A Multidimensional Markov Chain Model for Simulating Stochastic Permeability Conditioned by Pressure Measures


  • S Zein
  • V Rath
  • C Clauser




In this paper, we are interested in simulating a stochastic permeability distribution constrained by some pressure measures coming from a steady flow (Poisson problem) over a two-dimensional domain. The permeability is discretized over a regular rectangular gird and considered to be constant by cell but it can take randomly a finite number of values. When such permeability is modeled using a multidimensional Markov chain, it can be constrained by some permeability measures. The purpose of this work is to propose an algorithm that simulates stochastic permeability constrained not only by some permeability measures but also by pressure measures at some points of the domain. The simulation algorithm couples the MCMC sampling technique with the multidimensional Markov chain model in a Bayesian framework.


Z. Chen and J. Zou, "An Augmented Lagrangian Method for Identifying Discontinuous Parameters in Elliptic Systems", SIAM Journal on Control and Optimization Volume 37, Issue 3 (1999). https://doi.org/10.1137/s0363012997318602

W. Li, "Markov chain random fields for estimation of categorical variables", Mathematical Geology, vol. 39, no 3, pp. 321-335 (2007). https://doi.org/10.1007/s11004-007-9081-0

A. G. Journel, "Combining knowledge from diverse sources: An alternative to traditional data independence hypotheses", Mathematical geology, vol. 34, no 5, pp. 573-596 (2002).

A. Elfeki and M. Dekking, "A Markov chain model for subsurface characterization: Theory and applications", Mathematical geology, vol. 33, no 5, pp. 569-589 (2001). https://doi.org/10.1023/a:1011044812133

W. Li, "A fixed-path Markov chain algorithm for conditional simulation of discrete spatial variables", Mathematical Geology, vol. 39, no 2, pp. 159-176 (2007). https://doi.org/10.1007/s11004-006-9071-7

C. Robert and G. Casella, "Monte Carlo Statistical Methods", Springer-Verlag (2004).

A. Tarantola, "Inverse Problem Theory and Model Parameter Estimation." SIAM (2005).

J. P. Chilï¿ and P. Delfiner, "Geostatistics: Modeling Spatial Uncertainty", Wiley (1999).

J. R. Norris, "Markov Chains", Cambridge Series in Statistical and Probabilistic Mathematics (1998).

S. F. Carle and G. E. Fogg, "Modeling spatial variability with one and multidimensional continuous-lag Markov chains", Mathematical Geology, 29(7), 891-917, 1997.



How to Cite

Zein, S., Rath, V. and Clauser, C. (2010) “A Multidimensional Markov Chain Model for Simulating Stochastic Permeability Conditioned by Pressure Measures”, The International Journal of Multiphysics, 4(4), pp. 359-374. doi: 10.1260/1750-9548.4.4.359.