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## Final Review

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**Final Review**• Exam cumulative: incorporate complete midterm review**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**y = axn + bxp + cxq + … 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 ≈Dy/Dx**Derivative of Sine and Cosine**• sin(0) = 0 • period of both sine and cosine is 2p • d(sin(x))/dx = cos(x) • d(cos(x))/dx = -sin(x)**Partial Derivatives**• Functions of more than one variable • Example: h(x,y) = x4 + y3 + xy**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**Gradients**• del h (or grad h) • Darcy’s Law:**Watersheds**http://www.bsatroop257.org/Documents/Summer%20Camp/Topographic%20map%20of%20Bartle.jpg**Watersheds**http://www.bsatroop257.org/Documents/Summer%20Camp/Topographic%20map%20of%20Bartle.jpg**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**Ground Water Basics**• Porosity • Head • Hydraulic Conductivity**Porosity Basics**• Porosity n (or f) • Volume of pores is also the total volume – the solids volume**Porosity Basics**• Can re-write that as: • Then incorporate: • Solid density: rs = Msolids/Vsolids • Bulk density: rb = Msolids/Vtotal • rb/rs = Vsolids/Vtotal**Ground Water Flow**• Pressure and pressure head • Elevation head • Total head • Head gradient • Discharge • Darcy’s Law (hydraulic conductivity) • Kozeny-Carman Equation**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 = rghp • r density • g gravity • hpdepth**P = 0 (= Patm)**Pressure Head Pressure Head (increases with depth below surface) Elevation Head**Elevation Head**• Water wants to fall • Potential energy**Elevation Head**(increases with height above datum) Elevation Elevation Head Elevation datum Head**Total Head**• For our purposes: • Total head = Pressure head + Elevation head • Water flows down a total head gradient**P = 0 (= Patm)**Pressure Head Total Head (constant: hydrostatic equilibrium) Elevation Elevation Head Elevation datum Head**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 1803 - 1858 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**L2 L T-1**More on gradients**• Three point problems: h 412 m h 400 m 100 m h**More on gradients**h = 10m • Three point problems: • (2 equal heads) 412 m h = 10m 400 m CD • Gradient = (10m-9m)/CD • CD? • Scale from map • Compute 100 m h = 9m**More on gradients**h = 11m • Three point problems: • (3 unequal heads) Best guess for h = 10m 412 m h = 10m 400 m • Gradient = (10m-9m)/CD • CD? • Scale from map • Compute CD 100 m h = 9m**Types of Porous Media**Isotropic Anisotropic Heterogeneous Homogeneous**Hydraulic Conductivity Values**K (m/d) 8.6 0.86 Freeze and Cherry, 1979**Layered media (horizontal conductivity)**Q4 Q3 Q2 Q1 Q = Q1 + Q2 + Q3 + Q4 K2 b2 Flow K1 b1**Layered media(vertical conductivity)**R4 Q4 Flow R3 Q3 K2 b2 R2 K1 b1 Q2 Controls flow R1 Q1 Q ≈ Q1 ≈ Q2 ≈ Q3 ≈ Q4 The overall resistance is controlled by the largest resistance: The hydraulic resistance is b/K R = R1 + R2 + R3 + R4**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**T2 (or K2) b2 (or h2) i = 2 k2 d2 T1 b1 i = 1 k1 d1**T2 (or K2)**k2 T1 k1 Pumped Aquifer Heads b2 (or h2) i = 2 d2 b1 i = 1 d1**T2 (or K2)**k2 T1 k1 Heads h2 - h1 b2 (or h2) h2 i = 2 d2 h1 b1 i = 1 d1**T2 (or K2)**k2 T1 k1 Flows h2 h2 - h1 b2 (or h2) h1 i = 2 d2 qv b1 i = 1 d1**Terminology**• Derive governing equation: • Mass balance, pass to differential equation • Take derivative: • dx2/dx = 2x • PDE = Partial Differential Equation • CDE or ADE = Convection or Advection Diffusion or Dispersion Equation • Analytical solution: • exact mathematical solution, usually from integration • Numerical solution: • Derivatives are approximated by finite differences