Introduction

Rock physics describes the relationship between geophysical data such as compressional and shear wave velocities with reservoir rock characteristics such as pore types, lithology, fluid type, saturation, and other reservoir characteristics. Furthermore, it helps to establish relationships between engineering parameters (e.g., porosity, permeability) and seismic attributes (e.g., bulk modulus, velocities). Pore types are one of the main parameters needed for rock physics modeling in carbonate rocks. They control both petrophysical properties of reservoir rocks, including fluid content, porosity, fluid flow, permeability, electrical resistivity, and pore pressure (Tao and King 1993; Xu and White 1995; Xu and Payne 2009) and seismic properties including velocities and impedances. Based on different laboratory studies on carbonate rocks, a wide range of scattering has been observed in their measured velocity-porosity crossplots, and this scattering has been attributed to their pore-type variations (Brie et al. 1985). Carbonates, which have more than fifty percent of the world’s hydrocarbon reserves tend to have very complex and diverse pore types (Adams and MacKenzie 1998) . Since pore-type variations control fluid flow in a porous media, they are essential in permeability prediction and reservoir modeling. Therefore, identifying pore types from micro to macro is essential for optimizing the number of wells that need to be drilled in a field and estimating the recoverable oil and gas reserves (Gharechelou et al. 2015). Generally, it has been determined that pores, such as vugs and molds, are round and resist against pressure changes. On the contrary, thin pen-shaped cracks have less stiffness and less resistance against pressure changes. Carbonate rocks are often modeled assuming their pores as ellipsoids characterized by their aspect ratios (minor axis divided by major axis). Using long-wavelength first-order scattering theory, Kuster and Toksöz (1974) derived a relation for the effective modulus for porous media composed of particles with different pore geometries. The main limitation of this theory is that the ratio of porosity to aspect ratio must be less than or equal to 1. In other words, for elliptical cracks with an aspect ratio close to 0.01, the mentioned theory can compute the properties of the effective medium up to 1% porosity. To address this issue, the differential effective medium (DEM) (Berryman 1992; Mavko et al. 2020) theory divides total porosity into different groups and adds them into a matrix sequentially. Xu and White (1995) used DEM in a workflow to model the velocities of a sand and shale mixture. They divided porosities into two classes: sand and shale pore types. Moreover, Sain et al. (2008) divided porosities into three types: stiff (vug and moldic), reference (interparticle), and crack, and Xu and Payne (2009) extended the Xu and White (1995) concept for carbonate rocks by dividing the total porosity into four groups (cracks, reference, stiff and clay). Finally, Golsanami et al. (2021) determined different pore types (micro, meso, and macro) using the nuclear magnetic resonance (NMR)-based study in the laboratory. One of the main challenges within these approaches is having a good estimation for different pore types.

Different studies have provided various methods for pore-type characterization. Kumar and Han (2005) introduced a workflow to calculate pore aspect ratios based on measured sonic logs. Then, Sun et al. (2012) applied DEM-Gassmann rock physics models to compute the volume ratio of various pore types of complex carbonate reservoirs. Zhao et al. (2013) used a method to quantitatively describe the geophysical pore distribution in carbonate rocks using well logs and seismic data. Wu and Chen (2014) calculated rock pore types quantitatively using petrophysics and rock physics. A study of determining the pore types using NMR log, and seismic data in mixed clastic-carbonate formation was carried out by Gharechelou et al. (2015). Furthermore, ** et al. (2017) proposed the frame flexibility factor in rock physics models for deeply buried carbonate reservoirs to characterize pore types and used it for implications on determining porosity and permeability. Reservoir evaluation was carried out by Ali et al. (2019) using petrophysical analysis of well logs. Saberi (2020) used geology information to modify calculated pore types from sonic velocities. Moreover, Mirkamali et al. (2020) and Emami Niri et al. (2021) applied petrophysical logs, rock physics models, and seismic data to calculate the distribution of different pore types in heterogeneous carbonate rocks. Zhu et al. (2018) and Li et al. (2022) presented an approach to describe the complex pores within a carbonate system by creating a 3D rock physics model. They used this 3D rock physics model to extract pore system parameters and reservoir properties simultaneously from seismic data and well logs. Finally, permeability was estimated by Khoshdel et al. (2022) using pore-type classification on seismic inversion results in a carbonate reservoir. Kasha et al. (2023) proposed a method to reduce the uncertainty in closure correction by pore throat sizes and macro pores from thin sections. Also, Alipour and Sakhaee-Pour 2023) used contact angle and interfacial tension to measure pore throat size in shale.

