Numerical Analysis for Relative Permeability Hysteresis Models in Reservoir Simulation of CO2 Trapping in Underground Carbon Storage
Authors: Al Ali Ammar 
Published: February 2022

Abstract
There are many reservoir simulation applications for multiphase flow in porous media where hysteresis or path-dependence of both relative permeability and capillary pressure functions are crucial to capture. The formation of a residual non-wetting phase saturation due to capillary trapping in a hysteretic manner carries significant implications to some major petroleum development processes such as EOR or water-alternating-gas (WAG), as well as environmental processes, such as geologic CO2 storage. In this paper, we focus on accurately quantifying how much of the injected CO2 gets trapped underground due to relative permeability hysteresis only and the most efficient way to model this physical phenomenon.
Over the years, multiple methods for implementing hysteresis into reservoir simulators were introduced to capture the trapping phenomenon. However, these complex methods created numerical difficulties especially when flow reversal happens, creating nonlinear solver convergence issues due to discontinuous derivatives. A new technique has been introduced recently with a claim of smoother behavior and better non-linear solver performance. The main goal of this study is to assess this new technique by looking at both nonlinear solver performance as well as the method accuracy compared to previous standard models. Here, three models are implemented in an implicit state-of-the-art simulator especially developed for this study. This is equipped with nonlinear-convergence-enhancing techniques such as Appleyard saturation chopping and different upstream weighting.
The hysteresis models are implemented for relative permeability of the non-wetting phase only and has been ignored in the wetting phase, and the study also neglects the capillary pressure hysteresis. The paper presents the theoretical background of the models and their implementations as well as the significance of accounting for hysteresis in such applications. Then, simulation results and numerical analyses are presented for a 1D gravity segregation case in a hypothetical CO2 storage setting. The results show that the new model proved to offer a better numerical handle of the hysteresis in reservoir simulation. This improvement is particularly significant in normal moderate CFL number scenarios, while in the very low or very high scenarios, the improvement is modest. All models can produce similar results if their relative permeability curves have been fitted well. It is important to keep in mind that even though the numerical differences are not huge in this simple test case, these results show indication of where difficulties can arise from when this simple test case is taken into more complicated settings.
Capturing the accurate physics for such processes, namely underground CO2 storage, is vital as studies show that this accounts for a great deal of the CO2 trapped underground; however, this may be a difficult task for most commercial simulators. In this work, we analyze different models to capture such physics and introduce new way with enhanced efficiency compared to existing techniques as evident by numerical analysis results.