Exploitation of the Inherent Parallel-in-Time Undrained Split Coupling Scheme for Solving Coupled Flow and Geomechanics Problems

Abstract

Summary A more efficient formulation of the undrained split iterative coupling scheme is derived for solving the coupled fluid flow and geomechanics problem. In the standard undrained split scheme, the mechanics problem is solved first then followed by the fluid flow problem within every iterative coupling iteration (Mikelic et al. 2013). The scheme will not advance to the next timestep until the current iterative coupling iteration is converged. In this work, we will exploit the inherent parallel-in-time nature of the mechanics solve only ( Borregales et al. 2018 ), which leads to a partially parallel in time scheme. This is a natural extension of the multirate (i.e. multiple timestep sizes) formulation of the undrained split iterative coupling scheme (Kundan et al. 2016, Almani et al. 2019 ) in which the fluid flow problem takes a sequence of fine timesteps within one relatively coarser geomechanics timestep. However, in the proposed approach, the fluid flow problem takes a sequence of fine timesteps, and the mechanics problem assumes the same sequence of fine timesteps, which are solved in parallel within one coupled iteration. In this work, a rigorous theoretical investigation of the proposed algorithm is carried out and the convergence analysis is established for heterogeneous poro-elastic media. The approach has shown to be contractive, and the localized contraction estimates are determined, and compared against the standard multirate undrained split scheme. This new formulation uses the same timestep size for both fluid flow and geomechanics without sacrificing the computational efficiency of the multirate scheme. Moreover, it will result in a more accurate solution due to the fact that both fluid flow and geomechanics share the same timestep size. This is the first time in literature the convergence of a partially parallel in time formulation of the undrained split iterative coupling scheme is established. Mikelic A., and Wheeler M. F. Convergence of iterative coupling for coupled flow and geomechanics. Computational Geosciences. 2013; 17(3):455–461. Kumar K., Almani T., Singh G., and Wheeler M.F. Multirate undrained splitting for coupled flow and geomechanics in porous media, Springer International Publishing, Cham 2016; pp. 431–440. Almani T., Manea A., Kumar K., and Dogru A. Convergence of the undrained split iterative scheme for coupling flow with geomechanics in heterogeneous poroelastic media”. Computational Geomechanics, 2019. Borregales M., Kumar K., Radu F. A., Rodrigo C., and Gaspar FJ. A partially parallel-in-time fixed-stress splitting method for Biot’s consolidation model. Comput Math Appl. 2018; 77(6):1466–1478.