Saturation from Resistivity Log

Specific electrical resistivity $\begin{array}{l}R_t\end{array}$ or specific electrical conductivity $\begin{array}{l}\sigma_t = \frac{1}{R_t}\end{array}$ of formations is defined by mineralization of the rock matrix and saturating fluids which are in due turn depend on formation water-saturated shaliness $\begin{array}{l}V_{sh}\end{array}$ , formation porosity $\begin{array}{l}\phi_e\end{array}$ and water saturation volumetric share $\begin{array}{l}s_w\end{array}$ .

Archie Model

Specific electrical resistivity $\begin{array}{l}R_t\end{array}$ is defined:

(1)	$\begin{array}{l}\displaystyle \frac{1}{R_t} = \frac{\phi_e^m \, s_w^n }{A R_w} \quad \Rightarrow \quad s_w = \Big ( \frac{A}{\phi_e^m} \; \frac{R_w}{R_t} \Big) ^{1/n}\end{array}$

where

$\begin{array}{l}R_w\end{array}$	specific electrical resistivity of formation water
$\begin{array}{l}A\end{array}$	dimensionless constant, characterizing 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

Archie model is usually used in:

high permeable clean sands with low or no shaliness
high permeable clean limestones with low or no shaliness

Indonesia Model (Poupon-Leveaux)

Indonesia model is the same Archie model:

(2)	$\begin{array}{l}\displaystyle \frac{1}{R_t} = \frac{\phi_e^m \, s_w^n }{A R_w} \quad \Rightarrow \quad s_w = \Big ( \frac{A}{\phi_e^m} \; \frac{R_w}{R_t} \Big) ^{1/n}\end{array}$

where constant $\begin{array}{l}A\end{array}$ is defined by formation shaliness $\begin{array}{l}V_{sh}\end{array}$ :

$\begin{array}{l}\displaystyle \frac{1}{A} = 1 + \Big( \frac{V_{sh}^{2-V_{sh}}}{\phi_e} \, \frac{R_w}{R_{sh}} \Big)^{1/2}\end{array}$

where

$\begin{array}{l}R_{sh}\end{array}$	specific electrical resisitvity fo shale

Simandeux Model

Simandeux model suggest a more complicated correlation between resitvity $\begin{array}{l}R_t\end{array}$ and water saturation $\begin{array}{l}s_w\end{array}$ :

(3)

$\begin{array}{l}\displaystyle \frac{1}{R_t} = \frac{\phi_e^m \, s_w^n}{A R_w (1-V_{sh})} + \frac{V_{sh}}{R_{sh}} s_w^{n/2} \quad \Rightarrow \quad s_w^{n/2} = \frac{A R_w (1-V_{sh})}{2 \phi_e^m} \, \Big( \sqrt{ \Big( \frac{V_{sh}}{R_{sh}} \Big)^2 + \frac{4\phi_e^m}{a R_t R_w (1-V_{sh}) } } - \frac{V_{sh}}{R_{sh}} \Big)\end{array}$

with default value $\begin{array}{l}A = 0.8\end{array}$ .

Dual-Water Model (DW)

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:

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

Waxman-Smits-Thomas Model (WST )

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

$\begin{array}{l}\displaystyle \frac{1}{R_t} = \phi_t^m s_{wt}^n \, \Big[ \frac{1}{R_w} +\frac{B Q_V}{s_{wt}} \Big]\end{array}$

which is similar to dual-water with complex parameter $\begin{array}{l}B Q_V\end{array}$ relating to:

$\begin{array}{l}\displaystyle B Q_V = s_{wb} \Big( \frac{1}{R_{wb}} - \frac{1}{R_w} \Big)\end{array}$

In some opractical cases, the kaboiratiry data is available on $\begin{array}{l}B\end{array}$ and $\begin{array}{l}Q_V\end{array}$ separately, but still need calibration on core data.

Page tree

Archie Model

Indonesia Model (Poupon-Leveaux)

Simandeux Model

Dual-Water Model (DW)

Waxman-Smits-Thomas Model (WST )

References