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:**

**HOW IS EXPONENT n RELATED TO EXPONENT m**

The saturation exponent *n* is the most misunderstood parameter in formation resistivity interpretation. But, its purpose in electrical resistivity interrpretation is quite straightforward, as is the purpose of exponent *m*. The saturation exponent *n* is related to both pore geometry and the interference to electrical current flow within the complex water-filled paths remaining in the pores *after displacement by hydrocarbon has taken place*. In Figure 1 and Eq. (3b) it is shown that *n* is a geometrical element similar to *m*. In **Figure 1**, it can be seen that *slope n is what slope m becomes* after hydrocarbon has migrated into the pores and has displaced a fraction of the water.

Historically, and in the absence of better information, the usual default value for *n* in *water-wet- *and oil-wet rocks has been the same as for *m* ( *n* = *m* ). Also see discussion in **APPENDIX (B) (4)**. It has been cited in petrophysics literature that *n* often is < *m*. However, the presence of insulating oil at any saturation displaces some water volume and produces some electrical interference and, therefore, the value of *n* must be greater than *m* and must decrease the electrical effectiveness of the remaining bulk volume of water. The presence of oil cannot increase the electrical conductivity of water to a value more than 100%, and can neither increase the volume of electrically conductive water paths nor increase the effectiveness of the electrically effective water paths more than actually exists. Much to the contrary. Because of the increased electrical interference by the presence of hydrocarbon, the result will be to increase the minimum value of *n* to some value higher than the value of *m*, all other things remaining the same. Because *n* is what *m*_{1} has become after hydrocarbon has displaced a fraction of water, exponent *n* cannot have a lower value than the actual value of *m*_{1}. A detailed explanation appears in **APPENDIX (D)**, based on **Figure 6**, why the actual value of *n* cannot be lower than the actual value of *m*_{1} in the same sample, in situ or in the laboratory. Exponent *n* can and will increase over *m*_{1} in the presence of oil or gas as the presence of hydrocarbon decreases the volume of electrically conductive water and changes the dimensions of electrical paths, or otherwise impedes the flow of the electrical-survey current. Additionally, there can be multiple values for both *m* and *n* as oil saturation changes and/or wettability to oil changes. For oil-wet and partially-oil-wet rocks this effect can be quite significant. When oil is present, in partially oil-wet and oil-wet rocks, for any given water saturation the saturation exponent *n* can vary from as low as *m* to as high as 8.0 or 9.0 or more depending on the degree of, and effectiveness of, wettability to oil, physical distributions of oil and water, oil properties, and rock-framework surface properties and characteristics. In **Figure 1**, it can be seen that at any constant value of *R*_{t} , if the redistribution of a constant fraction of oil causes the electrical interference to change, then the slope representing exponent *n* ( and *m*_{2}* *) will rotate to a different intercept H with *R*_{t }, and this will result in a corresponding change in the value of *S*_{wt }.

The exaggerated influence due to the presence of oil will increase both the usual exponent *n* for *S*_{wt} and the combination exponent *m*_{2} for bulk volume water *S*_{wt}*ϕ*_{t} . For any given porosity and any given oil saturation, the exponents *m*_{2} and *n* will increase with those properties of the rock and pore walls that when covered with adhesive oil films increase the interference to the flow of electrical current through the conductive paths. These factors increase in severity with the increase in wettability to oil, finer grained sandstone (increased surface area), increased efficiency of packing, increased number of grain-to-grain contacts, finer pores and pore throats, properties of oil (increase in viscosity of oil), interfacial tension between oilfield brine and crude oil, isolation of pores, and the physical saturation distributions of both the wetting- and nonwetting- phases whether oil or water. All these influences act in concert at their respective levels of severity to cause or alter interference to electrical current flow.

