Introduction

The petrophysical discrimination of reservoir rocks using terms such as rock-typing, flow unit, and electrofacies is essential for building 3D models and reservoir characterization. These models should represent the reservoir's geological characteristics, petrophysical, and dynamic performance (Shehata et al. 2021; Kassem et al. 2022). Carbonate sequences are one of the main hydrocarbon resources worldwide (Radwan et al. 2022). Each reservoir rock type is deposited under comparable stratigraphic conditions and undergoes similar diagenetic modifications. Moreover, each rock type contains a unique porosity–permeability relationship, pore throat geometry, capillary pressure (Pc) profile, and water saturation (Sw). On the other hand, the flow unit term regards reservoir layers with stratigraphically continuous trends that maintain characteristics of rock types and show unique reservoir process speed (Gunter et al. 1997; Wibowo and Permadi 2013; Enayati-Bidgoli et al. 2014; Nouri-Taleghani et al. 2015). The terms rock type and flow unit could be interchanged whenever the productivity term is preferred (Taghavi et al. 2007).

The heterogeneity in petrophysical properties and rock facies of carbonate rocks presents a persistent challenge. This widespread heterogeneity introduces various uncertainties for petroleum engineers and geologists. The interpretation of the reservoir's facies helps distinguish micro-scale variations in mineral composition, fabric, and pore geometry, affecting the main petrophysical characteristics of rocks. Facies bear sedimentary basin imprints, although they generally undergo numerous evolutionary events during their formation, causing the original fabric of the facies to be altered. These alterations are known as diagenetic changes. In this context, Abdel-Fattah et al. (2022) and Mebrouki et al. (2024) concluded that diagenetic features and microfacies control reservoir quality. For instance, overload-compaction, cementation, and secondary mineralization, such as chlorite and kaolinite, are reducing factors. Conversely, fracturing and dissolution improve petrophysical properties.

Considering that each flow unit/rock type comprises a unique pore geometry and if the wettability distribution of a porous medium is a function of pore size, knowing the relationship between flow unit distribution, reservoir heterogeneity, and wettability variation enables us to understand the origin of some challenges facing the flow units’ classification. Some authors (i.e., Donaldson and Siddiqui 1989) found a linear relationship between wettability and saturation in sandstone rocks. They had no mention of the role of pore size.

Hwang et al. (2006) showed that different regions of a porous medium can have varying wettability preferences due to factors such as mineral composition and surface roughness. Therefore, the wettability of porous systems can be classified as either uniform or non-uniform. In recent years, Zhao et al. (2018) investigated the effect of wettability heterogeneity on relative permeability (Kr), which is influenced by microscale fluid topology, fluid phase connectivity, and fluid cluster morphology. They concluded that the velocity of fluid flow is affected by the cluster size of the fluid, with small clusters being restrained by capillary force and trapped in pore spaces. Additionally, the distribution of flow resistance in porous media is mainly controlled by the wettability-related microscale fluid distribution.

Numerous methods have been introduced to identify flow units, but only a few have been widely used. The first method was derived from the Winland R35 equation (Kolodzie 1980):

$${\text{Log}}R35 = 0.732 + 0.588({\text{Log }}K) - 0.864({\text{Log}}\,\varnothing )$$
(1)

The R35 index corresponds to the pore radius of a rock imbibed by 35% mercury saturation. In Eq. (1), permeability (K) is input in millidarcy (mD), porosity (Ø) is in percent (%), and R35 in microns (µm). Winland empirically developed this index and has been proven to be very valuable as a cut-off standard to represent commercial hydrocarbon reservoirs and classify flow units. Martin et al. (1997) and El Sayed et al. (2021) introduced four flow units by categorizing R35 as follows:

  1. 1.

    Megaport-flow units: R35 > 10 microns

  2. 2.

    Macroport-flow units: 2 < R35 < 10microns

  3. 3.

    Mesoport-flow units: 0.5 < R35 < 2 microns

  4. 4.

    Microport- flow units: R35 < 0.5 micron

Another method proposed by Lucia (1995, 1999) divided the pore spaces in carbonate rocks into interparticle and vuggy porosity. Non-vuggy carbonate rocks were classified into three groups based on rock-fabric/petrophysical characteristics and the size of pore space. In contrast, vuggy pore spaces were classified into two categories: separated and touching vug. The current method is exclusively derived from the non-vuggy samples. Moreover, the carbonate fabrics were described as grain-dominated and mud-dominated instead of grain-supported and mud-supported, as introduced by Dunham (1962). Integrating the Lucia method with the R35 technique for unraveling these complex carbonates would be interesting, but it is beyond the scope of this study (Martin et al. 1999).

Amaefule et al. (1993) recommended the third popular approach. This method modifies Kozeny’s (1927) and Carmen’s (1937) equations, explaining the theory of the mean hydraulic unit radius (rmh) concept for a circular, cylindrical capillary tub as follows:

$$rmh=\frac{r}{2}$$
(2)

where rmh is in (µm).

The mean hydraulic radius can be related to the surface area per unit grain volume (SG) and effective porosity (Øe) as follows:

$$SG=\frac{2}{r}\left(\frac{\varnothing e}{1-\varnothing e}\right)=\frac{1}{rmh}\left(\frac{\varnothing e}{1-\varnothing e}\right)$$
(3)

in which SG is in (µm−1) and Øe is a fraction. The generalized form of the Kozeny-Carmen function is given by Eq. (4).

$$K=\frac{{\varnothing e}^{3}}{{(1-\varnothing e)}^{2}}\left[\frac{1}{Fs{\tau }^{2} {(SG)}^{2}}\right]$$
(4)

where Fs is the shape factor, which is equal to 2 for a circular cylinder and τ is the tortuosity. In addition, the term \(Fs{\tau }^{2}\) is referred to as the Kozeny constant. Dividing both sides of Eq. (4) by Øe and taking the square root of both sides will be

$$\sqrt{\frac{K}{\varnothing e}}=\left[\frac{\varnothing e}{1-\varnothing e}\right]\left[\frac{1}{\surd Fs\tau SG}\right]$$
(5)

where K is in µm2. In case permeability is presented in mD, (In SI unit: 1 mD = 0.000986923 µm2), then a new parameter will be defined as follows:

$$RQI=0.0314\surd \frac{K}{\varnothing e}$$
(6)

In this context, the reservoir quality index (RQI) is measured in micrometers (µm). The constant value of 0.0314 represents the conversion coefficient from square root millidarcy (√mD) to micrometers (µm). The following equation defines the normalized porosity (ØN):

$$\varnothing N=\left(\frac{\varnothing e}{1-\varnothing e}\right)$$
(7)

The flow zone indicator (FZI) is given by

$$FZI=\frac{1}{\surd Fs\tau SG}=\frac{RQI}{\varnothing N}$$
(8)

where FZI is in µm.

