Robert C. Ransom



What are Archie’s Basic Relationships

The Graphical Model

What is Meant by the Plot of Rt versus Swtϕt

Summary of Equations

Parallel Resistivity Equations Used in Resistivity Interpretation

What is the Formation Resistivity Factor

The m Exponents

How is Exponent n Related to Exponent m

The a Coefficient

The Saturation Evaluation

Challenging Well-Log Examples

Observations and Conclusions from Figure 10 about Exponent n

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




Symbols Defined



All Figures

About the Author

Table of Retrievable Contents:


Historically, the a coefficient always has appeared in the numerator of the modified Formation Resistivity Factor, and has been perpetuated in industry with no defined purpose. In this model there is no support for the appearance of an a with a constant value in the formation factor. The formation factor is intrinsic to the rock at 100% water saturation. The a coefficient is not an intrinsic property, but is dependent on variables not related to the electrically inert rock or the physical geometries of its pores. The a is related neither to volume nor shape imparted to the network of electrically-conductive water paths. The a coefficient contributes nothing to the efficiency of the conversion of water resistivity to resistance, or vice versa. In the model, the shift in resistivity corresponding to a always is found in the resistivity axis, and the shift varies from data to data with depth. Coefficient a emerges as an inherent and inseparable factor of the complex resistivity relationship, as in Eqs.(1a) and (1b), and is related to the proportions of all secondary electrically-conductive constituents and influences that vary the relative proportions of Rw and Rwb .

It can be seen in the graphics of Figure 2 that Rwe is the product of a and Rw , therefore aRw= Rwe .   Rwe varies from depth to depth, and so does coefficient a. It can be seen in this relationship that a varying a coefficient technically can be used in single-water single-porosity methods where Rwe is not calculated. But, in dual-water dual-porosity methods where Rwe is calculated, the a coefficient cannot be used as an independent parameter. If the a is used in single-water single-porosity methods, it must be calculated properly. And that is difficult to do when the correct value of a varies with rock constituents and depth, and only one water and one porosity is available to work with. That is one of the reasons why dual-water dual-porosity methods were conceived.

When a is associated with Rw , as in the equation developments of (1a) and (1b), and as a multiplier or reduction factor for Rw , the a coefficient more readily can be recognized as being an indispensable factor that accounts for those heterogeneities that produce additional electrical conductivity. Coefficient a is the composite factor that relates Rw to the resistivity value of the combination of Rw and Rwb , and all other natural electrically-conductive influences, in such proportions to result in Rwe .

In the modified formation factor used in single-water single-porosity methods, the a coefficient appeared in the numerator. In the modified formation factor the a became a multiplier of the Rw in Archie’s saturation equation. The aRw became a single-water single-porosity equivalent to Rwe .

Secondary conductive influences, inherent to the rock, that produce coefficient a can be: clay shale, surface conductance, or solid semi- conductors such as pyrite and siderite as separate influences or in concert. One influence that is neither electrically conductive nor intrinsic is hydrocarbon saturation. These in turn, and perhaps others, can produce a change in rock resistivity.

The occurrence of one additional electrically-conductive influence would add one more term to Eqs.(1a) and (1b), or their equivalent, and if there were no change in Swtϕt , the value of the compound coefficient a would be reduced. These influences, all, can change Rw to an apparent or pseudo value Rwe with a corresponding change in R0 to R0  corrected at their respective porosities.

Technically, in dual-water dual-porosity methods, the coefficient a should be transposed from the modified Formation Resistivity Factor term to the water resistivity relationship. For example:

R0  corrected = Ft Rwe = ( a / ( ϕtm ) ) Rw = ( 1.0 / ( ϕtm ) ) aRw = ( 1.0 / ( ϕtm ) ) Rwe

The question might be asked, "What is the difference?" There are a number of reasons, four of which are:

1.  The a coefficient is a required proportionality factor, related to all factors influencing the electrical conductivity of the rock, that convert resistivity Rw to Rwe , and is calculated along with Rwe at each depth increment.

2.  To prevent duplication by the user in the correction for conductive influences. Duplication occurs when both a and Rwe appear in the same water-saturation equation, or when a appears as an independent variable in dual-water dual-porosity methods. Any saturation relationship involving resistivity can employ either coefficient a or Rwe , but not both.

3.  To prevent the use of a constant artificial and unwarranted correction factor.

4.  Although the formation factor is meaningless at 100% porosity, the value of Ft always must be 1.0 when both porosity and water saturation in the Ft equation are 1.0.

As explained above, and in Figure 2,

                                                                           a = Rwe / Rw                                                       (1c)

After substituting Eq.(1c) into Eq.(1b), a relationship is derived showing the proportionality terms in coefficient a for the usual shaly sand:

1 / a = ( Sweϕe ) / ( Swtϕt ) + ( ϕne / ( Swtϕt ) ) ( Rw / Rwb )

It can be seen here that coefficient a is a function of water saturation as well as porosities. See APPENDIX (A) for further explanation. This is the evaluation for the a coefficient appearing in Figure 2.

This relationship is shown for comparison purposes or informative purposes only. It is not to be calculated and used independently in dual-water dual-porosity methods because it already has been incorporated in the calculation of Rwe as can be seen in Eq.(1b).



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