Page 286 - Petrophysics 2E

-DERIVED EVALUATION OF SHALY 259

For clean sands, C, = 0, and X becomes the reciprocal of the Archie
formation resistivity factor defined as C,/C,. Thus, by analogy, for shaly
sands, where C, = 0,

(4.1 22)

where F* is the shaly (clayey) sand formation resistivity factor, related to
porosity according to the Archie or Humble-type relationship, i.e.,

a*
F* = - (4.1 23)
om*

where a* and m* are the Humble coefficient and the cementation factor,
respectively, for shaly sand. For the straight-line portion of the shaly sand
conductivity (Figure 4.39, the equation is:

where: Cscl = specific conductance of the clay counterions for the
straight line portion of the C, -vs. -C, curve, mhokm.
Qv = volume concentration of sodium exchange cations
associated with the clay, equivfi.
h~~ = maximum equivalent ionic conductance of the sodium
exchange ions, cm2equiv-'ohm-'.

Equation 4.124 can be modified to include the curved portion of the
conductivity curve in the low C, region, by assuming an exponential
rise of the counterion mobility up to its constant and maximum mobility.

cs = tjCSC1 (4.125)

with

(4.126)

where: y = constant, determined by the rate of increase of the
counterion mobility ftom that at zero water conductivity
up to its constant value at the higher water
conductivities, mho/cm.
a = dimensionless constant, 1 - Ga/hNa.
Aha = equivalent ionic conductance of the exchange cations
at Cw = 0.