Calculus Review
description
Transcript of Calculus Review
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Calculus Review
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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
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Exponents
• x0 = 1
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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
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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) + …
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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) + …
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Numerical Derivatives
• ‘finite difference’ approximation
• slope between points
• dy/dx ≈ y/x
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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)
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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
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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
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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
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WHY?
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Gradients
• del h (or grad h)
• Darcy’s Law:
y
h
x
hh
ji
hKq
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Equipotentials/Velocity Vectors
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Capture Zones
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Capture Zones
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Hydrologic Cycle/Water Balances
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Earth’s Water
• Covers approximately 75% of the surface
• Volcanic emissions
http://earthobservatory.nasa.gov/Library/Water/
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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/
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Fresh Water
Ice caps, Glaciers, & PermanentSnow
Groundwater
Soil Moisture
Ground Ice & Permafrost
Lakes
Atmosphere
Swamp Water
Rivers
Biological Water
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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
http://earthobservatory.nasa.gov/Library/Water/
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Hydrologic Cycle
http://earthobservatory.nasa.gov/Library/Water/
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.)
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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
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Water (Mass) Balance
• Principal components:– Precipitation– Evaporation– Transpiration– Runoff
• P – E – T – Ro = Change in Storage
• Units?
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Ground Water (Mass) Balance
• Principal components:– Recharge– Inflow– Transpiration– Outflow
• R + Qin – T – Qout = Change in Storage
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Water Balance Components
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http://www.srs.fs.usda.gov/gallery/images/5_rain_gauge.jpg
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DBHydroRainfall Stations
• Approximately 600 stations
0
200000
400000
600000
800000
1000000
1200000
1400000
1600000
1800000
2000000
0 200000 400000 600000 800000 1000000 1200000
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Spatial Distribution of Average Rainfall
http://sflwww.er.usgs.gov/sfrsf/rooms/hydrology/compete/obspatialmapx.jpg
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Voronoi/Thiessen Polygons
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Evaporation Pan
www.photolib.noaa.gov/ historic/nws/wea01170.htm
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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
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Watersheds
http://www.bsatroop257.org/Documents/Summer%20Camp/Topographic%20map%20of%20Bartle.jpg
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Watersheds
http://www.bsatroop257.org/Documents/Summer%20Camp/Topographic%20map%20of%20Bartle.jpg
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Stage
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Stage Recorder
http://gallatin.humboldt.edu/~brad/nws/assets/drum-recorder.jpg
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River Hydrograph
http://cires.colorado.edu/lewis/epob4030/Figures/UseandProtectionofWaters/figures/ColoradoRiverHydrograph.gif
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Well Hydrograph
http://wy.water.usgs.gov/news/archives/090100b.htm
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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
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http://www.nws.noaa.gov/om/hod/SHManual/SHMan040_rating.htm
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Ground Water Basics
• Porosity
• Head
• Hydraulic Conductivity
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Porosity Basics
• Porosity n (or )
• Volume of pores is also the total volume – the solids volume
total
pores
V
Vn
total
solidstotal
V
VVn
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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
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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
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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
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Effective
Porosity
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Effective
Porosity
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Porosity Basics
• Volumetric water content ()– Equals porosity for
saturated system total
water
V
V
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Sand and Beads
Courtesey C.L. Lin, University of Utah
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Aquifer Material Aquifer Material (Miami Oolite)(Miami Oolite)
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Aquifer MaterialAquifer MaterialTucson recharge siteTucson recharge site
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Aquifer Material Aquifer Material (Keys limestone)(Keys limestone)
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Aquifer Material Aquifer Material (Miami)(Miami)
Image provided courtesy of A. Manda, Florida International University and the United States Geological Survey.
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Aquifer Material (CA Coast)Aquifer Material (CA Coast)
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Aquifer Material (CA Coast)Aquifer Material (CA Coast)
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Aquifer Material (CA Coast)Aquifer Material (CA Coast)
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Karst (MN)
http://course1.winona.edu/tdogwiler/websitestufftake2/SE%20Minnesota%20Karst%20Hydro%202003-11-22%2013-23-14%20014.JPG
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Karst
http://www.fiu.edu/~whitmand/Research_Projects/tm-karst.gif
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Ground Water Flow
• Pressure and pressure head
• Elevation head
• Total head
• Head gradient
• Discharge
• Darcy’s Law (hydraulic conductivity)
• Kozeny-Carman Equation
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Multiple Choice:Water flows…?
