Probability in Petroleum and Environmental Engineering

George V Chilingar, Leonid F. Khilyuk, Herman H. Reike

• Publisher: Gulf Publishing Company
• Year: 2012
• ISBN:
  9780127999708
• Topic: Technology & Engineering
• Subtopic: Environmental Management
• Index: Technology & Engineering

Written by three of the world's most renowned petroleum and environmental engineers, Probability in Petroleum and Environmental Engineering is the first book to offer the practicing engineer and engineering student new cutting-edge techniques for prediction and forecasting in petroleum engineering and environmental management. The authors combine a rigorous, yet easy-to-understand, approach to probability and how it is applied to petroleum and environmental engineering to solve multiple problems that engineers or geologists face every day.

CHAPTER 1

INTRODUCTION

Environmental issues have the highest possible priority in contemporary science. There is no need to discuss the reasons for this situation in detail—they are almost obvious. The purpose of applied science is to satisfy the demands of daily life, and today’s realities are such that the deteriorating environment threatens the quality of life in industrialized countries and people’s existence in developing nations.

THE APPROACH

Many efforts by concerned scientists and engineers have been made in environmental health and safety, and to correct ecological disturbances during the last 30 years, ever since the Rome’s Club (Meadows et al., 1974) identified the main survival problem as adaptation to The Limits to Growth. Many useful, but many more unsuccessful, environmental programs and projects were developed and implemented. A new generation of progressive politicians was raised, who devoted their professional careers to the themes of environmental protection and improvement. Owing to the news media, general awareness of the educated public about possible disasters has grown dramatically. Meanwhile, globally, the environmental problems are getting worse every day.

This does not mean, however, that the developed and implemented programs were useless. The tendency toward deterioration of the environment is increasing because the rate of deterioration is considerably higher than that of improvement. It is obvious that one cannot count on rapid progress in the solution of environmental problems. Only continuous, persistent, and meticulous work can overcome the negative environmental tendencies.

The leading and organizing role in this work belongs to education. Many colleges and universities worldwide instituted academic programs to prepare specialists in environmental protection. Almost all the programs, however, were highly specialized owing to the prevailing scientific interests of a particular institution. Many different environmental specialties have appeared, and there are hundreds of different titles of positions, occupants of which can hardly interact or even communicate effectively with each other.

To overcome these obstacles, leading environmental scientists borrowed from the industrial-military planning and development process and introduced the interdisciplinary concept of the systems analysis approach. This approach systematically uses mathematical models of various forms for analysis and synthesis of suitable formal descriptions of a studied system together with their informal interpretation. It is noteworthy that the form and complexity of the models developed depend on the purpose of their application and available resources that can be used for the solution of the problem.

For environmental problems, the outcome of a monitored process frequently depends on numerous uncontrolled factors that have a random nature. The results of monitoring the environmental processes can be presented as outcomes of some stochastic experiments (real or conceptual). These outcomes can be conveniently interpreted as corresponding random events, which occur in some stochastic experiments. This opens a wide area of applications for the concepts and models based on probability theory in the field of environmental issues.

In this book, the authors combined a rigorous and yet easy axiomatic approach to probability theory with numerous examples of environmental applications. The book is written as a treatise on basic probabilistic concepts and methods. In each chapter the probabilistic concepts are considered together with examples of environmental applications. There is no doubt that such an approach can provide students and practicing environmentalists with a convenient, practical guide to the theoretical issues, and simultaneously present specific conceptual approaches in the development of useful environmental applications.

For this book, the key word is chance, often associated with odds, occasion, event, accident, and luck. The situations in which one uses these terms are endless in variety, but the common feature is the presence of uncertainty. In playing roulette one puts a token on say rouge without being sure of success. The general feeling about any new business is that it is risky, and it is customary to evaluate the chances for success before starting it.

Evaluation (or estimation) of chances for occurrence of an event of interest is even more important for environmental issues. Possible atmospheric events, for example, are predicted with probabilistic estimates of their occurrence. Consequences of environmental projects cannot even be expressed without the use of probabilistic terms.

In these situations, the events of interest may or may not occur. It is natural to call the events of this kind random. For random events, it is usually desirable to evaluate the chances of their appearance (which requires some measure to express the chances). It is customary to use probability of event as such a measure.

