This paper describes the extension of the concepts of connectedness and
conservation of connectedness that underlie the generalized Archie's law for

Currently, there is no well-accepted physical interpretation of the saturation exponent other than qualitatively as some measure of the efficiency with which electrical flow takes place within the water occupying a partially saturated rock. Some might say that the meaning is not important as long as one can reliably obtain the water saturation of reservoir rocks with sufficient accuracy to calculate reserves. According to the 2016 BP Statistical Review of World Energy (BP, 2016), the world had proven oil reserves at the end of 2015 of 1.6976 trillion (million million) barrels (Tbbl.), slightly down on the value at the end of 2014 (1.7 Tbbl.) and significantly above the respective values at the end of 1995 (1.1262 Tbbl.) and 2005 (1.3744 Tbbl.). The same source lists proven natural gas reserves of 186.9 trillion cubic metres (Tcm) at the end of 2015, slightly lower than at the end of 2014 (187.0 Tcm) and significantly and progressively higher than the values at the end of 1995 (119.9 Tcm) and 2005 (157.3 Tcm). This represents combined oil and gas reserves of approximately USD 78.4 trillion at end December 2015 prices (using WTI crude and Henry Hub).

Even a tiny uncertainty of, say, 0.01 in a saturation exponent of 2 (i.e.
0.5 % or 2

Within the hydrocarbon industry it is extremely common to assume that the saturation exponent is about 2 for most rocks. However, it is worthwhile thinking about the USD 254 billion global shortfall in revenue if it really is equal to 2.01 instead. These frightening, large financial values make it extremely important that the physical interpretation of the saturation exponent in the classical Archie's law is well understood. This paper attempts to provide a new theoretical and physical interpretation.

The classical Archie's laws (Archie, 1942) link the electrical resistivity
of a rock to its porosity, to the resistivity of the water saturating its
pores, and to the fractional saturation of the pore space with the water.
They have been used for many years to calculate the hydrocarbon saturation
of the reservoir rock and hence hydrocarbon reserves. The classical Archie's
laws contain two exponents,

Like the cementation exponent, and despite its importance to reserves
calculations, the physical meaning of the saturation exponent is difficult to
understand from a physical point of view, which leads to petrophysicists not
giving it the respect it deserves. It is common, for example, to hear that,
in the absence of laboratory measurements, the saturation exponent has been
taken to be equal to 2, which it has just been noted is bound to lead to
gross errors. While it is true that there seems to be a strong preference for
values of saturation exponent near 2

When a saturation exponent is derived from laboratory measurements, it is
commonly done by fitting a straight line to resistivity data where the

It is clear that the physical understanding of the saturation exponent needs to be improved. The purpose of this paper is to investigate the elusive physical meaning of the saturation exponent, where it is shown that the saturation exponents are intimately linked to the phase exponents in the generalized Archie's model.

Considering the classical form of Archie's laws; the first Archie's law relates
the formation factor

Archie's second law considers that the rock is not fully saturated with a
conductive fluid but is partially saturated with a fractional water
saturation

The two laws may be combined to give

Archie's laws require that both the rock matrix and all but one of the fluid phases that occupy the pores have infinite resistivity. Hence, it is a model for the distribution of one conducting phase (the pore water) within a rock sample consisting of a non-conducting matrix and other fluids which also have zero or negligible conductivity. Problems arise when there are other conducting phases in the rock, such as clay minerals. These problems have generated a huge amount of research in the past (e.g. Waxman and Smits, 1968; Clavier et al., 1984), which is reviewed in Glover (2015). The classical Archie's laws were based upon experimental determinations. However, there has been progressive theoretical work (Sen et al., 1981; Mendelson and Cohen, 1982) showing that for at least some values of cementation exponent, Archie's law has a theoretical pedigree, while hinting that the law may be truly theoretical for all physical values of cementation exponent. A study has recently shown that the Winsauer et al. (1952) modification to Archie's law is only needed to compensate for systematic errors in the measurement of its input parameters and has no theoretical basis (Glover, 2016). Meanwhile, independent modifications to the original Archie's law have allowed it to be used when both the pore fill and the matrix have significant electrical conductivities (Glover et al., 2000a; Glover, 2009), such as the case when a rock melt occupies spaces between a solid matrix in the lower crust (Glover et al., 2000b). This has culminated in a generalized Archie's law which is valid for any number of conductive phases in the three-dimensional medium and which was published in 2010 (Glover, 2010).

