Stability Criteria for the Thermal Adaptive Implicit method

Abstract

Abstract The fully implicit method (FIM) is widely used due to its unconditional stability (even with possibly large time steps). However, FIM is computationally expensive per time step, especially for large number of components. IM-PES (Implicit Pressure, Explicit Saturations), on the other hand, is computationally inexpensive, but only conditionally stable. For large-scale heterogeneous models, the allowable stable time step of IMPES may be extremely small. In the Adaptive Implicit Method (AIM), only a subset of the primary variables is treated implicitly. AIM offers a balance between FIM and IMPES by employing implicit treatment only where necessary. The challenge with AIM is to find a robust and sharp stability criteria that can be used to choose an optimal time step size. The objective here is to derive and verify the stability criteria for a given time step selection in simulations based on a Thermal Adaptive Implicit Method (TAIM). Our linear stability analysis accounts for mass and heat convection, heat conduction, capillary mixing and compressibility. The TAIM stability criteria are obtained using a von Neumann approach. We found the stability criteria to be sharp. That is, mild violations of the stability limits lead to unstable solutions both in the saturation and temperature profiles. The instabilities lead to significant deviations from reference solutions for small CFL (Courant-Friedrichs-Lewy) violations and oscillations for larger CFL violations. Our analysis indicates that, overall, a TAIM-base approach is a promising candidate for simulation of thermal displacement processes of practical interest.