Narrow-spaced resistivity-measuring systems used in electric logging of wells respond mostly to the resistivities of the mud in the well bore and that of the mud filtrate invaded zone. After the necessary corrections are made for hole size, electrode spacing, and mud resistivity, one obtains the true resistivity of the invaded zone. The latter is a function of mud resistivity and of its characteristics as well as of the porosity of the formation and of other lithological factors such as the cementation and saturation exponents. These exponent values are reasonably characteristic of well defined reservoir rocks. If reasonable assumptions are made on the resistivity of connate water, the degree of its flushing by mud filtrate invasion, and on residual oil and/or gas, porosi y may be calculated at various levels in the well bore.

Limitations must necessarily be imposed on the approach since the presently accepted mathematical petrophysical expressions apply only to intergranular porosity and uniformly distributed fissures and vugs.

An application of the approach to the interpretation of the limestone sonde (LS 32^Prime) illustrates the method.

Narrow-spaced resistivity-measuring systems of electrodes used in electric logging of wells respond mostly to the resistivities of the mud in the well bore (^rgr_m) and to that of the mud filtrate invaded zone (^rgr_i). By known methods of reduction for hole size (d), electrode configuration, and mud resistivity it is possible to obtain the true resistivity of the invaded zone.

The electrode configurations which are under consideration in this study are designated under the following terminology.

Short normal        AM = 8^Prime to 18^Prime
Correlation curve   4iZ 18^Prime
Limestone log       LS 32^Prime

The determination of porosity from the foregoing resistivity curves is readily made, provided the validity of Archie's empirical relations is assumed and the basic lithological constants of the rocks are known, together with the resistivity of the connate water. Reasonable assumptions on the degree of flushing and mixing of mud filtrate and connate water must also be made.

In the case of the limestone sonde log, however, certain features of the resistivity curve are desirable from the point of view of porosity calculation. This study deals primarily with this special type of log.

The limestone device, generally LS 32^Prime, has a very short radius of investigation which is of the order of 10^Prime. Therefore, when located in front of porous formations which are invaded by mud filtrate of resistivity ^rgr_m (approximately equal

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to mud resistivity at that depth and temperature) the limestone logging device responds almost altogether to the resistivity of the invaded zone of resistivity ^rgr_i. This resistivity may be obtained to a first approximation by Archie's formula under the assumption of 100 per cent saturation in mud filtrate:

EQUATION (1)

where ^phgr is the porosity of the formation. The cementation exponent m = 2 is deemed applicable to most carbonate rocks (in which the limestone sonde is most commonly used) where the porosity is of the intergranular type. Expression 1 is still deemed applicable when the porosity is made up of uniformly distributed fissures and vugs. Accordingly, the porosity profile may be calculated from:

Fig. 1. Electrode configuration for 32^Prime limestone sonde.

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EQUATION (2)

provided ^rgr_m/^rgr_i may be obtained from the log.

Since measurements of formation resistivity are affected by mud resistivity, bore-hole size, and electrode spacing, only an apparent resistivity ^rgr_a is measured which must be corrected in order to obtain ^rgr_i. This is done by means of departure curves, some of the peculiarities of which are here explained for the limestone sonde.

LIMESTONE SONDE DEPARTURE CURVES

Consider Figure 1 where a current I is supplied at electrode A and let the potential difference ^DgrV created across MN be measured. It is obtained by,

EQUATION (3)

where ^rgr_a is the combined resistivity read by the limestone device in the vicinity of the electrode system; r₁ and r₂ are the distances from the current electrodes to the pick-up electrodes.

