In saturated CAC, diffusion and electromigration are two driving forces that cause the transport of free ions in the pore solution, which can be described using the Nernst–Planck equation: J i = − D i [ ∇ c i + z i c i F R T ∇ ϕ ] (6) where J _i (mol/m²/s) denotes the flux of free ion i; D _i (m²/s) is the diffusion coefficient of free ion i in CAC, which will be discussed in detail later; c _i (mol/m³) and z _i (−) denote the ionic concentration and electric charge number, respectively; ϕ (V) denotes the electric potential. The electric neutrality in CAC should be satisfied when subjected to chloride ingress, and hence, F ∑ i z i J i = 0 (7)

Combining Eqs 6, 7, the following can be deduced: J i = − D i ( ∇ c i − R T F ∑ i z i D i ∇ c i ∑ i z i 2 D i c i ) (8)

According to the continuity equation, we can obtain: ∂ c i ∂ t + ∇ ⋅ J i = Q i (9) where Q _i denotes the source term, which is determined by the physical and chemical interactions between the pore solution and cement hydrate, which is calculated via thermodynamic modelling in Physical and Chemical Interactions Between Pore Solution and Cement Hydrate.

The diffusion coefficient (D _i ) of free ion i in the pore solution of a porous material is completely different from that in free water because it is affected by the constrictivity and tortuosity of the pore structure (Appelo et al., 2010). Considering that the diffusion coefficient (D _Cl) of chloride in saturated cement-based materials can be easily measured by rapid chloride migration testing, the diffusion coefficient (D _i ) of free ion i in CAC is expressed using the following equation: D i = D Cl D Cl_free   ⋅   D i _ free (10) where D Cl D Cl_free represents the effect of tortuosity and constrictivity of the pore structure (Tang and Nilsson, 1995), including the effect of the porous property of the coral aggregate. D _{i_free} denotes the diffusion coefficient of ion i in free water, which can be measured experimentally, and their values for Ca²⁺, Na⁺, K⁺, OH⁻, SO₄ ^2-, and Cl⁻ are 0.791 × 10^–9 m²/s, 1.33 × 10^–9 m²/s, 1.96 × 10^–9 m²/s, 5.27 × 10^–9 m²/s, 1.07 × 10^–9 m²/s, and 2.03 × 10^–9 m²/s at 20°C (Haynes et al., 2014).

Reactive transport modelling is divided into two modules using the operator splitting algorithm, i.e., the reaction and transport modules (Nardi et al., 2014). The PHREEQC program, a geochemical simulator developed by the United States Geological Survey (USGS), can solve the reaction module (Parkhurst and Appelo, 2013). COMSOL Multiphysics, a commercial finite element method (FEM) software, can solve the transport module (COMSOL, 2021). Hence, a COMSOL-PHREEQC interface must be developed to achieve the interaction between these two modules (Guo et al., 2018; Nardi et al., 2014). COMSOL Multiphysics with MATLAB can connect COMSOL Multiphysics to the MATLAB script (COMSOL, 2021). The IPhreeqc COM allows PHREEQC to be used as a COM server for MATLAB (Charlton and Parkhurst, 2011). Therefore, in this study, the MATLAB scripting environment was selected to establish the COMSOL-PHREEQC interface. The main structure and workflow of the interface are illustrated in Figure 1.

The hydrated cement compounds and concentrations of species in the pore solution are necessary to solve the reactive transport model in cement-based materials and can be obtained from experiments or calculation of GEMS (Kulik et al., 2013; Lothenbach and Winnefeld, 2006; Lothenbach and Zajac, 2019). However, the pore solution and hydrated cement compounds from experiments or GEMS may not be in an equilibrium state in PHREEQC. Thus, as shown in Figure 1, the first step is to initialize such that they are in an equilibrium state in PHREEQC. Simultaneously, the COMSOL model needs to be established, which includes building geometric model, setting material parameters and physical fields, and defining boundary conditions. Then, the concentrations of the free ions from the initialization are output to the COMSOL model as the initial values to calculate the transport process in the first time step. COMSOL is based on FEM, and the concentrations of free ions at each node can be obtained. The values are saved in matrix and then passed to PHREEQC to solve the thermodynamic equilibrium at each FEM node with the hydrated cement compounds from the initialization. When completing the thermodynamic equilibrium calculations at all the nodes, the concentrations of the free ions at the nodes are updated, and then sent back to the COMSOL model as initial values via linear interpolation function to calculate the transport process in the second time step. Meanwhile, the hydrated cement compounds at all the nodes are saved in the form of a matrix for calculating the thermodynamic equilibrium in the third time step. The above procedure is repeated until the calculation ends.

