Calculus Review

Slope

• Slope = rise/run • = y/x • = (y2 – y1)/(x2 – x1)

• Order of points 1 and 2 abitrary, but keeping 1 and 2 together critical

• Points may lie in any quadrant: slope will work out

• Leibniz notation for derivative based on y/x; the derivative is written dy/dx

Exponents

• x0 = 1

Derivative of a line

• y = mx + b• slope m and y axis intercept b• derivative of y = axn + b with respect to x:• dy/dx = a n x(n-1) • Because b is a constant -- think of it as bx0 -- its

derivative is 0b-1 = 0 • For a straight line, a = m and n = 1 so• dy/dx = m 1 x(0), or because x0 = 1, • dy/dx = m

Derivative of a polynomial

• In differential Calculus, we consider the slopes of curves rather than straight lines

• For polynomial y = axn + bxp + cxq + …

• derivative with respect to x is

• dy/dx = a n x(n-1) + b p x(p-1) + c q x(q-1) + …

Example

a 3 n 3 b 5 p 2 c 5 q 0

0

2

4

6

8

10

12

14

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x

y

y = axn + bxp + cxq + …

-5

0

5

10

15

20

25

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x

y

dy/dx = a n x(n-1) + b p x(p-1) + c q x(q-1) + …

Numerical Derivatives

• ‘finite difference’ approximation

• slope between points

• dy/dx ≈ y/x

Derivative of Sine and Cosine

• sin(0) = 0 • period of both sine and cosine is 2• d(sin(x))/dx = cos(x) • d(cos(x))/dx = -sin(x)

-1.5

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6 7

Sin(x)

Cos(x)

Partial Derivatives

• Functions of more than one variable• Example: h(x,y) = x4 + y3 + xy

1 4 7

10 13 16 19S1

S7

S13

S19

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

X

Y

2.5-3

2-2.5

1.5-2

1-1.5

0.5-1

0-0.5

-0.5-0

-1--0.5

-1.5--1

Partial Derivatives

• Partial derivative of h with respect to x at a y location y0

• Notation ∂h/∂x|y=y0

• Treat ys as constants• If these constants stand alone, they drop

out of the result• If they are in multiplicative terms involving

x, they are retained as constants

Partial Derivatives

• Example: • h(x,y) = x4 + y3 + x2y+ xy • ∂h/∂x = 4x3 + 2xy + y

• ∂h/∂x|y=y0 = 4x3 + 2xy0+ y0

1 4 7

10 13 16 19

S1

S7

S13

S19

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

X

Y

Gradients

• del h (or grad h)

• Darcy’s Law:

y

h

x

hh

ji

hKq

Equipotentials/Velocity Vectors

Capture Zones

Hydrologic Cycle/Water Balances

Earth’s Water

• Covers approximately 75% of the surface

• Volcanic emissions

http://earthobservatory.nasa.gov/Library/Water/

One estimate of global water distribution

Volume

(1000 km3)

Percent of Total Water

Percent of Fresh Water

Oceans, Seas, & Bays 1,338,000 96.5 -

Ice caps, Glaciers, & Permanent Snow

24,064 1.74 68.7

Groundwater 23,400 1.7 -

Fresh (10,530) (0.76) 30.1

Saline (12,870) (0.94) -

Soil Moisture 16.5 0.001 0.05

Ground Ice & Permafrost 300 0.022 0.86

Lakes 176.4 0.013 -

Fresh (91.0) (0.007) .26

Saline (85.4) (0.006) -

Atmosphere12.9 0.001 0.04

Swamp Water 11.47 0.0008 0.03

Rivers 2.12 0.0002 0.006

Biological Water 1.12 0.0001 0.003

Total 1,385,984 100.0 100.0

Source: Gleick, P. H., 1996: Water resources. In Encyclopedia of Climate and Weather, ed. by S. H. Schneider, Oxford University Press, New York, vol. 2, pp.817-823.http://earthobservatory.nasa.gov/Library/Water/

Fresh Water

Ice caps, Glaciers, & PermanentSnow

Groundwater

Soil Moisture

Ground Ice & Permafrost

Lakes

Atmosphere

Swamp Water

Rivers

Biological Water

Hydrologic Cycle

• Powered by energy from the sun• Evaporation 90% of atmospheric water• Transpiration 10% • Evaporation exceeds precipitation over oceans• Precipitation exceeds evaporation over

continents• All water stored in atmosphere would cover

surface to a depth of 2.5 centimeters• 1 m average annual precipitation


Hydrologic Cycle


In the hydrologic cycle, individual water molecules travel between the oceans, water vapor in the atmosphere, water and ice on the land, and underground water. (Image by Hailey King, NASA GSFC.)

