2. Final Exam May 10, 5 – 7:30 pm, ESS 081

3. Energy Transformation 1 Caloria of heat = energy necessary to raise the temperature of one gram of pure water from 14.5 – 15.5 o C Latent Heat of vaporization Hv = 597.3 – 0.564T (Cal./g) Latent Heat of condensation

4. Energy Transformation, Cont. Latent heat of fusion – Hf – 1 g of ice at 0 o C => ~80 cal of heat must be added to melt ice. Resulting water has same temperature. Sublimation – Water passes directly from a solid state to a vapor state. Energy = Hf + Hv => 677 cal/g at 0 o C. Hv > 6Hf > 5 x amt. to warm water from 0 o C -> 100 o C

5. Hydrologic Equation Inflow = outflow +/- Changes in storage Equation is simple statement of mass conservation

7. Condensation Condensation occurs when air mass can no longer hold all of its humidity. Temperature drops => saturation humidity drops. If absolute humidity remains constant => relative humidity rises. Relative humidity reaches 100% => condensation => Dew point temperature.

8. Cool, moist Warm, dry Limited soil-moisture storage

9. All infiltrate some water always on the surface Puddles and overland flow

10. Q0Q0

11. Determining ground water recharge from baseflow (1) Meyboom method (Seasonal recession method): utilizes stream hydrographs from two or more consecutive years. Assumptions: the catchment area has no dams or other method of streamflow regulation; snowmelt contributes little to the runoff.

12. Determining ground water recharge from baseflow (2) Rorabaugh method (Recession curve displacement method): utilizes stream hydrograph during one season.

13. Aquifer Properties: Porosity, specific yield, specific retention. Potential: Transmissivity, storativity. Types: confined, unconfined. Hydraulic conductivity, Physical Laws controlling water transport.

15. d 60 d 10

16. Sediment Classification Sediments are classified on basis of size of individual grains Grain size distribution curve Uniformity coefficient C u = d 60 /d 10 d 60 = grain size that is 60% finer by weight. d 10 = grain size that is 10% finer by weight. C u = 4 => well sorted; C u > 6 => poorly sorted.

17. Specific Yield and Retention Specific yield – S y : ratio of volume of water that drains from a saturated rock owing to the attraction of gravity to the total volume of the rock. Specific retention – S r : ratio of the volume of water in a rock can retain against gravity drainage to the total volume of the rock. n = S y + S r. S r increases with decreasing grain size.

18. Darcy’s Law Q = -KA(dh/dl). dh/dl = Hydraulic gradient. dh = change in head between two points separated by small distance dl.

19. Laminar flow (Small R < 10) Turbulent flow (Large R) Flow lines Darcy’s Law: Yes Darcy’s Law: No

20. Hydraulic conductivity K = hydraulic conductivity (L/T). K is also referred to as the coefficient of permeability. K = -Q[A(dh/dl)] [ L 3 /T/[L 2 (L/L)] = L/T] V = Q/A = -K(dh/dl) = specific discharge or Darcian velocity.

21. Intrinsic Permeability Intrinsic permeability K i = Cd 2 (L 2 ). K = K i (γ/μ) or K = K i (ρg/ μ) Petroleum industry 1 Darcy = unit of intrinsic permeability K i 1 darcy = 1 cP x 1 cm 3 /s / (1 atm/ 1 cm). cP – centipoise - 0.01 dyn s/cm 2 atm – atmospheric pressure – 1.0132 x 10 16 dyn/cm 2 1 darcy = 9.87 x 10 -9 cm 2 ~ 10 -8 cm 2

22. Aquifer Aquifer – geologic unit that can store and transmit water at rates fast enough to supply amounts to wells. Usually, intrinsic permeability > 10 -2 Darcy. Confining layer – unit with little or no permeability … < 10 -2 Darcy. aquifuge – absolutely impermeable unit. aquitard - a unit can store and transmit water slowly. Also called leaky confining layer. Raritan formation on Long Island. -- all these definitions are in a relative sense.

