
Publication History:
This article is based on
Chapter 10 of "The Log Analysis Handbook" by E. R. Crain, P.Eng., published by Pennwell Books 1986 Updated 2004.
This
webpage version is the copyrighted intellectual
property of the author.
Do not copy or distribute in any form without explicit
permission. 
Relative Permeability and
Water Cut
Since we are interested in how much oil or gas will flow from
a well, and would rather not have any water flow, it is necessary
to calculate relative permeability and water cut to determine
how good a well might be.
Relative
permeability for water.
1: Krw = (Sw  SWir) / (1  SWir)
Relative
permeability for oil.
2: Kro = (0.9  Sw) / (0.9  SWir)
Relative
permeability for gas.
3: Krg = (1  (Sw  SWir) / (1  SWir)) * (1  ((Sw  SWir) /
(1  Sw)) ^ 0.25 * Sw ^ 0.25)^0.5
Water
oil ratio.
4: KRwo = Krw / Kro
5: WOr = BO * (VISO / VISW) * (KRwo)
Water
gas ratio.
6: KRwg = Krw / Krg
7: WGr = BG * (VISG / VISW) * (KRwg)
Water
cut for oil wells.
8: WCo = WOR / (1 + WOR)
Water
cut for gas wells:
9: WCg = WGR / (1 + WGR)
Where:
BG = volume factor for gas (fractional)
BO = volume factor for oil (fractional)
Krg = relative permeability to gas (fractional)
Kro = relative permeability to oil (fractional)
Krw = relative permeability to water (fractional)
KRwg = water to gas permeability ratio (fractional)
KRwo = water to oil permeability ratio (fractional)
Sw = water saturation (fractional)
SWir = irreducible water saturation (fractional)
VISG = viscosity of gas (cp)
VISO = viscosity of oil (cp)
VISW = viscosity of water (cp)
WCg = water cut for gas well (fractional)
WCo = water cut for oil well (fractional)
WGr = water gas ratio (fractional)
WOr = water oil ratio (fractional)
COMMENTS:
Charts for solving the relative permeability relationships and for water cut
below. Viscosity
data required for wateroil and watergas ratio are also given for oil and gas.
Relative
permeability from log analysis without core data as a control
is probably pointless.
RECOMMENDED
PARAMETERS:
See Figures.
Relative permeability
Water cut
Viscosity
Viscosity of water and pseudo critical gas properties
NUMERICAL
EXAMPLE:
1. Assume data from Sand C
PHIe = 0.30
Sw = 0.50
PHIxSW = 0.06
SWir = 0.06 / 0.30 = 0.20
Krw = (0.50  0.20) / (1  0.20) = 0.375
Kro = (0.9  0.50) / (0.9  0.20) = 0.571
Krg = (1  (0.5  0.3) / (1  0.2)) * (1  ((0.5  0.3) / (1 
0.5)) ^ 0.25 * 0.5 ^ 0.25) ^ 0.5
= (0.75 * (1  0.4 ^ 0.25 * 0.5 ^ 0.25) ^ 0.5 = 0.431
KRwo = 0.375 / 0.571 = 0.657
KRwg = 0.375 / 0.431 = 0.870
2.
Assume oil well data:
BO = 0.8
oil gravity = 40 degrees API
VISO = 1.2 cp
FT = 160 degrees F
VISW = 0.5 cp
 salinity = 120,000 ppm
WOr = 0.8 * (1.2 / 0.5) * 0.657 = 1.26
3.
Assume gas well data:
BG = 0.8
VISG = 0.015 cp
VISW = 0.5 cp
dry gas
pressure = 1800 psi
salinity = 120,000 ppm
WGr = 0.8 * (0.015 / 0.5) * 0.870 = 0.021
4.
Water Cut:
WCo = 1.26 / (1 + 1.26) = 0.557
WCg = 0.021 / (1 + 0.021) = 0.021
Thus,
if this is an oil zone, it will produce over 55% water on initial
completion. If it is a gas well, it will produce only 2% water.