Water (Mass) Balance

• In – Out = Change in Storage– Totally general– Usually for a particular time interval– Many ways to break up components– Different reservoirs can be considered

Water (Mass) Balance

• Principal components:– Precipitation– Evaporation– Transpiration– Runoff

• P – E – T – Ro = Change in Storage

• Units?

Ground Water (Mass) Balance

• Principal components:– Recharge– Inflow– Transpiration– Outflow

• R + Qin – T – Qout = Change in Storage

Water Balance Components

http://www.srs.fs.usda.gov/gallery/images/5_rain_gauge.jpg

DBHydroRainfall Stations

• Approximately 600 stations

0

200000

400000

600000

800000

1000000

1200000

1400000

1600000

1800000

2000000

0 200000 400000 600000 800000 1000000 1200000

Spatial Distribution of Average Rainfall

http://sflwww.er.usgs.gov/sfrsf/rooms/hydrology/compete/obspatialmapx.jpg

Voronoi/Thiessen Polygons

Evaporation Pan

www.photolib.noaa.gov/ historic/nws/wea01170.htm

http://www.photolib.noaa.gov/historic/nws/wea01170.htm

Pan Evaporation

• Pan Coefficients: 0.58 – 0.78

• Transpiration

• Potential Evapotranspiration– Thornwaite Equation

0

200000

400000

600000

800000

1000000

1200000

1400000

1600000

1800000

0 200000 400000 600000 800000 1000000 1200000

Watersheds

http://www.bsatroop257.org/Documents/Summer%20Camp/Topographic%20map%20of%20Bartle.jpg

Stage Recorder

http://gallatin.humboldt.edu/~brad/nws/assets/drum-recorder.jpg

River Hydrograph

http://cires.colorado.edu/lewis/epob4030/Figures/UseandProtectionofWaters/figures/ColoradoRiverHydrograph.gif

Well Hydrograph

http://wy.water.usgs.gov/news/archives/090100b.htm

Stream Gauging• Measure

velocity at 2/10 and 8/10 depth

• Q = v*A

• Rating curve:– Q vs. Stage

http://www.co.jefferson.wa.us/naturalresources/Images/StreamGauging.jpg

http://www.nws.noaa.gov/om/hod/SHManual/SHMan040_rating.htm

Ground Water Basics

• Porosity

• Head

• Hydraulic Conductivity

Porosity Basics

• Porosity n (or )

• Volume of pores is also the total volume – the solids volume

total

pores

V

Vn

total

solidstotal

V

VVn

Porosity Basics

• Can re-write that as:

• Then incorporate:• Solid density: s

= Msolids/Vsolids

• Bulk density: b

= Msolids/Vtotal • bs = Vsolids/Vtotal

total

solidstotal

V

VVn

total

solids

V

Vn 1

s

bn

1

Cubic Packings and Porosity

http://members.tripod.com/~EppE/images.htm

Simple Cubic Body-Centered Cubic Face-Centered Cubic n = 0.48 n = 0. 26 n = 0.26

FCC and BCC have same porosity

• Bottom line for randomly packed beads: n ≈ 0.4

http://uwp.edu/~li/geol200-01/cryschem/

Smith et al. 1929, PR 34:1271-1274

http://wulff.mit.edu/crystp/

http://wulff.mit.edu/crystp/

Effective

Porosity

Porosity Basics

• Volumetric water content ()– Equals porosity for

saturated system total

water

V

V

Sand and Beads

Courtesey C.L. Lin, University of Utah

Aquifer Material Aquifer Material (Miami Oolite)(Miami Oolite)

Aquifer MaterialAquifer MaterialTucson recharge siteTucson recharge site

Aquifer Material Aquifer Material (Keys limestone)(Keys limestone)

Aquifer Material Aquifer Material (Miami)(Miami)

Image provided courtesy of A. Manda, Florida International University and the United States Geological Survey.

