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 + KT1) / KT5
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:
KT1 = 6.8 for English units
KT1 = 21.5 for Metric units
KT5 = 83.8 for English units
KT5 = 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:
When
Vsh = 0, then CEC = Qv = 0 and equation reverts to the Archie
model.
The product B * Qv is available on modern lab reports of
electrical properties, but a relationship between BQv and a well
log property such as Vsh is still needed.
This set of electrical properties is very detailed but is not
sufficient to apply to a log analysis without many more data samples
covering a wider spread of reservoir properties.
An
alternative is to calculate CEC from a clay
mineral analysis based on elemental capture spectroscopy (ECS) log:
9: CEC = Sum(CECi * Vclayi)
Reference:
1. Electrical Conductivities in Oil Bearing Shaly Sands, M. Waxman, L. Smits, SPEJ, June 1968.
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 lack of 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:
In the
absense of measured shaly sand electrical properties, use A =
1.00
M* = 2.00
N* = 2.00.
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.
