In Trog w Course Note Outline - [PDF Document] (2024)

7/29/2019 In Trog w Course Note Outline

1/27

Introduction to Groundwater Hydrology

Ye Zhang

Dept. of Geology & Geophysics

University of Wyoming

c Draft date February 26, 2012

7/29/2019 In Trog w Course Note Outline

2/27

Contents

Contents i0.1 This Class . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . vi

0.1.1 Textbook . . . . . . . . . . . . . . . . . . . . . . . . .. . vii0.1.2 Tools . . . . . . . . . . . . . . . . . . . . . . . .. . . . . viii

0.1.3 Questions and Answers . . . . . . . . . . . . . . . . . .. viii0.1.4 Homework, Labs, Exams and Grades . . . . . . . . . . .. ix

0.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . xi0.2.1 Groundwater Science . . . . . . . . . . . . . .. . . . . . . xi0.2.2 Groundwater Resources . . . . . . . . . . . .. . . . . . . xi0.2.3 Groundwater Quality . . . . . . . . . . . . .. . . . . . . xi0.2.4 Groundwater in Wyoming . . . . . . . . . . .. . . . . . . xii0.2.5 What Groundwater Scientists Do . . . . . . .. . . . . . . xii

0.3 Homework 1 . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . xiii

1 Basic Math 1

1.1 Scientific Notation . . . . . . . . . . . . . . . . . . . .. . . . . . 11.2 Decimal Places . . . . . . . . . . . . . . . . . .. . . . . . . . . . 11.3 Dimension Analysis . . . . . . . . . . . .. . . . . . . . . . . . . . 21.4 Logarithm & Exponential . . .. . . . . . . . . . . . . . . . . . . 31.5 Areas, Volumes,Circumferences . . . . . . . . . . . . . . . . . . . 41.6 AnalyticGeometry & Trigonometric Functions . . . . . . . . . . 41.7Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 41.8 Slope . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 51.9 Differentiation . . . . . . . . . . . . . . .. . . . . . . . . . . . . 61.10 Functions of One Variable . . . . .. . . . . . . . . . . . . . . . . 9

1.11 Functions of two or more variables . . . . . . . . . . . .. . . . . 101.12 Scalar, Vector, Matrix . . . . . . . . . . . . . .. . . . . . . . . . 101.13 The Summation Sign . . . . . . . . . . .. . . . . . . . . . . . . . 131.14 Test 1 . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13

2 Introduction 152.1 The Hydrologic Cycle . . . . . . . . . . .. . . . . . . . . . . . . 152.2 Fluxes Affecting Groundwater . . .. . . . . . . . . . . . . . . . . 17

2.2.1 Infiltration and Recharge . . . . . . . . . . . . . . . .. . 17

i

7/29/2019 In Trog w Course Note Outline

3/27

ii CONTENTS

2.2.2 Evapotranspiration . . . . . . . . . . . . . . . . . . . .. . 192.2.3 Groundwater Surface Water Interaction . . . . . . . . .. 192.2.4 Hydrologic Balance . . . . . . . . . . . . . . . . . . .. . . 19

2.3 Properties of Water . . . . . . . . . . . . . . . . . . . .. . . . . . 202.3.1 Molecular Properties . . . . . . . . . . . . .. . . . . . . . 202.3.2 Density and Compressibility . . . . . . . .. . . . . . . . . 212.3.3 Viscosity . . . . . . . . . . . . . . . .. . . . . . . . . . . 212.3.4 Surface Tension & Capillarity . .. . . . . . . . . . . . . . 22

2.4 Properties of Porous Media . . . . . . . . . . . . . . . . .. . . . 232.4.1 Porosity . . . . . . . . . . . . . . . . . . . . .. . . . . . . 232.4.2 Grain Size of Unconsolidated Sediments . . .. . . . . . . 25

2.5 Fluid Mechanics Background . . . . . . . . . . . . . . . . .. . . 282.5.1 Pressure Variation in a Static Fluid . . . . . . . .. . . . 282.5.2 Bernoullis Equation for a Dynamic Fluid . . . . . .. . . 30

2.6 Groundwater Hydrostatics . . . . . . . . . . . . . . . . . .. . . . 312.6.1 Hydrostatic Versus Hydrodynamic Conditions . . . .. . . 32

2.7 Measurement of Hydraulic Head . . . . . . . . . . . . . . .. . . 322.7.1 What it is that a well measures? . . . . . . . . . .. . . . 342.7.2 Well Schematics . . . . . . . . . . . . . . . . . .. . . . . 35

2.8 Homework 2 . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 382.9 Test 2 . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 38

3 Aquifer and its properties 393.1 Shallow Groundwater . . . . .. . . . . . . . . . . . . . . . . . . 393.2 A Few Definitions . . .. . . . . . . . . . . . . . . . . . . . . . . 413.3 Classificationof Aquifers . . . . . . . . . . . . . . . . . . . . . . . 413.4Aquifer Properties (Optional) . . . . . . . . . . . . . . . . . . .. 453.5 Test 3 . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 46

4 Principles of Groundwater Flow 474.1 Introduction . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . 474.2 Darcys Law .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3Hydraulic Conductivity . . . . . . . . . . . . . . . . . . . . . .. 494.4 Darcy Flux & Average Linear Velocity . . . . . . . . .. . . . . . 504.5 Darcys Law in Three Dimensions . . . . . . . . .. . . . . . . . . 51

4.5.1 Darcy Flux of This Course . . . . . . . . . . . . . . . .. 564.6 Intrinsic Permeability . . . . . . . . . . . . . . . . . .. . . . . . 58

4.7 Homework 3 . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 614.8 Test 4 . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 614.9 Limitation of the Darcys Law . . . . .. . . . . . . . . . . . . . . 61

4.9.1 Continuum Assumption . . . . . . . . . . . . . . . . . . .614.9.2 Laminar Flow . . . . . . . . . . . . . . . . . . . . . . .. . 64

4.10 Darcys Law Derived from the Navier-Stokes Equation(Optional) 644.11 Heterogeneity & Anisotropy . . . . . . . . .. . . . . . . . . . . . 664.12 Flow Across Interface . . . . . . .. . . . . . . . . . . . . . . . . 724.13 Equivalent HydraulicConductivity . . . . . . . . . . . . . . . . . 72

