Multilevel techniques lead to accurate numerical upscaling and scalable robust solvers for reservoir simulation

Max la Cour Christensen, Umberto Villa, Panayot Vassilevski

Research output: Chapter in Book/Report/Conference proceedingArticle in proceedingsResearchpeer-review


This paper demonstrates an application of element-based Algebraic Multigrid (AMGe) technique developed at LLNL (19) to the numerical upscaling and preconditioning of subsurface porous media flow problems. The upscaling results presented here are further extension of our recent work in 3. The AMGe approach is well suited for the solution of large problems coming from finite element discretizations of systems of partial differential equations. The AMGe technique from 10,9 allows for the construction of operator-dependent coarse (upscaled) models and guarantees approximation properties of the coarse velocity spaces by introducing additional degrees of freedom associated with non-planar interfaces between agglomerates. This leads to coarse spaces which maintain the specific desirable properties of the original pair of Raviart-Thomas and piecewise discontinuous polynomial spaces. These coarse spaces can be used both as an upscaling tool and as a robust and scalable solver. The methods employed in the present paper have provable O(N) scaling and are particularly well suited for modern multicore architectures, because the construction of the coarse spaces by solving many small local problems offers a high level of concurrency in the computations. Numerical experiments demonstrate the accuracy of using AMGe as an upscaling tool and comparisons are made to more traditional flow-based upscaling techniques. The efficient solution of both the original and upscaled problem is also addressed, and a specialized AMGe preconditioner for saddle point problems is compared to state-of-the-art algebraic multigrid block preconditioners. In particular, we show that for the algebraically upscaled systems, our AMGe preconditioner outperforms traditional solvers. Lastly, parallel strong scaling of a distributed memory implementation of the reservoir simulator is demonstrated.
Original languageEnglish
Title of host publicationSPE Reservoir Simulation Symposium 2015
Number of pages12
PublisherSociety of Petroleum Engineers
Publication date2015
Article numberSPE-173257-MS
Publication statusPublished - 2015
EventSPE Reservoir Simulation Symposium 2015 - Houston, TX, United States
Duration: 23 Feb 201525 Feb 2015


ConferenceSPE Reservoir Simulation Symposium 2015
CountryUnited States
CityHouston, TX


  • Modeling and Simulation
  • Geochemistry and Petrology
  • Algebra
  • Degrees of freedom (mechanics)
  • Differential equations
  • Finite element method
  • Memory architecture
  • Partial differential equations
  • Petroleum engineering
  • Petroleum reservoir evaluation
  • Porous materials
  • Software architecture
  • Approximation properties
  • Block preconditioners
  • Finite element discretizations
  • Multicore architectures
  • Non-planar interfaces
  • Numerical experiments
  • Saddle point problems
  • Systems of partial differential equations
  • Petroleum reservoir engineering

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