Bergische Universität Wuppertal Fachbereich Mathematik und Naturwissenschaften Angewandte Mathematik - Numerische Analysis

People Research Publications Teaching

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Mathematical Modelling and Numerical Simulation of Oil Pollution Problems Editor Matthias Ehrhardt (Bergische Universität Wuppertal) Series "The Reacting Atmosphere", Vol. 2, Springer Verlag, Heidelberg, March 2015, 166 p. 58 illus., 44 illus. in color. ISBN: 978-3-319-16458-8

Contents: Written by outstanding experts in the fields of marine engineering, atmospheric physics and chemistry, fluid dynamics and applied mathematics, the contributions in this book cover a wide range of subjects, from pure mathematics to real-world applications in the oil spill engineering business.
Offering a truly interdisciplinary approach, the authors present both mathematical models and state-of-the-art numerical methods for adequately solving the partial differential equations involved, as well as highly practical experiments involving actual cases of ocean oil pollution.
It is indispensable that different disciplines of mathematics, like analysis and numerics, together with physics, biology, fluid dynamics, environmental engineering and marine science, join forces to solve today's oil pollution problems.

Academic Level: The book will be of great interest to researchers and graduate students in the environmental sciences, mathematics and physics, showing the broad range of techniques needed in order to solve these pollution problems; and to practitioners working in the oil spill pollution industry, offering them a professional reference resource. The book is designed to be consisting of a collection of contributed chapters. Outstanding experts working successfully in this challenging research area will be invited to contribute each a chapter of roughly 30-40 pages to this volume.

Preface

This second volume is edited by the applied mathematics group organized at the Institute of Mathematical Modelling, Analysis and Computational Mathematics (IMACM). There is a strong interest at the IMACM in tackling the mathematical problems arising in the environmental sciences like the modelling of oil spills, especially the analysis of mathematical models of these spills in seas, design of numerical algorithms for transport of oil spills, and creating methods for determining the location, time and total power of oil emissions ('inverse problems'), i.e., for the simulation and detection of oil spill emitters.

Let me emphasize that there are close links between oil pollution problems and air pollution models. First, on a modelling level the oil pollution on the sea surface is naturally coupled to pollution in the lower layers of the atmosphere. Secondly, from a purely mathematical point of view, the models, i.e., the partial differential equations exhibit a very similar structure. Finally, from a socio-economic viewpoint, a major part of the pollution in both areas is caused by humans and is often the result of economic decisions or behaviour.

The collected chapters in this book cover a wide range of subjects, from pure mathematics to real-world applications in the oil spill engineering business. The reader will quickly recognise the increasingly interdisciplinary nature of these works. It is indispensable that different disciplines of mathematics, like analysis and numerics, together with physics, biology, fluid dynamics, environmental engineering and marine science join forces to solve today's oil pollution problems.

The principal audience of this book is graduate and Ph.D. students in the environmental sciences, mathematics & physics, lecturers in the environmental sciences, mathematics & physics, and researchers working in the oil spill pollution industry, offering them a professional reference resource.

  Equilibrium theory of bidensity particle-laden flows on an incline (13 pages)
     by Sungyon Lee, Department of Mechanical Engineering, Texas A&M University, USA
     and Jeffrey Wong, Department of Mathematics, University of California Los Angeles, USA
     and Andrea L. Bertozzi, Department of Mathematics, University of California Los Angeles, USA

Abstract: The behavior 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.

  Operational oil spill modelling: from science to engineering applications in the presence of uncertainty (27 pages)
     by Ben R. Hodges, Dept of Civil, Architectural and Environmental Engineering University of Texas at Austin, USA,
     and Alejandro Orfila, Institut Mediterrani d'Estudis Avançats-IMEDEA (CSIC-UIB), Mallorca, Spain
     by Juan Manuel Sayol,
     and Xianlong Hou, Dept of Civil, Architectural and Environmental Engineering University of Texas at Austin, USA,

Abstract: 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.

