1. FLUID STATICS No flow Surfaces of const P and  coincide along gravitational equipotential surfaces h = head = scalar; units of meters = energy/unit weight (energy of position)

2. P =1 atm surface P ~1.3 atm @10 feet P ~1.6 atm @20 feet P ~ 2 atm @33 feet   P  0.1 bar/m

3.  P = 0.1 bar/m 0.6 1.2 2.4 1.8 3.0

5. h P L > P h  P h = 0.1 bar/m 0.6 1.2 1.8 2.4 3.0

6. FLUID DYNAMICS in PERMEABLE MEDIA Consider flow of homogeneous fluid of constant density Fluid transport in the Earth's crust is dominated by Viscous, laminar flow, thru minute cracks and openings, Slow enough that inertial effects are negligible. What drives flow within a permeable medium? Down hill? Down Pressure? Down Head?

7. What drives flow through a permeable medium? Consider: Case 1: Artesian well Case 2: Swimming pool Case 3: Convective gyre Case 4: Metamorphic and Magmatic Systems

8. Humble Texas Flowing 100 years Hot, sulfur-rich, artesian water http://www.texasescapes.com/ TexasGulfCoastTowns/Humble-Texas.htm

9.  P = 0.1 bar/m 0.6 1.2 1.8 2.4 3.0

10.  P = 0.1bar/m    0.6 1.2 1.8 2.4 3.0

11. Criss et al 2000

12. What drives flow within a porous medium? RESULTS: Case 1: Artesian well Fluid flows uphill. Case 2: Swimming pool Large vertical P gradient, but no flow. Case 3: Convective gyre Ascending fluid moves from high to low P Descending fluid moves from low to high P Case 4: Metamorphic and Magmatic Systems Fluid flows both toward heat source, then away, irrespective of pressure

14. Darcy's Law Henry Darcy (1856) Sanitation Engineer Public water supply for Dijon, France. Filtered water thru large sand column; attached Hg manometers Observed relationship bt the volumetric flow rate and the hydraulic gradient Q  (h u -h l )/L where(h u -h l ) is the difference in upper & lower manometer readings L is the spacing length

15. Q = KA(h u -h l )/L

16. Rewrite Darcy's Law Specific Discharge: q= Q/A = -K ∆h/∆L = -K ∂h/∂L = -Ki q= - K  h  "Darcy Velocity" where q Volumetric flux; m 3 /m 2 -sec units of velocity, but is a macroscopic quantity  h hydraulic gradient; dimensionless   i ∂/∂x + j ∂/∂y + k ∂/∂z K hydraulic conductivity, units of velocity (m/sec)  

17. GRADIENT LAWS q= - K  h Darcy’s Law J= - D  C Fick’s Law of Diffusion f= - K  T Fourier’s Law of Heat Flow i= (1/R)V  Ohm’s Law Negative sign: flow is down gradient   

18. Actual microscopic velocity (  )  = q/  = Darcy Velocity/effective porosity Clearly,  > q HYDRAULIC CONDUCTIVITY, K m/s K = k  g/      kg/  units of velocity Proportionality constant in Darcy's Law Property of both fluid and medium see D&S, p. 62

19. HYDRAULIC POTENTIAL (  ): energy/unit mass cf. h = energy/unit weight  = g h = gz + P/  w Consider incompressible fluid element @ elevation z i = 0 pressure P i  i and velocity v = 0 Move to new position z, P,  v Energy difference: lift mass + accelerate + compress (=  VdP) = mg(z- z i ) + mv 2 /2 + m  V/m) dP latter term = m  (1/  dP Energy/unit mass  g z + v 2 /2 +   /  dP For incompressible fluid  = const) & slow flow (v 2 /2  0), z i =0, P i = 0 Energy/unit mass:  g z + P/  = g h Force/unit mass =  = g -  P/  Force/unit weight =  h = 1 -  P/  g

20. Rewrite Darcy's Law : Hubbert (1940, J. Geol. 48, p. 785-944 ) q m  Fluid flux mass vector (g/cm 2 -sec)  k  rock (matrix) permeability (cm 2 )    fluid density (g/cm 3 )  [.....]  Force/unit mass acting on fluid element  1/  where  Kinematic Viscosity  =  cm 2 /sec 

