The Lattice Boltzmann Method for Ocean Oil Spill Propagation Modeling and Simulation—A Comparison Study of the Navier–Stokes Model and the Advection Diffusion Model

Ocean oil spills have devastating impacts on marine ecosystems and human society in the surrounding coastal areas. The 2010 Deep Water Horizon spill in the Gulf of Mexico lasted 87 days and is estimated to have released over 3 million barrels of oil. It impacted over 1,600 miles of coastline, killed over 8000 marine animals/seabirds and caused direct economic loss from fishing and tour industries estimated at tens of billions of dollars [1]. In addition, the impacts to long-term public health and quality of life of millions of people are still unknown.

The transport of oil spilled into the ocean is a complex process that depends in a critical way on the current, wind, temperature and chemical composition of the oil and seawater [2]. Weathering further compounds the complexity of this process, a phenomenon which involves evaporation, emulsification dissolution, oxidation and microbial processes. Spilled oil in ocean undergoes these physical, chemical and biological processes will be transformed into substances with physical and chemical characteristics that differ from the original source material [3]. The authors in [3] describe the fate of oil spilled into oceans as going through three major phases: (i) after oil introduced into the oceans; (ii) transport the resulting degradation oil away from the source; and (iii) incorporate the residual substances into compartments of the earth’s surface system. One process may play a dominating role over the others in certain phases during the lifespan of spilled oil. Therefore, it is important to use suitable models in different phases that reflect the underlying behaviors of spilled oil.

Many models use Advection-Diffusion Equation (ADE) with various ways of obtaining the ocean current, wind and tide data, to predict oil slick transport [4]. GNOME (General NOAA Operational Modeling Environment) is an ADE based modeling tool which is used to predict the possible route, or trajectory of pollutants. While solutions of the ADE provide changes of oil concentration over time and space, it does not compute the advection field, but uses an external ocean surface velocity field as input. ADE based models can work well for modelling the fate of spilled oil at certain phase, such as phase (ii). But it may not work well for other phases of oil spills, such as phase (i), where the oil concentration is very high in the vicinity of initial spill location. Therefore, it is important to consider the impact of velocity generated by rapid distribution of oil density. In such cases, we may need to take advantage of the Navier-Stokes Equation (NSE) that does conserve both mass and momentum while solving the velocity field.

We investigated the feasibility of using the Lattice Boltzmann Method (LBM) as a framework to model and simulate ocean oil spill propagation at surface level. Previous research suggests that (a) LBM has certain advantages compared to traditional analytical and numerical approaches for solving complex nonlinear ocean flow dynamic problems with reasonable accuracy and computational complexity [5]; (b) LBM can be a NSE or an ADE solver providing numerical solutions for ocean fluid problems that fit the governances of NSE or ADE [6]; (c) LBM provides a flexible structure to model multi-scale, multi-fluid ocean oil spills with a variety of boundary conditions [7]. Therefore, LBM can be a promising framework for modelling ocean oil spill fate and transport at different phases.

We develop a LBM model and simulation that is capable of providing numerical solutions for both NSE and ADE based models. To validate typical models, one of the most commonly used benchmarks for NSE solvers is Poiseuille Flow and for ADE solvers is Gaussian Hill with a simplified velocity field. However, the ocean surface current is a much more complex velocity field that is temporal-spatial dependent. The major contributions of our work are: (a) in addition to the most common benchmarks for LBM NSE and LBM ADE solvers, we tested the LBM ADE solver against a Finite Difference Method (FDM) ADE solver using a perturbation of the Taylor-Green velocity field. To the best of our knowledge, no such benchmark have been done in the past for an LBM ADE model using a velocity field as complex as the perturbed Taylor-Green field. Our test results show that the LBM ADE and the FDM ADE agree closely; (b) we experimented coupling a LBM NSE solver with a LBM ADE solver using the NSE to solve the ocean surface velocity field, then feeding the velocity to the ADE solver for tracking oil concentration in slicks and their transport. Results show that the LBM provide a simple velocity projection schema that allows LBM NSE solvers to integrate several forces (ocean current, winds, tides, gravity) into a single ocean surface velocity field, which feeds into LBM ADE solvers to enhance its accuracy. As of future work, we will extend our work to use LBM model for multi-fluids with weathering effects and to use LBM NSE solvers to assimilate ocean sensor data with some existing ocean surface velocity models.

References:

[1] “Deep Water – The Gulf Oil Disaster and the Future of Offshore Drilling”, Report to the President by National Commission on the BP Deepwater Horizon Oil Spill and Offshore Drilling. January 2011.

[2] K. Mishra, G. S. Kumar, “Weathering of Oil Spill: Modelling and Analysis”, International Conference on Water Resources, Coastal and Ocean Engineering (ICWRCOE 2015).

[3] National Research Council 2003. “Oil in the Sea III: Inputs, Fates, and Effects.” The National Academies Press. https://doi.org/10.17226/10388.

[4] Cucco et al, “A High-Resolution Real-Time Forecasting System for Predicting the Fate of Oil Spills in the Strait of Bonifacio (western Mediterranean Sea)”. Marine Pollution Bulletin 64 (2012) 1186-1200L..

[5] L. Lou, “The lattice-gas and lattice Boltzmann methods: Past, present, and future.” Proceedings of Applied CFD 4, Beijing, October 16-19, 2000, pp. 52-83.

[6]Timm Kruger et al, “The Lattice Boltzmann Method – Principles and Practice.” Springer, 2017.

[7] Shan and H. Chen, “Lattice Boltzmann model for simulating flows with multiple phases and components.” Physical Review, Vol. 47, No. 3, March 1993.