The computation or estimation of probabilities of the events of interest is the main focus of probability theory. In brief, probability theory is a collection of methods for the evaluation of probabilities of the events of interest, based on given probabilities of some set of primary events. Conceptual basis of the theory was shaped during the last 70 years, and now it is possible to select its most fruitful and applicable concepts and methods. Simultaneously, it is possible to develop the most convenient pedagogic methods of presenting the theory keeping in mind that the book is addressed to students, scientists, managers, and engineers specializing in environmental issues.

Taking these circumstances into consideration, the authors have chosen the direct way of presenting the main probabilistic concepts, using examples of corresponding environmental issues as illustrations and sources of probability problems. This book can be regarded as a necessary introductory guide to probability theory and its logic for the students and professionals who will have to ameliorate and/or manage environmental messes. The authorsbelieve that numerous exercises and examples of environmental applications will convince the students and practicing environmentalists of the necessity and usefulness of the probabilistic methodology.

OVERVIEW

The main goals and overview of the book are presented in Chapter 1. The primary concepts of experiment and event are discussed at the beginning of Chapter 2. The material that follows in Chapter 2 is, in considerable degree, traditional for the contemporary probabilistic courses and relates to the operations with events and various systems of events. Chapter 2 is concluded with a specially developed scheme of interpretation of an environmental phenomenon as a set of events of a particular stochastic experiment.

An essential innovation is introduced in Chapter 3. It relates to the notion of an elementary event, the basic theoretical concept that allows constructing strict mathematical models for probabilistic compositions. The elementary event is strictly defined based on a general definition of event and the relation of implication. Notation ⊂ is used for this relation throughout the book. The expression A ⊂ B means: “If event A occurs, then event B occurs,” or “event A implies event B.”

Any event of a particular experiment can be defined as a proper combination of elementary events. The space of elementary events is considered to be a union of all possible elementary events. It contains all possible outcomes of a specific stochastic experiment. In the last section of Chapter 3, examples of composition of the spaces of elementary events for specific environmental problems are considered in detail.

Chapter 4 covers the main probabilistic concepts. It introduces the idea of probabilistic space and presents various methods of its construction and possible definitions of probability. The examples of their application for description of environmental uncertainties are discussed at the end of Chapter 4.

The concept of conditional probability forms the core of Chapter 5. Closely related issues, such as the Formula of Total Probability and Bayes’ Formula are discussed as the expansions of this concept. Independence of random events is one of the most applicable concepts for the probability computation in composite probabilistic spaces. It is defined and discussed in terms of conditional probability (if–then propositions).

Sequences of independent trials and related probability distributions are discussed in Chapter 6. Before the calculation of any related probabilities, one needs to compose a probabilistic space for the sequence of trials based upon the spaces of the individual trials. The authors explain in detail how one can construct a composite space for a multistage stochastic experiment. Inasmuch as the Bernoulli framework of independent trials can be applied to numerous practical problems, the writers also included the traditional material on Bernoulli (binomial) distribution and supporting theoretical information.

Random variables and their distribution functions are introduced in Chapter 7. Two numerical characteristics of distributions—moments and quantiles—are considered in Chapters 8 and 9. All theoretical concepts are illustrated with applications addressing ecological issues.

The most commonly used probability distributions for discrete and continuous random variables are described in Chapters 10 and 11. The normal distribution should be studied carefully, because it forms the theoretical foundation of many applied mathematical models, and it has numerous applications in the evaluation of probabilities of events associated with random variables in the environmental issues. The last section of Chapter 11 contains a detailed discussion and examples of the role of the lognormal distribution in modeling random variables characterizing concentrations of pollutants in various media.

Chapter 12 contains general information on and examples of the probabilistic limit theorems of two kinds: the law of large numbers and central limit theorem. The authors discuss the conditions of validity of these theorems and examples of their applications for stochastic processes in the environment.

Methods of decision making under uncertainty are discussed in Chapter 13. They include techniques of assessing environmental risks and methods of estimating the properties of a random variable based on statistical sampling.

The book is designed for students, scientists, and engineers who have completed a two-semester introductory calculus course. The authors wish to assure the readers that with the basic background, the world of probability, with all its important applications in science as well as everyday life, is accessible to them. Given its mix of abstrac...