Reservoir Rock Typing of Heterogeneous Sandstones Using Machine Learning, Petrophysics, and Core Characterization: A Case Study of the Nubia Sandstone, Gulf of Suez, Egypt

Abstract

Pre-Cenomanian Nubia sandstone is recognized one of the most prolific reservoirs in the Gulf of Suez, Egypt. Accurately determining its reservoir rock type (RRT) is crucial for reservoir characterization and modeling, especially when the reservoir is extremely heterogeneous. This study addresses the critical challenge of characterization in extremely heterogeneous reservoirs by introducing a novel integrated workflow that bridges the gap between traditional sedimentological geology, traditional x-y approaches, and advanced machine learning methods. To achieve this, this study utilizes sedimentological core description, routine core analysis, and conventional well log data from two wells (well A and well B) located in the southern Gulf of Suez, Egypt. The results demonstrate that the complete Nubia interval in the southern Gulf of Suez can be separated into seven distinct lithofacies (LF1–LF7). The first six lithofacies comprise various types of sandstone, while the seventh is composed of shale. The traditional techniques used to predict the RRTs show that the normalized reservoir quality index (NRQI) was the most effective method for predicting the Nubia rock types. The machine learning K–means clustering and self-organizing map (SOM) techniques utilizing raw log data and principal component analysis (PCA) can properly predict the Nubia reservoir rock types. The reservoir quality ranges from poor to very good; well A is dominated by moderate reservoir quality, while well B exhibits predominantly very good reservoir quality. This discernible difference in reservoir quality between the two wells is probably attributed to post-depositional diagenetic processes and variations in sandstone texture.

1. Introduction

Reservoir rock typing is crucial for the reservoir characterization process. For carbonates and heterogeneous clastic rocks, rock typing is essential for reservoir simulation and management. Core analysis is the basis for the direct determination of rock type, whereas well logs are used for indirect determination. Although there is no consistent definition of rock type, Gunter et al. [1] introduced the most acceptable rock type definition. They defined rock type as “units of rock deposited under similar conditions that experienced similar diagenetic processes resulting in a unique porosity-permeability relationship, capillary pressure profile, and water saturation for a given height above free water in a reservoir.” This definition can be seen as a modified form of the rock type definition proposed by Archie [2], who described a rock type as “a formation whose parts have been deposited under similar conditions and have undergone similar processes of later diagenesis.” Determining reservoir rock type involves several methods. These methods may be based on geological attributes (depositional facies and diagenesis), petrophysical attributes (porosity, permeability, grain size, and capillary pressure curves), or large-scale production dynamic data [3].

Two other primary terms are utilized in this study: lithofacies and hydraulic flow unit. Perez et al. [4] defined lithofacies as “mappable stratigraphic units, laterally distinguishable from the adjacent intervals based upon lithologic characteristics such as mineralogical, petrographical, and paleontological signatures that are related to the appearance, texture, or composition of the rock”. The hydraulic flow unit (HFU) concept was developed to identify and characterize rock types based on geological and physical parameters at the pore scale. Ebanks [5] defined hydraulic flow units as “a mappable portion of the total reservoir, within which geological and petrophysical properties that affect the flow of fluids are consistent and predictably different from the properties of other reservoir rock volumes.” Consequently, lithofacies define the original geological rock types based on the visible physical, mineralogical, and biological features observed in core samples, which diagenesis modifies into distinctive reservoir rock types (RRTs). These RRTs are then mathematically grouped into HFUs.

Based on capillary pressure curves, several authors investigated the relationship among permeability, porosity, and pore throat radius [6,7,8]. The authors of the first two references formulated empirical equations known as R₃₅ (the effective pore system that dominates flow through a rock corresponds to a mercury saturation of 35%), and equations were formulated for R₂₀ by the author of the third (Table 1). Additionally, Buckles [9] demonstrated that multiplying the initial water saturation (S_wi) by porosity (ϕ) yields a constant (C) that characterizes a distinct rock type. The Buckles plot can be displayed at either an arithmetic or logarithmic scale.

A major breakthrough in permeability prediction and HFU delineation was achieved by Amaefule at al. [10]. Based on the Kozeny–Carman equation, they introduced a theoretical relationship (Table 1) that related porosity to permeability. Using core porosity (ϕ) and permeability (k), they introduced the terms “reservoir quality index (RQI)” and “flow zone indicator (FZI)”. Each average value of FZI delineates a distinct HFU. Additionally, they introduced an equation that uses the average value of FZI to predict permeability in uncored intervals. To facilitate classification, the discrete rock type (DRT) is usually used to discretize continuous FZI values.

Subsequently, Amaefule et al.’s [10] approach was modified by several researchers [11,12,13,14,15]. Based on the value of FZI, Corbett and Potter [16] introduced a type curve known as the global hydraulic element (GHE). They plotted porosity against permeability in a semi-log plot, resulting in ten distinct GHEs; each GHE is distinguished by its respective color. Additionally, Wibowo and Permadi [17] introduced another type curve plot to determine rock types. This plot utilizes a log–log plot of the pore geometry (k/ϕ^0.5) against the pore structure (k/ϕ³).

Rezaee et al. [18] introduced the term electrical flow unit (EFU) to define zones with similar electrical properties. In their approach, they use the terms “current zone indicator (CZI)” and “the electric radius indicator (ERI)” (Table 1). Subsequent modifications to this approach were proposed by several researchers [19,20,21,22] (Table 1). This approach and its modifications depend principally on the availability of a comprehensive dataset of measured formation resistivity factors.

Table 1. The most common models for rock typing.

Eq. No.	Authors	Type	Equations
(1)	Winland [6]	Empirical	Log (R35) = 0.732 + 0.588 log (k) − 0.864 log(ϕ)
(2)	Kolodzie [7]	Empirical	Log (R35) = 0.9058 + 0.5547 log (k) − 0.9033 log(ϕ)
(3)	Pittman [8]	Empirical	Log (R20) = 0.218 + 0.519 log (k) − 0.303 log(ϕ)
(4)	Amaefule et al. [10]	Theoretical	$\mathrm{F}\mathrm{Z}\mathrm{I}={\displaystyle \frac{\mathrm{R}\mathrm{Q}\mathrm{I}}{{\mathsf{\varphi }}_{\mathrm{z}}}}={\displaystyle \frac{0.0314\sqrt{\frac{\mathrm{k}}{\mathsf{\varphi }}}}{\frac{\mathsf{\varphi }}{(1-\mathsf{\varphi })}}}$
(5)	Nooruddin et al. [12]	Theoretical	$\mathrm{F}\mathrm{Z}\mathrm{I}\mathrm{M}={\displaystyle \frac{0.0314\sqrt{\frac{\mathrm{k}}{\mathsf{\varphi }}}}{\frac{{\mathsf{\varphi }}^{\mathrm{m}}}{(1-\mathsf{\varphi })}}}$
(6)	Rezaee et al. [18]	Theoretical	$\mathrm{C}\mathrm{Z}\mathrm{I}={\displaystyle \frac{\mathrm{E}\mathrm{R}\mathrm{I}}{{\mathsf{\varphi }}_{\mathrm{z}}}}={\displaystyle \frac{\sqrt{\frac{\mathsf{\varphi }}{\mathrm{F}}}}{{\mathsf{\varphi }}_{\mathrm{z}}}}$, ${\mathsf{\varphi }}_{\mathrm{z}}={\displaystyle \frac{\mathsf{\varphi }}{1-\mathsf{\varphi }}}$
(7)	Mohammadi et al. [19]	Theoretical	$\mathrm{E}\mathrm{Z}\mathrm{I}={\displaystyle \frac{\mathrm{E}\mathrm{Q}\mathrm{I}}{{\mathsf{\varphi }}_{\mathrm{z}}}}={\displaystyle \frac{\sqrt{\frac{1}{\mathrm{F}\mathsf{\varphi }}}}{{\mathsf{\varphi }}_{\mathrm{z}}}}$
(8)	Mohammadi et al. [21]	Theoretical	$\log ({\displaystyle \frac{\mathrm{k}}{\mathsf{\varphi }}})=\mathrm{l}\mathrm{o}\mathrm{g}{\mathsf{\varphi }}_{\mathrm{H}\mathrm{E}\mathrm{I}}+\mathrm{l}\mathrm{o}\mathrm{g}({(\mathrm{F}\mathrm{Z}1)}^2\ast {(\mathrm{E}\mathrm{Z}\mathrm{I})}^2)$ ${\mathsf{\varphi }}_{\mathrm{H}\mathrm{E}\mathrm{I}}=1014{\displaystyle \frac{{\mathsf{\varphi }}^6}{{(1-\mathsf{\varphi })}^4}}$
(9)	Omrani et al. [22]	Practical	$\mathrm{C}\mathrm{n}=\sqrt{{\displaystyle \frac{\mathrm{l}\mathrm{n}\mathrm{k}}{\mathsf{\varphi }}}}\ast {\mathrm{S}}_{\mathrm{w}\mathrm{i}\mathrm{r}}$

