PRE Book 07 Standard handbook of petroleum and natural gas engineering
Standard Handbook of
Petroleum & Natural Gas
Engineering Second EditionThis page intentionally left blank
Standard Handbook of Petroleum & Natural Gas Engineering Second Edition Editors William C. Lyons, Ph.D., P.E. Gary J. Plisga, B.S.
AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO
Gulf Professional Publishing is an imprint of Elsevier Gulf Professional Publishing is an imprint of Elsevier 200 Wheeler Road, Burlington, MA 01803, USA Linacre House, Jordan Hill, Oxford OX2 8DP, UK Copyright © 2005, Elsevier Inc. All rights reserved.
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Librar y of Congress Cataloging-in-Publication Data
Standard handbook of petroleum & natural gas engineering.—2nd ed./ editors, William C. Lyons, Gary J. Plisga. p. cm. Includes bibliographical references and index.
ISBN 0-7506-7785-6
1. Petroleum engineering. 2. Natural gas. I. Title: Standard handbook of petroleum and natural gas engineering. II. Lyons, William C. III. Plisga, Gary J. TN870.S6233 2005 665.5–dc22
2004056285
British Librar y Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
ISBN: 0-7506-7785-6 For information on all Gulf Professional Publishing publications visit our Web site at www.gulfpp.com 10 9 8 7 6 5 4 3 2 1 Printed in the United States of America
Contents Contributing Authors vii Preface xi
5.2 Formation Evaluation 5-53
5.8 Secondary Recovery 5-177
5.7 Reserve Estimates 5-172
5.6 Decline Curve Analysis 5-168
5.5 Material Balance and Volumetric Analysis 5-161
5.4 Mechanisms & Recovery of Hydrocarbons by Natural Means 5-158
5.3 Pressure Transient Testing of Oil and Gas Wells 5-151
5.1 Basic Principles, Definitions, and Data 5-2
5.10 Estimation of Waterflood Residual Oil Saturation 5-201
5 Reser voir Engineering 5-1
4.21 Offshore Drilling Operations 4-558
4.20 Environmental Considerations for Drilling Operations 4-545
4.19 Corrosion in Drilling and Well Completions 4-501
4.18 Tubing and Tubing String Design 4-467
4.17 Well Cementing 4-438
4.16 Casing and Casing String Design 4-406
5.9 Fluid Movement in Waterflooded Reservoirs 5-183
5.11 Enhanced Oil Recovery Methods 5-211
4.14 Well Pressure Control 4-371
6.9 Environmental Considerations in Oil and Gas Operations 6-406
1 Index
Appendix: Units, Dimensions and Conversion Factors
7.2 Estimating the Value of Future Production 7-15
7.1 Estimating Producible Volumes and Future of Production 7-2
7 Petroleum Economic Evaluation 7-1
6.11 Industry Standards for Production Facilities 6-443
6.10 Offshore Operations 6-424
6.8 Corrosion in Production Operations 6-371
6 Production Engineering 6-1
6.7 Gas Production Engineering 6-274
6.6 Oil and Gas Production Processing Systems 6-242
6.5 Stimulation and Remedial Operations 6-218
6.4 Sucker Rod Pumping 6-120
6.3 Natural Flow Performance 6-89
6.2 Flow of Fluids 6-40
6.1 Properties of Hydrocarbon Mixtures 6-2
4.15 Fishing and Abandonment 4-378
4.13 Selection of Drilling Practices 4-363
1 Mathematics 1-1
1.9 Computer Applications 1-38
2.5 Geological Engineering 2-69
2.4 Thermodynamics 2-32
2.3 Strength of Materials 2-27
2.2 Fluid Mechanics 2-19
2.1 Basic Mechanics (Statics and Dynamics) 2-2
2 General Engineering and Science 2-1
1.8 Applied Statistics 1-31
2.7 Chemistry 2-100
1.7 Numerical Methods 1-20
1.6 Analytic Geometry 1-16
1.5 Differential and Integral Calculus 1-10
1.4 Trigonometry 1-8
1.3 Algebra 1-6
1.2 Geometry 1-2