Spiesz et al. (2012) reported that the time to achieve a new thermodynamic equilibrium between the pore solution and cement hydrate is approximately 10 days. Therefore, this 10 was selected as the time step in this model.

To investigate chloride ingress in CAC, CAC cubic specimens with the size of 100 mm × 100 mm × 100 mm were exposed to 3.5% NaCl solution in the literature (Huang, 2020; Huang et al., 2020a), and the free chloride concentrations in the CAC were measured at 30, 60, 90, 120, and 180 days. The coral aggregates presented in Figure 2 were used in their experiment. The chloride ion contents of coral fine and coarse aggregates are 0.6284% and 0.6375, respectively (Wang et al., 2019). In this study, the CAC specimen C30F15 in their experiment was taken as a benchmark example to show the implementation of the reactive transport modelling of chloride ingress in CAC. The experimental results about the free chloride concentration at different exposure times were used to verify the numerical model. The mix proportion, chloride diffusion coefficients, porosity accessible to water, and slump are listed in Table 2. Other details regarding the preparation procedure of CAC and the experiments to measure these characteristics can be found in the literature (Huang et al., 2020a; Huang, 2020). Among the parameters listed in Table 2, the mix proportion is used to calculate the hydrated cement compounds and pore solution, which will be further discussed in the next paragraph. Because CAC itself contains a certain amount of chloride, the rapid chloride migration method cannot measure the chloride diffusion coefficient. Hence, the chloride diffusion coefficient of CAC fitted at the minimum exposure time in Huang et al. (2020a) is used although it is a little unreasonable, as given in Table 2.

The chemical compositions of the cement and FA used in the literature (Huang, 2020; Huang et al., 2020a) are presented in Table 3. According to the mix proportion in Table 2, the hydrated cement compounds and the pore solution can be calculated by employing the GEMS software with the CEMDATA18 and Nagra/PSI TDB thermodynamic databases (Kulik et al., 2013; Lothenbach et al., 2019). It is assumed that the CAC is completely hydrated, and the effect of ageing is neglected. The effect of CO₂ in the water or atmosphere on cement hydration is also neglected. The hydrated cement compounds that are considered to participate in the dissolution/precipitation reactions of hydrate phases are calcium silicate hydrate (C−S−H), portlandite (CH), monosulfoaluminate (AFm), ettringite (AFt), Friedel’s salt (Fs), Kuzel’s salt (Ks), and hydrotalcite. Free ions (including Ca²⁺, K⁺, Na⁺, OH⁻, SO₄ ^2-, and Cl⁻) and aqueous complexes (such as AlO₂ ⁻, HSiO₃ ⁻, etc.) in the CAC pore solution are taken into account. Therefore, the initial hydrated cement compounds and pore solution obtained by GEMS calculation are given in Tables 4, 5. It has been reported that when the free chloride concentration in pore solution is low, the chemical binding of chloride in cement-based materials is primarily through Kuzel’s salt formation (Mesbah et al., 2011). However, the chloride concentration in CAC is high because coral aggregates contain a considerable amount of chloride. In the experimental study by Wang et al. (2019), Friedel’s salt in CAC was found, which is in line with the results in Table 4. In the study of Huang et al. (2020a), when the exposure time was 30d, the chloride penetration depth was less than 18 mm. The free chloride concentration in the interior of the CAC specimen where the external chloride cannot penetrate should be nearly equal to its initial concentration, and it is 0.17% of concrete weight. Then, according to the mix proportion of CAC and the porosity of CAC, it is 89.29 mol/L of pore solution. The calculation result of the chloride concentration in Table 5 is 105.99. There is a small and acceptable gap between the calculation and experimental results.

The specific surface area and surface site density of C−S−H are two key parameters for calculating the surface complexation reactions, and their values are 500 m²/g and 4 × 10³ mol/g, respectively (Pointeau et al., 2006).