Table 1. The most common models for rock typing.

Eq. No.	Authors	Type	Equations
(1)	Winland [6]	Empirical	Log (R35) = 0.732 + 0.588 log (k) − 0.864 log(ϕ)
(2)	Kolodzie [7]	Empirical	Log (R35) = 0.9058 + 0.5547 log (k) − 0.9033 log(ϕ)
(3)	Pittman [8]	Empirical	Log (R20) = 0.218 + 0.519 log (k) − 0.303 log(ϕ)
(4)	Amaefule et al. [10]	Theoretical	$\mathrm{F}\mathrm{Z}\mathrm{I}={\displaystyle \frac{\mathrm{R}\mathrm{Q}\mathrm{I}}{{\mathsf{\varphi }}_{\mathrm{z}}}}={\displaystyle \frac{0.0314\sqrt{\frac{\mathrm{k}}{\mathsf{\varphi }}}}{\frac{\mathsf{\varphi }}{(1-\mathsf{\varphi })}}}$
(5)	Nooruddin et al. [12]	Theoretical	$\mathrm{F}\mathrm{Z}\mathrm{I}\mathrm{M}={\displaystyle \frac{0.0314\sqrt{\frac{\mathrm{k}}{\mathsf{\varphi }}}}{\frac{{\mathsf{\varphi }}^{\mathrm{m}}}{(1-\mathsf{\varphi })}}}$
(6)	Rezaee et al. [18]	Theoretical	$\mathrm{C}\mathrm{Z}\mathrm{I}={\displaystyle \frac{\mathrm{E}\mathrm{R}\mathrm{I}}{{\mathsf{\varphi }}_{\mathrm{z}}}}={\displaystyle \frac{\sqrt{\frac{\mathsf{\varphi }}{\mathrm{F}}}}{{\mathsf{\varphi }}_{\mathrm{z}}}}$, ${\mathsf{\varphi }}_{\mathrm{z}}={\displaystyle \frac{\mathsf{\varphi }}{1-\mathsf{\varphi }}}$
(7)	Mohammadi et al. [19]	Theoretical	$\mathrm{E}\mathrm{Z}\mathrm{I}={\displaystyle \frac{\mathrm{E}\mathrm{Q}\mathrm{I}}{{\mathsf{\varphi }}_{\mathrm{z}}}}={\displaystyle \frac{\sqrt{\frac{1}{\mathrm{F}\mathsf{\varphi }}}}{{\mathsf{\varphi }}_{\mathrm{z}}}}$
(8)	Mohammadi et al. [21]	Theoretical	$\log ({\displaystyle \frac{\mathrm{k}}{\mathsf{\varphi }}})=\mathrm{l}\mathrm{o}\mathrm{g}{\mathsf{\varphi }}_{\mathrm{H}\mathrm{E}\mathrm{I}}+\mathrm{l}\mathrm{o}\mathrm{g}({(\mathrm{F}\mathrm{Z}1)}^2\ast {(\mathrm{E}\mathrm{Z}\mathrm{I})}^2)$ ${\mathsf{\varphi }}_{\mathrm{H}\mathrm{E}\mathrm{I}}=1014{\displaystyle \frac{{\mathsf{\varphi }}^6}{{(1-\mathsf{\varphi })}^4}}$
(9)	Omrani et al. [22]	Practical	$\mathrm{C}\mathrm{n}=\sqrt{{\displaystyle \frac{\mathrm{l}\mathrm{n}\mathrm{k}}{\mathsf{\varphi }}}}\ast {\mathrm{S}}_{\mathrm{w}\mathrm{i}\mathrm{r}}$

Recently, machine learning (ML) has become a key player in the rock typing process. ML approaches are broadly categorized into supervised or unsupervised [23]. The main difference between the two types is that unsupervised machine learning algorithms use unlabeled datasets, while supervised learning uses labeled datasets [24]. Supervised learning uses labeled data to train models to predict outcomes, where each input has a corresponding output. On the other hand, unsupervised learning uses unlabeled data to discover hidden patterns, structures, or relationships within the data, without predefined outputs. Doveton [25] noted that unsupervised learning has the ability to identify electrofacies whose lithofacies cannot be identified in the core. Supervised algorithms are typically classified into classification or regression [26]. Unsupervised learning algorithms encompass clustering (e.g., hierarchical clustering and K-means), dimensionality reduction (e.g., principal component analysis), association rule learning (e.g., the Apriori algorithm), outlier detection (e.g., k-nearest neighbors), and density estimation (e.g., kernel density estimation).

Numerous studies have successfully implemented these methodologies in reservoir characterization. For instance, Man et al. [27] used an unsupervised learning algorithm to identify HFUs and a supervised learning algorithm to predict them using integrated core and well log data (

Table 2. Selected machine learning algorithms models for rock typing.

Authors	Applied Algorithm	Input Parameters	Prediction Target
Man et al. [27]	SOM, ANN, SVM, BT, RF	GR, DT, LLD, LLS, RHOB, NPHI	HFU
Mohammadian et al. [28]	XGB	K, Swc, R35, ϕ,	K, PRT
Mohammadinia et al. [29]	SVM, ANN, LB, RF, LR	Rt, DT, NPHI, RHOB, CAL, PEF.	HFU
Astsauri et al. [31]	RF, XGB, ANN, SVM	GR, DT, LLD, LLS, RHOB, NPHI,	HFU
Amosu et al. [32]	K-means clustering, K-medians clustering, hierarchical clustering, GMM clustering.	GR, RHOB, LLD	Electrofacies

). Similarly, Mohammadian et al. [28] used both supervised and unsupervised learning to detect the optimum number of petrophysical rock types as well as to predict k and R₃₅. To identify HFUs, Mohammadinia et al. [29] used various supervised learning algorithms, including support vector machine (SVM), artificial neural network (ANN), and random forest (RF), comparing their predictive performance. To enhance recognition of lithofacies, Dong et al. [30] used kernel methods utilizing conventional well logs. Astsauri et al. [31] used unsupervised learning (K-means) and several supervised learning methods to predict FZI and identify reservoir HFUs. They concluded that the use of ML in reservoir characterization is valuable and cost-effective, and offers precise results. Furthermore, Amosu et al. [32] employed unsupervised learning to predict electrofacies, thereby enabling them to differentiate between electrofacies that can produce hydrocarbons better than others. Finally, Wang and Hou [33] and Sadrikhanloo et al. [34] utilized advanced algorithms such as RF and support vector regression (SVR) for accurate porosity and permeability prediction.