Substituting the above variables into Eq. (5) and taking the logarithm of both sides results in

$$\log RQI = \log \varnothing N + \log FZI$$
(9)

When plotting the Reservoir Quality Index (RQI) against the normalized porosity (ØN) on a log–log scale, each flow unit is expected to produce a straight line with a unit slope. The intercept of this line with the ØN = 1 slope is known as the Flow Zone Index (FZI), which serves as a unique parameter for each hydraulic unit. These indices are associated with routine core data, such as porosity and permeability. Alam et al. (2011) suggested that the FZI is suitable for rocks with high permeability and small surface areas, rather than chalky rocks. On the other hand, Riazi (2018) encountered difficulties in identifying flow units using the FZI index.

The fourth well-known approach is a graphical method based on the stratigraphic modified Lorenz plot (SMLP) introduced by Gunter et al. (1997). This method quickly assesses flow units based on a geological and petrophysical framework, such as pore types, storage capacity, flow capacity, and reservoir speed manner. This method identifies the flow units based on their geological framework and the stratigraphic order. The initial flow units should be verified according to the geologic framework, R35, and petrophysical attributes. Nabawy (2021) improved the stratigraphic modified Lorenz plot (I.S.M.L.P) by finding a relation between the tangent dip of flow units using graphical and/or mathematical methods corresponding to their contribution to the total reservoir flow. The introduced equation to mathematically calculate of flow units’ dip is as follows:

$$\tan \theta = \frac{{K.h2 - K.h1}}{{\varnothing .h2 - \varnothing .h1}}$$
(10)

where tan θ is the slope of the flow unit line segment of the start point (Ø.h1, K.h1) and endpoint (Ø.h2, K.h2) representing the (x,y) coordinate flow capacity (K.h) and storage capacity (Ø.h) on the Y-axis and the X-axis, respectively. The proposed method has been verified using six carbonate and clastic wells drilled in the Sudan and Egypt.

Skalinski and Kenter (2015) conducted a study on integrating conventional rock type approaches. They argued that the failure of these approaches is due to the lack of incorporating diagenetic alterations, inaccurate transfer of rock types from core to log domain, and neglecting the fracturing effect. The authors proposed a comprehensive procedure to determine the petrophysical rock type (PRT), which could control the reservoir's static and dynamic properties. This procedure consists of eight steps based on integrating data scenarios, depositional facies and pore type prediction, petrophysical characteristics, and validation. Additionally, the instability of carbonates against diagenetic and tectonic events led to the classification of petroleum reservoirs into four types according to the flow controller (i.e., depositional, diagenetic, fractures, and hybrid).

Mirzaei-Paiaman et al. (2015) conducted recent research and found that all forms of the RQI-FZI (Amafule et al. 1993; Nooruddin & Hossain 2011; and Izadi and Ghalambor 2013) derived by the mean hydraulic radius are not successful in classifying flow units. This is because the FZI was derived based on grain diameter characteristics instead of a function of pore size (Mirzaei-Paiaman et al. 2018). This inaccuracy is intensified in rocks with complex pore geometry. They suggested that the base form of the K-C equation is preferred. In this regard, the FZI formulation was modified and called FZI* (FZI star) as follows:

$${\text{FZI}}^{*} = \frac{rmh}{{\sqrt {Fs\tau } }}$$
(11)

in which FZI* is in µm\(.\)

The numerical values of FZI* for each plug can be calculated by using the following equation.

$${\text{FZI}}^{*} = { }0.0314\sqrt {\frac{K}{\varnothing }} = \frac{rmh}{{\sqrt {Fs\tau } }}$$
(12)

Based on this approach, all samples with equal FZI* values will lie on a straight line provided the values have been shown on the log–log plot of 0.0314√K versus√∅. However, the accuracy of this approach depends on the pore structure network. If all reservoir rocks contain similar pore geometry or the same Fsτ, the FZI* would be more successful in linking pore geometry with representative PC profile (Mirzaei-Paiaman et al. 2018). However, this ideal assumption may not be predictable, particularly within highly heterogeneous rocks due to the diagenesis alteration effect on the carbonate rocks. The authors also recommended various cluster techniques (e.g., Discrete rock type (DRT), probability plot, etc.) as an influencing factor on the accuracy of rock typing.

Numerous authors have attempted to identify flow unit/rock types in different reservoirs (Beiranvand et al. 2007; Kadkhodaie-Ilkhchi and Amini 2009; Rahimpour-Bonab et al. 2012; Kadkhodaie-Ilkhchi et al. 2013; Irajian et al. 2017; Nabawy et al. 2018; Rafiei and Motie 2019; Al‑Jawad and Hassan Saleh 2020; Azadivash et al. 2020; Ranjbar‑Karami et al. 2021; Al-Dujaili et al. 2021; Abu-Hashish et al. 2022; Al-Dujaili 2023). Moradi et al. (2017) employed Sw alongside various static measurements such as Neutron, Density, Sonic, Gamma-ray, and lithologic indices. This approach focused on Sw rather than irreducible water saturation (SWIR) and did not consider the transition zone in the reservoir. Due to the dynamic nature of water saturation, different rock types along the transition zone of the reservoir may display varying saturations, even if static parameters remain consistent.

Dakhelpour‑Ghoveifel et al. (2018) introduced a rock typing sketch with fixed ranges of SWIR due to failed flow unit delineation attempts in Iranian reservoirs. Ali et al. (2019) utilized well-log data for reservoir characterization and Bulk volume of water (BVW) to define grain size. Hussain et al. (2022) used a clustering technique called the self-organizing map (SOM) and the Buckles plot (Buckles 1965) to identify reservoir rock types and the reservoir zones that achieved SWIR conditions. The Buckles plot displays curved lines obtained by multiplying SWIR and porosity. Plotting these numbers on a log–log graph produces a series of sloping lines. The constant BVW is associated with zones containing equal pore geometry and the same rock type (Riazi 2018). This phenomenon occurs when rocks originate from above the reservoir's transition zone. A lower BVW typically indicates a larger grain size and superior petrophysical quality. In contrast, rocks within the transition zone may span multiple BVW lines. Similar outcomes can be observed when water is extracted from the reservoir through fractures.

Rebelo et al. (2022) conducted a comprehensive study and found that the R35 method is more reliable than FZI for application in complex carbonate reservoirs. Some authors have integrated flow unit methods with specific petrophysical attributes, such as Nuclear Magnetic Resonance (NMR) (Wang et al. 2022; Al-Dujaili et al. 2023). Then, Hussain et al. (2023) recommended the use of neural networks (ANN) and fuzzy logic (FL) for porosity prediction.