21. Rewrite Darcy's Law : Hubbert (1940; J. Geol. 48, p. 785-944 ) q v  Fluid volumetric flux vector (cm 3 /cm 2 -sec) = q m   k  rock (matrix) permeability (cm 2 )  [.....]  Force/unit vol. acting on fluid element  1/  where  Kinematic Viscosity  =  cm 2 /sec  

22. STATIC FLUID (NO FLOW) Force/unit mass = 0 for q m =0 ∂P/∂z =  g ∂P/∂x =0 ∂P/∂y = 0 Converse: Horizontal pressure gradients require fluid flow 

23. STATIC FLUID (NO FLOW) Force/unit mass = 0 for q m =0 ∂P/∂z =  g ∂P/∂x =0 ∂P/∂y = 0 Converse: Horizontal pressure gradients require fluid flow  0

24. Darcy's Law: Isotropic Media: q = - K  h OK only if K x = K y = K z Darcy's Law: Anisotropic Media K is a tensor Simplest case (orthorhombic?) where principal directions of anisotropy coincide with x, y, z Thus

25. General case: Symmetrical tensor Kxy =Kyx Kzx=Kxz Kyz =Kzy

26. End

27. Relevant Physical Properties for Darcy’s Law Hydraulic conductivity K  kg/ cm/s Density  g/cm 3 Kinematic Viscosity cm 2 /sec Dynamic Viscosity   poise Porosity  dimensionless Permeability kcm 2 q v = - K  h

28. DENSITY (  ) g/cm 3 also, Specific weight (weight density)  g  = f(T,P) Thermal expansivity Isothermal Compressibility where

29. DYNAMIC VISCOSITY  A measure of the rate of strain in an imperfectly elastic material subjected to a distortional stress. For simple shear  =  ∂u  ∂y Units  (poise; 1 P = 0.1 N sec/m 2 = 1 dyne sec/cm 2 Water 0.01 poise (1 centipoise) KINEMATIC VISCOSITY  =   m 2 /sec or cm 2 /sec Water: 10 -6 m 2 /sec = 10 -2 cm 2 /sec Basaltic Magma 0.1 m 2 /sec Asphalt @ 20°C or granitic magma10 2 m 2 /sec Mantle 10 16 m 2 /sec see Tritton p. 5; Elder p. 221)

30. Darcy's Law : Hubbert (1940; J. Geol. 48, p. 785-944 ) where: q v  Darcy Velocity, Specific Discharge or Fluid volumetric flux vector (cm/sec) k  permeability (cm 2 ) K = kg/  hydraulic conductivity (cm/sec)  Kinematic viscosity, cm 2 /sec 

31. POROSITY (  or n) dimensionless Ratio of void space to total volume of material  = V v / V T Dictates how much water a saturated material can contain Important influence on bulk properties of material e.g., bulk , heat cap., seismic velocity…… Difference between Darcy velocity and average microscopic velocity Decrease with depth: Shales  =  o e -cz exponential Sandstones:  =  o - c z linear

32. Fractured Basalt crystalline rocks Limestone karstic & Dolostone Shale Sandstone Siltstone Gravel Sand Silt & Clay FCC BCC Simple cubic 26% 32% 47.6%  Pumice

33. Domenico & Schwartz (1990) Shales ( Athy, 1930) Sandstones (Blatt, 1979)

34. PERMEABILITY (k) units cm 2 Measure of the ability of a material to transmit fluid under a hydrostatic gradient Most important rock parameter pertinent to fluid flow Relates to the presence of fractures and interconnected voids 1 darcy  0.987 x 10 -8 cm 2 .987 x  10 -12 m 2 (e.g., sandstone) Approximate relation between K and k K m/s  10 7 k m 2  10 -5 k darcy

35. 2 10 1nd 1  d 1 md 1 d 1000d Clay Silt Sand Gravel Shale Sandstone argillaceous Limestone cavernous Basalt Crystalline Rocks

36. GEOLOGIC REALITIES OF PERMEABILITY (k) Huge Range in common geologic materials > 10 13 x Decreases super-exponentially with depth k = Cd 2 for granular material, where d = grain diameter, C is complicated parameter k = a 3 /12L for parallel fractures of aperture width “a” and spacing L k is dynamic (dissolution/precipitation, cementation, thermal or mechanical fracturing; plastic deformation) Scale dependence: k regional ≥ k most permeable parts of DH >> k lab; small scale )

