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:


Archie’s classic relationships usually are considered to be clean-sand relationships. In shaly sands where clay shales produce additional electrical conductivity, Archie’s relationships are said not to apply. Relative to Archie’s original concept and its popular use, this pronouncement is correct. In the concept born of this model, an overall version of Archie’s relationships emerges that shows that Archie’s concept is a dual-water dual-porosity concept that also applies to shaly sands and other heterogeneous rocks that exhibit uniformity within the depth of investigation of the logging tools. This model is used to derive and define each of the terms in Archie’s relationships, and the parameters derived and defined apply tany other methodology that makes use of Archie’s parameters.

Figure 1 is an illustration of the fundamental resistivity model serving as the basis of this concept. This model first was introduced in Ransom (1974), again in Ransom (1995), and will be shown in greatest detail herein . This figure is right-facing in left-to-right format. A Figure 1 in reversed format, right-to-left, is furnished for the convenience of readers who are more familiar with left-facing diagrams. Figure 1 is not intended to be a working graphical procedure, it is extremely informative and explanatory at a basal level. This figure illustrates how bulk-volume water (Swtϕt ) is related to a unit volume of rock with resistivity (Rt ). The model in Figure 1 is illustrative in nature and is not drawn to scale. The line drawing in the X - axis has been expanded so that detail can be observed and discussed. The figure is designed primarily to illustrate the electrical behavior of a volume of formation water as its environment changes with variations of insulating rock and insulating fluid.

In Figure 1, the origin of the diagram is represented by Rwe (and Ft = 1.0) when ϕt is 1.0 or 100%. Where it commonly has been believed there are only two slopes in Archie’s concept, the diagram demonstrates that there actually are three slopes, each representing exponents in Archie's concept. They are m1, m2 , and n. The first is m1 that pertains to the single parameter ϕt . This is the familiar porosity exponent m known in industry. Sometimes for the sake of clarity the term m1 will be used instead of m. The second exponent is m2 which pertains to two parameters, ϕt and Swt , and is the exponent for the product Swtϕt . The third is the exponent n that pertains only to Swt , and is the saturation exponent commonly known in industry. Each of these exponents pertains to a resistivity gradient, the rate that resistivity changes as the volume of water in rock changes. The minimum resistivity gradient, or minimum value, for the saturation exponent n is the value of the porosity exponent m. A value for n that is lower than the value of m often is used by petrophysicists in their literature. This is contrary to physics. For the value of n to be lower than the value of m, the displacement of water by hydrocarbon must make the remaining water more electrically conductive, a violation of physics. It has never been explained in literature how the displacement of electrically-conductive water by hydrocarbon, under either in situ conditions or laboratory conditions, can increase the electrical conductivity of the remaining formation water.

From the diagram, the total fractional volume of water in the rock is represented by the projection on the X-axis under the two slopes, representing m and n (or m2), drawn from 100% water at Rwe to the intersection of the line representing specific slopes extrapolated to intercept resistivity level Rt . The fraction of water in total pore volume, Swt , and the fraction of water in the total rock volume, Swtϕt , are depicted on the X-axis by logϕt and logSwt . The fraction of total rock volume that is water provides the conductivity to the rock resulting in resistivity Rt . As oil or gas displaces water, the water saturation, Swt , decreases to the right (in the right-facing Figure 1) as the saturation of oil or gas (1.0 - Swt ) increases.

It is illustrated in the model in Figure 1 that there can be conditions related to the presence of hydrocarbon, oil in particular, that cause in situ rock resistivities to increase to extraordinarily high values. In oil-wet rocks the values of n will increase greatly (Keller, 1953; Sweeney and Jennings, 1960). The presence of oil in both water-wet and oil-wet rocks produces an increase in rock resistivity, but, in oil-wet rocks, the presence of oil causes very exaggerated interference to the flow of electrical-survey current. Resistivity Rt then will be increased correspondingly with the wettability to oil and its resulting electrical interference. The right-facing Figure 1 also shows that, under these conditions, when the commonly used default values of n = m are employed, the line representing the slope of exponent n will be extended far to the right to intersect the level of the measured or derived value of Rt at a location H that would suggest a low value of Swt . The lowest water saturations and the highest corresponding values of oil saturation, that can be calculated for the input data, occurs at point H. Values for saturation exponent n that are lower than the values of porosity exponent m are commonly seen in literature and are perpetuated by conventional wisdom. However, the employment of such values for n would cause the slope or resistivity gradient to decrease and the extended slope to intersect the Rt level beyond point H at artificially low water saturations. The saturation range for reasonable calculated water saturations is between irreducible water saturation and irreducible hydrocarbon saturations. How reasonable the estimated water saturation will be depends on the validity of exponent n or exponent m2 .

This illustrates the resistivity interpretation problem in oil- bearing rocks where resistivity is exaggerated by the properties of oil. Here, to repeat, when the default value of n is lower than m, the extrapolated slope for n intersects the resistivity level Rt far to the right beyond point H in the model in the right-facing Figure 1, suggesting that water saturation is very low. When the common default value for n or any other unusually low value for n is employed, the derived saturations from the basic Archie equation, or from any more comprehensive equation, might not be correct and should be used with extreme caution in any field or reservoir description. This is a problematic but common occurrence when using the usual Archie-based resistivity interpretation methods by both private and commercial organizations. In the presence of oil, particularly in oil-wet rock where resistivity is high, the value of exponent n always is greater than the value of either exponent m1 or m2.

An exploration of the graphic model will promote a better understanding of resistivity behavior in rocks. This exploration will be accompanied by parallel algebraic development of the terms and parameters in the basic Archie relationships. What each term or parameter represents is the key to understanding where to begin to solve interpretation problems.

This exploration will be concluded with a discussion of three unusual, but vintage, resistivity well logs, with special attention devoted to exponent n. Not only will exponent n be shown to be a resistivity gradient, but also is shown to be a measure of effectiveness of the electrical resistivity interference caused by the presence, distribution, and wettability to oil at high and low water saturation levels.



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