The novelty of this study is to reveal the role of various clustering techniques in improving flow unit classification and identifying influential reasons for the failure of conventional flow unit classification methods. Additionally, it aims to classify flow units through an alternative approach that incorporates SWIR and porosity, with potential applications in similar settings and accommodating the reservoir's saturating transition zone. Finally, the paper will characterize the influential geological and petrophysical factors in the distribution and production of the flow units. The current research emphasizes the clustering approach to resolve common issues affecting flow unit classification in heterogeneous carbonate reservoirs (Table 1).

Table 1 The list of flow unit classification methods employed in this research
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Geological setting

Structural framework

The Zagros Basin stretches across a vast area from northwest to southern Iran, and many Iranian oilfields are located in the southwestern part of the Zagros structural zones, specifically in the Dezful Embayment, which is structurally divided into northern and southern sedimentary basins. The Gulkhari (GL) Oilfield was discovered in the southern Dezful Embayment (S-D-EMD) in 1964, and its structure has a tortuous shape, with a production history of only 36 years. It has a spread of approximately 45 km long and 7 km wide. Two regional studies focusing on magnetometry data (Farahzadi et al. 2019) and the major structural lineaments (Seraj et al. 2020) have indicated paleo-highs' role in the basement architecture along the Dezful embayment, implying variation in basement depth in the western and central areas of the Gulkhari Oilfield (Fig. 1).

Fig. 1
figure 1

Index map: Colored contours show the magnetic basement depth of the Dezful Embayment. Red lines show the location of major faults. Abbreviations: N-D-EMB (North-Dezful Embayment), S-D-EMB (South-Dezful Embayment), HD-F (Hindijan-Nowrouz Fault), KG-M-F (Kazerun-Kharg-Mish Fault). KD: Kabud; LS: Labe-safid; DH: Dehluran; MIS: Masjed-Soliman; AS: Kohe-Asmari; AZ: Ahwaz; AT: Abtymour; AJ: Aghajari; PZ: Pazanan; RS: Rage-sefid; TAG: Tangou; GS: Gachsaran; BH: BiBi hakimeh; BK: Binak; GL: Gulkhari; NR: Nargesi; DR: Dara. (Adopted after Murris 1977; Farahzadi et al. 2019)

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Stratigraphic framework and petrophysical nature

The Asmari (Oligo-Miocene) and Jahrum (Middle Eocene) successions collectively constitute a shared reservoir, containing all the hydrocarbons produced in the Gulkhari Oilfield (Fig. 2). A sedimentary study conducted by Mohseni et al. (2016) revealed that the Asmari (As) primarily accumulated in the inner-mid ramp, while the Jahrum (Ja) succession consists of deeper sediments and was deposited in mid-outer ramp settings. The Jahrum Formation underlies the Asmari Formation, with a disconformity demarcating their boundary in numerous regions of the Zagros Basin (Motiei 1993; Ahzan et al. 2020; Van Buchem et al. 2010).

Fig. 2
figure 2

The simplified stratigraphic column of the Asmari and Jahrum formations in the eastern margin of the Dezful Embayment is shown. The wavy dashed line indicates the regional disconformity that affected the Asmari and Jahrum formations. Please refer to the table on the right to determine the lithology index

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A regional study has classified the onshore and offshore reservoirs in southwestern Iran into six groups based on their sedimentary and diagenetic history (Esrafili-Dizaji and Rahimpour-Bonab 2019). According to this classification, the creation of these reservoirs has been generally influenced by paleoclimate conditions, resulting in various types of reservoirs such as reefal carbonates, karstified, dolomitized, fractured carbonates, and sandstones within the related stratigraphic sequence. The production performance of these groups is influenced by different facies, with ooidal grainstones dominating the Asmari, Khami (Jurassic), and Dehram (Permian–Triassic) groups. Additionally, minor facies like Rudist-rudstone/grainstone exhibit high oil storage and production capacity within the upper Cretaceous reservoirs, such as the Sarvak Formation. Moreover, the Jahrum succession in certain oilfields, such as Gulkhari and Nargesi, hosts the main volume of produced hydrocarbons.

In the Asmari succession of the Zagros Basin, tectonic forces resulting from compaction and lateral stress imposed by the collision, obduction, and thrusting of the Arabian Plate on the Eurasia Plate have generated distinct fractured reservoirs. The northeast (NE) movement of the Arabian Plate is attributed to its rifting away from the African Plate along an active divergent ridge system that forms the Red Sea and the Gulf of Aden, leading to the simultaneous arid climate during the Cenozoic (Alsharhan and Nairn 2003).

A comprehensive logging presentation (Fig. 4) from the studied wells in the GL Oilfield reveals the lithologic differences between the Asmari and Jahrum formations. The Asmari notably contains dolostone, calcareous dolostone, anhydride patches, and thin shale layers, while the Jahrum predominantly consists of limestone and dolomitic limestone. Layers with the highest petrophysical potential (i.e., High Ø, Low Sw) are found adjacent to disconformity (i.e., lower As and upper Ja), possibly due to the effect of the dissolution phase during sea level fall. Another reason can be attributed to the mid-ramp facies at the top of the Jahrum Formation. The characteristics of these facies will be explained in the following sections. Additionally, some areas in the middle Asmari and Jahrum formations exhibit fair porosity content, although their frequency is not constant.

Material and methods

The data utilized in this study are derived from three wells drilled in the Gulkhari Oilfield (Gl-10ST-1, Gl-11, and Gl-13), which include core samples, well log suites, and thin sections (Fig. 3). The sample counts in the mentioned boreholes are as follows: 333, 234, and 530 core plugs, collectively providing 500 thin sections. Additionally, the reservoir stratigraphy was categorized into ten qualitative lithologies (Litholog) (Table 2) based on interpreted porosity and mineral contents (e.g., limestone, dolomite, anhydrite, and claystone). The previously identified facies and their interpretation (Mohseni et al. 2016) were reassessed using petrographic results from thin sections. The reservoir layering (zonation) was also depicted according to internal reports from the Exploitation Company (NISOC) (Fig. 4). Furthermore, the intensity of heterogeneity was assessed using the methods of Fitch et al. (2015) and El-Deek et al. (2017) to reveal characteristic variations throughout the reservoir (Figs. 5, 6 and Table 3). Subsequently, the density (Rhob) log was employed for porosity calculation, and the Archie equation (Schlumberger 1989) was used for water saturation determination. The available core analysis data for this study are as follows:

  1. 1.

    Helium porosity and air permeability measurements were obtained from wells Gl-10ST-1, Gl-11, and Gl-13 to classify the flow units.

  2. 2.

