Adaptive Time Stepping with the Modified Local Error Method for Coupled Flow-Geomechanics Modeling
Authors: An Cheng , Zhou Peng , Yan Bicheng , Wang Yuhe , Killough John 
Published: May 2017

Abstract
The coupled flow-geomechanics model is required to investigate the stress change, rock-compaction behavior, and stress-dependent properties in many reservoir scenarios. However, the coupled model for large-scale or three-dimensional simulation problems usually encounters large matrix system and high computation expenses, where the time stepping is a crucial factor for numerical stability and computational efficiency. In this paper, an adaptive time stepping with the modified local error method was presented to reduce iteration time and improve the computation efficiency for the coupled flow-geomechanics modeling. Firstly, the iterative coupling approach with the fixed-stress method was introduced, where the flow and geomechanics equations are sequentially solved at each time step. Secondly, due to updating geomechanics module consumes the major computing time for the coupled problems, the modified local error method was mainly used for geomechanics module, where fewer geomechanics time steps are needed after implementation. Specifically, the geomechanics module will be updated until a given local error of displacement is reached, and the time step size will be automatically adapted based on the change of displacement, which is more efficient than the constant time step method. Finally, a synthetic two-dimensional coupled production problem is established to apply the proposed adaptive time stepping approach, where the numerical results including the computing efficiency are compared with the results from regular sequential method and the fully coupled model. The sensitivity about the local error tolerance was also investigated.
The geomechanical responses regarding mechanical loading is validated by comparing with the analytical solution of Terzaghi's consolidation problem. The numerical results about the pressure and displacement change on the two-dimensional coupled model are compared with the results from the regular sequential method and the fully-coupled method. The modified local error method, which adjusts time step size for both flow and geomechanics module, not only yields a higher-order solution for better accuracy, but also significantly reduces the iteration number and computation time, especially for the cases with low truncation error requirement. The error tolerance of displacement is critical on when the step size will be adapted. Small error tolerance can maintain the accuracy while it needs more iteration computing. The strategy about how to modify step size plays an important role in the stability and computing efficiency for the modified local error method. A large increase or cut on one-step size could bring the oscillation results. Overall, the adaptive time stepping approach can both effectively reduce total computation time and simultaneously maintain the accuracy for the coupled flow-geomechanics problems. It is useful for large-scale or three-dimensional coupled problems, where the high computational efficiency is required.