7/29/2019 In Trog w Course Note Outline

4/27

CONTENTS iii

4.14 Transmissivity . . . . . . . . . . . . . . . . . . . . . .. . . . . . 754.14.1 Isotropy in Transmissivity . . . . . . . . . .. . . . . . . . 764.14.2 Anisotropy in Transmissivity . . . . . . .. . . . . . . . . 77

4.15 Measuring Hydraulic Conductivity . . . . . . . . . . . . .. . . . 774.15.1 Grain Size Analysis . . . . . . . . . . . . . . .. . . . . . 774.15.2 Laboratory Measurements With Permeameter . . .. . . . 784.15.3 Other Tests . . . . . . . . . . . . . . . . . . .. . . . . . . 81

4.16 Variable-Density Groundwater Flow (Optional) . . . . . . .. . . 824.16.1 Pressure Formulation . . . . . . . . . . . . . . . .. . . . 824.16.2 Ghyben-Herzberg Relation . . . . . . . . . . . . .. . . . 84

4.17 H omework 4 . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 864.18 Test 5 . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 87

5 Geology and Groundwater Flow 895.1 Aquifers and ConfiningLayers . . . . . . . . . . . . . . . . . . . . 895.2 Recharge andDischarge . . . . . . . . . . . . . . . . . . . . . . . 90

5.3 Water Table & Potentiometric Surface . . . . . . . . . .. . . . . 945.4 Groundwater Surface Water Interaction . . . . . . .. . . . . . . 955.5 Groundwater Flow in Geological Processes . . .. . . . . . . . . . 95

5.5.1 Topography-Driven Flow . . . . . . . . . . . . . . . . . .975.5.2 Water/Rock Interaction . . . . . . . . . . . . . . . . . .. 98

5.6 Groundwater in Unconsolidated Deposits . . . . . . . . . . .. . 995.7 Groundwater in Sedimentary Rocks . . . . . . . . . . . .. . . . 1025.8 Homework 5 . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 1035.9 Midterm Exam . . . . . . . . . . . . . . .. . . . . . . . . . . . . 103

6 Deformation, Storage, and General Flow Equations 1076.1Effective Stress . . . . . . . . . . . . . . . . . . . . . . . . .. . . 1076.2 Excavation Instability and Liquefaction . . . . . . .. . . . . . . 1106.3 Matrix Compression . . . . . . . . . . . . . .. . . . . . . . . . . 1116.4 Aquifer Storage . . . . . . . . . . .. . . . . . . . . . . . . . . . . 114

6.4.1 Elastic Storage in A Confined Aquifer . . . . . . . . . .. 1166.4.2 Water Table Storage in An Unconfined Aquifer . . . . . .117

6.5 General Groundwater Flow Equations . . . . . . . . . . . . .. . 1206.5.1 3D General Flow Equation (Confined Aquifer) . . . . .. 1216.5.2 2D Planeview Flow Equation (Confined & Unconfined) .126

6.6 Modeling Overview . . . . . . . . . . . . . . . . . . . . .. . . . . 130

6.7 Homework 6 . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 1336.8 Test 6 . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 134

7 Steady-State Flow Analysis 1357.1 Steady Flow in a ConfinedAquifer . . . . . . . . . . . . . . . . . 136

7.1.1 Uniform Flow . . . . . . . . . . . . . . . . . . . . . . .. . 1367.1.2 Radial Flow to A Well . . . . . . . . . . . . . . . .. . . . 1407.1.3 Superposition of Simple Solutions . . . . . . . .. . . . . . 141

7.2 Wells Near Straight Constant Head Boundary . . . . . . . . .. . 143

7/29/2019 In Trog w Course Note Outline

5/27

iv CONTENTS

7.2.1 With Background Uniform Flow . . . . . . . . . . . . . .1467.3 Wells Near Straight Impermeable Boundary . . . . . . . . . .. . 1487.4 Flow Net (Optional) . . . . . . . . . . . . . . . . . .. . . . . . . 150

7.4.1 Total Discharge Through a Flow Net . . . . . . . . . . . .1527.4.2 How to Draw a Flow Net . . . . . . . . . . . . . . . . . .1537.4.3 Anisotropic System . . . . . . . . . . . . . . . . . . . .. . 1557.4.4 Another Use of Coordinate Transform . . . . . . . . .. . 158

7.5 Steady Flow in an Unconfined Aquifer . . . . . . . . . . . .. . . 1607.5.1 Uniform Flow . . . . . . . . . . . . . . . . . . . .. . . . . 1607.5.2 Radial Flow to a Well . . . . . . . . . . . . .. . . . . . . 1617.5.3 Superposition & Image Well . . . . . . .. . . . . . . . . . 161

7.6 Homework 7 . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 1637.7 Test 6 . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 165

8 Transient Flow Analysis 1678.1 Radial Flow in a ConfinedAquifer . . . . . . . . . . . . . . . . . 167

8.1.1 Limitations of Thesis Formula . . . . . . . . . . . . . .. . 1708.1.2 Use Thesis Formula to Predict Drawdown . . . . . . . .. 1738.1.3 Use Thesis Formula for Parameter Estimation . . . . . .. 173

8.2 Jacob Late Time Approximation of the Theis Solution . . . .. . 1778.2.1 Semi-Log Drawdown vs. Time . . . . . . . . . . . . . .. 1778.2.2 Semi-Log Drawdown vs. Distance . . . . . . . . . . . . .180

8.3 Additional Considerations . . . . . . . . . . . . . . . . .. . . . . 1818.3.1 Superposition in Space . . . . . . . . . . . . .. . . . . . . 1838.3.2 Superposition in Time (Optional) . . . . . .. . . . . . . . 185

8.4 Homework 8 . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 1878.5 Final Exam . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 189

9 Appendix A Coordinate Transform for Laplace Equation 191

10 Laboratories 19310.1 Porosity and Related Parameters . . . .. . . . . . . . . . . . . . 193

10.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . .. . . 19310.1.2 This Lab . . . . . . . . . . . . . . . . . . . . .. . . . . . 19410.1.3 Assignments: . . . . . . . . . . . . . . . .. . . . . . . . . 19610.1.4 Final Thoughts . . . . . . . . . . . .. . . . . . . . . . . . 198