1.1 General 1-2
2.6 Electricity 2-91
2.8 Engineering Design 2-129
4.12 Directional Drilling 4-356
4.4 Mud Pumps 4-95
4.11 MWD and LWD 4-300
4.10 Downhole Motors 4-276
4.9 Underbalanced Drilling and Completions 4-259
4.8 Drilling Mud Hydraulics 4-255
4.7 Bits and Downhole Tools 4-192
4.6 Drill String: Composition and Design 4-124
4.5 Drilling Muds and Completion Fluids 4-103
4.3 Rotary Equipment 4-82
3 Auxiliar y Equipment 3-1
4.2 Hoisting System 4-9
4.1 Drilling and Well Servicing Structures 4-2
4 Drilling and Well Completions 4-1
3.4 Compressors 3-48
3.3 Pumps 3-39
3.2 Power Transmission 3-17
3.1 Prime Movers 3-2
1
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Contributing Authors Egill Abrahamsen
Houston, Texas Jason Fasnacht
Schlumberger
Balikpapan, Indonesia Tracy Darr van Reet
Chevron—retired
El Paso, Texas Robert Desbrandes
Louisiana State University
Baton Rouge, Louisiana Aimee Dobbs
Global Santa Fe
Houston, Texas Patricia Duettra
Consultant in Applied Mathematics and Computer Analysis
Albuquerque, New Mexico Ernie Dunn
Weatherford International Limited
Houston, Texas Michael Economides
University of Houston
Boart Longyear
Weatherford International Limited
Salt Lake City, Utah Joel Ferguson
Weatherford International Limited
Houston, Texas Jerr y W. Fisher
Weatherford International Limited
Houston, Texas Robert Ford
Smith Bits International
Houston, Texas Kazimierz Glowacki
Consultant in Energy and Environmental Engineering
Krakow, Poland Bill Grubb
Weatherford International Limited
Houston, Texas Mark Heironimus
El Paso Production
El Paso, Texas Matthew Hill
Unocal Indonesia Company
Houston, Texas Heru Danardatu
Las Vegas, Nevada Robert B. Coolidge
Weatherford International Limited
Consultant in Petroleum Engineering
Houston, Texas Chip Abrant
Weatherford International Limited
Houston, Texas Bo Anderson
Weatherford International Limited
Houston, Texas Robert P. Badrak
Weatherford International Limited
Houston, Texas Frederick Beck
Consultant
Denver, Colorado Susan Beck
Weatherford International Limited
Houston, Texas Joe Berr y
Varco Incorporated
Houston, Texas Daniel Boone
Houston, Texas Gordon Bopp
Consultant in Geology and Geophysics
Environmental Technology and Educational Services Company
Richland, Washington Ronald Brimhall
Consultant
College Station, Texas Ernie Brown
Schlumberger
Sugarland, Texas Tom Carlson
Halliburton Energy Services Group
Houston, Texas William X. Chavez, Jr.
New Mexico Institute of Mining and Technology
Socorro, New Mexico Francesco Ciulla
Weatherford International Limited
Houston, Texas Vern Cobb
Consultant
Robert Colpitts
Jakarta, Indonesia
John Hosford
Weatherford International Limited
Houston, Texas Carroll Rambin
Weatherford International Limited
Lawrence, Kansas Toby Pugh
University of Kansas
Albuquerque, New Mexico Floyd Preston
Consultant in Hydrocarbon Properties
Houston, Texas Gar y J. Plisga
Bandung, Indonesia Jim Pipes
Houston, Texas Bharath N. Rao President, Bhavya Technologies, Inc. Richard S. Reilly
Chevron Texaco
Global Santa Fe
Houston, Texas Abdul Mujeeb
Henkels & McCoy, Incorporated
Blue Bell, Pennsylvania Bob Murphy
Weatherford International Limited
Houston, Texas Tim Parker
Weatherford International Limited
Weatherford International Limited
Houston, Texas
Institut Teknologi Bandung
New Mexico State University
El Paso, Texas Phillip Johnson
University of Alabama
Tuscaloosa, Alabama Harald Jordan BP America, Inc.
