Computational Geosciences

, Volume 23, Issue 1, pp 107–126 | Cite as

A heterogeneous multiscale MPFA method for single-phase flows in porous media with inertial effects

  • Sergey AlyaevEmail author
  • Eirik Keilegavlen
  • Jan M. Nordbotten
Open Access
Original Paper


Upscaling of flow from pore to Darcy scale is a long-standing research field within flow in porous media. It is well known that non-linearities can occur in near-well regions and high-porosity or fractured media. At the same time, the upscaled non-linear effects associated with high flow rates are hard to quantify a priori with single-scale models. Advances in pore-scale imaging combined with increased computational have made flow simulations in small pore-scale domain feasible, but computations on domains larger than at most a few centimeters are still elusive. In this work, we present a multiscale simulation framework that automatically adapts to non-linear effects as they arise. We formulate a control volume heterogeneous multiscale method (CVHMM) by coupling of a Darcy-scale control volume method with a constitutive relation that is captured based on the fine-scale physics. While the CVHMM formulation works with arbitrary upscaled laws, we emphasize its ability to be applied in fully discrete multiscale context, in particular when a finite element solver is used for solving Navier-Stokes equations on the fine-scale pore geometry. Previous versions of CVHMM are consistent only when the coarse grid is aligned with the upscaled permeability. Herein, we generalize CVHMM by introducing a new coarse solver, thus significantly improving the applicability of the method. The presented method is applied to study flow in near-well regions, as well as media with fractures and irregular grain shapes. The examples show that the method successfully copes with general grids and pore geometries and handles flows with varying degree of non-linearities even outside the domain of applicability of classical upscaled models. In terms of computational efficiency, the method seamlessly localizes computations to regions where non-linear effects are important.


Pore scale Numerical upscaling Hybrid multiscale method Near-well flow 



SA acknowledges the Research Council of Norway and the industry partners; ConocoPhillips Skandinavia AS, BP Norge AS, Det Norske Oljeselskap AS, Eni Norge AS, Maersk Oil Norway AS, DONG Energy A/S, Denmark, Statoil Petroleum AS, ENGIE E&P NORGE AS, Lundin Norway AS, Halliburton AS, Schlumberger Norge AS, Wintershall Norge AS of The National IOR Centre of Norway for support. SA also thanks Anna Kvashchuk, Trine Solberg Mykkeltvedt and Kundan Kumar for helpful discussions during the preparation of the manuscript. The authors thank the reviewers for constructive criticism that helped to improve and better frame the paper.


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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.NORCE Norwegian Research CentreBergenNorway
  2. 2.Department of MathematicsUniversity of BergenBergenNorway
  3. 3.Department of Civil and Environmental EngineeringPrinceton UniversityPrincetonUSA

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