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:


Examine the curves drawn in Figure 10.

All calculations of exponent n shown in Figure 10 are based on triangle CDG of the model in Figure 1. Triangle CDG is the graphic depiction of the dual-water Archie two-exponent water saturation Eq. (4b).

The values of the calculated n from the logic of triangle CDG in the model and from Eq.(4b) are summarized in the type curves drawn in Figure 10.

How to Interpret Figure 10. Caution. Do not think of the curves in Figure 10 as profiles.

Each type curve represents a relationship between exponent n and saturation Swt in a specific reservoir bed or same sample of rock at a deep radial depth beyond the invaded zone.

Imagine each of the curves as a series of smoothed bar graphs for exponent value of n over a full range of water saturations at a constant value of Rt . At any specific water saturation, the n value at that water saturation represents the degree of, or the effectiveness of, the interference by oil to the flow of electrical survey current in oil-wet- or water-wet rocks.

Not all of the chart is applicable. The values of exponent n are meaningful only at oil saturations greater than irreducible oil saturation. The values of n can be calculated in the region of irreducible oil and in the flushed zone only if the appropriate Rt is known and employed.

The curves are applicable only where a constant value of Rt remains valid. The curves are not valid where water saturations are high remains and exponent n is steep because a high value of Rt cannot be supported at high water saturations. At high values of water saturation, exponent n will begin a decline and continue to decline as water saturations increase until it reaches the value of exponent m . In Figure 1, resistivity Rt also will decline along with the decline of n until it reaches the value of R0 .

As a result, the steep upward mathematical sweep of n values probably does not exist in nature. The high values of n can be calculated, but at high values of water saturation Rt must ultimately return to R0 , thus n will return to m.

One of the first things to be observed in Figure 10 is that in both Zones A and B in Wells 1 and 2, as the resistivity gradient described by exponent n increases, predicted water saturation increases and the predicted oil saturation decreases. And, where the oil saturation is the greatest, exponent n is the lowest. Puzzling? This is in precise conformity with the prediction by n illustrated by slope CG of triangle CDG in Figure 1.

For example:

1.  If the voids in the rock are filled to a high oil saturation, an increase of 10% oil will decrease the volume of electrically conductive water correspondingly.

2.  If the voids in the rock are filled to a low oil saturation, an increase of 10% oil will displace the same bulk volume of water as at high saturations.

3.  Wherever the oil phase is continuous, whether at low saturations or high saturations, the electrical interference caused by adhesive insulating oil films on the pore surfaces will be nearly the same.

4.  Although the electrical interference by oil in items 1 through 3, immediately above, is the same wherever the oil exists in a continuous phase, Figure 10 and the model in Figure 1 illustrate that the effectiveness is not the same.

5.  The effectiveness of the interference by oil is greater at higher water saturations than at lower water saturations as seen by the increased values of n. This phenomenon is corroborated by the model in Figure 1, Equation (4b), and the type curves in Figure 10.

In Figure 10 what do the curve crossovers mean at point H? Figure 10 illustrates that everywhere oil is present, the true value of exponent n is greater than exponent m, but some calculated values of n are lower than m. Why?

The calculated oil saturation is highest where exponent n approaches exponent m. In both Figure 1 and Figure 10 that maximum calculated oil saturation level is seen at point H. In Zones A in Wells 1 and 2, that maximum occurs at about 90% oil saturation (disregarding irreducible water saturation) where the curves for Zones A meet the assumed value of 1.6 for exponent m.

In Figure 10, because the n curves for Well 1 Zone A and Well 2 Zone A, intercept the value of m at high oil saturations, it would imply that there is more oil to be produced from this reservoir. For Zone B, the highest oil saturation that can be calculated is about 50% where the curves for zones B also meet the value of exponent m at a point H. This behavior is illustrated in Figure 1 where line CG (slope n) of triangle CDG collapses to line CH that represents slope m. Any calculated value of n lower than the value of m is imaginary.

