PERMEABILITY
Some slides in this section are from NExT PERF Short Course Notes, 1999.
Some slides appear to have been obtained from unknown primary
sources that were not cited by NExT. Note that some slides have a notes section.
Permeability
• Definition (ABW, Ref: API 27)
– … permeability is a property of the porous medium and
is a measure of the capacity of the medium to transmit
fluids
– … a measure of the fluid conductivity of the particular
material
• Permeability is an INTENSIVE property of a
porous medium (e.g. reservoir rock)
Sources for Permeability Determination
• Core analysis
• Well test analysis (flow testing)
– RFT (repeat formation tester) provides small well tests
• Production data
– production logging measures fluid flow into well
• Log data
– MRI (magnetic resonance imaging) logs calibrated via
core analysis
Examples, Typical PermeabilityPorosity Relationship
From Tiab and Donaldson, 1996
Darcy’s
Apparatus for
Determining
Permeability
A
h1-h2
q
A
h1
(Sand Pack Length) L
•Flow is Steady State
•q = KA (h1-h2)/L
•K is a constant of
proportionality
•h1>h2 for downward flow
q
h2
PERMEABILITY (k)
• Darcy’s “K” has a unit of m/s and is called
hydraulic conductivity
• Permeability can be easily obtained from K:
Kμ
k
g
• or hydraulic conductivity can be easily obtained
from k
k g
μ
K
Review - Derived Units
• Consider Newton’s 2nd Law for constant
mass, F=ma
– SI Units are Absolute and Coherent
• Absolute: Force is a derived unit, 1 N = 1 (kg*m/s2)
• Coherent (consistent): No conversion constants
needed
– So:
F[N] = m[kg]*a[m/s2]
Review - Derived Units
• What if we want F to be in [lbf ]?
– Then, F[lbf]*4.448[ N/lbf] =
m[kg]*a[m/s2]
– Adding the conversion factor restores the
original formula (a true statement in SI units)
Review - Derived Units
• For: F in [lbf ], m in [lbm ], a in [ft/s2]
– Then, F[lbf ]*4.448222[N / lbf] =
m[lbm ]*0.4535924[kg/lbm] *
a[ft/s2]*0.3048[m/ft]
– Or, F[lbf ] = m[lbm ] * a[ft/s2] /
32.17405[(lbmft)/(lbf s2)]
– Remember: 1[N] / 1[kg*m/s2] = 1; dimensionless
Permeability
Dimensions & Units
• Permeability is a derived dimension
• From Darcy’s equation, the dimension of permeability is
length squared
3
qμ L  L P  T L 1 1 
k
;  
  2    L2
A Δp  T 1 1 L P 
 
– This is not the same as area, even though for example, it is m2 in SI
units
• In Darcy and SI Units, this equation is coherent
– Oilfield units are non-coherent, a unit conversion constant is
required
Darcy Units (Obsolete)
•
•
•
•
•
•
Peremebility in d
viscosity in cp
length in cm
volume in cm3
pressure in atm
time in s
k p
vs  
 x
SI Units
• Permeability has a derived dimension L2 based on Darcy’s
Equation:
k p
vs  
 x
m
 s 

 m2
Pa 



Pa

s
m


• The SI unit of permeability is 1 m2
– The oilfield unit is millidarcy
– 1 md = 9.87×10-16 m2
• If you have 1 m long core with 9.87×10-16 m2 permeability and
you apply 105 Pa pressure difference to the two ends, the Darcy
velocity of water (μ=0.001 Pa·s) will be vs = 9.87×10-8 m/s
• If the core has a cross sectional area 0.1 m2, the flow rate will be
q = A vs = 9.87×10-9 m3/s
English Units
p
v s  7.324  10  
 x
-7
ft / s
k
md / cp
psi / ft
• If you have 3 ft long core with 1 md
permeability and you apply 14.7 psi pressure
difference to the two ends, the Darcy velocity
of water (μ = 1 cP) will be vs = 3.59×10-7 ft/s
• If the core has a cross sectional area 1 ft2, the
flow rate will be 3.59×10-7 ft3/s
Field Units
k p
q  1.127  10  A 
 x
3
bbl / day
ft2
md /cP
psi / ft
• If you have 3 ft long core with 1 md
permeability and you apply 14.7 psi
pressure difference to the two ends, and
the flowing fluid is water (μ = 1 cP) and
the core has a cross sectional area 1 ft2,
the flow rate will be 0.00552 bbl/day
Differential Form - Darcy’s Law
• Darcy’s Equation rearranged as Darcy velocity (volumetric
flux)
vs = q/A = (k/) (p/L)
• This equation applies for any L,  as L0
vs = q/A = -(k/) (dp/ds)
where,
vs Darcy velocity, (volumetric flux)
s distance along flow path (0s  L), in the direction of
decreasing pressure (note negative sign)
• The differential form is Darcy’s Law
Flow Potential
s
• The generalized form of Darcy’s Law includes pressure
and gravity terms to account for horizontal or nonhorizontal flow
q
k  dp
dz 
vs 
s
A

 g

  ds
ds 
– The gravity term has dimension of pressure / length
• Flow potential includes both pressure and gravity terms,
simplifying Darcy’s Law
q
k
vs 

A
μ
 dΦ 
 ds 
  = p - gZ/c ; Z+; Z is elevation measured from a datum
  has dimension of pressure