This study provides a multi-layered, critical evaluation of reservoir rock types in an extremely heterogeneous reservoir by comparing sedimentological, petrophysical, and machine learning methods. Specifically, it highlights the impact of utilizing raw log data, PCA, and CDF normalization alongside a critical assessment of Amaefule et al.’s [10] approach and its modifications to identify hidden rock types and optimize prediction accuracy. The primary objective of this study is to delineate the RRTs of the highly heterogeneous Nubia sandstone reservoir using the sedimentological core description, traditional x-y crossplot approaches, and machine learning methods. To achieve this, the study addresses several key research questions, as follows: (1) it investigates whether the lithofacies revealed by sedimentological core descriptions are sufficient, or if there are hidden rock types that can be detected via well logs; (2) it examines the most influential logs and how their inclusion or elimination impact the accuracy of RRT prediction; (3) it critiques the widely used approaches, such as that of Amaefule et al. [10], and clarifies their applicability and limitations in the studied area; (4) it evaluates whether advanced machine learning methods outperform conventional methods; (5) it comprehensively compares the effects of using raw well log data against data standardized via PCA and data normalized via CDF on the resulting RRT prediction. Ultimately, this comprehensive workflow establishes practical recommendations and guidelines for future research on reservoir rock type prediction.

2. Geologic Setting

The Gulf of Suez (GOS), as a mature basin, is considered the most prolific hydrocarbon province in Egypt. It can be understood as the northern extension of the Red Sea (Figure 1). The Oligocene–Early Miocene rift basin provides a valuable opportunity to investigate the interaction among several geological (depositional environment, accommodation, diagenesis), tectonic (subsidence, uplift), and eustatic factors controlling its evolution. The stratigraphic column extended from the Cambrian to the Pleistocene periods (Figure 2). The deposition of the Nubia sandstone unconformably over the basement rock is the result of the transgression phase that was predominant during the Paleozoic. Its age extended from the Cambrian to the Lower Cretaceous. Nubia sandstone is regarded as one of the most significant pay zones in the GOS. Nubia sandstone was deposited in various environments, including shallow marine, braided fluvial, and aeolian conditions. The degree of heterogeneity in the Nubia interval is primarily dictated by variations in depositional conditions, sediment sources, and the impacts of post-depositional processes (compaction and diagenesis). El Sharawy [35] characterized the Nubia interval as exhibiting strong to extreme heterogeneity, which can be traced vertically and laterally. Generally, the thickness of the Nubia sandstone decreases southward. Shale streaks can be interspersed throughout the Nubia interval. The diagenesis of the Nubia sandstone played a major role in controlling reservoir quality. Several scholars [36,37] identified six diagenetic processes, mostly leading to a reduction in both porosity and permeability. These diagenetic processes are compaction, pressure solution, recrystallization, the development of authigenic clay minerals, replacement, and cementation by silica, iron oxides, and carbonates.

A marine transgression cycle from the Upper Cretaceous to Oligocene resulted in the deposition of the main source rocks as well as several secondary reservoirs in the GOS. During the Miocene, critical petroleum system elements, including reservoirs, source rocks, and seal rocks, were formed. The Lower–Middle Miocene represents the main rift stage in GOS’ tectonic evolution [35,37]. Thick deposits can be encountered in the depocenters, reaching several kilometers and several hundred meters in the peripheral margins [37]. The Miocene deposits in the southern GOS consist of clastics, carbonates, and evaporites.

GOS’ tectonic evolution progressed through several stages [39,40,41]. These stages are pre-rift, syn-rift, and post-rift. The pre-rift stage, extended from the Cambrian to the Oligocene, was characterized by tectonic quiescence in most parts. However, Younes and McClay [42] concluded that the lithological and structural properties of the basement significantly influenced subsequent rifting, in which basement rocks influenced the quality and volume of clastic material delivered to the Neogene basin as well as the Clysmic and Duwi trends inherited from basement structural discontinuities. With the opening of the Neo–Tethys during the Mesozoic era, as well as the absence of Triassic sediments in the Gulf of Suez area, Patton et al. [43] argued that the northern Gulf of Suez experienced extension and tectonic subsidence during the Lower and Middle Jurassic. There was also a renewed uplift movement during the Late Jurassic–Early Cretaceous. The preservation of the pre-rift sequence disputes the occurrence of substantial doming prior to the Suez rift. During the Late Oligocene–Middle Miocene Suez rifting, the tectonic stages can be categorized into pre-rift and syn-rift. The syn-rift stage is characterized by the intensive tectonic activities that resulted in the subsidence of the basin and the uplift of the shoulders. As a result of these tectonic activities, accommodated thick deposits occurred, especially in the depocenters. The subsidence rate reached its peak during the deposition of the Lower–Middle Rudeis Formation. According to Bosence et al. [44], fault-block rotation acts as an important control on platform morphology, facies associations, and the development of depositional sequences. They stated that carbonate platforms commonly develop on fault blocks in the late syn-rift to post-rift stage of basin evolution. Montenat et al. [39] argued that all fault trends have a large vertical throw. This means that the former tilted blocks are split into a mosaic of smaller blocks, with carbonates and reefs on the crests and muds on the grabens. During the Late Miocene, the deposition of a thick evaporite sequence was triggered by the transition of environmental conditions from open marine to restricted conditions. Deformation of the Plio–Pleistocene deposits as a consequence of the South Gharib salt movement is the most important tectonic event during the post-rift stage, especially at the southern Gulf of Suez. This movement led to the formation of salt diapirs at the central axis of the rift, which were ultimately reduced to salt pillows toward the gulf margins. The salt diapirs functioned as an effective structural seal for hydrocarbons.

Structurally, the GOS is dominated by trending normal faults and tilted fault blocks, which were formed during the Miocene [45]. The GOS is subdivided into three provinces separated by two major accommodation zones [46]. The northern and southern provinces are characterized by southwest dipping fault blocks and northeast dip faults, whereas the central province is dominated by northeast dipping fault blocks and southwest dip faults.

Four major fault trends were identified in the Gulf of Suez [39,43]. These are the Clysmic trend NW-SE, the N-oblique or Aqaba trend NNE–SSW, the NW–oblique or Duwi trend N100–120 and the cross or transverse fault WSW–ENE. The Aqaba and Duwi trends are oblique to the Clysmic trend. The association of these trends resulted in a zigzag fault pattern.

Based on volume of hydrocarbon production, the Egyptian General Petroleum Corporation (EGPC) [47] classified the GOS reservoirs into three main categories: primary reservoirs such as the Middle Miocene Kareem Formation and the pre-Cenomanian Nubia sandstone; secondary reservoirs such as the Upper Cretaceous Nezzazat Group; and tertiary reservoirs such as the fractured basement. The GOS reservoirs are primarily sourced from the Campanian Duwi Formation and alongside other organic-rich shale intervals within the Eocene, Paleocene, and Miocene. Trapping mechanisms are predominantly combination structural–stratigraphic traps. Effective sealing is provided through several ways, including the structural juxtaposition of permeable strata against impermeable rocks, thick shale layers, evaporite sequences, and salt diapirs.

