We present a new practical constrained pressure residual (CPR) preconditioner including well degrees of freedom (DOFs).

The action of the new CPR preconditioner applies only to the reservoir DOFs of the linear system, and includes well-reservoir coupling by solving an extended linear system for the pressure. This extended pressure system is similar to the one proposed in Zhou et al. (Comp. geosci 17(2), 2013). The preconditioner is suitable for a linear system which only includes reservoir DOFs as unknowns and where the effect of the wells is included by Schur complement in the linear operator, without explicit fill-in in the matrix.

The main feature is that the pressure system is extended to include well DOFs. The full preconditioner then combines block ILU0 on the reservoir matrix as fine smoother with the new extended pressure CPR system, using standard AMG cycles on the latter. The new preconditioner has been implemented in the open-source reservoir simulator OPM Flow. The approach is compared with several different CPR approaches on conceptual and real-field cases including open benchmark cases, looking at accuracy, tolerances, performance and parallel scalability.

Compared to applying CPR to the reservoir matrix without well fill-in, the new method yields lower linear iteration counts, similar to those that result from applying CPR to the reservoir matrix with well fill-in (explicit Schur complement). However, each iteration is less costly since one avoids the fill-in, which adds k2 extra nonzero matrix elements for a well with k perforations.

An advantage of the approach is that the structural complexity introduced with the well system is included only in the scalar CPR pressure system in a way suitable for algebraic multigrid (AMG) preconditioning. All other complexity of the wells is handled in the linear operators used in the Krylov solvers.

The new method can be implemented in reservoir simulators by building on existing preconditioner components, and can improve simulation times for complex cases, in particular those with many wells and perforations.

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