Abstract
We describe a new nonlinear solver for immiscible two-phase transport in porous media, where viscous, buoyancy, and capillary forces are significant. The ‘fractional-flow’, F, is a nonlinear function of saturation and typically has inflection points and can be non-monotonic. The non-convexity and non-monotonicity of F are major sources of difficulty for nonlinear solvers. A modified Newton algorithm that employs trust-regions of the flux function to guide the Newton iterations is proposed. The flux function is divided into saturation trust regions. The delineation of these regions is dictated by the inflection, unit-flux, and end points. The saturation updates are performed such that two successive iterations cannot cross any trust-region boundary. If a crossing is detected, we ‘chop back’ the saturation value to the appropriate trust-region boundary. Our trust-region Newton solver has excellent convergence properties across the parameter space of viscous, buoyancy and capillary effects, and it represents a significant generalization of the inflection-point approach of Jenny et al. (JCP, 2009) for viscous dominated flows.
We analyze the nonlinear transport equation using low-order finite-volume discretization with phase-based upstream weighting. Then, we prove unconditional convergence of the trust-region Newton method irrespective of the timestep size for single-cell problems. For one-dimensional transport, numerical results across the full range of the parameter space of viscous, gravity and capillary forces indicate that our trust-region scheme is unconditionally convergent. That is, for any choice of the timestep size, the unique discrete saturation solution is found independently of the initial guess. For problems dominated by buoyancy and capillarity, the trust-region Newton solver overcomes the often severe limits on timestep size associated with existing methods. We use complex 3D reservoir models to demonstrate the effectiveness of the proposed nonlinear solver. Specifically, we use the top zone of the SPE 10 model (Tarbert formation) and the full SPE 10 model. Compared with state-of-the-art Newton-based nonlinear solvers, our trust-region solver results in superior convergence performance, and it reduces the total Newton iterations by more than an order of magnitude, which leads to a comparable reduction in the overall computational cost.