Let us study the limiting case where ^rgr_i=^infinity or where the measurements are made facing a dense non-porous limestone. In that case, the potential difference ^DgrV will be maximum and results entirely from the potential drop in the mud column of resistivity ^rgr_m. It corresponds with the recording of a maximum apparent resistivity (^rgr_a)_max. Hence:

EQUATION (4)

Another expression for ^DgrV_max may be obtained by application of Ohm's law in the mud column:

EQUATION (5)

where the half current (I/2) is used because of its separation in two equal parts, d is the diameter of the well bore and d_s that of the sonde (85 mm.). Equating 4 and 5 we obtain after solving for (^rgr_a)_max:

EQUATION (6)

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In the foregoing expression, (^rgr_a)_max is the maximum resistivity value which will be read on the log facing a bed of infinite resistivity, i.e., non-porous.

In view of the small distance MN (^cong4^Prime), the product r₁ r₂ may be replaced by the square of the designated spacing for the limestone device which is normally 32^Prime.

On the limestone sonde log and for formations of finite resistivity, the resistivity-curve deflections are a function of ^rgr_a/^rgr_m. These deflections depend on mud resistivity, electrode spacing, well-bore diameter, and formation resistivity in the immediate vicinity of the bore hole.

Fig. 2a. Departure curves for limestone device. Corrected for diameter of sonde (85 mm.). d=8 3/4 inches.

Fig. 2b. Departure curves for limestone device. Corrected for diameter of sonde (85 mm.). d=7 7/8 inches.

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Fig. 2c. Departure curves for limestone device. Corrected for diameter of sonde (85 mm.). d=6½inches.

Fig. 2d. Departure curves for limestone device. Corrected for diameter of sonde (85 mm.). d=4½inches.

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The maximum values obtainable for (^rgr_a)_max^rgr_m corresponding with various hole sizes are readily obtained from 6 and are given in the following table for LS 32^Prime:

d
(inches)   (^rgra)max/^rgrm
4½                        910
6½                        264
7 7/8                     160
8 3/4                     125

The foregoing figures are essentially the maximum values given on the departure curves of Figure 2a, b, c, and d. For formations around the bore hole other than infinitely resistive, smaller deflections are obtained corresponding with the true ^rgr_i/^rgr_m given on the departure curves. It is observed from a departure curve comparison that the deflections ^rgr_a/^rgr_m are very sensitive to hole size. Therefore, adjustments are required in the determination of porosity from different hole sizes. Similarly corrections may be needed for limestone devices of length other than 32^Prime. These may readily be calculated from the Schlumberger Departure Curves (Document No. 3), using the lateral device for thick beds without invasion. The procedure to etermine other desired departure curves similar to Figure 2 but for other spacings and hole sizes, is to compute the maximum value for ^rgr_a/^rgr_m by equation (6), to read this value on the R_t/R_m=^infinity line, to draw a straight line parallel with the ordinate and plot the intersection values R_t/R_m=^rgr_i/^rgr_m as shown in Figure 2.

POROSITY CALIBRATION

As inferred from the departure curves, should mud resistivity and scale be shown on all limestone device logs, it would be possible approximately to calculate porosity (^phgr) from resistivity measurements by means of formula (2) in view of the small investigation radius of the limestone device.

However, the essential data, such as resistivity scale and mud resistivity, are generally missing from most old limestone device logs and it became desirable to devise a method for the evaluation of porosity which does not require a knowledge of the scale nor a reliable determination of mud resistivity. For this purpose, it is necessary that a hard non-porous limestone zone should have been logged. The resistivity value recorded at that level is chosen as a base line of zero deflection. The line of maximum deflection is chosen as the mud resistivity at bottom-hole conditions. Actually this line is extremely close to the zero line of the log and, unless the mud is very fresh, there is no error in selecting this zero line as the line of maximum deflection. Calibration of the method is a rived at by plotting percentage deflection as measured from the base line of zero deflection versus the corresponding porosity as obtained from core analysis. The correlation must extend over the full range from zero to maximum deflection on the log.

^rgr_i/^rgr_m = ^infinity corresponds with zero porosity.

^rgr_i/^rgr_m = 1 corresponds with 100 per cent porosity.

