PERMEABILITY

Flow of Liquids in Porous Media

Linear Flow, Incompressible Liquid• 1-D Linear Flow System Assumptions

• steady state flow

• incompressible fluid, q(0s L) = constant

• d includes effect of dZ/ds (change in elevation)

• A(0s L) = constant

• Darcy flow (Darcy’s Law is valid)

• k = constant (non-reactive fluid)

• single phase (S=1)

• isothermal (constant )

Lq

A

1

2

Linear Flow, Incompressible Liquid

• Darcy’s Law:

• q12 > 0, if 1 > 2

• Use of flow potential, , valid for horizontal, vertical or inclined flow

Lq

A

1

2

ds

dΦ

μ

k

A

qvs

dΦμ

kAdsq

2

1

dΦμ

kAdsq

L

0

21Lμ

Akq

Radial Flow, Incompressible Liquid• 1-D Radial Flow System Assumptions

• steady state flow• incompressible fluid, q(rws re) = constant• horizontal flow (dZ/ds = 0 = p)• A(rws re) = 2rh where, h=constant• Darcy flow (Darcy’s Law is valid)• k = constant (non-reactive fluid)• single phase (S=1)• isothermal (constant )• ds = -dr

q

rerw

Radial Flow, Incompressible Liquid

• Darcy’s Law:

• qew > 0, if pe > pw

ds

dΦ

μ

k

A

qvs

dpμ

kdr

rh2π

q

q

rerw

w

e

w

e

p

p

r

r

dpμ

kh2πdr

r

1q

wewe

pp)/rln(rμ

kh2πq

Flow Potential - Gravity TermSI

= p - gZ Z+ Z is elevation measured from a datum

has dimension of pressure In SI Z is measured in m, both and p are in

Pa, ρ is in kg/m3 and g is 9.81 m/s2

The second term can also be written as S.G * 9810 Pa where S.G. is specific gravity

w.r.t. water

Flow Potential - Gravity TermField

In Field units: = p – Z ρ g / conversions = p – Z (S.G.) c

Z+ Z is elevation measured from a datum

Both and p are measured in psi Z is measured in ft, ρ is in lbm/ft3 and g is 32.7 ft/s2 conversions are complicated S.G. is specific gravity with respect to water and is

dimensionless c is 0.433 psi/ft

Flow Potential - Darcy’s Experiment

Discuss ABW, Fig. 2-26 (pg. 68) Confirm that for the static (no flow) case, the flow

potential is constant (there is no potential gradient to cause flow)

top of sand pack bottom of sand pack

Flow Potential - Example Problem

Discuss ABW, Example 2-8 (pg. 75) Solve this problem using flow potential

Permeability Units

Discuss ABW, Example 2-9 (pg. 79) 2 conversion factors needed to illustrate

permeability units of cm2

cp Pas atm Pa