When scientists apply Archie's first law they often include an extra
parameter

In petroleum engineering, Archie's first law (Archie, 1942) is used as a tool
to obtain the cementation exponent of rock units. This exponent can then be
used to calculate the volume of hydrocarbons in the rocks, and hence
reserves can be calculated. Archie's law is given by the equation:

However, at least nine out of ten reservoir engineers and petrophysicists do
not use Archie's first law in this form. Instead, they use a slightly
modified version which was introduced 10 years later by Winsauer et al. (1952), and which has the
form:

A problem arises, however, when we consider the result that the Winsauer et
al. (1952) modification to Archie's equation gives when

While most scientists fit Eq. (2) to measurements made on a group of data
from core plugs from the same geological unit or facies type on a log
formation factor vs. log porosity plot, some petrophysicists prefer to
calculate cementation exponents for individual core plugs than calculate a
mean and standard deviation for a given group of measurements. This approach
has been considered justified (e.g. Worthington, 1993), but runs the risk
of including samples from more than one facies type by accident or
oversight, whereas the use of a plot allows the uniformity and relevance of
the data from all of the samples to be judged during the derivation of the
cementation exponent. Moreover, plug-by-plug calculation of the cementation
exponent is carried out with the equation:

The rest of this paper examines the apparent paradox that whereas Eq. (1) has a longer and theoretically better pedigree, Eq. (2) is the version that is overwhelmingly more commonly applied because it fits experimental data better. We show that, while the original Archie's law is the most correct physical description of electrical flow in a clean porous rock that is fully saturated with a single brine, the Winsauer et al. (1952) variant is the most practical to apply because it compensates to some extent for systematic errors that are present in the experimental data.

Table 1 shows typical ranges of values for the cementation exponent and the

Typical ranges of cementation exponent and the

In a paper such as this, the dataset is very important. The inferences made at the end of this paper have a bearing on the quality of data measurement. First of all, the dataset should be typical of its type within the oil industry, and preferably represent the best or close to the best practice within the industry. Generally service companies have very well-developed protocols for making the best and most reliable as well as the most repeatable measurements possible within tight financial constraints. Consequently, the data are often of high quality, but not as high as it might be if the measurement were carried out in an academic environment with no pressures of time or funding.

All of the data analysed in this paper were provided by service companies, and
it is understood that the great majority come from a single service company.
Routine core analysis within the service company would normally follow a
very clear protocol. In this case, the samples would have been received as
preserved core or core plugs, and would have been subsampled if required.
The core plugs would have been cleaned, commonly with a Soxhlet approach,
then dried in a humidity-controlled oven at a temperature low enough to
ensure the preservation of most clay structures. The porosity measurements
here are all made using helium pycnometry, and are likely to have been made
on an automated basis. Such porosity measurements do not have the high
accuracy of those made in academic petrophysics laboratories, but have a
very good repeatability, and are usually accurate to

In this work all of the data are from relatively clean clastic reservoirs whose dominant mineralogy is quartz, exhibiting a low degree of surface conduction. However, there is no reason why the arguments made in this paper should not apply equally well to carbonates (e.g. Rashid et al., 2015a, b) or indeed any reservoir rock for which Archie's parameters might be useful in determining their permeability (e.g. Glover et al., 2006; Walker and Glover, 2010).

The question why the practice of using an equation that is not theoretically correct remains commonly applied in industry is worth asking. The answer is that the variant form of Archie's law (Eq. 2) generally fits the experimental data much better than the original form (Eq. 1).

We have carried out analysis of a large dataset using the two equations and by calculating the cementation exponents for individual core plugs. Figure 1 shows formation factors (blue symbols) and cementation exponents (red symbols) of the fully saturated rock sample as a function of porosity for 3562 core plugs drawn from the producing intervals of 11 unattributable clean sandstone and carbonate reservoirs. The formation factor data have been linearised by plotting the data on a log axis against the porosity, also on a log axis. Best fits were made by linear regression from both the Winsauer et al. (1952) variant of the first Archie's law (Eq. 2, solid lines) and the theoretically correct first Archie's law (Eq. 1, dashed lines). In addition, the individually calculated cementation exponents were calculated by inverting Eq. (3) (red symbols).

