Keywords: Bio-détérioration, Canalisations d'égout, Béton, Acide sulfurique, Biofilm, Modélisation du transport réactif, Portlandite, C-S-H, Gypse Bio-deterioration, Sewer pipe, Concrete, Sulfuric acid, Biofilm, Reactive transport modelling, Portlandite, C-S-H, Gypsum V

Seule la diffusion des espèces aqueuses est considérée. Différents coefficients de diffusion sont utilisés pour divers ions et l'équation de Nernst-Planck est implémentée. L'effet, pendant la détérioration, de la modification de la microstructure sur les propriétés de transport est aussi considéré. Pour les biofilms et les matériaux cimentaires, les équations d'équilibre de masse totale de chaque atome (Ca, Si, S, K, Cl) sont utilisées pour coupler les équations de transport et les réactions (bio) chimiques.

Le modèle est implémenté dans un code volumes finis, Bil. Grâce à l'introduction de la méthode des volumes finis, on illustre le couplage du processus bio-chimie dans les biofilms et le processus de la chimie des matériaux cimentaires. Par ce modèle, certaines expériences rapportées dans la littérature, dont des tests d'immersion chimiques (condition de la solution statique et condition de la solution d'écoulement) et des simulations microbiologiques, sont simulées. Les résultats numériques et les observations expérimentales sont comparés et discutés. L'influence des propriétés des matériaux cimentaires (porosité initiale, couche carbonatée, etc.) et les facteurs d'environnement (concentration de H 2 SO 4 quantité de H 2 S etc) sont aussi étudiés par ce modèle. En outre, une prédiction à long terme est menée.

Résumé

La biodétérioration du béton, très courante dans les systèmes d'égouts et de traitement des eaux usées, entraîne une dégradation significative de la structure. Normalement, le processus peut être décrit par les deux étapes suivantes : 1) Des réactions biochimiques produisent des espèces agressives dans les biofilms qui tapissent la surface du béton. L'un des plus importants acides biogéniques que l'on trouve dans les canalisations d'égout est l'acide sulfurique (H 2 SO 4 ) que est produit par des bactéries sulfo-oxydante (BSO) à partir de l'hydrogène sulfuré (H 2 S). 2) Les réactions chimiques entre les espèces agressives biogéniques et les produits d'hydratation du ciment sont responsables de la détérioration du béton.

Un modèle de transport réactif est proposé afin de simuler les processus des détériorations chimique et biochimique des matériaux cimentaires en contact avec les BSO et le H 2 S ou une solution d'acide sulfurique. L'objectif de ce modèle est de résoudre simultanément le transport et la biochimie / chimie dans les biofilms et les matériaux cimentaires par une approche globale couplée.

7.1 Experiment setup of immersion test of biodeterioration: (a) [START_REF] Vincke | Recent developments in research on biogenic sulfuric acid attack of concrete[END_REF]; (b) (De Belie et al., 2004). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Experiment setup of biodeterioration by simulation reactors: (a) [START_REF] Mori | Interactions of nutrients, moisture and ph on microbial corrosion of concrete sewer pipes[END_REF]; (b) [START_REF] Herisson | Biodétérioration des matériaux cimentaires dans les ouvrages d'assainissement: étude comparative du ciment d'aluminate de calcium et du ciment Portland[END_REF] [START_REF] Mori | Interactions of nutrients, moisture and ph on microbial corrosion of concrete sewer pipes[END_REF]. . . . . . During the service life of materials, biodeterioration is an important cause of the reduction of the performance and durability of materials as well as physical and chemical deterioration. Hueck [START_REF] Hueck | The biodeterioration of materials-an appraisal[END_REF]) first proposed that biodeterioration is "any undesirable change in the properties of a material caused by the vital activities of organisms".

