1 Introduction

Classification of reservoir rocks into different rock types, called reservoir rock typing, is an essential tool in drilling, production and especially reservoir studies. A petrophysical rock type is presented as a group of rock samples that have similar petrophysical and geological properties that influence fluid flow (Stolz and Graves 2003). Generally, petrophysical rock typing is categorized into two separate classes which are petrophysical static rock typing (PSRT) and petrophysical dynamic rock typing (PDRT). PSRT is defined as a group of rocks with a similar capillary pressure curve in the drainage process, whereas PDRT is described as a set of rocks that shows similar fluid flow behavior (Mirzaei-Paiaman et al. 2018). Proper application of rock typing provides more real dynamic reservoir behavior in simulation models (Attar et al. 2015; Saboorian-Jooybari et al. 2015, 2016). Several techniques have been reported in the literature to determine reservoir rock types. These techniques can be classified into two general groups: the theoretical and the empirical methods. The theoretical methods (such as rock quality index (RQI)/flow zone indicator (FZI) (Amaefule and Altunbay 1993), shale zone indicator (SZI) (Jongkittinarukorn and Tiab 1997; Nooruddin and Hossain 2011), modified FZI (Nooruddin and Hossain 2011), FZI* (FZI-star) (Mirzaei-Paiaman et al. 2015) and FZI** (FZI-double star) (Mirzaei-Paiaman and Saboorian-Jooybari 2016) and PSRTI (Mirzaei-Paiaman et al. 2018)) are basically derived from the well-known Kozeny–Carman equation. Empirical methods, such as Winland (Kolodzie Jr 1980; Pittman 1992; Aguilera 2002), generate relationships between porosity, permeability and a specific size of pore throat which is taken from mercury injection capillary pressure tests.

Generally, rock typing approaches are performed at a pressure other than reservoir pressure, especially at atmospheric pressure. However, changes in pressure can alter the pore structure of the rock. When pressure is applied to different rocks, they respond differently, consequently a rock sample in a rock type that was determined at the atmospheric pressure may shift to another rock type when pressure changes (see Fig. 1). It should be noted that the effect of overburden pressure on rock is considered in commercial simulators by the rock compressibility (Cr) parameter. This parameter describes the change of porosity with pressure. However, the effect of pressure on permeability is not given as input data into the simulator. In other words, the effect of pressure on the pore structure of rock is considered by using Cr, whereas permeability and porosity change in different ways. Therefore, the effects of pressure on the process of rock type determination must be taken into consideration.

Fig. 1
figure 1

Schematic of effect of pressure on rock typing process (RT1: Rock type 1, RT2: rock type 2 and RT3: rock type 3). Each circle represents a rock sample

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In this work, 113 core samples from one of the carbonate oil reservoirs in the Middle East have been categorized into different rock types using RQI/FZI, Winland and FZI* methods at five different pressures to investigate the effects of pressure on the process of rock type determination. It is worth mentioning that the effect of pressure in the rock typing process has not yet been investigated. In other words, most of the research examines rock type determination at a specific pressure.

In this study, first reservoir rock typing is defined. Then, three selected methods of rock typing are applied to classify studied rock samples at different overburden pressures, and finally, the results of the three methods are discussed thoroughly.

2 Reservoir rock typing

Various techniques have been suggested for classification of reservoir rocks into rock types such as the J-function method, RQI/FZI technique, capillary pressure approach and the Winland method (Soleymanzadeh et al. 2018). Among these methods, RQI/FZI and Winland approaches are the most widely used techniques of rock typing (Winland 1972; Abbaszadeh et al. 1996; Svirsky et al. 2004; Biniwale 2005; Obeida et al. 2007; Shenawi et al. 2007; Chekani and Kharrat 2009; Ye et al. 2011; Riazi 2018). However, as it is concluded from Mirzaei-Paiaman et al. (2018), the RQI/FZI method completely fails in complicated cases such as heterogeneous rocks. Therefore, they suggested that using FZI* gives more reliable results. These methods are described here briefly, and pressure effects on these techniques of rock typing are examined in following sections.

2.1 RQI/FZI approach

RQI/FZI has been derived from Kozeny–Carman equation which is based on assuming a porous medium as a bundle of capillary tubes. It is obtained by combining Poiseuille’s equation and Darcy’s law (Zhao et al. 2016; Chen and Yao 2017). The generalized form of the Kozeny–Carman relationship is given by the following equation:

$$k = \frac{{\phi^{3} }}{{\left( {1 - \phi } \right)^{2} }}\left[ {\frac{1}{{F_{\text{s}} \tau^{2} S_{\text{gv}}^{2} }}} \right]$$
(1)

where k is permeability (mD), ϕ is porosity, Fs is the shape factor, τ is tortuosity, and Sgv is the surface area per unit grain volume.