3. Data and Methods

The dataset utilized in this study was derived from two boreholes (well A and well B) located in the southern GOS, Egypt. Both wells produce hydrocarbons from the encountered Nubia interval. In well A, fluid density variations were detected using repeat formation tester (RFT) results, which indicate the presence of three types. The fluid gradient ranges from 0.334 to 0.584 g/cc, while the pressure gradient ranges from 3257 Pa/m (indicating gas) to 5655 and 6333 Pa/m (indicating light oil). According to RFT data for well A, the GOC (gas–oil contact) is located at nearly 3285.7 m measured depth, with no OWC (oil–water contact) observed. Another change in fluid density occurs at a depth of about 3353 m. The dataset includes geophysical electric well logs, routine core analysis (RCAL), and special core analysis (SCAL). The electric geophysical logs include gamma ray (GR), density (RHOB), neutron (NPHI), sonic (DT), and true formation resistivity (Rt). Only well A has U, Th, and K natural gamma ray readings. The RCAL data include horizontal and vertical permeabilities, helium and fluid porosities, grain density, and water and hydrocarbon saturations, as well as a sedimentological description of the core lithofacies. For well A, the cored interval is about 183 m thick, covering the whole drilled Nubia section except for the upper 45.7 m. The cored interval in well A represents about 85% of the whole Nubia section. On the other hand, in well B, the cored interval is about 55 m thick, covering the middle part of the encountered interval and representing approximately 50% of its Nubia section. A statistical summary of the core analyses is presented in

Table 3. The statistical summary of the petrophysical parameters of the studied wells (Averg: average, FZI: flow zone indicator, RQI: reservoir quality index).

Well	No. of Sample	Permeability, md
		St.dev.	Min.		Max.	kA		kH	kG		kA/kH
A	519	126.65	0.01		1050	68.61		0.543	11.48		126.35
B	155	352.2	0.07		1568	422		3.27	207.71		129.16
Total	674
		Helium porosity
		St.dev.		Min.			Max.			Averg.
A	536	3.39		0.016			0.21			0.133
B	158	2.29		0.08			0.24			0.162
Total	694
		FZI, um
A	519	2.12		0.12			11.47			2.79
B	155	3.29		0.34			15.98			7.1
Total	674
		RQI, um
A	519	0.433		0.011			2.43			0.48
B	155	0.63		0.03			3.09			1.41
Total	674

St.dev. = standard deviation; k_A= arithmetic mean; k_H = harmonic mean; k_G = geometric mean.

. The SCAL data include measurements of formation resistivity factor (F) (5 samples for well A and 10 samples for well B) and the formation resistivity index (n). These measurements are crucial in the accurate determination of water saturation (S_w).

The flowchart illustrates the workflow utilized in this study (Figure 3). Prior to analyzing the log data, environmental correction and depth matching were performed. Data standardization and normalization were applied to overcome the variations in vertical logs’ resolution and depth of investigation. The depth matching between logs and core data is the second step. In this step, a vertical plot between core permeability and gamma ray log or density porosity was constructed for calibration. Figure 4 shows the crossplot of core permeability versus density porosity in well B before and after depth matching. In this study, the logging data were used in three different forms: as raw log data; as principal component analysis (PCA) data to minimize the effect of different scales and in of the logging data; and as cumulative distribution function (CDF) data to normalize the logging data.

It is common for well logs (GR, RHOB, etc.) to have skewed distributions. Thus, the results will be biased when raw well log data are input into clustering algorithms. Subsequently, the cumulative distribution function (CDF) was implemented, which involves the conversion of each log value to a cumulative probability between 0 and 1. Therefore the following equation was used [48]:

F(X) = norm.dist(x; µ; σ; cumulative)

(10)

where (X) represents the accumulated probability up to a specific point, x is the random variable, µ is the mean, σ is the standard deviation, and cumulative is a logical value that determines the form of the function. If the cumulative is true, norm.dist returns the cumulative distribution function; if false, it returns the probability density function.

Principal component analysis (PCA) represents another effective method to mitigate the biased distribution of well log values. Before applying PCA, raw log data must undergo pre-processing steps. These steps include remove outliers, log transformations, and standardization. The primary objective of PCA is to perform a dimensionality reduction in statistical data analysis. The output data are presented by PC1, PC2, etc. In order to identify the optimal number of RRTs, the most representative principal components were chosen as input data for the subsequent cluster analysis. In this workflow, DT, RHOB, GR, NPHI, and ILD were used as the input variables for the PCA, yielding five components (PC1 through PC5). PCA training demonstrated that the first three PCs account for 89% of the total data variability. Specifically, the first component (PC1) reveals a strong correlation with RHOB; the second component (PC2) correlates highly with DT and NPHI; and the third component (PC3) exhibits a stronger correlation with GR compared to the first two components (Figure 5). The first three PCs’ transformations have the following equations:

PC1 = −0.4146 (DT) + 0.39201 (GR) + 0.57816 (RHOB) + 0.29701 (NPHI) − 0.50195 (ILD)

PC2 = 0.5942 (DT) + 0.33054 (GR) − 0.3107 (RHOB) + 0.6282 (NPHI) − 0.21962 (ILD)

PC3 = −0.02895 (DT) + 0.79028 (GR) + 0.0385 (RHOB) − 0.16405 (NPHI) + 0.5884 (ILD)

The following methods were used to predict the RRT.

Buckles [9], at the irreducible water saturation (S_wir), introduced the following equation:

BVW = S_wir*ϕ

(11)

where BVW is the bulk volume water. Each RRT has a constant value of BVW. Based on capillary pressure, the R₃₅ equation (Winland Equation) was used:

Log R₃₅ = 0.732 + 0.588 log k − 0.864 log ϕ

where R₃₅ is the pore aperture radius corresponding to the 35th percentile of mercury saturation in micron, k is permeability in mD and ϕ is porosity as a percentage. The core samples of a given petrophysical flow unit have similar R₃₅ values, which are used to define HFU as the following:

•

Megaport units with R₃₅ > 10 μm.

•

Macroport units with 2 < R₃₅ < 10.

•

Mesoport units with 0.5 < R₃₅ < 2.

•

Microport units with 0.1 < R₃₅ < 0.5.

•

Nanoport units with R₃₅ < 0.1.

Stratigraphic modified Lorenz (SML) plot was used to detect the inflection which defines a new RRT. It is a plot of cumulative flow capacity versus cumulative storage capacity. It can be calculated using the following equation [49]:

(k_h)cum = k₁(h₁ − h₀) + k₂(h₂ − h₁)+ …………+k_n(h_n − h_n−1)

(12)

where k is permeability (mD) and h is the thickness of the sample interval.

A similar equation is used to determine a single cumulative storage capacity value [49]:

(ϕ_h)cum = ϕ₁(h₁ − h₀) + ϕ₂(h₂ − h₁)+ …………+ϕ_n(h_n − h_n−1)

(13)

where ϕ is fractional porosity.

Determination of hydraulic flow unit (HFU) based on flow zone indicator (FZI) and reservoir quality index (RQI) can be carried out using the following equation [10]:

$$FZI= {\displaystyle \frac{1}{\sqrt{F_s}t{S}_{gv}}}={\displaystyle \frac{RQI}{{\mathsf{\varphi }}_z}}$$

(14)

where F_s is the shape factor, S_gv is the surface area per unit grain volume, and t is the tortuosity. RQI can be defined as follows:

$$\mathit{R}\mathit{Q}\mathit{I} = 0.0314\sqrt{{\displaystyle \frac{\mathit{k}}{\mathit{\varphi }}}}$$

and the normalized porosity (${\mathsf{\varphi }}_{\mathbf{z}}$) can be defined as the following:

$${\mathsf{\varphi }}_{\mathbf{z}}={\displaystyle \frac{\mathsf{\varphi }}{(1-\mathsf{\varphi })}}$$

The FZI model can be converted to 3D discrete rock type (DRT) by using the following equation [50]:

DRT = round (2log (FZI) + 10.7)

(15)

In this study, three machine learning methods were used to predict the possible RRT in Nubia sandstone in the southern Gulf of Suez. These methods are presented as follows

1—Ward’s Hierarchical Clustering Method:

In this method, core k, core ϕ, FZI, ϕ_z, and RQI were used as input data, whereas the other two ML methods (K-means and SOM) relied on well log data as inputs. Consequently, Ward’s method services as a benchmark calibration tool against which the predictive accuracy of alternative methods can be evaluated and adjusted. Ward’s method is a criterion commonly applied in unsupervised hierarchical cluster analysis. This agglomerative technique organizes data into a nested sequence of clusters, ultimately constructing a tree-like dendrogram. In Ward’s method, the optimal value of an objective function is used to determine the pair of clusters to merge at each step. The initial step of the procedure is to calculate a distance matrix that measures pairwise dissimilarities between every data point. The two clusters with the smallest dissimilarity are merged at each iteration. Using linkage criteria, the dissimilarity between clusters can be determined. The linkage criteria could be single linkage (minimum distance), complete linkage (maximum distance), and average linkage (mean distance). A common way to measure dissimilarity is the Euclidean distance, according to the following equation [51]:

$$d\left(x, y\right)=\sqrt{{\sum }_{i=1}^n(x_i-y_i{)}^2}$$

(16)

where x and y are two data points in n-dimensional space.

