Summary: PHYSICS-BASED PRECONDITIONERS FOR POROUS MEDIA FLOW
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 ill-conditioning to this subblock of the system matrix. The remaining of the matrix will
then be well-conditioned if certain heuristics about the permeability field are satisfied. In our two-stage
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 two-phase flow scenarios in reservoir simulation
applications. We demonstrate that our two-stage preconditioners are more effective and robust compared
to deflation methods. Due to their algebraic nature, they support flexible and realistic reservoir topology.
Keywords. Preconditioning, two-stage, Schur complement, porous media flow, Krylov subspace, GMRES,
iterative solver, reservoir simulation, deflation preconditioners.