37. MEANS: (D&S, p. 66-70) Arithmetic MeanM =   X i /N X i = data points, N = # samples Geometric MeanG = { X 1 X 2 X 3.....X N } 1/N Harmonic MeanH = N/   X i ) Commonly (always?), M > G > H Example: N = 3 samples: X i = 2, 4, 8 M = 4.6667 G = 4.0 H = 3/(7/8) = 3.428

38. In general, both K and k are tensors, and the direction of fluid flow need not coincide with the gradient in hydraulic head

39. Stratigraphic Sequence K x > K z

40. So: Horizontal K is simple mean, weighted by layer thickness Horizontal Flow

41. Stratigraphic Sequence

42. So Vertical Flow thru Stratigraphic Sequence K z is Harmonic Mean, weighted by layer thickness

43. Stratigraphic Sequence

44. PERMEABILITY ANISOTROPY Justification: For vertical flow, Flux must be the same thru each layer! (see F&C, p. 33-34) q = K z,bulk (∆h/m) = K 1 (∆h 1 /m 1 ) = K 2 (∆h 2 /m 2 ) =....... = K n (∆h n /m n ) => K z,bulk = q m/ ∆h = q m/ (∆h 1 + ∆h 2 +.... + ∆h n ) = q m/ (q m 1 /K 1 + q m 2 /K 2 +.... + q m n /K n ) = = m /  m i /K i ) => For horizontal flow, the most permeable units dominate, but For vertical flow, the least permeable units dominate! Anisotropy Ratio: K x / K z ~ 1 to 10x, for typical layer (e.g., because of preferred orientation, schistosity...) Anisotropy Ratio: K x / K z up to 10 6 or more, for stratigraphic sequence In general, for layered anisotropy: K x > K z However, for fracture-related anisotropy, commonly K z > K x

45. End

46. Aquifers Saturated geologic formations with sufficient porosity  and permeability k to allow significant water transmission under ordinary hydraulic gradients. Normally, k ≥ 0.01 d e.g., Unconsolidated sands & gravels; Sandstone, Limestone, fractured volcanics & fractured crystalline rocks Aquitard Geologic formations with low permeability that can store ground water and allow some transmission, but in an amount insufficient for production. Less permeable layers in stratigraphic sequence; = Leaky confining layer e.g., clays, shales, unfractured crystalline rocks Aquiclude Saturated geologic unit incapable of transmitting significant water Rare.

47. Unconfined Aquifer: aquifer in which the water table forms upper boundary. = water table aquifer e.g., Missouri R.; Mississippi R., Meramec River valleys Hi yields, good quality e.g., Ogalalla Aquifer (High Plains aquifer)- CO KS NE NM OK SD QT Sands & gravels, alluvial apron off Rocky Mts. Perched Aquifer: unconfined aquifer above main water table; Generally above a lens of low-k material. Note- there also is an "inverted" water table along bottom! Confined Aquifer: aquifer between two aquitards. = Artesian aquifer if the water level in a well rises above aquifer = Flowing Artesian aquifer if the well level rises above the ground surface. e.g., Dakota Sandstone: east dipping K sst, from Black Hills- artesian) Hydrostratigraphic Unit: e.g. MO, IL C-Ord sequence of dolostone & sandstone capped by Maquoketa shale

48. after Driscoll, FG (1986) http://www.uwsp.edu/water/portage/undrstnd/aquifer.htm

49. after Fetter http://www.uwsp.edu/water/portage/undrstnd/aquifer.htm Unconfined Aquifer

50. after Fetter http://www.uwsp.edu/water/portage/undrstnd/aquifer.htm Perched and Unconfined Aquifers

51. after Fetter http://www.uwsp.edu/water/portage/undrstnd/aquifer.htm Confined Aquifer