24. Transmissivity The amount of water that can be transmitted horizontally through a unit width by the full saturated thickness of the aquifer under a hydraulic gradient of 1. T = bK T = transmissivity. b = saturated thickness. K = hydraulic conductivity. Multilayer => T 1 + T 2 + … + T n

25. Specific Storage Specific storage S s = amount of water per unit volume stored or expelled owing to compressibility of mineral skeleton and pore water per unit change in head (1/L). S s = ρ w g(α+nβ) α = compressibiliy of aquifer skeleton. n = porosity. β = compressibility of water.

26. Storativity of confined Unit S = b S s S s = specific storage. b = aquifer thickness. All water released in confined, saturated aquifer comes from compressibility of mineral skeleton and pore water.

27. Storativity in Unconfined Unit Changes in saturation associated with changes in storage. Storage or release depends on specific yield S y and specific storage S s. S = S y + b S s

28. Volume of water drained from aquifer V w = SAdh V w = volume of water drained. S = storativity (dimensionless). A = area overlying drained aquifer. dh = average decline in head.

29. Hydraulic head, h Hydraulic head is energy per unit weight. h = v 2 /2g + z + P/gρ. [L]. Unit: (L; ft or m). v ~ 10 -6 m/s or 30 m/y for ground water flows. v 2 /2g ~ 10 -12 m 2 /s 2 / (2 x 9.8 m/s 2 ) ~ 10 -13 m. h = z + P/gρ. [L].

31. Flow lines and flow nets A flow line is an imaginary line that traces the path that a particle of ground water would flow as it flows through an aquifer. A flow net is a network of equipotential lines and associated flow lines.

32. Boundary conditions No-flow boundary – flow line – parallel to the boundary. Equipotential line - intersect at right angle. Constant-head boundary – flow line – intersect at right angle. Equipotential line - parallel to the boundary. Water-table boundary – flow line – depends. Equipotential line - depends.

33. Estimate the quantity of water from flow net q’ = Kph/f. q’ – total volume discharge per unit width of aquifer (L 3 /T; ft 3 /d or m 3 /d). K – hydraulic conductivity (L/T; ft/d or m/d). p – number of flowtubes bounded by adjacent pairs of flow lines. h – total head loss over the length of flow lines (L; ft or m). f - number of squares bounded by any two adjacent flow lines and covering the entire length of flow.

37. Water table Water table = undulating surface at which pressure in fluid in pores = atmospheric pressure. Water table

38. Our purpose of well studies Compute the decline in the water level, or drawdown, around a pumping well whose hydraulic properties are known. Determine the hydraulic properties of an aquifer by performing an aquifer test in which a well is pumped at a constant rate and either the stabilized drawdown or the change in drawdown over time is measured.

39. Drawdown T = Q/ 4  (h 0 -h)G(u) G(u) = W(u) - completely confined. W(u,r/B) – leaky, confined, no storage. H(u,  ) – leaky, confined, with storage. W(u A,u B,  ) - unconfined.

41. Aquifer test Steady-state conditions. Cone of depression stabilizes. Nonequilibrium flow conditions. Cone of depression changes. Needs a pumping well and at least one observational well.

42. Aquifer tests T = Q/ 4  (h 0 -h)G(u) G(u) = W(u) - completely confined. W(u,r/B) – leaky, confined, no storage. H(u,  ) – leaky, confined, with storage. W(u A,u B,  ) - unconfined.

46. Slug test Overdamped – water level recovers to the initial static level in a smooth manner that is approximately exponential. Underdamped – water level oscillates about the static water level with the magnitude of oscillation decreasing with time until the oscillations cease.