Farmington, New Mexico Mike Juenke
Weatherford International Limited
Houston, Texas Reza Kashmiri
International Lubrication and Fuel, Incorporated
Rio Rancho, New Mexico William Kersting, MS
Las Cruces, New Mexico Murty Kuntamukkla
Weatherford International Limited
Westinghouse Savannah River Company
Aiken, South Carolina Doug LaBombard
Weatherford International Limited
Houston, Texas Julius Langlinais
Louisiana State University
Baton Rouge, Louisiana William Lyons
New Mexico Institute of Mining and Technology
Socorro, New Mexico James Martens
Houston, Texas Pudji Permadi
F. David Martin
University of Tulsa
Tammoak Enterprises, LLC
Houston, Texas
Weatherford International Limited
Houston, Texas Ron Schmidt
National Oil Well
Socorro, New Mexico Eddie Scales
New Mexico Institute of Mining and Technology
Grand Junction, Colorado Jorge H.B. Sampaio, Jr.
Consultant in Environmental Engineering
Los Alamos, New Mexico Chris Russell
Socorro, New Mexico Cher yl Rofer
Huntington Beach, California Stefan Miska
New Mexico Institute of Mining and Technology
Consultant
Albuquerque, New Mexico George McKown
Smith Services
Houston, Texas David Mildren
Dril Tech Mission
Fort Worth, Texas Mark Miller
Pathfinder
Texas Richard J. Miller
Richard J. Miller and Associates, Incorporated
Tulsa, Oklahoma Tom Morrow
Ardeshir Shahraki Dwight’s Energy Data, Inc.
Smith Bits International
Louisiana State University
Albuquerque, New Mexico Andrzej Wojtanowicz
Sandia National Labs
Jack Wise
Consultant in Computer and Mathematics
Houston, Texas Sue Weber
Houston, Texas Bill Wamsley
Richardson, Texas Paul Singer
Weatherford International Limited
Houston, Texas Adrian Vuyk, Jr.
Houston, Texas Mark Trevithick T&T Engineering Services, Inc.
Weatherford International Limited
Socorro, New Mexico Jack Smith
New Mexico Institute of Mining and Technology
Baton Rouge, Louisiana
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Preface
Several objectives guided the preparation of this second edition of the Standard Handbook of Petroleum and Natu-
ral Gas Engineering
. As in the first edition, the first objective in this edition was to continue the effort to create for the worldwide petroleum and natural gas exploration and pro- duction industries an engineering handbook written in the spirit of the classic handbooks of the other important engi- neering disciplines. This new edition reflects the importance of these industries to the modern world economies and the importance of the engineers and technicians that serve these industries.
The second objective of this edition was to utilize, nearly exclusively, practicing engineers in industry to carry out the reviews, revisions, and any re-writes of first edition mate- rial for the new second edition. The third objective was, of course, to update the information of the old edition and to make the new edition more SI friendly. The fourth objective was to unite the previous two volumes of the first edition into a single volume that could be available in both book and CD form. The fifth and final objective of the handbook was to maintain and enhance the first edition objective of hav- ing a publication that could be read and understood by any up-to-date engineer or technician, regardless of discipline.
The initial chapters of the handbook set the tone by inform- ing the reader of the common language and notation all engineering disciplines utilize. This common language and notation is used throughout the handbook (in nearly all cases consistent with Society of Petroleum Engineers publi- cation practices). The 75 contributing authors have tried to avoid the jargon that has crept into petroleum engineering literature over the past few decades.
The specific petroleum engineering discipline chapters cover drilling and well completions, reservoir engineering, production engineering, and economics (with valuation and risk analysis). These chapters contain information, data, and example calculations directed toward practical situa- tions that petroleum engineers often encounter. Also, these chapters reflect the growing role of natural gas in the world economies by integrating natural gas topics and related subjects throughout the volume.
The preparation of this new edition has taken approxi- mately two years. Throughout the entire effort the authors have been steadfastly cooperative and supportive of the editors. In the preparation of the handbook the authors have used published information from both the American Petroleum Institute and the Society of Petroleum Engineers. The authors and editors thank these two institutions for their cooperation. The authors and editors would also like to thank all the petroleum production and service company employees that have assisted in this project. Specifically, edi- tors would like to express their great appreciation to the management and employees of Weatherford International Limited for providing direct support of this revision. The editors would also like to specifically thank management and employees of Burlington Resources Incorporated for their long term support of the students and faculty at the New Mexico Institute of Mining and Technology, and for their assistance in this book. These two companies have exhibited throughout the long preparation period exemplary vision regarding the potential value of this new edition to the industry.