10.2 Grain Size Distribution and Hydraulic Properties . . . . .. . . . 200

10.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . .. . . 20010.2.2 This Lab . . . . . . . . . . . . . . . . . . . . .. . . . . . 20010.2.3 Assignments: . . . . . . . . . . . . . . . .. . . . . . . . . 20510.2.4 Final thoughts . . . . . . . . . . . .. . . . . . . . . . . . 209

10.3 Darcy Test . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 21010.3.1 Introduction . . . . . . . . . . . . . . . .. . . . . . . . . 21010.3.2 A Single Test . . . . . . . . . . . . .. . . . . . . . . . . . 21010.3.3 Multiple Tests . . . . . . . . .. . . . . . . . . . . . . . . 21310.3.4 This Lab . . . . . . . . .. . . . . . . . . . . . . . . . . . 213

7/29/2019 In Trog w Course Note Outline

6/27

CONTENTS v

10.3.5 Assignments . . . . . . . . . . . . . . . . . . . . . . .. . 21410.3.6 Final thoughts . . . . . . . . . . . . . . . . . . .. . . . . 215

10.4 Equivalent Hydraulic Conductivity . . . . . . . . . . . . .. . . . 21610.4.1 Introduction . . . . . . . . . . . . . . . . . .. . . . . . . 21610.4.2 This Lab . . . . . . . . . . . . . . . . .. . . . . . . . . . 21610.4.3 Assignments: . . . . . . . . . . . .. . . . . . . . . . . . . 219

10.5 Regional Flow Analysis of the Powder River Basin . . . . .. . . 22210.5.1 Introduction . . . . . . . . . . . . . . . . . . .. . . . . . 22210.5.2 A Modeling Workflow . . . . . . . . . . . . .. . . . . . . 22310.5.3 Assignments . . . . . . . . . . . . . . . .. . . . . . . . . 228

7/29/2019 In Trog w Course Note Outline

7/27

vi CONTENTS

This is a course note written & assembled by Ye Zhang for anintroductoryGroundwater Hydrology class.

Spring 2011

GEOL 4444/ GEOL 5444

4 CREDITSGRADING: A-F

Prerequisite: Calculus I & II

Location: GE318

Class days & times: Tues + Thurs (9:35 am 10:50 am)Officehours: M(4:005:30 pm), F(3:004:30 pm), GE 318Email:[emailprotected]

Phone: 307-766-2981

********** No field trip in September **********

Lab days & times: Tues (3:10 pm 5:00 pm), GE318Labinstructor: Mr. Guang Yang (aka Kelvin)

Office hours: TBA by TA in the first lab.

Email: [emailprotected]

Phone: TBA by TA in the first lab.

NOTE: The lecture notes will be periodically posted on theWyowebcourse website, usually before a class so students canfamiliarize them-selves with the martials. Please make a habit ofchecking for notesand other materials from the site.

NOTE: The lecture notes do not include: (1) solutions to theexer-cises and homework; (2) proofs to theories and equationderivations.These will be presented during the lectures only. Thus,do not relyon the notes for everything attendance and in-classparticipationare key. Due to time limitation, not allproofs/derivations can be pre-

sented. Some advanced derivations (though noted as detailsgivenin class in notes) will be posted on Wyoweb, typically under afoldercalled Advanced Folder.

0.1 This Class

In this class, a fairly rigorous mathematical treatment ispresented, which de-viates from the typical introductory classesthat emphasize the more applied

7/29/2019 In Trog w Course Note Outline

8/27

0.1. THIS CLASS vii

aspects. I feel that mastering of such skills would comenaturally after a stu-dent has first developed a firm grasp of thefundamentals. This course is thusdesigned at the upper undergradand graduate level, appropriate for the levelof mathematical rigorcontained herein. To fully comprehend the materials pre-sented, astudent should have a sufficient knowledge in college math, i.e.,Calcu-lus (and preferably differential equations). If in doubt,please review Chapter 1,Basic Math to determine if a preliminaryclass may be necessary before takingthis class.

Throughout the course, many formulations and equations aredevelopedusing differential equations and integration. The emphasisis on under-standing how these equations are obtained and theirunderlying as-sumptions. However, you will rarely be tested onequation derivations inexams/quizes (those few that you may betested on Ill let you know), nor isit necessary to memorize a largenumber of specific formulas or solutions (typi-cally, the exam/quizwill provide the necessary formulas so understanding whatthese meanand how to use them is key). Therefore, do not unduly dwellupon

the derivation details, almost all homework and exam questionscan be solvedusing a pencil and a calculator.

0.1.1 Textbook

The textbook for this course is Groundwater Science by C. R.Fitts, pub-lished in 2002 by the Academic Press. Many exercises,homework, and readingassignments are selected from this book. It isrecommended that you own acopy. For some homework problems, thebook provides the final answers (onpage 415). However, yourcompleted assignment must provide the entire anal-ysis rather thana single number. Some of the answers occasionally contain

typographical errors, so do not entirely rely on them to judgeyour own results.There are a few other minor mistakes in thecurrent version of the book. Seean Errata list:

http://www.academicpress.com/groundwater.Make sure you make theappropriate corrections before using the textbook.To allow anin-depth analysis of the steady state and transient flowsystems

within a one-semester time frame, the chapters in this book onwater chemistryand groundwater contaminations are not included.However, your money shouldnot be wasted as these materials can beuseful in subsequent classes on ground-water modeling. Tosupplement the textbook, additional materials are compiled

& assembled based on several books, each with its ownspecial emphasis:

General Overview: Groundwater, Freeze & Cherry, 1979,Prentice Hall, p604.

General Overview: Applied Hydrogeology, Fourth Edition, C. W.Fetter,2001, Prentice Hall, p 598.

Practical Problems: Practical Problems in Groundwater Hydrology,Bair& Lahm, 2006, Prentice Hall.

7/29/2019 In Trog w Course Note Outline

9/27

viii CONTENTS

Mathematical Treatment: Dynamics of Fluids in Porous Media, J.Bear,1988, Dover Publications, p 784.

Mathematical Treatment: Quantitative Solutions in Hydrogeologyand Ground-water Modeling, Neven Kresic, 1997, Lewis Publishers, p461 (it addresses

a variety of analytical problems with clear graphics: anexcellent self-studybook after you take this class).

Mathematical Treatment: Groundwater Hydraulics, (Chinese) Y-QXueet. al (Editors), 1986, Geology Publishing House of P.R. China,p 345.

Other materials are obtained from course notes prepared by otherprofessors(references are listed in the notes).

