Multiscale finite element methods for stochastic porous media flow equations and application to uncertainty quantification

P. Dostert, Y. Efendiev*, T. Y. Hou

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

45 Scopus citations


In this paper, we study multiscale finite element methods for stochastic porous media flow equations as well as applications to uncertainty quantification. We assume that the permeability field (the diffusion coefficient) is stochastic and can be described in a finite dimensional stochastic space. This is common in applications where the coefficients are expanded using chaos approximations. The proposed multiscale method constructs multiscale basis functions corresponding to sparse realizations, and these basis functions are used to approximate the solution on the coarse-grid for any realization. Furthermore, we apply our coarse-scale model to uncertainty quantification problem where the goal is to sample the porous media properties given an integrated response such as production data. Our algorithm employs pre-computed posterior response surface obtained via the proposed coarse-scale model. Using fast analytical computations of the gradients of this posterior, we propose approximate Langevin samples. These samples are further screened through the coarse-scale simulation and, finally, used as a proposal in Metropolis-Hasting Markov chain Monte Carlo method. Numerical results are presented which demonstrate the efficiency of the proposed approach.

Original languageEnglish (US)
Pages (from-to)3445-3455
Number of pages11
JournalComputer Methods in Applied Mechanics and Engineering
Issue number43-44
StatePublished - Aug 1 2008
Externally publishedYes

Bibliographical note

Funding Information:
We would like to thank Wuan Luo for many helpful comments and suggestions. The research is supported by DOE Grant DE-FG02-05ER25669. T.Y.H is partially supported by the NSF ITR Grant ACI-0204932 and the NSF FRG Grant DMS-0353838. The computations are performed using TAMU parallel computers funded by NSF Grant DMS-0216275. We are grateful to reviewers for their valuable comments and suggestions which helped to improve the paper.


  • Multiscale
  • Porous media
  • Two-phase flow
  • Uncertainty quantification
  • Upscaling

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • General Physics and Astronomy
  • Computer Science Applications


Dive into the research topics of 'Multiscale finite element methods for stochastic porous media flow equations and application to uncertainty quantification'. Together they form a unique fingerprint.

Cite this