To determine the optimal number of RRT, the sum of squared errors (SSE) was plotted against the number of HFUs, in which the SSE decreased as HFU increased. At a certain HFU number, the SSE is nearly constant. After this number, SSE is almost constant with small variations, which can be neglected. This number signifies the optimal number of HFUs for the reservoir model.

2—K-Means Clustering:

The K-means clustering algorithm is a widely implemented unsupervised learning technique designed to partition a dataset into k discrete clusters, where k is a predetermined number. The primary objective of this algorithm is to minimize the within-cluster sum of squared errors, which serves as a metric for cluster compactness [52]. Based on the mathematical relationships among the input variables, K-means clustering facilitates the identification of natural groupings or inherent patterns within the dataset. The SSE can be determined using the following equation [53]:

$$\mathrm{SSE}={\sum }_{k=1}^K{\sum }_{x\in Ck}{\Vert x-\mu k\Vert }^2$$

(17)

where Ck represents the set of points in cluster k, μk is the centroid of cluster k, and $\Vert x-\mu k\Vert$² denotes the Euclidean distance between a point x and the cluster centroid μk. In K-means, the initialization phase begins by selecting the centroids. These centroids are chosen randomly. The iteration phase begins immediately after the initialization phase proceeds. The subsequent iteration phase includes two steps: assignment and update [52]. These two steps are repeated until convergence. Convergence occurs when the centroids stabilize or a predefined stopping criterion is met [52]. In this study, we begin with a number of clusters equal to 20. Then, a cluster randomness plot is established to detect the optimal number of clusters, using the minimized within-cluster sum of the squares distance clustering method.

According to Bradley and Fayyad [54] and Amosu et al. [32], K-means has notable limitations, such as being sensitive to the choice of initial centroids, being sensitive to outliers, and assuming spherical clusters and equal cluster sizes. These limitations can result in a significant distortion of centroid positions.

3—Self-Organizing Map (SOM):

Self-organizing map (SOM), also known as the Kohonen map, was introduced by Kohonen [55], and he reviewed and collected its basic material in 1990 [56]. According to Cottrell et al. [57], SOM is used as a powerful clustering algorithm, which considers a neighborhood structure among clusters. Self-organizing maps are an unsupervised machine learning technique used to produce a low-dimensional (typically two-dimensional) representation of a higher-dimensional data set while preserving the topological structure of the data. SOM is a type of artificial neural network [56], but it is trained using competitive learning. For example, a data set with p variables measured in n observations could be represented as clusters of observations with similar values for the variables. These clusters then could be visualized as a 2-D map. This can make high-dimensional data easier to visualize and analyze. Self-organizing maps, like most artificial neural networks, operate in two modes: training and mapping. First, a lower-dimensional representation of the input data is trained using an input data set. Second, mapping uses the created map to classify new input data. In this study, a random initialization method was used with a grid size of 400 nodes. The number of training iteration loops was set to 60,000 to ensure that all the input data were reflected in the map. The training iteration will stop when it reaches this number. The initial learning rate was 0.1.

4. Results

4.1. Lithofacies Based on Core Description

The cored interval in well A presents an excellent opportunity to identify and characterize the lithofacies of the Nubia sandstone. The core description for this well is primarily based on the work of Gameel and Darwish [36], information in the composite log, and well log data. However, thin-section photomicrographs were unavailable for this study. A description of the cores reveals that the Nubia interval consists of three lithologies: sandstones (main constituents), conglomerates, and shales (minor constituents). Moreover, the cored interval in well A exhibits seven distinct lithofacies.

4.1.1. Dark Gray to Black Quartzarenite Lithofacies (LF1)

LF1 represents the lowermost interval of the Nubia sandstone and attains a thickness of about 7.6 m. It unconformably overlies the fractured basement and is characterized by high GR readings as a result of the presence of radioactive minerals (Figure 6). According to Gameel and Darwish [36], the lowermost 0.76 m is composed of thick black sandstone with basalt granules, cemented by silica, and plugged by clay minerals, indicating a deposition in shallow marine conditions. The upper part is composed of light-tan, very fine-to fine-grained, moderately sorted sandstone, which is cemented by kaolinite. LF1 is characterized by an average porosity of 13.6%, poor permeability averaging 0.68 mD, and an average true formation resistivity of 5.3 ohm-m. Furthermore, LF1 shows a very weak relationship between ϕ and k, with r² = 0.02 (Figure 7). This poor relationship suggests that the authigenic clay minerals plug the pore spaces; consequently, the relatively high porosity (ϕ_avg. = 13.6%) exerts no discernible impact on the permeability values.

4.1.2. Black-Colored Coarse Pebbly Sandstone Lithofacies (LF2)

LF2 was likely deposited in a distributary channel and is composed of laminated and cross-bedded white and black lithic quartzarenite. It attains a thickness of approximately 30.5 m and is capped by a thin layer of shale. LF2 can be dissected into two distinct units based on log responses: the lower unit exhibits R_t around 10 ohm-m, whereas the upper unit ranges between 10 and 100 ohm-m (Figure 6). This vertical variation in the resistivity log response is mirrored by both the GR and DT logs. Lithologically, LF2 is composed of dark brown to dark tan, fine- to medium-grained, occasionally coarse-grained, moderately sorted, and cemented sandstone. It exhibits an average porosity of 14.7%, and an average permeability of 18.8 mD. The relationship between k and ϕ is very weak, yielding r² = 0.023. According to Gameel and Darwish [36], a geochemical reaction between pyritic sulfur content and the hydrocarbons led to the formation of pyrobitumen; this byproduct fills the pore spaces, thereby significantly reducing the permeability.

4.1.3. Brown Pebbly Sandstone Lithofacies (LF3)

LF3 exhibits a coarsening upward trend, as indicated by the GR curve. The resistivity log shows high values (R_t > 1000 ohm-m), except for two thin intervals in which R_t drops to less than 100 ohm-m. These two intervals correspond to shale layers (Figure 6). Similar to LF2, LF3 is capped by a thin layer of shale. Lithologically, LF3 is composed of dark tan, fine- to coarse-grained, moderately to well-sorted, moderately to well-cemented sandstone, which is argillaceous and pebbly in parts. The porosity and permeability recorded average values of 14.4% and 96.3 mD, respectively. The porosity–permeability shows a relatively weak relationship, with r² = 0.415.

4.1.4. Brown Sandstone Lithofacies (LF4)

LF4 is interbedded between two impermeable layers. The lower layer is very fine-grained siltstone with probable dolomite cement, whereas the upper layer is called “marker shale,” which can be easily correlated within the Nubia interval in adjacent wells [36]. The highest potassium readings were recorded within this marker shale (Figure 7). The sandstone of LF4 is light to dark tan, fine- to medium-grained, moderately sorted, subangular to subrounded, and moderately to well-cemented. The resistivity log exhibits intervals with R_t > 1000 ohm-m intercalated with zones with R_t < 100 ohm-m. Porosity and permeability recorded average values of 15.3% and 165.3 mD, respectively. In contrast to the preceding lithofacies, the porosity–permeability relationship of LF4 exhibits the strongest correlation among all the studied lithofacies, with r² = 0.795 (Figure 7). According to Gameel and Darwish [36], the latter three lithofacies (LF2–LF4) were deposited in distributary channels and bars.

