## Introduction

One of the big challenges in building static and dynamic models for simulation in naturally fractured and vuggy carbonates is the porosity partitioning (matrix and secondary porosity).

This post describes different secondary porosity calculation methods applied to one specific field, using all the available data from micro to macro-scale (thin section petrography, electrical images, core porosity measurements in whole diameter cores and plugs, conventional logs and nuclear magnetic resonance logs). By analyzing the results of the different methods, a correlation of secondary porosity as a function of total porosity was derived. This correlation was used to calculate a secondary porosity distribution in the simulation grid.

As expected, the secondary porosity derived from logs and core data was found to be extremely variable within few millimetres of depth. But, when this property was upscaled into the geo-cellular grid with cell heights of few meters, the value of the upscaled secondary porosity in every cell was similar to the value obtained by applying the derived correlation.

The structure of the described field corresponds to an anticline and reservoirs consist of Mesozoic carbonates with vuggy porosity and natural fractures.

## SECONDARY POROSITY CALCULATION

**1 – Thin-Section Petrography**

The result of the quantitative characterization of the porosity in thin-sections using an automated point-counting device under the microscope was the starting point for the analysis of secondary porosity. The thin section petrographic analysis was developed using 65 samples from 12 wells distributed in the field (Fig. 1).

**Fig. 1 **Examples of thin-sections used in the secondary porosity quantification.

It was observed that although the total thin-section porosity is lower in magnitude than the measured total porosity in core plugs of the same interval, a good correlation exists between them. Higher porosities in thin sections correspond to higher porosity measurements in core plugs (Figure 2).

**Fig. 2 **The X-axis of each graph represents different samples and the Y-axis the total porosity. The red line is the total porosity measured in the thin-sections, the black line is the plug porosity, and the blue line is the whole-diameter core porosity.

The laboratory reported the percentage of porosity of each thin-section, differentiating the type of porosity: intergranular, inter-crystalline, intrabioclastic, microporosity, porosity in fractures, and vugular porosity. For purpose of this study the vugular and fracture porosity are named as secondary porosity and the rest as matrix porosity. The sum of these two porosities is total porosity.

One interesting result of this analysis was the good correlation between the total and secondary porosity, expressed in mathematical terms with a correlation coefficient of 0.9 (Figure 3). It is important to note that the correlation used data from all the formations and 12 wells distributed throughout the entire field.

**Fig. 3** Correlation between the total and secondary porosity derived from thin-section petrography.

Some carbonate studies have shown that some characteristics or observable relations on micro-scale can be repeated at larger scales. Several analysis techniques are based on this principle, for example the use of fractals to characterize the secondary porosity (Montaron, n.d.). Using this premise, the secondary porosity was analyzed on macro-scale to verify the mathematical correlation between the total and secondary porosity observed at micro-scale (thin-sections) .

**2 – Borehole Electrical Images**

Borehole electrical images, with a vertical resolution of about 5mm and azimuthal borehole coverage of approximately 80% in a well of 8.5 in, provided the information necessary to quantitatively solve the different components of a heterogeneous porous system. Even using hi-res density-neutron logs, with one inch vertical resolution, a vug not azimuthally aligned with the tool sensors, is not registered correctly. In the same interval, an electrical image tool is able to obtain 2000 measurements in several directions throughout the borehole and therefore with this information it is possible to quantify the distribution of the porosity (Newberry et al., 1996) .

Using the technique described by Newberry et al. (1996), calibrated electrical images are converted to porosity with the Archie’s equation assuming that the readings are obtained from the invaded zone,

Φ image = sqrt(Rmf / Rimage)

Where, Φimage is the porosity converted from the measured resistivity in each sensor of the tool, Rmf is the resistivity of the mud filtrate, and Rimage is the numerical value of the electrical image in a particular point.

Additionally, Newberry et al. (1996) developed another algorithm to convert the electrical image into porosity using an external porosity and a resistivity measurement with medium depth investigation, preferably a shallow Laterolog, LLS. This algorithm was used in this work to calculate the secondary porosity

Φ image = Φ ext * sqrt(LLS / Rimage)

Where, Φext is the effective porosity obtained from the nuclear magnetic resonance log (NMR) or calculated from the density-neutron logs.

Taking a vertical window of 1.2″ with the converted porosities from the resistivity measurements of each sensor of the tool in that window (approx. 2400 data), a porosity histogram is generated. When the porosity distribution of the system is uniform, the histogram shows only one peak (matrix). For a more complex porosity system, several peaks can be observed. Thus, it is possible to statistically identify the point that marks the separation between the distribution of primary and secondary porosity (Figure 4) .

