The single-cell transport problem for two-phase flow with polymer

Abstract

Polymer injection is a widespread strategy in enhanced oil recovery. Polymer increases the water viscosity and creates a more favorable mobility ratio between the water and the oil phase. Pressure-transport splitting of the equations combined with reordering strategies can be used to significantly increase the computational speed of two-phase flow simulations with polymer ([10]). Such scheme relies on a robust single-cell solver, which computes the saturation and polymer concentration of a cell, given the total flux and the saturation and polymer concentration of the neighboring cells. In this paper, we consider a standard but relatively comprehensive polymer model and show that, in the case of a two-point flux approximation, the single-cell problem always admits a solution and that the solution is unique. The uniqueness part of the proof is essentially based on monotonicity arguments while the existence part is based on bracketing. In particular, it shows that also the pressure equation must be chosen with care. For the segregation part, we observe that, even if the polymer belongs to the water phase, water-phase upwinding of the polymer concentration leads to ill-posedness and we present an alternative which guarantees well-posedness.

Robust Nonlinear Newton Solver With Adaptive Interface-Localized Trust Regions

“…One such approach is to reduce the fully implicit system by a priori estimation of nonzero update regions before each Newton iteration (Sheth and Younis 2017). Another approach is to reorder the grid cells either according to fluid potential (Kwok and Tchelepi 2007) or by using topological traversal of the interface flux graph (Natvig and Lie 2008) to develop highly efficient iterative Gauss-Seidel solvers that repeatedly sweep through the cells in a predefined order and solve single-cell problems (Lie et al 2014;Raynaud et al 2016;Klemetsdal et al 2018).…”

“…Polymer Example 1: Subset of SPE 10 Model 2. As an example of an EOR process, we consider polymer flooding described by a Todd and Longstaff (1972) model for miscible flow as specified in the ECLIPSE commercial simulator (Schlumberger 2013). We pick a horizontal layer from SPE 10 Model 2 (Christie and Blunt 2001), which describes a weakly compressible waterflood problem, and inject 1 PV of water over a period of 2,000 days from an inverted-five-spot well pattern with the four producers in the corners operating at constant BHP.…”

“…A number of methods have been proposed to improve the convergence rates of nonlinear transport solvers, such as use of Appleyard chopping (Schlumberger 2013) to safeguard updates, or more-sophisticated approaches such as localized nonlinear iterations (Younis et al 2010). Jenny et al (2009) noted that inflection points in the flux function might send the Newton update to different contraction regions and effectively result in oscillations and convergence issues.…”

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

“…We present a new approach to develop accompanying transport solvers, which is best described as dynamic coarsening, in which lower spatial resolution is introduced temporarily in parts of the domain. The resulting method is conceptually similar to local grid refinement and dynamic versions thereof (Heinemann et al 1983;Quandalle and Besset 1983;Ewing et al 1990;Boyett et al 1992;Deimbacher and Heinemann 1993;Edwards 1996;Mlacnik et al 2001;Sammon 2003;Batenburg et al 2011;Lavie and Kamp 2011;Terekov and Vassilevski 2013;Kheriji et al 2014;Hoteit and Chawathé 2016). However, whereas dynamic refinement usually introduces a structured subdivision of cells in a structured grid, coarsening methods can easily form nonuniform and more irregular blocks to adapt to flow patterns or features in the reservoir model (Durlofsky et al 1997;Aarnes et al 2007;Hauge et al 2012;Durlofsky 2014, 2018;Guion et al 2019;Klemetsdal et al 2019a).…”