q v = - K h = kg/ν units of velocity Darcy's Law: K = kρg/µ HYDRAULIC CONDUCTIVITY, K Proportionality constant in Darcy's Law

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1 Darcy's Law: q v - K h HYDRAULIC CONDUCTIVITY, K m/s K kρg/µ kg/ν units of velocity Proportionality constant in Darcy's Law Property of both fluid and medium see D&S, p. 62

2 HYDRAULIC POTENTIAL (Φ): Φ g h gz + P/ρ w energy/unit mass cf. h energy/unit weight Consider incompressible fluid elevation z i 0 pressure P i ρ i and velocity v 0 Move to new position z, P, ρ, v Energy difference: lift mass + accelerate + compress ( VdP) mg(z- z i ) + mv 2 /2 + m V/m) dp latter term m (1/ρ)dP Energy/unit mass Φ g z + v 2 /2 + (1/ρ)dP For incompressible fluid (ρ const) & slow flow (v 2 /2 0), z i 0, P i 0 Energy/unit mass: Φ g z + P/ρ g h Force/unit mass Φ g - P/ρ Force/unit weight h 1 - P/ρg

3 Rewrite Darcy's Law: Hubbert (1940, J. Geol. 48, p ) q m ρ q v kρ ν g P ρ kρ ν force/unit mass [ ] q m Fluid flux mass vector (g/cm 2 -sec) k rock (matrix) permeability (cm 2 ) ρ fluid density (g/cm 3 ) [...] Force/unit mass acting on fluid element 1/ ν where ν Kinematic Viscosity µ/ρ cm 2 /sec

4 Rewrite Darcy's Law: Hubbert (1940; J. Geol. 48, p ) q v k ρν ρg P [ ] k ρν force/unit vol [ ] q v Fluid volumetric flux vector (cm 3 /cm 2 -sec) q m /ρ k rock (matrix) permeability (cm 2 ) [...] Force/unit vol. acting on fluid element 1/ ν where ν Kinematic Viscosity µ/ρ cm 2 /sec

5 Rewrite Darcy's Law: Hubbert (1940; J. Geol. 48, p ) q v k ρν ρg P [ ] k ρν ρg h [ ] [force/unit vol] kg ν h K h

6 STATIC FLUID (NO FLOW) q m kρ ν g P ρ

7 STATIC FLUID (NO FLOW) 0 q m kρ ν g P ρ Force/unit mass 0 for q m 0 P/ z ρg P/ x 0 P/ y 0 Converse: Horizontal pressure gradients require fluid flow

8 Darcy's Law: Isotropic Media: q - K h OK only if K x K y K z Darcy's Law: Anisotropic Media K, k are tensors Direction of fluid flow need not coincide with the gradient in hydraulic head

9 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 q q x K xx h x i K xx K yy K zz q y K yy h y j i h x j h y k h z q z K zz h z k

10 General case: Symmetrical tensor Kxy Kyx KzxKxz Kyz Kzy q K xx K xy K xz K yx K yy K yz K zx K zy K zz i h x j h y k h z q x K xx h x q y K yx h x q z K zx h x K xy h y K yy h y K zy h y K xz h z K yz h z K zz h z

11 Relevant Physical Properties for Darcy s Law Hydraulic conductivity K kg/ν cm/s Density ρ g/cm 3 Kinematic Viscosity ν cm 2 /sec Dynamic Viscosity µ ν*ρ poise Porosity φ dimensionless Permeability k cm 2 q v - K h q v k ρν [ ρg P] q m ρ q v

12 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 q v - K h q v kg ν P 1 ρg q m ρ q v > K kg/ν and where h 1 - P/ρg

13 DENSITY (ρ) g/cm 3 Fluid property Specific weight (weight density) γ ρ g ρ f(t,p) where ρ T,P ρ o 1 α(t T o ) +β(p P o ) Thermal expansivity α 1 V V T P 1 ρ ρ T P for small α,β because dρ ρ dv V Isothermal Compressibility β T 1 V V P T 1 ρ ρ P T

14 DYNAMIC VISCOSITY µ Fluid property Geo. Stokes Law: 4πr 3 3 ( ρ s ρ f )g 6πrµu Gravitational Force Frictional Force 6πrµu Mass 4πr 3 ρ s 3 Units of µ : poise; 1 P 0.1 N sec/m 2 1 dyne sec/cm 2 Water 0.01 poise (1 centipoise) 4πr 3 3 ( ρ s ρ f )g

15 DYNAMIC VISCOSITY µ Fluid property A measure of the rate of strain in an imperfectly elastic material subjected to a distortional stress. For simple shear τ µ u/ y Newtonian fluid Units (poise; 1 P 0.1 N sec/m 2 1 dyne sec/cm 2 Water 0.01 poise (1 centipoise) KINEMATIC VISCOSITY ν Fluid property ν µ/ρ m 2 /sec or cm 2 /sec Water: Basaltic Magma 20 C or granitic magma Mantle 10-6 m 2 /sec 10-2 cm 2 /sec 0.1 m 2 /sec 10 2 m 2 /sec m 2 /sec see Tritton p. 5; Elder p. 221)

16 Darcy's Law: Hubbert (1940; J. Geol. 48, p ) q v kg ν P 1 - kg gρ ν [ h ] K h where: q v Darcy Velocity, Specific Discharge or Fluid volumetric flux vector (cm/sec) k permeability (cm 2 ) K k(g/ν) hydraulic conductivity (cm/sec) ν Kinematic viscosity, cm 2 /sec

17 POROSITY (φ, or n) dimensionless Rock property Ratio of void space to total volume of material φ 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 φ φ o e -cz exponential Sandstones: φ φ o - c z linear

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

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

20 PERMEABILITY (k) cm 2 Measure of the ability of a material to transmit fluid under a hydrostatic gradient Differences with Porosity?

21 PERMEABILITY (k) cm 2 Measure of the ability of a material to transmit fluid under a hydrostatic gradient Differences with Porosity? Different Units Styrofoam cup: High φ, Low k Uniform spheres: φ f(dia); k ~ dia 2

22 PERMEABILITY (k) 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 x 10-8 cm x m 2 (e.g., sandstone) Approximate relation between K and k (for cool water): K m/s 10 7 k 2 m 10 3 k 2 cm 10-5 k darcy K cm/s 10 5 k 2 cm 10-3 k darcy factor g/ν K ft/y k cm 2

23 1nd 1µd 1 md 1 d 1000 d Clay Silt Sand Gravel Shale Sandstone B argillaceous Limestone cavernous Basalt Crystalline Rocks PERMEABILITY, cm 2

24 GEOLOGIC REALITIES OF PERMEABILITY (k) Huge Range in common geologic materials > x Decreases super-exponentially with depth k Cd 2 k a 3 /12L for granular material, where d grain diameter, C is complicated parameter 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: k regional k most permeable parts of drill holes >> k lab; small scale