47. Cooper-Bredehoeft-Papadopulos Method (confined aquifer) H/H 0 = F( ,  ) H – head at time t. H 0 – head at time t = 0.  = T t/r c 2  = r s 2 S/r c 2

49. Underdamped Response Slug Test Van der Kamp Method – confined aquifer and well fully penetrating. H(t) = H 0 e -  t cos  t H(t) - hydraulic head (L) at time t (T) H 0 - the instantaneous change in head (L)  - damping constant (T -1 )  - an angular frequency (T -1 )

50.  = ln[H(t 1 )/H(t 2 )]/ (t 2 – t 1 )  = 2  /(t 2 -t 1 )

51. Underdamped Response Slug Test (cont.) T = c + a ln T c = -a ln[0.79 r s 2 S(g/L) 1/2 ] a = [r c 2 (g/L) 1/2 ] / (8d) d =  /(g/L) 1/2 L = g / (  2 +  2 )

55. Mass transport of solutes Diffusion – both ionic and molecular species dissolved in water move from area of higher concentration (chemical activity) to areas of lower concentration. Advection – moving water carries it dissolved solutes.

56. Diffusion – Fick’s laws Fick’s first law F = -D dC/dx F = mass flux of solute per unit area per unit time. D = diffusion coefficient (area/time) C = solute concentration (mass/volume) dC/dx = concentration gradient (mass/volume/distance). D ranges from 1 x 10 -9 to 2 x 10 -9 m 2 /s, for the major cations and anions.

57. Diffusion – Fick’s laws (cont.) Fick’s second law  C/  t = D  2 C/  x 2 D = diffusion coefficient (area/time) C = solute concentration (mass/volume) t = time

58. Advection Advecting contaminants travel at the same rate as the average linear velocity of ground water v x = -(K/n e ) dh/dl v x = average linear velocity K = hydraulic conductivity n e = effective porosity dh/dl = hydraulic gradient

59. Mechanical Dispersion Longitudinal dispersion: if the mixing occurs along the pathway of fluid flow - it moves faster through the center of the pore; - some of the fluid will travel in longer pathways; - fluid travels faster through larger pore. Transverse or lateral dispersion: if the mixing occurs normal to the pathway of fluid flow. - flow paths can split and branch out to the side.

60. Hydrodynamic Dispersion Hydrodynamic dispersion: D L = D* + a L v x D L = longitudinal coefficient of hydrodynamic dispersion D* = effective molecular diffusion coefficient a L = dynamic dispersivity v x = average linear ground-water velocity

61. Advection-dispersion Equation D L  2 C/  x 2 – v x  C/  x =  C/  t D L  2 C/  x 2 – dispersion (diffusion + dispersivity). v x  C/  x – Advection

62. Solute Transport by Advection- Dispersion C = C 0 /2{erfc[(L-v x t)/2(D L t) 1/2 ] + exp(v x L/D L )erfc[(L-v x t)/2(D L t) 1/2 ] } C = solute concentration (M/L 3, mg/L) C 0 = initial concentration (M/L 3, mg/L) L = flow path length (L; ft/m) v x = average ground velocity (L/T) t = time since release of the solute (T) D L = longitudinal dispersion coefficient (L 2 /T)

64. Retardation Factor Retardation factor = 1 + (  b /  )(K d )  b = dry bulk mass density of the soil (M/L 3 ; gm/cm 3 )  = volumetric moisture content of the soil (dimensionless). K d = distribution coefficient for solute with the soil (L 3 /M; mL/g)

65. Solute Movement with Retardation v c = v x /[1+ (  b /  )(K d )] v c = velocity of the solute front. In one- dimensional column the solute concentration is one-half of the original value (L/T; ft/day or m/day). v x = average linear velocity (L/T; ft/day or m/day).

66. x = -y/tan(2  Kbiy/Q) Q - pumping rate K - conductivity b – initial thickness i – initial h gradient x 0 = -Q/(2  Kbi) y max =  Q/(2Kbi) Confined

67. Static fresh and slat water Ghyben-Herzberg principle