Abstract
Locality is inherent to all subsurface flow and transport phenomena. While the spatial locality in the temporal evolution of state variables is well-studied, little is known about the locality that is present over the course of individual Newton iterations that arise in the solution of an implicit timestep. This work derives a priori sharp and conservative estimates for the Newton update before its corresponding linear system has been solved. While the current focus is the sequential implicit procedure for general nonlinear and heterogeneous two-phase flow in multiple dimensions, we outline the extension to fully-coupled multicomponent models. The estimates are universal to any accurate spatial discretization. There are numerous applications of these estimates, including their use towards characterizing the convergence rate of Newton's method as a function of timestep size, or towards the development of scalable linear preconditioning strategies. This work focuses on applying the estimates to reduce the size of the linear system that is to be solved at each Newton iteration.
The key to the derivation of these estimates is in forming and solving the infinite-dimensional Newton iteration for the semidiscrete residual equations. In an implicit simulator, the Newton updates are accurate approximations to the continuous in space infinitedimensional updates. While the infinite dimensional updates are obtained by solving linear Partial Differential Equations (PDE) analytically, the discrete approximations are obtained by inverting a linear system. For flow, the analytical estimate for the Newton update is the solution of a linear constant coefficient screened Poisson equation. For transport, the estimate is the solution of a first-order linear PDE. The analytical solutions for these estimates are derived, and they can be applied computationally to any discrete problem.
In a simulation, at each nonlinear iteration, the estimates are evaluated over parts of the domain and they are subsequently used to identify the unknowns and cells that will undergo a Newton update that is larger than any prescribed tolerance. The reduced linear system is then formed for these unknowns only, and is solved to obtain the Newton update. Since the estimates are conservative, there is no degradation of the nonlinear convergence rate.
We present comprehensive large-scale computational results. The performance improvement using the proposed method directly depends on the extent of locality present as dictated by the physics. For problems with slight compressibility, the localization of the linear systems results in an order of magnitude of improvement in computational performance. On the other hand, incompressible models involve global pressure updates, and the locality is exploited in the transport part only.