In the detailed preparation of this new edition, the authors and editors would like to specifically thank Raven Gary. She started as an undergraduate student at New Mexico Institute of Mining and Technology in the fall of 2000. She is now a new BS graduate in petroleum engineering and is happily work- ing in the industry. Raven Gary spent her last two years in college reviewing the incoming material from all the authors, checking outline organization, figure and table organization, and references, and communicating with the authors and Elsevier editors. Our deepest thanks go to Raven Gary. The authors and editors would also like to thank Phil Carmical and Andrea Sherman at Elsevier for their very competent preparation of the final manuscript of this new edition. We also thank all those at Elsevier for their support of this project over the past three years.
All the authors and editors know that this work is not per- fect. But we also know that this handbook has to be written. Our greatest hope is that we have given those that will follow us in future editions of this handbook sound basic material to work with.
William C. Lyons, Ph.D., P.E.
Socorro, New Mexico and Gary J. Plisga, B.S. Albuquerque, New Mexico
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1 Mathematics Contents
1.1 GENERAL 1-2
1.2 GEOMETRY 1-2
1.3 ALGEBRA 1-6
1.4 TRIGONOMETRY 1-8
1.5 DIFFERENTIAL AND INTEGRAL CALCULUS 1-10
1.6 ANALYTIC GEOMETRY 1-16
1.7 NUMERICAL METHODS 1-20
1.8 APPLIED STATISTICS 1-31
1.9 COMPUTER APPLICATIONS 1-38
1-2 MATHEMATICS
A trapezoid has one pair of opposite parallel sides. A par-
similar
if their corresponding angles are congruent and cor- responding sides are proportional. A segment whose end points are two nonconsecutive vertices of a polygon is a
diagonal . The perimeter is the sum of the lengths of the sides.
1.2.3 Triangles A triangle is a three-sided polygon. The sum of the angles of a triangle is equal to 180
◦
. An equilateral triangle has three sides that are the same length, an isosceles triangle has two sides that are the same length, and a scalene triangle has three sides of different lengths.
A median of a triangle is a line segment whose end points are a vertex and the midpoint of the opposite side. An angle
bisector
of a triangle is a median that lies on the ray bisect- ing an angle of the triangle. The altitude of a triangle is a perpendicular segment from a vertex to the opposite side.
Two triangles are congruent if one of the following is given (where S = side length and A = angle measurement): SSS,
1.2.4 Quadrilaterals A quadrilateral is a four-sided polygon.
allelogram
. Angles are congruent if they have the same measure- ment in degrees and line segments are congruent if they have the same length. A dihedral angle is formed by two half-planes having the same edge, but not lying in the same plane. A plane angle is the intersection of a perpendicular plane with a dihedral angle.
has both pairs of opposite sides congruent and parallel. The opposite angles are then congruent, and adja- cent angles are supplementary. The diagonals bisect each other and are congruent. A rhombus is a parallelogram whose four sides are congruent and whose diagonals are perpendicular to each other.
A rectangle is a parallelogram having four right angles; therefore, both pairs of opposite sides are congruent. A rectangle whose sides are all congruent is a square.
1.2.5 Circles and Spheres If P is a point on a given plane and r is a positive number, the circle with center P and radius r is the set of all points of the plane whose distance from P is equal to r. The sphere with center P and radius r is the set of all points in space whose distance from P is equal to r. Two or more circles (or spheres) with the same P but different values of r are
concentric .
A chord of a circle (or sphere) is a line segment whose end points lie on the circle (or sphere). A line which intersects the circle (or sphere) in two points is a secant of the circle (or sphere). A diameter of a circle (or sphere) is a chord containing the center, and a radius is a line segment from the center to a point on the circle (or sphere).
The intersection of a sphere with a plane through its center is called a great circle. A line that intersects a circle at only one point is a tangent to the circle at that point. Every tangent is perpendicular to the radius drawn to the point of intersection. Spheres may have tangent lines or tangent planes.
Pi (p) is the universal ratio of the circumference of any circle to its diameter and is approximately equal to 3.14159. Therefore, the circumference of a circle is pd or 2pr.
1.2.6 Arcs of Circles A central angle of a circle is an angle whose vertex is the center of the circle. If P is the center and A and B are points, not on the same diameter, which lie on C (the circle), the
minor arc
AB is the union of A, B, and all points on C in the interior of <APB. The major arc is the union of A, B, and all points on C on the exterior of <APB. A and B are the end points of the arc and P is the center. If A and B are the end points of a diameter, the arc is a semicircle. A sector of a circle is a region bounded by two radii and an arc of the circle.
