Prediction of cementation factor using different methods of reservoir rock classification in heterogeneous carbonate reservoirs

Cementation exponent or factor is one of most crucial factors in the Archie’s equations that should be estimated precisely in order to accurately determine relevant petrophysical characteristics of the reservoir. Inaccurate estimation of this factor leads to incorrect petrophysical analysis and inaccurate determination of water saturation which is highly important for the economic forecasting of hydrocarbon reservoirs. An alternative to the cementation factor precise estimation is to classify rocks based on a common petrophysical property such as permeability. This research aims to investigate the cementation factor in different rock types and eventually to provide a suitable model for estimating this parameter based on different approaches of rock classification. In this paper, first, conventional methods of rock classification (e.g., current zone indicator (CZI), flow zone indicator (FZI), permeability and electrical efficiency) have been used for cementation factor estimation. Then, the data, taken from Regnet et al. (J Geophys Res Solid Earth 120:790–811, 2015), have been analyzed by these methods in order to find the best way of rock classification. The analyzed data show that only electrical efficiency method is accurate enough to classify rocks into different groups as high values of determination coefficient (i.e., 0.9828 and 0.9725) are obtained as a result of classification by this method in which the data are classified into distinct groups. The obtained values of determination coefficient as a result of classification by permeability, FZI and CZI methods are 0.805, 0.809 and 0.7568 respectively. Due to this result and high scattering of the classified data, it can be concluded that these three methods have not been precise enough for rock classification of the data. In addition to the above-mentioned existing methods, a new rock classification method based on tortuosity coefficient or factor has also been proposed in this paper. This new method has been proven to be accurate due to low scattering of the data results and falling them into distinct groups in addition to obtaining high values of determination coefficient (i.e., 0.9996, 0.9986 and 0.9951) in the classification by this new method.


Introduction
There is a huge volume of hydrocarbons trapped in unconventional resources but their geology is ubiquitous and such a nature restricts the hydrocarbon flow in the reservoirs. The petrophysical properties of these resources also differ from the conventional reservoirs and their complex lithology make challenges in evaluating the exact porosity, permeability and water (hence oil) saturation of the formation (Muther et al. 2022). Water saturation is an essential parameter in forecasting the development of hydrocarbon reservoirs (Rezaee et al. 2007;Al-Gathe et al. 2009).
First attempt to estimate water saturation using well log data was made by Archie in 1942. This method required laboratory measurements of cementation factor but due to high variation of the cementation factor along depth of wellbore due to complex lithology in unconventional reservoirs, assuming a constant value for the cementation factor leads to an error in determination of water saturation (Heydari Gholanlo et al. 2018). Moreover, laboratory measurements for obtaining the cementation factor values are expensive and time-consuming (Mahmoodpour et al. 2021).
In order to develop an appropriate relationship for the cementation factor, many authors have proposed a variety of equations. However, in most of these equations, the cementation factor has been assumed to be constant. Assuming a constant value for the cementation factor will lead to an imprecise petrophysical evaluation. In order to eliminate the inaccuracy, the cementation factor should not be assumed constant (Soleymanzadeh et al. 2018;Ara et al. 2001;Hasanigiv and Rahimi 2008;Focke et al. 1987).
Due to complexity in carbonate reservoirs, especially the heterogeneities of these reservoirs, and also, the presence of various rock types, to determine the precise cementation factor, for each class of rock in question, different equations are needed. This conduces to new methods of reservoir rock classification and rock typing (Focke et al. 1987).
In this paper, the data, taken from Regnet et al. (2015), have been analyzed by different proposed methods (e.g., CZI, FZI, permeability and electrical efficiency) and the most accurate rock classification has been proposed. In addition to already existing methods, a new rock classification method has also been proposed. This new method, which is based on tortuosity coefficient or factor, has been proposed to classify rocks into different classes according to their electrical properties that improves the cementation factor estimation, and as a result, a more precise water saturation will be obtained.

