OBSERVATIONS AND CONCLUSIONS FROM FIGURE 10 ABOUT EXPONENT n

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 S_wt 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 R_t . 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. It can be seen in Figure 10, except at very high water saturations, that the lower the value of n, the lower is the cumulative interference by oil or gas, and the higher is the oil or gas saturation for the prevailing resistivity.

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 value of R_t is known for that saturation, and employed.

The curves are applicable only where a constant value of R_t remains valid. The curves are not valid where water saturations are high remains and exponent n is steep because a high value of R_t 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 R_t 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 R_t 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 locations 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 calculated oil saturation is predicted by the intersection of the extrapolated slope of m and the level of R_t 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 values of n along the steep slope of the curves in Figure 10, and the values of n lying below slope m in Figures 1 and 10, is mathematics. But, the values for n falling only to the limiting values of slope m, from whence they came, 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 S_wt observed between one bed and another at any constant saturation value.

As R_t 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 S_wt 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 R_t 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 at 100% water saturation (at 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, R_w first must be calculated from Equation(1b) for each iterated value of S_wt . Along with R_we , exponent n must be solved from triangle CDG in Figure 1 or from the dual-water Archie equation, Eq.(4b):

                 S_wtⁿ = (1.0 / ( ϕ_t^m1 ) ) ( R_we / R_{t measured} ) = ( R₀ / R_{t measured} ) . . . (4b)

where from triangle CDG of Figure 1,

n = ( logR_t - logR₀ ) / ( log1 - logS_wt ) = ( logR₀ - logR_t ) / ( logS_wt )

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

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

Solve for n by iterating S_wt 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 / R_{t 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 / R_{t measured} might be 10/1,000 or 10/10,000. It can be seen in Equation(4b) that as R_t inceases (or R_we decreases), exponent n increases. The ratio R₀ / R_t in this order of magnitude moves the type curve shapes upward in Figure 10 because exponent n increases everywhere. As R_t 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? In Figure 10, like the values of n found below the value of exponent m, the extreme values in the steep upward sweep of the curves are imaginary. They can be calculated, as reported above and seen in Figure 10. They actually would exist if the high values of R_t actually existed at the high water saturations, but they do not. At high values of S_wt the bulk volume of water is electrically more conductive, and high values of R_t used in the calculations are not co-existent, and do not exist in real life. Hence, the upward sweep in n at these saturations is artificial. The n 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 R_t until the water saturation becomes 100%, n becomes m, and R_t 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 R_t . As R_t increases, the ratio R₀ / R_t (in Eq.(4b)) decreases and exponent n increases. As n increases, the apex becomes narrower and sharper from which n declines approaching and ultimately returning 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 R_t 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 R_we 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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