Aquifer Material (CA Coast)Aquifer Material (CA Coast)

Karst (MN)

http://course1.winona.edu/tdogwiler/websitestufftake2/SE%20Minnesota%20Karst%20Hydro%202003-11-22%2013-23-14%20014.JPG

http://course1.winona.edu/tdogwiler/websitestufftake2/

Karst

http://www.fiu.edu/~whitmand/Research_Projects/tm-karst.gif

Ground Water Flow

• Pressure and pressure head

• Elevation head

• Total head

• Head gradient

• Discharge

• Darcy’s Law (hydraulic conductivity)

• Kozeny-Carman Equation

Multiple Choice:Water flows…?

• Uphill

• Downhill

• Something else

Pressure

• Pressure is force per unit area• Newton: F = ma

– Fforce (‘Newtons’ N or kg ms-2)– m mass (kg)– a acceleration (ms-2)

• P = F/Area (Nm-2 or kg ms-2m-2 =

kg s-2m-1 = Pa)

Pressure and Pressure Head

• Pressure relative to atmospheric, so P = 0 at water table

• P = ghp

– density– g gravity

– hp depth

P = 0 (= Patm)

Pre

ssur

e H

ead

(incr

ease

s w

ith d

epth

bel

ow s

urfa

ce)

Pressure Head

Ele

vati

on

Head

Elevation Head

• Water wants to fall

• Potential energy

Ele

vatio

n H

ead

(incr

ease

s w

ith h

eigh

t ab

ove

datu

m)

Eleva

tion

Head

Ele

vati

on

Head

Elevation datum

Total Head

• For our purposes:

• Total head = Pressure head + Elevation head

• Water flows down a total head gradient

P = 0 (= Patm)

Tot

al H

ead

(con

stan

t: h

ydro

stat

ic e

quili

briu

m)

Pressure Head

Eleva

tion

Head

Ele

vati

on

Head

Elevation datum

Head Gradient

• Change in head divided by distance in porous medium over which head change occurs

• A slope

• dh/dx [unitless]

Discharge

• Q (volume per time: L3T-1)

• q (volume per time per area: L3T-1L-2 = LT-1)

Darcy’s Law

• q = -K dh/dx– Darcy ‘velocity’

• Q = K dh/dx A– where K is the hydraulic

conductivity and A is the cross-sectional flow area

• Transmissivity T = Kb– b = aquifer thickness

• Q = T dh/dx L– L = width of flow field

www.ngwa.org/ ngwef/darcy.html

1803 - 1858

http://www.ngwa.org/ngwef/darcy.html

Mean Pore Water Velocity

• Darcy ‘velocity’:q = -K ∂h/∂x

• Mean pore water velocity:v = q/ne

Intrinsic Permeability

g

kK w

L T-1 L2

Kozeny-Carman Equation

1801

2

2

3md

n

nk

More on gradients1 2 3 4 5 6 7 8 9 10 11

2.445659 2.445659 2.937225 3.61747 4.380528 5.182307 5.999944 6.817582 7.619361 8.382418 9.062663 9.554228 9.5542283.399753 3.399754 3.685772 4.152128 4.722335 5.348756 5.99989 6.651023 7.277444 7.847651 8.314006 8.600023 8.6000234.067833 4.067834 4.253985 4.582937 5.007931 5.490497 5.999838 6.509179 6.991744 7.416737 7.745689 7.931838 7.9318384.549766 4.549768 4.679399 4.917709 5.235958 5.605464 5.999789 6.394115 6.76362 7.081868 7.320177 7.449806 7.4498064.902074 4.902077 4.99614 5.172544 5.412733 5.695616 5.999745 6.303874 6.586756 6.826944 7.003347 7.097408 7.0974085.160327 5.160329 5.230543 5.363601 5.546819 5.764526 5.999705 6.234885 6.452591 6.635808 6.768864 6.839075 6.8390755.348374 5.348377 5.402107 5.504502 5.646422 5.815968 5.999672 6.183375 6.35292 6.494838 6.597232 6.650959 6.6509595.482701 5.482704 5.52501 5.605886 5.718404 5.853259 5.999644 6.146028 6.280883 6.393399 6.474273 6.516576 6.5165765.574732 5.574736 5.609349 5.675635 5.768053 5.879029 5.999623 6.120216 6.231191 6.323607 6.389891 6.424502 6.424502

