The Archie equation was the first empirical model built (1942) to estimate the water saturation in
non conductive matrix rocks. It usually works well with clean
clastic sandstones and carbonate rocks. Typical parameters for the Archie equation
for consolidated sandstones are a=0.81 (tortuosity), m=1.7 (cementation exponent),
and n=2.0 (saturation exponent). For carbonates, typical
parameters are a=1.0; m=n=2.0.
When the rock matrix has some electrical conductivity, the resistivity is not only
a function of the water resistivity R_{w} through their free dissolved ions,
but also depends upon the matrix rock minerals beside the non conductive quartz and
calcite matrix grains. The most common cases happen on clastic shaly rocks with
important content of clay minerals.
In these shaly rocks the Archie law overestimates the water saturation.
Many models consider the Shale Volume (Vshale or Volume of Shale) in the matrix to account for
the excess of conductivity. The Simandoux equation (1963) is among the most used ones.
It reduces mathematically to the Archie equation when n=2 and V_{sh}=0.
Below are the expressions for the general Archie and Simandoux equations:
Other popular models that deal with shaly sands are the Fertl (1975) equation, and the 1971
PouponLeveaux (Indonesia) equation.
The Indonesia equation
may work well with fresh formation water. The parameter R_{shale}
(resistivity of shale) is usually taken from the resistivity reading of a nearby
pure shale, assuming that the clay cements & silt, and the shale nature, are
similar to those of the shaly sand.
The Fertl (1975) equation for shaly sands has the advantage that
does not depend upon R_{shale}.
It uses instead a reservoir dependent empirically
adjusted 0.25 ≤ α ≤ 0.35 parameter α:
There are two equations that occasionally yields reasonable estimates for the water saturation
—even if there are no estimates for porosity—
when the resistivity of the flushed zone R_{xo}, and the mud filtrate resistivity R_{mf}
are available:
The former equation is called the Archie SW ratio,
and requires a clean, nonshaly, nonconductive matrix to work.
For the cases or moderately shaly matrixes, we introduced
on April 2020 the following approximate correction to deal with the excess of matrix conductivity:
∎
Notice that all the electrical equations shown above to estimate SW require to know the value for the formation water resistivity Rw.
This is usually the most important parameter to estimate SW.
The table below shows the most popular techniques aimed to estimate either salinity or its companion Rw value at
the reservoir temperature:

NaCl or Rw Technique

Reliability

Source

Remarks

1

Ionic Water Analysis

Best

Water sample

Water sample must be representative. Independent, log free technique

2

Rw from water bodies and pockets

Good

Logs

Needs to find 100% water bodies to work, like aquifers or water pockets

3

Hingle Plot

Good

Logs

Same math as Rw from H2O body, but from a crossplot.

4

Core SW vs. log SW match

Medium

Logs and Core

Move salinity until match. Needs accurate lab SW measurements.

5

Pickett Plot

Poor

Logs

Rw & m estimation. Does not work if Phi is almost constant in water body

6

SP Spontaneous Potential

Worst

Logs

Last resource to try. Seldom yields accurate or usable Rw estimates

GeolOil has +19 builtin models for water saturation: Archie, Fertl, Simandoux,
Schulumberger, PouponLeveaux (Indonesia), SW ratio, laminar shales, Dual Water, Juhasz,
WaxmanSmits, Archie flushed zone ratio, irreducible low bound, irreducible Timur,
saturation height through capillary pressure, and others.
All water saturation equations yield similar results to the Archie equation
for moderately clean sands (see the aqua color clean sand zone
in the picture below). However, the results differ in the case of shaly sand
(see the pink color shaly sand zone), where the Archie law clearly
overestimates the water saturation (too much water, so a pay zone could be easily
missed if the Archie equation is the only model used).
The two following tables help to figure out how the water saturation SW and the hydrocarbon saturations
SO and SG, move by trend increases or decreases on the electrical parameters
a, m, n, Salinity, Rw, Rt, Vsh, Rsh, φ, α, and Cation Exchange Capacity Qv.
As each parameter changes, the remaining ones are kept constant for sensibility comparisons.

↑
Parameter
increases

SW

SO or SG

1

a

↑

↓

2

m

↑

↓

3

n

↑

↓

4

Rw

↑

↓

5

Salinity (Rw ↓)

↓

↑

6

Rt

↓

↑

7

φ

↓

↑

8

Vsh (φ_{e} ↓)

↑

↓

9

Vsh (Sat. correction)

↓

↑

10

Rsh

↓

↑

11

Fertl α

↓

↑

12

Grain density (φ ↑)

↓

↑

13

CEC Qv

↓

↑



Parameter
decreases
↓

SW

SO or SG

1

a

↓

↑

2

m

↓

↑

3

n

↓

↑

4

Rw

↓

↑

5

Salinity (Rw ↑)

↑

↓

6

Rt

↑

↓

7

φ

↑

↓

8

Vsh (φ_{e} ↑)

↓

↑

9

Vsh (Sat. correction)

↑

↓

10

Rsh

↑

↓

11

Fertl α

↑

↓

12

Grain density (φ ↓)

↑

↓

13

CEC Qv

↑

↓


The figure below compares the results of Archie, Simandoux, Indonesia, and Fertl models

The photograph to the left shows at the top, a classical pure gray shale
with VSH = 1.00 (VCL = 0.83, and Vsilt = 0.17).
This shale is an effective seal that blocks the upwards migration of oil.
The zone at the bottom, shows an oil impregnated, shaly sand. The oil impregnation is seen
as the black stains on regular light. You are welcome to see the
photograph in full size.
If the Archie model would be applied to the oily shaly sand, it would forecast too much water.
Hence overlooking the searching for oil, and underestimating resources.
A correction to take into
account the excess of conductive clays in the shale is needed. All the models shown above that
incorporate shale adjustments would yield more realistically, less water saturation and more oil.

The figure below shows the panel to compute water saturation using Archie, Fertl, & Simandoux equations
The figure below shows the calculation of water saturation using PouponLeveaux (Indonesia) & Dual Water Model algorithms

Related articles:
