Robert C. Ransom

What are Archie’s Basic Relationships

What is Meant by the Plot of *R*_{t} versus *S*_{wt}*ϕ*_{t}

Parallel Resistivity Equations Used in Resistivity Interpretation

What is the Formation Resistivity Factor

How is Exponent *n* Related to Exponent *m*

Observations and Conclusions from Figure 10 about Exponent *n*

Are There Limitations to Archie's Relationships Developed in this Model?

**Table of Retrievable Contents:**

**PARALLEL RESISTIVITY EQUATIONS USED IN RESISTIVITY INTERPRETATIONS**

The basic premise is that the host rock is an insulator. Equations will be demonstrated for the case of shaly sand with a uniform distribution of constituents and pore space. Shale is a term used to refer to fine grained, fissile, sedimentary rock. The term shale is a descriptive property of rock, not a mineral, therefore, shale may be referred to as "clay shale" when the predominant mineral constituent is clay. The shale in this presentation is not a source for oil or kerogen, but is shale or mudstone that has been compacted by overburden to the degree that any form of porosity has become noneffective, and migrating oil cannot or has not penetrated the void space.

The term "connate", con-nate, often misused and ill-defined in petrophysics literature and glossaries, is from the Latin meaning: together at birth. In petrophysics it means together at time of disposition. Connate water is water entrapped within the pores or spaces between the grains or particles of rock minerals, muds, and clays *at the time of their **deposition*. The water is derived from sea water, meteoric water, or ground surface water. Other investigators have shown that as clay shales and mudstones are compacted, a fresh water component of the original connate water is expelled and an ion-concentrated component remains in the voids. Therefore, water saturation in the voids remains 100% and *S*_{w } is 1.0 in *ϕ*_{ne }.

The relationship in Eq.(1a) below, presented in Ransom (1977), was developed to show that conductivity in water-filled voids in the host rock and the conductivity in clay shale can be represented by parallel conductivity relationships commonly used in basic physics. The relationship applies to heterogeneous rocks with a uniform distribution of minerals and porosity. In resistivity and conductivity relationships the dimensional units for a reservoir bed (*m*/*m*^{2}) must be made to be unity so that bed dimensions will not be a factor.

For shaly sand it was shown that

1 /*R*_{we} = (*ϕ*_{e }/*ϕ*_{t }) 1 / *R*_{w } + (*ϕ*_{ne }/*ϕ*_{t }) 1 / *R*_{wb } . . . ( 1a )

where *ϕ*_{t} = *ϕ*_{e } + *ϕ*_{ne }.

For 100% water-bearing rock the conductivity relationship in Eq.(1a) yields 1 / *R*_{w } (in clean rock) where *ϕ*_{ne}_{ } is 0.0, and yields 1 / *R*_{we } (in shaly sands) when conductive clay is present and *ϕ*_{ne}_{ } is greater than 0.0.

It should be noted that in this equation there are no limitations on the values of formation water *R*_{w} and bound water in shale, *R*_{wb }. However, nature does set limits. The bound water resistivity *R*_{wb } in the mudstone and clay shale is related to the original connate water. Waters entrapped at the time of deposition do vary, but usually do not vary greatly from shale to shale. Sea water has an average salinity of about 35,000 ppm, other surface waters probably are significantly lower. However, as indicated above, the ion concentration in clay-bound water can increase as fresh water is expelled from the clays with increasing depth and compaction.

Unlike connate water, interstitial-water resistivity *R*_{w } in the reservoir can vary over a wide range. Interstitial water might have undergone many changes through the dissolution and/or precipitation of minerals throughout geologic history. It can vary from supersaturated salt solutions to very fresh potable waters. Waters in the Salina dolomite (Silurian) in Michigan have specific gravities of 1.458 and a reported salinity of 642,798 ppm. In other formations, the waters vary to as little as a few hundred parts per million. Most often *R*_{w} will be greater than *R*_{wb }; and *R*_{we }, a mixture of *R*_{w} and *R*_{wb }, will be lower than *R*_{w }. But, it is quite common for *R*_{w} to have a lower value than *R*_{wb }, in which case *R*_{we } will have some value greater than *R*_{w }.