Due to time limitation, this course cannot hope to cover everyaspect ofthe subject as presented in the books. For example,materials on GroundwaterManagement, Groundwater Chemistry, SoluteTransport, Vadose-Zone Hydrol-ogy, and Field Methods are notcovered. Some of these topics can be understoodby independentstudy, others may be the contents of more advanced or special-izedclasses. Thus, most of this class is devoted to the study ofsingle-phase(water), uniform-density flow moving throughnon-deforming porous media(e.g., groundwater aquifers that are notgoing through compaction or subsidencethough we may briefly touchon these topics). The relevant physical laws andmathematicalequations are developed in detail. Immiscible, multiphase flu-idsare not covered. Note, however, the basic approaches in formulationandderivation are similar.

0.1.2 Tools

For this class, besides some simple Excel exercises, we will notgenerally docomputer-aided modeling analysis (this will be formallytaught in a separateclass I teach on groundwater modeling). Forsome labs, a personal laptop mightbe ideal to facilitate repetitivecalculations.

Since theoretical vigor in the mathematical development isemphasized, Iexpect that students develop a good understanding onthe fundamental aspectsof the topics. Besides a few importantformulations (where memorizing them willserve you well), youre notexpected to memorize a large number of equations.

Most homework and exercises can be solved with pencil, ruler,and a calculator.Make sure you have them during both lectures andexams.

0.1.3 Questions and Answers

Students can ask questions: (1) during office hour; (2) duringlectures. Asa rule, email is a last resort since I receive a lot ofthem every day and yourmessage stands a chance of being overlookedby mistake.

7/29/2019 In Trog w Course Note Outline

10/27

0.1. THIS CLASS ix

0.1.4 Homework, Labs, Exams and Grades

When working on homework/lab/exam problems, read thedescriptions of theproblem carefully. Read them twice if you haveto. Do not skip anything, oryou may find that later questions willnot make sense to you. Of course, youshould always point out to theinstructor/TA if there is anything unclear in the

descriptions.Concerning homework, 4 points must be emphasized:(1) For problem

sets involving equations, if appropriate, provide a completeanalysis rather thana single number. (2) Be professional in yourpresentations: write down the unitfor your numerical results andround off the final number to 1 or 2 decimal point.If the probleminvolves a short essay, give it some thoughts and then write itoutclearly, precisely, and concisely. (3) You can discuss theproblem sets with fellowstudents, but complete your assignments byyourself. Copying others work isconsidered cheating and no pointswill be given for that homework (even if youonly copy one problemout of a total of 6). (4) Hand in the homework on time.

Unless otherwise stated, the general timeline is to hand overthe homework inthe beginning of the class a week from when thehomework is assigned.Further, if the homework is not handed in ontime, for every day it is delayed,10 points will be taken out ofthe 100 points assigned to each homework untilno points remain.

Exams include (1) multiple quizzes given throughout thesemester; (2) mid-term; (3) final. The above homework rules (1) and(2) apply. All exams must behanded in at the end of the class.Cheating will incur 0 point for that exam/quizand a student caughtcheating may receive a F for the course. A study guidewill beprovided before the mid and final exams, but not for the quizzeswhich

test the materials you just learned.

Labs discuss several important topics covered in depth. Thesetopics mayalso appear in lectures. For TA contact info, office hourand other relatedinfo, ask TA during Lab one. Unless otherwisem*ntioned, the lab assignmentsare expected to be handed in at theend of the lab. For a few big homeworkassignments, lab time is alsoused.

Grades: The final grades will be given based on your homework,labs, quizzesand exams. The appropriate percentage is shown:

Homework 24% (3% 8 homework) Quiz 24% (4% 6 quizzes)

Lab 20% (4% 5 labs)

Midterm 16%

Final 16%

7/29/2019 In Trog w Course Note Outline

11/27

x CONTENTS

Table 1: Letter versus numerical gradeA B C D F

90-100 80-89 70-79 60-69 < 60

Note that each homework/lab/exam has a stand-alone grade basedon 100

points. When determining the final grade, these will benormalized reflectingthe percentage distribution above. Forexample, if you receive the grades below:

-Homework: (1) 80; (2) 0; (3) 95; (4) 70; (5) 80; (6) 90; (7)75; (8) 89Quizzes: (1) 70; (2) 90; (3) 60; (4) 55; (5) 80; (6)95;Labs: (1) 80; (2) 90; (3) 75; (4) 78; (5) 89Mid term: 80Final:90You final numerical grade out of 100 points will be:

(80 + 0 + 95 + 70 + 80 + 90 + 95 + 89) 3.0% +(70 + 90 + 60 + 55+ 80 + 95) 4.0% +

(80 + 90 + 75 + 78 + 89) 4% +80 16% +

90 16%= 80.01

For a final grade of 80.01, you will get a B. The correspondingletter gradeis shown in Table 1. Your grade therefore reflects yourperformance throughout

the semester. Since the graduate students (the 5444 group) willhave gen-erally better preparation than the undergraduate students(the 4444 group),the final grading will be done separately. Theundergraduates will be gradedamong themselves and the grades scaledaccordingly (the grade earned by thetop undergraduate will be usedas the yardstick to assign grades to the otherundergrads). Thegraduate students will be held at a higher standard and willbegraded based on their absolute performance inhomework/lab/quiz/exam(rather than scaled among themselves whichwill not be fair).

Finally, I set high expectation in this class. If youreinterested in getting agood grade, be prepared to come to everyclass, pay full attention, participate

in exercises, work out the homework by yourself, hand inhomework on time,write professionally (clear, concise, precise,logical), and finally be helpful toyour fellow students.

Final thoughts:

The subject of groundwater hydrology is a challenging one thoughat thesame time rewarding. It solves real-world problems usingphysical principlesand mathematical formulations youve been taughtever since grade school. Itis rewarding in the sense that your pasttraining can help you understand and

7/29/2019 In Trog w Course Note Outline

12/27

0.2. INTRODUCTION xi

solve new problems. Besides, understanding natural processesusing physics andmathematics is in itself very interesting. Thoughyou may encounter unfamiliarequations and concepts, keep in mindthat your primary goal in this class is tolearn something usefulrather than getting a grade. Thus, consider this courseas a chanceto challenge yourself.

