Adaptive finite element method for multiphase flow and poroelasticity

Numerical simulation of subsurface flow for applications such as carbon sequestration and nuclear waste deposit has always been a computational challenge. The main reason points to the strong nonlinearity inherited in the governing equations that describe the multiphysics phenomena. The enormous number of unknowns and small timesteps required for stable Newtonian convergence make this type of problems computationally exhaustive. To address this issue, we introduce adaptive finite element approaches guided by a posteriori error estimators to improve computational efficiency.

A space-time discretization scheme with temporal and spatial mesh adaptivity is formulated for multiphase flow system. The solution algorithm adopts a geometric multigrid procedure that starts with solving the system in the coarsest resolution and locally refines the mesh in both space and time. Error estimators that measure the spatial and temporal discretization error are employed to guide such an adaptivity. These estimators provide a global upper bound on the dual norm of the residual and the non-conformity of the numerical solution. Results from two-phase immiscible and three-phase miscible flow are presented to confirm solution accuracy and computational efficiency as compared to the uniformly fine timestep and fine spatial discretization solution. We also resolve the common issue of high frequency residuals in multigrid methods by local residual minimization and dynamic advection-diffusion coupling to achieve additional computational speedup and stability.

In addition to the multiphase flow models, we also study the Biot system that couples poromechanics with flow. A posteriori error estimators are derived with the flow and mechanics solved by mixed finite element formulation and continuous Galerkin respectively in a fixed-stress split algorithm. The effectivity of such estimators is validated by Mandel’s problem which enable us to compute the a priori error with its analytical solution. We demonstrate the efficiency of the estimators for adaptive mesh refinement using a fractured porous media example. The validity of the novel stopping criterion which balances the fixed-stress algorithm error with the discretization error is confirmed afterwards. We aim to ultimately provide efficient computations for high fidelity models from carbon sequestration and underground hydrogen storage scenarios.