REE2: Solvers for petroleum reservoir simulation

Researcher:  James Lottes
Team Leader(s): Dr Andy Wathen
Collaborators: Dr Andreas Papadopoulos, Schlumberger
Dr Gareth Shaw, Schlumberger


Project report to follow


As oil reserves are depleted, decisions about how best to develop less-readily accessed residual reserves are guided by more and more sophisticated computational petroleum reservoir models. More detail is required as the petroleum resources become depleted, but of course the more scarce the resource becomes, the more valuable it is.

The resulting complexity and greater need for detail in the representation of reservoirs demands more large scale and complicated computational models. The required computational results can only be achieved at the additional cost of significantly more discrete variables. This has a knock-on effect, since the algorithms used in the reservoir simulation software must still be capable of delivering solutions within reasonable time scales in order to guide reservoir engineers in their choice of extraction strategy.

The key bottleneck is the iterative solver for the relevant linearised systems of matrix equations. The oil industry has been one of the prime application areas which has driven the development of more and more efficient solvers. However, existing solvers do not scale linearly, in that a model of twice the size takes longer than twice the time to solve. A method with linear CPU time is required.

Techniques and Challenges

One of the prime methodologies on which to build a general solver for reservoir problems which have the desired linear complexity is the algebraic multigrid (AMG) technology which has been moving forward over the past few years for simpler partial differential equation (PDE) problems. The aim of this project is to develop the AMG paradigm into an optimal solver suitable for the non-symmetric block matrices which arise in reservoir simulation. The key aspects being addressed are non-normality and anisotropy (some initial work on this has already been done by researchers at the Oxford Centre for Collaborative Applied Mathematics, OCCAM) as well as really effective ways to use the known block structure.

The Future

The project will involve investigation of the elements which comprise the AMG approach, concentrating on appropriate coarsening strategies and interpolation or “grid-transfer” operators. The research methodologies will be the numerical analysis of iterative AMG methods using mathematical guidance to indicate how best to achieve the fastest possible convergence. The associated extensive numerical testing will be facilitated by our close links with the nearby Schlumberger Petroleum Reservoir Simulation arm with whom we have had much previous contact. This collaboration should be of considerable help in obtaining appropriate test problems at the relevant stage.