CALCULATING ELECTRICAL PROPERTIES (A, M, and N)
electRICAL
PROPERTIES BASICS
Electrical properties are terms used in the Archie water
saturation equation, and in its shale corrected cousins, to
calibrate water saturation to the physical properties of the
pore geometry of the rock. The parameters required are:
1: tortuosity constant A
2: cementation exponent M
3: saturation exponent N
These three terms are usually
written in lower case font (a, m, n) but in keeping with the policy
of this Handbook that input parameters be capitalized, they are
referred to here and elsewhere in upper case
M
is defined as the slope of the best fit line drawn through a graph
of formation factor (F = R0 / RW) versus porosity, and A is
the intercept of that line on the F axis. A sample of such a plot is
show at the left and is discussed in more detail later in this
Chapter.
N is defined as the slope of the
best fit line on a graph of resistivity index (RI = RT / R0) versus
water saturation. A sample plot is shown at the left and discussed
in more detail later..
Although there is not a lot of
evidence to support the assertion, N is often taken as equal to M,
probably because M can sometimes be found from available log data in
the absence of core data, and N cannot.
Pore geometry in sandstones varies
with lithology, grain size, sorting, shape, roughness, and shale
volume. In carbonates, it varies with pore type and how well the
pores are connected. Some examples are shown below, and combinations
of these pore geometries are common. For example, fractured, moldic,
intercrystalline porosity is relatively common.
Examples of different pore geometry models
The cementation exponent has been studied the most and the
following illustrations, keyed to the pore geometry shown above,
give some idea of the complexity of the problem. The two images
below are from "Cementation Exponents in Middle Eastern Carbonate
Reservoirs" by J. W. Focke and D. Munn, SPE Paper 13175, June 1987.
Formation factor plot and M versus porosity plot for intergranular
and sucrosic pore geometry.
A = 1.00, M = 2.00, and do not vary with porosity.
Similar plots for a moldic limestone. Formation factor data is
scattered and M increases with
increasing porosity. A variable M method is required for a
satisfactory water saturation calculation.
The equation for M is: M = 1.40 + 6.57 * PHIe where PHIe is a
decimal fraction.(permeability
regime 0.1  1.0 md)
The
slope and intercept of the variable M plots will of course vary from
one reservoir to another, and vary with the permeability regime,
which is an indicator of the connectedness of the moldic porosity.
There are a number of methods for finding and adjusting the
electrical properties of rocks. The common approaches are
listed here.
tortuosity
CONSTAnt (A) from
Log and RW Data
This
is easy if there is a clean water zone on the log which also tested
water. The formula is:
1: A = (RW@FTlog) / (RW@FTdst)
WHERE:
A = cementation exponent (unitless)
RW@FTlog = RW@FT with A = 1.0 and M = 2.0 calculated from a
water zone on logs (ohmm)
RW@FTdst = RW@FT from DST measured in the laboratory (ohmm)
and converted to formation temperature
Data
from several wells may be averaged to obtain a good value. The
parameter A varies with grain size, so wells used should come
from similar geologic settings (depth, distance from source of
sediment, etc.).
Cementation Exponent (M) from Special Core Data
Plot
porosity vs lab measured formation factor on loglog axes. Fit
regression or eyeball line to data. Slope of line is M. Intercept
at PHIe = 1 is A. The line forcefitted through F = PHIe = 1.0
is called a pinned line. Some people prefer the pinned line but
most data sets do not support this approach. Strictly speaking,
the line must pass through F = 1 = PHIe, so the line must be nonlinear
approaching this point on the graph. An example is shown below.
Find A and M from special core data (electrical properties
data)  M is slope of best fit line
(pinned or free regression  your choice), A is intercept at
PHIe = 1.0.
saturation exponent (N) from Special Core Data
Plot
brine saturation from core analysis versus formation resistivity
index (RESD / R0) on loglog axes. Draw line through the data
to intercept at SW = 1.0. The slope of this line
is N. Data from several wells may have to be combined to get a
reasonable fit.
Find N from special core data (electrical properties
data). Slope is N and line must pass
through Sw = 1.0 at RI = 1.0..
Cementation Exponent (M) from Log Data
A variety of
approaches are available to assess the electrical property M
from log data. There
is no direct method for finding N from log data, but there is an
indirect approach. If you are satisfied that A, M, and RW are
correct, you can compare the log analysis SW with the capillary
pressure SW. Any difference can be repaired by adjusting
N. Changing N will make no difference in a water zone, so it
helps to be able to calibrate A, M, and RW in the water zones
first.