    Ten mercury injection tests representing various rock fabrics were conducted to calculate the variety of pore size distribution, measured by AutoPore 3–9420 equipment (Fig. 7a).

  3. 3.

    The centrifuge capillary pressure (Pc) profiles belonging to Gl-10ST-1 and Gl-11 were analyzed using Beckman (L8-60 M/P) equipment (Fig. 7b) to validate the classified flow units. Additionally, oil–water relative permeability (Kro-w) tests were conducted on 40 plugs gathered from Gl-11.

  4. 4.

    Flow unit classification methods and clustering techniques were integrated to accurately link the pore geometry of rock with the related Pc profile.

  5. 5.

    The Pc data were utilized to derive the reservoir transition zone. An optimal regression was established between SWIR and porosity in the above interval of transition zone, and the resulting function was applied to predict SWIR throughout the entire intervals of the studied wells.

  6. 6.

    The self-organizing map (SOM) technique was employed to cluster, predict, and propagate the finalized flow units throughout the studied boreholes using GEOLOG 7.2 software.

  7. 7.

    Wettability (w) and saturation exponent (n) were compared with related pore size using Amott and resistivity index procedures, measured on 12 samples, to recognize the wettability heterogeneity throughout the porous media of the reservoir. All tests were performed by the Core Laboratory of RIPI in Tehran, Iran. The executed procedure in this study is stepwise presented in Fig. 8.

Fig. 3
figure 3

Structural contour map displaying studied wells’ locations on the top of the Asmari reservoir. The contours and colors represent the sub-sea depth in meters. Scale: 1:200,000

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Table 2 Qualitative stratigraphic column (litholog) based on the value of 10% porosity and mineral component types used in this study (for more information please refer to text and Fig. 2)
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Fig. 4
figure 4

Composite log showing cored interval, interpreted facies, reservoir zonation (Layering) and petrophysical nature of studied wells. List of tracks from left to right (Gamma-ray, Formation’s age, Formations name, Depth, Reservoir’s zonation, Cored interval, Facies, Log (blue curve) and Cored (dot red) Porosity, Oil (red) and Water (blue) saturation curves, Density (Rhob)-Neutron (Nphi) logs, Qualitative stratigraphy (litholog). The wavey dash line indicates the regional disconformity affected on the formations. The table below of the figure introduces the facies colors. For identification of stratigraphy (litholog) please refer to Fig. 2

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Fig. 5
figure 5

Histogram comparing the heterogeneity within the studied wells. A distinct color separates the reservoir's zones

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Fig. 6
figure 6

Dykstra-Parsons coefficient of core permeability variation (VDP) suggests a high degree of reservoir heterogeneity among the studied wells

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Table 3 Heterogeneity Calculation within the studied wells separated by the reservoir’s zonation
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Fig. 7
figure 7

a Mercury injection capillary pressure (MICP) procedure performed by Auto pore 3-9420 equipment, b Centrifuge capillary pressure procedure performed by Beckman (L8-60 M/P) equipment

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Fig. 8
figure 8

The stepwise tasks to classify flow units. Please refer to the text for more details

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Results

Descriptive studies

Heterogeneity analysis

Three statistical techniques were utilized to estimate reservoir heterogeneity of porosity and permeability. The high-scale heterogeneity calculation for the reservoir's zonation is presented in Table 3 and Fig. 5, while the total heterogeneity for the studied wells is determined using the Dykstra-Parsons method, as shown in Fig. 6. A comparison of the calculated statistical values with the classifications used by Fitch et al. (2015) and El-Deek et al. (2017) reveals that heterogeneity is prevalent and extensive throughout the reservoir. This implies that the reservoir's pore geometry and pore-to-throat connection are anisotropic, leading to the presence of diverse flow units.

Facies interpretation

Essentially, the pore geometry of rocks is formed by sedimentary fabric and diagenesis overprints. When interpreting thin sections, the blue dye impregnation technique was used to enhance the visibility of pore geometry. Figure 9 displays a photomicrograph taken from thin sections of representative facies. Based on microscopic information, changes in the mineral composition (e.g., anhydrite, dolomite, and calcite) of the facies, shape, size, and related distribution of pores, grain sizes, and their forms are observed. Further details about sedimentological description and related information are tabulated in Table 4.

Fig. 9
figure 9

Photomicrograph of rock groups introduced by this study. The number illustrated in the top-left corner of the images indicates the sample number. The predominant rock fabric characteristic is mentioned by black words. For more information (e.g. well number, formation, and description) please refer to Table 4 and the text

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Table 4 The general information, facies, predominant diagenetic alteration, lithology (Litho), porosity type, and depositional environment for representative rock groups
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Relationship of facies with permeability and porosity distribution

According to the core data, the variation in porosity can be classified into three clusters based on the multimodal shape of the histogram (Fig. 10a). The contribution of each facies varies within the porosity clusters, depending on their related pore geometry. The first cluster, with Ø < 0.05, is predominantly spread through facies 1 (Peritidal) and 2 (Lagoon). This cluster is also found in facies 3 (Bar), albeit less frequently. The second porosity cluster has contributions from all facies containing Ø = 0.05–0.10; however, facies 5 (Mid-outer ramp) is only observed in this cluster. The third cluster, with Ø > 0.10, is chiefly distributed through facies 4 (Mid-ramp) and is formed by various interconnected effective porosities (e.g., intergrains, fractures, and touching-vugs) (Fig. 9). This facies mainly comprises a plentiful amount of echinoid debris and nummulite deposited under unstable water energy conditions (Table 4). The related matrix is mainly composed of echinoid debris, with the rare cementing phase, except around some bioclasts. High volumes of crushed echinoid bodies were distributed throughout the middle-ramp margin during stormy conditions when the mud content was washed out. Under calm conditions, micro-sized sediments settled between coarse grains, forming a packstone rock. However, some volume of the third cluster uncommonly congregated in facies 1 and 2, where the dissolution effect improved the original micro-sized pore network.

Fig. 10
figure 10

Core’s porosity (a) and permeability (b) distribution classified by facies through studied wells

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The diverse nature of the reservoir, resulting from various diagenetic alterations, leads to permeability heterogeneity (Fig. 10b). While the permeability distribution is spread across three clusters, facies do not solely control the permeability classification. For instance, the first cluster (K < 0.1 mD) contains facies 1, 2, and 5, while clusters 2 (K = 0.1–1.0 mD) and 3.0 (K > 1.0 mD) consist of facies 1, 2, 3, and 4. Facies 5 and 4 exhibit distinctive permeability, likely influenced by sedimentary factors such as rock fabric (e.g., mud content) and burial diagenesis (e.g., physical and chemical compaction) (Fig. 9, sample 136V, 150H, 195H). Therefore, the pore geometry heterogeneity of the related clusters substantially increased due to the influence of fracture and dissolution on facies 1 and 2. The role of facies 3 can be negligible due to the low frequency. Additionally, Fig. 11 shows the overlapping distribution of Ø and K in all facies, corresponding to pore geometry complexity throughout the reservoir.