4.1.5. Conglomeratic to Argillaceous Sandstone Lithofacies (LF5)

LF5 exhibits a distinct log response on both resistivity and sonic logs. Resistivity values are around 100 ohm-m, with no extremely high R_t values, as in LF3 and LF4. The sandstone ranges in color from light tan to dark brown, consisting of fine- to medium- grained sand that can occasionally be very fine or coarse-grained. It is fairly to poorly sorted, and contains isolated pebbles, siliceous cement and traces of authigenic pyrite. Porosity and permeability recorded average values of 10.7% and 34.6 mD, respectively. LF5 displays a strong correlation between porosity and permeability, with r² = 0.763 (Figure 7). This lithofacies was likely deposited under coastal conditions.

4.1.6. Siliceous and Argillaceous Sandstone Lithofacies (LF6)

LF6 constitutes the uppermost part of the Nubia interval and is characterized by a high true formation resistivity (R_t > 1000 ohm-m). This lithofacies attains the thickest interval among all identified units in well A. When combined with the uppermost uncored interval, the thickness of LF6 reaches approximately 106.7 m. Lithologically, LF6 is composed mostly of sandstone. The sandstone is brown to dark brown, medium- to coarse-grained, semi-friable, with siliceous cement, fair to poorly sorted (locally well-sorted), and occasionally with pebble grains. Porosity and permeability recorded average values of 12.5% and 67.4 mD, respectively. Similar to LF4 and LF5, the porosity–permeability relationship of LF6 is strong, with r² = 0.792 (Figure 7).

4.1.7. Shales (LF7)

Shale intervals occur as thin, streaked layers throughout the Nubia sandstone sequence. These shales are explicitly observed at depths of 3352.8 m, 3327.8 m, and 3319.9–3322.9 m (Figure 6). Mineralogically, illite and kaolinite comprise the dominant clay minerals between 3319.9 and 3323 m depth, whereas cryptocrystalline amorphous silica is encountered at 3327.8 and 3353 m depth. Overall, kaolinite and illite constitute the dominant clay minerals throughout the studied sequence.

4.2. Traditional Methods for Determination of RRT

Buckles [9] introduced a method by which RRTs can be differentiated. Figure 8 exhibits several rock types that can be identified in well A. LF6 and LF4 show one cluster around C = 0.004, with some scattered points toward decreasing reservoir quality. The LF6 and LF4 clusters indicate good reservoir quality, as implied by the C value. On the other hand, LF2 exhibits a wide range of C values ranging between C = 0.02 and 0.06. Unlike the other lithofacies, LF2 shows a limited variation in porosity but wide variation in water saturation. LF3 shows a wide range of variation in both S_w and ϕ, with one cluster around C = 0.004 and considerable scattered points between C = 0.004 and C = 0.03. These scattered points illustrate a downward trend toward enhanced reservoir quality. On the other hand, poor reservoir quality is restricted in LF1, while fair reservoir quality can be observed in LF2 and LF5. In well B, one cluster centered on C = 0.002 represents the bulk of the encountered Nubia interval in this well.

Applying the R₃₅ equation (Equation (1), Table 1), well A displays five RRTs. LF6 and LF5 span across all five categories, with an obvious proportional trend between k and ϕ (Figure 9). LF4 shows mostly macro- and mega-porous rock types, with an obvious proportional trend between k and ϕ. Similarly, the majority of LF3 is composed of macro- and mega-porous rock types; however it lacks a discernible trend between k and ϕ, demonstrating that an increase in k is not accompanied by an increase in ϕ for this unit. LF2 is predominantly a macro-porous rock type, like LF3, and lacks any correlation between k and ϕ. Conversely, LF1 is primarily distributed between nano- and micro-porous rock types and shows no apparent relationship between k and ϕ. In well B, the data forms a single cluster that falls between mega- and macro-porous rock types, with a pore throat size exceeding 5 µm. In this well, there is no obvious trend between k and ϕ, as k escalates from 100 to 1000 mD without any corresponding change in ϕ.

The stratigraphic modified Lorenz (SML) plot provides a straightforward approach for identifying RRTs by using cumulative porosity as storage capacity and cumulative permeability as flow capacity. By plotting the storage capacity against flow capacity, eight HFUs were identified in well A (Figure 10a). The reservoir quality and thickness of the resultant flow units vary from one flow unit to another. While HFU4 represents the lowest reservoir rock quality, HFU3 represents the highest quality among all units, as indicated by their respective slope. Incorporating depth as a parameter reveals that HFU3 accounts for approximately 60% of the total flow delivered from the interval between 3316.2 and 3372 m depth (Figure 10b). HFU3 corresponds to LF3 and LF4, which are dominated by a high-reservoir-quality facies with a minimal proportion of low-quality rock types (Figure 9). For well B, five HFUs can be delineated using the SML plot (Figure 11a). Similar to well A, HFU3 represents both the thickest and the highest-quality flow unit in this well, supplying approximately 60% of the total flow from a depth interval spanning 3291.8 to 3314 m (Figure 11b).

Amaefule et al. [10] presented a theoretical framework for the classification reservoir rocks into HFUs by plotting RQI versus ϕ_z, which is based on the Carman–Kozeny equation. Figure 12 (well A) demonstrates the difficulty of accurately determining HFUs using this method, as the identification process relies heavily on manual intervention due to the narrow range of the normalized porosity values. The same behavior is observed in well B (Figure 12). Furthermore, the loss of depth information represents another critical drawback of this approach. To mitigate this, the DRT is utilized to facilitate the determination of the log–log line slopes. Across the various HFUs in well B, distinct changes in these slopes are observable (Figure 12), which may be attributed to the variations in the cementation factor and its subsequent impact on HFUs [14]. Alternatively, the normalized RQI (NRQI) method provides a robust technique for delineating HFUs. As shown in Figure 13, eleven HFUs were identified using the NRQI method in well A, where LFI–LF4 correspond directly to HFU1–HFU4, LF5 corresponds to HFU6 and HFU7, and the uppermost lithofacies (LF6) corresponds to HFU8–HFU11. As noted from Figure 13, the increasing slope of the plotted NRQI line corresponds to the increasing quality of the rock reservoir. In contrast, a vertical line signifies poor or non-reservoir rock quality. Consequently, LF1 and LF7 are classified as having the lowest reservoir rock quality, followed by HF5. In well B, six HFUs were delineated using this approach (Figure 13); here, HFU4 and HFU6 act as barriers due to their vertical slopes, while HFU3 and HFU5 represent the highest-quality reservoir rock type in this well.

The global hydraulic element (GHE) method simplifies the identification of the number of HFUs based on the FZI approach. Figure 14 (well A) shows that the plotted k-ϕ data of well A span from GHE1 to GHE7, with the majority of data points clustered within the GHE4 to GHE6 intervals. In well B, most data points cluster between GHE6 and GHE7, suggesting high reservoir quality (Figure 14). The type curve developed by [17] resulted in interference in the RRTs because the lines representing different rock types have very small gaps between them. Therefore, this type of curve can only be used with limited and wide distribution data. A more trustworthy approach should be used to adjust the results obtained by the curve shown in [17].