The generalized Archie's law (Glover, 2010) extends the classical Archie's
law to a porous medium containing

In the 2009 paper the connectedness was defined as

The generalized Archie's law was derived by Glover (2010) and is given by

In the generalized law the phase exponents can take any value from 0 to

It is clear that the classical and generalized laws share the property that the exponents modify the volume fraction of the relevant phase with respect to the total volume of the rock. However the exponents in the generalized law differ from the classical exponent because some of them have values which are not measurable because their phases are composed of materials with negligible conductivity. Despite this, each phase has a well-defined exponent providing (i) it has a non-zero volume fraction and (ii) the other phases are well-defined.

It should be noted that higher phase exponents tend to be related to lower phase fractions, although this relationship is not implicit in the generalized Archie's law as it is currently formulated.

The generalized Archie's law as formulated by Glover (2010) hinges upon the
proposal that the sum of the connectednesses of the phases in a
three-dimensional

Considering a two-phase system, Eq. (1) gives

Another, more intuitive way of looking at this is as follows. It has already
been shown that the connectedness of a system that contains only one phase is
unity as a result of Eq. (1); i.e. if there is one phase,

Distribution of a four-phase clay-rich, water-wet sandstone
saturated with water and oil (quartz – orange; clay – brown; water – blue; oil – grey) represented by a 2-D slice through a 3-D medium. The left-hand
column differs from the right-hand column by the addition of a single grain
of quartz with its associated surface water, labelled Q. Consequently, the
figure should be read vertically comparing the two columns:

Figure 1 is an illustrative example of the idea of a fixed amount of connectedness, using a 2-D slice for simplicity and clarity. Hence, Fig. 1 shows a two-dimensional slice through a 3-D four-phase water-wet medium composed of detrital quartz grains, a string of clay, and a porosity that is partially filled with water, at near irreducible saturation and oil. The figure should be read in two columns. The left-hand column shows an arbitrary arrangement of the four phases that together completely make up the medium (Fig. 1a). In this case I have chosen to represent the detrital quartz as sub-angular detrital grains with a grain size distribution, the clay as a stringer, the near-irreducible water as covering the quartz grain surfaces and the oil as occupying the centre parts of the pores as these geometries can be found in typical water-wet shaly sandstone reservoirs. It should be noted, however, that the equations make no such distinction and what follows is true for any geometrical set of four phases composing the 3-D medium completely. Reading downwards, panels (c), (e), (g), and (i) show each of the quartz, clay, water, and oil phases alone and respectively. One can imagine that each phase has a certain phase fraction and a certain connectedness. Some of the phases look disconnected in the figure, but it should be remembered that there will be a greater connectedness in reality because there will be connection in the third dimension that is not shown in the figure. If we imagine hydraulic flow or electrical flow from the bottom to the top of the medium, the quartz seems to have a relatively high phase fraction and a moderate connectedness, the clay seems to have a moderate phase fraction and a high connectedness, the water seems to have a low phase fraction but a relatively high connectedness due to the multiple pathways formed by the thin “ribbons” of water, and the oil has a moderate phase fraction but a relatively low connectedness as the patches of oil are relatively isolated. The right-hand part of the figure represents the same medium but with the small addition of a quartz grain, labelled “Q”, and its accompanying thin film of surface water. The addition of this makes a minuscule increase in the phase fractions of the detrital quartz and water phase fractions, and, literally, an equally small decrease in the phase fractions of the clay and oil. Reading the distributions for the quartz, clay, water, and oil phases alone (panels (d), (f), (h), and (j)) shows that the addition has made a significant increase in the connectedness of the quartz as well as some increase in that of the water, which was well connected anyway. The low connectedness of the oil will have changed little, but the addition has blocked the main pathway through the clay, leaving only a minor secondary pathway and consequently resulting in a significant decrease in the clay connectedness. Consequently, Fig. 1 shows the principle behind the idea of the conservation of connectedness given in Eq. (4) but not a proof, the latter of which is considered in Glover (2010).

In summary, both the sum of the volume fractions and the sum of the connectednesses of the phases composing a 3-D medium is equal to unity. The corollary is that connectedness is conserved; if the connectedness of one phase diminishes, there must be an increase in the connectedness of one or more of the other phases to balance it.