Rearrangement of Eq. (1) results in:

$$\sqrt {\frac{k}{\phi }} = \frac{\phi }{{\left( {1 - \phi } \right)}}\left[ {\frac{1}{{\sqrt {F_{\text{s}} } \tau S_{\text{gv}} }}} \right]$$
(2)

Amaefule and Altunbay (1993) presented FZI as Eq. (3):

$${\text{FZI}} = \frac{1}{{\sqrt {F_{\text{s}} } \tau S_{\text{gv}} }}$$
(3)

Also, RQI is defined as follows:

$${\text{RQI}} = 0.0314\sqrt {\frac{k}{\phi }}$$
(4)

where k is permeability in mD. Normalized porosity (ϕz) is calculated from Eq. (5):

$$\phi_{\text{z}} = \frac{\phi }{1 - \phi }$$
(5)

Substituting Eqs. (3) to (5) into Eq. (2) gives:

$${\text{RQI}} = \phi_{\text{z}} {\text{FZI}}$$
(6)

Taking logarithms of both sides of Eq. (6) leads to:

$$\log {\text{RQI}} = \log \phi_{\text{z}} + \log {\text{FZI}}$$
(7)

where RQI and FZI are in µm, and ϕz is dimensionless. Equation (7) shows that a log–log plot of RQI versus ϕz results in a straight line with unit slope. This means that all rock samples with similar FZI value lie on an individual straight line. Therefore, the presence of different straight lines implies different rock types. Each of this rock type is denoted by its intercept at ϕz = 1.

The rock typing methods were frequently used to classify reservoir rocks at atmospheric pressure. It is obvious that values of porosity and permeability in reservoir conditions are different from their values at atmospheric pressure. Therefore, using data at atmospheric pressure may result in an incorrect rock typing process and inaccurate reservoir performance prediction. A proper solution for considering pressure effect on rock type is to perform the RQI/FZI method at reservoir pressure.

2.2 Winland method

Winland performed mercury injection experiments on a large set of sandstone and carbonate rock samples to correlate porosity, permeability and the size of the pore throats. His multiple regression analysis for various mercury saturations revealed that the best correlation coefficient (R2) is related to 35% mercury saturation. The corresponding pore throat radius of 35% mercury saturation was denoted by r35. The Winland correlation is as follow:

$${\text{log }}r35 = 0.732 + 0.588{\text{log }}k - 0.864 {\text{log }}\phi$$
(8)

where r35 is in µm, k is uncorrected air permeability in mD, ϕ is porosity in percentage.

r35 can be used as a basis to classify a reservoir into different rock types. All rock samples with similar r35 constitute a single rock type and lie on an iso-pore throat curve.

2.3 FZI* method

The base form of Kozeny–Carman equation is obtained by combining Poiseuille’s equation and Darcy’s law, as noted in Sect. 2.1. This form of Kozeny-Carman equation is as follows:

$$k = \phi \frac{{r_{\text{mh}}^{2} }}{{F_{\text{s}} \tau }}$$
(9)

where rmh is the effective or mean hydraulic unit radius. Mirzaei-Paiaman et al. (2015) introduced FZI* from Eq. (9):

$${\text{FZI}}^{*} = 0.0314\sqrt {\frac{k}{\phi }}$$
(10)

Herein, FZI* is in µm which can be calculated for each sample from measurement of its porosity and permeability. Hence, rocks with the same FZI* lie within an individual group. The fluid flow behavior of this group is assumed to be the same. Taking logarithm from both sides of Eq. (10) leads to Eq. (11).

$$\log \left( {0.0314\sqrt k } \right) = \log \sqrt \phi + \log {\text{FZI}}^{*}$$
(11)

It is inferred from Eq. (11) that in a log–log scale, the plot of \(0.0314\sqrt k\) versus \(\sqrt \phi\) for an individual rock type shows a straight line with the slope of unity and intercept of FZI* at the ϕ = 1.

3 Description of rock samples

In this work, permeability–porosity data related to 113 carbonate core samples from a carbonate reservoir were used. These data have been obtained at different confining pressures (atmospheric pressure, 2000, 4000, 5000 and 6000 psia). Porosity and permeability of these rock samples at atmospheric conditions are illustrated in Fig. 2.

Fig. 2
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Porosity and permeability of 113 core samples at atmospheric conditions

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The value of r35 at atmospheric conditions is calculated for all samples from Eq. (8) (see Fig. 3).

Fig. 3
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r35 of all samples at atmospheric pressure

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Table 1 summarizes the average (Ave) and median (Med) of permeability, porosity and r35, FZI and FZI* at five different overburden pressures.