1.2.7 Concurrency Two or more lines are concurrent if there is a single point that lies on all of them. The three altitudes of a triangle (if taken as lines, not segments) are always concurrent, and their point of concurrency is called the orthocenter. The angle bisectors of a triangle are concurrent at a point equidistant from their sides, and the medians are concurrent two thirds of the way along each median from the vertex to the opposite side. The point of concurrency of the medians is the centroid.
1.2.8 Similarity Two figures with straight sides are similar if corresponding angles are congruent and the lengths of corresponding sides are in the same ratio. A line parallel to one side of a triangle divides the other two sides in proportion, producing a second triangle similar to the original one.
1.2.2 Polygons A polygon is a closed figure with at least three line segments that lies within a plane. A regular polygon is a polygon in which all sides and angles are congruent. Two polygons are
◦
1.1 GENERAL See Reference 1 for additional information.
. The radian system of measurement uses the arc length of a unit circle cut off by the angle as the measurement of the angle. In this system, a circle is measured as 2p radians, a straight line is p radians and a right angle is p/2 radians. An angle A is defined as acute if 0
1.1.1 Sets and Functions A set is a collection of distinct objects or elements. The inter- section of two sets S and T is the set of elements which belong to S and which also belong to T. The union (or inclusive) of S and T is the set of all elements that belong to S or to T (or to both).
A function can be defined as a set of ordered pairs, denoted as (x, y) such that no two such pairs have the same first element. The element x is referred to as the independent variable, and the element y is referred to as the dependent variable. A function is established when a condition exists that determines y for each x, the condition usually being defined by an equation such as y = f(x) [2].
References
1. Mark’s Standard Handbook for Mechanical Engineers, 8th Edition, Baumeister, T., Avallone, E. A., and Baumeister III, T. (Eds.), McGraw-Hill, New York, 1978.
1.2 GEOMETRY See References 1 and 2 for additional information.
1.2.1 Angles Angles can be measured using degrees or with radian mea- sure. Using the degree system of measurement, a circle has 360
◦
, a straight line has 180
◦
, and a right angle has 90
◦
◦
or are supplementary if their sum is 180
< A < 90
◦
, right if A = 90
◦
, and obtuse if 90
◦
< A < 180
◦
. Two angles are complemen-
tary
if their sum is 90
◦
1.2.9 Prisms and Pyramids A prism is a three-dimensional figure whose bases are any congruent and parallel polygons and whose sides are paral-
GEOMETRY 1-3 any polygon and with triangular sides meeting at a point in the Pythagorean theorem a plane parallel to the base.
2
2 v a
= + b Prisms and pyramids are described by their bases: a trian-
gular prism
has a triangular base, a parallelpiped is a prism For every pair of vectors (x
1 , y 1 ) and (x 2 , y 2 ), the vector sum
whose base is a parallelogram and a rectangular parallelpiped is given by (x
1 + x 2 , y 1 + y 2 ). The scalar product of the vector
is a right rectangular prism. A cube is a rectangular par- P = (x, y) and a real number (a scalar) r is rP = (rx, ry). allelpiped all of whose edges are congruent. A triangular
Also see the discussion of polar coordinates in the Section
pyramid
has a triangular base, etc. A circular cylinder is a “Trigonometry” and Chapter 2, “Basic Mechanics.” prism whose base is a circle and a circular cone is a pyramid whose base is a circle.
1.2.13 Lengths and Areas of Plane Figures For definitions of trigonometric functions, see “Trigonome-
1.2.10 Coordinate Systems try.” Each point on a plane may be defined by a pair of numbers. ●
Right triangle
(Figure 1.2.1) The coordinate system is represented by a line X in the plane (the x-axis) and by a line Y (the y-axis) perpendicular to line X
A
in the plane, constructed so that their intersection, the origin, is denoted by zero. Any point P on the plane can be described by its two coordinates, which form an ordered pair, so that c P(x
1 , y 1 ) is a point whose location corresponds to the real
b numbers x and y on the x-axis and the y-axis. If the coordinate system is extended into space, a third axis, the z-axis, perpendicular to the plane of the x
1 and y
1 c B
axes, is needed to represent the third dimension coordinate a defining a point P(x
1 , y 1 , z 1 ). The z-axis intersects the x and
2
2
2
y axes at their origin, zero. More than three dimensions c ( Pythagorean theorem) = a + b are frequently dealt with mathematically but are difficult to
2
cot A area = 1/2 • ab = 1/2 • a visualize.
