GEO-SCI HYDROGEOLOGY LAB 1 - MATH REVIEW AND PLOTTING

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1 GEO-SCI HYDROGEOLOGY LAB 1 - MATH REVIEW AND PLOTTING In this lab we will do three things: 1. Become familiar with the units commonly used in groundwater and, in particular, become familiar with how to make unit changes. 2. Review basic geometry, trigonometry, calculus, differential equations and other "tricks of the trade" that are fundamental to hydrogeology. 3. Provide some general exercises for practice. Part 1 - Units and Conversions Units, Abbreviations, Definitions, etc. - Handout #1 is a list of units, abbreviations, definitions and symbols used in the groundwater industry. Become familiar with these symbols. I will try to be consistent and use these symbols throughout the course. Conversions and Physical Properties of Water - Handout #2 is a collection of tables listing common English and metric conversions for lengths, areas, volumes, and time. You should become familiar with them and use them as a resource. These can also be found in Fetter [Applied Hydrogeology] (Appendices 7,8,9, and 10). How to Perform Unit Conversions - The best way to perform unit conversions is to identify all the conversions you will need to make the required changes and then write an expression which allows you to cancel out units. Let's demonstrate this with an example. Suppose you wanted to convert from a value of 10 m/sec (meters per second) to its equivalent value in ft/day (feet per day). Step 1 - From the textbook appendices we know that 1 m = ft; 60 sec = 1 min; 60 min = 1 hr; 24 hr = 1 day. Step 2 - Write an expression for the units as follows: 10 m/sec x {3.281 ft/m x 60 sec/min x 60 min/hr x 24 hr/day} = 2,834,780 ft/day Notice that the meters, sec, min, and hours cancel out and you are left with ft/day. The value in brackets = and is the value you would multiply m/sec by to obtain the answer in ft/day. Setting up unit conversions this way is very useful because it provides a check on whether you are making the proper conversions. If the units do not cancel, then you have probably done something wrong and should recheck your work. Long and involved equations can also be checked by looking at the units. If you know your answer should 1

2 be in units of meters and you get m/sec, then it's probably a sure bet that you made an error in evaluating the equation or the equation is incorrect. Let's do another example: Suppose you want to convert from 10,000 gpd/ft (gallons per day per foot) to m 2 /day (meters squared per day). These are both units of transmissivity, one is in English units, the other metric. Proceed as follows: gal/day-ft x 1 ft 3 /7.48 gal x ft/1 m x 1 m 3 /(3.281) 3 ft 3 x 1 day/24 hrs x 1 hr/60 min x 1 min/60 sec = m 3 /sec-m = m 3 /sec/m = m 2 /sec Notice that if the unit is cubed (ie. ft 3 ), squared or whatever, the conversion must also be cubed (ie. (3.281) 3, squared etc. Also notice that m 3 /sec-m, m 3 /sec/m, and m 2 /sec are all equivalent statements. Part 2 - Math Review Constants - The following constants may be useful in hydrogeology. Gravitational acceleration (g) average m/sec ft/sec 2 pi (π) Unit Weight of fresh water Unit Weight of sea water 62.4 pounds/ft 3 (pcf) 1.0 grams/cm 4 C 64.0 pcf grams/cm 3 Basic Geometry - The following are useful relationships in hydrogeology. Circumference of Circle πd or 2πr (where d=diameter; r=radius) Area of Circle πr 2 Surface Area of Sphere 4πr 2 Volume of Sphere (4/3) πr 3 Area of triangle (1/2)bh (where b=base; h=height) Area of trapezoid 1/2 x sum of parallel sides x height Area of square, rectangle or parallelogram Volume of pyramid Volume of cone Surface area of cone Volume of cylinder Surface area of cylinder base x height area of base x 1/3 the height (1/3)πr 2 h πr x slant height πr 2 h 2πrh 2

