This paper deals with a time stepping method for the finite element simulation of two phase flow hydraulic and mechanical (H2M) coupled processes in porous media, which is a common phenomenon in geological applications such as CO2 storage facilities. The computation task arising from the numerical modeling of H2M coupled processes in such real geological application is intensive. Therefore, the high performance computing is of interest to the corresponding researchers. In the present study, we present a time stepping method with PI (proportional and integral feedback) automatic control to improve the computation efficiency of the modeling of H2M coupled processes. We apply the PI control to solve the nonlinear coupled partial differential equations with a first order finite difference scheme for time discretization and the Picard method for linearization. The efficiency of the present method is demonstrated by applying it to a CO storage benchmark.
To reduce anthropogenic greenhouse gas emissions into the atmosphere, the carbon dioxide capture and storage (CCS) concept is introduced by some researchers as an emerging transition technology [1,2]. The study of CCS is therefore under active consideration recently. According to various studies, deep saline aquifers provide the most substantial carbon dioxide storage capacity [3,4,5,6,7], and are often located near possible CO2 sources such as coal-fired power plants.
To ascertain migration and trapping of CO2 in the formations and assess the capacity and the safety (possible leakage) of the reservoir, the numerical simulation of injection and spreading of carbon dioxide in the underground is essential for understanding the physical and chemical processes at different length and time scales. In the numerical analysis of time dependent thermo-hydraulic processes in porous media, the time stepping is a crucial issue for numerical stability and computational efficiency. Practically, the fixed time step size does not often satisfy the stability and efficiency requirements in solving problems that exhibit complexity in geometry and nonlinearity in material properties. Therefore, adaptive time stepping methods including high-order integration have been developed and are widely applied . Among the available adaptive time stepping methods, the well-known techniques for prediction of the time step size h are e.g. Courant number approach based on Courant-Friedrichs- Lewy condition  for the finite difference method, primary variable based prediction (e.g. those presented in ref. [10,11] and local error control methods (cf.[8,12,13]). The local error control methods especially those based on theoretical control ideas are problem independent for any numerical methods for ODEs [8,12,13,14]. For nonlinear equations, the theory based automatic controls such as P (proportional feedback) or PI (proportional and integral feedback) permit stable and efficient time stepping [8,14] for numerical solver. The present work is subjected to apply the adaptive time stepping with automatic control for the finite element modeling of two phase flow hydraulic and mechanical coupled processes in CO2 storage facilities.
To this purpose, we present an approach of PI (proportional and integral feedback)  automatic time stepping for modeling the problems with different coupled physical processes. Within the context of the presented time stepping approach, each process uses the time step size predicted by the PI control itself to guarantee the stability of the simulation of each process under coupling.