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)  Force/unit mass = 0 for qm =0 ∂P/∂z = rg ∂P/∂x =0 ∂P/∂y = 0 Converse: Horizontal pressure gradients require fluid 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 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. 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

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

29. DYNAMIC VISCOSITYm • 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 • 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)

30. 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 = kg/n hydraulic conductivity (cm/sec) • Kinematic viscosity, cm2/sec 

31. POROSITY(f, or n) dimensionless Ratio of void space to total volume of material f = Vv/VT Dictates how much water a saturated material can contain Important influence on bulk properties of material e.g., bulk r, heat cap., seismic velocity…… Difference between Darcy velocity and average microscopic velocity Decrease with depth: Shales f= foe-czexponential Sandstones: f= fo - cz linear

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

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

34. PERMEABILITY (k) units 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 Km/s@ 107 k m2 = 10-5 kdarcy

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

36. 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) Scale dependence: kregional ≥ kmost permeable parts of DH >> klab; small scale )

37. 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

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 Kx > Kz

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

41. Stratigraphic Sequence

42. Vertical Flow thru Stratigraphic Sequence So Kz 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 = 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! Anisotropy Ratio: Kx / Kz ~ 1 to 10x, for typical layer (e.g., because of preferred orientation, schistosity...) Anisotropy Ratio: Kx / Kz up to 106 or more, for stratigraphic sequence In general, for layered anisotropy: Kx > Kz However, for fracture-related anisotropy, commonly Kz > Kx

45. End

46. Aquifers Saturated geologic formations with sufficient porosity f 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. Unconfined Aquifer after Fetter http://www.uwsp.edu/water/portage/undrstnd/aquifer.htm

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