Please keep all course materials (notes, exercises, homework,exams, labs) toyourself and do not pass them on to future students.They must, as you have,work to earn the credit.

0.2 Introduction

0.2.1 Groundwater Science

Groundwater hydrology studies the movement of underground waterand its dis-solved chemical species. It is an importance subject ofthe applied natural sci-

ences. It emerges from an early engineering root (development ofundergroundwater resources) to become, in recent decades, afull-fledged environmental,engineering and geological science. Theprinciples of groundwater hydrologyare intimately related to otherfields, e.g., petroleum engineering, aqueous geo-chemistry, soilphysics, agricultural engineering, to name just a few, whereflow,transport, and reaction through porous media play afundamental role.

0.2.2 Groundwater Resources

From a practical point of view, securing water of suitablequality is one ofthe leading environmental concerns of the 21stcentury. It is estimated thataround 80% of diseases and 33% ofdeaths in the world are related to consump-tion of contaminatedwater. With the continued population growth, demandsof freshwatersupplies are expected to grow, while groundwater is thelargestreadily-available freshwater reservoir. In many areas of theworld (e.g., Nan-tucket Island, Massachusetts, Saharan desert), itis the only source of freshwater.In the US, its estimated thatabout 1/3 of the country is underlain by potableaquifers that canproduce at a rate of 50 gallons/minute. In the westernUS,groundwater is heavily used for irrigatione.g., the pivotalirrigation systemin the High Plains aquifer covering parts ofNebraska, Wyoming, Colorado,Kansas, Oklahoma, Texas and NewMexico.

0.2.3 Groundwater Quality

In the industrialized world, the mid-1900 chemical revolutionhas introducedpetroleum-derived synthetic chemicals into thenatural environment. The earlydisposal of these chemicals was notreally regulated. Such chemicals migrateunderground, dissolvinginto groundwater and creating contamination widely(see EPAs websiteon superfund: http://www.epa.gov/superfund/). The con-taminatedwater often flows away from the source zones, transportingchemi-

7/29/2019 In Trog w Course Note Outline

13/27

xii CONTENTS

cals to wells or other water bodies (lake, stream, reservoirs).These chemicalscan be harmful to humans. Famous examples includethe Love Canal site,e.g., http://en.wikipedia.org/wiki/Love Canal,and the Woburn contaminationin Massachusetts, e.g.,http://www.civil-action.com/.

0.2.4 Groundwater in WyomingIn Wyoming, groundwater is not onlyused for domestic purposes, it is also in-tricately linked to theenergy industry. For example, in the Power River Basin,groundwaterpumped from the gas-bearing formations contains significantcon-centrations of dissolved salts; its disposal has become anincreasing problem forthe economic development of coal bed methane(http://seo.state.wy.us/cbng.aspx).In Laramie, groundwatercontamination has also occurred at the Union PacificRailroad (UPRR)Tie Plant site, see the information on the cleanup effortsinhttp://deq.state.wy.us/shwd/N UPRailroad z03.asp.

0.2.5 What Groundwater Scientists Do

As groundwater scientists, we study a number of issues, roughlydivided intothese categories (below are excerpts from Fitts,2002):

1. Water Supply. Water supply wells for drinking water,irrigation, andindustrial use. Assemble data on the hydrogeology ofthe site, e.g., drillingdata, well data, and geologic maps. Testwells are drilled and hydraulictesting is used to estimate theaquifer capacity to produce water. Waterchemistry is checked toensure that the water is suitable for its intendeduse. If all arefavorable, a production well is designed and installed.

2. Water Resource Management. To manage aquifers, make decisionssuchas who is allowed to pump water, how much can be pumped,wherewells can be located, where potential contaminant sources tothe aquiferare located. The potential impact of surface waterprojects (location ofdams, diversions for irrigation, sewer systememplacement) on groundwa-ter level/quality is also considered.

3. Engineering & Construction. Dewatering: when a deepexcavation is madefor a building or tunnel, groundwater flows intothe pit. Dewatering canreduce the local water table, causing landsubsidence. In dam stability

analysis, seepage rate and pore water pressure are estimated.Groundwa-ter study is also part of a study on siting landfill andsubsurface wastestorage locations.

4. Environmental investigation and clean-up. Remediation caninvolve con-struction of trenches where contaminants are captured,pump-and-treat,injecting air, chemical solvents, or bacteria, andother schemes.

5. Geologic Processes. Better understand the process involved inthe originof oil, gas, and mineral deposits. Work are also sheddinglight on past

7/29/2019 In Trog w Course Note Outline

14/27

0.3. HOMEWORK 1 xiii

climates, earthquake generation mechanisms, and geologic hazards(e.g.,land slides).

The applications of groundwater science are ofteninterdisciplinary, bridginggeology, engineering, environmentalsciences, chemistry, biology, and resourcemanagement. Problems aretypically addressed by a team of people from various

disciplines.

0.3 Homework 1

Please hand in your homework in the beginning of the class aweek from today.1. In a few sentences, describe what you alreadyknow about groundwater

hydrology and what you hope to learn in this class.2. Among thevarious fields described in What Groundwater Scientists Do,

select two categories of your interest. For each category,present a short essay(< 200 words) describing a case study: whatkind of problem transpired and

what groundwater scientists did to solve the problem. For this,the best way isto do online or library research. The case studiesmay be from Laramie locally,or, some of the more well known casesin the country, e.g., the Love Canal (seeyour textbook), the Woburncase upon which the movie Civil Action isbased(http://www.geology.ohio-state.edu/courtroom/images.htm).

Please write the essays yourself and provide references asappropriate. Di-rectly copying from somewhere (e.g., someone elseswriting or a report down-loaded from the internet) will not do. Iwill check your writing against suchpotential sources to make surethere is no plagiarism.

7/29/2019 In Trog w Course Note Outline

15/27

Chapter 1

Basic Math

The following is a review of some basic math you need to knowbefore taking

this class (it is in no way complete more may be added as weproceed inthe course). If you find the math incomprehensible,please first take CollegeAlgebra and Calculus. This course is not amath class. Most equations Iwill present are derived assuming youknow the basic math already.

1.1 Scientific Notation

Scientific Notation is commonly used when dealing with verylarge and/or verysmall numbers.

