
Publication History:
This article is based on
Chapter 8 of "The Log Analysis Handbook" by E. R. Crain, P.Eng., published by Pennwell Books 1986 Updated 2004,
2016, 2018.
This
webpage version is the copyrighted intellectual
property of the author.
Do not copy or distribute in any form without explicit
permission. 
WAXMANSMITS Saturation (CEC) Method
Another popular method, based on laboratory measured values of
cation exchange capacity versus shale content, was developed by
Waxman and Smits. It uses the same response equation as in other
saturation models, but finds the value for 1/Fsh differently. The method requires
a formula for the value of cation exchange capacity, such as the
one below:
1:
IF PHIe > 0.0
2: THEN CEC = 10 ^ (1.9832 * Vsh  2.4473)
The
above relationship must be derived for each particular area by
curve fitting the laboratory data. Some authors have related CEC
to porosity in certain areas, but there is no physical reason
why this should be true, since specific CEC values depend on shale
volume and clay type, and not porosity. The only time this might work is when
porosity is strictly a function of shale volume and there are
no other mineral variations. Others have tried to relate CEC to
some other log data, such as the SP (which of course is a shale
indicator), with limited success. CEC data from laboratory
measurements are now routine.
The balance of the equations
do not need further modification.
3: RW2 = (RW@FT) * (FT + KD1) / KD5
4: B = 4.6 * (1  0.6 exp (0.77 / RW2))
5: F = A / (PHIe ^ M)
6: Qv = CEC * (1  PHIe) * DENSMA
/ PHIe
7: Swc = 0.5 * (( B * Qv * RW2) + ((B * Qv * RW2) ^ 2 + 4 * F
* RW@FT /
RESD) ^ 0.5) ^ (2 / N)
8: OTHERWISE Swc = 1.0
Where:
KD1 = 6.8 for English units
KD1 = 21.5 for Metric units
KD5 = 83.8 for English units
KD5 = 46.5 for Metric units
A = tortuosity exponent (unitless)
B = equivalent conductance of clay cation (mS/m)
CEC = cation exchange capacity of shale (meq/gm)
DENSMA = matrix density (gm/cc or kg/m3)
F = formation factor (unitless)
FT = formation temperature (degrees Fahrenheit or Celsius)
M = cementation exponent (unitless)
N = saturation exponent (unitless)
PHIe = effective porosity (fractional)
Qv = counter ion concentration (meq/gm)
RESD = deep resistivity log reading (ohmm)
RW2 = water resistivity at 77 degrees Fahrenheit (ohmm)
RW@FT = water resistivity at formation temperature (ohmm)
Swc = water saturation from CEC method (fractional)
Vsh = shale volume (fractional)
COMMENTS:
Reference:
1. Electrical Conductivities in Oil Bearing Shaly Sands,
M. Waxman, L. Smits, SPEJ, June 1968.
This lengthy procedure does not lend itself to a graphical solution.
Review the references on this method before attempting to use
it.
Good
CEC data is still hard to come by. CEC measured on core and sample
chips often do not correlate well with either effective porosity
or shale content, most likely due to the fact that more than one
clay mineral is present, each in varying proportions. Thus a pragmatic
fit of CEC to a log derived porosity or shale volume is usually
necessary. This field specific approach is commonly applied by
those who insist on using the WaxmanSmits approach even when
the data does not support its use.
Some
analysts use density porosity (PHID), uncorrected for shale, to
predict CEC. Some use PHID in the saturation equations instead
of PHIe. Others call PHID the “total porosity”, which
is wrong, since the standard definition of total porosity is (PHIN
+ PHID) / 2. These terminology problems stem from shortcuts used
in specific areas before sophisticated computer programs made
it easy to do better work. Unfortunately, younger analysts learn
the tricks of the trade from older analysts who have long forgotten
that the shortcut was ever taken.
RECOMMENDED
PARAMETERS:
for
carbonates A = 1.00
M = 2.00
N = 2.00 (Archie Equation as first
published)
for sandstone A = 0.62
M = 2.15
N = 2.00 (Humble Equation)
A = 0.81 M = 2.00 N = 2.00 (Tixier Equation 
simplified version of Humble Equation)
Asquith (1980 page 67) quoted other authors, giving values for A
and M, with N = 2.0, showing the wide range of possible values:
Average sands A = 1.45 M = 1.54
Shaly sands
A = 1.65 M = 1.33
Calcareous sands
A = 1.45 M = 1.70
Carbonates
A = 0.85 M = 2.14
Pliocene sands S.Cal. A = 2.45 M = 1.08
Miocene LA/TX
A = 1.97 M = 1.29
Clean granular
A = 1.00 M = 2.05  PHIe
NUMERICAL
EXAMPLE:
Data for Sand "D"
RESD = 1.0 ohm
PHIe = 0.11
Vsh = 0.33
A = 0.62
M = 2.15
N = 2.00
RSH = 4.0 ohmm
RW@FT = 0.015 ohmm
DENSMA = 2650 kg/m3
FT = 43 degrees Celsius
CEC = 10 ^ (1.9832 * 0.33  2.4473) = 0.0161
RW2 = 0.015 * (43 + 21.5) / (83.8  37.3) = 0.0208
B = 4.6 * (1  0.6 * exp(0.77 / 0.0208)) = 4.6
F = 0.62 / (0.11 ^ 2.15) = 71.35
Qv = 0.0161 * (1  0.11) * 2.650 / 0.11 = 0.3452
Swc = 0.5 * ((4.6 * 0.3452 * 0.0208) + ((4.6 * 0.3452 * 0.0208)^2+
4 * 71.35 * 0.015 / 1.0)^0.5)^(2 / 2.0)
= 0.5 * (0.0330 + (0.0011 + 4.281) ^ 0.5) ^ (2 / 2)
Swc = 1.05
If
Qv or Vsh were higher the saturation would be lower.