Most methods mentioned above use velocity data to invert for pore type and then use that information for velocity modeling. In this study, we attempt to determine different pore types (clay, micro, meso, and macro porosities) using NMR T2 distribution. Then, we modeled the sonic velocities using Xu and Payne’s (2009) rock physics model, and confirmed the results against the measured sonic velocities. Here, we are modeling pore types independently from sonic velocity data and applying a multi-physics approach to model velocities. Therefore, the mentioned pore type result is a better representative of the actual pore types in the subsurface rock and can be used for purposes other than velocity modeling.

Methodology

This paper describes using petrophysical logs and rock physics models on a carbonate reservoir. Data used in this study come from a reservoir in the Main Ilam and Sarvak carbonate formations. Figure 1 describes how clay, micro, meso, and macro pore types are characterized using NMR log T2 distribution by applying appropriate cut-offs combined with the mineralogy and fluid content from petrophysical evaluation. The results are used in Xu and Payne’s (2009) rock physics model to model sonic velocity and bulk density; then, absolute percentage error is calculated from modeled and measured velocity and bulk density. Finally, the pore-type results were compared with thin sections, scanning electron microscopy photographs and mercury injection capillary pressure core data.

Fig. 1
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Flowchart of petrophysical data and rock physics modeling to calculate pore type in the formations

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Geological setting and data analysis

The reservoir under study is in the northern part of the Abadan Plain structural zone and 60 km from the west-southwest of Ahvaz in Iran. Due to the lack of outcrop formations, geophysical methods have been used to discover this structure. It is a stratigraphic-structural form in the Main Ilam and Sarvak formations. The lithology of the Main Ilam and Sarvak formations includes mostly calcite and a small amount of clay minerals. These are the most essential oil reservoirs in the Iran’s Cretaceous formations of Zagros. Figure 2 shows the location and lithology column of the studied field. Available data includes fullset well logs, NMR log, mercury injection capillary pressure (MICP) data, thin sections, and scanning electron microscopy (SEM) images of one vertical well (well A). Due to the lack of core data in the Sarvak Formation, only core data in the Main Ilam Formation have been used to validate the results. The following section explains the well-log interpretation for lithology, fluid, and pore-type determination. Then, rock physics modeling will be demonstrated in more details, and in the end, the results of the entire workflow will be shown and discussed.

Fig. 2
figure 2

(Reproduced from Du et al. 1999). The connection between them can be written as Eq. (1):

$$\frac{1}{{T_{2} }} = \frac{1}{{T_{2surface} }} + \frac{1}{{T_{2bulk} }} + \frac{1}{{T_{2diffusion} }}$$
(1)

where T2 is the NMR relaxation time, \(T_{2surface }\) is the surface relaxation time affected by the grain surface of the rock, \(T_{2bulk}\) is the bulk relaxation time controlled by the physical properties of the fluid (it can be ignored due to its long relaxation time), and \(T_{2diffusion }\) is diffusion relaxation time that is used to detect gas, light oil, water, and some oils with medium viscosity and is close to zero in a homogeneous magnetic field. In this regard, in fully water-saturated pores, the transverse relaxation time T2 is equal to the surface relaxation time. Therefore, Eq. (1) can be simplified to Eq. (2):

$$\frac{1}{{T_{2} }} = \frac{1}{{T_{2surface} }} = \rho \frac{s}{v}$$
(2)

where ρ is the surface relaxivity, \(\frac{S}{V}\) is the ratio of pore surface to fluid volume.