Table 1 Average and median of porosity, permeability and r35 of the rock samples at different pressures
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4 Results and discussion

The classical approach to reservoir rock typing, a semi-log plot of permeability versus porosity (Abbaszadeh et al. 1996), leads to undesirable results in heterogeneous reservoirs such as most carbonated reservoirs. It is noted that there is not any mathematical support for this traditional method of rock typing (Mirzaei-Paiaman et al. 2015). Fig. 4 depicts log K versus ϕ for 113 core samples at ambient pressure. This figure confirms the inappropriateness (R2 = 0.4882) of the mentioned traditional technique of reservoir rock typing. Therefore, it is concluded that these data should be classified into distinct rock types.

Fig. 4
figure 4

Classical method of rock typing: log K versus ϕ

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The first step of the rock typing process is data clustering. There are different clustering techniques can be used in rock typing processes, such as discrete rock type (DRT), histogram, parabolic plots and global hydraulic element (Abbaszadeh et al. 1996; Corbett and Potter 2004). The DRT method was used in this work.

In order to investigate the effect of pressure on rock type determination, the RQI/FZI method was applied to determine rock types at different pressures. These rock types are illustrated in Fig. 5. Comparing rock types at different pressures reveals that the rock type of core samples changes in various ways:

Fig. 5
figure 5

RQI/FZI methods at different overburden pressures

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  1. a)

    Increasing trend (shift from lower rock type to upper one): such as core No. 51 which has been denoted by symbol in Fig. 5.

  2. b)

    Decreasing trend (shift from upper rock type to lower one): for example, core No. 47 which has been shown by symbol in Fig. 5.

  3. c)

    Fluctuating trend: such as core No. 19 which has been indicated by symbol in Fig. 5.

  4. d)

    No change: major part of studied samples remained in the same RQI/FZI rock type.

Table 2 presents the number of samples for each mentioned trend.

Table 2 Four different trends due to pressure change based on the RQI/FZI method
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In order to clarify the abovementioned trends, for each trend, some samples were selected and their FZI values were plotted versus pressure in Fig. 6. In fact, each trend in Table 2 was named according to the change in FZI value versus pressure as it is shown in Fig. 6.

Fig. 6
figure 6

Change in FZI during increasing pressure for different trends

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Figure 7 shows the number of rock samples in each rock type. This figure reveals that the number of samples in the rock types with low values of FZI (EX1, RT7, RT8 and RT9) increased by increasing pressure. It should be emphasized that rock types EX1 and EX2 did not exist at atmospheric pressure and were added to the other rock types when pressure was increased. It means that by increasing pressure the number of rock types increases.

Fig. 7
figure 7

Number of samples in each rock types based on the RQI/FZI method

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Since, permeability mostly depends on pore throat size rather than pore size, the authors believe that using the Winland method which contains pore throat size (r35) leads to a clearer description of the effect of pressure changes on the rock type determination. Whereas the RQI/FZI method is based on the Kozeny–Carman model in which the pore radius and pore throat are considered equal. In order to investigate the effect of pressure on the rock type determination by the Winland method, this method was applied to rock samples at five different pressures: 14.7, 2000, 4000, 5000 and 6000 psia (see Figs. 8, 9, 10, 11, 12).

Fig. 8
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Flow units based on the Winland method at 14.7 psia

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Fig. 9
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Rock types based on the Winland method at 2000 psia

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Fig. 10
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Rock types based on the Winland method at 4000 psia

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Fig. 11
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Rock types based on the Winland method at 5000 psia

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Fig. 12
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Rock types based on the Winland method at 6000 psia

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Figures 8, 9, 10, 11 and 12 show that most of the rock samples shift to the left and downward simultaneously. In other words, this leads to change in the number of rock types and also changing a rock sample from one rock type to another one. In addition, these figures indicate that the number of data in the low k–ϕ zone (blue circle) increases with an increase in pressure.

Figure 13 depicts the number of rock samples in each of the rock types at different pressures. Three points are inferred from this figure: (1) an increase in pressure increases the number of rock types: two rock types were added to the rock types at atmospheric pressure which are indicated by EX1 and EX2 in Fig. 13. In other words, increasing pressure exacerbates the degree of heterogeneity of this dataset; (2) Increasing pressure increases the number of rock samples in the lower rock types (EX1, RT1, RT2 and RT3), and (3) for pressures greater than 4000 psia, the number of rock samples in upper rock types (RT4, RT5, RT6, EX2) remains constant.

Fig. 13
figure 13

Frequency of rock types based on the Winland method

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The shift of the rock samples between different rock types (based on the Winland method) during pressure changes was examined, and results are reflected in Table 3. Indeed, Table 3 reveals that 37% of rock samples jump from one rock type to another one due to change in pressure. This means that ignoring the effect of pressure on the determination of rock types and considering k-ϕ at atmospheric pressure, make large errors in subsequent processes in a reservoir study.