4.3. Machine Learning (ML) Methods

Ward’s hierarchical algorithm provides high accuracy and, unlike the FZI approach, operates independently of user bias. In this method, core k, core phi, ϕ_z, FZI, and RQI were used as input data. In Ward’s method, each data point forms a discrete cluster, and similar clusters are then merged together to form super clusters. A dendrogram is the output of the hierarchical clustering. The sum of squared error (SSE) was employed to detect the optimal number of RRTs. Applying this methodology to well A yielded eight distinct RRTs, designated sequentially from RRT1 to RRT8 (Figure 15). Notably, an inverse relationship exists between the RRT index number and reservoir rock quality: RRT1 and RRT2 are restricted to very good reservoir rock; good reservoir rock was primarily represented by RRT3, RRT4, and RRT5; RRT6 is mostly moderate-quality rock; and RR7 and RR8 are restricted to fair and poor reservoir rock, respectively. Some RRTs may experience interference, although this interference is usually minimal. The distribution of RRT1 and RRT2 (good to very good reservoir) represented less than 10% of the total cored interval, while RRT7 and RRT8 (fair to poor reservoir quality) represented about 45% of the total cored interval. Following the same procedure as that implemented in well A, six RRTs were identified in well B (Figure 16). In contrast to well A, poor reservoir quality (RRT6) represents about 6% of the cored Nubia interval in well B. RRT1–RRT4 exhibit very good reservoir quality, accounting for about 80% of the cored interval. Meanwhile, RRT5 shows moderate to good reservoir quality. The merging these first four high-quality RRTs results in three principal petrophysical rock types that are dominated by superior reservoir properties with minor interference from poor and moderate facies.

5. Discussion

5.1. Well-to-Well Correlation

Tilted fault blocks which characterize the Suez rift resulted in a considerable variation in the thickness of encountered drilled sequences, such as the Nubia interval, within the studied wells. The thickness varies from about 213 m thick in well A to about 110 m thick in well B. This variation in thickness is attributed to fault-cutting rather than a non-deposition or erosion process. Well-to-well correlation confirms this structural interpretation, demonstrating that while the upper section of the Nubia interval is preserved in well B, the lower interval is completely missing (Figure 17). The correlation between the two wells was determined using GR and DT logs. Additionally, the resistivity log provided valuable assistance in this correlation. As indicated by the dip log (Figure 17), the Nubia interval unconformably overlies the basement rock and is unconformably overlain by the Upper Cretaceous Matulla Formation. Moreover, internal unconformity surfaces are detectable between the distinct lithofacies of Nubia sandstone. The correlated section between the two wells (Figure 17) reveals poor-to-good reservoir rock types in well A and predominantly very good reservoir rock types in well B, indicating a significant difference in reservoir quality between the two wells. The correlated section in the two wells is characterized by a high true formation resistivity of up to 2000 ohm-m. However, high reservoir quality is not guaranteed by this extremely high R_t. This might be explained by the rock texture and post-deposition diagenesis, such as the presence of authigenic clay minerals, high-viscosity asphaltic matter, compaction, and the development of siliceous overgrowths. Texturally, the quartz grains in well A are predominantly fine- to medium-grained and poorly to moderately sorted, whereas the quartz grains in well B are primarily moderate- to coarse-grained and moderately to well-sorted.

5.2. Porosity—Permeability Relationship

Tiab and Donldson [58] divided reservoir quality into five categories according to the permeability values:

• Poor reservoir: k < 1mD.
• Fair reservoir: 1< k <10 mD.
• Moderate reservoir: 10 < k < 50 mD.
• Good reservoir: 50 < k < 250 mD.
• Very good reservoir: k > 250 mD.

Figure 18 demonstrates that LF1 is mostly a poor reservoir rock type. Conversely, LF2 is the only lithofacies that completely lacks a poor reservoir rock classification, being instead predominantly composed of fair and moderate RRTs. LF3 exhibits an upward trend of increasing reservoir quality, whereas LF4 is distinguished by a high percentage of very good RRTs. In contrast to LF3, LF5 exhibits a prominent upward decrease in reservoir quality. Lastly, the reservoir quality of LF6 is highly variable, ranging primarily from fair to good RRTs.

Routine core analysis provides two types of porosities: helium porosity (ϕ_h) and fluid porosity (ϕ_f) (Figure 19a). For well A, the relationship between the two types of porosities is moderate, with r² = 0.609. The most robust correlation was observed in LF5, which was recorded as r² = 0.82. For the lowermost three lithofacies (LF1-LF3), the values of helium core porosity are mostly restricted to between 12% and 18% (Figure 19a). The helium core porosity values of the uppermost three lithofacies (LF4–LF6) are primarily restricted to a broader range of 6% to 18%, featuring notable variations in LF4 where ϕ exceeds 18%. The total–log-derived porosity exhibits a stronger correlation with helium core porosity than with fluid core porosity, with r² = 0.541. This indicates that helium core porosity provides a more accurate representation of the true formation porosity. The complex, heterogeneous nature of the reservoir in well A accounts for this moderate correlation between log- and core-derived porosity. In well B, a good relationship between core- and log-derived porosity was observed. The porosity of well B is mostly restricted to between 12% and 18%. Some variations can be observed in the upper interval, in which ϕ > 18% is recorded. Visually, this vertical porosity profile is characterized by an alternating, cyclic-crescent-shaped pattern.

There is no discernible downward trend in the porosity with depth, except for the uppermost two lithofacies (LF5 and LF6), which exhibit a general compaction-driven reduction in porosity with increasing depth (Figure 20). The magnitude of k within the uppermost three lithofacies mirrors the variations in porosity values, suggesting that porosity acts as the primary factor influencing k values. As noted in Figure 20, a change in ϕ corresponds directly to a change in k. The relationship between k and ϕ in these three lithofacies shows that r² ranged between 0.763 and 0.795. Conversely, the lowermost three lithofacies (LF1-LF3) display poor to negligible tracking between k and ϕ. The relationship between k and ϕ across these three lower lithofacies yields r² values that fall within a range of 0.02 to 0.415. This weak correlation implies that secondary petrophysical factors, rather than primary porosity, play the dominant role in determining the magnitude of k.

Statistically, a general upward increase in average porosity and permeability is observed from LF1 to LF4 (

Table 4. Average values of porosity and permeability for facies of well A.

Facies	Average Porosity	Average Permeability, mD	r2
LF1	0.136	0.68	0.02
LF2	0.147	18.8	0.023
LF3	0.144	96.3	0.415
LF4	0.153	165.3	0.795
LF5	0.107	34.6	0.763
LF6	0.125	67.4	0.792

), with a localized decrease in LF5 before resuming the upward trend in LF6. This localized decrease corresponds to a distinct, abrupt change in compressional wave velocity (V_p), as illustrated in Figure 6.

The relationship between horizontal permeability (k_h) and core porosity is moderate for the entire interval of well A, with r² = 0.55, but increases to 0.602 in well B. In the case of vertical permeability (k_v), the relationship between with ϕ becomes weaker, with r² = 0.45 for well A and 0.473 for well B. The relationship between k_h and k_v can be expressed using the following equation, with r² = 0.719 (Figure 19b):

k_v = 0.959*k_h^0.824

(18)

Incorporating porosity in the form of $\sqrt{{\displaystyle \frac{{\mathrm{k}}_{\mathrm{h}}}{\mathsf{\varphi }}}}$ or $\sqrt{{\mathrm{k}}_{\mathrm{h}}\ast \mathsf{\varphi }}$ yielded no improvement in the strength of the relationships between k_v and k_h. The coefficient of determination decreased to 0.713 and 0.729, respectively. In well B, a significant drop in r² was noted from 0.823 to 0.777 when incorporating porosity in Equation (18). This may suggest that porosity has a limited impact on the magnitude of permeability. The relationship between k_h and ϕ based on reservoir quality (poor, fair, etc.) reveals very weak relationships with r² ranging from 0.076 to 0.247. The porosity–permeability relationship can be interpreted in the light of the Nelson [59] plot. According to this plot, cementation and clay content are the primary parameters influencing the permeability reduction; both have a negative impact on the permeability values. The response to cementation and clay content varies across lithofacies. For example, LF6 shows an obvious trend between k and ϕ as a result of the increase in cementation, consolidation, diagenesis, and clay content (Figure 21). Conversely, LF2 does not exhibit this trend, as a significant decrease in k was not accompanied by a comparable increase in ϕ (Figure 21). In well B, on the other hand, the reservoir characteristics were enhanced through low cement, low clay content, and an increase in grain size and degree of sorting. These enhancing reservoir characteristics led to good to very good reservoir quality with high k and ϕ (Figure 21).