Fig. 11
figure 11

Porosity versus permeability distribution classified by facies. Data belong to three wells (Gl-10ST-1, Gl-11, Gl-13). The latter well contains only RCAL data. The black squares show the representative mercury injection samples used in this study, units: permeability (mD), porosity (fraction)

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Petrophysical study

Information based on log readings

Porosity and water saturation

Porosity and water saturation are two key parameters that reveal the petrophysical quality of a reservoir. Porosity (Ø) was derived using the density (Rhob) log, while water saturation (Sw) was calculated using Archie's equation (Schlumberger 1989). The equations are as follows:

$$\varnothing = \frac{{\rho_{m} - \rho_{b} }}{{\rho_{m} - \rho_{f} }}$$
(13)
$$SW = \left( {\frac{a*Rw}{{Rt*\varnothing^{m} }}} \right)^{\frac{1}{n}}$$
(14)

where ρm = matrix density (Kg/m3), which was individually chosen equal to 2860 for the As Formation and 2740 in the Ja Formation according to core analysis. ρf = fluid density (mud filtrate) that was chosen as 850 (Kg/m3) because of using oil-based mud. ρb = formation bulk density (log value) in Kg/m3. Finally, the result was compared by the core’s porosity (Fig. 4, 8th track). The related symbols to estimate Sw (Fig. 4, 9th track) are represented as follows: Rt = true resistivity (ohm.m) collected by log reading. Rw = formation water resistivity (ohm m) at formation temperature, which corresponds to 0.018 in our case. m = cementation exponent, which was acquired in the As and the Ja equal to 1.8 and 1.5, respectively. a = tortuosity factor that was individually used in the As (1.5) and in the Ja (2.5). n = saturation exponent, which was determined equal to 2.45 for both formations based on core analysis. The value used is an optimal mean value (Fig. 15) because there were not enough adequate samples from the As and Ja formations. The result was compared based on core porosity (Fig. 26, 9th track).

Information based on coring data

Pore geometry

Figure 12 shows the classification of rocks into six groups based on individual PTS and the mode of curves. The first and second groups have macro-mesopore throats, a porosity of more than 0.1, and a permeability of more than 0.1mD. However, the pore throats decrease until the nano in size. The effective porosity distribution in these groups is dominantly formed by the interconnected vug and intergranular pores (Fig. 9; Table 4). These rocks were deposited in the proximal mid-ramp basin. The fragmentation of bioclasts occurred due to reworking wave processes in a relatively shallow water environment, likely above the storm wave base.

Fig. 12
figure 12

Pore throat radius distribution showing 6 rock groups according to pore throat size (micron), porosity (%), and permeability (mD)

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Groups 3, 4, and 5 contain micro to nanopore throats, and the porosity is between 0.05 and 0.01. The predominant pore type in these groups is controlled by the matrix with occasional dispersed moldic spaces, which would be connected through the muddy matrix of the host rock. The sedimentary environment of these groups is different. For example, group 3 shows mid-outer ramp features, whereas group 4 composes a series of biological allochems indicating the mid-ramp basin influenced by physical compaction. The sedimentary environment of group 5 reflects a sea-quiet energy condition (e.g., lagoon) because of the micritic fabric existence and a few imperforate benthic foraminiferas. In addition, the anhydritic cement fills the spaces between grains. Group 4 shows less variance in PTS than other groups among the above RGs.

Group 6 has nanopores with the lowest porosities, less than 0.05. The microscopic interpretation suggests that destructive diagenetic alteration, including anhydritization, physical compaction, and chemical compaction, has caused a tight pore network, resulting in no effective porosity in this group. The dissolved fluid is likely released into the surrounding environment and settles as a filling cement phase. It is worth noting that rocks were deposited in distinct basins, although their pore network is comparatively similar due to destructive diagenetic alteration. For instance, the sample of 40H is formed by a peritidal basin composing a dolomudstone fabric. Still, the sample of 195H illustrates a fossiliferous packstone fabric precipitated in the mid-outer ramp basin.

Data visualization on a semi-log scale (Fig. 13) indicates that the reservoir's rock groups can be merged into three classes based on the Winland R35 index and pore size distribution.

Fig. 13
figure 13

The Semi-Log plot of Mercury capillary pressure represents six rock groups and, the R35 size of their pore throat radius. The vertical red arrow shows the point of 35% mercury saturation called R35

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Buckles (BVW) values

Figure 14 presents two scenarios. Initially, the reservoir rocks exhibit varying pore geometry, as determined by pore geometry analysis in the preceding chapter. Additionally, the rocks likely originate from the reservoir's transition zone. Since the SWIR values are derived from Pc tests, the influence of water charging through fractures is not considered. This is also a factor in repealing the transition zone impact.

Fig. 14
figure 14

Buckles Plot displays the BVW numbers of all used Pc samples obtained through centrifuge test

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Relationship between pore size and wettability

The Amott and saturation tests were used to investigate the impact of pore size on the wettability variation of the reservoir. The samples have porosity ranging from 8 to 22% and permeability from 0.2 to 30.0 mD (Fig. 15). The results show that oil wettability increases with increasing pore size (√K/Ø) and saturation, indicating that regions or rocks with high porosity tend to be oil-wet. Consequently, these inherent characteristics and the fracture network are likely to contribute to oil production with a water cut. Production tests support this prediction. It appears that vuggy porosity should be considered an exception that does not exhibit the expected performance.

Fig. 15
figure 15

The relationship between wettability (w), saturation exponent (n), and pore size (√K/Ø). The oil wettability increases as pore size and saturation increase

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The results from flow unit classification and clustering methods

The study employed various flow unit delineation methods, including Lucia (1995), RQI/FZI (Amafule et al. 1993), Winland R3 (Martin et al. 1997), FZI* (Mirzaei-Paiaman et al. 2015; Ranjbar-Karimi et al. 2021), and I.S.M.L.P (Gunter et al. 1997; Nabawy 2021), which were integrated into four clustering schemes. These schemes consist of DRT, probability plot, standard chart (specifically for Winland R35), and global hydraulic elements (GHE) (specifically for the FZI method), as introduced by Corbett and Potter (2004). The DRT function applied in this study is as follows:

$${\text{DRT}}\;{\text{no}}. = {\text{ROUND}}\;\left( {{\text{2LN}}\;\left( {{\text{index}}} \right) + {\text{A}},\;0} \right)$$
(15)

in which: A is a constant. In our case A is equal to 10.6 for FZI, and for R35 and FZI* it is set to 7.1 and 10.1, respectively. Then, the flow units classified were compared with capillary pressure profiles. The outcomes achieved are individually presented through relevant figures using the following methods.