10 10 = 100 = 102 = 1.0 102

10 102

= 101+2

= 103

= 1.0 103

5 104 + 60 104 = 65 104 = 6.5 1051 103 1 104 = 1.0 1071103

1104 = 1 1034 = 1 101 = 0.1(1 103)4 = 1.0 1012(1 104)1/2 = 1.01020.0000209 = 2.09 105Note that we often write 1 102 as 1.0E2, or2.09 105 as 2.09E5, etc.

1.2 Decimal Places

In this class, when dealing with non-integer numbers, we adopt(at least) 1decimal place for the final result, see the exampleshown in Figure 1.1. Duringthe computation, keep all decimalplaces. Round the result only in the last step.

We can first write: V = 5.234 3.72 4.1 = 79.828968 m3, thenround-ing:

V = 80.0 m3 (1 decimal place required minimum)V = 79.83 m3 (2decimal places preferred)

1

7/29/2019 In Trog w Course Note Outline

16/27

2 CHAPTER 1. BASIC MATH

5.234 m

3.72 m

4.1 m

Figure 1.1: What is its volume (V)?

1.3 Dimension Analysis

In the physical sciences, there are 7 Basic SI Units (or themetric system allnumbers are related by 10). In this class, weuse:

Name Dimension Unitlength [L] meter(or m)mass [M] kilogram (orkg)time [T] second (or s)

Other units are derived from the basic units:

Force[F] (N) = m (kg) a (m/s2

)Pressure (P a) = Force [N]Area [m2] kg/(m s2)

In Dimension Analysis, the type and dimensions of units at bothsides ofan equation must be the same. For example, in equation:Distance [L] =V elocity [L/T] Time [T], the unit type anddimensions are the same on bothsides [L]. The rule applies to morecomplicated equations as well. This isimportant for checking theconsistency of the equations.

There is also the issue of unit conversion, e.g., between the SIunits andother non-standard units, e.g., 1 m = 3.28084 feet, 1liter = 0.001 m3, 1 bar =

105

P a. Make sure you know how its done. If you work on a computerwith aninternet connection, try using:

Online Unit ConverterFor many exercises in this class, to ensurethat the units are con-

sistent, you need to double-check the units of all relevantquantitiesbefore calculation. Sometimes, you can convert thequantities to theSI unit before computation, other times you canwork with the En-glish unit. I will usually give you some commentsto help you withthese exercises. A convenient conversion table isalso provided in Appendix

7/29/2019 In Trog w Course Note Outline

17/27

1.4. LOGARITHM & EXPONENTIAL 3

0.5 m3

A gasoline drum

A bucket

0.022 m3

Figure 1.2: What many buckets does it take to fill the drum?

A of your textbook (Fitts, 2002).In doing the exercise of thisstudy, there is yet another type of unit you

must comprehend. In Figure 1.2, do you know what unit you shoulduse whenpresenting the result?

The answer is: 0.50.022m3

(m3/bucket) = 22.7 (buckets).

1.4 Logarithm & Exponential

The logarithm of a number is the exponent to which a base numbermust beraised to yield the value of the number. There are twocommon bases: 10 ande=2.718... (natural logarithm).

102 = 100; log10100 = 2 or log100 = 2102 = 0.01; log0.01 = 2100= 1; log1 = 0e0 = 1; ln1 = 0251/2 = 5; log525 = 1/2

If a, b are constants:axay = ax+y

a0 = 1 (if a = 0)(ab)x = axbx

(ax)y = axy

ax = 1/ax

ax/ay = axy

Logarithms have the following relationships:b = ea lnb = ab =10

a

logb = alne = 1ln1 = log1 = 0log10 = 1ln(ab) = lna + lnb (sameapplied to log(ab))ln(a/b) = lna lnbln(ab) = blnaln(1/b) = lnbln(a)2.30log10a

7/29/2019 In Trog w Course Note Outline

18/27

4 CHAPTER 1. BASIC MATH

y

a=tg()

y=ax+b

x

b

(a) (b) A right triangle

b

ac

-b/a

Figure 1.3: A line function (a) and a right triangle (b).

Exponential functions are defined as f(x) = ax + B, where a is areal con-stant and B is any expression. Commonly, exponentialfunction is e-based, forexample, f(x) = ex is an exponentialfunction (a = e), and, ln f(x) = x.

1.5 Areas, Volumes, CircumferencesBe familiar with thecalculation of different areas (A) and volumes.

Area of a circle with radius r: A = r2, pi = 3.14159265 . ..Circumference of a circle with a diameter d: dArea of a trianglewith a base b and altitude h: A = (1/2)bhArea of a parallelogramwith sides a and b and an included angle : A =

a b sin.Volume of a cylinder with radius r and height h: V =r2h.

1.6 Analytic Geometry & Trigonometric Func-tions

Equation of a straight line in rectilinear coordinates (Figure1.3a):y = ax + ba is the slope, b is the intercept of the y axis.InFigure 1.3b, for a right triangle, we have:c2 = a2 + b2; c =

a2 + b2

Common trigonometric functions:sin = a/c

cos = b/ctg = a/b

1.7 Integration

Most analytic solutions for hydrologic problems are developedbased on integra-tion and differentiation. You should know thebasics from your calculus class.For example, to find the areabeneath a function (Figure 1.4), we can use:

7/29/2019 In Trog w Course Note Outline

19/27

1.8. SLOPE 5

y

a

y=f(x)

xb

?

Figure 1.4: What is the area underneath f(x), between a andb?

Area =b

af(x)dx

What is the area of the grey right triangle is Figure 1.3a?

Area =0b/a(ax + b) dx = (

a2x

2 + bx)|0b/a =

( a2x2 + bx)|x=0 ( a2x2 + bx)|x=b/a =

0 ( a2 b2

a2 + b(b/a)) = ( b2

2a b2

a ) =b2

2a

From the area relation for right triangle, we can directlycalculate: (b/a) b/2 = b2/2a

Some simple rules of integration (all rigorously derivable!): xis an inde-pendent variable, c,a,b are expressions not involving x(e.g., constant), n is an

integer.Indefinite integral:1dx = xdx = xd[f(x)] = f(x)

Definite integral:ba

dx = x|ba = b aba

cdx = cx|ba = c(b a)ba [f(x) + g(x)]dx =

ba

f(x)dx +b

ag(x)dx

ba

xndx = xn+1

n+1 |ba =

bn+1

n+1 an+1

n+1ba (1/x)dx = ln x|ba = lnb lna = ln(b/a)

