Trust-region Newton solver for multiphase flow and transport in porous media [electronic resource]
Keywords: Energy Resources Engineering

The simulation of immiscible fluid displacement in subsurface porous media remains an important and challenging problem with application to oil-recovery processes and subsurface CO2 sequestration. The saturation equations, which describe the transport of the fluid phases in space and time, are highly nonlinear. They are characterized by non-convex flux functions that can also be non-monotonic in the presence of strong buoyancy forces. The saturation field is strongly coupled to the phase velocities, which are in turn strongly dependent on the distribution of the phase pressures. In order to model immiscible processes in natural porous media efficiently, it is critical to deal effectively with the nonlinear coupling between flow (pressures and velocities) and transport (saturation). 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 the nonlinear solver. 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 trust-region 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 in the total Newton iterations by more than an order of magnitude, which leads to a comparable reduction in the overall computational cost. We then describe a nonlinear solution algorithm for coupled flow and transport in heterogeneous porous media where both the viscous and buoyancy forces are significant. We show that flow reversals between Newton updates (or timesteps) are the primary source of nonlinear convergence problems for coupled multiphase flow and transport. For a given flux function, the combinations of saturation and phase-flow direction that lead to flow-reversal are enumerated. These `flip points', which can be computed a-priori, identify the interfaces that experience a flow reversal as the saturation fields evolves over the current iteration, or timestep. This flow-reversal information is used to update the flow field to ensure consistency of the residual equations of both flow and transport and the associated Jacobian. If flow reversal is detected anywhere in the model for a given timestep, we switch from the sequential-implicit method (SIM) to the fully-implicit method (FIM) using the latest estimates of pressure and saturation as initial guesses. Numerical evidence shows that the proposed nonlinear solver is able to converge for timesteps that are much larger than what the SIM can handle. In addition, for very large timestep sizes, the new nonlinear solver yields better convergence performance than standard FIM. A preconditioning strategy to overcome convergence difficulties in the nonlinear solver that are associated with the propagation of saturation fronts into regions that are at, or near, the residual saturation is proposed. The convergence difficulties are due to unphysical mass accumulation in certain grid blocks during the Newton iterations. The unphysical mass accumulation, which is proportional to the throughput over the timestep, propagates in the computational domain quite slowly with Newton iterations - as slow as one grid block per iteration. As a result, convergence is often not possible, and when it occurs, it can be quite slow, especially for large throughput (i.e., large timesteps). We propose a strategy that guarantees that mass in any grid block moves no slower than in any of its upwind grid blocks. The preconditioning strategy leads to monotonic iterative updates of the saturation field, resulting in rapid convergence. Numerical examples show that this strategy accelerates the convergence of existing nonlinear solvers quite significantly, especially for aggressively large timesteps.