What about the imaginary values of n in Figure 10? The oil saturation where the calculated n crosses the value of exponent m at location H represents the highest oil saturation that can be calculated for that data and that well depth. It does not imply that the specific saturation at H is the predicted saturation for water or oil. Examine the model in Figure 1. Remember: Exponent m becomes exponent n after some of the interstitial water has been displaced by oil. So, in Figure 1 when gradient CG collapses along angle δ to lie on gradient m (now n = m) it cannot collapse any further. The maximum oil saturation is predicted by the intersection of the extrapolated slope of m and the level of Rt wherever intersection H resides. This is the same H as appears on the effectiveness chart in Figure 10. The calculated values of n lying below the value of m do not exist. The imaginary values absolutely can be calculated, but cannot be reproduced in the laboratory or in nature. Such values published in petrophysics literature are disputable. The model exhibited in Figure 1 representing the dual-porosity Archie Equation, Eq.(4b), is mathematics, but the slope CG collapsing to the limiting value of the extended slope m is physics.

Exponent n is higher in Zone A than in Zone B at the same water saturation in both wells. It can be seen in Figure 10 there is no mathematical relationship between exponent n and Swt observed between one bed and another at any constant saturation value.

As Rt changes, the slope representing values for n might or might not change. Exponent n can vary independently of water saturation. For the value of n to increase above that of m, it is necessary only for oil to be present. Water saturation influences n, but, except for coincidences, there is no curvilinear relationship between Swt and n except within the same bed, and the bed exhibits uniform distributions of oil, water, porosity, and other factors influencing electrical conductivity, as the individual curves in Figure 10 illustrate.

What if the imaginary values are used in interpretation? Slope m in Figure 1 is a resistivity gradient representing the value of R0 at 100% electrical efficiency in the selected rock at all porosities. If slope n were to lie below slope m , resistivity Rt would lie below the extrapolated values of R0 , and oil saturations would be artificially overestimated over all porosities and saturations. This is a commonplace occurrence in literature that has become a factoid of conventional wisdom. The presence of oil cannot increase the electrical conductivity of water more than 100% efficiency (m). See Figure 6 and APPENDIX (D).

Figure 9. Description of Well 3. This well log, too, is a vintage resistivity log. This log was chosen because of its simplicity. In this figure is a resistivity log in a water-wet rock. Like the two examples above, the resistivity curves show great contrast between an oil-bearing zone and a water-bearing zone. This example shows a dual induction log with an SP curve. At the top of the bed between X151 and X161 is an oil-bearing interval marked Zone A. Near the bottom of the bed is a water-bearing interval between X183 and X192 marked Zone B.

The calculated values of n relative to water saturations for this bed also are shown as a curve in Figure 10.

A moderately high value of saturation exponent n up to perhaps 8 or 9 might not be a factor in distinguishing between water-wet and oil-wet rocks.

As it can be seen in Figure 10, as water saturation in Well 3 increases, the calculated curve for n will increase again the same as the curves for Wells 1 and 2. This, again, is corroborated by the slope CG representing exponent of triangle CDG in the model in Figure 1.

Examine the effectiveness curve from Zone A of Well 3 in Figure 10. This curve is plotted from values of exponent n in water-wet rock. It can be seen that the curve is very similar to the curves from oil-wet rock. Why is this so?

It is a simple matter for a computer to solve the necessary equations and iterations required to compute the results appearing in Figure 10. Let’s examine how exponent n is calculated:

In the calculation of each exponent n, Rw first must be calculated from Equation(1b) for each iterated value of Swt . Along with Rwe , exponent n must be solved from triangle CDG in Figure 1 or from the dual-water Archie equation, Eq.(4b):

                 Swtn = (1.0 / ( ϕtm1 ) ) ( Rwe / Rt measured ) = ( R0 / Rt measured ) . . . (4b)

where from triangle CDG of Figure 1,

n = ( logRt - logR0 ) / ( log1 - logSwt ) = ( logR0 - logRt ) / ( logSwt )

It can be seen in Eq.(4b) above, that the ratio ( R0 / Rt measured ) varies little for each Zone A in the Wells 1, 2, and 3. Therefore, the equation for each curve in Figure 10 is:

Swtn = A near constant for each depth of interest depending on the value of the ratio.

