Calculus Review

Slope Slope = rise/run =  y/  x = (y 2 – y 1 )/(x 2 – x 1 ) 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 x 0 = 1

Derivative of a line y = mx + b slope m and y axis intercept b derivative of y = ax n + b with respect to x: dy/dx = a n x (n-1) Because b is a constant -- think of it as bx 0 -- its derivative is 0b -1 = 0 For a straight line, a = m and n = 1 so dy/dx = m 1 x (0), or because x 0 = 1, dy/dx = m

Derivative of a polynomial In differential Calculus, we consider the slopes of curves rather than straight lines For polynomial y = ax n + bx p + cx q + … derivative with respect to x is dy/dx = a n x (n-1) + b p x (p-1) + c q x (q-1) + …

Example y = ax n + bx p + cx q + … 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)

Partial Derivatives Functions of more than one variable Example: h(x,y) = x 4 + y 3 + xy

Partial Derivatives Partial derivative of h with respect to x at a y location y 0 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) = x 4 + y 3 + x 2 y+ xy ∂ h/ ∂ x = 4x 3 + 2xy + y ∂ h/ ∂ x| y=y 0 = 4x 3 + 2xy 0 + y 0

WHY?

Gradients del h (or grad h) Darcy’s Law:

Equipotentials/Velocity Vectors

Capture Zones

Hydrologic Cycle/Water Balances

Earth’s Water Covers approximately 75% of the surface Volcanic emissions

One estimate of global water distribution Volume (1000 km 3 ) Percent of Total Water Percent of Fresh Water Oceans, Seas, & Bays1,338, Ice caps, Glaciers, & Permanent Snow 24, Groundwater23, Fresh(10,530)(0.76)30.1 Saline(12,870)(0.94)- Soil Moisture Ground Ice & Permafrost Lakes Fresh(91.0)(0.007).26 Saline(85.4)(0.006)- Atmosphere Swamp Water Rivers Biological Water Total1,385, 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

Fresh 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 + Q in – T – Q out = Change in Storage

Water Balance Components

DBHydro Rainfall Stations Approximately 600 stations

Spatial Distribution of Average Rainfall

Voronoi/Thiessen Polygons

Evaporation Pan historic/nws/wea01170.htm

Pan Evaporation Pan Coefficients: 0.58 – 0.78 Transpiration Potential Evapotranspiration –Thornwaite Equation

Watersheds

Watersheds

Stage

Stage Recorder

River Hydrograph

Well Hydrograph

Stream Gauging Measure velocity at 2/10 and 8/10 depth Q = v*A Rating curve: –Q vs. Stage

Ground Water Basics Porosity Head Hydraulic Conductivity

Porosity Basics Porosity n (or  ) Volume of pores is also the total volume – the solids volume

Porosity Basics Can re-write that as: Then incorporate: Solid density:  s = M solids /V solids Bulk density:  b = M solids /V total  b  s = V solids /V total

Cubic Packings and Porosity Simple Cubic Body-Centered Cubic Face-Centered Cubic n = 0.48 n = n = 0.26

FCC and BCC have same porosity Bottom line for randomly packed beads: n ≈ Smith et al. 1929, PR 34:

Effective Porosity

Porosity Basics Volumetric water content (  ) –Equals porosity for saturated system

Sand and Beads Courtesey C.L. Lin, University of Utah

Aquifer Material (Miami Oolite)

Aquifer Material Tucson recharge site

Aquifer Material (Keys limestone)

Aquifer Material (Miami) Image provided courtesy of A. Manda, Florida International University and the United States Geological Survey.

Aquifer Material (CA Coast)

Karst (MN) SE%20Minnesota%20Karst%20Hydro% % %20014.JPG

Karst

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 –F  force (‘Newtons’ N or kg ms -2 ) –m mass (kg) –a acceleration (ms -2 ) P = F/Area (Nm -2 or kg ms -2 m -2 = kg s -2 m -1 = Pa)

Pressure and Pressure Head Pressure relative to atmospheric, so P = 0 at water table P =  gh p –  density –g gravity –h p depth

P = 0 (= P atm ) Pressure Head (increases with depth below surface) Pressure Head Elevation Head

Elevation Head Water wants to fall Potential energy

Elevation Head (increases with height above datum) Elevation Head Elevation Head Elevation datum

Total Head For our purposes: Total head = Pressure head + Elevation head Water flows down a total head gradient

P = 0 (= P atm ) Total Head (constant: hydrostatic equilibrium) Pressure Head Elevation Head Elevation 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: L 3 T -1 ) q (volume per time per area: L 3 T -1 L -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 ngwef/darcy.html

Mean Pore Water Velocity Darcy ‘velocity’: q = -K ∂h/∂x Mean pore water velocity: v = q/n e

Intrinsic Permeability L T -1 L2L2

Kozeny-Carman Equation

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 = 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, K (m/d)

Layered media (horizontal conductivity) Q1Q1 Q2Q2 Q3Q3 Q4Q4 Q = Q 1 + Q 2 + Q 3 + Q 4 K1K1 K2K2 b1b1 b2b2 Flow

Layered media (vertical conductivity) Controls flow Q1Q1 Q2Q2 Q3Q3 Q4Q4 Q ≈ Q 1 ≈ Q 2 ≈ Q 3 ≈ Q 4 R1R1 R2R2 R3R3 R4R4 R = R 1 + R 2 + R 3 + R 4 K1K1 K2K2 b1b1 b2b2 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, ft 2 /d, m 2 /d

Schematic i = 1 i = 2 d1d1 b1b1 d2d2 b 2 (or h 2 ) k1k1 T1T1 k2k2 T 2 (or K 2 )

Pumped Aquifer Heads i = 1 i = 2 d1d1 b1b1 d2d2 b 2 (or h 2 )k1k1 T1T1 k2k2 T 2 (or K 2 )

Heads i = 1 i = 2 d1d1 b1b1 d2d2 b 2 (or h 2 )k1k1 T1T1 k2k2 T 2 (or K 2 ) h1h1 h2h2 h 2 - h 1

Leakance Leakage coefficient, resistance (inverse) Leakance From below: From above:

Flows i = 1 i = 2 d1d1 b1b1 d2d2 b 2 (or h 2 )k1k1 T1T1 k2k2 T 2 (or K 2 ) h1h1 h2h2 h 2 - h 1 qvqv

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

Derivation of Poisson Equation (q x | x - q x | x+  x )  b  yρ  t + R  x  yρ  t =0

General Analytical Solution of 1-D Poisson Equation

Water balance Q in + R  x  y – Q out = 0 q in b  y + R  x  y – q out b  y = 0 -K dh/dx| in b  y + R  x  y – -K dh/dx| out b  y = 0 -T dh/dx| in  y + R  x  y – -T dh/dx| out  y = 0 -T dh/dx| in + R  x +T dh/dx| out = 0

2-D Finite Difference Approximation