7.17

In biodeterioration, the organisms vary widely, such as marine borers (including mollusks and crustaceans), insects, fungi, prokaryotes (including bacteria and archea) and plants [START_REF] Sanchez-Silva | Biodeterioration of construction materials: state of the art and future challenges[END_REF]. They form specific communities that interact with materials and external environment in many different ways. As organisms exist everywhere, they can accelerate deterioration of a wide range of materials (as shown in Fig. 1.1), including inorganic minerals [START_REF] Mitchell | Changes in the biofilm microflora of limestone caused by atmospheric pollutants[END_REF], concrete [START_REF] Gu | Biodeterioration of concrete by the fungus f usarium[END_REF][START_REF] Cwalina | Biodeterioration of concrete[END_REF], stones [START_REF] Warscheid | Biodeterioration of stone: a review[END_REF][START_REF] Perry | Biodeterioration of stone[END_REF], metals [START_REF] Cragnolino | The role of sulphate-reducing and sulphur-oxidizing bacteria in the localized corrosion of iron-base alloys. a review[END_REF], and natural and synthetic polymers [START_REF] Guezennec | Bacterial colonization of artificial substrate in the vicinity of deep-sea hydrothermal vents[END_REF][START_REF] Flemming | Relevance of biofilms for the biodeterioration of surfaces of polymeric materials[END_REF]. Specifically, microorganisms play major role in deterioration. According to US estimation, the contribution of microbiologically induced corrosion (MIC) to the deterioration of materials as a whole is about 30% [START_REF] Sand | Microbial corrosion and its inhibition[END_REF]. In many cases, MIC is due to the presence of a surface layer of microorganisms and their products. Such microbial layers are known as biofilms and are defined as surface accumulations of the organic products of biological activity. Generally, biofilms are in direct contact with the material rather than dispersed in the surrounding environment. Thus, the aggressive products of microorganisms are concentrated and result in severe damage of materials. Some examples are shown in Table 1.1. Table 1.1: Some problems associated with the presence of biofilms on materials [START_REF] Allsopp | Introduction to biodeterioration[END_REF] .

Biofilm location Effects

Teeth Tooth decay, caries Sewer pipe Decreased durability, reduced flow Cooling towers Degradation of material, reduced heat transfer Drinking water distribution systems Decreased water quality, health risks Oil industry pipelines

Blockage and corrosion

As one of the most widely used materials, concrete suffers from biodeterioration. The first mention of microbiologically induced concrete corrosion (MICC) in literatures was by [START_REF] Olmstead | Converting portions of the los angeles outfall sewer into a septic tank[END_REF]. Most of MICC were detected in constructions like sewer pipes [START_REF] Diercks | Microbial corrosion of concrete[END_REF][START_REF] Mori | Interactions of nutrients, moisture and ph on microbial corrosion of concrete sewer pipes[END_REF][START_REF] Cho | A newly isolated fungus participates in the corrosion of concrete sewer pipes[END_REF], waste water treatment facilities [START_REF] Redner | Evaluation of protective coatings for concrete exposed to sulfide generation in wastewater treatment facilities[END_REF], cooling towers [START_REF] Zherebyateva | Microbiological corrosion of concrete structures of hydraulic facilities[END_REF], gas and oil platforms [START_REF] Edyvean | Biodeterioration problems of north sea oil and gas production-a review[END_REF], marine structures [START_REF] Hughes | Microscopic study into biodeterioration of marine concrete[END_REF], and many others where various microorganisms (like bacteria, microscopic fungi, algae and lichen) are usually present at high concentrations [START_REF] Viitanen | Moisture and bio-deterioration risk of building materials and structures[END_REF].

The most rapid cases of deterioration always occur in the places with high H 2 S concentration, moisture, and oxygen in the atmosphere. Such conditions are commonly found in sewage collection systems (as shown in Table 1.2).

Table 1.2: Biodeterioration rate of concrete in sewage systems .

Location of samples Biodeterioration rate (mm/year) Reference

Sewer pipe in treatment plant 4.3-4.7 [START_REF] Mori | Interactions of nutrients, moisture and ph on microbial corrosion of concrete sewer pipes[END_REF]) Manhole in treatment plant 5.7-7.6 [START_REF] Hudon | Biodeterioration of concrete sewer pipes: State of the art and research needs[END_REF]) Urban sewer pipe in Houston 3.1 [START_REF] Davis | Analysis of concrete from corroded sewer pipe[END_REF]) Urban sewer pipe in Hamburg 8-10 [START_REF] Milde | Thiobacilli of the corroded concrete walls of the hamburg sewer system[END_REF] Besides sewer pipes, approximately 40% of a wastewater system is made up of concrete structures.

About 40% of the damage in concrete pipeline is caused by biodeterioration [START_REF] Kaempfer | Estimation of service life of concrete pipes in sewer networks[END_REF].

As shown in Fig. 1.2, MICC in sewerage system can lead to fragmentation of pipe surface, debonding of concrete, collapse of the physical and mechanical properties of structures. Thus, huge money is being spent on the repair and maintenance of sewerage collection system. For Germany, a cost of approximately 100 billion Euros is estimated for maintenance and repair of private and public sewage systems [START_REF] Kaempfer | Polymer modified mortar with high resistance to acid to corrosion by biogenic sulfuric acid[END_REF]). ASCE's report also estimates that the United States will need 390 billion dollars during the next 20 years to repair and replace the existing wastewater infrastructure and construct new facilities, including collection systems (Gutiérrez-Padilla et al., 2010).

