One of the saturation from resistivity models.

The dual-water model accounts for the fact that different shales have different shale-bound water saturation $\begin{array}{l}s_{wb}= \frac{V_{wb}}{V_t}\end{array}$ :

$\begin{array}{l}\displaystyle \phi_t = \phi_e + \phi_t s_{wb}\end{array}$

so that formation water saturation $\begin{array}{l}s_w\end{array}$ is related to total water saturation $\begin{array}{l}s_{wt} = \frac{V_{wb} + V_w}{V_t }\end{array}$ as:

$\begin{array}{l}\displaystyle s_w = \frac{s_{wt} - s_{wb}}{ 1 - s_{wb}}\end{array}$

Expand for math

Rock volume $\begin{array}{l}V\end{array}$ is a sum of rock matrix volume $\begin{array}{l}V_m\end{array}$ and total pore volume $\begin{array}{l}V_t\end{array}$ :

$\begin{array}{l}\displaystyle V = V_m + V_t = (1-\phi_t) V + \phi_t V\end{array}$

where

$\begin{array}{l}\displaystyle \phi_t = \frac{V_t}{V}\end{array}$

Total pore volume $\begin{array}{l}V_t\end{array}$ is a sum of shale-bound water $\begin{array}{l}V_{wb}\end{array}$ and free fluid volume $\begin{array}{l}V_e\end{array}$ (water and hydrocarbons):

$\begin{array}{l}\displaystyle V_t = \phi_t V = V_e + V_{wb} = \phi_e V + s_{wb} V_t\end{array}$

where

$\begin{array}{l}\displaystyle V_e = V_t (1 - s_{wb})\end{array}$

and therefore:

$\begin{array}{l}\displaystyle \phi_e = \phi_t (1 - s_{wb})\end{array}$

Total volume of water is a sum of shale-bound water $\begin{array}{l}V_{wb}\end{array}$ and free water $\begin{array}{l}V_{wf}\end{array}$ :

$\begin{array}{l}\displaystyle V_{wt} = V_{wb} + V_{wf}\end{array}$

and relates to $\begin{array}{l}V_t\end{array}$ as:

$\begin{array}{l}\displaystyle s_{wt} V_t = s_{wb} V_t + s_w V_e = s_{wb} V_t + s_w V_t (1 - s_{wb})\end{array}$

or

$\begin{array}{l}\displaystyle s_{wt} = s_{wb} + s_w (1 - s_{wb})\end{array}$

which gives an explicit formula for formation water saturation:

$\begin{array}{l}\displaystyle s_w = \frac{s_{wt} - s_{wb}}{ 1 - s_{wb}}\end{array}$

Formation resistivity $\begin{array}{l}R_t\end{array}$ is given by the following correlation:

$\begin{array}{l}\displaystyle \frac{1}{R_t} = \phi_t^m s_{wt}^n \, \Big[ \frac{1}{R_w} + \frac{s_{wb}}{s_{wt}} \Big( \frac{1}{R_{wb}} - \frac{1}{R_w} \Big) \Big] \quad \Rightarrow \quad s_w = \frac{s_{wt} - s_{wb}}{ 1 - s_{wb}}\end{array}$

where

$\begin{array}{l}s_{wb} = \frac{V_{wb}}{V_t}\end{array}$	shale-bound water saturation
$\begin{array}{l}s_{wt} = \frac{V_{wb} + V_w}{V_t}\end{array}$	total water saturation (shal-bound water and free-water)
$\begin{array}{l}R_{wb}\end{array}$	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:

(1)	$\begin{array}{l}\displaystyle s_{wb} = \zeta_{wb} V_{sh}\end{array}$

$\begin{array}{l}s_w\end{array}$	formation water saturation
$\begin{array}{l}s_{wb}\end{array}$	bound water saturation
$\begin{array}{l}\phi_e\end{array}$	effective porosity
$\begin{array}{l}V_{sh}\end{array}$	shaliness
$\begin{array}{l}R_t\end{array}$	total measured resistivity from OH logs
$\begin{array}{l}R_w\end{array}$	formation water resistivity
$\begin{array}{l}R_{sh}\end{array}$	wet clay resistivity
$\begin{array}{l}A\end{array}$	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
$\begin{array}{l}m\end{array}$	formation matrix cementation exponent	1.5 ÷ 2.5, default value is 2
$\begin{array}{l}n\end{array}$	formation matrix water-saturation exponent	1.5 ÷ 2.5, default value is 2

In some practical cases, the clay resisitvity $\begin{array}{l}R_{sh}\end{array}$ can be expressed as:

(2)	$\begin{array}{l}\displaystyle \frac{1}{R_{sh}} = B \cdot Q_V\end{array}$

where

$\begin{array}{l}B\end{array}$	conductance per cat-ion (mho · cm²/meq)
$\begin{array}{l}Q_V\end{array}$	Cation Exchange Capacity (meq/ml)

and both can be measured in laboratory.

The other model parameters still need calibration on core data.

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Dual Water Model (DW ) @model

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