Theoretical background
Cementation factor estimation Archie (1942) was the first who discovered a relationship between formation resistivity factor (F) and porosity (φ) as a result of illustrating them on a logarithmic cross plot. Then, he proposed a correlation between them in the following form: He figured out that in Eq. (1), the slope of formation resistivity factor plot against porosity on a logarithmic scale is cementation factor, which is a key parameter in petrophysics.
Later on, Winsauer et al. (1952) added another parameter to Archie's equation named tortuosity coefficient (a): Equation (2) is called Archie-Winsauer correlation. Humble's correlation, expressed in the following form, is one of the most common correlations, for estimating the formation resistivity factor: Humble's correlation has been developed by Winsauer et al. (1952).
Afterwards, Carothers attained Eq. (4), using 793 sandstone samples (Mabrouk et al. 2015): Thereafter, Timur et al. (1972 suggested the following correlation after analyzing 1800 sandstone samples (Tiab et al. 2015): Tixier et al. (1965 approached a Humble-like correlation, for predicting the cementation factor: For estimating the cementation factor value based on porosity in non-fractured carbonate reservoir rocks, Shell correlation can be used as follows (Kadhim et al. 2013). Borai (1987) proposed the following correlation as a result of analyzing 64 carbonate samples to predict the cementation factor: Gomez was the first scholar who obtained a correlation to estimate the cementation factor and tortuosity coefficient for each surface area (Rivero et al. 1978;Watfa et al. 1987): Assadollahi et al. (2008 proposed the following correlation that can be used to estimate the cementation factor in Iranian carbonate reservoirs: Thus, many correlations have been proposed by various authors to estimate the cementation factor, but only few of them will lead to an accurate value of the cementation factor. Petrophysical analysis and geological studies of many rock samples show that the classification of this samples according to their mutual characteristics (i.e., permeability, pore type, FZI, and electrical efficiency) will give a better and more accurate value of the cementation factor (Rezaee et al. 2007;Soleymanzadeh et al. 2018;Soleymanzadeh et al. 2021).

Rock typing
Different methods of classification have been suggested, to estimate more accurate correlation between porosity and formation resistivity factor. Hasanigiv and Rahimi (2008) studied on 155 core samples from Asmari, Ilam, and Sarvak Formations in southwest of Iran. Based on rock types and pore types, they classified the samples into 6 different classes described in Table 1.
After some investigations in heterogeneous carbonate reservoirs, Focke and Munn (1987) realized that the main reason of uncertainty for estimation of water saturation is the cementation factor. Afterwards, they classified limestone samples based on their permeability values, and demonstrated formation resistivity factor against porosity on a logarithmic cross plot. Finally, they proposed the following equations for each class of permeability: The results of this analysis indicate that although the classes have been illustrated separately, it cannot give a good fit for each class. This might be for the weak relation between permeability and formation resistivity factor (Rezaee et al. 2007).
Generally, there is a weak relation between porosity and permeability. To find a better relation between porosity and permeability, and to interpret the hydraulic flow unit (HFU), Amaefule et al. (1993) proposed the following equation: FZI, k and φ are, respectively, the flow zone indicator (μm), permeability (mD), and porosity (fraction), and φ Z is the matrix volume ratio (PMR) that can be calculated from the following equation: This equation shows the relation between pores or void spaces and their geometric distribution. Rezaee et al. (2007) classified samples from a reservoir based on their FZI values obtained from Eq. (15), and then, they separated them into different HFUs via FZI logs. The FZI classes separating the HFUs are: Figure 1 demonstrates the formation resistivity factor data versus porosity. As can be seen from the Figure, the data are scattering for each HFU, meaning that this method cannot classify the data well. This implies that hydraulic and electrical paths are not the same and the hydraulic tortuosity is much larger than the electrical tortuosity (Rezaee et al. 2007). Table 1 Cementation factor correlation for each class of samples studied by Hasanigiv and Rahimi (2008) Classification m Intracrystalline and vuggy pore types = 2.25 + 0.6 Irregular, medium vugs and rare molds = 2.48−0.048

