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Bayesian inversion for steady flow in fractured porous media with contact on fractures and hydro-mechanical coupling

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Abstract

The paper is motivated by a strong interest in numerical analysis of flow in fractured porous media, e.g., rocks in geo-engineering applications. It follows the conception of porous media as a continuum with fractures which are represented as lower dimensional objects. In the paper, the finite element discretization of the flow in coupled continuum and fractures is used; fluid pressures serve as the basic unknowns. In many applications, the flow is connected with deformations of the porous matrix; therefore, the hydro-mechanical coupling is also considered. The fluid pressure is transferred to the mechanical load in both pores and fractures and the considered mechanical model involves elastic deformations of the porous matrix and opening/closing of the fractures with the non-penetration constraint. The mechanical model with this constraint is implemented via the technique of the Lagrange multipliers, duality formulation, and combination with a suitable domain decomposition method. There is usually lack of information about problem parameters and they undergo many uncertainties coming e.g. from the heterogeneity of rock formations and complicated realization of experiments for parameter identification. These experiments rarely provide some of the asked parameters directly but require solving inverse problems. The stochastic (Bayesian) inversion is natural due to the mentioned uncertainties. In this paper, the implementation of the Bayesian inversion is realized via Metropolis-Hastings Markov chain Monte Carlo approach. For the reduction of computational demands, the sampling procedure uses the delayed acceptance of samples based on a surrogate model which is constructed during a preliminary sampling process. The developed hydro-mechanical model and the implemented Bayesian inversion are tested on two types of model inverse problems.

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Acknowledgments

This work was supported by the Czech Science Foundation (GAČR) through project No. 19-11441S and the project CZ.1.05/1.1.00/02.0070 and LQ1602 funded by the Ministry of Education, Youth and Sports of the Czech Republic. The participation of the first three authors at the CouFrac 2018 conference in Wuhan supported by Grant No. Z016001 of the State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences in Wuhan is also greatly acknowledged.

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Authors and Affiliations

  1. Institute of Geonics of the Czech Acad. Sci., Ostrava, Czech Republic

    R. Blaheta, M. Béreš, S. Domesová & D. Horák

  2. FEECS, Department of Applied Mathematics, VŠB – Technical University of Ostrava, Ostrava, Czech Republic

    M. Béreš, S. Domesová & D. Horák

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  1. R. Blaheta
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  4. D. Horák
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Correspondence to M. Béreš.

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Blaheta, R., Béreš, M., Domesová, S. et al. Bayesian inversion for steady flow in fractured porous media with contact on fractures and hydro-mechanical coupling. Comput Geosci 24, 1911–1932 (2020). https://doi.org/10.1007/s10596-020-09935-8

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