1. Well Logging Course (1st Ed.)

2. 1. Spontaneous Potential
A. membrane potential
B. Application
C. Log Example of The SP

3. 1.
2.
3.
4.
5.
6.
Early Electric Log Interpretation
Formation Factor
Water saturation
The Porosity Exponent, m
The Saturation Exponent, n
A Thought Experiment For A Logging
Application

5. usefulness of measuring
the resistivity of earth formations
the water saturation Sw
Is the desired petrophysical parameter
from resistivity measurements
In this lecture the empirical basis
for the interpretation of resistivity measurements is reviewed.
For many years, at the outset of well logging,
it was not possible to address
the water saturation question any more precisely than
whether the resistivity of a formation was high or low.
It was through the work of Leverett and Archie
that it became possible to be more quantitative about
the interpretation of a formation resistivity measurement and
to link resistivity to
formation water resistivity, porosity (ϕ), and water saturation (Sw)
Fall 13 H. AlamiNia
Well Logging Course (1st Ed.)
5

6. a log of SP and formation resistivity
made prior to 1935
The scale “Ohms
m3” presumably
refers to
ohm-m.
It seems
possible, noting
the higher
resistivity,
that zone a-A
contains more
oil
(has a lower Sw)
than zone B-b.
 But how can
this be
verified?
An early resistivity-SP log.
Fall 13 H. AlamiNia
Well Logging Course (1st Ed.)
6

7. Early SP log interpretation
The “standard”
procedure at the time
[for verification] was
to take a core sample,
representative of the
zones in question, and
to make laboratory
measurements of
its resistivity under
different conditions of
water saturation.
Fall 13 H. AlamiNia
Resistivity measurements of
two core samples as a function of
water saturation
for use in electric log interpretation.
Well Logging Course (1st Ed.)
7

9. Leverett experiments
M. C. Leverett was conducting experiments
with unconsolidated sands,
to determine
the relative permeability of oil and water (Kro & Krw)
as a function of the water saturation (Sw).
As a by-product of his research,
he measured the conductivity of
the material in a sample chamber,
after a calibration of the system constant,
in order to conveniently determine the fraction of
kerosene and water in his permeable samples.
Fall 13 H. AlamiNia
Well Logging Course (1st Ed.)
9

10. Leverett conclusions
Calibration
curve of
Leverett’s
core holder
with sand
pack,
showing
variation of
relative
conductivity
as a function
of water
saturation.
Fall 13 H. AlamiNia
Well Logging Course (1st Ed.)
10

11. Archie experiments
Shortly after the publication of Leverett’s work,
G. E. Archie of Shell
was making electrical measurements on core samples,
with the aim of
relating them to permeability.
His measurements consisted of
completely saturating core samples
with saltwater of known resistivity Rw and
relating the measured resistivity Ro of
the fully saturated core to the resistivity of the water.
Fall 13 H. AlamiNia
Well Logging Course (1st Ed.)
11

12. Relation between water resistivity
(Rw) and formation resistivity (Ro)
Archie found that,
regardless of the resistivity of the saturating water,
the resultant resistivity of a given core sample
was always related to the water resistivity (Rw)
by a constant factor F.
He called this the formation factor,
and his experiments are summarized by the following relation:
Next slide is an example of Archie work on cores
from two different locations.
Fall 13 H. AlamiNia
Well Logging Course (1st Ed.)
12

13. the formation factor F vs. k and,
almost as an afterthought, porosity, ϕ
Examples of
Archie
attempts to
correlate
the electrical
F
with K and ϕ
for watersaturated rock
samples from
two regions.
F vs. K and phi (phi on a much compressed scale)
Fall 13 H. AlamiNia
Well Logging Course (1st Ed.)
13

