Robust Nonlinear Newton Solver With Adaptive Interface-Localized Trust Regions

Abstract

Summary The interplay of multiphase-flow effects and pressure/volume/temperature behavior encountered in reservoir simulations often provides strongly coupled nonlinear systems that are challenging to solve numerically. In a sequentially implicit method, many of the essential nonlinearities are associated with the transport equation, and convergence failure for the Newton solver is often caused by steps that pass inflection points and discontinuities in the fractional-flow functions. The industry-standard approach is to heuristically chop timesteps and/or dampen updates suggested by the Newton solver if these exceed a predefined limit. Alternatively, one can use trust regions (TRs) to determine safe updates that stay within regions that have the same curvature for numerical flux. This approach has previously been shown to give unconditional convergence for polymer- and waterflooding problems, also when property curves have kinks or near-discontinuous behavior. Although unconditionally convergent, this method tends to be overly restrictive. Herein, we show how the detection of oscillations in the Newton updates can be used to adaptively switch on and off TRs, resulting in a less-restrictive method better suited for realistic reservoir simulations. We demonstrate the performance of the method for a series of challenging test cases ranging from conceptual 2D setups to realistic (and publicly available) geomodels such as the Norne Field and the recent Olympus model from the Integrated Systems Approach for Petroleum Production (ISAPP) optimization challenge.

Dynamic Coarsening and Local Reordered Nonlinear Solvers for Simulating Transport in Porous Media

“…However, careful examination of several numerical examples conducted on real field models indicates that even in cases with strong gravity effects, cells belonging to a connected component only amount to a small fraction of the total number of grid cells. Some of these results are reported in Klemetsdal et al (2019bKlemetsdal et al ( , 2019d. One can also use a nonlinear Gauss-Seidel solver Lie et al 2014) in LRNTS, in which the nodes in a connected component are solved sequentially in a predefined order with variables in all other nodes fixed.…”

“…One can also use a nonlinear Gauss-Seidel solver Lie et al 2014) in LRNTS, in which the nodes in a connected component are solved sequentially in a predefined order with variables in all other nodes fixed. This is repeated a number of times until all nodes in the component have converged and has shown promising results (Lie et al 2014;Klemetsdal et al 2019d). Dynamic coarsening can be combined with LRNTS in a straight-forward manner.…”

“…This is admittedly not sufficient to describe real reservoir fluids but greatly simplifies the interpretation of numerical results. More complex cases involving compositional simulations have been demonstrated for an early version of the dynamic coarsening method in Klemetsdal et al (2019b), and the efficiency of LRNTS has been demonstrated for black oil and compositional simulations in Klemetsdal et al (2019bKlemetsdal et al ( , 2019d. All results reported in the following are based on a proofof-concept implementation in the MRST (Lie 2019).…”

“…Likewise, one can permute the transport equations into a block-triangular form that can be solved efficiently by a (nonlinear) back-substitution method by reordering the grid cells according to fluid-phase potential (Appleyard and Cheshire 1982;Kwok and Tchelepi 2007;Hamon and Tchelepi 2016), or based on the total flux from the pressure equation (Aarnes et al 2006;Natvig and Lie 2008). This has been demonstrated for a wide range of scenarios, including linear transport equations (Natvig et al 2007;Eikemo et al 2009; Rasmussen and Lie 2014), two-phase flow (Natvig and Lie 2008), and polymer flooding (Lie et al 2014;Raynaud et al 2016), and more recently for compressible black oil (Klemetsdal et al 2019d) and compositional problems (Klemetsdal et al 2019b).…”

Efficient reordered nonlinear Gauss–Seidel solvers with higher order for black-oil models

“…For the transport equations, a number of methods aim to arXiv:2001.01630v1 [math.NA] 6 Jan 2020 accelerate computations by utilizing inherent locality and co-current flow properties of hyperbolic equations. Examples include streamline simulation [9,4], a priori estimation of nonzero update regions [30], and use of interface-localized trust regions to determine safe saturation updates [23,15].…”