One of the saturation from resistivity models.

The dual-water model accounts for the fact that different shales have different shale-bound water saturation :

\phi_t = \phi_e  + \phi_t s_{wb}

so that formation water saturation is related to total water saturation as:

s_w = \frac{s_{wt} - s_{wb}}{ 1 - s_{wb}}

Rock volume is a sum of rock matrix volume and total pore volume :

V = V_m + V_t = (1-\phi_t) V + \phi_t V

where

\phi_t = \frac{V_t}{V}

Total pore volume is a sum of shale-bound water and free fluid volume (water and hydrocarbons):

V_t = \phi_t V = V_e + V_{wb} = \phi_e V + s_{wb} V_t

where

V_e = V_t  (1 - s_{wb})

and therefore:

\phi_e = \phi_t  (1 - s_{wb})

Total volume of water is a sum of shale-bound water and free water :

V_{wt} = V_{wb} + V_{wf}

and relates to as:

s_{wt} V_t = s_{wb} V_t + s_w V_e = s_{wb} V_t + s_w V_t  (1 - s_{wb})

or

s_{wt} = s_{wb} + s_w (1 - s_{wb})

which gives an explicit formula for formation water saturation:

s_w = \frac{s_{wt} - s_{wb}}{ 1 - s_{wb}}

Formation resistivity 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}}

where

	shale-bound water saturation
	total water saturation (shal-bound water and free-water)
	specific electrical resisitvity of shale-bound water

In simple case when all shales have the same properties, the shale-bound water saturation can be expressed through the shaliness as:

s_{wb} = \zeta_{wb} V_{sh}

	formation water saturation
	bound water saturation
	effective porosity
	shaliness
	total measured resistivity from OH logs
	formation water resistivity
	wet clay resistivity
	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
	formation matrix cementation exponent	1.5 ÷ 2.5, default value is 2
	formation matrix water-saturation exponent	1.5 ÷ 2.5, default value is 2

In some practical cases, the clay resisitvity can be expressed as:

\frac{1}{R_{sh}} = B \cdot Q_V

where

	conductance per cat-ion (mho · cm²/meq)
	Cation Exchange Capacity (meq/ml)

and both can be measured in laboratory.

The other model parameters still need calibration on core data.

See Also