In water-filled dirty sands, Eq.(1a) applies. In clean sands, Eq.(1a) becomes

1 / *R*_{we} = *(**ϕ*_{e }/*ϕ*_{t }) 1 / *R*_{w } = 1 / *R*_{w}

and the corrected *R*_{0} in either event is

*R*_{0}_{ corrected} = *F*_{t }*R*_{we} . . . (2a)

where, from Figure 1,

*F*_{t} = 1.0 / ( *S*_{wt}*ϕ*_{t }) ^{m}2 = 1.0 / (*ϕ*_{t}^{m}1 ) . . . (3a)

For the value of *R*_{0}_{ corrected} the value of *S*_{wt}*ϕ*_{t} becomes *ϕ*_{t } because *S*_{wt}_{ }has become equal to 1.0. Therefore, the exponent *m*_{2} becomes *m*_{1} as described above and in Figure 1. In Eq.(3a), the *F*_{t} that formerly pertained to *S*_{wt}*ϕ*_{t} now pertains only to *ϕ*_{t }.

However, in the presence of oil or gas, water saturations are lower than 1.0 and the conductive water-filled volumes are reduced by the displacement of water volume by the hydrocarbon, and Eq.(1a) becomes

1 / *R*_{we} = ( ( (*S*_{w}_{e}*ϕ*_{e}) / (*S*_{wt}*ϕ*_{t }) ) 1 / *R*_{w} + (*ϕ*_{ne}_{ }/ (*S*_{wt}*ϕ*_{t }) ) 1 / *R*_{wb} . . . (1b)

and *R*_{t }_{ calculated} = *F*_{t }*R*_{we} . . . (2b)

where *R*_{we} is determined from Eq.(1b), and

*F*_{t} = 1.0 / ( *S*_{wt}*ϕ*_{t }) ^{m}2 . . . (3b)

Here *S*_{wt} is less than 1.0 and exponent *m* again becomes *m*_{2 }._{ } And, it will be shown in **Appendix (B) (3) **that

( *S*_{wt}*ϕ*_{t }) ^{m}2 = ( *S*_{wt} )^{n} ( *ϕ*_{t} )^{m}1 . . . (3c)

As it will be seen later that Eq.(3b) is the key in dual-water dual-porosity interpretations. Equation (3b) is a direct progression from the model in Figure 1 and will be further corroborated in the development from the common electrical resistance Eq.(3d), that will be developed later.

Now that the importance of *R*_{we} in the above equations has been established, just what is *R*_{we} and can waters represented by *R*_{w } and *R*_{wb} actually be combined as seen in Eq.(1b)?

Water with resistivity *R*_{w} is interstitial water. Water with resistivity *R*_{we} is not interstitial water and does not exist in nature at 100% water saturation, i.e. as *R*_{0}_{ corrected} . Resistivity *R*_{0}_{ corrected} is a hypothetical water mixture of dual waters, influenced by the presence of oil, that exists at 100% water saturation only in a dual-water concept. In electrical equations the dual waters can be mixed together, physically and in nature they cannot. A dual-water mixture with resistivity *R*_{we} cannot be recovered in a production test.

To continue, a very similar relationship to Eq.(2b), involving Eqs.(1b) and (3b), is seen in work published by Best et al (1980) and by Schlumberger (1989).

**[**An historical retrospection: Equations(1b)and (3b) result from early developments in resistivity well-log interpretation. Equation (1a) above was shown in Ransom (1977), and in the presence of hydrocarbon becomes Eq.(1b). The model in Figure 1 in this paper was first seen and described in Figure 2 on page 7 of Ransom (1974). From that model, *F*_{t} in Eq.(3b) above was first developed. Formation factor *F*_{t} of Eq.(3b) was shown as *F*_{3} in the explanation of Figure (2), page 7, of Ransom (1974). The equivalence relationship, ( *S*_{wt}*ϕ*_{t }) ^{m}2, is helpful for the use of Eq.(3b) in the determination of Archie’s dual-water saturation. Although the proof of the equivalence is found in **APPENDIX (B) (3)** in this paper, the equivalence was first seen as Equation (8) on page 7 of Ransom (1974). Archie's saturation equation of 1942 was first derived in Ransom (1974) on pages 7 and 8. Also, dual-porosity was suggested in Ransom (1974), where *ϕ*_{e }was described as hydrodynamically effective porosity and total porosity, *ϕ*_{t}_{ }, was described as electrically effective porosity. The difference between the two is hydrodynamically noneffective porosity, *ϕ*_{ne} . It was shown how to calculate all three porosities in Ransom (1977). Archie’s relationships of 1942 were derived in Figure 2 of Ransom (1974), again in Ransom (1995), and again, in considerably greater detail, in Figure 1 in this paper.