1.8 Slope

For a straight line: y = tan()x + b (Figure 1.5), the slope isdefined by:

slope=tan() = y(x+x)y(x)x

7/29/2019 In Trog w Course Note Outline

20/27

6 CHAPTER 1. BASIC MATH

y

x

y(x)x

x+ x

1.40x

0.65y(x+ x)-y(x)

y(x)

Slope=tan()>0y

xx

x+ x

1.40x

y(x)

Slope=tan()

7/29/2019 In Trog w Course Note Outline

21/27

1.9. DIFFERENTIATION 7

In this course, we use h to represent the hydraulic head (moreon this later).If the hydraulic head is a 1D function of x: h =h(x), its gradient is:

dh

dx h

x=

h(x + x) h(x)x

If the hydraulic head is a 3D function: h = h(x,y,z), itsgradient becomes:

h =

hxhyhz

Note that now the head gradient is a vector with 3 components.Eachcomponent is a partial derivative of h with respect to aparticular coordinateaxis. For example, by definition:

h

x= lim

x0

h(x + x,y,z) h(x,y,z)x

This derivative thus evaluates the head gradient along the xdirection, keepingy and z fixed. We can similarly write thedefinitions out for hy and

hz .

If we only evaluate h along a 2D vertical cross section (x,z),the hydraulicgradient has only two components:

h ={

hxhz

}

where the partial derivatives:

h

x h(x + x, z) h(x, z)

xh

z h(x, z + z) h(x, z)

z

Where hx is evaluated by looking at how head varies along x,keeping z fixed.hz is evaluated by looking at how head varies alongz, keeping x fixed. In Chp4,Exercise 5, well use the aboveformulation for a 2D flow analysis. When wedo that exercise, wellcome back to the gradient definition presented here. Youwill seewhy we chose those particular locations to calculate the particularhxand hz for that exercise.

1

What if our head is only varying along the 2D planeviewcoordinate (x,y)?

How would you write the hydraulic gradient vector and itscomponents?

h ={ h

xhy

}

1For example, to evaluate hx

h(x+x,z)h(x,z)x

, we choose two points in the aquifer:P1 and P2. If point P1lies at the same elevation (z) as point P2, but P1 occurs at ahighervalue along the x axis than P2 (this will depend on where thex axis is pointing), we can setthe horizontal distance from P2 toP1 as x, thus h(x + x) = h(P1), h(x) = h(P2), then

the partial derivative is evaluated as: hx

h(P1)h(P2)x

.

7/29/2019 In Trog w Course Note Outline

22/27

8 CHAPTER 1. BASIC MATH

where the partial derivatives are, by definition:

h

x h(x + x, y) h(x, y)

xh

y

h(x, y + y) h(x, y)

y

Their meanings and how well calculate them for a particularcoordinate systemare similar to above.

Some simple rule of differentiation: (a is a constant)dadx =0dxdx = 1d(ax)

dx =adxdx = a

d(xn)dx = nx

nl

e.g.,d(x2)

dx = 2x,d(1/x)

dx = (1/x2)d(lnx)

dx = 1/xd(ex)

dx = ex

d(ax)dx = a

x ln a

u, v are independent variables:

d(u + v) = du + dv

d(uv) = udv + vdu

d(u/v) = (vdu udv)/v2

If f(x) and g(x) are a function of x:d

dx [af(x)] = adf(x)

dxd

dx [f(x)g(x)] = g(x)df(x)

dx + f(x)dg(x)

dx

Commonly, chain-rule is applied:d

dx g(f(x)) =dgdf

dfdx

e.g.,d(eax)

dx = eax d(ax)

dx = aeax

d(ex3)

dx = ex3 d(x

3)dx = e

x3(3x2)

Exercise 1 Calculate the weight of freshwater in a cylindricaltank that hasa diameter of 4 ft and a height of 6 ft. (Note,convert everything to the SI unit.)

Exercise 2 Calculate the volume of water in an aquifer shown inFigure 1.6given a porosity of 0.1. This aquifer representation isoften used in this class:we see that the actual 3D geometry isrepresented by a vertical cross section inthe x-z plane and a unitaquifer thickness in the y direction.

7/29/2019 In Trog w Course Note Outline

23/27

1.10. FUNCTIONS OF ONE VARIABLE 9

100.0

99.9 z(x)=100-0.001x2

0.0 10.0x

z

y

y=1.0 m

Figure 1.6: A vertical slice of an aquifer.

1.10 Functions of One Variable

An example function of the one variable x is: h(x) = x2 + 2x1The first derivative of this function is: dh/dx = 2x + 2

The second derivative of this function is: d2h

dx2 =d

dx (dhdx ) = 2

In mathematics, the function h is also called a dependentvariable, while thespatial axis x called an independent variable.Some physical sense can be made

out of this terminology. For example, if the hydraulic head isvarying along ax-y plane: h = h(x, y) (draw a regional confinedaquifer and its potentiometricsurface on the board to help studentsvisualize that), head is changing as x and ychanges, so headdepends on x and y. HOWEVER, the spatial axes themselvescan changeirrespective of what head (or any other physical quantity) is.AND,x changes irrespective of y, vice versa. In a more generalcase, head is varyingin 3D and over time, so we have h =h(x,y,z,t), in this case, head depends onx, y, z, t, though each ofthese are independent of head and independent of oneanother. Theseare thus the independent variables.

An ordinary differential equation (ODE) is an equationcontaining deriva-

tives of a function of one variable. For example, h(x) = x2 +2x

1 is a solution

of the following ODE: d2h

dx2 =dhdx 2x (this can be verified by substituting the

pervious derivatives into the RHS and LHS of this ODE).

Some simple hydrological systems can be described along 1D axisand steadystate (it means the head does not change with time, moreon this later), inwhich case we typically solve a ODE to find,e.g., h(x). This solution can useanalytical methods (a sperateclass in Math department exists in just solvingODE) or numericalmethods. Such problems will be the topics of an advancedgroundwatermodeling class Ill teach in future.