Formation factor and cementation exponent of the fully
saturated rock sample as a function of porosity for 3562 core plugs drawn
from the producing intervals of 11 clean sandstone and carbonate reservoirs.
Blue symbols represent the formation factor for individual core plugs
calculated as

A first qualitative comparison of the fits in Fig. 1 shows that fitted lines
from both equations seem to describe the data very well and it would be
tempting to assume that either would be sufficient to use for reservoir
evaluation. The adjusted

Summary data from the 11 test reservoirs.

There is, however, an important difference in the values of cementation exponent that the two methods of fitting provide. The cementation exponents that are derived from each fit are shown in the regression equations given in each panel of Fig. 1 and are summarised in Table 2. It is clear that there is a significant difference in the cementation exponents derived from the two different equations in almost every case. The extent of the differences is clear in Fig. 2, where the cementation exponents calculated from Eq. (1) and from Eq. (2) are plotted as a function of the mean of the individual exponents calculated using Eq. (3), with the dashed line representing a 1 : 1 relationship. There is no significance in the almost perfect agreement between Eq. (1) and the mean of the individual core plug determinations as both measurements are based on the same underlying equation, that of Archie's original law. What is surprising is that the difference between the cementation exponents derived from using Eq. (2) differs significantly from the difference between those that used Eq. (1).

Cementation exponent derived from fitting Archie's (1942) law (Eq. 1, solid blue symbols) and the Winsauer et al. (1952) variant of Archie's law (Eq. 2, solid orange symbols) as a function of the cementation exponent derived as the mean of the cementation exponents calculated from data from individual core plugs using Eq. (3), which is based on Archie's original law. The dashed line shows a 1 : 1 relationship. Each symbol represents data from one of the 11 reservoirs analysed in Fig. 1.

Percentage difference between cementation exponents derived from
Eq. (2) with respect to that derived from the use of Eq. (1) (i.e.

The small, but apparently significant differences in adjusted

However, the calculated percentage difference between the cementation
exponents that have been derived from fitting Eq. (2) with respect to Eq. (1)
as a function of the

Consequently, statistical analysis of the 3562 data points analysed in this work shows that Eq. (2) provides a better fit than Eq. (1), confirming the experience of many petrophysicists. Equation (2) provides a better physical quality of fit to real data despite the data being theoretically flawed, and having a lower theoretical/mathematical quality model of the process. This paper will examine the implications of this observation, examine possible causes for the apparent paradox, and then make a number of recommendations.

We have compared the results of the calculated cementation exponents from
each of the equations using the 11 reservoirs that are summarised in Table 2.
The mean cementation exponent from fitting Eq. (2) to the whole dataset
is

Cross-plot of the cementation exponents calculated using Eqs. (1) and (2) for a database of 3562 core plugs drawn from the producing intervals of 11 unattributable clean sandstone and carbonate reservoirs. The solid line shows the least-squares regression and the dashed line shows the 1 : 1 ideal.

Hence, even though Eq. (2) provides only a marginally better fit than Eq. (1),
its application can give cementation exponents that are as much as

For example, if one assumes arbitrarily that the true cementation exponent is

In summary, apparent small differences in fit can cause significant
differences in the derived cementation exponent which will have important
implications for reserves calculations. Moreover, it is the Winsauer et al. (1953)
variant of Archie's equation which contains the theoretically
unjustified

Therefore, there is an apparent paradox: Eq. (2) is theoretically incorrect but
fits the data better than a theoretically correct form. There are two
possible reasons.

The theoretically correct form of the first Archie's law is wrong.

All of the experimental data are incorrect.

Furthermore, Table 1 and our analysis of 11 reservoirs shows that the

One of the possibilities for the observed behaviour is that the original
Archie's law is incorrect. If that is the case we can hypothesise that there
is an unknown mechanism

The effect occurs in clean rocks – Fig. 1 shows it operating in 11 reservoirs composed of clean sedimentary rocks.

Surface conduction can only decrease the resistivity of the saturated rock, whereas the mechanism for which we search must have the capability of both increasing and decreasing the resistivity of the fully saturated rock.

Surface conduction does not scale linearly with the pore fluid resistivity and is well described by modern theory (Ruffet et al., 1995; Revil and Glover, 1997; Glover et al., 2000; Glover, 2010).

It is not possible to generate the second scenario from any of the previous theoretical approaches to electrical conduction in rocks (Pride, 1994; Revil and Glover, 1997, 1998).

Finally, it is worth remembering that, although initially an empirical equation, Archie's first law now has a theoretical pedigree since its proof (e.g. Ewing and Hunt, 2006). It seems unlikely, therefore, that the theoretical equation is wrong in itself.