According to the experiment of Huang et al. (2020a), the size of the CAC specimen was 100 × 100 × 100 mm³; one surface was exposed to a 3.5% NaCl solution (approximately 0.6 mol/L), and the others were sealed with epoxy resin. In addition, to ensure a constant concentration of chloride, the NaCl solution was periodically replaced. This meant that the Na⁺ and Cl⁻ concentrations in the external solution nearly maintained 0.6 mol/L throughout the experimental process. In the numerical model, the operator splitting algorithm was used, and the initial values of the ionic concentration in the domain were redefined at each time step (see Figure 1). Thus, the initial values of the Na⁺ and Cl⁻ concentrations in the external solution were defined as 0.6 mol/L at each time step, instead of the fixed boundary condition widely used in the ionic transport modelling. To reduce the calculation time, the height of the domain of the solution should be as small as possible. The two-dimensional geometrical model was built as shown in Figure 3A, in which the height of the domain of the external solution was chosen as 0.02 m to balance the calculation time and the accuracy. A quadrilateral mesh was adopted, as shown in Figure 3B. The difference in the initial ionic concentrations between the external solution and the CAC specimen was significant, and hence, to improve the convergence of the numerical model, the boundary layer mesh was used to densify the finite element mesh at the interface between the CAC specimen and the external solution. All boundaries in the domain were insulated. In the study of Huang et al. (2020a) The longest time that the specimen was exposed to NaCl solution was 180 days. Herein, the period for simulating chloride ingress for CAC was setas 360 days, giving 36 time steps of 10 days.

When the exposure times are 30, 60, 90, 120, 180, and 360 d, the concentrations of free chloride in the pore solution are shown in Figures 4A‐F respectively. Coral aggregates contain chloride, and hence, free chloride exists in the pore solution of CAC before exposure to the NaCl solution. As presented in Table 5, the initial free chloride concentration is 0.106 mol/L. Thus, the minimum free chloride concentration in the entire domain is 0.106 mol/L of pore solution. The free chloride concentration at the exposure side is the highest, and it gradually decreases with increasing distance from the exposed surface, becoming zero beyond a certain depth. The numerical results in Figure 4 clearly show that the penetration depth of chloride increases with prolonged exposure time. When the CAC specimen is exposed to the external solution for 360 days, the penetration depth of chloride is 23.85 mm (Figure 4F).

The numerical model predicts the pH of the pore solution when the exposure times are 30, 60, 90, 120, 180, and 360 d, which are shown in Figures 5A‐F respectively. Due to the presence of a concentration gradient, OH⁻ moves from high-concentration regions to low-concentration regions. The external solution is neutral, and OH⁻ will move from the pore solution to the external solution. Hence, the pH in the pore solution near the exposed side decreases with increasing exposure time. When the exposure time is 360 days, the pH near the surface is lower than 11.5 (Figure 5F). In addition, similar to the penetration depth of chloride, the regions with reduced pH of the pore solution become thicker with the elapse of time.

The physical and chemical binding of chloride has a significant effect on chloride transport in cement-based materials because it can slow down chloride transport, and prolong the time before chloride arrives at the steel surface. When the free chloride concentration decreases, the bound chloride is released into the pore solution as free chloride. The predicted chloride content physically bound by C−S−H at different exposure time is shown in Figure 6. Prior to the ingress of external chloride, some chloride ions in CAC have already been bound by C−S−H or chemically bound by producing Friedel’s salt. With increasing exposure time, increasing number of external chloride ions penetrate CAC, and hence, the amount of chloride bound by C−S−H increases, as shown in Figure 6. The amount of chloride physically bound by C−S−H is closely related to the free chloride concentration in the pore solution. The free chloride concentration at the exposed side is the highest and remains constant (Figure 4). Thus, the amount of chloride physically bound at the exposed side is also the most, which is barely affected by the exposure time.

The amount of chemically bound chloride is expressed in the form of the content of Friedel’s salt in the cement hydrate compounds, as shown in Figure 7. As is well known, when OAC is subjected to chloride attack, the chloride concentration in OAC is lower at the early stage, and some chlorides are bound by Kuzel’s salt formation (Mesbah et al., 2011; Tran et al., 2018). With increasing the chloride concentration in OAC from the external environment, Kuzel’s salt is converted into Friedel’s salt (Zibara, 2001), which means that chloride is bound by Friedel’s salt formation. However, because coral aggregates contain chloride, prior to chloride ingress, there are plenty of free chloride ions in the initial pore solution (Table 5 and Figure 4), and some chlorides are directly bound by Friedel’s salt formation (Figure 6). Thus, Kuzel’s salt does not appear in CAC during chloride ingress.