**Fig. 4 **Schematic Porosity distribution histogram generated with converted porosities from image resistivities in a 1.2” window.

In this technique, the window used for generation of the porosity histogram is moved throughout the interval of interest resulting in continuous curves of total and secondary porosity (Figures 5-6) .

**Fig. 5 ***Secondary porosity distribution calculated from a borehole electrical image. Tracks: (1) Depth, (2) Calibrated electrical image, (3) porosity distribution histogram from image, (4)Porosity distribution percentiles, and (5) Total porosity from the image in black, total porosity from density-neutron logs in red, and secondary porosity from the image in blue.*

The analysis of secondary porosity from electrical images was made in 4 wells distributed in the North, Center, and South of the structure, and covering all the producing formations.

**3 – Nuclear Magnetic Resonance (NMR)**

**3 – Nuclear Magnetic Resonance (NMR)**

The NMR logs use magnets to polarize the hydrogen nuclei in the water and hydrocarbons contained in the porous space of the reservoirs. When the magnets are removed, the hydrogen nuclei relax. The time of relaxation, called T2, depends on the pore-size distribution; generally, greater pores have longer times of relaxation (Akbar et al., 2000, p. 28). In carbonates with vuggy porosity, a second peak in the T2 distribution can frequently detect vugs since pores with diameter greater than 100 microns have T2 greater than ~1 second (Chang et al., n.d.).

In agreement with the mentioned concepts, the calculated secondary porosity from the electrical images was correlated with the curve of longer relaxation times of the nuclear magnetic resonance log showing a very good match (Figure 7).

**Fig. 7 **Good match between secondary porosity from images and NMR bin porosity at long relaxation times. Tracks: (1) Depth, (2) image, (3) porosity spectrum, and (4) shaded black is secondary porosity from image log, magenta is NMR bin porosity at long relaxation times, blue is total porosity from image, and green is total porosity from NMR log.

**4 – Sonic Log – Secondary Porosity Index**

In carbonates with vugular porosity, the porosity calculated from the sonic log depends on the intergranular primary porosity (Schlumberger, 1989). Consequently, the secondary porosity is calculated from the difference between the total porosity calculated from the density-neutron logs and the porosity calculated with the sonic log.

Φ secondary = Φ D-N – Φ sonic

When applying this methodology to the shallowest reservoir, the resulting values of secondary porosity are higher than expected. However, for the other three formations the index of secondary porosity calculated from the combination of the sonic-density-neutron logs had a very good correlation with the secondary porosity calculated from the electrical images (Figure 8).

Fig. 8 Last track shows the good match between secondary porosity from images and from Sonic-Density-Neutron logs in deeper reservoirs.

**5 – Whole diameter Core Porosity vs. Plug Porosity**

In this method it is assumed that the porosity measured in a plug of 1 to 1,5 inches of diameter is representative of the matrix, and that the porosity measured in the whole diameter core represents the total porosity (Matrix porosity plus vugs and fractures). Accordingly, the secondary porosity is calculated from the difference of the porosity measured in the whole-diameter core and the porosity measured in the plug (Figure 9).

Φ secondary = Φ core whole diameter – Φ core plug

**Fig. 9 **Schematic representation of the secondary porosity calculation using core measurements. Plug porosity is representative of the matrix and whole-diameter core porosity is representative of total porosity (matrix plus vugs and fractures).

Cores from 10 wells were used for this calculation. The analyzed fragments of whole-diameter have lengths that vary between 10 and 20 centimetres, and the plugs were taken from distances not greater than 30 centimetres from the whole-diameter analyzed. Tomographies were used for the selection of the plugs location. In 2 wells the plugs were taken within the same whole-diameter core whose porosity had been quantified previously and the correlation of the calculated secondary porosity in these samples has a very good match with the calculated secondary porosity from the electrical images.

Until this point, I think this could be an interesting post by itself, but the next section is food for thought on how the results were used in a wider pragmatic scope.

## SECONDARY POROSITY vs. TOTAL POROSITY

Understanding that each one of the described methods has his own assumptions and limitations, the electrical images provide a unique solution with the resolution, azimuthal borehole coverage, and the density of data required to characterize the complexity of a porous system with vugs and fractures. Additionally, the secondary porosity derived from images was validated with the results obtained from other methods.

Figure 10 illustrates the secondary porosity distribution histogram for 3 different formations (including data from 4 wells). In the top Mesozoic formation, the frequency and the magnitude of secondary porosity is much greater than in the other formations, in agreement with what was observed in the cores samples and tomographies. In the Middle formation, the frequency and the magnitude of the secondary porosity are smaller, and finally in the Lower formation, the secondary porosity is much smaller.