RQI/FZI method

The FZI values were calculated using Eq. (8) for all plugs, and the results were classified using clustering techniques, including DRT, probability plot, and GHE. Finally, the flow units and individual capillary pressure were compared (Fig. 16). Although the various clustering methods derive different classes of flow units, none can establish a proper link between capillary pressure profiles and the classified flow units. This occurs due to the assumption of a constant value for the Kozeny constant (Fsτ2) within a specific flow unit or a porous medium containing a layer of uniform particles (spheres) (Amafule et al. 1993).

Fig. 16
figure 16

Flow unit classification using FZI/RQI method, a Flow unit classes using the DRT technique, b Divided capillary pressure data using DRT, c Clustering based on probability, d Divided capillary pressure data based on probability, e FZI clustering using GHE, f Divided capillary pressure using GHE. Note The capillary data is not available for some flow units. The right column indicates the sample name and flow unit (fu) number

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Lucia method

The groups identified by the Lucia method were compared with the corresponding Pc measurements (Fig. 17). The widespread distribution of Pc profiles, as evidenced by the distinct colors, implies that the current method fails to identify comparable pore geometries in the analyzed rocks. This discrepancy can be attributed to the impact of fracturing and vuggy porosity, leading to inaccurate outcomes.

Fig. 17
figure 17

Flow unit classification using the Lucia method, a Cross plot of Rock-Fabric classification, b Individual capillary pressure segregated based on identified flow units

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Winland R35 method

There are three available approaches to calculate R35. The first method involves using Eq. (1), followed by classifying the obtained results using a clustering technique. The second approach is the mercury injection test, which determines the pore rock filled with 35% mercury. The third technique utilizes the standard charts introduced by Martin et al. (1999). In this study, three clustering techniques were used to classify the flow units, including the DRT, probability, and standard chart (Fig. 18). Based on the results, the standard chart shows a better outcome than the other methods. In contrast, the DRT has the worst performance. However, the classification of failed flow units is likely due to the dispersed vugs and fractures.

Fig. 18
figure 18

Flow unit classification using Winland R35 method, a Clustered flow units using R35 chart, b Capillarity pressure data divided by R35 Chart, c Clustered flow units based on the DRT method, d Capillarity pressure data divided by DRT, e Probability plot showing 5 classes of Flow units, f Capillarity pressure data divided by probability method. Note The R35 values used in the clustering of DRT and probability (c and e) were calculated according to Eq. 1

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FZI* method

The FZI values were calculated utilizing Eq. (12) for all samples, followed by clustering based on DRT and probability techniques. The resulting clusters were plotted on a logarithmic graph using 0.0314√K versus √∅ values. The identified flow units and capillary pressures were then compared (Fig. 19). The overlapping of flow unit classes in the irreducible water saturation (SWIR) range suggests that this method may not provide a clear distinction between different pore shapes, similar to previous approaches. Mirzaei-Paiaman et al. (2015, 2018) have identified several reasons for this inaccuracy.

Fig. 19
figure 19

Flow unit classification using FZI* method, a Clustered flow unit using DRT, b Capillarity pressure data divided by DRT method, C Clustered flow units based on the probability method, d Capillarity pressure data divided by probability plot

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Firstly, the Kozeny and Carmen equation's simplifications may lead to discrepancies. If the correct Kozeny constant is unavailable, the FZI* index cannot be accurate. Secondly, the lab measures the total connected porosity of the rock, which is not always the same as the effective porosity that contributes to fluid flow. This substitution can lead to significant errors in the calculation of FZI*, particularly in carbonate rocks undergoing diagenetic alteration. Thirdly, the presence of fractures can affect the results. Lastly, the loss of the original pore network due to high pressure during capillary pressure testing to fill all pores can also cause deviations.

In the subject of clustering, it seems that the probability plot comparably yields better results than the DRT in low to fair porosity rocks in the FZI* method.

Improved stratigraphic modified Lorenz plot

The first step involved stratigraphically ordering the data, followed by calculating the storage capacity and flow capacity for each sample. Subsequently, the obtained flow units were classified based on the corresponding tangent dip. Finally, a comparison was made between the flow unit and relevant capillary measurements (Fig. 20). The results indicate no expected separation between rocks with unique pore geometry. This failure may have occurred due to the influence of variables such as porosity and permeability, or possibly due to the impact of fractures or the distribution of vugs.

Fig. 20
figure 20

Flow unit classification using the I.S.M.L.P method (Nabawy 2021). a Identified flow units and their categorization based on tangent dip, b Distribution of the corresponding Pc. curves. Abbreviations: C: Conductor; B: Barrier; S.c: Semiconductor; Sp.c: Superconductor. Note Numbers on the I.S.M.L.P illustrate the flow units’ numbers

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Flow unit classification using SWIR

Estimation of the reservoir’s transition zone

The water saturation (Sw) is a dynamic characteristic of rock, representing the volume of residual water retained in porous media. This attribute depends on rock quality and the height of the rock above the free water level (FWL) (Martin et al. 1997). Additionally, irreducible water saturation and specific surface area (S) increase with decreasing permeability and porosity for most reservoirs (Basbug and Karpyn 2007; Mohammadi et al. 2020). Any change in the rock fabric leads to a change in S, and there is a direct relationship between S and SWIR, contrasting with the inverse relationship between S and permeability. Therefore, SWIR variations can directly reflect the dynamic trait of the rock, emphasizing the importance of SWIR for flow unit classification.

It's important to note that the reservoir will only gain SWIR above the transition zone (Sw = SWIR). Therefore, instead of using capillary pressure averages, it is recommended to apply petrophysical data from the area above the transition zone to achieve an equal oil-in-place volume in simulation models, considering porosity as a weight factor (Dakhelpour-Ghoveifel et al. 2018). The accurate porosity cutoff can be determined by interpreting core porosity, pore geometry, and the Winland R35 index, which expresses better results than other classifying methods in this study. This method can be applied to all reservoirs, as the data is predominantly available.