+0.042
In formation resistivity factor against porosity logarithmic cross plot, there might be high rate of scattering in the data. This can cause a smaller value for R 2 (coefficient of determination) and as a result, it will cause high errors in the estimation of the values m and a (Rezaee et al. 2007). Amaeful et al. (1993) defined reservoir quality index (RQI) by combining Poiseuille law for flow in cylindrical tubes, Darcy's law for fluid flow in porous medium, and Kozney-Carmen model: Higher values of RQI indicates better quality of the reservoir.
However, there is no general relation between formation resistivity factor and permeability, but, an inverse relation has been reported by many authors (e.g., Wong et al. 1984;Kostek et al. 1992;Nettelblad et al. 1995;Celzard et al. 2002).
The following equation can be used to define electrical radius indicator (ERI): Unlike RQI that gives an average value of hydraulic radius, ERI is dimensionless, and only compares the electrical radii of the samples. Equation (18) shows that with a given porosity, a decrease in the formation resistivity factor will increase the ERI or vice versa. Rezaee et al. (2007) compared the RQI and ERI values of 92 samples (Fig. 2). Incoherence between RQI and ERI in this Figure can indicate the difference between hydraulic and electrical paths.
ERI is the sign of electrical path for each sample. To separate samples with similar electrical flow, ERI should be divided to void volume, or more accurately, to the ratio of pore volume to matrix: where CZI is current zone indicator. This equation defines the relation between the void volume and electrical flow properties. This equation can divide samples with similar electrical current values, and for this, it is called current zone indicator or CZI (Rezaee et al. 2007). CZI is a factor that can be used to separate samples with rather similar m and a values, and where the change in formation resistivity factor is only affected by porosity. For each sample in a specific CZI range, units having similar electrical flow have been proposed. It is obvious that in an equal distance from reservoir with similar electrical properties, it shows close CZIs. This range with similar electrical flow properties is called electrical flow unit (EFU). Rezaee et al. (2007) used four classes of CZI to classify four EFUs: Figure 3 shows the logarithmic cross plot of formation resistivity factor against porosity for four CZI classes. Table 2 shows the classification of CZI, best power equation, m, a and R 2 between porosity and formation resistivity factor in each EFU. Using this approach, R 2 have increased significantly, and this will help to obtain more accurate values of m and a (Rezaee et al. 2007).
Some authors believe that tortuosity coefficient and cementation factor have straight effect on each other, meaning a rock sample with higher values of tortuosity coefficient will make a harder path to conduct the electrical flow. Rezaee et al. (2007) assumed that these two parameters were not in relation to each other necessarily. This means that the tortuosity coefficient can  (Rezaee et al., 2007) change in a rock sample, but no change occurs in the value of cementation factor and vice versa. In the model presented by Herric and Kennedy (1994), conductivity of the straight tubes C s is C w V w, where C w is the conductivity of water that has filled the tubes and V w is the pore volume that has been occupied by the tubes. V w Can be estimated from the expression S w φ . Thus, it can be said that: where E is the proportionality factor, and for the total conductivity of a clean rock, the following equation has been developed: E indicates the electrical flow efficiency for a rock sample having maximum electrical current and passing through straight tubes. For a rock completely saturated with water, the following correlation is used to estimate the electrical efficiency: (20) Electrical efficiency of a rock is not related to water volume and is only affected by the non-uniform current distribution. Soleymanzadeh et al. (2021) used this concept to provide a new parameter for rock classification, named electrical efficiency. It seems that electrical efficiency is an appropriate classification for porosity against formation resistivity factor that results in different classes. Taking logarithm from both sides of the equation, and rewriting Eq. (22), we will have: This shows that if we plot porosity against formation resistivity factor on a logarithmic scale, for all rock samples having the same electrical efficiency 1 η e , we obtain a straight line with the slope of -1, and as a result, we have a distinct electrical rock type. Existence of different parallel lines on this logarithmic cross plot implies different electrical classes of the rock samples. Intercept of each line is the inverse ofη e . Hence, the higher value of the intercept gives smaller electrical efficiency for that class of electrical rocks that results in a new equation between cementation factor and porosity for each class of electrical rocks (Soleymanzadeh et al. 2021): Merging Eq. (22) and (24), we obtain Eq. (25): In this correlation, for each ERI, a similar sample with similar electrical efficiency exists. The result of cementation factor against the reciprocal of electrical efficiency is a straight line with the slope of lnη e and the intercept of 1 (Soleymanzadeh et al. 2021).

Methodology
In this study, 112 rock samples, taken from limestones of the eastern part of the Paris Basin reported by Regnet et al. (2015), were analyzed. This dataset includes m, F, φ , a and k. Later, 18 data have been deleted due to the lack of information. Figure 4 illustrates the ranges of porosity and permeability values in the data. Table 3 indicates the interval of each parameter including m, F, φ and k, meaning that this dataset covers a wide range of data.
In order to find the best method of rock typing and estimate an accurate value for cementation factor, different rock classification methods are applied on the dataset.   indicates the methodology flowchart of the study. First, the data has been analyzed by the proposed method of Focke and Munn (1987) that divides the data into several groups based on their permeability values. Then, the method, proposed by Amaefule et al. (1993) has been applied on the data that leads to have different groups in the data based on their FZI classes. Afterwards, the data have been grouped based on their CZI classes. Eventually, the method, proposed by Soleymanzadeh et al. (2021), has been applied on the data that results in having various groups based on the electrical efficiency of the data. In this study, after applying these four methods on the dataset, which has been taken from Regnet et al. (2015), a new method has also been proposed and applied on the data based on the tortuosity coefficient classification of the data, and finally, the best way of rock classification has been proposed.