52. Hubbert (1940)

53. after Darton 1909 Potentiomtric Surface, Dakota Aquifer Black Hills

54. Unconfined Aquifer: Water table aquifer Aquifer in which the water table forms upper boundary. e.g., MO, Miss, Meramec River valleys. Hi yields, good quality e.g., Ogalalla Aquifer (High Plains aquifer) Properties: 1) Get large production for a given head drop, as Specific Yield Sy is large (~0.25). 2) Storativity S = Sy + Ss*h  Sy, commonly (eq 4.33 Fetter) 3) Easily contaminated 4) Artesian flow possible Confined Aquifer: Aquifer between two aquitards. Artesian aquifer if the water level in a well rises above aquifer Flowing Artesian aquifer if the water level in the well rises above the ground surface. e.g., Dakota Sandstone Properties: 1) Get large changes in pressure (head) with ~ no change in the thickness of the saturated column. Potentiometric sfc remains above the unit. 2) Get large head drop for a given amount of production, as Ss is very small. 3) Storativity S= Ss*m where Ss = specific storage Commonly, S ~ 0.005 to 0.0005 for aquifers

55. Darcy's Law : Hubbert (1940; J. Geol. 48, p. 785-944 ) where: q v  Darcy Velocity, Specific Discharge or Fluid volumetric flux vector (cm/sec) k  permeability (cm 2 ) K = kg/  hydraulic conductivity (cm/sec)  Kinematic viscosity, cm 2 /sec  = (k/  [force/unit mass]

56. Gravitational Potential  g

58. Flow Nets: Set of intersecting Equipotential lines and Flowlines Flowlines  Streamlines Instantaneous flow directions Pathlines = actual particle path; Pathlines ≠ Flowlines for transient flow. Flowlines | to Equipotential surface if K is isotropic Can be conceptualized in 3D

59. Fetter No Flow

60. Topographic Highs tend to be Recharge Zones h decreases with depth Water tends to move downward => recharge zone Topographic Lows tend to be Discharge Zones h increases with depth Water will tend to move upward => discharge zone It is possible to have flowing well in such areas, if case the well to depth where h > h@ sfc. Hinge Line: Separates recharge (downward flow) & discharge areas (upward flow). Can separate zones of soil moisture deficiency & surplus (e.g., waterlogging). Topographic Divides constitute Drainage Basin Divides for Surface water e.g., continental divide Topographic Divides may or may not be GW Divides

61. MK Hubbert (1940) http://www.wda-consultants.com/java_frame.htm?page17

62. Fetter, after Hubbert (1940)

63. Equipotential Lines Lines of constant head. Contours on potentiometric surface or on water table  map => Equipotential Surface in 3D Potentiometric Surface: ("Piezometric sfc") Map of the hydraulic head; Contours are equipotential lines Imaginary surface representing the level to which water would rise in a nonpumping well cased to an aquifer, representing vertical projection of equipotential surface to land sfc. Vertical planes assumed; no vertical flow: 2D representation of a 3D phenomenon Concept rigorously valid only for horizontal flow w/i horizontal aquifer Measure w/ Piezometers  small dia non-pumping well with short screen- can measure hydraulic head at a point (Fetter, p. 134)

64. Domenico & Schwartz(1990) Flow beneath Dam Vertical x-section Flow toward Pumping Well, next to river Plan view River Channel

65. after Freeze and Witherspoon 1967 http://wlapwww.gov.bc.ca/wat/gws/gwbc/!!gwbc.html Effect of Topography on Regional Groundwater Flow

66. for unconfined flow

67. Saltwater Intrusion Saltwater-Freshwater Interface: Sharp gradient in water quality Seawater Salinity = 35‰ = 35,000 ppm = 35 g/l NaCl type water  sw = 1.025 Freshwater < 500 ppm (MCL), mostly Chemically variable; commonly Na Ca HCO 3 water  fw = 1.000 Nonlinear Mixing Effect: Dissolution of cc @ mixing zone of fw & sw Possible example: Lower Floridan Aquifer: mostly 1500’ thick Very Hi T ~ 10 7 ft 2 /day in “Boulder Zone” near base,  ~30% paleokarst? Cave spongework

68. PROBLEMS OF GROUNDWATER USE Saltwater Intrusion Mostly a problem in coastal areas: GA NY FL Los Angeles Abandonment of freshwater wells; e.g., Union Beach, NJ Los Angeles & Orange Ventura Co; Salinas & Pajaro Valleys; Fremont Water level have dropped as much as 200' since 1950. Correct with artificial recharge Upconing of underlying brines in Central Valley