It is worth taking a little time to imagine the implications of this question. It implies that the majority or even all of the electrical measurements made in petrophysical laboratories around the world since 1942 have included significant systematic errors (random measurement errors are not the issue here). Given the importance of the calculation of the cementation exponent for reserves calculations, this statement will seem incredible and will have far-reaching implications.

It is hypothesised in this paper that there have been systematic errors in
the measurement of the electrical properties that contribute to the first
Archie's law. The result of these errors has been to make the version of the
first Archie's law given in Eq. (2) a better model for the erroneous data
than the theoretically correct model (Eq. 1), and implies that the
theoretically correct model would be a better fit to accurate experimental
data. If correct, it would also imply that most of the cementation exponents
that have been calculated historically are correct because the errors in the
experimental data have been compensated for by the parameter

There are at least three possible sources of systematic error in the relevant experimental parameters used in Archie's laws, and others may be realised in time. Each has the potential for ensuring that the Winsauer et al. (1952) variant of Archie's law will fit the data better than the classical Archie's law. These errors are associated with the measurement of porosity, fluid resistivity, and temperature, and will each be reviewed in the following subsections.

Let us assume that if instead of measuring the correct porosity

The calculated value of the parameter

We should examine the possible sources of systematic error in the porosity. The question should be what the correct porosity to use in the first Archie's law is. There is some doubt whether this question is possible to answer at the moment. If Eq. (1) is founded on good theoretical grounds as it seems to be, then the porosity required should be the measured porosity that best approaches the true total porosity which is fully saturated with the conducting fluid. However, if the sample's total porosity is only partially saturated with conducting fluid, the water saturated porosity would likely be a better measure.

There are many ways of measuring porosity, and it is well known that they
give systematically different results. Without being comprehensive, we
should consider at least three types of porosity measurements that are
commonly used as inputs to the first Archie's law for the calculation of the
cementation exponent.

is well known to give effective porosities that are systematically higher than other methods because the small helium molecules can access pores in which other molecules cannot fit. Hence, it is a good measure of the combined effective micro-, meso-, and macro-porosity of a rock.

Again, this method is well known to give effective porosities that are systematically lower than other methods because it takes extremely high pressures to force the non-wetting mercury into the smallest pores. Consequently, the micro-porosity is not commonly measured, even with instruments which can generate very high pressures.

This method relies on measuring the dry and saturated weights of a sample, and then using either caliper measurements or Archimedes' method for obtaining the bulk volume, from which the porosity may be calculated. Measurements made in this way generally fall between those made on the same sample using the helium and mercury methods. The problem here is one of saturation. If the sample is not fully saturated, the porosity will be underestimated. Since saturation in any laboratory is generally governed by its protocols, attainment of an only partially saturated sample would be systematic.

It is important to distinguish between (i) the bulk pore fluid resistivity and (ii) the resistivity of the fluid in the pores. The bulk pore fluid resistivity is that fluid which has been made in order to saturate the rock. It has a given pH and resistivity, which may be measured in the laboratory, but is sometimes calculated from charts, using software, or empirical models such as that of Sen and Goode (1992a, b). It is the resistivity of this fluid that petrophysicists have most commonly used in their analysis of data using the first Archie's law.

However, the first Archie's law is not interested in the bulk fluid resistivity, but the actual resistivity of the fluids in the pores. When an aqueous pore fluid is flowed through a rock sample, it changes. Precipitation and, more commonly, dissolution reactions occur until the pore fluid is in physico-chemical equilibrium with the rock sample.

We have carried out tests on three samples of Boise sandstone, and we find
that the fluid in equilibrium with the rock can have a resistivity up to
100 % less than the bulk fluid (and a pH that is up to

Percentage difference between the conductivity of the fluid in the pores and that of the bulk fluid originally used to saturate the rock as a function of the resistivity of the fluid in the pores for three samples of Boise sandstone.

The apparent clear difference between the resistivity of the bulk fluid, which is used as an input to Archie's first law, and the resistivity of the fluid, which should be used, is clearly the source of an invisible systematic error to which many petrophysical laboratories have succumbed.