  A strategy for bioremediation of marine shorelines polluted with oil by using several nutrient release points (28 pages)
     by David Parra-Guevara,
     and Yuri N. Skiba, Centro de Ciencias de la Atmósfera, Universidad Nacional Autónoma de México Circuito Exterior, Ciudad Universitaria. México.

Abstract: Crude oil is one of the most important organic pollutants in marine environments. It has been estimated that worldwide approximately 1.3×10⁶ tons of petroleum impact marine waters and estuaries annually. Massive releases from pipelines, wells and tankers receive the most public attention, but in fact these account for only a relatively small proportion of the total petroleum entering the environment. Almost 50% comes from natural seeps, and less than 9% emanates from catastrophic releases. Consumption and urban run-off is responsible for almost 40% of the input.
Independently of the source of pollution, a substantial number of smaller releases of petroleum occur regularly in coastal waters, as a result, oil stranded in shorelines has become a common problem which needs attention. It is well known that oil is comprised of many different toxic compounds which endanger the marine environment involved in a spill, however there are many natural, native microorganisms which are not only capable, but thrive on the decomposition of these toxic compounds. This process of using microorganisms for such cleanup efforts in shorelines is known as bioremediation, and it has proven to be a successful method for the cleanup of marine areas affected by oil spills.
There are two different techniques of bioremediation used for oil spill cleanup: bioaugmentation and biostimulation. Bioaugmentation is the addition of microorganisms capable of degrading the toxic hydrocarbons, in order to achieve a reduction of the pollutants. Biostimulation is the addition of nutrients needed by indigenous hydrocarbon degrading microorganisms in order to achieve maximum degradation of toxic compounds present in the oil. The degradation of hydrocarbons (biodegradation) begins by the conversion of the alkanes chain or polycyclic aromatic hydrocarbons (PAH) into alcohol. Oxidation then converts the compound to an aldehyde and then into an acid and eventually into water, carbon dioxide, and biomass.
In this chapter, an optimal strategy for the bioremediation (biostimulation) of marine shorelines polluted with oil is presented. Suppose that in a limited sea area along the shoreline there are N oil-polluted zones. In this area, N discharge points are selected in order to release a nutrient and reach critical concentration of the substance in the above mentioned zones. The strategy is optimal in the sense that the location of discharge points and their release rates are planned with the aim of minimizing the amount of the nutrient to be introduced into the aquatic system. This problem is solved in two stages, each of which represents a certain minimization procedure. In the first (prediction) stage, the variational problem is formulated and solved separately for each contaminated zone to determine an optimum discharge point and a basic temporal behavior of discharge rate. In the second (correction) stage, the amplitudes of basic discharge rates are corrected by using a quadratic programming problem in order to reach the critical concentration of nutrient simultaneously in all polluted zones. The existence and uniqueness of solutions of the minimization problems arising in each of these two stages are proved.
The new multi-point strategy is applied to the initial-boundary value problem for the three-dimensional advection-diffusion equation. It is shown that such problem is well posed in the sense of Hadamard, and its solution satisfies the mass balance equation. This problem is used together with its adjoint problems to simulate, estimate and control the dispersion of the nutrient in a limited area. It should be noted that the mean concentration of nutrient for each oilpolluted zone is determined by means of an integral formula in which the adjoint model solution serves as a weight function. The critical values of these mean concentrations are used as the constraints for the variational problem in the prediction stage as well as for the quadratic programming problem posed in the correction stage. In this way, the adjoint model solutions are essential elements in the development of the new bioremediation strategy.
Finally, with the help of numerical schemes by Marchuk and Crank-Nicolson, we construct an unconditionally stable and efficient numerical algorithm of second-order approximation in space and time for the solution of the dispersion model and its adjoint. The performance of the new strategy is shown by using numerical experiments on the remediation of several oil-polluted zones in a channel.

  Application of a numerical statistical model to estimate potential oil spill risk (14 pages)
     by Weijun Guo, College of Environmental Sciences and Engineering, Dalian Maritime University, Dalian 116026, China,
     and Tiaojian Xu, State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China.