3 Know this relationship: Rate x Time = Distance Basic Trigonometry Know your basic trigonometry and how to use it in practical situations. For right triangles: sinθ = side opposite θ/hypotenuse (opp/hyp) cosθ = side adjacent θ/hypotenuse (adj/hyp) tanθ = side opp θ/side adj θ (opp/adj) side c 2 = side a 2 + side b 2 For non-right triangles: Law of Cosines: c 2 = a 2 + b 2-2abcosθ Coordinate Systems Two coordinate systems used in groundwater - the cartesian system and cylindrical coordinates. Basic Functions 1. Logarithms - a. Definition: log b x is read "the logarithm to the base b of x". In practical terms this means "the power to which b must be raised to produce x" Example: log = 2 - the power that ten must be raised in order to produce a value of 100 is 2, that is 10 2 = 100 For this course log x means log 10 x b. Properties: These are called common logarithms. The important properties of common logarithms are: log 1 = 0 log x/y = log x - log y log xy = log x + log y log x 2 = 2 log x log 1/x = -log x c. Useful Qualities of Logs in Solving Problems - Example 1: y = x 3

4 2. Natural logarithms To solve for x, just take the log of both sides log y = log x = -6.3x Rearranging, x = (-log y)/6.3 Example 2: y = log 6x To solve for x, take the antilog (ie, 10 t ) where t is any function 10 y = 10 log 6x = 6x Rearranging, x = (10 y )/6 The natural logarithm is designated - ln (pronounced ell-en) It is a function used to solve 1/x dx and other integrals The natural logarithm is log e x (log to the base e) or ln x where e = ln x means "the power to which e must be raised to produce x" a. Properties of ln ln 1 = 0 ln x/y = ln x - ln y ln 1/x = -ln x ln xy = ln x + ln y ln x y = y ln x 3. Natural Exponential Function The natural exponential function is designated e x (read as "e to the x") It is an important function because it describes exponential decay. Radioactive decay follows an exponential decay curve. It also "undoes" natural logs. ln e x = x for all x e lnx = x for x>0 a. Properties of e x e 0 = 1 ln e = 1 e k+6 = e k x e 6 4. Error Functions and Complimentary Error Functions These functions are commonly used in solutions for contaminant transport problems. They are defined as follows: y 4

5 Error Function: erf y = (2/ π) e y² dy 0 this exponential integral can be solved numerically and values are tabulated on your 3 rd Handout. Complimentary Error Function erfc y = 1 - erf y a. Properties of error functions erf ( ) = 1 erfc ( ) = 0 erf (0) = 0 erfc (0) = 1 erf (- ) = -1 erfc (- ) = 2 erf (-y) = -erf (y) erfc (-y) = 1 + erf (y) Operations on the Computer Normal Syntax Operation Computer Syntax x Multiplication * Division / + Addition + - Subtraction - ln Natural log ln 10 2 Exponents ^ (10^2) log Common log base 10 log e x Exponential Function exp Square Root Sqrt Example: Normal Syntax Computer Syntax T = 464 r 2 /t T = (464*(r^2))/t When writing equations for the computer the order in which you tell the computer to perform operations is critical!!! When in doubt use lots of parentheses because operations begin in the innermost parentheses and move out. Note: Individual spreadsheet programs may have their own customized syntax for mathematical operations. Refer to the spreadsheet manuals for details. Calculus Review 1. Differentiation - Derivatives of a function are obtained through the process of differentiation and give us the instantaneous rate of change of a function at any point. This stuff is used frequently in groundwater because through differentiation we can determine the gradient of a flow field which gives us not only the direction of flow but also the magnitude. 5