• Uphill
• Downhill
• Something else
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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)
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Pressure and Pressure Head
• Pressure relative to atmospheric, so P = 0 at water table
• P = ghp
– density– g gravity
– hp depth
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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
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Elevation Head
• Water wants to fall
• Potential energy
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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
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Total Head
• For our purposes:
• Total head = Pressure head + Elevation head
• Water flows down a total head gradient
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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
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Head Gradient
• Change in head divided by distance in porous medium over which head change occurs
• A slope
• dh/dx [unitless]
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Discharge
• Q (volume per time: L3T-1)
• q (volume per time per area: L3T-1L-2 = LT-1)
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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
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Mean Pore Water Velocity
• Darcy ‘velocity’:q = -K ∂h/∂x
• Mean pore water velocity:v = q/ne
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Intrinsic Permeability
g
kK w
L T-1 L2
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Kozeny-Carman Equation
1801
2
2
3md
n
nk
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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
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More on gradients
• Three point problems:
h
h
h
400 m
412 m
100 m
![Page 76: Calculus Review](https://reader033.fdocuments.net/reader033/viewer/2022061522/568157c3550346895dc548cd/html5/thumbnails/76.jpg)
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
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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
![Page 78: Calculus Review](https://reader033.fdocuments.net/reader033/viewer/2022061522/568157c3550346895dc548cd/html5/thumbnails/78.jpg)
Types of Porous Media
Homogeneous Heterogeneous
Isotropic
Anisotropic
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Hydraulic Conductivity Values
Freeze and Cherry, 1979
8.6
0.86
K
(m/d)
![Page 80: Calculus Review](https://reader033.fdocuments.net/reader033/viewer/2022061522/568157c3550346895dc548cd/html5/thumbnails/80.jpg)
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
![Page 81: Calculus Review](https://reader033.fdocuments.net/reader033/viewer/2022061522/568157c3550346895dc548cd/html5/thumbnails/81.jpg)
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
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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
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Transmissivity
• T = Kb
gpd/ft, ft2/d, m2/d
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Schematic
i = 1
i = 2
d1
b1
d2
b2 (or h2)
k1
T1
k2
T2 (or K2)
![Page 85: Calculus Review](https://reader033.fdocuments.net/reader033/viewer/2022061522/568157c3550346895dc548cd/html5/thumbnails/85.jpg)
Pumped Aquifer Heads
i = 1
i = 2
d1
b1
d2
b2 (or h2)
k1
T1
k2
T2 (or K2)
![Page 86: Calculus Review](https://reader033.fdocuments.net/reader033/viewer/2022061522/568157c3550346895dc548cd/html5/thumbnails/86.jpg)
Heads
i = 1
i = 2
d1
b1
d2
b2 (or h2)
k1
T1
k2
T2 (or K2)
h1
h2
h2 - h1
![Page 87: Calculus Review](https://reader033.fdocuments.net/reader033/viewer/2022061522/568157c3550346895dc548cd/html5/thumbnails/87.jpg)
LeakanceLeakage coefficient, resistance (inverse)
• Leakance
• From below:
• From above:
d
k
1
11
i
iiiv d
khhq
1
11
i
iiiv d
khhq
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Flows
i = 1
i = 2
d1
b1
d2
b2 (or h2)
k1
T1
k2
T2 (or K2)h1
h2 h2 - h1
qv
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Boundary Conditions
• Constant head: h = constant
• Constant flux: dh/dx = constant– If dh/dx = 0 then no flow– Otherwise constant flow
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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
![Page 91: Calculus Review](https://reader033.fdocuments.net/reader033/viewer/2022061522/568157c3550346895dc548cd/html5/thumbnails/91.jpg)
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
![Page 92: Calculus Review](https://reader033.fdocuments.net/reader033/viewer/2022061522/568157c3550346895dc548cd/html5/thumbnails/92.jpg)
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
![Page 93: Calculus Review](https://reader033.fdocuments.net/reader033/viewer/2022061522/568157c3550346895dc548cd/html5/thumbnails/93.jpg)
Water balance
• Qin + Rxy – Qout = 0• qin by + Rxy – qout by = 0• -K dh/dx|in by + Rxy – -K dh/dx|out by = 0• -T dh/dx|in y + Rxy – -T dh/dx|out y = 0• -T dh/dx|in + Rx +T dh/dx|out = 0
BAxxT
Rh 2
2
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
![Page 94: Calculus Review](https://reader033.fdocuments.net/reader033/viewer/2022061522/568157c3550346895dc548cd/html5/thumbnails/94.jpg)
2-D Finite Difference Approximation
h|x,y h|x+x,y
x, y
y +y
h|x-x,y
x -x x +x
h|x,y-y
h|x,y+y
4,,,,
,
yyxyyxyxxyxx
yx
hhhhh