### 2.1 Deterioration of Sewage Pipelines Under Sulfate Attack

RC sewer systems in anaerobic-sulfate environments deteriorate with the growth of sulfate-reducing bacteria and the subsequent generation of hydrogen sulfide gas (Parker, 1945), as explained in Fig. 1. Anaerobic sulfate-reducing bacteria that live in the sediments, introduced into sewer pipes, generate a large amount of hydrogen sulfide (H_{2}S) gas by reducing sulfate (SO_{4}^{2−}) when decomposing and consuming organic sediments (De belie et al., 2004). Meanwhile the bacteria oxidize organic matter using the oxygen bonded with sulfur instead of molecular oxygen for protein synthesis and energy acquisition, as shown in Eq. (1). Thiosulfate (S_{2}O_{3}^{2−}) and tetrathionate (S_{4}O_{6}^{2−}) are generated together with H_{2}S, all of which decrease pH of the concrete structure, thereby inducing the reproduction of sulfur-oxidizing bacteria, which exhibit optimal growth efficiency in neutral and acidic environments. The polythionic acid (a sulfur-based chemical) formed during their growth further decreases pH as well. Sulfur-oxidizing bacteria form sulfuric acid by using thiosulfate and elemental sulfur (S) as intermediates in the reduction of the energy acquisition reaction for their growth. The generated sulfuric acid deteriorates the concrete structure through chemical reactions with the cement hydrates on the concrete pipeline surfaces that are in contact with microorganisms. Once an environment dominated by sulfuric acid has been created, the sulfuric acid reacts with cement hydrates and generates gypsum dihydrate (CaSO_{4}·2H_{2}O) and anhydrous gypsum (CaSO_{4}), as shown in Eqs. (2 and 3) (Monteny et al., 2000). Gypsum dihydrate, being water soluble, is easily dissolved from the cement matrix, creating coarse pores in the structure and accelerating the performance degradation of concrete. In addition, anhydrous gypsum expands through forming ettringite (3CaO·Al_{2}O_{3}·3CaSO_{4}·32H_{2}O) through reaction with aluminate (C_{3}A, 3CaO·Al_{2}O_{3}) in cement as listed in Eq. (4), which causes cracking due to the lack of dimensional stability (Aviam et al., 2004).

$$\eqalign{ & {\text{SO}}_4^{2 - } + {\text{ATP}} \to {\text{adenosylphospate - sulfate }}\left( {{\text{APS}}} \right) \to {\text{SO}}_3^{2 - }\left( { + {\text{AMP}}} \right) \cr & {{\text{S}}_{\text{3}}}{\text{O}}_6^{2 - } \to {{\text{S}}_{\text{2}}}{\text{O}}_3^{2 - }\left( { + {{\text{S}}_{\text{2}}}{\text{O}}_3^{2 - }} \right) \cr}$$

(1)

$${\text{Ca}}{\left( {{\text{OH}}} \right)_2} + {{\text{H}}_{\text{2}}}{\text{S}}{{\text{O}}_4} \to {\text{CaS}}{{\text{O}}_4} \cdot 2{{\text{H}}_2}{\text{O}}$$

(2)

$${\text{CaO}} \cdot {\text{Si}}{{\text{O}}_{\text{2}}} \cdot 2{{\text{H}}_2}{\text{O}} + {{\text{H}}_{\text{2}}}{\text{S}}{{\text{O}}_4} \to {\text{CaS}}{{\text{O}}_4} + {\text{Si}}\left( {{\text{O}}{{\text{H}}_{\text{4}}}} \right) + {{\text{H}}_2}{\text{O}}$$