6 The symbols used for differentiation are: d/dx [function of x] For example: y = x dy/dx = 2x This gives us the instantaneous rate of change in slope at any point for this function. Techniques of Differentiation 1. Constants d/dx [constant] = 0 d/dx[5] = 0 2. Power Rule d/dx [x n ] = nx n-1 d/dx [x 3 ] = 3x 2 3. Rule of Constants w/ d/dx [cx 2 ] = c d/dx [x 2 ] Differentiable Terms = 2cx 4. Addition d/dx [x 2 + x] = d/dx [x 2 ] + d/dx [x] = 2x + 1 Subtraction is the same as addition 5. Product Rule d/dx [f(x)g(x)] = f(x)d/dx[g(x)] + g(x)d/dx[f(x)] d/dx [(x 2-2)(x 3-1)] (x 2-2)(3x 2 ) + (x 3-1)(2x) 3x 4-6x 2 + 2x 4-2x 5x 4-6x 2-2x 6. Quotient Rule d/dx[f(x)/g(x)] = {g(x)d/dx[f(x)] - f(x)d/dx[g(x)]} / (g(x)) 2 7. Higher Order Derivatives y = 5x 4 /x 3 dy/dx = {(x 3 )(20x 3 ) - (5x 4 )(3x 2 )}/(x3) 2 (20x 6-15x 6 )/x 6 = 5x 6 /x 6 = 5 Second Order Derivatives d 2 y/dx 2, the 2 means 2nd order and all it means is that you do the differentiation twice. 6

7 Example d 2 y/dx 2 of 6x 3-4x 2 + 2x + 3 dy/dx = 18x 2-8x + 2 d 2 y/dx 2 = 36x Special Cases d/dx [ln x] = 1/x d/dx [e x ] = e x 9. Chain Rule dy/dx = dy/du x du/dx Example: y = (x 2 + 1) 10 let u = x so, y = u 10 dy/du = 10u 9 du/dx = 2x Therefore, dy/dx = (10u 9 )(2x) substitute back in for u dy/dx = 20x(x 2 + 1) Partial Derivatives Partial derivatives are used with functions where the dependent variable depends on two or more independent variables. For example: Area of a triangle A = 1/2bh A depends on b and h Volume of a Box V = lwh V depends on l, w, and h In groundwater flow problems we often want to specify hydraulic head or water levels as functions of three space coordinates, x,y, and z and time, t. The symbol used to represent partial derivatives is: δx/δt rather than dx/dt How to take partial derivatives Example: h = x 2 y + 2y 2 x 1. hold y constant and take derivative w/r to x δh/δx = 2xy + 2y 2 2. then hold x constant and take derivative w/r to y δh/δy = x 2 + 4yx 3. Now combine 7

8 δh = (2xy + 2y 2 ) δx + (x 2 + 4yx) δy Integration Integration is the opposite of differentiation and is used to determine areas and volumes within specific limits. 1. There are three types of integrals a. Indefinite integral - no limits imposed x dx = 1/2x 2 + C where C is the constant of integration b. Definite Integral - determines area or volume over specific limits 2 2 x dx = 1/2 x 2 = 1/2(2) 2-1/2(1) 2 = 2-1/2 = c. Improper Integrals - integrals with infinite interval of integration - integrand becomes infinite within the interval of integration + u u dx/x 2 = dx/x 2 = -1/x = -1/u - (-1) = 1-1/u Now take the limit as u +, 1/u 0, therefore (1-1/u) = 1 Techniques of Integration 1. Power Rule x n dx = (x n+1 /n+1) + C where n does not equal -1 3x 2 dx = x 3 + C 2. Rule of Constants w/differentiable Terms ax dx = a x dx = a/2 x 2 + C 3. Addition (2x + 1) dx = 2x dx + dx = x 2 + x + C 8

9 Subtraction is similar, just subtract. 4. ln x and ex dx/x = ln x + C e x dx = e x + C 5. u-substitution Example: (x 2 + 1) 50 (2x) dx a. Make a choice for a u-substitution. In this case, let u = x b. Compute du/dx = 2x, so du = 2x dx c. Make the substitution u 50 du d. Evalulate integral using power rule - 1/51u 51 + C d. Replace u with function for answer - 1/51(x 2 + 1) 51 + C 6. Integration by Parts - this technique is also used in many groundwater problems 7. Partial Fractions - Another technique often used in groundwater. 8. Higher order integrations Example: d 2 h/dx 2 = 0 a. In this case just integrate twice b. 1st integration use the rule for constants [constant = 0] = C 1 1st integration c. 2nd integration is just the power rule for x 0 dx (ie, 1 dx) C 1 dx = C 1 x + C 2 2nd integration - Note there are two constants, one for each integration Differential Equations 9

