Specific electrical resistivity
R_t or specific electrical conductivity
\sigma_t = \frac{1}{R_t} of formations is defined by mineralization of the rock matrix and saturating fluids which are in due turn depend on formation water-saturated shaliness
V_{sh} , formation porosity
\phi_e and water saturation volumetric share
s_w.
Archie Model
Specific electrical resistivity R_t is defined as:
(1) | \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} |
where
R_w | specific electrical resistivity of formation water | |
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A | 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 |
m | formation matrix cementation exponent | 1.5 ÷ 2.5, default value is 2 |
n | 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) | \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} |
where constant A is defined by formation shaliness V_{sh}:
\frac{1}{A} = 1 + \Big( \frac{V_{sh}^{2-V_{sh}}}{\phi_e} \, \frac{R_w}{R_{sh}} \Big)^{1/2} |
where
R_{sh} | specific electrical resisitvity fo shale |
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Simandeux Model
Simandeux model suggest a more complicated correlation between resitvity R_t and water saturation s_w:
where
s_w
\phi_e
V_{sh}
R_t
R_w
R_{sh}
A 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
m
n 1.5 ÷ 2.5, default value is 2
(3)
\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}
formation water saturation effective porosity shaliness specific electrical resistivity from OH logs specific electrical resistivity of formation water specific electrical resistivity of wet shales formation matrix cementation exponent 1.5 ÷ 2.5, default value is 2 formation matrix water-saturation exponent
(4) | \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} \, \left( \sqrt{ \left( \frac{V_{sh}}{R_{sh}} \right)^2 + \frac{4\phi_e^m}{a R_t R_w (1-V_{sh}) } } - \frac{V_{sh}}{R_{sh}} \right) |
with default value A = 0.8.
Dual-Water Model (DW)
The dual-water model accounts for the fact that different shales have different shale-bound water saturation
s_{wb}= \frac{V_{wb}}{V_t}:
\phi_t = \phi_e + \phi_t s_{wb} |
so that formation water saturation s_w is related to total water saturation s_{wt} = \frac{V_{wb} + V_w}{V_t } as:
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}} |
where
s_{wb} = \frac{V_{wb}}{V_t} | shale-bound water saturation |
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s_{wt} = \frac{V_{wb} + V_w}{V_t} | total water saturation (shal-bound water and free-water) |
R_{wb} | 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:
(5) | s_{wb} = \zeta_{wb} V_{sh} |
Waxman-Smits-Thomas Model (WST )
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{B Q_V}{s_{wt}} \Big] |
which is similar to dual-water with complex parameter B Q_V relating to:
B Q_V = s_{wb} \Big( \frac{1}{R_{wb}} - \frac{1}{R_w} \Big) |
In some opractical cases, the kaboiratiry data is available on B and Q_V separately, but still need calibration on core data.