Let us assume that instead of using the resistivity of the fluid in the
pores

If there is a

Temperature also affects the pore fluid resistivity that we use in the first
Archie's law. The resistivity of an aqueous pore fluid changes by about
2.3 % per

Resistivity of an aqueous solution of NaCl as a function of
temperature for a number of different pore fluid salinities using the method
of Sen and Goode (1992a, b). Dashed lines show the change in conductivity
resulting from a difference in temperature between 20 and 25

Modelling of the calculated cementation exponent and

If we measure the pore fluid resistivity, or calculate it using the model at
25

The systematic error can be removed by measuring the resistivity of the fluid emerging from the rock sample at the same time or just after the resistivity of the bulk rock has been measured because the bulk rock and the emerging fluids should both have the same temperature. Providing the pore fluid has been equilibrated properly with the sample, this procedure also removes any errors associated with using the resistivity of unequilibrated bulk fluids in Archie's law calculations.

Equations (6) to (11) mathematically imply that the use of the

First, let us assume that we are using Archie's law to calculate the
cementation exponent of a single core plug. Equation (1) can be used in its
rearranged form (Eq. 3). If we assume that the measurements of porosity,
fluid resistivity, and core plug resistivity are all accurate, then Eq. (3)
will give an accurate value of cementation exponent. Conversely if there is
an error in any of the input parameters, they will be in error in the
cementation exponent. Equation (2) can also be rearranged for the calculation
of cementation exponent, but since there is no a priori knowledge of the value of
the

Now let us examine the calculation of cementation exponent from a population of core samples by the fitting of the original or Winsauer et al. variants of Archie's law. Three possibilities will be analysed, of which the results of two are shown in Fig. 8. The three possibilities are that there is a random error (which is not shown in the figure), a systematic error resulting in the measured porosity of each sample being overestimated or underestimated by a constant value (Fig. 8a), and a systematic error resulting in the measured porosity of each sample being overestimated or underestimated by a constant fraction of the real value (Fig. 8b). Both graphs are given as a function of the real error-free porosity.

The figure assumes that the real fluid resistivity is 10

Now let us use Eq. (2). In practice the value of the

Figure 8b shows that exactly the same process was implemented for a systematic
error which is a constant multiple of the real porosity, which in this
example is a factor of 1.2, but could be much higher than unity or much less
than unity with the same general effect. Once again the use of Eq. (1) with
the error-prone data provides an erroneous calculated cementation exponent
varying between

Similar analyses can be done with the fluid resistivity variable and as a
function of temperature, with exactly the same results. The individual
changes in the

The remaining error type to consider is random error. However, these errors are automatically removed from any fitting that is carried out using either equation by the least-squares operation, and so are not considered further in this work.

In summary, examination of the sources of error described above allows us to
make the following two statements. The first is that if Eq. (2) has been
used, the systematic measurement errors do not affect the calculated value
of the cementation exponent because the fitted value of the

It is possible to make recommendations for the improvement of the accuracy of data used in the first Archie's law, as follows.

The saturation of samples should be as close to 100 % as possible. Vacuum
and pressure saturation followed by flow under back-pressure can be
recommended. Full saturation can be improved by flooding the sample with
CO

There is some ambiguity about what is the “correct” porosity to use with the first Archie's law. Until this is resolved, it is recommended in this paper that the porosity calculated from the saturation of the sample with the reservoir water by dry and saturated weights is carried out, and Archimedes' method is used to measure the bulk volume of the core plug. Other sources of porosity should be avoided.

The resistivity of both the bulk fluid and the fluid in equilibrium with the rock sample should be measured, with the latter being used in the first Archie's law to calculate the cementation exponent. This implies that fluid is flowed through the core until equilibrium is attained. This process may take several days.

All measurements of bulk fluid resistivity, equilibrium resistivity, and effective resistivity of the saturated rock sample should be either made at the same temperature, or all corrected to a standard temperature.

The value of

This paper shows that the commonly applied Winsauer et al. (1952) variant of the first Archie's law is incorrect theoretically, yet paradoxically fits data better than the classical, and formally correct, Archie's law.

The apparent paradox can be explained by systematic errors in the majority of all previous data. Errors in porosity, pore fluid salinity, and temperature can all explain the effect and may combine to produce the observed results.

Consequently, cementation exponents which have been calculated historically
using the Winsauer et al. (1952) variant of the first Archie's law (Eq. 2)
will be accurate because the

A range of recommendations have been made to improve the accuracy of
calculations of the cementation exponent using the first Archie's law.
Furthermore, the parameter

While the raw data used in this work remain confidential, the summary and generic data that are shown in the figures have been provided as a Supplement to this publication.