Mathematical Modelling and Numerical Simulation of Oil Pollution Problems

Satellite observations and their derived products played a key role during the Deepwater Horizon oil spill monitoring efforts in the Gulf of Mexico in April–July 2010. These observations were sometimes the only source of synoptic information available to monitor and analyse several critical parameters on a daily basis. These products also complemented in situ observations and provided data to assimilate into or validate model. The ocean surface dynamics in the Gulf of Mexico are dominated by strong seasonal cycles in surface temperature and mixing due to convective and storm energy, and by major currents that include the Loop Current and its associated rings. Shelf processes are also strongly influenced by seasonal river discharge, winds, and storms. Satellite observations were used to determine that the Loop Current exhibited a very northern excursion (to approximately 28\(^{\circ }\)N) during the month of May, placing the core of this current and of the ring that it later shed at approximately 150 km south of the oil spill site. Knowledge gained about the Gulf of Mexico since the 1980s using a wide range of satellite observations helped understand the timing and process of separation of an anticyclonic ring from the Loop Current during this time. The surface extent of the oil spill varied largely based upon several factors, such as the rate of oil flowing from the well, clean up and recovery efforts, and biological, chemical, and physical processes. Satellite observations from active and passive radars, as well as from visible and infrared sensors were used to determine the surface extent of the oil spill. Results indicate that the maximum and total cumulative areal extent were approximately 45 \(\times \) 10\(^3\) km\(^2\) and 130 \(\times \) 10\(^3\) km\(^2\), respectively. The largest increase of surface oil occurred between April 22 and May 22, at an average rate of 1.3 \(\times \) 10\(^3\) km\(^2\) per day. The largest decrease in the extent of surface oil started on June 26, at an average rate of 4.4 \(\times \) 10\(^3\) km\(^2\) per day. Surface oil areas larger than approximately 40 \(\times \) 10\(^3\) km\(^2\) occurred during several periods between late May and the end of June. The southernmost surface oil extent reached approximately 85\(^{\circ }\)W 27\(^{\circ }\)N during the beginning of June. Results obtained indicate that surface currents may have partly controlled the southern and eastern extent of the surface oil during May and June, while intense southeast winds associated with Hurricane Alex caused a reduction of the surface oil extent at the end of June and beginning of July, as oil was driven onshore and mixed underwater. Given the suite of factors determining the variability of the oil spill extent at ocean surface, work presented here shows the importance of data analyses to compare against assessments made to evaluate numerical models.

In this chapter, a strategy for the bioremediation of marine shorelines polluted with oil is presented. Several discharge points are chosen in a limited region in order to release a nutrient and reach critical concentration of this substance in the oil-polluted shorelines. The strategy is optimal in the sense that the location of the discharge points and the release rates are planned so as to minimize the amount of the nutrient introduced into the aquatic system. To accomplish this task, a variational problem is solved to find the location of the discharge point in each oil-polluted zone, and to determine a basic (preliminary) shape of its release rate. After that, a quadratic programming problem is solved to specify the strength of these release rates in order to reach the critical concentration in all the polluted zones during the same time interval. An initial-boundary value 3D advection-diffusion problem and its adjoint problems are considered in a limited area to model, estimate and control the dispersion of the nutrient. It is shown that the advection-diffusion problem is well posed, and its solution satisfies the mass balance equation. In each oil-polluted zone, the mean concentration of nutrient is determined by means of an integral formula in which the adjoint model solution serves as the weight function showing the relative contribution of each source. Critical values of these mean concentrations are used as the constraints for the variational problem as well as for the quadratic programming problem. The ability of new method is demonstrated by numerical experiments on the remediation in oil-polluted channel using three control zones.

The formation of water-in-oil emulsions, a major complication in oil spills, is described. Research has shown that asphaltenes are the prime stabilizers of water-in-oil emulsions and that resins are necessary to solvate the asphaltenes. It has also been shown that many factors play a role, including the amount of saturates and the oil viscosity. Two schemes are given to describe the formation of emulsions using the characteristics of starting oils including the resin and asphaltene contents and the viscosity. Essentially, water droplets injected into the oil by turbulence or wave action can be stabilized temporarily by the oil viscosity and on a longer-term basis by resins and then asphaltenes. Depending on the starting oil properties, four types of water-in-oil types are created: meso-stable and stable emulsions, entrained water-in-oil and unstable or those-that-do-not-form types. Each type is described and has unique properties. For most oils, loss of lighter components by evaporation is necessary before the oils will form a water-in-oil type. It was noted that variability in emulsion formation is, in part, due to the variation in types of compounds in the asphaltene and resins groups. Certain types of these compounds form more stable emulsions than others within the same asphaltene/resin groupings. A review of numerical modelling schemes for the formation of water-in-oil emulsions is given. A recent model is based on empirical data and the corresponding physical knowledge of emulsion formation. The density, viscosity, asphaltene and resin contents were correlated with a new stability index. A simplified screening approach is also described. Although of lesser accuracy, the approach is simple to implement.

The behaviour of inhomogeneous suspensions in a viscous oil is relevant in the context of oil spill and other oil-related disasters which may lead to the unwanted mixture of sand grains and oil. This warrants the fundamental study of the dynamics of solid particles in a thin film of viscous fluid. Specifically, sheared concentrated suspensions in a viscous fluid are subject to a diffusive mechanism called shear-induced migration that consists of “drift diffusion” and “self or tracer diffusion”. Drift diffusion causes particles to move from high to low concentrations, while tracer diffusion dictates mixing between particles of the same size. The latter mechanism becomes important in polydisperse slurries. In this chapter, we incorporate the effects of shear-induced migration and sedimentation to develop a model for the gravity-driven thin film of bidensity suspensions. We use this mathematical model to validate recent experimental results.

Quantifying uncertainties in real-time operational oil spill forecasts remains an outstanding problem, but one that should be solvable with present science and technology. Uncertainties arise from the salient characteristics of oil spill models, hydrodynamic models, and wind forecast systems, which are affected by choices of modelling parameters. Presented and discussed are: (1) a systems-level approach for producing a range of oil spill forecasts, (2) a methodology for integrating probability estimates within oil spill models, and (3) a multi-model system for updating forecasts. These technologies provide the next steps for the efficient operational modelling required for real-time mitigation and crisis management for oil spills at sea.

Both deterministic and probabilistic strategies are employed in numerical oil spill model to estimate potential oil spill risk. The deterministic model simulates transport and weathering processes by means of a particle tracking method. While a Monte Carlo stochastic simulation approach is run for multiple scenarios, spill size, oil type, and environmental conditions (meteorological and hydrological data) combinations, to characterize the consequences of spills for a specified potential spill location. The statistically-defined oil spill map does not demonstrate the probabilities of oil-slick presence for each grid area, but also provide the information of the shortest arrival time which is quite vital for oil contingency plan.