2
2 The slope m of a line segment in a plane with end points
sin 2A = 1/2 • b tan A = 1/4 • c
P
1 (x 1 , y 1 ) and P 2 (x 2 , y 2 ) is determined by the ratio of the ● Any triangle
(Figure 1.2.2) change in the vertical (y) coordinates to the change in the horizontal (x) coordinates or
A
)/( )
2 − y 1 x 2 − x
1
m = (y except that a vertical line segment (the change in x coor-
c b dinates equal to zero) has no slope (i.e., m is undefined). h
A horizontal segment has a slope of zero. Two lines with the same slope are parallel and two lines whose slopes are negative reciprocals are perpendicular to each other.
B C
Because the distance between two points P (x , y ) and a
1
1
1 P (x , y ) is the hypotenuse of a right triangle, the length
2
2
2
area = 1/2 base • altitude = 1/2 • ah = 1/2 • ab sin C (L) of the line segment P P is equal to
1
2
y y ) = ± 1/2 • {(x
1 2 − x
2
1
2
2
( ) ) y y ) x
2 − x 1 + (y 2 − y
1 + (x
2 3 − x
3
2 L =
y y )}
- (x
3 1 − x
1
3 where (x , y ), (x , y ), (x , y ) are coordinates of vertices.
1
1
2
2
3
3
1.2.11 Graphs ●
Rectangle
(Figure 1.2.3) A graph is a set of points lying in a coordinate system and a graph of a condition (such as x = y + 2) is the set of all points that satisfy the condition. The graph of the slope-
intercept equation
, y = mx+b, is a straight line which passes
u
through the point (0, b), where b is the y-intercept (x = 0) b and m is the slope. The graph of the equation
D D
2
2
2
(
- (y − b) = r x − a) is a circle with center (a, b) and radius r. a
2
sin u area = ab = 1/2 • D
1.2.12 Vectors ● where u = angle between diagonals D, D.
Parallelogram
A vector is described on a coordinate plane by a directed seg- (Figure 1.2.4)
ment
from its initial point to its terminal point. The directed
b
segment represents the fact that every vector determines a magnitude and a direction. A vector v is not changed when
h
moved around the plane, if its magnitude and angular ori- entation with respect to the x-axis is kept constant. The
a u
initial point of v may therefore be placed at the origin of
D D
1
2
the coordinate system and v may be denoted by
C
v = a, b where a is the x-component and b is the y-component of the D sin u area = bh = ab sin c = 1/2 • D
1
2
1 D
1 D
2
2
(Figure 1.2.10)
2
Hyperbola
6
4
2
(x/a) length of perimeter of ellipse = p(a + b)K, where K = (1 + 1/4 • m
−1
area of ellipse = pab area of shaded segment = xy + ab sin
x b y y b a a
(Figure 1.2.9)
Ellipse
rad A where rad A = radian measure of angle A ● s = length of arc = r rad A
2
= 1/2 • r
2 A/360 ◦
area = 1/2 • rs = pr
r A s
- 1/64 • m
- 1/256 • m
- . . . ) m = (a − b)/(a + b) ●
1 D
2
2
1 and D
1-4 MATHEMATICS ●
Trapezoid
(Figure 1.2.5)
a u b h
D
2
area = 1/2 • (a + b)h = 1/2 • D
1 D 2 sin u
where u = angle between diagonals D
2 and where bases a and b are parallel. ● Any quadrilateral
Sector
(Figure 1.2.6)
b c u d a D
2
area = 1/2 • D
2
sin u
Note:
a
2
2
(Figure 1.2.8)
- b
- c
- d
- D
- 4 m
(y/a) = a
where m = distance between midpoints of D
2
a b b A
P y y a x
For any hyperbola, shaded area A = ab • ln[(x/a) + (y/b)] For an equilateral hyperbola (a = b), area A = a
2
sinh
−1
) = p(D
and D
2
cosh
−1
(x/a) where x and y are coordinates of point P. ●
Parabola
(Figure 1.2.11)
A h c
1
= D
1
area = pr
2 . ● Circles
− r
− d
2
)/4 = 2pR
′
b where R
′
= mean radius = 1/2 • (R + r) ●
2
area = p(R
2
R r d b D
(Figure 1.2.7)
Annulus
where r = radius d = diameter C = circumference = 2pr = pd. ●
2
= 0.785398 d
2
= 1/2 • Cr = 1/4 • Cd = 1/4 • pd
2
2
GEOMETRY 1-5
2
shaded area A = 2/3 • ch volume = pr h = Bh lateral area = 2prh = Ph where B = area of base ● P = perimeter of base