(3)

$${\text{3CaO}} \cdot {\text{A}}{{\text{l}}_2}{{\text{O}}_{\text{3}}}{\text{ + 3}}\left( {{\text{CaS}}{{\text{O}}_{\text{4}}} \cdot 2{{\text{H}}_2}{\text{O}}} \right){\text{ + 26}}{{\text{H}}_2}{\text{O}} \to 3{\text{CaO}} \cdot {\text{A}}{{\text{l}}_2}{{\text{O}}_{\text{3}}} \cdot 3{\text{CaS}}{{\text{O}}_{\text{4}}} \cdot 32{{\text{H}}_2}{\text{O}}$$

(4)

### 2.2 Background of Service Life and Repair Cost Evaluation Method

#### 2.2.1 Deterministic and Probabilistic Service Life Evaluation

Several studies on service life prediction under sulfate ingress have been performed and some models have handled complicated chemical reactions of sulfate ion with calcium hydroxide and calcium silicates, which generated gypsum and ettringite, however cracking and simultaneous intrusion of sulfate ion are still difficult for actual durability design in engineering level. As previously mentioned, several service life evaluation methods in engineering level adopt simplified patterns such as multi-layer diffusion in the surface layer (Yang et al., 2020a), the linear deterioration depth with diffusion and cement compositions (Atkinson & Hearne, 1989), and relative strength reduction rate (Zhang et al., 2018). Among the models, the second is predominantly used since it can handle chemical component in cement which reacts with sulfate ion, diffusion characteristics in material, exposure concentration of sulfates, and roughness of surface. This model considers the deterioration depth as a linear function of exposure period by assuming the penetration of sulfate ions into concrete through diffusion, reactions between sulfate and aluminum hydrates, and volumetric expansion confined to the surface. Equation (5) shows the deterioration rate by sulfate (Atkinson & Hearne, 1989; Lee et al., 2013):

$$R = \frac{{E \cdot B^{2} \cdot c_{0} \cdot C_{E} \cdot D_{i} }}{{\alpha \cdot {\gamma_{f}} (1 - \nu )}},$$

(5)

where \(E\) is the elastic modulus of concrete (MPa), \(B\) is the linear deformation coefficient by 1 mol of sulfate ions reacting in a unit volume (1.8 × 10^{−6} m^{3}/mol), \(c_{0}\) is the concentration of sulfate ions (mol/m^{3}), \(D_{i}\) is the sulfate ion diffusion coefficient (m^{2}/s), \(\alpha\) is the roughness coefficient, \(\gamma_{f}\) is the concrete fracture energy (= 10 J/m^{2}), \(\nu\) is the Poisson’s ratio of concrete, and \(C_{E}\) is the sulfate ion concentration reacting with ettringite (mol/m^{3}).

In the method, the service life of structure is determined when the increasing deterioration depth with exposure period exceed to the design cover depth. Material reduction factors and environmental factors are considered for marginal durability safety in the design process.

Unlike deterministic durability design, in the probabilistic service life evaluation, the probability of exceeding the critical condition during the target service life is defined and service life is evaluated based on the critical probability. For chloride attacks and carbonation, service life limit conditions and target durability failure probabilities are defined (EN 1991, 2000; Stewart & Mullard, 2007), however, for sulfate attack, no clear target failure probability has been proposed. Assuming that the limit condition is the time for the deterioration depth by sulfate penetration to reach the cover depth, the governing equation can be written as Eq. (6) for probabilistic method:

$${ p_{f}} \left[ {\frac{{E \cdot B^{2} \cdot c_{0} (\mu ,\sigma ) \cdot C_{E} \cdot D_{i} (\mu ,\sigma )}}{{\alpha (\mu ,\sigma ) \cdot {\gamma_{f}} (1 - \nu )}}(t) > C_{d} (\mu ,\sigma )} \right] > p_{d} ,$$

(6)

where \(p_{f} (t)\) is the durability failure probability for the deterioration depth, which increases with time, \(C_{d} (\mu ,\sigma )\) is the probability distribution for cover depth, and \(p_{d}\) is the target durability probability (maximum allowable probability during intended service life). In Eq. (6), the external sulfate ion concentration (\(c_{0}\)), diffusion coefficient (\(c_{0}\)), roughness coefficient (\(\alpha\)), and cover depth (\(c_{d}\)) are random variables.