One molar of Friedel’s salt contains two molars of chloride, which can be seen from the formula in Table 4. The amount of Friedel’s salt in CAC directly reflects the amount of chemically bound chloride. The amount of bound chloride at the exposed side decreases with prolonging the exposure time. When the exposure time exceeds 60 days, the amount of chloride chemically bound at the exposed side is not the highest in the whole specimen although the free chloride concentration is the highest. Some experimental results indicate that the stability of Friedel’s salt is dependent on the pH of the pore solution, and a decrease in the pH can lead to the dissolution of Friedel’s salt (Suryavanshi and Swamy, 1996). The pH at the exposed side is the lowest in the entire specimen owing to the OH⁻ movement from the pore solution to the external solution, and it decreases with increasing the exposure time (Figure 5). This is unfavourable for the formation of Friedel’s salt. Therefore, the amount of Friedel’s salt on the exposed side decreases, which differs from the physical binding of chloride. However, the variation in the amount of Friedel’s salt in the interior region of the specimen is similar to that of the physical binding.

Huang et al. (2020a) measured the free chloride concentration in CACs with different proportions at different exposure time. Their experimental results of C25F15, C30F15 and C40F15 CAC at 60 days and 180 days are used to verify the numerical model. In Numerical Implementation, C30F15 CAC is used a benchmark example to illustrate the implementation of the numerical model and how to obtain the required parameters from the study of Huang et al. (2020a). Similarly, the required parameters of C25F15 and C40F15 can be obtained. The initial parameters are listed in Tables 6, 7. The diffusion coefficients of chloride in C25F15 and C40F15 CAC used in the numerical model are 15.28 × 10^–12 and 12.71 × 10^–12, respectively.

Comparisons between the experimental and numerical results for free chloride in C25F15, C30F15 and C40F15 are shown in Figures 8A‐C respectively. The results for C30F15 CAC are presented in Figure 4, which are represented in Figure 8B using traditional 2D plots. It is noted that the free chloride concentration is versus the CAC weight in their experimental results, and the free chloride concentration in the numerical results needs to be converted according to the mix proportion of CAC and the porosity of CAC.

By comparing the numerical and experimental results in Figure 8, it is clear that the numerical model overestimates the free chloride concentration in the surface layer of the specimen, especially when the distance from the exposed side is less than 5 mm. However, the numerical results agree well with the experimental results in the interior of the specimen. Doubt may arise as to whether the developed numerical model can accurately predict the chloride transport in CAC. In the experiment of Huang et al. (2020a), free chloride was obtained from CAC powder using distilled water extraction. At the end of the exposure, all the specimens were placed indoors for 7 days and then dried for 24 h to remove the moisture in the surface layer of CAC before they were milled using a concrete pulveriser. These processes to remove moisture significantly affect the chloride transport, especially in the surface layer of the CAC specimen. It should be noted that these processes occurred at the end of the chloride ingress for CAC, and it is unnecessary to consider them in the reactive transport modelling of chloride ingress for saturated CAC. In other words, the post-processing about the experimental results of (Huang et al., 2020) was not included by the numerical model. This is the primary reason why the numerical model does not agree with the concentration of the free chloride ion in the surface layer of the CAC specimen. However, the processes hardly affect the free chloride concentration measurement in the interior of the CAC specimen, and the chloride ingress is predicted well with the numerical model in this region. Thus, the numerical model can predict the chloride ingress for saturated CAC.

The diffusion coefficient of chloride in the pore solution of CAC is a key input parameter for simulating ionic transport, and this parameter is constant in this study. However, Friedel’s salt formation during chloride ingress causes a less porous structure of cement-based materials (Yuan et al., 2009). Thus, chloride ingress results in a decrease in the diffusion coefficient of chloride owing to the increase in the content of Friedel’s salt (see Figure 7). In practice, chloride ingress into saturated CAC generally occurs in an underwater environment, which is accompanied by the decalcification of cement hydrates. This can also be seen from the benchmark example, in which the numerical model predicts the CH content in CAC at different exposure time, as shown in Figure 9. It is clear that the CH amount decreases with prolonged exposure time, especially in the vicinity of the exposed side. Decalcification would result in a more porous structure of CAC, leading to an increase in the diffusion coefficient of chloride in the pore solution of CAC. However, the positive effect of decalcification and the negative effect of Friedel’s salt formation on the diffusion coefficient of chloride only appears in the local region close to the external solution. These effects are more significant with time, and the local region will increase. Thus, the diffusion coefficient of chloride varies with time and space. However, it is difficult to consider this in the reactive transport modelling. This is the limit of the model developed in this study. This may also be another reason for the gap between the numerical and experimental results shown in Figure 8.