Any prism or cylinder
(Figure 1.2.15)
P y P u
T O F M
In Figure 1.2.12,
h
length of arc OP = s = 1/2 • PT + 1/2 • p • ln [cot(1/2 • u) ] Here c = any chord p = semilatus rectum
PT = tangent at P
Note:
OT = OM = x
1.2.14 Surfaces and Volumes of Solids ●
Regular prism
(Figure 1.2.13) volume = Bh = Nl lateral area = Ql where l = length of an element or lateral edge
B = area of base N = area of normal section ● Q = perimeter of normal section
Hollow cylinder
(right and circular)
2
2
volume = ph(R − r ) = phb(D − b) = phb(d + b) =
′
p hbD = phb(R + r)
h
where h = altitude r, R (d, D) = inner and outer radii (diameters) b = thickness = R − r
′
D ● = mean diam = 1/2 • (d + D) = D − b = d + b
Sphere
3
3
3
volume = V = 4/3 • pr = 4.188790r = 1/6 • pd =
3
0.523599d
2
2 r
area = A = 4pr = pd
a a
where r = radius 3 √ 3
a
6V/p = 1.24070 3 √ = A
V d = 2r = diameter =
A/p = 0.56419 volume = 1/2 • nrah = Bh ●
Hollow sphere
, or spherical shell lateral area = nah = Ph
3
3
3
3
2
) ) − r = 1/6 • p(D − d = 4pR volume = 4/3 • p(R t + where n = number of sides
1
3 B = area of base 1/3 • pt ● P = perimeter of base where R, r = outer and inner radii Right circular cylinder
(Figure 1.2.14)
D, d = outer and inner diameters t = thickness = R − r R
1 ● = mean radius = 1/2 • (R + r) Torus
, or anchor ring (Figure 1.2.16) r c
h r
2
2
cr volume = 2p
2
cr (proof by theorems of Pappus) area = 4pr
1-6 MATHEMATICS References and
3
3
2
2
3
1. Moise, E. E., and Downs, Jr., F. L., Geometry, Addison (
= a − 3a + 3ab − b a − b) Wesley, Melano Park, 1982.
(For higher-order polynomials, see the “Binomial Theo-
2. Graening, J., Geometry, Charles E. Merrill, Columbus,
n n
rem.”) a + b is factorable by (a + b) if n is odd, and 1980.
3
3
2
2
a )
- b = (a + b)(a − ab + b
n n
1.3 ALGEBRA
and a − b is factorable by (a − b), thus
n n n−1 n−2 n−2 n−1
See Reference 1.3 for additional information.
) a − b = (a − b)(a + a + b b + . . . + ab
1.3.1 Operator Precedence and Notation
1.3.4 Fractions Operations in an equation are performed in the following
The numerator and denominator of a fraction may be mul- order of precedence: tiplied or divided by any quantity (other than zero) without altering the value of the fraction, so that, if m = 0,
1. Parenthesis and grouping symbols
2. Exponents ma + mb + mc a + b + c
=
3. Multiplication or division (left to right) mx + my x + y
4. Addition or subtraction (left to right) To add fractions, transform each to a common denomina-
For example: tor and add the numerators (b, y = 0):
3
/ e a x ay bx a + b • c − d ay + bx
- = = will be operated upon (calculated) as if it were written b y by by by
3
)/ a + (b • c) − [(d e] To multiply fractions (denominators = 0): a x ax
The symbol |a| means “the absolute value of a,” or the
- = numerical value of a regardless of sign, so that b y by
| − 2| = |2| = 2 a ax
- x = The n! means “n factorial” (where n is a whole number) b b and is the product of the whole numbers 1 to n inclusive, so a x c axc
- that
= b y z byz 4! = 1 • 2 • 3 • 4 = 24
To divide one fraction by another, invert the divisor and multiply: 0! = 1 by definition a x a y ay
The notation for the sum of any real numbers a , a , . . . , a
1 2 n
- ÷ = = is
b y b x bx
n
a
i
1.3.5 Exponents
i=1 m n m n m+n m−n
a •a = a and a ÷ a = a and for their product
n
1
a and a = 1 (a = 0) = a a i
−m m i=1
a = 1/a The notation “x ∞ y” is read “x varies directly with y” or
m n mn
( a ) = a
“x is directly proportional to y,” meaning x = ky where k is some constant. If x ∞ 1/y, then x is inversely proportional √ √ n n
1/n m/n m
a = a and a = a to y and x = k/y.