In general, porosity derived from the NMR log has a good accuracy because it is not affected by other rock microstructure factors, such as the rock matrix. But it still suffers from the small depth of investigation (about one inch), which can make it erroneous in rugose wellbore wall, and difficulty in distinguishing fractures due to its small investigation scale (pore scale). Its acquisition also needs more calibration and parameter tuning, such as determining the logging tool’s speed and Lamor frequency. However, in practice, a classification should be done to better interpret NMR porosity. For this purpose, different cut-offs are applied to the T2 distribution to divide the porosity into various parts. For example, T2 distribution can be divided into six parts, and each part is called a bin, which represents a group of pores with a certain size (Ferrer et al. 2007). Usually, an interval of 0.3–3 ms is considered clay-bound water (CBW). Smaller amounts of 0.3 ms are related to matrix hydrogens, which are not measured by the NMR log measurement tool. A cut-off value from 3 up to 92 ms in carbonates is called bulk-volume irreducible (BVI) or capillary-bound water in small pores. Effective porosity is calculated after applying the cut-off value of more than 92 ms, and fluid in this range is called producible or free fluid.

Rock physics modeling

A porous reservoir rock comprises minerals, porosity, and one or more fluid phases. Rock physics modeling uses this information to calculate the velocity of a saturated rock. There are different rock physics models for such purposes, and one of the famous ones for carbonate rocks is the Xu and Payne’s (2009) model. In this model, four different types of porosity are defined, including porosity related to clay, interparticle porosity, micro-cracks, and vug porosity, as Eq. (3):

$$\phi_{ T} = \phi_{ clay} + \phi_{crack} + \phi_{IP} + \phi_{stiff}$$
(3)

where \(\phi_{T}\) is total porosity, \(\phi_{clay}\) is clay porosity, \(\phi_{crack}\) refers to the soft component of the porosity, which is typically crack porosity, \(\phi_{IP}\) is interparticle porosity, and \(\phi_{stiff}\) refers to the stiff parts of the pore system, which is vug or moldic porosity.

To model the compressional and shear velocities according to the Xu and Payne’s (2009) model, the following steps need to be taken:

  1. 1.

    Elastic moduli of the rock matrix are calculated based on the Voigt–Reuss–Hill average.

  2. 2.

    Clay-related pores are added to the rock matrix.

  3. 3.

    After the aspect ratio definition, three different pore types (stiff, interparticle, and crack) are added to the rock matrix using the DEM theory to compute dry elastic moduli.

  4. 4.

    The effective fluid modulus is modeled using Wood’s or Brie’s equations.

  5. 5.

    Saturated rock moduli are calculated using Gassmann’s fluid substitution model.

Berryman (1992) and Mavko et al. (2020) presented the DEM theoretical model to compute effective bulk and shear moduli for various pore types. In the DEM model, there are two distinct phases by adding inclusion to the mineral phase. The mineral starts as phase one and is changed at each stage by adding phase two (inclusions). These changes continue until it reaches the total porosity volume percentage. The differential equation can be written as Eq. (4):

$$\begin{gathered} \left( {1 - \phi } \right)\frac{d}{{{\text{d}}\phi }}\left[ {K*\left( \phi \right)} \right] = \left( {K_{2} - K*} \right)P^{{\left( {*2} \right)}} \left( \phi \right) \hfill \\ \left( {1 - \phi } \right)\frac{d}{{{\text{d}}\phi }}\left[ {\mu *\left( \phi \right)} \right] = \left( {\mu_{2} - \mu *} \right) Q^{{\left( {*2} \right)}} \left( \phi \right) \hfill \\ \end{gathered}$$
(4)

where the initial assumptions are \(K*\left( 0 \right) = K_{1}\) and \(\mu *\left( \phi \right) = \mu_{1}\), K1 and µ1 are the bulk modulus and shear modulus of the mineral phase, K2 and µ2 are the bulk and shear moduli of the inclusion phase. P and Q are geometric factors that are related to the aspect ratio of inclusions (pores).

