FLUID STATICS No flow

1. FLUID STATICS No flow Surfaces of const P and r 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. 0.6 1.2 1.8 2.4 3.0 P = 0.1 bar/m

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

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

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

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  (hu -hl)/L where (hu -hl) is the difference in upper & lower manometer readings L is the spacing length

15. Q = KA(hu-hl)/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; m3/m2-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hDarcy’s Law J = - DCFick’s Law of Diffusion f = - KTFourier’s Law of Heat Flow i = (1/R)VOhm’s Law    Negative sign: flow is down gradient

18. Actual microscopic velocity (u) • u = q/f = Darcy Velocity/effective porosity • Clearly, u > q • HYDRAULIC CONDUCTIVITY, K m/s • K = krg/m • =kg/n units of velocity • Proportionality constant in Darcy's Law • Property of both fluid and medium • see D&S, p. 62

19. HYDRAULIC POTENTIAL (F): energy/unit mass cf. h = energy/unit weight F = g h = gz + P/rw Consider incompressible fluid element @ elevation zi= 0 pressure Pi ri and velocity v = 0 Move to new position z, P, r , v Energy difference: lift mass + accelerate + compress (= VdP) = mg(z- zi) + mv2/2 + m V/m) dP latter term = m(1/r)dP Energy/unit mass F = g z + v2/2 +  (1/r)dP For incompressible fluid(r = const) & slow flow (v2/2 0), zi=0, Pi = 0 Energy/unit mass: F = g z + P/r= g h Force/unit mass = F= g - P/r Force/unit weight = h= 1 - P/rg

20. Rewrite Darcy's Law: Hubbert (1940, J. Geol. 48, p. 785-944)  • qm Fluid flux mass vector (g/cm2-sec) •  k rock (matrix) permeability (cm2) •  r fluid density (g/cm3) •  [.....] Force/unit mass acting on fluid element •  1/ • whereKinematic Viscosity • =  cm2/sec

21. Rewrite Darcy's Law: Hubbert (1940; J. Geol. 48, p. 785-944)   • qv Fluid volumetric flux vector (cm3/cm2-sec) =qm/ •  k rock (matrix) permeability (cm2) •  [.....] Force/unit vol. acting on fluid element •  1/ • whereKinematic Viscosity • =  cm2/sec

22. STATIC FLUID (NO FLOW)

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

24. Darcy's Law: Isotropic Media: q = - K h OK only if Kx = Ky = Kz Darcy's Law: Anisotropic Media K, k are tensors Direction of fluid flow need not coincide with the gradient in hydraulic head

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

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

27. End

28. qv = - Kh Relevant Physical Properties for Darcy’s Law Hydraulic conductivity K = kg/n cm/s Density r g/cm3 Kinematic Viscosity n cm2/sec Dynamic Viscosity m = n*rpoise Porosity f dimensionless Permeability k cm2 “Darcy” VersionHubbert Version

29. qv = - Kh Relevant Physical Properties for Darcy’s Law Hydraulic conductivity (K) cm/s Units of velocity Proportionality constant in Darcy’s Law Property of both fluid and medium “Darcy” VersionHubbert Version => K = kg/n

30. DENSITY (r) g/cm3 Fluid property Specific weight (weight density) g = r g r = f(T,P) where Thermal expansivity Isothermal Compressibility

31. DYNAMIC VISCOSITYm Fluid property Stokes Law: Gravitational Force = Frictional Force • Units of m : • poise; 1 P = 0.1 N sec/m2 • = 1 dyne sec/cm2 • Water 0.01 poise (1 centipoise)

32. DYNAMIC VISCOSITYm Fluid property • A measure of the rate of strain in an imperfectly elastic material • subjected to a distortional stress. • For simple shear t = m ∂u/∂y • Units (poise; 1 P = 0.1 N sec/m2 = 1 dyne sec/cm2 • Water 0.01 poise (1 centipoise) • KINEMATIC VISCOSITYn Fluid property • n = m/rm2/sec or cm2/sec • Water: 10-6 m2/sec = 10-2 cm2/sec • Basaltic Magma 0.1 m2/sec • Asphalt @ 20°C • or granitic magma 102 m2/sec • Mantle 1016 m2/sec see Tritton p. 5; Elder p. 221)