5.63216 5.632163 5.662024 5.719259 5.799151 5.895187 5.999608 6.10403 6.200064 6.279955 6.337188 6.367045 6.3670455.659738 5.659741 5.68733 5.740232 5.814114 5.902965 5.999601 6.096237 6.185087 6.258968 6.311867 6.339453 6.339453

5.659741 5.68733 5.740232 5.814114 5.902965 5.999601 6.096237 6.185087 6.258968 6.311867 6.339453 6.339453

1 2 3 4 5 6 7 8 9 10 11

S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

S11

S12

10.5-11

10-10.5

9.5-10

9-9.5

8.5-9

8-8.5

7.5-8

7-7.5

6.5-7

6-6.5

5.5-6

5-5.5

4.5-5

4-4.5

3.5-4

3-3.5

2.5-3

2-2.5

1.5-2

More on gradients

• Three point problems:

h

h

h

400 m

412 m

100 m

More on gradients

• Three point problems:– (2 equal heads)

h = 10m

h = 10m

h = 9m

400 m

412 m

100 m

CD • Gradient = (10m-9m)/CD

• CD?– Scale from map– Compute

More on gradients

• Three point problems:– (3 unequal heads)

h = 10m

h = 11m

h = 9m

400 m

412 m

100 m

CD • Gradient = (10m-9m)/CD

• CD?– Scale from map– Compute

Best guess for h = 10m

Types of Porous Media

Homogeneous Heterogeneous

Isotropic

Anisotropic

Hydraulic Conductivity Values

Freeze and Cherry, 1979

8.6

0.86

K

(m/d)

Layered media (horizontal conductivity)

M

ii

M

ihii

h

b

KbK

1

1

Q1

Q2

Q3

Q4

Q = Q1 + Q2 + Q3 + Q4

K1

K2

b1

b2

Flow

Layered media(vertical conductivity)

M

ihii

M

ii

v

Kb

bK

1

1

/

Controls flow

Q1

Q2

Q3

Q4

Q ≈ Q1 ≈ Q2 ≈ Q3 ≈ Q4

R1

R2

R3

R4

R = R1 + R2 + R3 + R4

K1

K2

b1

b2

Flow

The overall resistance is controlled by the largest resistance: The hydraulic resistance is b/K

Aquifers

• Lithologic unit or collection of units capable of yielding water to wells

• Confined aquifer bounded by confining beds

• Unconfined or water table aquifer bounded by water table

• Perched aquifers

Transmissivity

• T = Kb

gpd/ft, ft2/d, m2/d

Schematic

i = 1

i = 2

d1

b1

d2

b2 (or h2)

k1

T1

k2

T2 (or K2)

Pumped Aquifer Heads

i = 1

i = 2

d1

b1

d2

b2 (or h2)

k1

T1

k2

T2 (or K2)

Heads

i = 1

i = 2

d1

b1

d2

b2 (or h2)

k1

T1

k2

T2 (or K2)

h1

h2

h2 - h1

LeakanceLeakage coefficient, resistance (inverse)

• Leakance

• From below:

• From above:

d

k

1

11

i

iiiv d

khhq

1

11

i

iiiv d

khhq

Flows

i = 1

i = 2

d1

b1

d2

b2 (or h2)

k1

T1

k2

T2 (or K2)h1

h2 h2 - h1

qv

Boundary Conditions

• Constant head: h = constant

• Constant flux: dh/dx = constant– If dh/dx = 0 then no flow– Otherwise constant flow

Poisson Equation

• Add/remove water from system so that inflow and outflow are different

• R can be recharge, ET, well pumping, etc.• R can be a function of space• Units of R: L T-1

x y

qx|x qx|x+xb

R

x y

qx|x qx|x+x

x yx yx y

qx|x qx|x+xb

R

Derivation of Poisson Equation

x y

qx|x qx|x+xb

R

x y

qx|x qx|x+x

x yx yx y

qx|x qx|x+xb

R(qx|x- qx|x+x)byρt + Rxyρt =0

x

hKq

xRbx

hK

x

hK

xxx

T

R

x

xh

xh

xxx

T

R

x

h

2

2

General Analytical Solution of 1-D Poisson Equation

AxT

R

x

h

xAxT

Rx

x

h

BAxxT

Rh 2

2

T

R

x

h

2

2

xT

Rx

x

h2

2

BAxxT

Rh 2

2

Calculus Review

Documents

Transcript of Calculus Review