All equations referred to and all equations appearing in this paper support the fundamental certainty of the influence of the geometry of the bulk volume of water on the electrical resistance of rock. The bulk volume geometry concept was first introduced relative to resistivity analysis in Ransom (1974). This concept is best exemplified by the bulk volume of water term, *S*_{wt}*ϕ*_{t }, in Eqs.(1b) and (3b), and by water geometry represented by exponent *m*_{2 }, and therefore by both *m*_{1} and *n*, observed in **Figure ****1** and seen in Eq.(3b).**]**

In Eq.(3b) both the bulk volume of water, *S*_{wt}*ϕ*_{t }, and its electrical efficiency can be seen to change in the term ( *S*_{wt}*ϕ*_{t }) ^{m}2 as the saturation distribution of hydrocarbon changes the geometry of water. The basic Archie method is changed from a single-water single-porosity method to the dual-water dual-porosity methodology where the two waters have resistivities of *R*_{w } and pseudo *R*_{wb} . The former *R*_{w} of the single-water single-porosity method becomes *R*_{we} , as seen in Eqs.(1a) and(1b). The *R*_{wb} has been referred to as a pseudo *R*_{wb} because some of the conductivity attributed to the clay shale might not be in the form of water, but the usual resistivity-measuring logging tools do not know the difference.

In dual-porosity, dual-water methodology the numerator in Eq.(3b) always must be 1.0 because the compensation for all electrically conductive influences, along with formation water *R*_{w} ,_{ }should be relegated to terms in the *R*_{we} equation.

Equation (3b) applies to all cases where *S*_{wt} ≤ 1.0, and the term ( *S*_{wt}*ϕ*_{t }) in the denominator of the *F*_{t} relationship must be compatible with the same term in the denominator in the calculated *R*_{we}_{ } relationship in Eq.(1b) in whatever application it is used. The fractional volume of conductive water represented in the formation factor must always be the same fractional volume of water that exhibits the equivalent resistivity *R*_{we }. Also, note that both *S*_{wt } and *ϕ*_{t}_{ } have the same exponent *m* . In this case, the graphical model shows this m to be *m*_{2} because *m*_{2} is the combination exponent equivalent to the individual exponents *m*_{1} and *n*. It is illustrated in the trigonometry of Figure 1 that the same result would be realized if *S*_{wt} and *ϕ*_{t} were to use their individual exponents *n* and *m*_{1}, respectively, because calculated water saturation is related to the resistivity level of *R*_{t} and this remains unchanged. The algebraic proof that ( *S*_{wt}*ϕ*_{t }) ^{m}2 = ( *S*_{wt} )^{n} ( *ϕ*_{t} )^{m}1 is shown in **APPENDIX (B)(3)**.

In **Figure 2**, adapted from Ransom (1974), it is illustrated that the resistivity value of water-filled rock at any specific depth can be the result of a shift by the influence of a variable *a*, and the cause of that influence must be explained.

In these equations the accountability or compensation for the influence of clay shale in shaly sands resides in the proportionality terms involving the conductivity counterparts of *R*_{w} and *R*_{wb} observed in Eq.(1b). In this relationship the proportionality terms are functions of both porosity and saturation as well as clayiness. It further can be seen that for every value of clayiness or “shaliness” and resulting volume of bound water within *ϕ*_{ne}_{ }, the prevailing values of effective porosity and water saturation can vary the conductive water volume in effective pore space and, therefore, can vary the relative proportions of *R*_{w } and *R*_{wb} for the mixture *R*_{we }. And, this can and does vary depth by depth.

It is further illustrated in **Figure 2** that as *R*_{w } is corrected to *R*_{we }, *R*_{0}_{ } is corrected to *R*_{0}_{ corrected }, both by the same factor *a*. Just as *ϕ*_{e} and *ϕ*_{t} are two intrinsic properties of the rock, the *m* exponent describes an intrinsic property of the rock and produces the parallelism seen in the figure. As a result, the slope represented by m is independent of the conductivity of the waters in the pores. In this figure, *R*_{w} is related to *R*_{we} by the factor *a*, or coefficient *a*, that varies depth by depth. As a consequence, when these relationships for *R*_{we } are implemented, or their derivatives or equivalent relationships are used in any resistivity-based interpretation, it is important to recognize that the accountability or compensation by the a coefficient must be removed from the modified Formation Resistivity Factor. And, the numerator of the Formation Resistivity Factor must always be equal to 1.0 to prevent duplication in the accountability for the secondary conductivity provided by clay shale or other conductive constituents. This perspective will be discussed in detail below.

**A CLARIFYING CONCEPT OF ARCHIE'S RESISTIVITY RELATIONSHIPS AND PARAMETERS.**

**A MODEL AND DISCUSSION**

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