In oil-wet and partially oil-wet rock, the effects of these factors become magnified and the electrical interference within the pore paths is increased. As a result, the saturation exponents *n *and* m*_{2}* * increase. In oil-wet rock, it might be thought that the higher the value of these exponents, the higher will be the oil saturation. To emphasize, this is not the case. Figure 1 shows that the saturation exponent *n* represented by slope CG of triangle CDG is the resisitivity gradient employed between *R*_{0 corrected} and *R*_{t} relative to changes in the saturation of oil (or water). Similarly, exponent *m*_{2} is the gradient between *R*_{we} and *R*_{t} relative to changes in bulk volume water, *S*_{wt}*ϕ*_{t} * .*

The effectiveness of the cumulative interference is related to the existing water saturation. Observe the curves in **Figure 10**. It can be seen that interference is more effective at higher water saturations.

The maximum value for *n* in any specific water-wet rock is that value where the presence of oil or gas has the greatest effectiveness in producing electrical interference. This occurs at high water saturations. Interference and how effective that interference can be are not the same thing. The effectiveness of the cumulative interference to increase resistivity is greatest at higher water saturations, and is least effective to increase resistivity at low water saturations. Read that carefully. Again, study the effectiveness curves in **Figure 10**. The greatest electrical interference occurs in rock with the greatest wettability to oil and depends on the distribution of the oil and its viscosity. The minimum value for *n* occurs where the interference caused by the presence of hydrocarbon exhibits the least effectiveness in producing electrical interference. The greater the value of *n* or *m*_{2} for any given value of resistivity, the greater will be the calculated water saturation. But, these statements should not imply that there is a strong relationship between the value of *n* or *m*_{2} and the value of either water or oil saturation. There is none. That is because of the many different factors that can influence the presence and distribution of oil and its properties, and thereby influence *R*_{t }. There is no mathematical relationship between the value of exponents *n* or *m*_{2} and the value of oil saturation, except when within the same bed with uniform electrically effective constituents. To the contrary, it will be seen later in this paper, and can be seen in Figure 1, that if all other things remain constant, as exponent *n* increases along angle Ƴ (or **δ **) or *m*_{2} increases along angle β, water saturation increases with the possibility of increased water cut; and/or, water production takes place without significant oil. In the model, it can be seen that when *R*_{t} is constant, if *n* or *m*_{2} increases, the slope representing *n* or *m*_{2} becomes more vertical *(n* might increase to 8 or 9) and the downward projection of slope *n* is toward increased water saturation, *S*_{wt} , and the downward projection of *m*_{2}* _{ }* is to an increased bulk volume of water,

That having been said, it must be understood that the two exponents *n* and *m _{2}* not only represent gradients, but represent efficiency or effectiveness. And, effectiveness of oil to interfere with the flow of electrical survey current is greatest at high water saturations where oil saturation is the lowest. This is corroborated by the slope CG representing exponent

These features relative to the presence of oil, and sometimes gas, must be recognized. Is there any exception? Theoretically, it might be possible to hypothecate a condition whereby the value of *n* could have a value lower than *m*, but it is not likely. In literature, the value of exponent *n* with a value lower than *m* is commonplace and overwhelmingly is accepted as conventional wisdom; but, it violates physics and it has never been explained in the same literature how a valid *n* < *m* can occur. In the laboratory *n* < *m* is a common occurrence due to sample damage or degradation. In situ, *n* < *m* would be a violation of physics.

See discussion under **APPENDIX (D)**. For a comprehensive treatment of *n*, please study the explanations under **OBSERVATIONS AND CONCLUSIONS FROM FIGURE 10 ABOUT EXPONENT *** n*. The discussions following the Observations and Conclusions absolutely corroborate triangle CDG of the model in

Gas usually does not have the same exaggerated effect on *n* unless the reservoir has been filled with oil at some former time in geologic history and an adhesive film of remnant oil precedes the occupation by gas. The resistivity of a gas-bearing zone can increase, however, due to the decrease in irreducible water saturation. This, too, can be demonstrated in **Figure 1**. The primary exaggeration in *n* is with partially oil-wet and oil-wet rocks that are filled with oil or have been filled with oil at a former time whether as a reservoir or as a migration path.

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

**A MODEL AND DISCUSSION**

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