Past Event: Oden Institute Seminar

Multi-Scale Meshless Numerical Analysis of Involution-Constrained PDEs

Alexander A. Lukyanov, Visiting Research Fellow, University of Delft

Abstract

The goal of this talk is to describe various aspects of multiscale meshless numerical modeling of subsurface multiphase fluid flow. The existing fundamental issues as well as evolving theoretical and algorithmic approaches for PDEs with involutions have received a significant interest in the oil-gas industry. Meshless multiscale method for fully-implicit (FIM) simulations of multiphase flow in porous media is presented. Meshless multiscale methods are employed as a preconditioner to obtain efficient and accurate solutions of the diffusion pressure equation. Preconditioning can be used to damp slowly varying error modes in the linear solver residuals, corresponding to extreme eigenvalues. Meshless multiscale solver uses a sequence of aggressive restriction, coarse-particle correction and prolongation operators to handle low-frequency modes on the coarse particle distribution. High-frequency errors are then resolved by employing a smoother on fine grid. The prolongation and restriction operators in this method are based on a SPH gradient approximation (instead of solving localized flow problems) commonly used in the meshless community for thermal, viscous, and pressure projection problems. The multiscale prediction stage (first-stage) is coupled with a FIM corrector stage (second-stage), and the converged solution is obtained through outer iterations between these stages. The formulations and implementations are all done in an algebraic form. While the second-stage FIM stage is solved using a classical scheme, the meshless multiscale stage is investigated in full detail. Several choices for fine- scale pre- and post-smoothing along with finite volume options for meshless multiscale coarse system are considered, among which the optimum choices are highlighted for a wide range of heterogeneous three-dimensional cases. The meshless multiscale constrained pressure residual method is the first in its type, and extends the applicability of the so-far developed multiscale methods to displacements with strong coupling terms. The numerical results are presented, discussed and future studies are outlined.