Discussion and results

The proposed pore-type modeling approach started with determining different pore aspect ratios in the Main Ilam and Sarvak carbonate formations using the cross plot between total porosity and compressional velocity. Then, from the T2 distribution of the NMR log, utilizing the porosity cut-off, the volume fraction of different pore types (clay, micro, meso, and macro porosities) was calculated. Furthermore, elastic properties of the Main Ilam and Sarvak formations have been modeled using these pore types (pore aspect ratio and their volume fractions) in Xu and Payne’s (2009) model. Finally, the modeling results are compared with the measured log and are analyzed statistically.

The other researchers’ methods to calculate different pore types are based on using sonic logs, thin sections, MICP data or NMR study of carbonate core sample in the laboratory. However, our proposed approach, which uses the NMR log to determine the pore types, can be considered as a new method independent of sonic logs, thin sections, and core data. Contrary to the operational and financial limitation in coring, which reduces the coring intervals and subsequently limits the determination of pore types, the method used in this study can continuously cover the entire reservoir. Furthermore, the calculated volume fraction of different pore types (clay, micro, meso, and macro porosities) by NMR log can help to distinguish the reservoir’s productive and non-productive zones. They can even help to reduce uncertainty in constructing static and dynamic models since the fluid in clay and micro-pore types cannot be produced.

Characteristics of pore types from rock physics template

In Fig. 4, we observe that, in the template of P-wave velocity versus total porosity, our data lies between the lower and upper limits of the Voight–Reuss–Hill and modified Hashin–Shtrikman bounds. According to the time average equation, the line in the middle is called the reference line, which refers to the interparticle porosities. Since the stiff pore type increases the wave velocity, data points higher than the reference line should be linked to the stiff pore type. On the other hand, crack porosity decreases wave velocity, so data points lower than the reference line should be linked to the crack pore type. In this regard, Figs. 4a and c shows that the pore type in the Main Ilam Formation mainly consists of reference pore type, and both stiff and crack pore types are minimal. However, in the Sarvak Formation (Figs. 4b, d), pore types are mainly reference and stiff, while crack pore type is minimal.

Fig. 4
figure 4

The compressional velocity versus total porosity at well A: a and b Voigt–Reuss–Hill template in the Main Ilam and Sarvak formations, respectively; c and d Hashin–Shtrikman template with aspect ratio varies from 0.01 to 1 in the Main Ilam and Sarvak formations, respectively

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Pore-type volume fraction determination using nuclear magnetic resonance log

The porosity obtained from the nuclear magnetic resonance log has a good accuracy because it is free from the influence of other rock microstructure factors, such as the rock matrix (lithology). In the NMR log, the porosity is calculated from the T2 distribution curve. T2 distribution also indicates the size of the pores. In this regard, a short T2 distribution indicates small pores, and a long T2 distribution has large pores (Coates et al. 1999; Liang and Wei 2008). In this study, the T2 distribution of the NMR log has been used to determine the different types of pores. For this purpose, a different cut-off (Table 2) is applied to the T2 distribution to divide total porosity into distinct parts (each part is called magnetic resonance bin porosity). The standard defaults T2 distribution cut-off for clay-bound water (clay) and capillary-bound water (micro) are 3 and 92 ms, respectively, in carbonates (Chang et al. 1994; Coates et al. 1999). According to elastic parameters error in rock physics modeling, a cut-off of 300 ms is considered to calculate meso and macro porosities. Figures 5 and 6 shows the calculated bins and pore types in the Main Ilam and Sarvak formations in well A, located in one of the fields within the Abadan Plain. In the Main Ilam Formation, despite a big washout in the entire formation due to proper contact between the NMR tool and borehole wall, the quality of the NMR log was acceptable. However, in the Sarvak Formation, there was no washout or rugosity in the borehole wall; thus, the quality of the NMR log was quite good. Based on the applied T2 distribution cut-offs in the NMR log (Table 2), it was found that in the Main Ilam Formation, pore types mainly consist of micro- and meso-, while the macro-pore type is minimal. At interval 2892–2901 m, there are no meso- and macro-pore types, and most pores are micro-pore. In the Sarvak Formation, in addition to micro- and meso-pore types, there is also macro-pore type. The Main Ilam and Sarvak formations are relatively clean and contain low clay pore type.

