Fully implicit non-uniform algebraic dynamic multilevel method as an approximate solver and as a preconditioner for the numerical simulation of two-phase fluid flow in highly heterogeneous petroleum reservoir

Purpose This study aims to improve the efficiency and accuracy of fully implicit (FI) multiphase flow simulations in highly heterogeneous petroleum reservoirs through advanced multiscale numerical strategies. Design/methodology/approach Two strategies based on the non-uniform algebraic dynamic multilevel (NU-ADM) method are proposed and investigated in new contexts. The first, denoted FINU-ADM, extends the NU-ADM methodology (originally developed for IMPES schemes) to an FI formulation, using NU-ADM as an approximate nonlinear solver within a Newton-based framework. The second, CPR-FINU-ADM, integrates NU-ADM into a constrained pressure residual (CPR) approach, using it as a multilevel preconditioner for the pressure subsystem to accelerate the fine-scale solution. Both strategies rely on adaptive non-uniform multilevel meshes that preserve fine-scale resolution in critical regions, such as high-pressure gradients and saturation fronts. Operator construction follows the original NU-ADM algorithm, reformulated for FI context. Findings The proposed methods enable the use of larger time steps and eliminate post-processing steps required for constructing conservative velocity fields. Numerical results from challenging benchmarks confirm that both FINU-ADM deliver accurate pressure and saturation fields while significantly reducing computational cost compared to classical fine-scale simulations or multiscale methods. The CPR-FINU-ADM improves preconditioning and speeds up the fine-scale solution. Originality/value To the best of the authors’ knowledge, this is the first work to extend the NU-ADM method to the FI flow simulation framework, both as a standalone multiscale solver and as a preconditioner in combination with the CPR approach. These formulations enhance scalability and numerical robustness, offering a promising path for large-scale reservoir simulation with adaptive resolution and reduced overhead.