10 Many groundwater articles will show flow and transport equations written as differential equations. For example, δ 2 h/δx 2 + δ 2 h/δy 2 = S/T δh/δt is the 2-dimensional flow equation for an isotropic, homogeneous confined aquifer This equation has the following characteristics: 1. It is a 2nd order equation - because the partials have a 2 in them. 2. It is inhomogeneous rather than homogeneous - because the equation is subject to external forces (in other words, the equation has a time variable in it, therefore it does not equal 0; a homogeneous equation would equal 0) 3. It is a linear differential equation - because the dependent variable, h (which appears in the numerator in a differential equation), does not appear as a function with a power higher than 1 or as a trig function. 4. It is a partial differential equation rather than an ordinary differential equation - because it is multivariable. h varies with x and y spatial coordinates and time, t. If h was dependent on only one independent variable it would be an ordinary differential equation. Definition of a differential equation - A differential equation is an equation which contains one or more derivatives of some unknown function. The goal of hydrogeologists who do groundwater modeling is to determine what that unknown function is and come up with an explicit solution. For example, d 2 h/dx 2 = 0 is a 1 dimensional, 2nd order, linear, homogeneous, ordinary differential equation. The general solution for this equation is: h = C 1 x + C 2 In groundwater modeling, the game is to design a physical model that describes some natural system (ie., a confined aquifer in a narrow bedrock valley), develop a set of mathematical equations (differential equations) that describe the flow in that system, and then solve these equations so that we can describe the flow anywhere in the model at any time. Developing the physical model and flow equations is relatively simple. The difficult part is in selecting and using different methods to solve these equations. Analytical Models - solve the equations explicitly with an exact solution. Numerical Models - provide solutions to the equations using numerical approximations. 10

11 Part 3 - Exercises Units and Conversions - Show all calculations. 1. How many gallons in one cubic foot? 2. How many cubic centimeters are there in one liter? 3. How much does one liter of water weigh in grams? 4. How many feet are there in one meter? 5. How many meters are there in one mile? 6. How many square feet are there in one acre? 8. A pond has 4 acre-feet of water in it. How many gallons does this represent? How many liters are in this pond? If the pond is an average of 1 foot deep how many acres does this pond occupy? If the pond is an average of 4 feet deep how many acres does this pond occupy? 9. Two inches of rain fell on Amherst. Amherst has an area of 36 square miles. How many acre-feet of water does this represent? 11

12 10. An aquifer has a hydraulic conductivity of 3.2 x 10-7 cm/sec. What is this value in ft/day and gpd/ft 2? 11. An unconsolidated sand has an intrinsic permeability (k i ) of 2.66 x cm 2. The relationship between hydraulic conductivity (K) and intrinsic permeability (k i ) is: K = k i ρg/µ where, ρ = density of water g = gravitational acceleration µ = viscosity of water What is the value of K for 15 C? (Hint: you must convert centipoises to poises to solve this problem; 1 centipoise (cp) = 0.01 poises (p); a poise has units of g/sec-cm). 12. Water moves through a soil with an average linear velocity of 4.55 x 10-6 cm/sec. How many meters will the water move in one year? Basic Geometry - Show all your work. 13. A 4-foot diameter concrete pipe is discharging from a large watershed. You determine that the depth of flow, d, measured with your ruler at the center of the pipe is 1 foot. What is the cross sectional area of flow in the pipe? Make a sketch of the problem and use trig to solve the problem. 12

13 14. In perfectly flat terrain you measure the water table in a well and find the depth to the water table is 33 feet below the ground surface. You also know that the water table dips 5 to the east. At a point 6000 feet due east of your present position, what will be the depth to the water table? Draw a sketch and show your calculations. 15. A well screen is an open cylindrical screen placed in the ground through which groundwater is pumped. Suppose you have a well that is 4 feet in diameter and has a screen length of 30 feet. What is the surface area of the well screen? 16. A 2-inch diameter well has 10 feet of water in it. How many gallons of water are there in the well? Typically, when you collect water samples from a well for water quality analyses, you want to remove the entire volume inside the well at least three times to be sure you are acquiring a fresh sample. Your bailer, which you use to evacuate the water from the well, has a volume capacity of 0.15 gallons. How many bails will you have to make in order to be assured of obtaining a fresh water sample? 17. ph is defined as -log [H+] where [H+] is the hydrogen ion concentration in mg/l (milligrams per liter). If the hydrogen ion concentration in a sample of water is , what is the ph of the water? 13

