Methodological Models for Optimal Control of Marine Oil Spill
Chapter One
Aim and Objectives of the Study
The aim of this study is to formulate mathematical abstractions of the control processes where decisions would be taken at several stages through an optimal control path to achieve the following specific objectives:
1. minimizing the uncertainty in the remote sensing data to reduce the high number of false alarms (oil slick look-a-likes) phenomena,
2. minimizing the apparent toxicological effect of clean-up technique like chemical disperssants,
3. determining the control measure that would cause a process to satisfy the physical constraints of chemical dispersants applications, and at the same time
4. optimizing some performance criteria for all future earnings from marine biota.
Chapter Two
Review of Related Literature
Oil Spill Control Modelling
In the modern Control literature about marine oil spills, substantial consideration has been given to modelling the transport and fate of oil spilled in water as a method of control, with the assumption that spill fate and transport modelling can help to facilitate short-term tactical forecasting in making decisions to contain, control and clean up accidental spill, and all involved linking mathematical formulations or algorithms to represent the oil transport and
fate processes, with the transport computation used to determine the oil movement in space and time, and the fate trajectory used to estimate oil transport between various environmental compartments (ASCE, 1996; Reed et al., 1999).
Although earliest approaches to oil spill control modelling used semi-empirical formulae for the slick area evaluation (Fay, 1969, 1971; Lehr and Cekirge, 1980), the complexity of oil spill fate and transport in water governed by interrelated physicochemical processes that depend on oil properties, hydrodynamic conditions, and environmental conditions triggered a new developments of computational technique in which many elements of mathematical
methods like partial differential equation and variational calculus dominated oil spill transport and fate modelling(Spaulding, 1988; Steinberg et al., 1997). With these alternatives, oil slick models were formulated via the Navier-Stokes equations.
Elaborate information on the use of Navier-Stokes equations that accounted for major phenomena of oil physics in an aquatic environment was given by Tkalich et al. (2003), where a multiphase oil spill model that accounted for major phenomena of oil physics in an aquatic environment was formulated. In formulating the multiphase model, six state variables were computed simultaneously, which included oil slick thickness on the water surface, and
concentration of dissolved, emulsified and particulate oil phases in the bottom sediments at
Singapore Strait. The effectiveness of different oil combating techniques was also evaluated, which included recovery of spilled oil from the sea surface with mechanical devices, ispersion of oil into the water column with chemical dispersants; and sinking of oil with heavier-than-water materials.
Other contributors to marine oil spill modelling include Xiaobo et al. (2003)who formulated mathematical model via simulation that was capable of predicting the oil particle concentration distribution in water body at the Singapore Straits. Satellite images, field observations of oil slicks on the water surface, and measurements of the vertical concentration of oil particles in flume were conducted to validate the model.
Du et al (2005) used Random Walk Method in a Monte Carlo simulation framework to track the transport of oil due to the effects of waves, buoyancy, and turbulent diffusion. The method of moments was used to derive the spreading parameters of oil under regular waves.
Wang et al. (2008) developed a three-dimensional numerical simulation model for transport of oil spills in sea. The amount of oil released at sea was distributed among a large number of particles tracked individually. A Lagrangian discrete particle algorithm was applied to simulate the transport and fate of the oil slick, which was treated as an ensemble of a large number of small particles whose discrete path and mass were followed and recorded as
functions of time.
Mendes et al.(2009) conducted a study aiming at the estimation of dispersion through Ria de Aveiro of an offshore oil-spill that reached the Barrier/lagoon mouth. The approach used consisted of the Lagrangian modelling of passive particle emissions. The Lagrangian model was coupled to a hydrodynamic model, previously calibrated and validated for this lagoon.
This model solved the shallow waters equations, obtained by vertical integration of the continuity and Navier-Stokes equations, representing the fundamental principles of mass and momentum conservation in a fluid (assumed Newtonian).
Guo et al.(2009) proposed a numerical method to simulate oil spill trajectories, which were affected by the combination of advection, turbulent diffusion and mechanical spreading process, based on a particle tracking algorithm. Thus, in modelling the diffusion process, a discrete method was employed for the generation of fractional Brownian motion (fBm) to illustrate super diffusive transport.
Conversely, a study by Redondo et al. (2008a) also showed that oil spill fate and transport modelling were only a first step in marine oil spill control paradigm. This was further sustained by Chigbu and Bassey (2010). A mathematical formulations to control the use of combating techniques like chemical dispersant which required the addition of another substance to the water, which in some cases, may be toxic to marine biota has just began.
Chapter Three
The Methodological Models
This chapter focused on the design and development of methodological control models via Operational Research formalism. The first section involves the introduction and development of a framework for optimal control theory of marine oil spill, which is a new strategy for combination methodology. The next section involves the introduction and formulation of an optimal control theory in marine oil spill management, as well as the development of decision
rules for optimal decision via sequential optimization models and Markovian decision processes.
Methodological Model for Combination Methodology Recent literature on oil spill problem of observability and detectability in water made references to the use of remote sensing technique. The solution module of the remote sensing application only expressed a situation whereby potential regions affected by oil spill within
marine environment could be captured via remote sensor (Karathanassi et al., 2006). This section presents a new theoretical strategy for optimization of oil slick look-a-likes phenomena associated with optical detectors (remote sensing).
The Conceptual Model
Starting from selection of locations within the spill region for the deployment of containment, we developed a dynamic model of a new strategy based on a diffusion process, which makes the distinction between two types of optimization objectives: increasing awareness and changing.
Chapter Four
Optimal Control Theory for Marine Oil Spill Problem
This section focused on formulation of mathematical expression for theoretical formulation of a typical marine oil spill problem.
Problem Statement
We considered hypothetical situation where a typical region was assumed to be exposed to a serious oil spill that resulted in series of conflicts between the host community in the exploration region and the oil firm. We also assumed that the problem for the oil firm operating in the area in collaboration with the Government was to determine the optimal response and remediation regime in order to maximize the present value of all future earnings
associated with the given area of their operation in every exploration, drilling or transportation. This followed the fact that, whenever there was a spill incidence in that region, aside from the cost of compensation, the affecting community will cause a temporary suspension of operation in the area. This temporary suspension could always result in the loss of lots of millions of dollars.
Chapter Five
Concluding Remarks
Conclusions
Emphasis of earlier studies on marine oil spill modelling has primarily been on the formulation of mathematical models of oil transport and fate processes to forecast the trajectories of spilled oil. This study introduced a new marine oil spill research direction where an optimal control theory could be used to enhance spill detection and evaluation of clean-up techniques applications. Focus was typically, under relatively restrictive assumption that the monitoring tool was a remote sensor with element of uncertainty in its signal data, while the clean-up tool was chemical dispersants with conflicting priorities in its applications.
The study showed that the problem of data uncertainty with remote sensing and conflicting priorities with chemical dispersant could be solved with some OR techniques known as coherent pluralism and sequential optimization. The methodological models describing these techniques were developed and validated theoretically as a critical first step in objectifying the proposed optimal control theory in marine oil spill modelling. The study also showed, through
a philosophical illustration, that with the assimilation of the proposed research direction, objective functional could be abstracted for any marine oil spill scenario as an optimization problem.
Consequently, statistical relevance of the models was given as an analogy while the extension of the study to a numerical implementation is recommended for further studies.
Contribution to Knowledge
Over the past three decades, there have been a plethora of theoretical developments with optimal control models and their applications to practical problems in management science and engineering. An extension of this development to marine oil spill control to deal with uncertainty in optical sensor data and toxicity of chemical dispersants application is an innovation in marine oil spill literature.
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