n n n
( ab) = a b
1.3.2 Rules of Addition
n n n
( / a/b) = a b a + b = b + a (commutative property) ( a + b) + c = a + (b + c) (associative property)
Except in simple cases (square and cube roots), radical a − (−b) = a + b and signs are replaced by fractional exponents. If n is odd,
√ √ a − (x − y + z) = a − x + y − z n n a
−a = − (i.e., a minus sign preceding a pair of parentheses operates but if n is even, the nth root of −a is imaginary. to reverse the signs of each term within if the parentheses are removed)
1.3.6 Logarithms The logarithm of a positive number N is the power to
1.3.3 Rules of Multiplication and Simple Factoring N which the base must be raised to produce N. So, x = log b a • b = b • a (commutative property)
x
means b = N. Logarithms to the base 10, frequently used
( ab)c = a(bc) (associative property) in numerical computation, are called common or denary log- a(b + c) = ab + ac (distributive property)
arithms
, and those to base e, used in theoretical work, are a(−b) = −ab and − a(−b) = ab called natural logarithms and frequently notated as ln. In
2
2
( − b a + b)(a − b) = a any case,
2
2
2
( a + b) = a + 2ab + b log(ab) = log a + log b and
2
2
2
( = a − 2ab + b log(a/b) = log a − log b a − b)
3
3
2
2
3
( a + b) = a + 3a + 3ab + b
ALGEBRA 1-7
n 1/2
) log(a = n log a , etc., obtained with corresponding formulas for (1 − x) by reversing the signs of the odd powers of x. Provided
( log b b) = 1, where b is either 10 or e
|b| < |a|:
n
b log 0 = −∞
n n
( = a a + b) 1 + a log 1 = 0
n
2
3 n−1 n−2 n−3
= a + n
1 a 2 a b + n 3 a b + . . .
b + n log
10 e = M = 0.4342944819. . . , so for conversion
where n
1 , n 2 , etc., have the values given earlier.
log x
10 x = 0.4343 log e e )
and since 1/M = 2.302585, for conversion (ln = log
1.3.8 Progressions In an arithmetic progression, (a, a + d, a + 2d, a + 3d, . . . ), x ln x = 2.3026 log
10
each term is obtained from the preceding term by adding a constant difference, d. If n is the number of terms, the last term is p = a + (n − 1)d, the “average” term is 1/2(a + p)
1.3.7 Binomial Theorem and the sum of the terms is n times the average term or Let s = n/2(a + p). The arithmetic mean between a and b is (a + b)/2.
2
3
n
1 = n
In a geometric progression, (a, ar, ar , ar , . . . ), each term is obtained from the preceding term by multiplying by a con- n(n − 1)
n−1
n
2 = stant ratio, r. The nth term is ar , and the sum of the first n n
2! n terms is s = a(r − 1)/(r − 1) = a(1 − r )/(1 − r). If r is a
n
fraction, r will approach zero as n increases and the sum of n(n − 1)(n − 2) n
3 = n terms will approach a/(1 − r) as a limit.
3! The geometric mean, also called the “mean proportional,”
√ and so on. Then for any n, |x| < 1, between a and b is ab. The harmonic mean between a and b is 2ab/(a + b).
n
2
3
( = 1 + n
1 2 x + n 3 x + . . .
1 + x) x + n
1.3.9 Sums of the First n Natural Numbers If n is a positive integer, the system is valid without restriction
n ●