14 18. The following drawdown data were recorded in an observation well located 90 meters from a pumping well. Time (min) Drawdown (m) Time (min) Drawdown (m) Use the attached 3x5 cycle log paper and plot the data. Put drawdown on the y axis and time on the x axis. Place origin in lower left corner of graph. 14

15 Plotting and Online Data Resources Today we will explore the USGS website for water resources and have some fun plotting various types of data. This website is a great resource for all sorts of surface-water and groundwater data. I encourage all of you to spend some additional time looking around at all the resources on this site. Navigate with Mozilla to Click on Science in your watershed in the upper right Click on Map Your Watershed Find the watershed for where we are located right now -- You should end up being in watershed Once there, click on additional information for this watershed Explore the Real-time Streamflow Stations in Middle Connecticut o Find the closest station to us (Hint: We are in Hampshire County) o What is the current streamflow (ft 3 /s)? o Using the output feature of the website plot the last 7 days of stream discharge in Excel. Hint: Use tab-separated and save file using save as feature to your desktop. Open excel and open the file and import. Use the chart feature and plot with XY scatter option with data points connected by lines without markers. Label axes and print and include with your lab. o Consult plot. Do you have a hypothesis as to the cause of the peak in discharge on February 2 nd or 5 th? What is your explanation for the sharp rise and falling limb of the stream hydrograph on the 2 nd and 3 rd? o Make a similar plot but this time plot the streamflow in meters per second. o What is the maximum, minimum, and average streamflow for this site? Use excel (min, max, average) commands. From the real-time data for Middle Connecticut use navigational bar at top right of page to navigate to groundwater data in Massachusetts. Make sure to select groundwater and then the state of Massachusetts. Again, find the closest well to us here in Hampshire county. o You should be at well MA-PDW 23. o What type geologic formation is this well located in? o How deep is this well? o Using the graph feature on the webpage, examine the last 31 days of water table data. What could the small amplitude oscillations in the water table be caused by? Think outside forcing. What are possible sources change in water level in the last couple of days? o Using the drop down selector adjacent to the text Available data for this site choose groundwater levels. 15

16 Examine graph. What is the cause of the large water table decline in 1998? Given the historical data, does this look like a natural phenomena? Use the Output Formats above the graph and output water level data in a Tab-separated format. Plot water levels from 1992 to 2002 in excel as we did before. Is there any pattern to the timing of the highest and lowest groundwater levels? Explain. What is the overall trend in water level in this well? What could explain this trend. What is the overall slope of this trend? An estimate will do. Lastly use the realtime data explorer to navigate to any well located in Eastern Massachusetts. Use the plotting feature on the USGS web to examine a similar time frame of data. Is this trend consistent with the Pellham well above? Why? Examining Real Data Finally we will explore some data that the last Hydrogeology class collected out near the CT river at our well site. We will visit this site throughout the semester. I have placed a dataset on the computers in the DML in a folder called WellAndRiverHead. The River data was collected from early February until April using a pressure transducer located in the river. We had set up a transducer in the pumping well (a 2 inch well screened at 16 ft below the surface in coarse lag deposit on top of the varved clay) but due to an equipment issue we lost all the data. Your first task will be to determine if the upstream gage (the discharge) at Montague (as found on the USGS website) is a reliable predictor of the elevation we measured. The other dataset located in the folder is head data from the pumping well at the field site (located 300 m from the river) collected from April through July. Please plot this data as a function of Time (real or elapsed your choice) for the length of the dataset. If, you find above that the discharge is reasonably correlated to the to our site, determine if there is any relationship between the mean daily discharge and the mean daily head. 16