## 1 Introduction  
Carbonate rock reservoirs are believed to contain nearly 50% of the world's hydrocarbons. The carbonate rocks are dominated by a few carbonate minerals (mainly calcite and dolomite) and contain additional traces of minerals such as silica, detrital grains, phosphate and glauconite. On the other hand, carbonate rocks may contain organic residues and some cementing material. Petrography helps identify grains, leading to the detailed classification of rocks, the determination of the types of deposition and the indicators of historical post-depositional alteration (diagenesis). It also helps determine the timing of porosity development. Comparing these grains with photographs allows us to readily identify the essential properties/features that can potentially predict the amount of petroleum under adequate trap conditions in the oil and gas industry. In this study, the pore/fracture space component using multi-objective classification is based on microscopic studies, particularly of Malaysian carbonate rocks. Here we focus on the image analysis of rock image samples to automatically reach a conclusion on pores or fractures contained in the image sample instead of illustrating the quantitative petrophysical method. Hence, we will not work on overall petrography analysis but only its segment related to image analysis.  
Porosity is widely considered a fundamental parameter for reservoir characterisation and is used in many hydrogeological and geophysical calculations. For crystalline rock, it can be calculated as the sum of micro-fractures, irregular micro-pores and fluid inclusions [1]. Porosity also determines the storage capacity of a specific reservoir. To our knowledge, most hydrocarbon reservoirs belong to sedimentary rock formations in which porosity values vary between 10% and 40% for sandstones and 5% and 25% for carbonates [2]. The micro-pores are spherical or nearly spherical [3]. In contrast, fractures are assumed to maintain near linearity (i.e. their length is longer than their width). Total porosity can be divided into two types: connected (characterised using flow and diffusion) and unconnected (because of the presence of isolated pores) [1]. Unconnected spherical or near-spherical pores along with micro-porosity can effectively change the elastic properties of the rock frame, making porosity an important parameter for reservoir characterisation in terms of estimating elastic moduli [3].  
The current study aims to propose a novel meta-heuristic image analysis approach using the multi-objective algorithm approach to alleviate the previous difficulty in identifying pores and fractures (natural and induced, even at the micro-level) in the wells of a hydrocarbon reservoir. Furthermore, we present better identification accuracy in the grayscale sample rock images in this work.  
The high-level idea of the proposed methodology is multi-objective, i.e. the primary objective is to find the neighbouring dark pixels of every discovered dark pixel within a specific region and create a bag of contiguous pixels. A secondary objective is to classify the shapes obtained from the pixel bag into pores or fractures based on a certain chosen hyper-parameter. This methodology was compared with the CNN (convolution neural network)-based [4; 5] approach, which is considered a state-of-the-art framework for image classification. CNNs comprise pixel-to-pixel and end-to-end architectures, which use a large image dataset with features and tuneable neural network parameters. Our proposed method can obtain more accurate results than a deep learning CNN, which thus far has been considered the most efficient method for identifying pores and fractures in hydrocarbon reservoirs.