To define the saturation values above the transition zone, the capillary pressure data obtained from mercury injection (MICP) and centrifuge tests were converted from laboratory conditions to subsurface reservoir conditions (Jennings 1987; Vavra et al. 1992). Subsequently, the height (H) above the free water level was calculated for both cases using the following method:

$$CF = \frac{{\left( {{\text{T}}_{Hg} } \right)(\cos \theta_{Hg} )}}{{({\text{T}}_{O/W} \left( {{\text{cos }}\theta_{O/W} } \right)}}$$
(16)
$$H=\frac{Pc}{(Ww-Wo)(CF)}*\frac{1}{A}$$
(17)

where CF = conversion factor from an air-mercury to the oil–water condition, \({\text{T}}_{Hg}\)= interfacial tension of mercury to the air of 480 (dynes/cm), \({\text{T}}_{O/W}\)=interfacial tension of oil to water in reservoir condition that in the Gulkhari is 15 (dynes/cm) considering API = 27.5 and reservoir temperature 215 (F), \({\theta }_{Hg}\)= the contact angle of mercury to air = 140 (degree), \({\uptheta }_{O/W}\)= contact angle of oil to the water that in this study was assumed to be 0°, H = height of oil column (m), Pc = capillary pressure (psi), Ww = density gradient of formation water (0.48 psi/ft), Wo = density gradient of oil (0.35 psi/ft), and A = a constant to converting depth from feet to meter that is equal to 3.281.

Therefore, after applying the above variables in Eqs. 16 and 17, the height of the oil column corresponding to mercury injection and centrifugation was determined: \(\text{H}=\frac{\text{Pc}}{3.2} \text{and} \text{H}=\frac{\text{Pc}}{0.13}\), respectively (Fig. 21). In Fig. 21b, the capillary pressure curve of all samples predominantly exhibits a vertical shape at a height exceeding 250 m. Consequently, the optimal height of the reservoir transition zone should be 250 m. This change is not visible in the samples of mercury injection (Fig. 21a) due to the very high pressure commonly applied by the laboratory to infiltrate the mercury in nanometer-sized pores.

Fig. 21
figure 21

a Hydrocarbon column calculated by MICP method, b Hydrocarbon column calculated by centrifuge method. The blue arrow indicates the height of the transition zone

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SWIR derivation through wells

In the upper area of the transition zone, the Sw and Ø values were used to estimate an optimal regression between the variables in the training wells (Gl-10ST-1 and Gl-13) (Fig. 22). The derived function and correlation coefficient (CC) between the two variables are as follows:

$${\text{SWIR}} = \left( {1.24317 - 12.9885\left( \varnothing \right) + 43.7095\left( \varnothing \right)^{2} - 57\left( \varnothing \right)^{3} } \right)\;\;{\text{CC}} = 0.78$$
(18)
Fig. 22
figure 22

Water saturation (Sw) versus porosity (Ø) cross-plot for above transition zone in training wells. The red curve shows the regression line

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Then, the above equation was used to predict a synthetic SWIR log through the entire column of studied wells.

Flow units’ clustering and propagation

The synthetic SWIR and porosity were incorporated into the self-organizing map (SOM) clustering algorithm to intelligently predict the flow units using GEOLOG 7.2 software. The SOM is an unsupervised machine-learning technique that operates in two modes: training and mapping, similar to most artificial neural networks. During the training process, an input data set generates a lower-dimensional representation of the input data. Subsequently, the mapping phase classifies additional input data using the generated map.

In the first step of this task, the model was adjusted to accommodate porosity values between 0 and 0.25 and SWIR values from 0.1 to 1.0, as shown in Figs. 10a and 22. Two boreholes, Gl-10ST-1 and Gl-13, were utilized for training, while Gl-11 was kept aside for testing. Nine electrofacies (efac) were initially selected to represent cored and uncored reservoir rocks (Fig. 23a). These electrofacies exceeded the maximum clusters identified by conventional flow unit methods outlined in Chapter 4.3. Subsequently, the flow units were categorized into three groups based on porosity thresholds: less than 0.05, between 0.05 and 0.10, and greater than 0.10. The thresholds were set using the rock groups in Figs. 12 and 13 and the predominant facies in the porosity clusters (Fig. 10a) and SWIR boundaries (Fig. 23b and c). The selected low porosity limit corresponds to the conventional cutoff utilized by numerous petroleum operations companies, to eliminate rocks that are not productive. These finalized flow units were then applied to the training wells (Fig. 24a). Subsequently, the model was tested in the validation well (Fig. 24b), and the results were compared with the representative Pc profiles. The successful identification of flow units in both the training and testing wells indicates the successful training of the model.

Fig. 23
figure 23

a Result of the SOM neural network method in the test wells, b Finalized flow units according to porosity cutoff and SWIR, c The SWIR and porosity distribution of finalized flow units that were propagated through the studied wells

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Fig. 24
figure 24

a Drainage capillary pressure of the training well classified by SWIR and porosity, b Drainage capillary pressure of the testing well, c The representative oil–water relative permeability of classified flow units

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This result indicates that SWIR and porosity can classify reservoir flow units more accurately than previous methods. However, two samples from the training wells do not support this approach, possibly due to errors during laboratory measurements. A representative of the relative water–oil permeability corresponding to each flow unit is depicted in Fig. 24c.

The principal petrophysical information of identified flow units is tabulated in Table 5. It indicates that the fu-3 has a saturation of over 50%, which is the typical cutoff used by many petroleum companies to filter out unproductive reservoir rocks, considering porosity.

Table 5 The petrophysical properties of identified flow units
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Avg average.

Sedimentary facies and flow unit correlation

The study compares the frequency of flow units and depositional facies to identify the primary geological factors influencing flow units (Fig. 25). The histogram reveals that the majority of fu-1 is found in the proximal middle-ramp facies, characterized by intergranular and vuggy porosity, uncompacted fabric, and rare cementation (Fig. 9). Conversely, fu-2 dominates the main volume of the Mid-outer ramp facies. Facies 1, 2, and 3 exhibit a more heterogeneous distribution of flow units.

Fig. 25
figure 25

The relative frequency of reservoir flow units within the depositional facies

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The correlation plot (Fig. 26) illustrates the distribution of flow units (8th track) across the entire interval drilled in the studied wells, taking into account the successions (i.e., Asmari and Jahrum), reservoir zonation, facies, and lithology type (i.e., litholog). Despite minor lateral and vertical variations in flow unit distribution, the plot attributes the primary control on the petrophysical quality of flow units to facies type and rock fabric influenced by diagenetic alteration.

Fig. 26
figure 26

The composite plot shows a comparison between wells 10ST-1, 11, and 13. Tracks name from left to right: 1. GR, 2. Age, 3. Formation, 4. Depth, 5. Reservoir zonation, 6. Rhob & Nphi, 7. Sedimentary Facies group, 8. Flow units, 9. Core and borehole porosity, 10. Lithology column (litholog). The wavy dash line indicates the regional disconformity. The green squares on the Gl-11 show the MICP samples used in this study. The table below of the figure illustrates the facies colors. The blue arrow indicates the transition zone. TVD: True vertical depth; OWC: Oil–water contact

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Discussion

Various information resulting from facies interpretation (Fig. 9), coring porosity and permeability (Fig. 10), MICP (Fig. 12), and statistical methods (Figs. 5, 6 and Table 3) indicate that rock attributes such as pore shape and size, mineral composition, grain packing, arrangement, and sorting are highly heterogeneous. These diversities commonly occur in micro-scale and macro (broad) scales due to the vigorous chemical instability of the carbonate rocks in facing the diagenetic alterations.

