Abstract
In subsurface characterization using a history matching algorithm subsurface properties are reconstructed with a set of limited data. Here we focus on the characterization of the permeability field in an aquifer using Markov Chain Monte Carlo (MCMC) algorithms, which are reliable procedures for such reconstruction. The MCMC method is serial in nature due to its Markovian property. Moreover, the calculation of the likelihood information in the MCMC is computationally expensive for subsurface flow problems. Running a long MCMC chain for a very long period makes the method less attractive for the characterization of subsurface. In contrast, several shorter MCMC chains can substantially reduce computation time and can make the framework more suitable to subsurface flows. However, the convergence of such MCMC chains should be carefully studied. In this paper, we consider multi-MCMC chains for a single–phase flow problem and analyze the chains aiming at a reliable characterization.
F. Pereira—The research by this author is supported in part by the National Science Foundation under Grant No. DMS 1514808, a Science Without Borders/CNPq-Brazil grant and UT Dallas.
A. Rahunanthan—The research by this author is supported by the National Science Foundation under Grant No. HRD 1600818.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Brockwell, A.: Parallel Markov Chain Monte Carlo simulation by pre-fetching. J. Comput. Graph. Stat. 15(1), 246–261 (2006)
Brooks, S., Gelman, A.: General methods for monitoring convergence of iterative simulations. J. Comput. Graph. Stat. 7, 434–455 (1998)
Brooks, S., Roberts, G.: Convergence assessments of Markov Chain Monte Carlo algorithms. Stat. Comput. 8, 319–335 (1998)
Cowles, M.K., Carlin, B.: Markov Chain Monte Carlo convergence diagnostics: a comparative review. J. Am. Stat. Assoc. 91, 883–904 (1996)
Dagan, G.: Flow and Transport in Porous Formations. Springer, Heidelberg (1989). https://doi.org/10.1007/978-3-642-75015-1
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)
Ginting, V., Pereira, F., Rahunanthan, A.: Multiple Markov Chains Monte Carlo approach for flow forecasting in porous media. Procedia Comput. Sci. 9, 707–716 (2012)
Ginting, V., Pereira, F., Rahunanthan, A.: A prefetching technique for prediction of porous media flows. Comput. Geosci. 18(5), 661–675 (2014)
Lee, H., Higdon, D., Bi, Z., Ferreira, M., West, M.: Markov random field models for high-dimensional parameters in simulations of fluid flow in porous media. Technical report, Technometrics (2002)
Loève, M.: Probability Theory. Springer, Berlin (1977). https://doi.org/10.1007/978-1-4684-9464-8
Mengersen, K.L., Robert, C.P., Guihenneuc-Jouyaux, C.: MCMC convergence diagnostics: a review. In: Bernardo, M., Berger, J.O., Dawid, A.P., Smtith, A.F.M. (eds.) Bayesian Statistics, vol. 6, pp. 415–440. Oxford University Press, Oxford (1999)
Neal, R.M.: MCMC Using Hamiltonian Dynamics. Chapman and Hall/CRC Press, Boca Raton (2011)
Pereira, F., Rahunanthan, A.: Numerical simulation of two-phase flows on a GPU. In: 9th International Meeting on High Performance Computing for Computational Science (VECPAR 2010), Berkeley, June 2010
Pereira, F., Rahunanthan, A.: A semi-discrete central scheme for the approximation of two-phase flows in three space dimensions. Math. Comput. Simul. 81(10), 2296–2306 (2011)
Vrugt, J.: Markov Chain Monte Carlo simulation using the DREAM software package: theory, concepts, and MATLAB implementation. Environ. Model. Softw. 75, 273–316 (2016)
Wong, E.: Stochastic Processes in Information and Dynamical Systems. McGraw-Hill, New York (1971)
Acknowledgments
The authors would like to thank the Department of Mathematics and Computer Science of the Central State University for allowing to run the MCMC simulations on the NSF-funded CPU-GPU computing cluster.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Mamun, A., Pereira, F., Rahunanthan, A. (2018). Convergence Analysis of MCMC Methods for Subsurface Flow Problems. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2018. ICCSA 2018. Lecture Notes in Computer Science(), vol 10961. Springer, Cham. https://doi.org/10.1007/978-3-319-95165-2_22
Download citation
DOI: https://doi.org/10.1007/978-3-319-95165-2_22
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-95164-5
Online ISBN: 978-3-319-95165-2
eBook Packages: Computer ScienceComputer Science (R0)