Sequential fully implicit Newton method for reservoir simulation
Keywords: Energy Resources Engineering.

Numerical simulation of compositional flow and transport is critical for numerous applications in reservoir simulation. The nonlinear coupling between thermodynamics, reservoir heterogeneity, and gravitational effects makes developing an efficient nonlinear solver for compositional modeling very challenging. The Fully Implicit (FIM) discretization scheme is extensively employed in the industry, where at each Newton step, a global Jacobian matrix is generated, then the corresponding linear system is solved. FIM can often be extremely costly for field-scale compositional simulations when a large and poorly conditioned linear system needs to be solved with tiny timesteps. The Sequential Fully Implicit (SFI) method has been proposed as a promising alternative solution strategy in which the coupled problem is separated into subproblems and solved implicitly in sequence. The SFI method is more flexible and can adopt different temporal and spatial discretizations, as well as specialized linear solvers and customizable nonlinear formulations for each subproblem. Additionally, it permits the development of multiscale simulation for flow and transport, which is viewed as a computationally more efficient alternative to the single-level fine-scale approach, potentially offering significant performance improvements. However, the SFI method has a critical limitation: slow convergence of the outer loop when the subproblems are tightly coupled. This constraint prevents SFI from achieving robust performance for general flow and transport simulations, limiting the method's applicability. This thesis describes numerical approaches based on the Sequential Fully Implicit Newton (SFIN) method for accelerating the SFI method's outer loop convergence, with the goal of enabling the SFI scheme to attain robust nonlinear performance comparable to the FIM method within tightly coupled scenarios. We first develop the Sequential Fully Implicit Newton (SFIN) method for compositional flow and transport simulation, with the proper consideration of the multi-phase and multi-component phase equilibrium and the fixed total volumetric flux constraint between flow and transport problems. The SFIN method's critical step is a global Newton update at the end of each sequential iteration. The pressure and transport variables are updated concurrently during this outer Newton step to better resolve the coupling. A Krylov solver is used for the associated linear system without constructing explicit Jacobians, relying solely on matrix-vector multiplications. We show that matrix-vector multiplications can be calculated by efficiently reusing previously computed Jacobian matrices and their preconditioners' information. Second, we extend the SFIN method to address its major constraint: the choice of the primary variables for each subproblem has to be fixed during the outer iteration, which includes performing Newton iteration loops for each subproblem. We propose a zero-out strategy to deal with the variables switch, specifically when it happens between flow and transport problems. The extended SFIN method obtains significantly improved nonlinear acceleration performance for natural black oil formulation even in situations with frequent phase changes. After that, we focus on further decreasing the additional computational cost associated with the SFIN method. We propose a reduced SFIN method, which requires solving a reduced SFIN transport linear system with an appropriate existing preconditioner rather than the original SFIN linear system, followed by a single update for the pressure variables, which avoids one of the SFIN method's major costs: inversion of the transport Jacobian matrix in the Krylov solver iterations. Finally, we demonstrate how the SFIN method can be extended to other multi-physics problems. We consider coupled reservoir and multi-segment well simulations and develop a similar SFIN method for the coupling problem. We present several challenging two- and three-dimensional cases and demonstrate that the SFIN approach can dramatically reduce the number of outer and inner loop iterations when compared to the standard (non-accelerated) SFI method