Results and discussion
In this section, the above five mentioned methods have been applied on the data reported by Regnet et al. (2015). For classifying the data based on their permeability values (i.e., the method proposed by Focke and Munn (1987)), the data has been separated into 3 distinct permeability intervals (0.01 < k < 0.05, 0.05 < k < 0.5, 0.5 < k < 0.1), and then, F-φ logarithmic plot has been obtained for these three groups. Figure 6 illustrates the F-φ logarithmic plot for these groups classified by their permeability values.  Table 4 shows the F-φ equation, a, R 2 , and m for each of the three permeability intervals that have been extracted from Fig. 6. Table 5 shows the m-φ equation and R 2 extracted from Fig. 7 for each of the three permeability groups.
As can be seen from Fig. 6 and Table 4, although the data have low level of scattering and high values of R 2 (i.e., 0.8583, 0.805, and 0.9569) have been obtained, due to not falling the data into distinct groups, permeability cannot be an accurate way of rock classification. The reason for this is that permeability as a main petrophysical property is related to hydraulic behavior; thus, it cannot illustrate a good electrical classification for limestones.
In this section, in order to group the data by their FZIs, first FZI of each rock sample should be estimated using Eq. (15). Figure 8 demonstrates the probability plot of log FZI. This plot has different spots that form broken lines with different slops. Each line depicts a group of rocks that has slop of -1 on RQI-φ z plot. After illustrating the probability plot, each line will be considered as an interval for the F-φ logarithmic plot. Then, if each group falls into a distinct group, it shows that FZI, which separates the hydraulic paths, will also separate the electrical paths. However, this does not usually occur in carbonate rocks, and the fluids, which pass through the electrical paths, have different characteristics, and thus, carbonate rocks cannot be classified based on their FZIs. Figure 9 indicates the F-φ logarithmic plot for each of log FZI intervals obtained from the probability plot in Fig. 8.
As shown in Fig. 9, the data scattering in each FZI group is high. In addition, a low R 2 value like 0.2913 has been obtained for the first group, which has the lowest FZI range in the FZI data. As can be seen from this Figure, none of the data has fallen into distinct groups. The reason for this is the complexity of carbonate rocks, and also, FZI is based on permeability that both describe the fluid hydraulic behavior.   Hence, it cannot be accurate for electrical rock typing. It also shows that the hydraulic path is different from the electrical path, and therefore, FZI is not a proper way of electrical classification for limestones. For classifying rocks based on their CZI values, first, we have calculated the CZI for each rock sample using Eq. (19). Afterwards, the data has been grouped into 3 CZI classes, and subsequently, F-φ logarithmic plot has been drawn for each CZI class. In Fig. 10, the F-φ logarithmic plot has been shown. Figure 11 is the m-φ plot for each CZI interval or class that shows the cementation factor data against porosity values is highly scattered.
The F-φ equation and values of m, a, and R 2 obtained for each CZI group, based on the formation resistivity factor plot against porosity on a logarithmic scale (Fig. 10), are given in Table 6. Table 7 indicates the m-φ equation and R 2 for each CZI interval based on the cross plot of cementation factor against porosity (Fig. 11).  Table 4 The F-φ equation, a, R 2 , and m extracted from the F-φ logarithmic plot shown in Fig. 6 Table 5 The m-φ equation and R 2 extracted from the m-φ plot shown in Fig. 7 for each permeability interval As can be seen from Fig. 10, illustrating the data on the F-φ logarithmic plot based on their CZI intervals yields almost distinct groups although the data shows some fluctuations and different values of R 2 (i.e., 0.7568, 0.8863 and 0.9871). Despite this, CZI can almost classify limestones into distinct groups. The reason is that the main equation for CZI (i.e., Eq. (19)) has the term φ z indicating the solid part of the rock (namely, matrix) that has the highest values of electrical resistivity. Therefore, CZI cannot completely demonstrate a good classification for limestones but taking the above points into account, it is an almost (not completely) accurate way of rock classification and is still an efficient way of separating reservoirs with same values of a and m.
For classifying rocks based on their electrical efficiency values, the first step is to estimate the quantity 1 η e using Eq. (22) for each rock sample, and then, to indicate a probability plot for it. In this regard, the closest data to each other are considered to fall into the same group. Then, the F-φ logarithmic plot should be depicted for each group that has been obtained from the probability plot as an interval. In Fig. 12, the probability plot has been demonstrated for the inverse of electrical efficiency or 1 η e data. As a result, 5 different groups of 1 η e have been determined. Figure 13 illustrates the F-φ logarithmic cross plot for each of 1 η e groups. Table 8 indicates the F-φ equation and values of m, R 2 and a for each inverse electrical efficiency interval based on the logarithmic plot of formation resistivity factor against porosity (Fig. 13).
As indicated in Fig. 13 and Table 8, each group has fallen into perfectly distinguished classes. Based on the high values of obtained R 2 (i.e., 0.9007, 0.9828, 0.9266, 0.9565 and 0.9725) and the low scattering of the data, we can infer that electrical efficiency is a precise method to classify limestones into different electrical groups. The reason for accurate classification of rocks based on their electrical efficiency values is that the reservoir rock is a porous medium and it  Table 6 The F-φ equation, m, R 2 , and a, obtained from the F-φ logarithmic plot shown in Fig. 10 Table 7 The m-φ equation and R 2 , obtained from the m-φ plot shown in Fig. 11 12 Probability plot for the inverse of electrical efficiency data has different tortuosity coefficients and different paths for each porosity value. Therefore, the electrical efficiency of each path will be different. Finally, we have applied our new proposed method on the data, taken from Regnet et al. (2015). In this method, the data has been classified based on their tortuosity coefficient values. As a result, the data have been classified into 10 different intervals. Then, the data have been illustrated on the F-φ logarithmic plot as seen in Fig. 14. This Figure shows the F-φ logarithmic plot for each tortuosity coefficient group or interval. Table 9 shows the F-φ equation and values of m, R 2 and a for each tortuosity coefficient interval based on the logarithmic plot of formation resistivity factor against porosity (Fig. 14).
As can be seen from Fig. 14, each tortuosity coefficient group is distinct from other groups and falls into a distinct group with low scattering in the data, and as shown in Table 9, high values of R 2 (i.e., 0.9996, 0.9937, 0.9986 and 0.9951) have been obtained. This low scattering in the data means that classifying rocks based on their tortuosity coefficient values is an accurate way of rock typing. The reason for this is that the tortuosity coefficient is a parameter that indicates the electrical path of the fluid, and therefore, it is an accurate way of classification for limestones.

