题目：Simulation of Geological Reservoirs at Various Scales:A unified approach to design stable schemes by generalized 

gradient flow

报告人：致远数学讲堂：Prof. Shuyu Sun (King Abdullah University of Science and Technology, Kingdom of Saudi Arabia)

时间：2023年12月18日 16:00-17:30

地点：致远楼108室

Abstract:  Multi-phase flows in geological porous media are central to a wide range of natural and industrial processes, 

including enhanced oil recovery, water infiltration into soil, subsurface hydrogen storage, as well as the sequestration 

of CO2 in geological formations, such as depleted reservoirs, deep saline aquifers, or basalt. Stable and efficient 

numerical algorithms are essential for the simulation of flows in geological porous media. To design stable and efficient 

numerical algorithms, we formulate a generalized gradient flow framework, where the dynamics is modeled using 

thermodynamic driving forces (the negative energy gradient) and a kinetic rate tensor (the inverse of the resistance tensor).  

Many geo-energy applications, in particular most multi-phase flow and transport problems, can be formulated using this 

generalized gradient flow framework.  A good portion of these problems are challenging majorly due to the nonlinear 

coupling among various physics, where tiny time steps are needed due to stability concern instead of maintaining accuracy 

in temporal discretization. In these scenarios, it is crucial to design unconditional energy stable schemes to increase 

its computational efficiency and to enhance its robustness. In this talk, we present our work on unconditionally energy 

stable schemes for flows in porous media at various scales.  We will first quickly review our work in the Darcy scale 

(our stable, phase-wise conservative, and bound-preserving semi-implicit and fully-implicit algorithms), the pore scale 

(pore-network models, PDE-based models, and kinetic theory-based models), the molecular scale (the MC speedup of MD 

and the thermodynamic extrapolation of MC), and the sub-atomic scale (our stable algorithms for DFT calculations), 

and we will then focus on the following two stories with details:

1) Navier-Stokes-Cahn-Hilliard (NSCH) simulation for two-phase flow at the pore scale: We present a pioneering study on 

the design of an unconditionally energy stable SPH (Smoothed Particle Hydrodynamics) discretization of the NSCH model 

for incompressible two-phase flows based on a number of novel techniques: subtle treatment of capillary forces at the 

discrete level, particles’ divergence-free projection, energy factorization, and discretization with physical 

consistency.

2) Density functional theory (DFT) calculations of the structural, physical and chemical properties of reservoir fluid 

mixture: We design an unconditionally energy stable, orthonormality-preserving scheme for the Kohn-Sham gradient 

flow-based model in the electronic structure calculation. The scheme is fully robust and it does not contain any tuned 

parameters.Unconditional stability of the scheme allows us to use large time step sizes and thus achieves great 

computational efficiency.The scheme is also fully robust; it converges for any initial guesses; it overcomes the 

non-convergence issue faced by SCF methods for certain difficult electronic structure calculations (e.g., some of 

open shell problems).  

The first story above is based on the joint work [1] with Xiaoyu Feng (KAUST), Zhonghua Qiao (HK PolyU), and 

Xiuping Wang (KAUST). 

The second story above is based on the joint work [2] with Xiuping Wang (KAUST), Huangxin Chen (Xiamen U), 

Jisheng Kou (Shaoxing U).

[1] X Feng, Z Qiao, S Sun, and X Wang, “An energy-stable Smoothed Particle Hydrodynamics discretization of the 

Navier-Stokes-Cahn-Hilliard model for incompressible two-phase flows”, Journal of Computational Physics, 479, 

111997, 2023.

[2] X Wang, H Chen, J Kou, and S Sun, “An unconditionally energy-stable and orthonormality-preserving iterative 

scheme for the Kohn-Sham gradient flow-based model”, Journal of Computational Physics, accepted, 2023.

欢迎各位参加！