**Fig. 10.** Secondary porosity distribution histogram (4 wells) for 3 different formations, in contrast with the core tomographies of other well. Note that the maximum scale of the Y-axis is different for each histogram.

The secondary porosity histograms for each formation and well shown in figure 11 reflect the lateral and vertical variability of the distribution of this property .

**Fig. 11 **Histograms of secondary porosity per well, per formation, showing the lateral and vertical variability of the distribution of this property.

Although, the histograms show that the secondary porosity has high vertical and lateral variability, in the cross-plots presented in figure 12 it is observed that the correlation between total and secondary porosity has a similar trend in all the cases.

**Fig. 12 **Crossplots of total porosity versus secondary porosity per well, per formation. Even when porosity distribution has significant variability, the porosities ratio has follow a similar trend.

In figure 13 correlations of total porosity versus secondary porosity for each of the formations appear in red, and the general correlation using all the data is represented by the white line. It is important to note that a cutoff of 60 ohm was applied to discard low resistivity points associated with argillaceous laminations. In general terms, the correlations for the different formations are very similar and is important to remember that the same behaviour was observed in the micro-scale analysis using thin-section petrography.

**Fig. 13 **Crossplot of Total porosity vs. Secondary porosity using the 4 wells analyzed and including all the formations. Red lines represent individual correlations for each formation, and the white line is the correlation using all the data.

Based on the previous findings, it is understood that the magnitude and distribution of the secondary porosity can vary significantly, and that a proportional correlation between secondary and total porosity was observed at different scales. The concern with these findings is that the correlation coefficient found can be high, but the dispersion / standard deviation is also high. This means that at a specific total porosity, several values of secondary porosity can be found. The pragmatic approach stated in this part of the post is that even when the high resolution secondary porosity curve calculated from image logs is upscaled to the simulation grid (with cell heights of few meters), each cell value is similar to the one calculated using the general correlation derived as a function of total porosity. (Figure 14).

For this reservoir, the general equation of secondary porosity as a function of the total porosity found is: Φ secondary = 5.24E-4 + 0.194*Φ total + 0.544*Φ total^2

**Fig. 14** Crossplot of the original secondary porosity derived from images and upscaled into the simulation model vs. synthetic secondary porosity generated applying the general equation to the total porosity grid.

This equation is not intended to replace the secondary porosity calculation in the well, but to be used at a grid level in the simulation model defining the three-dimensional distribution of the secondary porosity (Figure 15).

**Fig. 15 **Secondary porosity grid generated by applying the general equation to the total porosity population.

As an additional information, Figure 16 shows a decent correlation between the general correlation of secondary porosity derived from the images and the secondary porosity calculated as the difference of porosity measured in whole-diameter core and porosity measured in plugs.

**Fig. 16 **Crossplot of Total vs. secondary porosity. Green points represent the core data and the blue line the electrical image data.

**FREE FLUID COMMENTS**

Combining the calculation of the secondary porosity from image logs with the results of the processing of the nuclear magnetic resonance log (NMR), the percentage of porosity of the matrix containing free fluid was estimated. If it is assumed that all the fluid within the secondary porosity (vugs and fractures) is free fluid. Therefore the difference between the free fluid measured by the NMR and the secondary porosity corresponds to the free fluid within the matrix. Additionally, the NMR log provides a calculation of capillary fluid in the matrix or irreducible water saturation.

In both the Upper and Middle formations there is significant storage of free fluid in the matrix, with an average near 60%. The Lower formation has a lower ratio of free fluid in the matrix with respect to the total free fluid and a decreasing tendency is observed downwards (53% to 39%).

**CONCLUSIONS**

- Borehole electrical images provide the resolution, azimuthal coverage, and density of data necessary to characterize the complexity of a porous system with vugs and fractures. The secondary porosity for the reservoirs was determined from the processing of electrical images. This secondary porosity calculation was validated using core porosity measurements (whole-diameter and plugs), nuclear magnetic resonance logs, and empirical techniques with sonic-density-neutron logs.
- Similarly, in the measurements of porosity from logs and cores, like in the measurements made in thin-section petrography, a mathematical correlation between the total porosity and the secondary porosity was observed, indifferent of the formation.
- A mathematical expression of secondary porosity as a function of the total porosity was generated. When applying this expression to the total porosity grid in the simulation model, the result for each well was very similar to that obtained when the calculated curve of secondary porosity from image logs was upscaled.

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