Shuyu Sun, Professor, King Abdullah University of Science and Technology (KAUST), Kingdom of Saudi Arabi

Shipeng Mao, Professor

Generalized Gradient Flow: Unconditionally Energy Stable Schemes and Applications to Geo-Energy Problems at Various Scales

15:00-16:00 August 16 (Wednesday), Z311

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 cases, it is crucial to design unconditional energy stable schemes for increasing its computational efficiency and enhancing its robustness. In this talk, we review our work on unconditionally energy stable schemes for geo-energy problems at various scales by presenting 4 stories:

1) Navier-Stokes-Cahn-Hilliard (NSCH) simulation for two-phase flow at the pore scale (micrometers’ 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) Darcy-scale simulation for two-phase flow (meters to kilometers’ scale): Conventional two-phase modeling consists of conservation laws and extended Darcy’s laws do not have a background dissipated energy.  With a single primary pressure variable, it fails to work in the degenerated single-phase regions, and it also difficult to be coupled with geomechanics due to capillarity.  We derive a general and thermodynamically consistent formulation of two-phase flow using our gradient flow framework, so the total Helmholtz energy is rigorously dissipating and it also yields a well-defined effective pressure. This new framework allows us to propose an energy stable numerical method, preserving energy dissipation, conservation for both fluids and pore volumes, and positivity of porosity and saturations.

3) Phase behavior calculation (scale independent): Conventional flash calculation methods based on fixed point iterations or Newton’s method is known to lack robustness. To design a fully robust scheme for flash, we first derived a gradient flow model for VT flash that preserves both Onsager's reciprocal principle and the energy dissipation. We then design unconditional stable yet linear semi-implicit scheme to update the moles and volume. Then, with the convex splitting approach and the component-wise iteration, our scheme becomes fully explicit, but still maintaining unconditional stability.  

4) Density functional theory (DFT) calculations of the structural, physical and chemical properties of reservoir fluid mixture (Angstroms’ scale): This is a very recent and still on-going work. From algorithms’ perspective, we design an unconditionally energy stable, but 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 completely removes the non-convergence issued faced by SCF methods for open shell problems. Our initial numerical test on the electronic structure calculation of the Lithium hydride (LiH) and the methane (CH4) molecules indicates that the new algorithm can speed up 100 or even 1000 times as compared to other orthonormality-preserving schemes for the Kohn-Sham gradient flow-based model in the literature.  

Bio: Shuyu Sun is a founding Professor of Earth Science and Engineering at King Abdullah University of Science and Technology (KAUST); he is also jointly affiliated with the Program of Applied Mathematics and Computational Science and the Program of Energy Resource and Petroleum Engineering at KAUST. He obtained his Ph.D. degree in computational and applied mathematics from The University of Texas at Austin in 2003.  He has published 340+ refereed journal articles, plus numerous conference papers and technical reports as well as a few book chapters. Based on Google Scholar, his total citation number (as of July 1 2023) is 9104 with a h-index of 46.  He has 4 papers recognized as “Highly Cited in Field in Web of Science”, and recently he published a book with Elsevier entitled “Reservoir Simulation: Machine Learning and Modeling”. Currently he is the president of InterPore (International Society for Porous Media) Saudi Chapter. He is also an editorial board member for Computational Geoscience, Gas Science and Engineering, and Journal of Computational Physics, three top journals in his field.