33. Darcy's Law: Hubbert (1940; J. Geol. 48, p. 785-944) • where: • qv Darcy Velocity, Specific Discharge • or Fluid volumetric flux vector (cm/sec) • k= permeability (cm2) • K = k(g/n) hydraulic conductivity (cm/sec) • Kinematic viscosity, cm2/sec 

34. POROSITY(f, or n) dimensionless Rock property Ratio of void space to total volume of material f = Vv/VT Dictates how much water a saturated material can contain Large Range: <0.1% to >75% Strange behaviors Important influence on bulk properties of material e.g., bulk r, heat capacity, seismic velocity…… Difference between Darcy velocity and average microscopic velocity Decreases with depth: Shales f= foe-czexponential Sandstones: f= fo - cz linear

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

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

37. PERMEABILITY (k) cm2 Measure of the ability of a material to transmit fluid under a hydrostatic gradient Differences with Porosity? Differences with Porosity?

38. PERMEABILITY (k) cm2 Measure of the ability of a material to transmit fluid under a hydrostatic gradient Differences with Porosity? Different Units Styrofoam cup: High f, Low k Uniform spheres: f ≠ f(dia); k ~ dia2

39. PERMEABILITY (k) cm2 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 cm2 = 0.987 x10-12 m2 (e.g., sandstone) Approximate relation between K and k (for cool water):factor = g/n Km/s@ 107 k m2 = 103 k cm2 = 10-5 kdarcy Kcm/s@ 105 k cm2 = 10-3 kdarcy Kft/y@ 1011 k cm2

40. 1nd 1md 1 md 1 d 1000 d Clay Silt Sand Gravel Shale Sandstone argillaceous Limestone cavernous Basalt Crystalline Rocks 10 10 10 10 10 10 10 10 10 2

41. GEOLOGIC REALITIES OF PERMEABILITY (k) Huge Range in common geologic materials > 1013 x Decreases super-exponentially with depth k = Cd2 for granular material, where d = grain diameter, C is complicated parameter k = a3/12L for parallel fractures of aperture width “a” and spacing L k is dynamic (dissolution/precipitation, cementation, thermal or mechanical fracturing; plastic deformation) K is very low in deforming rocks as cracks seal (marbles, halite) Scale dependence: kregional ≥ kmost permeable parts of drill holes >> klab; small scale

42. MEANS: (D&S, p. 66-70) Arithmetic Mean M = SXi/N Xi = data points, N = # samples Geometric Mean G = {X1 X2 X3 .....XN}1/N Harmonic Mean H = N/S (1/Xi) Commonly (always?) , M > G > H Example: N = 3 samples: Xi = 2, 4, 8 M = 4.6667 G = 4.0 H = 3/(7/8) = 3.428

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

44. Stratigraphic Sequence Kx > Kz

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

46. Stratigraphic Sequence

47. Vertical Flow thru Stratigraphic Sequence So Kz is Harmonic Mean, weighted by layer thickness

48. Stratigraphic Sequence

49. PERMEABILITY ANISOTROPY Justification: For vertical flow, Flux must be the same thru each layer! (see F&C, p. 33-34) q = Kz,bulk (∆h/m) = K1 (∆h1/m1) = K2 (∆h2/m2) = ....... = Kn (∆hn/mn) => Kz,bulk = q m/ ∆h = q m/ (∆h1 + ∆h2 + .... + ∆hn) = q m/ (q m1/K1 + q m2/K2 + .... + q mn/Kn) =m / S(mi/Ki ) => For horizontal flow, the most permeable units dominate, but For vertical flow, the least permeable units dominate! For layered anisotropy: Kx > Kz For fracture-related anisotropy, commonly Kz > Kx Anisotropy Ratio Kx / Kz ~ 1 to 10x for a typical layerbecause of preferred orientation, schistosity... to 106 or more for stratigraphic sequence

50. Regional GW System USGS Circ 1186