Solve for n by iterating Swt in Eqs.(1b) and (4b).

Can oil-wet zones be distinguished from water-wet zones by high values of n? Sometimes and sometimes not. In Well 1, Zone A, an oil-wet zone, the ratio of ( R0 / Rt measured ) is approximately 10/100 = 1/10 or even smaller. In Well 2, Zone A, also an oil-wet zone, the ratio again is approximately 10/100 = 1/10 or thereabout (smaller if R0 is taken directly from Zone C). And in Well 3, Zone A, a water-wet zone, the ratio is about 1/10. So, it doesn’t matter whether the zone is oil wet or water wet the calculated values of n in these cases can be nearly the same for the same water saturations. When Equation(4b) is solved for exponent n it can be seen in these ratios that there is little difference in Equation (4b) whether the rock is oil wet or water wet. Although, at high n values, the rock probably is oil wet, but it is not always the case.

In some oil-wet rocks, the ratio might be very small. The ratio of R0 / Rt measured might be 10/1,000 or 10/10,000. It can be seen in Equation(4b) that as Rt inceases (or Rwe decreases), exponent n increases. The ratio R0 / Rt in this order of magnitude moves the type curve shapes upward in Figure 10 because exponent n increases everywhere. As Rt increases, point H on both Figure 1 and Figure 10 will be shifted to the right toward higher calculated maximum oil saturations.

The values for exponent n are viable only at higher oil saturations than irreducible oil saturations. This must be taken into consideration. The calculated bulk volume of oil consists of producible oil and irreducible oil; and the calculated bulk volume of water consists of water that can be produced, irreducible water, and water, or pseudo water, contained in dynamically non-effective rock constituents.

What about the steep upward sweep of n at high water saturations? Like the values of n found below the value of exponent m, they too are imaginary. They can be calculated, as reported above and seen in Figure 10. They actually would exist if the specific value of Rt actually existed at the high water saturations, but it does not. At high values of Swt the high values of Rt used in the calculations not co-existent, and do not exist in real life. Hence, the upward sweep in n at these saturations is artificial. The values are imaginary.

The actual values of n must gradually return to the value of m at high water saturations so that at 100% water saturation and 0% oil saturation exponent n becomes m again. From some location on the effectiveness curve, as water saturation increases, the value of n will increase to a maximum whereupon, as water saturation continues to increase, the value of n declines followed by a decline of Rt until the water saturation becomes 100%, n becomes m, and Rt becomes R0 .

The dashed curve in Figure 10 is one of a number of artful conceptions that exponent n might take as oil migrates into or out of water-wet- or oil-wet rock, simultaneously changing Rt . As Rt increases, the ratio R0 / Rt (in Eq.(4b)) decreases and exponent n increases. As n increases, the apex becomes narrower and sharper from which n declines to approach and ultimately return to exponent m at 100% water saturation.

In Figure 10, the effectiveness curves under these conditions might resemble normal frequency curves skewed heavily toward high water saturations, and slope CG in triangle CDG in Figure 1 might correspondingly morph into a shape similar to a lazy S as high water saturations approach 100%.

Both the model in Figure 1 and the curves in Figure 10 would be modified as Rt changes to correspond with the reasoning set forth above.

What about exponent m2? This is a single saturation exponent equivalent to the two exponents m and n for saturation calculations. It is derived from triangle AEG in the model in Figure 1. If the effectiveness curve for m2 were shown in the chart in Figure 10, the curve would lie between the constant exponent value of m and the curves for n. A close inspection of Figure 1 will show that the slope representing m2 is greater than that of slope m but smaller than that of slope n. It also can be seen that triangle AEG has its origin at Rwe instead of R0 , triangle AEG is more tolerant of errors. Because of the greater length between AG over CG, slope AG will change less than slope DG for the same error in porosity or saturation.

Everything that has been said about exponent n applies to exponent m2 with the exception that the effects on or by m2 will be smaller. On both Figure 1 and Figure 10, where the slope of exponent n meets the slope of m at point H, so does the slope of exponent m2 meet slope m at point H.



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