Pickett Plot Method
Pickett
proposed that the Archie formation factor and saturation equations
be rearranged as follows:
2: log(RESD) =  M * log(PHIe) +
log(A*RW@FT)
3: log(RESS) =  M * log(PHIe) + log(A*RMF@FT)
In
water zones:
4: M = (log(A*RW@FT)  log(RESD)) / log(PHIe)
In
invaded water or hydrocarbon zones:
5: M = (log(A*RMF@FT)  log(RESS)) / log(PHIe)
WHERE:
A = tortuosity exponent (unitless)
M = cementation exponent (unitless)
PHIe = porosity from any source (fractional
RESD = deep resistivity of rock with water and oil (or mercury)
(ohmm)
RESS = shallow resistivity of rock with water and oil (or mercury)
(ohmm)
RW@FT = water resistivity at formation temperature (ohmm)
RMF@FT = mud filtrate resistivity at formation temperature (ohmm)
COMMENTS
Equation 1 represents a plot of deep resistivity vs effective
porosity FROM KNOWN WATER ZONES on loglog paper. Draw a line
through southwest data points. Slope of this line is M and intercept
at PHIe = 1 is A*RW@FT.
Equation
2 may work in hydrocarbon zones if invasion is well developed
and residual hydrocarbon is small. M will be too high if ROS is
high.
Find M and A*RW from Pickett plot
Make
a separate graph FOR EACH ROCK TYPE. Typical rock types in carbonates
are intergranular (clastic texture), intercrystalline (fine, medium,
or coarse), vuggy (fine, medium, or large), microporosity (unconnected
pores), oomoldic, and fractured zones (with any other rock type).
Rock typing is usually done from sample or core description.
M
can be made to vary by solving equation 1 or 2 for each data point
instead of fitting a line through the average of the data set.
Shell Method
Analysts
at Shell Oil proposed a formula to vary M in carbonates with porosity.
Other relationships could be found by fitting nonlinear curves
to the data used for the Pickett plot or by plotting individual
M values versus porosity:
6:
M = 1.87 + 1.9 * PHIe
Focke and Munn l Method
The
Focke and Munn paper referred to earlier shows a variety of
data sets, in which M increases with porosity, as shown
below:
7: M = 1.20 + 12.76 * PHIe (perm < 0.10)
8: M = 1.40 + 6.57 * PHIe (perm = 0.1  1.0)
9: M = 1.20 + 6.29 * PHIe (perm = 1.0  100)
10: M = 1.22 + 3.49 * PHIe (perm > 100) WHERE:
M = cementation exponent (unitless)
PHIe = porosity from any source (fractional
Nugentl Method
An
equation proposed by Nugent uses the secondary porosity concept:
11: M >= 2 * log(PHIsc) / log(PHIxnd)
PHIsc
represents the matrix porosity and PHIxnd represents the effective
porosity in the carbonate rock. Both should be shale corrected
as described in Chapter Seven.
Use
Nugent's method
in intergranular, intercrystalline, vuggy, and fossilmoldic rock
types. Results
are too low in oomoldic rock type.
PHIsc
must be calculated with a matrix value (DELTMA) that varies with
the rock lithology. This can be derived from the results of a
two or three mineral model or sample description with DELTMA =
V1 * DELTMA1 + V2 * DELTMA2 + V3 * DELTMA3.
Nurmi Method
In
oomoldic porosity, Nurmi proposed the following:
12: PHIvug = 2 * (PHIxnd  PHIsc)
13: PHIma = PHIxnd  PHIvug
14: M >= 2 * log(PHIma) / log(PHIxnd)
Use
Nurmi method in oomoldic rock type.
PHIsc
must be calculated with a matrix value (DELTMA) that varies with
the rock lithology. This can be derived from the results of a
two or three mineral model or sample description.
Rasmus Method
The
same techniques used to derive M for various carbonate rock types
can also be used to find M in fractured carbonates. A standard
Pickett plot in water zones, or a Pickett plot using a shallow
resistivity log in the invaded zone, will usually suffice. The
M value so derived will be the result of BOTH fractures and the
rock type in the zone covered by the crossplot. Normally, M is
chosen once for each fractured interval from Pickett plots over
welldefined rock type zones or layers.
However,
there is no reason to believe M is a constant in a zone because
fracture intensity probably varies dramatically from foot to foot
within the layer.
A
method proposed by Rasmus, based on secondary porosity concepts,
solves this problem:
11: Md = log((1  (PHIxnd  PHIsc)) * (PHIsc^Mb) + (PHIxnd  PHIsc))
/ log(PHIe)
Mb
is the formation factor exponent for the bulk matrix (unfractured)
rock and Md is the value for the combined matrix plus fracture,
or double porosity, porosity. Mb should be determined separately
in unfractured zones if possible.
"META/FRF" 
Electrical Properties Spreadsheet
This spreadsheet
provides a tool for summarizing Electrical Properties
dara and
includes crossplots to find
A, M, and N.
Electrical Properties Analysis,
includes crossplots to find A, M, and N.
English and Metric Units
Sample output from "META/FRF"
spreadsheet for summarizing electrical
properties.