The classification methods for flow units are based on some assumptions or simplified models, as discussed in the introduction and results chapters. Such diversities in pore geometry can lead to inaccuracies when attempting to correlate representative Pc profiles with related pore geometries. For instance, since the Kozeny constant can vary from 5 to 100 in actual reservoir rocks, measuring corresponding values for each plug sample is necessary. However, this task is not typically performed in many petroleum companies. Therefore, in the absence of accurate values for each plug, the validity of this assumption decreases, as in highly heterogeneous reservoir rocks, grain (particle) and pore sizes can change along a given plug (Fig. 9). This inaccuracy can occur while using FZI and FZI*, as both methods rely on the Kozeny constant values without considering the impact of containing terms of grain diameter or pore network properties in each one.

The flow units are distributed across all facies with varying frequencies, corresponding to the variation in pore geometry. Figures 10, 11, and 12 demonstrate the complex distribution of pores within the facies, which mainly overlap, in contrast to individual sedimentary basins. For instance, facies 1 and 2 exhibit a widespread and highly variable pore volume ranging from 0.01 to over 0.20, with a dominant frequency of less than 0.05. This realistic nature may result from various diagenetic alterations spread throughout the reservoir. The related pore-throat sizes (PTS) of these facies (samples 40H and 55H, Fig. 12) vary between micro-nano sizes. The pore variation is also observed in facies 3, indicating the most varying frequency of flow units. Notably, facies 1 and 2 show nanopore throat sizes, and some rocks (sample 195H) deposited in the mid-outer ramp (facies-5) acquire nano size due to chemical compaction. Under this circumstance, facies-5 occasionally can also provide the fu-3. As the SWIR variation in the reservoir depends on the height, the flow unit frequency can also change in facies, corresponding to the given height above the free water level (FWL).

The best flow unit is primarily distributed by facies 4, which has the most abundant macro pore network (Figs. 12 and 13). When the relevant pore network decreases due to sedimentary situation overturn, this facies causes the other flow units to appear (Figs. 25 and 26). The same explanation can be correlated with non-diagenetic rocks (sample 292H and 312V) (Fig. 7), belonging to facies 5. Because the original micro-size pores are preserved with low heterogeneity and less porosity variability (0.05–0.10) (Figs. 9 and 10a), these rocks mainly contain fu-2.

The interaction between rock and fluid, known as wettability, introduces another source of inaccuracy. The FZI* method, as noted by Mirzaei-Paiaman et al. (2015), does not account for wetting characteristics. Therefore, when attempting to correlate a class of rocks with the same primary drainage of Pc, it is essential to consider the wettability index for each plug sample. Figure 15 reveals that wettability along rock columns is influenced by pore geometry, a finding also mentioned by Faramarzi-Palangar et al. (2022). It appears that the rock's propensity to absorb more oil increases with the related pore size, leading to a more oil-wetted wettability. The presence of vugs or moldic pores disrupts the wetting ordering of the rock. Additionally, differences in mineral composition, such as anhydrite, dolomite, limestone, and claystone, contribute to non-uniform wettability and saturation, causing capillary continuity deflection.

The third significant factor affecting the quality of outcomes is the influence of fractures and vugs. Despite efforts to filter samples before and after measurement, some hairline cracks may remain. Moreover, fractures and vugs can significantly control the genetic characteristics of productive reservoirs and should not be overlooked. The Winland R35, Lucia, and I.S.M.L.P methods, which rely on porosity type and permeability, are particularly susceptible to errors caused by fractures and vugs.

The study demonstrates that different clustering techniques yield distinct classes but ultimately have little to no impact on regularizing the identified flow units. This could be attributed to the lack of similar hypotheses between the theories underlying flow unit derivation methods and clustering techniques, except for the Winland R35 standard chart. It appears that the DRT yields the highest number of flow unit classes despite producing the least favorable outcome. Conversely, the probability and standard chart result in relatively fewer classes but yield better results when applying the FZI* and the Winland R35 methods, respectively (Figs. 18 and 19). Given these reasons, log suite reading can be a suitable alternative for segregating rocks based on their respective pore geometry, as it minimally deviates from pore geometry complexity, crack lines, and void spaces. Furthermore, neural network techniques can intelligently enable the clustering of rocks with similar pore geometry. The results indicate that SWIR incorporated with porosity segregates the variable pore geometries more effectively than customary flow unit classification methods. In this way, neural network techniques (e.g., SOM) act as an intelligent clustering tool that finds individual pore geometries throughout the reservoir with an emphasis on the varying SWIR due to different given heights above FWL.

Conclusions

This study examined how well the various clustering techniques can improve the customary flow unit classification methods to properly link rock pore geometries with relevant capillary pressure (Pc) profiles in a heterogeneous carbonate reservoir case. The novel findings of this research are as follows:

  1. 1.

    Clustering techniques do not improve the relationship between rock and Pc profiles as their role is conditioned by the flow unit classification schemes. Despite its poor performance, the DRT clustering technique delivers the maximum number of flow units.

  2. 2.

    Among the flow unit approaches, the Winland R35 method demonstrates better performance in correlating high and low-porosity rocks with their respective Pc profiles. However, it fails to associate medium porosity rocks with attributed Pc profiles. The FZI* method appropriately links low porosity rocks and Pc profiles but fails when applied to porous rocks. The Lucia and Improved Stratigraphic Modified Lorenz plot methods fail to find a proper relationship between Pc profiles and varying porosity rocks and facies types. The same result is achieved by the FZI method despite applying all clustering techniques.

  3. 3.

    It seems that some simplifying assumptions to derive conventional flow unit methods are not credible in highly heterogeneous reservoirs due to non-uniform wettability and pore network diversity.

  4. 4.

    The proposed method for flow unit classification, based on simultaneous clustering of irreducible water saturation (SWIR) and porosity, categorizes rocks into three groups by considering their pore size distribution and capillary pressure profiles. This approach accommodates the saturating transition zone of the reservoir and can be effectively utilized in reservoirs where conventional flow unit classification methods fail.

  5. 5.

    The pore geometry network controls the distribution of flow units, influenced by the main destructive and the most constructive diagenetic alteration and depositional facies.

  6. 6.

    Reservoir rocks tend to absorb more oil as their pore size increases, and vice versa. The presence of vuggy or void porosity can distort the distribution of wettability.