Conclusions
Accurate estimation of cementation factor is essential as it greatly affects some petrophysical properties. Rock typing is one of the methods that yields precise measurement of cementation factor. In this study, the data, reported in Regnet et al. (2015), have been analyzed using conventional methods, in addition to a new proposed method, of rock classification. The results of this research can be summarized in the following: • The rock classification based on permeability and flow zone indicator cannot be considered as accurate ways of rock classification due to high scattering of the data analyzed by these methods.   Fig. 14 F-φ logarithmic plot for each group of rocks that has been classified by their tortuosity coefficient range values Table 9 The F-φ equation, m, R 2 , and a, obtained from the F-φ logarithmic plot shown in Fig. 14 • Current zone indicator is a parameter, which can seldomly be used as an accurate method of rock classification. • Electrical efficiency data analysis show that the rock classification method based on the electrical efficiency is considered as an accurate way of rock classification. • Classifying rocks based on their tortuosity coefficient values is a new rock classification method that has been proposed in this research paper. It is an alternative method for classifying rocks into different classes, and can also be considered as an accurate way of rock classification. After analyzing the data using the equation obtained from the tortuosity factor classification, it has been observed that the values of formation resistivity factor were perfectly similar to the values of the initial formation resistivity factor data. Thus, it can be concluded that this method is an accurate way of rock classification in order to predict precise cementation factor.
Funding The authors have no relevant financial or non-financial interests to disclose. Moreover, no funding was received to conduct this research work.

Conflict of interest
On behalf of all the co-authors, the corresponding author states that there is no conflict of interest.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.