Table 3 Four different trends due to pressure change based on the Winland method
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Further investigations imply that 60% of rock samples which had remained in the same rock type during changes in pressure are dolomitic. This may be due to lower compressibility of dolomite rock samples with respect to limestone samples. Also, 82% of rock samples which shift from upper curves to lower curves are limestones. It should be noted that most of these samples contain vugs. It seems that the high compressibility of these vuggy limestone samples is the main reason of this trend of Table 3. A few samples (2%) jump from lower curves to upper curves which may be related to generation of fractures in the pore structure of these samples due to an increase in pressure. The fluctuating trend in Table 3 can be attributed to the generation of induced fractures and closeness of some pores in successive steps of pressure changes. It is worth mentioning that 80% of samples with a fluctuating trend contain anhydrite. Further investigation is required to explain the effect of anhydrite content on the fluctuating trend of a rock sample. Figures 14, 15, 16 and 17 illustrate four trends of Table 3: no change, decreasing, increasing and fluctuating, respectively. In these four figures, arrow direction indicates the path of change of rock types during pressure changes.

Fig. 14
figure 14

Samples remain in the same rock type during pressure changes (no change trend)

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Fig. 15
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Samples jump from upper curves to lower curves during pressure changes (decreasing trend)

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Fig. 16
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Samples shift from lower curves to upper curves during pressure changes (increasing trend)

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Fig. 17
figure 17

Samples fluctuate between different rock types during pressure changes (fluctuating trend)

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The value of r35 at different pressures was used to explain the observed trends in Fig. 18. This figure shows the value of r35 at different pressures for four different trends. (Each part of the figure is related to one trend in Table 3.) It is inferred from this figure that, all trends in Table 3 can be interpreted based on the change in r35 during pressure changes.

Fig. 18
figure 18

r35 changes due to an increase in pressure for four different trends of Table 3

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Using the FZI* method, the number of rock types increased from six to eight with an increase in overburden pressure (see Fig. 19). Rock samples move to the left and downward simultaneously, which obviously implies that the quality of the majority of the rocks reduces with an increase in pressure. Comparison of rock types at five different pressures shows that rock types change in two trends: decreasing and fluctuating. Core 19 (symbol ) and Core 65 (symbol ) represent decreasing and fluctuating trends, respectively.

Fig. 19
figure 19

Rock typing at different pressures using the FZI* method

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The frequencies of FZI* are demonstrated in Fig. 20 which confirms the results obtained from the other two methods. Increasing pressure causes rock types EX1 and EX2 to be added to the rock types at atmospheric pressure.

Fig. 20
figure 20

Frequency of rock type based on the FZI* method

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Table 4 presents the effect of pressure on the rock typing process by the FZI* method and details of observed trends due to pressure change. This table shows that, similar to RQI/FZI and Winland methods, more than 50% of studied rock samples have remained in their rock types at atmospheric pressure. Furthermore, a decreasing trend is the most common trend and vuggy limestone samples are majority of the rocks which fall within this trend, as observed in RQI/FZI and Winland methods.

Table 4 Four different trends due to pressure change based on FZI* method
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Finally, it is noted that having a clearer picture of the rock pore structure, such as from micro-computed tomography (Micro-CT) scans, improves the analysis of the effect of pressure on the determination of rock types.

5 Conclusions

The following conclusions arose from this work:

  1. (1)

    Studied samples were classified into different rock types using RQI/FZI, FZI* and Winland methods at five different pressures. Different behavior was observed for rock samples during changes in pressure. The majority of the samples remained in the same rock type during pressure increases. Some of the samples shifted from an upper curve to a lower curve, and a few samples change from a lower curve to an upper one. In addition, several of the rock samples showed fluctuating trends. These four different trends can be attributed to the mineralogy and change in pore structure of the studied samples.

  2. (2)

    Most of the rock samples which remained in the same rock type during pressure changes are dolomitic. It seems that this is related to the lower compressibility or higher density of this type of rock. In contrary, the drastic changes in rock types occur in the limestone rock samples which contain vuggy porosity. The higher compressibility of these samples is the main reason of this behavior.

  3. (3)

    In RQI/FZI, FZI* and Winland methods, it is observed that the number of rock types increases with an increase in pressure. Also, the number of rock samples in the lower rock types (the lower quality rock types) increases. Furthermore, generally, at pressures greater than a specific pressure (in this study, 4000 psia), the number of rock samples in the higher rock types (the higher quality rock types) remains constant.

  4. (4)

    The Winland method gives a clearer picture of changing rock samples between different rock types. This is because the Winland method has been developed based on the size of pore throats (r35).

  5. (5)

    The effect of pressure on the rock type determination implies that the process of reservoir rock typing should be performed at the reservoir conditions.