A Stable Multi-phase Nonlinear Transport Solver with Hybrid Upwind Discretization in Multiscale Reservoir Simulator

The nonlinearity of the coupled flow and transport problem results from various sources in the system consisting of heterogeneous reservoir geology and thermodynamic fluid properties under changing reservoir conditions. This generates a complex flow field in the multi-phase flow dynamics. Also, the mathematical discretization of the governing equations creates additional numerical instability in the solution. We have developed and implemented in a commercial reservoir simulator a robust numerical simulation engine that is computationally efficient and stable. The novelty of our approach relies on the flexibility of both spatial and temporal discretization schemes. We solve the parabolic pressure equation with the multiscale finite volume method and the hyperbolic transport equation in the sequential fully implicit scheme. With this framework, we can devise appropriate numerical algorithms accounting for the nonlinearity captured from different spatial scale resolutions and dynamic multi-phase flow transport phenomena respectively. The transport solver utilizes the hybrid upwind discretization, proposed by Lee et al. (2015), to stabilize the numerical convergence difficulty due to the interaction between viscous and gravity flows in the conventional phase potential upwind discretization. Specifically, we split the co-current viscous flow upwinding by total velocity direction and the counter-current gravity flow upwinding by phase density differences. This eliminates the discontinuity of the numerical flow in the transition from co-current to counter-current flows and guarantees the differentiable flow functions. Additionally, the Newton updates are safeguarded by the numerical trust region analysis previously proposed by Li and Tchelepi (2014) and further extended to the general three phase condition with a parameterization of the flow function derivatives along the solution direction. The superiority of the devised solver has been validated through application of the commercial reservoir simulator using synthetic and field scale models. This demonstrates stable larger timestep advancement and faster convergence compared to the conventional discretization and standard nonlinear solvers.