It is interesting to consider the role of percolation effects within the generalized model (see Glover, 2010, for a full treatment). In percolation theory, the bulk value of a given transport property is only perturbed by the presence of a given phase with a well-defined phase conductivity after a certain phase volume fraction has been attained. This critical volume fraction is called the percolation threshold. This works well for a two-phase system when one phase is non-conductive, with a percolation threshold occurring near the 0.3316 to 0.342 (Montaron, 2009). For such a system, consisting of one non-conducting and one conducting phase, the effective conductivity of the medium depends only on the conductivity of the conducting phase, its volume fraction, and how connected it is. It is intuitive, therefore, that there may exist a phase volume fraction below which the conducting phase is not connected and for which the resulting effective conductivity will be zero. The concept of a percolation factor becomes unclear if the matrix phase has a non-zero conductivity or one or more additional, either solid or fluid conducting phases are added. Under these circumstances a percolation threshold may not exist. Glover (2010) went further than this claiming that Eq. (4) in this work (which is Eq. 26 in Glover, 2010) contains enough information to make the explicit inclusion of percolation effects unnecessary.

Within the framework of the classical Archie's laws, it is possible to envisage the cementation exponent as controlling how the porosity is connected within the rock sample volume and to envisage the saturation exponent as controlling how the water is connected within that porosity. The cementation exponent is defined relative to the total volume of the rock, while the saturation exponent is defined relative to the pore space, which is a subset of the whole rock. This is an important concept for what follows.

The water is one of two phases within the porosity, while that porosity is one of two phases within the rock. Hence, there exists a three-phase system to which the generalized Archie's law can be applied. In fact, the generalized Archie's law can be used to show that the saturation exponents arise naturally and have a physical meaning: they are defined in the same way as the phase exponents but are expressed relative to the pore space instead of the whole rock.

By writing the generalized law (Eq. 4) for three defined phases – let us say
matrix, water, and hydrocarbon gas – and assuming that neither the matrix nor
the gas is conductive, i.e.

Accordingly, both equations provide a valid measure of the effective rock
conductivity, so they may be equated as

Now it is possible to write Eq. (6) in terms of connectednesses. The left-hand side of Eq. (6) is simply the connectedness of the pore space, as
defined by Eq. (1). It is the phase volume fraction of the pore space, i.e.
the classical porosity, raised to the power of the phase exponent that
contains the information about how that pore space is distributed, which is
the classical cementation exponent

However, there is nothing geometrically special about the entity we call the
pore space or any distinction between solid and fluid phases that compose the
whole rock. Consequently, Eq. (11) is only a partial generalization, and it
is possible to extend the result in Eq. (10) to any phase of

Equation (12) gives the connectedness of the

If one considers the whole 3-D

If a subset of a whole

The definition above is somewhat complex due to the requirement to be both
completely general and precise and due to the fact that there are two reference frames here.
The first is the whole 3-D

Sets and subsets of a three-phase medium using a 2-D slice to
represent the whole 3-D medium.

It should be noted that Eq. (14) is formally the same as Eq. (4) except that
Eq. (14) is valid for the reference subset of phases, while Eq. (4) is valid
for the whole

For a whole

The distinction between the phase exponent and saturation exponent becomes
trivial; they each control how connected the phase is relative to the
reference volume fraction. In other words, the transformation

The pore space may be occupied by any number of miscible or immiscible
fluids. Let us assume there are two immiscible fluids completely occupying the
pores, which are water and oil and which we will assign the names Phase 3
and Phase 4. Figure 2b shows this situation. Once again, the phase fraction
and connectedness of each of the three phases that compose the medium can be
defined as phase fractions

However, it is possible to use a different reference medium for calculations. For example, the classical Archie's second law is expressed in terms of
saturations and uses the pore space as a reference space in order to
express the amount of water and hydrocarbons not with respect to the total
volume of the rock but as a fraction of the pore space. Let us, therefore, also take the pore space as a convenient reference sub-space of the whole
medium. This situation is shown in Fig. 2c, where the dotted line delineated
the extent of the reference space. In this space, (i) what was the whole
medium, represented by unity in the transform given in Eq. (16), becomes the
volume fraction of the reference space

The transformation given in Eq. (16) is perhaps not immediately clear when
expressed in these most general terms. Let us take an illustrative example.
Imagine a three-dimensional five-phase medium where the phases are (i) detrital
quartz (dq), (ii) calcite cement (cc), (iii) distributed
clay (dc), (iv) saline water (sw), and (v) hydrocarbon gas
(hg), where the subscripts that will be used for each phase are
given in parentheses. First let us consider the whole medium (i.e.