Abstract: 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 methods. 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.

  Prediction of the Formation of Water-in-Oil Emulsions (24 pages)
     by Merv Fingas, Spill Science, Edmonton, Alberta, Canada.

Abstract: 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 graphical approach is also described. Although of lesser accuracy, the approach is simple to implement.

  Variability of the Deepwater Horizon Surface Oil Spill Extent and its Relationship to Varying Ocean Currents and Extreme Weather Conditions (24 pages)
     by G.J. Goni,
National Oceanic and Atmospheric Administration, Atlantic Oceanographic and Meteorological Laboratory, 4301 Rickenbacker Causeway, Miami, FL 33149, USA,
     and J.A. Trinanes,
National Oceanic and Atmospheric Administration, Atlantic Oceanographic and Meteorological Laboratory, 4301 Rickenbacker Causeway, Miami, FL 33149, USA.
University of Miami, Rosenstiel School of Marine and Atmospheric Science, Cooperative Institute for Marine and Atmospheric Studies, 4600 Rickenbacker Causeway, Miami, FL 33149, USA.
University of Santiago de Compostela Laboratory of Systems, Technological Research Institute, Campus Vida 15782, Santiago de Compostela, Spain.
National Oceanic and Atmospheric Administration, National Environmental Satellite Data and Information Service, CoastWatch, 5200 Auth Road, Camp Springs, MD 20746, USA,
     and A. MacFadyen,
National Oceanic and Atmospheric Administration, Office of Response and Restoration, Emergency Response Division, 7600 Sandpoint Way, Seattle, WA 98115, USA,
     and D. Streett,
National Oceanic and Atmospheric Administration, National Environmental Satellite Data and Information Service, Office of Satellite and Product Operations, Camp Springs, MD 20746, USA,
     and M.J. Olascoaga,
University of Miami, Rosenstiel School of Marine and Atmospheric Science, Ocean Sciences Department, 4600 Rickenbacker Causeway, Miami, FL 33149, USA,
     and M.L. Imhoff,
Pacific Northwest National Laboratorys Joint Global Change Research Institute, 5825 University Research Court, College Park, MD 20740, USA,
     and F. Muller-Karger,
University of South Florida, College of Marine Science, 140 7th Avenue South, St. Petersburg, FL 33701, USA,
     and M.A. Roffer,
Roffers Ocean Fishing Forecasting Service, Inc., 60 Westover Drive, West Melbourne, FL 32904, USA.

Abstract: 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. They 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° 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 45×10³ km² and 130×10³ km², respectively. The largest increase of surface oil occurred between April 22 and May 22, at an average rate of 1.3×10³ km² per day. The largest decrease in the extent of surface oil started on June 26, at an average rate of 4.4×10³ km² per day. Surface oil areas larger than approximately 40×10³ km² occurred during several periods between late May and the end of June. The southernmost surface oil extent reached approximately 85° W27° 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.

  Structural Analysis of Oil-Spill Booms (26 pages)
     by Frédéric Muttin, EIGSI La Rochelle & Casablanca, Engineering School, France

Abstract: Floating barriers, often named booms, are used to contain oil. They are a main device installed during pollution response and their efficient positioning is a critical question for both effective oil containment and structural material resistance.
A 3D non-linear finite-element model for static moored booms is forced by sea current hydrodynamic pressure. To improve the numerical convergence of the membrane equilibrium during the Newton-Raphson scheme we initialize the 3D solution by using a 2D non-linear cable model. The membrane stretched surface representing the booms permits to define the material stress and the boom underwater skirt angulation. Scale-one experimentations are realized on the European Atlantic coast to measure boom mooring tension and boom skirt geometry.
In this chapter, in-situ experimental method at coastal sea is given. Validation protocol of numerical results by experimental ones is described. Threshold values on boom tension, to avoid structural break, and skirt angle to evaluate the oil containment efficiency are discussed.
Finally, methodological aspect to combat oil pollution by using contingency planning based on such numerical modelling of booming structure is addressed.