5.3. Applicability of the Traditional RRT Approaches

The approach by Amaefule et al. [10] and its modification aim to identify the hydraulic flow units by using core porosity and permeability. On the other hand, Rezaee et al.’s [18] approach and its modification aim to identify the electrical flow units by using the formation resistivity factor (F). While Amaefule et al.’s [10] approach necessitates routine core analysis (RCAL), Rezaee et al.’s [18] method necessitates the use of SCAL data. There are three principal limitations that may discourage researchers from using these approaches. First, the availability of both types of data is rare; however, RCAL data are more frequently available than SCAL data. Even when measurements of the formation resistivity factor are available; these measurements are usually present at a small scale. For instance, in well A, RCAL provided more than 500 plug samples for core k and core ϕ, while SCAL only provided seven measurements for F. Consequently, Amaefule et al.’s [10] approach is more widely applicable than Rezaee et al.’s [18] method. The second significant limitation of these approaches is the loss of depth information, as illustrated in Figure 12 and Figure 14. Third, one of the biggest limitations with these approaches is that they depend on manual intervention to determine the number of RRTs. With a limited amount of data, the determination of RRTs will proceed without complications. However, with large datasets, like in well A, the accuracy when determining the number of RRTs becomes questionable.

5.4. Effects of Input Data on the Machine Learning Results

Raw log data, PCs, and CDF were used as input data for K-means clustering and SOM ML methods to evaluate which method performs best. The results of these machine learning methods were calibrated based on the previously identified lithofacies and reservoir quality categories (Figure 15 and Figure 16). LF1 is considered a distinctive lithofacies that is not repeated throughout the Nubia interval. It is characterized by very low k and high GR readings. The lowermost 0.76 m thick zone of LF1 is composed of thick black sandstone that contains basalt granules and has transgressed over the basement. This zone can be differentiated using K-means clustering and SOM methods based on raw logs and PCs as input data. On the other hand, the CDF input data failed to distinguish this zone (Figure 15). LF2 is characterized by a narrow distribution of porosity and a very weak correlation between k and ϕ. It is mostly composed of alternating fair and moderate reservoir quality, with a thin layer of good reservoir rock type. The K-means clustering method based on raw logs and PCs provided the best translation of the LF2 distribution. The SOM method based on raw logs and CDF succeeded in predicting the thin layer of good reservoir quality, but it was unable to predict the alternation between fair and moderate reservoir quality. LF3 is characterized by the first appearance of very good reservoir quality, with a low percentage of fair and poor reservoir rock types. Based on raw log data, SOM and K-means clustering methods effectively represented LF3 distribution. LF3 is characterized by the first appearance of two colors (turquoise and moss green), which represent the good to very good reservoir rock types. Among all lithofacies, LF4 shows the strongest correlation between k and ϕ. Similar to LF3, SOM and k-means clustering methods were conveyed for LF4 based on raw logging data. LF5 is mainly composed of alternating poor and fair reservoir rock types. SOM and K-means clustering methods based on CDF effectively differentiated LF5 but failed to detect the moderate to good reservoir rock type zones. LF6 is characterized by a good correlation between k and ϕ. It is composed of various alternating RRTs. The SOM and K-means clustering methods based on raw logs and PCs provide an accurate presentation for LF6. In conclusion, the rock types of the Nubia sandstone in well A can be reasonably predicted using SOM and K-means clustering methods based on raw log data and PCs. However, one of these methods’ limitations might be their inability to identify the thin zones. Another important note is related to the resistivity log. The rock type prediction was less accurate when a resistivity log was used as input data; conversely, prediction accuracy improved following the elimination of the resistivity log from the input datasets. This is likely because the resistivity log is highly sensitive the fluid type and its salinity rather than the rock type. Most of the Nubia interval in well B reads Rt = 2000 ohm-m, despite containing several RRTs. Similarly, in well A, LF6 reads Rt = 2000 ohm-m (Figure 13), yet it can be classified into four HFUs. Figure 4 provides an interpretation of this observation, demonstrating that there is no correlation between the resistivity log and any of the PCs.

In well B, the Nubia sandstone reservoir can be classified into six RRTs based on Ward’s clustering method (Figure 16). The first four RRTs (RRT1–RRT4) represent very good reservoir quality and can be merged into a single RRT. These rock types constitute the bulk of the Nubia section in this well, intercalated with some thin layers of poor (RRT6) to good (RRT5) reservoir quality. K-means clustering based on well log data was the most effective method for detecting the poor-reservoir-quality rock type (RRT6) (Figure 16). The remainder of the section was correlated with the high-quality rock types (RRT1–RRT4).

6. Conclusions

This study focused on identifying reservoir rock types (RRTs) in the pre-Cenomanian Nubia sandstone of the Gulf of Suez, Egypt. This Nubia sandstone is regarded as one of the most productive reservoirs in the Gulf of Suez. Based on sedimentological core descriptions, routine core analysis, and well logging data from two wells, the results indicate the following:

• The primary cause of the thickness variation between the two wells investigated is likely fault-cutting, rather than stratigraphic factors.
• The cored section in well A can be distinguished into seven distinct lithofacies (LF1–LF7). Six of these are represented by various types of sandstone, and the seventh one is composed of shale.
• The cored interval in well A is dominated by moderate reservoir rock quality, whereas the cored interval in well B is dominated by very good reservoir rock quality. This variation may be attributed to the post-depositional diagenetic processes and variations in sandstone texture.
• The normalized reservoir quality index (NRQI) method is arguably the most reliable traditional x-y crossplot method for predicting the Nubia rock types, especially when plotted against depth.
• The traditional x-y crossplot methods exhibit three major limitations: the limited availability of routine and special core analyses, the loss of depth information, and the reliance on manual intervention to determine the number of RRTs.
• Ward’s method, based on core permeability and porosity, identified eight RRTs in well A and six RRTs in well B. It is possible to merge the first four RRTs in well B into a single RRT, resulting in just three final RRTs dominated by very good reservoir quality.
• Based on raw log data and principal component analysis (PCA), the K-means clustering and self-organizing maps (SOM) methods provide reliable results for predicting the RRTs in the Nubia sandstone across both studied wells.

7. Limitations

While the proposed workflow successfully provides key insights into the reservoir, several data-driven and physical constraints must be acknowledged to contextualize the findings and guide future research:

• Limited Number of Wells: The available well dataset represents a localized subset of the asset. This sparse spatial sampling restricts the ability to fully capture lateral facies heterogeneity and regional structural variations across the study area.
• Lack of Thin-Section Evidence: The absence of a petrographic thin section prevents direct visual validation of micro-scale mineralogy, pore geometry, and diagenetic history.
• Limited SCAL Samples: Special core analysis (SCAL) data are sparse. This scarcity prevents the application of electrical flow unit approaches and consequently prevents comparison with hydraulic flow unit approaches.
• Workflow Transferability: Directly applying the exact introduced workflow to other fields (e.g., transitioning from siliciclastic rocks to carbonate rocks) requires empirical validation and practical expertise.