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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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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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  1. Bear, J., Cheng, A.H.D.: Modeling Groundwater Flow and Contaminant Transport. No. V. 23 in Theory and Applications of Transport in Porous Media. Springer, Dordrecht (2010)

    Google Scholar 

  2. Berre, I., Doster, F., Keilegavlen, E.: Flow in fractured porous media: a review of conceptual models and discretization approaches transp porous med. (2018)

  3. Blaheta, R., Béreš, M., Domesová, S., Pan, P.: A comparison of deterministic and Bayesian inverse with application in micromechanics. Appl.Math. 63(6), 665–686 (2018).

    Article  Google Scholar 

  4. de Borst, R.: Fluid flow in fractured and fracturing porous media: a unified view. Mech. Res. Commun. 80, 47–57 (2017).

    Article  Google Scholar 

  5. Béreš, M.: Karhunen-Loéve Decomposition of isotropic Gaussian random fields using a tensor approximation of autocovariance kernel. In: High Performance Computing in Science and Engineering, vol. 11087, pp 188–202. Springer International Publishing, Cham (2018).

  6. Christen, J.A., Fox, C.: Markov chain Monte Carlo using an approximation. J. Comput. Graph. Stat. 14(4), 795–810 (2005).

    Article  Google Scholar 

  7. Cui, T., Fox, C., O’Sullivan, M.J.: Bayesian calibration of a large-scale geothermal reservoir model by a new adaptive delayed acceptance Metropolis Hastings algorithm. Water Resour. Res 47(10). (2011)

  8. Cui, T., Marzouk, Y.M., Willcox, K.E.: Data-driven model reduction for the Bayesian solution of inverse problems. Int. J. Numer. Meth. Engng 102(5), 966–990 (2015).

    Article  Google Scholar 

  9. Dodwell, T.J., Ketelsen, C., Scheichl, R., Teckentrup, A.L.: A hierarchical multilevel Markov chain Monte Carlo algorithm with applications to uncertainty quantification in subsurface flow. SIAM/ASA J. Uncertain. Quant. 3(1), 1075–1108 (2015).

    Article  Google Scholar 

  10. Domesová, S: The Use of Radial Basis Function Surrogate Models for Sampling Process Acceleration in Bayesian Inversion. In: AETA 2018 - Recent Advances in Electrical Engineering and Related Sciences: Theory and Application, vol. 554, pp 228–238. Springer International Publishing, Cham (2020).

  11. Dostál, Z.: Optimal Quadratic Programming Algorithms: with Applications to Variational Inequalities. No. 23 in Springer Optimization and Its Applications. Springer, New York (2009)

    Google Scholar 

  12. Dostál, Z., Horák, D., Kučera, R.: Total FETI-an easier implementable variant of the FETI method for numerical solution of elliptic PDE. Commun. Numer. Meth. Engng. 22(12), 1155–1162 (2006).

    Article  Google Scholar 

  13. Dostál, Z., Kozubek, T., Sadowská, M., Vondrá, V.: Scalable algorithms for contact problems. No. 36 in advances in mechanics and mathematics. Springer, New York (2016)

  14. Efendiev, Y., Hou, T., Luo, W.: Preconditioning Markov chain Monte Carlo simulations using coarse-scale models. SIAM J. Sci. Comput. 28(2), 776–803 (2006).

    Article  Google Scholar 

  15. Formaggia, L., Fumagalli, A., Scotti, A., Ruffo, P.: A reduced model for Darcy’s problem in networks of fractures. ESAIM: M2AN 48(4), 1089–1116 (2014).

    Article  Google Scholar 

  16. Franceschini, A., Ferronato, M., Janna, C., Teatini, P.: A novel Lagrangian approach for the stable numerical simulation of fault and fracture mechanics. J. Comput. Phys. 314, 503–521 (2016).

    Article  Google Scholar 

  17. Girault, V., Kumar, K., Wheeler, M.F.: Convergence of iterative coupling of geomechanics with flow in a fractured poroelastic medium. Comput. Geosci. 20(5), 997–1011 (2016).

    Article  Google Scholar 

  18. Goodman, R.E.: Methods of geological engineering in discontinuous rocks. West Pub. Co, St Paul (1976)

  19. Haslinger, J., Blaheta, R., Hrtus, R.: Identification problems with given material interfaces. J. Comput. Appl. Math. 310, 129–142 (2017).

    Article  Google Scholar 

  20. Haslinger, J., Hlaváček, I., Nečas, J.: Numerical methods for unilateral problems in solid mechanics. In: Handbook of Numerical Analysis, vol. 4, pp. 313–485. Elsevier (1996)

  21. Hintermüller, M., Rösel, S.: A duality-based path-following semismooth Newton method for elasto-plastic contact problems. J. Comput. Appl. Math. 292, 150–173 (2016).

    Article  Google Scholar 

  22. Lewis, R. W., Schrefler, B.A., Lewis, R.W.: The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media, 2nd edn. Wiley, Chichester (1998)

    Google Scholar 

  23. Martin, V., Jaffré, J., Roberts, J.E.: Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26(5), 1667–1691 (2005).

    Article  Google Scholar 

  24. Marzouk, Y., Xiu, D.: A Stochastic collocation approach to Bayesian inference in inverse problems. CiCP 6(4), 826–847 (2009).

    Article  Google Scholar 

  25. Nečas, J., Hlaváček, I.: Mathematical Theory of Elastic and Elasto-Plastic Bodies: an Introduction., Studies in Applied Mechanics, 1st edn., vol. 3. Elsevier Science, Amsterdam (1981)

    Google Scholar 

  26. Robert, C.P., Casella, G.: Monte Carlo Statistical Methods, 2. Ed., Softcover Reprint of the Hardcover 2. Ed. 2004 Edn. Springer Texts in Statistics. Springer, New York (2010)

    Google Scholar 

  27. Stuart, A.M.: Inverse problems: a Bayesian perspective. Acta Numerica 19, 451–559 (2010).

    Article  Google Scholar 

  28. Sysala, S., Haslinger, J., Hlaváček, I., Cermak, M.: Discretization and numerical realization of contact problems for elastic-perfectly plastic bodies. PART, I - discretization, limit analysis: discretization and numerical realization of contact problems for elastic-perfectly plastic bodies. Z. Angew. Math. Mech. 95(4), 333–353 (2015).

    Article  Google Scholar 

  29. Toselli, A., Widlund, O.B.: Domain Decomposition Methods - Algorithms and Theory. Springer Series in Computational Mathematics, vol. 34. Springer, Berlin (2005)

    Book  Google Scholar 

  30. White, J.A., Castelletto, N., Tchelepi, H.A.: Block-partitioned solvers for coupled poromechanics: a unified framework. Comput. Methods Appl. Mech. Eng. 303, 55–74 (2016).

    Article  Google Scholar 

  31. Witherspoon, P.A., Wang, J.S.Y., Iwai, K., Gale, J.E.: Validity of cubic law for fluid flow in a deformable rock fracture. Water Resour. Res. 16(6), 1016–1024 (1980).

    Article  Google Scholar 

  32. Wriggers, P.: Computational Contact Mechanics, 2nd edn. Springer, Berlin (2006)

    Book  Google Scholar 

  33. Zoccarato, C., Ferronato, M., Franceschini, A., Janna, C., Teatini, P.: Modeling fault activation due to fluid production: Bayesian update by seismic data. Comput Geosci 23(4), 705–722 (2019).

    Article  Google Scholar 

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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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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).

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