Now consider the subset of the whole medium which comprises just its solid
parts. The reference fraction

There are two important aspects to note about Eq. (19). First, there are no terms for the saline water and hydrocarbon gas in the equation because these phases are not present in the reference subset. Second, the phase exponents that were used when considering the whole medium have been replaced by saturation exponents because we are now considering the distribution of each of the phases within the reference subset rather than within the whole medium. Third, both Eqs. (17) and (19) are simultaneously true and may be equated.

Equation (19) is clearly the same as Eq. (14). Under the transformation that
considers a subset of the whole medium (in this case the solid fractions
only) where

Both the phase (cementation) exponent and the saturation exponent control how the phase is connected. The phase exponent does this with reference to the whole rock, while the saturation exponent does it with reference to a subset of the whole rock. The underlying physical meaning of the saturation exponent is the same as that of the phase (cementation) exponent; it is only the reference frame that changes. The implication is that the general Archie's law replaces both of the classical Archie's laws. For an application to a sandstone gas reservoir, one would use a three-phase generalized Archie's law.

Equation (12) is easily transformed to provide a calculable value for the
saturation exponent by taking the logarithm of both sides of Eq. (12) and
rearranging the result before substituting Eq. (1) for the relevant
connectednesses and using the relationship

There is a reiterative symmetry in this transformation where both the whole-medium phase fractions and the reference subset saturations are both volume
fractions with respect to the whole medium and the reference subset,
respectively. Similarly, the phase exponents and the saturation exponents are
also defined with respect to the whole medium and the reference subset,
respectively. This would, therefore, allow the calculation of a reference
subset of a subset of a whole medium if required, and so on. There is of
course the possibility that the whole

This section provides a physical interpretation for the saturation exponent in a perfect analogy to that derived for the cementation exponent by Glover (2009).

The connectedness

We follow the approach of Glover (2009) in the analysis of the physical
interpretation of the cementation exponent. In this work Glover (2009) showed
that the cementation exponent was the differential of the connectedness with
respect to both porosity and pore connectivity. Following the same
methodology, differentiating the fractional connectedness with respect to the
phase saturation

Comparison of all the parameters in the classical and generalized Archie's laws.

The fractional connectedness is also the product of the saturation and the
connectivity with respect to the reference subset

For petrophysicists the reference subset has been the porosity, and there has
only been one conducting phase that partially saturates that porosity – the
pore water. Now we are not restricted to that model. The reference subset
could be, for example, the solid matrix, in which a number of separate mineral
phases can be defined, one of which might be, say, a target ore or a clay
phase. Let us take a four-phase medium as an example. Imagine a four-phase medium
composed of 65 % quartz matrix with a phase volume exponent of 0.3 and 15 % clay. Consequently, the medium's porosity is

The main conceptual steps in this paper are summarized as follows:

The classical Archie's saturation exponent arises naturally from the generalized Archie's law.

The saturation exponent of any given phase can be thought of as formally the
same as the phase (i.e. cementation) exponent, but with respect to a
reference subset of phases in a larger

The connectedness of each of the phases occupying a reference subset of an

The sum of the connectednesses of a 3-D

Connectedness is conserved in a 3-D

The sum of the fractional connectednesses (saturations) of an

Fractional connectedness is conserved in a 3-D

The saturation exponent may be calculated using the relationship

The connectivity of any phase with respect to the reference subset is given
by

The connectedness of a phase with respect to a reference subset (also called
the fractional connectedness) is given by

The rate of change of fractional connectedness with saturation

Hence, the saturation exponent is interpreted as being the rate of change of
the fractional connectedness with saturation and connectivity within the
reference subset,

This work is entirely theoretical and contains no data or supplements.

The author declares that he has no conflict of interest.

The author would like to thank Harald Milsch, Graham Heinson, and one anonymous reviewer for their detailed reading and constructive comments on the initial submission of this paper. Edited by: Charlotte Krawczyk Reviewed by: Graham Heinson, Harald Milsch, and one anonymous referee