One of the saturation from resistivity models:The dual-water model accounts for the fact that different shales have different shale-bound water saturation
s_{wb}= \frac{V_{wb}}{V_t}:
so that formation water saturation
s_w is related to total water saturation
s_{wt} = \frac{V_{wb} + V_w}{V_t } as: where
s_{wb} = \frac{V_{wb}}{V_t}
s_{wt} = \frac{V_{wb} + V_w}{V_t}
R_{wb} In simple case when all shales have the same properties, the shale-bound water saturation can be expressed through the shaliness as:
s_w
s_{wb} bound water saturation
\phi_e
V_{sh}
R_t
R_w
R_{sh}
A dimensionless constant, characterising the rock matrix contribution to the total electrical resistivity 0.5 ÷ 1, default value is 1 for sandstones and 0.9 for limestones
m
n 1.5 ÷ 2.5, default value is 2 In some practical cases, the clay resisitvity
R_{sh} can be expressed as: where
B conductance per cat-ion (mho · cm2/meq)
Q_V Cation Exchange Capacity (meq/ml) and both can be measured in laboratory.
\phi_t = \phi_e + \phi_t s_{wb}
s_w = \frac{s_{wt} - s_{wb}}{ 1 - s_{wb}}
Formation resistivity
R_t is given by the following correlation:
\frac{1}{R_t} = \phi_t^m s_{wt}^n \, \Big[ \frac{1}{R_w} + \frac{s_{wb}}{s_{wt}} \Big( \frac{1}{R_{wb}} - \frac{1}{R_w} \Big) \Big] \quad \Rightarrow \quad s_w = \frac{s_{wt} - s_{wb}}{ 1 - s_{wb}}
shale-bound water saturation total water saturation (shal-bound water and free-water) specific electrical resisitvity of shale-bound water
(1)
s_{wb} = \zeta_{wb} V_{sh}
formation water saturation effective porosity shaliness total measured resistivity from OH logs formation water resistivity wet clay resistivity formation matrix cementation exponent 1.5 ÷ 2.5, default value is 2 formation matrix water-saturation exponent
(2)
\frac{1}{R_{sh}} = B \cdot Q_V
The other model parameters still need calibration on core data.