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

qv = - K∇h Darcy's Law:

HYDRAULIC POTENTIAL (Φ): energy/unit mass cf. h = energy/unit weight

Φ = g h = gz + P/ρw

Consider incompressible fluid element @ elevation zi= 0 pressure Pi ρi and velocity v = 0

Move to new position z, P, ρ , v

Energy difference: lift mass + accelerate + compress (= ∫ VdP) = mg(z- zi) + mv2/2 + m ∫ V/m) dP latter term = m ∫(1/ρ)dP

Energy/unit mass Φ = g z + v2/2 + ∫ (1/ρ)dP

For incompressible fluid (ρ = const) & slow flow (v2/2→ 0), zi=0, Pi = 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

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) ∝ ρ ≡ fluid density (g/cm3) ∝ [.....] ≡ Force/unit mass acting on fluid element ∝ 1/ ν where ν ≡ Kinematic Viscosity = µ/ρ cm2/sec

→

€

qm = ρ qv = kρν

g − ∇Pρ

= kρν

force/unit mass[ ]

€

qv = kρν

ρg−∇P[ ]

= kρν

force/unit vol[ ]

Rewrite Darcy's Law: Hubbert (1940; J. Geol. 48, p. 785-944)

qv ≡ Fluid volumetric flux vector (cm3/cm2-sec) = qm /ρ

∝ k ≡ rock (matrix) permeability (cm2) ∝ [.....] ≡ Force/unit vol. acting on fluid element ∝ 1/ ν where ν ≡ Kinematic Viscosity = µ/ρ cm2/sec

→ →

€

qv = kρν

ρg−∇P[ ]

= kρν

ρg∇h[ ]

= kgν∇h

= K∇h

Rewrite Darcy's Law: Hubbert (1940; J. Geol. 48, p. 785-944)

[force/unit vol]

STATIC FLUID (NO FLOW)

€

qm = kρν

g − ∇Pρ

STATIC FLUID (NO FLOW)

€

qm = kρν

g − ∇Pρ

Force/unit mass = 0 for qm = 0

∂P/∂z = ρg ∂P/∂x =0 ∂P/∂y = 0

Converse: Horizontal pressure gradients require fluid flow

→

0

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

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

q = –

Kxx 0 00 Kyy 00 0 Kzz

i ∂h∂xj∂h∂yk∂h∂z

qx = – Kxx∂h∂x i qy = – Kyy

∂h∂y j qz = – Kzz

∂h∂zk

Thus

q = –

Kxx Kxy KxzKyx Kyy KyzKzx Kzy Kzz

i∂h∂xj∂h∂yk∂h∂z

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

qx = – Kxx∂h∂x – Kxy

∂h∂y – Kxz

∂h∂z

qy = – Kyx∂h∂x – Kyy

∂h∂y – Kyz

∂h∂z

qz = – Kzx∂h∂x – Kzy

∂h∂y – Kzz

∂h∂z

Relevant Physical Properties for Darcy’s Law Hydraulic conductivity K = kg/ν cm/s Density ρ g/cm3 Kinematic Viscosity ν cm2/sec Dynamic Viscosity µ = ν*ρ poise Porosity φ dimensionless Permeability k cm2

€

qv = kρν

ρg−∇P[ ]qv = - K∇h

€

qm = ρ qv

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

€

qv = kgν

1− ∇Pρg

€

qm = ρ qv

qv = - K∇h

=> K = kg/ν and where ∇h = 1 - ∇P/ρg

DENSITY (ρ) g/cm3

Fluid property

Specific weight (weight density) γ = ρ g

ρ = f(T,P)

α ≡

1V

∂V∂T P

= –1ρ

∂ρ∂T P

because dρρ = – dV

V

Thermal expansivity

β T ≡ – 1

V∂V∂P T

=1ρ

∂ρ∂P T

Isothermal Compressibility

ρT,P ≅ ρo 1 – α(T–To) +β(P–Po)for smallα,β

where

Units of µ : • poise; 1 P = 0.1 N sec/m2 •  = 1 dyne sec/cm2

Water 0.01 poise (1 centipoise)

€

4πr 3

3ρs − ρ f( )g = 6πrµu

DYNAMIC VISCOSITY µ Fluid property

Geo. Stokes Law:

Gravitational Force = Frictional Force

€

4πr 3

3ρs − ρ f( )g

€

6πrµu

€

Mass =4πr 3ρs3

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/m2 = 1 dyne sec/cm2 Water 0.01 poise (1 centipoise)

KINEMATIC VISCOSITY ν Fluid property •  ν = µ/ρ m2/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)

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/ν) hydraulic conductivity (cm/sec)

ν ≡ Kinematic viscosity, cm2/sec

→ €

qv = kgν

1− ∇Pgρ

= - kg

ν∇h[ ] = −K∇h

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 = φoe-cz exponential

Sandstones: φ = φo - c z linear

POROSITY (φ, or n) dimensionless Rock property

Ratio of void space to total volume of material

φ = Vv/VT

0 8 16 24 32 40 48 56 64

Porosity, %

Fractured Basalt crystalline rocks

Limestone karstic & Dolostone

Shale Sandstone Siltstone

Gravel Sand Silt & Clay

FCC BCC Simple cubic 26% 32% 47.6%

➙ Pumice

Non-uniform grain sizes

Domenico & Schwartz (1990)

Shales (Athy, 1930)

Sandstones (Blatt, 1979)

PERMEABILITY (k) cm2

Measure of the ability of a material to transmit fluid under a hydrostatic gradient

Differences with Porosity? Differences with Porosity?

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 φ, Low k

Uniform spheres: φ ≠ f(dia); k ~ dia2

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

Approximate relation between K and k (for cool water): factor = g/ν

Km/s ≅ 107 k m2 = 103 k cm2 = 10-5 kdarcy

Kcm/s ≅ 105 k cm2 = 10-3 kdarcy

Kft/y ≅ 1011 k cm2

1 darcy = 0.987 x 10-8 cm2 = 0.987 x 10-12 m2 (e.g., sandstone)

2

10 10 10 10 10 10 10 10 10 -18 -16 -14 -12 -10 -8 -6 -4 -2

B

PERMEABILITY, cm

1nd 1µd 1 md 1 d 1000 d

Clay Silt Sand Gravel

Shale Sandstone

argillaceous Limestone cavernous

Basalt

Crystalline Rocks

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