7/29/2019 In Trog w Course Note Outline

24/27

10 CHAPTER 1. BASIC MATH

1.11 Functions of two or more variables

If you differentiate a function of two or more variables withrespect to one of thevariables, you have a partial derivative.Consider the following function (Fitts,2002):

h = h(x, y) = 4x2 + 3y + 10xy2

The partial derivative with respect to x is evaluated just likethe derivativewith respect to x, treating y as though it were aconstant. The function h hasthe following partial derivatives:

hx = 8x + 10y

2

2hx2 =

x

hx =

x (8x + 10y

2) = 8hy = 3 + 20xy2hy2 =

y

hy =

y (3 + 20xy) = 20x

x

hy =

x (3 + 20xy) = 20y

y

hx =

y (8x + 10y

2) = 20y

The example function h is a solution of the following partialdifferentialequation (PDE):

2hy2 +

2hx2 = 20x + 8.

The idea of a dependent and independent variables holds heretoo. In hydrol-ogy, for more realistic flow configurations, we aregenerally interested in solvingthe properties (e.g., head,velocity, solute concentration) in higher dimensionsand evaluatingtheir changes over time, e.g., h = h(x,y,z,t). In these cases,wetypically solve a PDE using either analytical means (e.g.,turning PDE to ODE,Fourier Transform, and some other means), ornumerical means. The numericalmethods of solving such PDE will bedescribed in the modeling class.

1.12 Scalar, Vector, Matrix

Scalar is a mathematical quantity that has only 1 component,e.g., density,pressure, temperature. Vector is a quantity that has3 components in three-dimensional coordinate and 2 components intwo-dimensional coordinate. Vec-tor reduces to a scalar inone-dimensional coordinate.

A typical example of a vector is velocity of a canon propelledthrough space(Fig 1.7). If we choose to analyze its trajectory intwo-dimensions, the velocitymagnitude (|v| =

v2x + v

2z ) of the ball remains constant through time, but its

direction is changing. Thus, the two component velocities (vx,vz) are changing

in time. Notice the difference between vA and vC: why would theverticalcomponent of vC be negative? Answer: whether this value ispositive or negativedepends on the adopted coordinate system: the zaxis points upward, but thevertical component of vC points in theopposite direction (along -z axis), thatsway it is negative. Now,what happens when we rotate the axis so x faces theoppositedirection? Answer: The sign of the component vx changes; butthecomponent vz does not change and the velocity magnitude does notchange. So,in the new coordinate, vA = [1.0, 1.5], vB = [1.8, 0.0],vC = [1.0,1.5].Make sure you thoroughly understand thisconventionthroughout

7/29/2019 In Trog w Course Note Outline

25/27

1.12. SCALAR, VECTOR, MATRIX 11

trajectoryv

A

X

Z

vB

vC

Velocity & Trajectoryin Two-Dimensions

vA

=[vAx,vAz]=[1.0, 1.5] (m/s); |vA|=1.8 (m/s)

vB=[vBx,vBz]=[1.8, 0.0] (m/s); |vB|=1.8 (m/s)

vC=[vCx,vCz]=[1.0, -1.5] (m/s); |vC|=1.8 (m/s)

Figure 1.7: The trajectory and velocity of a ball at threelocations A, B, C.

this course, we are constantly adding or dropping the negativesignin response to the coordinate we use!

In 2D or 3D, the relationship between a vector and itscomponents (whichis just the normal projection of the vector ontoeach coordinate axis) is shownin Figure 1.8. Make sure youunderstand this.

An important property of vectors is the vector inner product ordot product.Ifa = {a1, a2, . . . , an}, b = {b1, b2, . . . , bn},their inner product is:

a b = a1b1 + a2b2 + . . . + anbn

Note that the result of two vector inner project is a scalarquantity.

In hydrology (as well as many other physical sciences), animportant vectoris the gradient vector :

=

x

y

z

The gradient vector is consider a mathematical operator. Twooperations areof particular relevance in this class: (1) operatingon a scalar gives a vector,

7/29/2019 In Trog w Course Note Outline

26/27

12 CHAPTER 1. BASIC MATH

x

y

z

q

qx

qy

qz

x

z q

qx

qz

Figure 1.8: Relation between a vector (q) and its components in2D and 3D.

e.g., the previous hydraulic gradient:

h =

hxhyhz

(2) vector product between and another vector gives a scalar.For ex-ample,the velocity vector in 3D has 3 components:q = {qx,qy, qz}, the vectorproduct between andq gives:

q =

x

y

z

qxqy

qz

=

x(qx) +

y(qy) +

z(qz )

Matrix

Now, what is matrix? Within the context of the linear algebra,it is a two-dimensional (2D) data structure. A square nn matrix cangenerally be writtenas:

Ann =

a11 a12 . . . a1na21 a22 . . . a2n. . . . . . . . . . . .an1 an2. . . ann

In linear algebra, there is an important relationship onmatrix-vector mul-tiplication (since we wont have time to reviewthe subject of Linear Algebra,you should know this relationship byheart):

(1.1)

[a11 a12a21 a22

]{b1b2

}=

{a11b1 + a12b2a21b1 + a22b2

}

We see that this multiplication outcome is a vector with 2components. Theabove relation can be extended to higher dimensions,e.g., we can write the

7/29/2019 In Trog w Course Note Outline

27/27

1.13. THE SUMMATION SIGN 13

vector outcome for a multiplication of a 33 matrix and 3 1vector: a11 a12 a13a21 a22 a23

a31 a32 a33

b1b2b3

=

a11b1 + a12b2 + a13b3a21b1 + a22b2 + a23b3a31b1 + a32b2 +a33b3

The resulting vector has 3 components or a 3 1 vector. This canbe extendedfor the multiplication of nn matrix and n1 vector.

1.13 The Summation Sign

We use the summation sign to condense the writing of longequations, for ex-ample:

x1 + x2 + . . . + x10 =10

i=1

xi

y1 + y2 + . . . + ym1 + ym =m

j=1

yj

(x1 a)2 + (x2 a)2 + . . . + (xn a)2 =n

k=1

(xk a)2

In these equations, i,j,k are called summation indexes. Theseare flexiblesymbols decided by us to use, e.g., we can use p,q,o asindexes instead. Lateron, we may briefly explain the convention ofEinstein Summation which issometimes used to condense longequations.

A constant can often be taken in and out of a summation sign,for example,below is a simple proof:

10i=1

axi = ax1 + ax2 + . . . + ax10 = a(x1 + x2 + . . . + x10) =a10

i=1

xi

Thus, we can write:

mi=1

b(xi c)3 = bm

i=1

(xi c)3

1.14 Test 1

This quiz will be given in the next class after Chapter 1 istaught.