From quasi-static to dynamics: A hierarchical asymptotic model and multigrid-in-time method for poroelasticity

Accurate numerical simulation of poroelastic models is essential across various fields, including seismology, soil mechanics, and biomechanics. A major challenge in modeling dynamic poroelastic systems lies in the significant computational cost required to resolve multi-physics coupling with fine temporal and spatial resolution. This is especially demanding for optimization applications which require repeated solutions across multi-dimensional parameter space. Existing simplified models of such systems often rely on assumptions and fail to systematically bridge the gap between quasi-static and fully dynamic regimes. To overcome this challenge, the present work introduces a systematic reduction framework based on dimensional analysis and asymptotic expansion for dynamic poroelasticity. The dimensional analysis reveals a key parameter that weighs the effect of inertia and damping, while the asymptotic expansion leads to a hierarchical reduced model that balances accuracy and efficiency. The reduced model is composed of two sub-models: namely a quasi-static limit model, which is computationally efficient and captures the averaged system behavior; and a corrector model, which reintroduces the lost dynamic behavior. The hierarchical structure of this model is utilized to build a finite-element based multigrid-in-time algorithm. This is then used to verify the performance and accuracy of the reduced model for a 1D dynamic consolidation problem.