14. A summary of an exhaustive set of
measurements of formation factor
Although Archie
was searching
for a correlation
with K, he finally
admitted that
 a generalized
relationship
between F and
K did not exist,
 although one
seemed to exist
for porosity.
His summary
graph shows
the
hopelessness of
a F/k correlation.
Fall 13 H. AlamiNia
Well Logging Course (1st Ed.)
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15. formation factor calculation
However it indicates that the F is a function of ϕ and
can be expressed as a power law of the form:
the exponent m is very nearly 2 for the data considered.
This empirical observation can be used
to describe the variation in F for
a fixed water resistivity when the ϕ changes:
the lower the porosity, the higher the resistivity will be.
The exponent m was soon named
the cementation exponent, as it was observed
to increase with the cementation of the grains.
In general, it was recognized that m increased
with the tortuosity of the electric path
through the pore space.
Fall 13 H. AlamiNia
Well Logging Course (1st Ed.)
15

18. A synthesis of various
resistivity/saturation experiments
The practical application
of resistivity
measurements
is for the determination
of water saturation (Sw).
This was made possible
by another observation
of Archie.
He noticed that the data
of Leverett and others
could be conveniently
parameterized after
having plotted the data in
the form shown in Figure.
Fall 13 H. AlamiNia
Well Logging Course (1st Ed.)
18

19. water saturation vs. relative resistivity
 On log–log paper, the data of
water saturation versus relative
resistivity plotted as a straight line,
suggesting a relationship of the
form:
 and with the porosity dependence,
the final form is:
 The exponent n,
called the saturation exponent,
is very nearly 2 for the data
considered.
 From this, an approximate
expression for the Sw is:
 which can be used for purposes of
estimation.
 However, a more general form, is:
 However,
the fully saturated resistivity Ro
(which is not usually accessible in
formation evaluation), can be
related to the water resistivity. So:
Fall 13 H. AlamiNia
 the constants a, m, and n need to
be determined for the particular
field or formation being evaluated.
Well Logging Course (1st Ed.)
19

20. interpret a resistivity measurement in
terms of water saturation (Sw)
in order to interpret
a resistivity measurement
in terms of water saturation (Sw),
two basic parameters need to be known:
the porosity φ and
the resistivity of the water
in the undisturbed formation (Rw).
As a starting point,
the value of the water resistivity Rw needs to be estimated.
• This can be done in water zones (resistivity logs).
Fall 13 H. AlamiNia
Well Logging Course (1st Ed.)
20

21. Example of interpretation of resistivity
measurement in terms of Sw
the value of the water resistivity Rw
needs to be estimated.
 From zone D’ or zone C’ (water zones)
 the porosity is about 28 p.u.
 so F is 1/(0.28)^2, or 12.8
 the apparent resistivity = 0.2 ohm-m
which is assumed to be
the fully water-saturated resistivity Ro,
 water resistivity = 0.2/12.8= 0.016 ohm-m
the increase in deep resistivity in
zone C to about 4 ohm-m
 correspond to a decrease in Sw
compared to zone C’
 the porosity is constant at 28 p.u.
 The saturation in zone C:
Fall 13 H. AlamiNia
Well Logging Course (1st Ed.)
21

22. Example of interpretation of resistivity
measurement in terms of Sw (Cont.)
zone of hydrocarbon (A) indicates
the same resistivity as zone C.
in zone A the porosity is much lower
and can be estimated about 8 p.u.
Thus F in zone A is 1/(0.08)2, or 156
If it were water-filled,
the resistivity would be expected
to be about 2.5 ohm-m compared
to the 4 ohm-m observed.
Thus the zone may contain
hydrocarbons, but the water
saturation can be expected to be
higher than in zone C.
Fall 13 H. AlamiNia
Well Logging Course (1st Ed.)
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23. Effective parameters
As Archie was aware,
his equations worked well in
rocks that have simple, uniform pore systems
filled with saline water.
Rocks with
heterogeneous-pore systems,
multiple-conduction mechanisms,
or that are oil-wet need a more complete solution.
The problems can be considered
with reference to Archie’s three equations:
the relation to porosity (m),
the relation to Sw (n),
and the definition of formation factor (F)
is mainly an issue of clay conductivity
Fall 13 H. AlamiNia
Well Logging Course (1st Ed.)
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25. The Porosity Exponent, m
Although Archie could fit his data with
a single parameter, m, in general a fit of F vs. φ throughout
 a reservoir will require two parameters, a and m.
In practice the error caused
by fitting with one parameter is often small.
In either case it would be better
 if the variations [of a and m] through the reservoir
could be related to some physical property,
 rather than relying on a general average.
Early efforts focused on
 finding a relation with porosity, the idea being that as porosity
decreased it was likely that the tortuosity, and hence m, increased.
 Many relations were developed but proved to be
 specific to particular reservoirs or areas, and not generally applicable.
Fall 13 H. AlamiNia
Well Logging Course (1st Ed.)
25

