Q922+log+l06 v1

Well Logging Course (2nd Ed.)

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1. Spontaneous PotentialA. membrane potential

B. Application

C. Log Example of The SP

1. Early Electric Log Interpretation

2. Formation Factor

3. Water saturation

4. The Porosity Exponent, m

5. The Saturation Exponent, n

6. A Thought Experiment For A Logging Application

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)

Spring14 H. AlamiNia Well Logging Course (2nd Ed.) 5

a log of SP and formation resistivity made prior to 1935The 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.Spring14 H. AlamiNia Well Logging Course (2nd Ed.) 6

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.


Resistivity measurements of

two core samples as a function of

water saturation

for use in electric log interpretation.

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.


Leverett conclusions

Calibration curve of Leverett’s core holder with sand pack, showing

variation of relative conductivity as a function of water saturation.


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.


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 samplewas 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.


the formation factor F vs. k and, almost as an afterthought, porosity, ϕExamples of

Archie attempts to correlate the electrical F with K and ϕ for water-saturated rock samples from two regions.

F vs. K and phi (phi on a much compressed scale)Spring14 H. AlamiNia Well Logging Course (2nd Ed.) 13

A summary of an exhaustive set of measurements of formation factorAlthough 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.


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.


A synthesis of various resistivity/saturation experimentsThe 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.


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:

The exponent n, called the saturation exponent, is very nearly 2 for the data considered.

From this, an approximate expression for the Sw is:

However, the fully saturated resistivity Ro (which is not usually accessible in formation evaluation), can be related to the water resistivity. So:

and with the porosity dependence, the final form is:

which can be used for purposes of estimation.

However, a more general form, is:

the constants a, m, and n need to be determined for the particular field or formation being evaluated.


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).


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:


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.


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


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.


the total effective m in a fractured reservoirClearer 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.


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.


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.


condition in which n can be significantly different than 2However, laboratory experiments

have highlighted some 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.


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


The Saturation Exponent in carbonatesCarbonates are particularly

heterogeneous, and also more likely to be oil-wet, so that the relation between

resistivity and Sw is likely to be complicated, with n not equal to 2 and also varying with saturation.


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.)


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.


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

1. Unfocused Devices:A. The Short Normal

B. Estimating the Borehole Size Effect

2. Focused Devices:A. Laterolog Principle

B. The Dual Laterolog

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