adaptive mechanism driven by an a posteriori error estimator has been developed. It will trigger locally the division of coarse faces and, hence, the addition of degrees of freedom.
These two approaches may also be combined to further improve the FV-MHMM.
Finally, different numerical tests are presented and different strategies to improve the multiscale FV-MHMM solution are discussed. For example, a compromise has to be found between gain in accuracy and the number of degrees of freedom added.
[1] Franc, J., Jeannin, L., Debenest, G., Masson, R. “FV-MHMM method for reservoir modeling.”, 21(5–6), 895–908, Comput. Geosc., 2017
[2] Harder, C., Parades, D., Valentin, F. “A family of multiscale hybrid-mixed finite element methods for the Darcy equation with rough coefficients”, Journ. of Comput. Phys., 2013