Table 2 Used cut-off values in NMR T2 distribution for pore type determination
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Fig. 5
figure 5

Track 1 is depth. Tracks 2–7 are calculated bin porosity from NMR T2 distribution, water saturation, porosity and caliper, lithology and fluid content, NMR T2 distribution, and zonation, respectively: a Main Ilam Formation, b Sarvak Formation

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Fig. 6
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Track 1 is depth. Tracks 2–5 are calculated pore-types. Track 6 is lithology and fluid content. Track 7 is zonation: a Main Ilam Formation; b Sarvak Formation

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Comparison of pore types in the NMR log and core data

NMR results on pore types are validated against thin sections, SEM photographs, and pore size distribution of MICP data within the Main Ilam Formation (Figs. 7, 8, and 9). There was no core data in the Sarvak Formation. Therefore, the NMR pore-type results within the Main Ilam Formation were only compared against core data. Table 3 shows a good match between calculated pore types from the NMR log and thin section images, SEM photographs, and pore size distribution from MICP data from the Main Ilam Formation cores. This reservoir has a stable tectonic regime. Hence, it has a few percentages of fractures, and most of which are filled with cement. Petrophysical evaluation in the Main Ilam Formation reveals that the formation has two sections named A and B, with different reservoir characteristics. Interval A is located within 2848–2892 m depth and has average porosity and water saturation of 16 and 36%, respectively. While interval B located within 2892–2910 m depth, and has average porosity and water saturation of 6 and 72%, respectively. From the formation test results (fullbore DST within the intervals of 2880–2889 m and 2852–2861 m), oil has been observed in both intervals. The core description and analysis show interparticle and micro-pore type in interval A which is also aligned with NMR log results of having more micro- and meso-pore type. Knowing the mixed-wet wettability character of the rock, it can be stated that as water is trapped in the micro-pores, only oil is produced, which refers to a meso-pore type within interval A. In interval B, the NMR log indicates a micro-pore type of porosity, but no testing was carried out in this interval. However, core data revealed that this interval has micro porosity, and the water in the pores is not movable. These results confirmed what was interpreted from the NMR log in this investigation.

Fig. 7
figure 7

Thin sections in carbonate reservoir of Main Ilam Formation, depths of a 2866.09 m, some vug, interparticle, and micro-pores; b 2889.2 m, crack filled with cement, interparticle, and micro-pores; c 2894.34 m micro-pores and, micrite; d 2904.08 m, micro-pores, micro-crack, and micrite

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Fig. 8
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SEM photomicrographs in the Main Ilam Formation of Well A, depths of a 2865.61 m, a view of the aggradation of smaller calcite micro-crystal to larger calcite crystal (red arrow) within moldic porosity (yellow arrow); b 2870.47 m, a view of the interparticle (yellow arrow) and micro porosity (red arrow); c 2893.5 m, a view of intraparticle porosity (red arrow), micro porosity and equant calcite cement (yellow arrow); and d 2897.6 m, a view of the equant filling cement within the bioclast chamber, micro porosity, and aggrading neomorphism of micritic crystals (red arrow)