#### 2.2.2 Deterministic and Probabilistic Repair Cost Evaluation

This section outlines the probabilistic repair cost evaluation based on previous studies. For probabilistic repair cost evaluation, variations in the extended service life through repairing and initial service life (MFP: maintenance-free period) are the primary factors. When the period during which the first deterioration depth reaches the cover depth is assumed to be \(T_{1}\) (the first service life), the number of repairs becomes zero for the period, which requires no repair. In this case, the initial condition can be given by Eq. (7) (Total Information Service Corporation, 2010; Yang et al., 2020b):

$$T_{1} \ge T_{end} ,$$

(7)

where \(T_{1}\) is the initial service life, and \(T_{end}\) is the final target service life of the structure to be used.

If the average value of the first repair timing is set to \(\overline{{T_{1} }}\), the safety index (\(\beta\)) and the probability that no repair is required (\(P_{1}\)) can be expressed through Eq. (8) and Eq. (9) (Jung et al., 2018; Kwon, 2017b; Yoon et al., 2021b):

$$\beta = \frac{{\left( {T_{end} - \overline{{T_{1} }} } \right)}}{{\sigma_{1} }},$$

(8)

$$P_{1} = \int_{{\beta_{1} }}^{\infty } {\frac{1}{{\sqrt {2\pi } }}\exp (} - \frac{{\beta^{2} }}{2})d\beta ,$$

(9)

where \(\sigma_{1}\) is the standard deviation of \(\overline{{T_{1} }}\) at the time of the first repair event.

In the theory, the condition for the number of repairs to be \(N\) is that \(T_{N}\) is smaller than the target service life (\(T_{end}\)), and the sum of \(T_{N + 1}\) and \(T_{N}\) (\(N\)-th repair timing) is larger than \(T_{end}\). In the condition, the safety index can be given by Eq. (10), and the probability (\(P_{N + 1}^{*}\)) that the sum of \(T_{N}\) and \(T_{N + 1}\) is larger than \(\overline{{T_{N} }}\) is as shown in Eq. (11):

$$\beta_{N} = \frac{{\left( {T_{\text{end}} - \left( {\overline{{T_{N} }} + \overline{{T_{N + 1} }} } \right)} \right)}}{{\sqrt {\sigma_{N}^{2} + \sigma_{N + 1}^{2} } }},$$

(10)

$$P_{N + 1}^{*} = 1 - \int_{ - \infty }^{{\beta_{N + 1} }} {f(\beta )d\beta = \int_{{\beta_{N + 1} }}^{\infty } {f(\beta )d\beta } = \int_{{\beta_{N + 1} }}^{\infty } {\frac{1}{{\sqrt {2\pi } }}\exp \left( { - \frac{{\beta^{2} }}{2}} \right)d\beta } } ,$$

(11)

where \(\sigma_{N}\) is the standard deviation of \(T_{N}\). The failure probability when the number of repairs is \(N\)(\(P_{N}\)) can be generalized as shown in Eq. (12). In addition, if the repair cost for the unit member (\(i\)) is constant at \(C_{i}\), the total repair cost can be shown as Eq. (13).

Details of the equations and conceptual diagrams can be found in previous studies (Jung et al., 2018; Kwon, 2017b; Yang et al., 2020a; Yoon et al., 2021c).

$$P_{N} = \left( {1 - \sum\limits_{k = 1}^{N - 1} {P_{N} } } \right) \times P_{N}^{*} ,$$

(12)

$$C_{T} = \sum\limits_{k = 1}^{N} {\left( {k \times C_{i} \times P_{k} } \right)} ,$$

(13)

where \(C_{T}\) is the total repair cost which considers the unit repair costs (\(C_{i}\)). Fig. 2 illustrates the schematic diagram of the probabilistic method and comparison with deterministic method for repair cost evaluation.