## 1 Introduction  
Concrete is the most widely used construction material in the world, due to its cost-effectiveness and structural performance. Over its designed service life, concrete develops cracks due to internal stresses and external conditions. The causes of these cracks can be broadly categorized into three main types. The first is chemical attack, where substances like sulfates and magnesium infiltrate the concrete, causing expansion and resulting in cracks. The second is physical actions, such as freezing-thawing cycles and wave action. The third is corrosion of the rebar, which occurs at the interface between the rebar and concrete due to chloride infiltration. Among these, rebar corrosion is one of the most common causes of durability issues in reinforced concrete structures. Especially when concrete structures are built in marine environments rich in chloride or when deicing salts are widely used during winter, the concentration of chloride ions reaches a critical level, initiating rebar corrosion. Once rust forms on the rebar surface, its volume expands, causing cracks in the concrete, and as it grows and spreads, it can lead to significant degradation in durability, reducing the structure's load-bearing capacity [1; 2; 3]. To reduce the deterioration and improve the durability performance of concrete, there are several techniques: (1) surface treatment, (2) corrosion inhibitors, (3) autonomic self-healing with expansive agents, admixtures, and fibers, and (4) mineral additives (called supplementary cementitious materials). In practical fields, these techniques are widely adopted to increase the design lifespan of concrete structures with the passing of penetration time. The design life takes into account chloride and carbon dioxide penetration rates, which are closely related to the protective ability against rebar corrosion. Both penetration rates rely on the porosity of the matrix, and the distribution of pore sizes in terms of the material aspect [4]. Additionally, for predicting the long-term durability of concrete structures, a mathematical model that characterizes the deterioration process needs to be developed. The process of various ions penetrating concrete has been studied by many researchers over several decades. [PERSON]'s law of diffusion is well known for describing the movement of molecules from a higher concentration to a lower concentration region. Chloride penetration follows [PERSON]'s law in fully saturated concrete, where chloride penetrates due to the concentration gradient. In partially saturated concrete, the degree of moisture diffusion is a significant factor affecting chloride ingress [5; 6; 7]. [PERSON] et al. presented a mathematical model and numerical results for the simulation of electrochemical chloride transport in saturated concrete. The concentration profiles of ion species charged at various current densities in the electric field were demonstrated, considering the activities of ions in the concentration solution [8]. Subsequently, the transport behavior of ions in saturated concrete was simulated using a finite element model. This model was developed based on the Nernst-Planck equation, related to the interactions of multiple ions in the concrete pore solution. The effects of boundary conditions, current density, and the diffusion coefficient of concrete were integrated into the study [9; 10; 11; 12]. An electrochemical chloride removal model was also proposed for concrete that considers electrostatic coupling of ions, adsorption, porosity of the pore structure, and tortuosity effects [13; 14]. [PERSON] et al. investigated the influence of C-S-H tortuosity and surface electrical properties on chloride transport in slag-blended cement systems through a combination of modeling and experimental analysis. The simulation results showed good agreement with the experimental data, and the findings contribute to a better understanding of corrosion risk in reinforced concrete incorporating slag-blended cement [15].