 
Summary: PHYSICSBASED PRECONDITIONERS FOR POROUS MEDIA FLOW
APPLICATIONS
BURAK AKSOYLU, HECTOR KLIE, AND MARY F. WHEELER
Abstract. Eigenvalues of smallest magnitude have known to be a major bottleneck for iterative solvers.
Such eigenvalues become a more dramatic bottleneck when the underlying physical properties have severe
contrasts. These contrasts are commonly found subsurface geological properties such as permeability and
porosity. We intend to construct a method as algebraic as possible. In particular, we propose an algebraic
way of using the underlying permeability field to mark certain degrees of freedom as high permeable when
they exceed a certain threshold. This marking process will define a permutation matrix which allows
us to collect the degrees of freedom that causes the smallest eigenvalues in a subblock. We claim the
responsibility of illconditioning to this subblock of the system matrix. The remaining of the matrix will
then be wellconditioned if certain heuristics about the permeability field are satisfied. In our twostage
preconditioning approach, the first stage comprises the process of collecting small eigenvalues and solving
them separately; the second stage deals with the remaining of the matrix possibly with a deflation strategy
if needed. Numerical examples are shown for one and twophase flow scenarios in reservoir simulation
applications. We demonstrate that our twostage preconditioners are more effective and robust compared
to deflation methods. Due to their algebraic nature, they support flexible and realistic reservoir topology.
Keywords. Preconditioning, twostage, Schur complement, porous media flow, Krylov subspace, GMRES,
iterative solver, reservoir simulation, deflation preconditioners.
1. Introduction