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Under the assumption of 100 per cent displacement efficiency of reservoir fluids by mud filtrate we would have

[EQUATION]

and:

[EQUATION]

then:

[EQUATION]

Assumptions of incomplete displacement are made in the theoretical justification of the method.

This correlation was experimentally arrived at in Stanolind-University "AJ" No. 4 in the Three Bar field, West Texas, for the prevailing hole diameter of 8 3/4 inches (Fig. 3). The reduction to hole diameters of 4½, 6½, and 7 7/8 inches was made by means of the corresponding departure curves shown in Figure 2. The procedure required to establish the bore-hole reduction for wells of other diameters than 8 3/4 inches is as follows.

Choose a ^rgr_a/^rgr_m such as 61 which corresponds with ^rgr_i/^rgr_m=100 for the 8 3/4-inch hole in Figure 2a. This corresponds with a percentage deflection read on a linear scale of 51.0 per cent and a porosity of 14 per cent as read on the calibration chart (Fig. 3b). It is to be noted that zero deflection is the ^rgr_i/^rgr_m = ^infinity line and maximum deflection is the ^rgr_i/^rgr_m = 1 line. Should a 6½-inch well have been drilled, the identical value of ^rgr_i/^rgr_m=100 would apply but the percentage linear deflection read on the left scale of the 6½-inch departure curve (Fig. 2c) would have been 62.5 per cent. This percentage deflection is plotted on the correspond ng porosity of 14.0 per cent in Figure 4. This gives a point on the 6½-inch calibration curve. Other points on the curve were obtained by a similar procedure which permitted drafting the 6½-inch calibration curve.

A similar procedure was followed for the calibration reduction of the 4½-inch and 7 7/8-inch well bores.

Figure 4 is the porosity-deflection chart which may be used to evaluate porosity in four different size holes: 8 3/4, 7 7/8, 6½, and 4½ inches. It may also be used for other hole sizes by interpolation. In order to interpolate the calibration curve for another hole size, it is first necessary to construct the corresponding departure curve as previously explained before proceeding with the hole-size reduction of the calibration curve.

THEORETICAL JUSTIFICATION

As a justification of the method of porosity determination from the limestone sonde log, one may reason as follows.

Porosity (^phgr) is related to formation factor (F) by the empirical equation:

[EQUATION]

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Fig. 3a. Composite log. Stanolind Oil and Gas Company, Three Bar field, Andrews County, Texas. University AJ No. 4.

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Fig. 3b. Correlation of porosity with limestone sonde deflections from maximum. University AJ No. 4.

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For the rocks of interest in this study, m = 2. Hence,

EQUATION (7)

On the other hand, plotting (log ^rgr_i/^rgr_m) versus ^rgr_a/^rgr_m for a 8 3/4-inch hole as shown in Figure 5, it can be observed that for values of 20<^rgr_a/^rgr_m<100 the plot is nearly a straight line, the equation of which may be written:

EQUATION (8)

In the invaded zone of porous oil-bearing formations, we may expect about 80 per cent replacement of oil by mud filtrate and the following Archie equation may be written:

EQUATION (9)

in which ^sgrw_i, is the saturation in water in the invaded zone. Hence,

EQUATION (10)

Fig. 4. Porosity determination chart from LS 32^Prime logs, for various bore-hole sizes.

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Substituting 10 in 7 we obtain:

EQUATION (11)

Substituting 8 in 11:

EQUATION (12)

The theoretical curve corresponding with Equation (12) for an 8 3/4-inch hole has been calculated and plotted as a dashed line in Figure 3 to show its relation

Fig. 5. Replot of departure curve scale for LS 32^Prime logs in 8 3/4-inch hole.

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to the empirical calibration curve. The results of the computation are shown on the following table and it is observed that the region wherein the agreement is most satisfactory is for percentage deflection comprised between 0 and 80 per cent, which corresponds with porosities varying from 0 to 30 per cent.

