In this paper, we introduce a physics-based nonlinear preconditioner, based on the Inexact Newton method, to accelerate the highly nonlinear thermal-hydraulic-mechanical (THM) simulation of fractured reservoirs. Inexact Newton method has become a popular iterative solver for solution of partial differential equations (PDE). Instead of solving the PDEs exactly with the expensive Newton method, the Inexact Newton method finds a direction for the iteration and solves the equations inexactly. The Inexact Newton method is very efficient when the initial guess is close to the objective solution. However, when the equations are not smooth enough, especially when local discontinuities exits, the Inexact Newton method may be slow or even stagnant.
Local discontinuities are commonly encountered during oil and gas flow in reservoirs. One problem that involves lots of local discontinuities is the simulation of multiphase flow in fractured reservoirs. Fractures in petroleum reservoirs are typically sensitive to the change of pressure and stress. The thermal-hydraulic-mechanical (THM) effects of injection and production can dramatically change the properties of fractures, resulting in a huge variation in the permeability, which adds nonlinearity to the governing partial differential equations. Considering these characteristics of fractured reservoirs, applying the Inexact Newton method directly to them faces severe difficulties. In this work, we have proposed and studied a nonlinear preconditioner to resolve the above problem. In this nonlinear preconditioner, a restricted additive Schwarz approach is used to coarsen the problem and the Inexact Newton method is used as a global iterative solver. We have developed a physics-based strategy to adaptively identify the highly nonlinear zones by computing and comparing the gradient of the stress/temperature-permeability correlations of the fractured zones. These nonlinear zones are treated as a subspace problem, which is solved locally. The results of the subspace problem are used to modify the global residual. By conducting the above operations, the local nonlinearity is eliminated and the global iteration is accelerated. An adaptive strategy is adopted to dynamically choose between the Inexact Newton method and the Newton method.
The above algorithm has the advantage of remarkable scalability and is easy to implement in massively parallel reservoir simulators. We have programed the algorithm and implemented it into our fully coupled, fully implicit THM reservoir simulator to study the effects of cold water injection on fractured reservoirs. The numerical and parallel framework of the simulator has been described by Wang et al. (2014). Previously, the cold water injection problem suffered from slow convergence at the injection zone where fracture permeability changes rapidly. The results of this work show that after the implementation of this nonlinear preconditioner, the iterative solver has become significantly more robust and efficient.