﻿ Structure-Preserving Algorithms for Two-Phase Flow at Various Scales-武汉大学数学与统计学院

Structure-Preserving Algorithms for Two-Phase Flow at Various Scales

Prof. Shuyu Sun (King Abdullah University of Science and Technology (KAUST), Kingdom of Saudi Arabia)

Two-phase and multi-phase flows in porous media are central to a wide range of natural and industrial processes, including geologic carbon sequestration, enhanced oil recovery, and water infiltration into soil.  Petroleum engineers use reservoir simulation models to manage existing petroleum fields and to develop new oil and gas reservoirs, while environmental scientists use subsurface flow and transport models to investigate and compare for example various schemes to inject and store carbon dioxide in subsurface geological formations, such as depleted reservoirs and deep saline aquifers. Darcy-scale multi-phase flow simulation in subsurface reservoirs has routinely used by reservoir engineers over half a century. During the recent decade, digital Rock Physics (DRP) and pore-scale flow simulation have also become a complementary part in reservoir characterization over the past 3 decades as non-destructive methods used to determine absolute/relative permeability, capillarity, effective elastic rock parameters and other porous media properties.

In this talk, we present our work in structure-preserving (especially bound-preserving and unconditionally energy-stable) algorithms for porous media flow at various scales by highlighting two specific topics: 1) structure-preserving algorithms for pore-scale two-phase flow, and 2) structure-preserving algorithms for Darcy-scale two-phase flow.  First, we present a novel particle method for pore-scale simulations. The Navier–Stokes–Cahn–Hilliard (NSCH) system has widely used as the standard model for the Direct Numerical Simulation (DNS) of incompressible immiscible two-phase flow, with finite volume methods (FVM), finite element methods (FEM), and Lattice Boltzmann methods (LBM) as popular discretization schemes.  However, structure-preserving (especially unconditionally energy-stable) particle methods for NSCH have not widely used nor thoroughly studied yet.  Here, a novel, efficient and structure-preserving Smoothed Particle Hydrodynamics (SPH) method is proposed and implemented for pore-scale two-phase fluid flow modeled by the Navier–Stokes–Cahn–Hilliard (NSCH) system of equations.  In addition to preserve the conservation of mass, the conservation of linear momentum and the conservation of angular momentum in the discrete solution, our scheme also preserves the conversion between kinetic energy and interfacial energy exactly and moreover, it is unconditionally energy-stable.  It is more flexible, more powerful and more accurate than conventional, mesh-based simulation methods, in particular for the treatment of convection (in fact, the numerical treatment of linear convection can be made to be exact). In order to enhance efficiency, we decouple the NSCH system to simplify the calculation into a few linear steps while still maintaining unconditional energy stability. We prove that our SPH method inherits mass and momentum conservation and the energy dissipation properties from the PDE level to the ODE level, and then to the fully discrete level. As a result, it also helps increase the stability of the numerical method; in particular, its time step size can be much larger than that of the traditional SPH methods. Numerical experiments are carried out to show the performance of the proposed energy-stable SPH method for the two-phase flow. The inheritance of mass and momentum conservation and the energy dissipation properties are verified numerically. The numerical results also demonstrate that our method captures the interface behavior and the energy variation process well.

In the second part of this talk, we present two new semi-implicit algorithms for incompressible two-phase in porous media with multiple capillary pressure functions, one in each subdomain; the two new semi-implicit algorithms include a conditionally-stable one and an unconditionally-stable one, each having certain unique advantages. The two proposed algorithms are locally mass conservative for both phases.  They are able to accurately reproduce the spatial discontinuity of saturation due to different capillary pressure functions, and they correctly ensure that the total velocity is continuous in the normal direction.  Moreover, the new schemes are unbiased with regard to the phases and the saturations of both phases are bounds-preserving (under certain conditions). The methods can be shown to be conditionally bound-preserving as well. The proposed semi-implicit algorithms are derived from our novel splitting of variables based on the physics of two-phase flow.  A few interesting examples are presented to demonstrate the efficiency and robustness of the new algorithms.