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.  

报告人简介：孙树瑜于2009年作为创校教授之一加入了沙特阿卜杜拉国王科技大学（KAUST，目前在US News排名全球前一百，2015到2020年连续五年位居QS世界大学排名-“单位教职的论文引用数”世界第一），现任该校计算传递现象实验室首席科学家，地球科学系教授，石油工程系教授，和数学系教授。在此之前孙树瑜曾任教于美国克莱姆森大学。他有两个博士学位，他于2003年在美国德州奥斯汀大学获得计算与应用数学博士学位，指导教师是美国国家工程院院士Mary Wheeler；他于1997年在天津大学获得化学工程专业博士学位，指导教师是中国科学院院士余国琮。孙树瑜的研究领域主要包括多孔介质渗流和对流扩散及反应的数值模拟，已发表SCI期刊学术论文370余篇，出版专著9本，总计被引超过10000余次，H指数51，发表期刊包括Journal of Computational Physics(23篇), Computer Methods in Applied Mechanics and Engineering (14篇)，SIAM Journal on Scientific Computing(7篇), SIAM Journal on Numerical Analysis(5篇) 等计算数学领域顶级期刊和Fuel(中国科学院1区，12篇), Journal of Petroleum Science and Engineering(中国科学院1区，12篇) 等工程领域顶级期刊。他先后以首席科学家身份主持美国能源部、加拿大CMG基金委员会、沙特自然科学基金、中国自然科学基金、陶氏化学公司及沙特阿美石油公司等项目课题24项，总经费超过1300万美元。先后组织承办SIAM-GS, ICCS, ICCES, InterPore, ICAE, ASCHT等各类学术会议及特设分会等40多个，受邀作大会主旨报告、特邀报告及邀请报告200余次。先后指导硕士、博士研究生共50余名（已毕业博士10名），博士后研究人员20余位。目前兼任 Journal of Computational Physics，Gas Science and Engineering，InterPore Journal 和 Computational Geoscience 等国际知名期刊的主编/副主编，以及 Applied Energy，Applied Thermal Engineering，Fuel，Computer Methods in Applied Mechanics and Engineering 等国际知名期刊客座主编。孙教授主持创立了国际多孔介质学会沙特分会并当选首任主席。