26. the total effective m
in a fractured reservoir
Clearer relations can be obtained
if the reservoir contains vugs or fractures.
Fractures offer a straight path for current,
with minimum tortuosity.
in a fractured reservoir,
if we can measure
the proportion of porosity due to fractures and
if we assume that
the conductive paths through the fractures and
the intergranular porosity
are in parallel, with no interaction between them,
we can calculate the total effective m.
Fall 13 H. AlamiNia
Well Logging Course (1st Ed.)
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27. Methods of m Calculation
More generally,
whatever the cause of the variations in m,
m can be measured directly
from resistivity and porosity in a water zone, and then
assumed to be the same in the hydrocarbon zone.
Alternatively,
if the water saturation can be measured
by another means in addition to resistivity,
One such method
uses dielectric measurements in the invaded zone.
then either m or n can be calculated
from Archie’s equation.
Fall 13 H. AlamiNia
Well Logging Course (1st Ed.)
27

29. saturation exponent measurement
It takes much longer to complete
the type of experiment that
leads to the saturation exponent (n)
than to measure m.
Each core sample must be measured
at several saturation states.
Displacing water with oil or gas takes time,
especially in low permeability samples.
Unlike m,
it is not possible to derive n from logs in a water zone.
As a result there is much less data on n, and
values other than 2 are less often used.
Fall 13 H. AlamiNia
Well Logging Course (1st Ed.)
31

30. 1st condition in which
n can be significantly different than 2
However, laboratory experiments
have highlighted two main conditions in which
n can be significantly different than 2.
The first is related to wettability.
in an oil-wet cores
• the oil coats the grains and starts blocking the pore throats
when even small volumes are introduced.
• The result is a sharp increase in resistivity and a high n
• So n values much larger than 2
In a water-wet core
• the water coats the grains and
provides a continuous conduction path down
to water saturations of 20% or less.
Fall 13 H. AlamiNia
Well Logging Course (1st Ed.)
32

31. Resistivity-saturation measurements
Resistivity-saturation
measurements
on carbonates
that have been flushed
to make them
water-wet or
oil-wet,
a large increase in n
Fall 13 H. AlamiNia
Well Logging Course (1st Ed.)
33

32. 2nd condition in which
n can be significantly different than 2
The second condition occurs
in rocks in which the pore space is no longer uniform but
consists of an irregular mixture of different sized pores.
When oil or gas is introduced into such rocks,
the water in some pore types displaced more easily
For example the oil should easily displace the water in fractures,
but may not do so in vugs if they are poorly connected.
Carbonates are
particularly heterogeneous, and
also more likely to be oil-wet,
so that for both reasons the relation between
resistivity and Sw is likely to be complicated,
with n not equal to 2 and also varying with saturation.
Fall 13 H. AlamiNia
Well Logging Course (1st Ed.)
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34. setup for measuring the resistivity of a
homogeneous formation
It consists of
 a current source of intensity
 and a voltage-measurement
electrode M
at some distance r
 from the current emission
at point A.
The resistivity of the
homogeneous medium is
Rt, so its conductivity σ is
given by σ = 1/Rt .
 (Conductivity is usually
written as σ in measurement
physics, and
as C in log interpretation.)
Fall 13 H. AlamiNia
Well Logging Course (1st Ed.)
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35. determining formation resistivity
the value of Rt is found to be:
The setup can be considered as a rudimentary
monoelectrode measurement device for
determining formation resistivity.
For this device the tool constant k is seen to be 4πr ,
where r is the spacing between
the current electrode and the measurement point.
Knowing the injected current and
the resultant voltage,
the resistivity of the homogeneous medium Rt
may then be found.
Fall 13 H. AlamiNia
Well Logging Course (1st Ed.)
37

36. 1. Ellis, Darwin V., and Julian M. Singer, eds. Well
logging for earth scientists. Springer, 2007.
Chapter 4