69. Craig et al 1996 Union Beach, NJ Water Level & Chlorinity

70. Air Fresh Water   =1.00 hfhf Fresh Water-Salt Water Interface? Sea level  Salt Water  =1.025 ? ? ?

71. Salt Water Fresh Water hfhf z Ghyben-Herzberg P Sea level z interface 

72. Ghyben-Herzberg Analysis Hydrostatic Condition  P -  g = 0 No horizontal P gradients Note: z = depth  fw = 1.00  sw = 1.025

73. Salt Water Fresh Water hfhf z Ghyben-Herzberg P Sea level z interface 

74. Physical Effects Tend to have a rather sharp interface, only diffuse in detail e.g., Halocline in coastal caves Get fresh water lens on saline water Islands: FW to 1000’s ft below sea level; e.g., Hawaii Re-entrants in the interface near coastal springs, FLA Interesting implications: 1) If  is 10’ ASL, then interface is 400’ BSL 2) If  decreases 5’ ASL, then interface rises 200’ BSL 3) Slope of interface ~ 40 x slope of water table

75. Hubbert’s (1940) Analysis Hydrodynamic condition with immiscible fluid interface 1) If hydrostatic conditions existed: All FW would have drained out Water table @ sea level, everywhere w/ SW below 2) G-H analysis underestimates the depth to the interface Assume interface between two immiscible fluids Each fluid has its own potential h everywhere, even where that fluid is not present! FW potentials are horizontal in static SW and air zones, where heads for latter phases are constant

76. Ford & Williams 1989 ….…...

77. after Ford & Williams 1989 ….…... Fresh Water Equipotentials  Fresh Water Equipotentials 

78. For any two fluids, two head conditions: P sw =  sw g (h sw + z) and P fw =  fw g (h fw + z) On the mutual interface, P sw = P fw so: ∂z/∂x gives slope of interface ~ 40x slope of water table Also, 40 = spacing of horizontal FW equipotentials in the SW region Take ∂/∂z and ∂/∂x on the interface, noting that h sw is a constant as SW is not in motion

79. after USGS WSP 2250 Saline ground water 0 0 0 Fresh Water Lens on Island Saline ground water 0

80. Confined Unconfined Fetter

81. Saltwater Intrusion Mostly a problem in coastal areas: GA NY FL Los Angeles From above analysis, if lower  by 5’ ASL by pumping, then interface rises 200’ BSL! Abandonment of freshwater wells- e.g., Union Beach, NJ Can attempt to correct with artificial recharge- e.g., Orange Co Los Angeles, Orange, Ventura Counties; Salinas & Pajaro Valleys; Water level have dropped as much as 200' since 1950. Correct with artificial recharge Also, possible upconing of underlying brines in Central Valley FLA- now using reverse osmosis to treat saline GW >17 MGD Problems include overpumping; upconing due to wetlands drainage (Everglades) Marco Island- Hawthorn Fm. @ 540’: Cl to 4800 mg/l (cf. 250 mg/l Cl drinking water std)

82. Possible Solutions Artificial Recharge (most common) Reduced Pumping Pumping trough Artificial pressure ridge Subsurface Barrier

83. End

84. USGS WSP 2250

87. Potentiometric Surface defines direction of GW flow: Flow at rt angle to equipotential lines (isotropic case) If spacing between equipotential lines is const, then K is constant In general K 1 A 1 /L 1 = K 2 A 2 /L 2 where A = x-sect thickness of aquifer; L = distance between equipotential lines For layer of const thickness, K 1 /L 1 = K 2 /L 2 (eg. 3.35; D&S p. 79)

88. Hubbert 1957

89. 76.1 mi 2

102. FLUID DYNAMICS Consider flow of homogeneous fluid of constant density Fluid transport in the Earth's crust is dominated by Viscous, laminar flow, thru minute cracks and openings, Slow enough that inertial effects are negligible. What drives flow within a porous medium? Down hill? Down Pressure? Down Head? Consider: Case 1: Artesian well- fluid flows uphill. Case 2: Swimming pool- large vertical P gradient, but no flow. Case3: Convective gyre w/i Swimming pool- ascending fluid moves from hi to lo P descending fluid moves from low to hi P Case 4: Metamorphic rocks and magmatic systems.

103. after Toth (1963) http://www.uwsp.edu/water/portage/undrstnd/topo.htm

104. Fetter, after Toth (1963)

105. Salt Water Fresh Water hfhf z Ghyben-Herzberg P Sea level 