Michael L. Fripp

Acoustic telemetry for oilfield operations

The continuing need for a reliable data path up to 10 km long with reasonable data rate is a critical issue for any number of petroleum industry applications. This article gives a brief introduction to well completions; casing and tubing use in finished wells; and considerations affecting a commercial system addressing the data rate problem in a finished well. The Measure-While-Drilling application necessarily has a toolstring in the hole, which greatly complicates the already complex task of data communications.

Probability and Statistics

This chapter summarizes the probability that states the ratio of the number of ways a certain desired thing can happen to the total number of possible outcomes. The distribution function is essentially the frequency function integrated over the possible outcomes. This probability of the null event gets smaller and smaller as the pre-determined number of tries increases, but it never equals to zero. Contingency table is a table that lists all possible outcomes of the data and then counts the desired events and compares that to all possible events. Using probability theory greatly increases the range of statistics. If probability distributions are understood, then the data can be approximated and analyzed, and the reliability of the distribution is tested, and the data required for investigating the problem is also determined.

Math — The Basics

This chapter focuses on the basics of math. Math is a thought process tool created by human creatures. It is also a language that scientists and engineers use to describe nature. When two or more sets of numbers are together, then the combined number is called the sum or addition. The negative addition is called subtraction. Multiplication is just adding the number a bunch of times, and division is the inverse of multiplication. Zero is a special number, and any number added or subtracted to it remains same; any number multiplied by it gives zero and zero divided by any number gives zero. If the numerator has a lower magnitude than the denominator, the expression is called a fraction. Laws of addition, associative laws of multiplication, commutative laws of addition and multiplication, and distributive laws are explained in this chapter. Rational numbers, irrational numbers, imaginary numbers, complex numbers, absolute value, exponents, and logarithms are also discussed.

Chapter 8 – Differential Equations

Publisher Summary This chapter summarizes differential equations, which is the language of the engineer and scientist and stands alone as a mathematical entity. The general linear differential equation is of order “n,” the highest-order derivative in the equation. If h(t)=0, then the equation is said to be homogeneous and if h(t) is not equal to zero, then it is a non-homogeneous differential equation. Linear differential operator is also valid, as the same differential equation describes different physical phenomena. Linear differential equations in which all of the coefficient functions, that is all of the fi(t)s are constant, is called the linear differential equation with constant coefficients. The exponential solution in a linear differential equation with constant coefficient, the factor out of the exponential function, and the algebraic polynomial remaining are the characteristic equation of differential equations.

Chapter 3 – Algebra

Publisher Summary This chapter summarizes algebra, which is fundamental to engineering, and is the intellectual instrument which has been created for rendering clear quantitative aspects of the world. The symbol “x” represents all sorts of things. If the right hand side of the equation is zero, then the left hand side must be zero as well. Algebra is the study of mathematics in which the operations and procedures of addition and multiplication are applied to variables than to specific numbers. The powers of “x” are kept together in a combined term. First and second degree polynomials are discussed in the chapter.

Chapter 9 – Vector Calculus

Publisher Summary This chapter provides an overview of vector calculus, where a vector is a representation of a quantity and a direction. The dot product is also called the scalar product since it turns two vectors into a scalar quantity. Another lesser-used name for the dot product is the inner product. The dot product of a vector with itself produces the magnitude, squared of that vector. When one row is a linear combination of another row, the determinant is equal to zero. Two parallel vectors are linear combinations of each other. Hence, the triple scalar product is zero if any two vectors are parallel. When two rows are interchanged, the sign on the determinant changes. The surface integral is useful in many calculations ranging from how much sound comes from a surface to strength of attraction between two magnets. The surface integral is moving the standard integral into two dimensions. The energy required to overcome the friction along the path is a dissipative energy, an energy primarily in the form of heat which is not readily put to useful work. The units of the electrical field are newtons/coulomb.

Chapter 5 – Trigonometry

Publisher Summary This chapter provides an overview of trigonometry, which means triangle measurement. Any two angles that sum to a right angle are called complementary angles. An angle smaller than a right angle is called an acute angle, and an angle larger than a right angle is called an obtuse angle. Angles add just like other sets of numbers. Any two angles which, when added together, make a straight line are called supplementary angles. A vector is a representation of both a quantity and a direction. The sum of forces at any place and at any time must equal to zero. Laws of trigonometry and periodicity are also discussed in this chapter.

Chapter 2 – Functions

Publisher Summary This chapter focuses on different types of mathematical functions. A function is a correspondence, transformation, or mapping of a chosen set of variables into another set of values. A graph of a function is the visual representation of a function, and it consists of all points, “x” and “f(x),” where “x” is in the independent variable set of the function and “f(x)” is the corresponding value in the dependent variable set. Coordinate points, distance and slope are also explained in the chapter.

Chapter 11 – Chaos

Publisher Summary This chapter focuses on chaos, which is the function of engineering and engineering math, all along to determine the behavior of physical systems. It describes that long-term mathematical predictions of the behavior of chaotic systems are no more accurate than random chance, whereas short-term predictions, can be accurate. The idea that such small perturbations in initial conditions can have such large effects later on is sometimes called the Butterfly Effect. Complex fluid flow systems include an equation for the conservation of momentum, another one for the conservation of energy, and a third for the conservation of material in the flow.

Chapter 6 – Calculus

Publisher Summary This chapter focuses on calculus, which is a practical tool of mathematics. Differential calculus, derivatives of compound functions, the importance of zero value velocity, and integral calculus are explained in the chapter. The fundamental theorem of calculus shows the relationship between differential calculus and integral calculus. Different examples on semiconductor photon detector, economics, pressure, automobile acceleration, Fourier analysis, Root Mean Square (RMS) are also illustrated in the chapter.