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Fig. 9
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Pore-size distribution from MICP data in the Main Ilam Formation, depths of a 2868.77 m, b 2871.5 m, c 2893.5, and d 2902.6 m

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Table 3 Results of core data and NMR log pore types in the Main Ilam Formation
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Elastic properties using NMR pore types and rock physics modeling

Utilizing the results of the petrophysical evaluation of the Main Ilam and Sarvak formations (Fig. 3), including the percentage of calcite, clay content, and water saturation, the minerals are combined, and average matrix elastic properties were obtained using the Hill average method. Then, the clay, micro-, meso-, and macro-pore types from the NMR log with aspect ratios of 0.06, 0.09, 0.15, and 0.8 were added to the average matrix using the DEM method to compute the elastic properties of the dry rock. Then, Brie’s method was used to mix different fluids, and the average elastic characteristics of the fluids were obtained. Finally, the elastic characteristics of the saturated rock were modeled, and calculated by adding the modeled fluids into the dry rock using Gassmann’s method. Furthermore, the modeled elastic properties were compared with the measured ones in well A for the Main Ilam and Sarvak formations (Fig. 10). Correlation coefficient and mean absolute percentage error are presented in Fig. 11 and Table 4.They show a match of modeled and measured RHOB and P-wave velocity in the Main Ilam and Sarvak formations.

Fig. 10
figure 10

Track 1 is depth. Measured (red) and modeled (blue) of RHOB, and Vp are compared in tracks 2 and 3. Tracks 4 and 5 are the calculated absolute percentage error of RHOB, and Vp. Track 6 is lithology and fluid content. Track 7 is zonation: a Main Ilam Formation, and b Sarvak Formation

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Fig. 11
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Crossplots of modeled and measured Vp (a), and RHOB (b) in the Main Ilam Formation; Vp (c), and RHOB (d) in the Sarvak Formation

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Table 4 Statistical parameters of modeled and measured RHOB and P-wave velocity
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Conclusions

Pore-type characterization in carbonate rocks is needed to understand better reservoir parameters (e.g., porosity, water saturation, permeability), and reservoir quality (impermeable zones, reservoir dynamic behavior). In this study, we proposed a workflow to use the interpretation of nuclear magnetic resonance log and conventional logs to model sonic velocities using a carbonate dataset from the Main Ilam and Sarvak formations in Iran. Interpretation of the conventional well logs provides lithology, porosity, and fluid content information of the reservoir, while nuclear magnetic resonance data help determine clay, micro, meso, and macro pore types. Results were, furthermore, confirmed against the available core data. Then, we used them as input in Xu and Payne’s (2009) model to predict P-wave velocity. The essential results are as follows:

  • According to P-wave velocity versus total porosity of Voight–Reuss–Hill and modified Hashin–Shtrikman bounds, pore type in the Main Ilam Formation mainly consists of reference pore type, and both stiff and crack pore types are minimal. However, in the Sarvak Formation, pore types are mainly reference and stiff, while crack pore type is minimal.

  • Based on NMR data, pore types in the Main Ilam Formation mostly consist of micro- and meso-, while the macro-pore type is minimal and in the Sarvak Formation, in addition to micro- and meso-pore types, there is also macro-pore type.

  • The correlation coefficient of P-wave velocity, 92 and 64 percent, respectively, in the Main Ilam and Sarvak formations indicated a good accuracy between the modeled and measured velocity.

  • Instead of sonic logs, core data, and thin sections, NMR data can be used for pore-type characterization.

  • NMR data can give a reasonable estimate of pore types, which can also be used to generate an accurate sonic velocity prediction.

  • Determination of pore type using NMR data is independent of lithology type and other geology phenomena, such as compaction, cementation, dissolution, dolomitization, and fracturing. Therefore, it is possible to define the pore types without having much knowledge of the sedimentary environment.