Table

The theoretical calculations of porosity from the limestone curve are necessarily based on a number of assumptions and simplifications in order to permit the reduction of the problem to a mathematical formulation. The theoretical deductions are only a justification of the empirical approach to determine porosity from the limestone log and an indication that the observed correlation trend is as should be expected. The mathematical deductions also indicate the generality of the approach and its possible application to other limestone reservoirs without further calibration, provided the porosity is of the intergranular type or of the uniformly fissured and vugular type.

PRACTICAL APPLICATIONS AND LIMITATIONS

In the application of the limestone sonde log to the determination of the porosity profile of wells and of their average porosity, various complications and limitations are encountered.

Determination of line of zero porosity or infinity line:
On many limestone sonde logs, the infinity line (R_t/R_m = ^infinity) is not indicated and in many instances it is impossible to find in the log a massive limestone bed for which it may be ascertained that porosity is non-existent. There are two ways by which a zero porosity line may be established.

1. By ascribing a low but finite porosity to some recognizable massive limestone, such as 2 or 3 per cent pore space. The deflection corresponding with this porosity is calculated for the existing well diameter by means of the calibration curve (Fig. 4).

2. By computing, by means of departure curves for the spacing used for the normal curve (AM) in conjunction with the limestone device, the ratio (R_t/R_m) which prevails in a well defined and sufficiently thick formation. The same ratio (R_t/R_m) must prevail for the limestone log. Through the use of the appropriate departure curves of Figure 2, or other computed curves, the deflection

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for a bed of infinite resistivity may be computed for the prevailing log sensitivity.

Elimination of shale and shaly sections:
It is obvious that shale and shaly sections will be reflected on the log as porous sections and it is imperative that they be properly eliminated in the evaluation of the effective pay thickness. This may be done through the S.P. deflections evaluation and better yet through selective S.P. measurements. However, in salty muds, they are relatively unreliable. The microlog and contact logs may also be of some help in eliminating shale sections as well as the gamma-ray log.

Lack of sufficient mud filtrate invasion:
A primary requirement for the proper working of this method is the existence of a sufficiently effective invasion by conductive mud filtrate. It is therefore obvious that a well drilled with oil or oil base mud, then spotted with fresh mud, will not satisfy these requirements. One way of ascertaining if sufficient mud invasion has taken place is by establishing the existence of a mud cake by running a microlog or contact log, or a caliper log. It has also been observed that wells in which there is considerable variations in fluid saturation with depth may give erroneous results, such as is the case through the transition zone from oil to water. While water saturation is not by itself a deterrent to the proper working of the method when sufficient mud filtrate invasion has taken place, th results are highly misleading when this requirement is not satisfied. The method works best in the reservoir rock sections lying high above the water-oil contact.

Type of mud:
The type of mud, as already suspected, will influence greatly the quality of the results obtained. Muds of high resistivity and especially oil-emulsion muds seem to give best results. As suggested, the conductive filter loss through the mud cake must be adequate to provide the required invasion.

Adequate correlation with core analysis log for purpose of calibration:
The method is predicated upon adequate correlation of core analysis results with the limestone log. Various reasons may exist why the correlation is insufficient. Core and log depths are often in disagreement over part if not all of the reservoir section of interest. This results from incomplete core recovery and misplacement of cores. Errors may also result from the differences in sampling volume between the log and the cores. The sample volume of the limestone log is roughly a sphere 10 inches in radius from which a cylindrical core of diameter equal to well-bore diameter has been taken out, whereas the sample volume of the core is either that of a full-size core of a standard permeability plug. The problem of the inadequacy of the method in highly fractured and vuggy reservoirs has al eady been covered at the outset. The method is adequate only for intergranular type porosity, and for uniformly distributed small vugs and fractures. For erratic distribution of fractures and vugs, it is doubtful if core data are representative of formation conditions and if the petrophysical relations between porosity and permeability apply to the type of formation.

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