Figure 1.2: Examples of MICC of sewer pipe Thus, there is great interest in predicting the corrosion rate and the service life of sewer pipes. To simulate the biodeterioration process, it is necessary to understand the mechanisms of biodeterioration of sewer pipe including the bio-activity in biofilm as well as the chemical reactions between concrete and biogenic sulfuric acid.

Mechanisms of biodeterioration of sewer pipes

Because of the alkalinity of concrete and the acidity of environment in sewer pipes, the main cause of biodeterioration of these structures is biogenic sulphuric acid attack (BSA). According to experimental observations, the process of BSA of cementitious materials in sewer pipes can be divided into two parts which are schematically indicated in Fig. 1.3.

Figure 1.3: Schematic of biogenic sulphuric acid attack in a sewerage [START_REF] Herisson | Toward an accelerated biodeterioration test to understand the behavior of Portland and calcium aluminate cementitious materials in sewer networks[END_REF]).

A) Hydrogen sulfide (H 2 S) formation in wastewater.

Sulfate is commonly present in wastewater. At the bottom of pipe where water is anoxic, sulfate can be converted into hydrogen sulfide (H 2 S). This conversion is performed by sulfate-reducing bacteria (SRB), which are heterotrophic bacteria found in almost any environmental sample [START_REF] Barton | Characteristics and activities of sulfate-reducing bacteria[END_REF].

SRB can use oxidized sulfur compounds as electron acceptor by following reactions [START_REF] Wood | Halothiobacillus neapolitanus strain oswa isolated from "the old sulphur well" at harrogate (yorkshire, england)[END_REF].

SO 2- 4 + 4H 2 O + 8e -SRB ---→ S 2-+ 8OH - (1.1a) SO 2- 4 + organic substance SRB ---→ S 2-+ H 2 O + CO 2 (1.1b) HS -⇋ H + + S 2- (1.1c) H 2 S (aq) ⇋ H + + HS - (1.1d)
In normal sewage, pH is between 5 and 6. Since H 2 S is poorly soluble (pK= 7.05), it is degassed in the arcal part of the pipe following Henry's Law.

B) Biodeterioration of concrete in the arcal part of pipes. The deterioration process can be described by the following three steps as shown in Fig. 1.4.

Figure 1.4: Theoretical changes in the biological and physical properties of concrete with time during the deterioration process, [START_REF] Islander | Microbial ecology of crown corrosion in sewers[END_REF].

Step 1: Abiotic pH reduction of concrete.

The pH of fresh concrete (11-13) is too high for the sulfur-oxidizing bacteria (SOB) to survive. CO 2 and H 2 S are both present in the arcal part of sewer pipes. Experiment observations [START_REF] Ismail | Effect of carbonation on microbial corrosion of concrete[END_REF][START_REF] Joseph | Surface neutralization and h2s oxidation at early stages of sewer corrosion: Influence of temperature, relative humidity and h2s concentration[END_REF] revealed that such acidic gases can reduce the pH of concrete surface to less than 9 as described in reaction (1.2a) and (1.2b).

CH + CO

2 ⇋ CaCO 3 + H 2 O (1.2a) CH + 2H 2 S ⇋ Ca(HS) 2 + 2H 2 O (1.2b)
Step 2: Microbial colonization and production of H 2 SO 4 by biofilm.

Once the pH of the surface of concrete is reduced to 9, with sufficient nutrients (element sulfur), moisture and oxygen, some species of SOB like T hiobacillus sp. can grow on the concrete surface and form biofilms [START_REF] Rigdon | Corrosion of concrete by autotrophes[END_REF][START_REF] Mori | Interactions of nutrients, moisture and ph on microbial corrosion of concrete sewer pipes[END_REF]. Although the bio-activity in biofilms is not fully understood, it is generally believed that firstly H 2 S in biofilms is oxidized to element sulfur by metal catalyst [START_REF] Zivica | Acidic attack of cement-based materials-a review part 2. factors of rate of acidic attack and protective measures[END_REF] or neutrophilic sulfur-oxidizing microorganisms (NSOM) [START_REF] Bielefeldt | Bacterial kinetics of sulfur oxidizing bacteria and their biodeterioration rates of concrete sewer pipe samples[END_REF] as follows:

H 2 S + 1 2 O 2 NSOM --------→ metalcatalyst S 0 + H 2 O (1.3)
Not only element sulfur, but also a few sulfuric acid is produced by NSOM. After pH has decreased to 4-5, acidophilic sulfur-oxidizing microorganisms (ASOM) like T.thiooxidans can use element sulfur to produce large amounts of sulfuric acid as reaction (1.4), which makes the main contribution to the biodeterioration of sewer pipe [START_REF] Parker | The corrosion of concrete[END_REF][START_REF] Parker | Mechanics of corrosion of concrete sewers by hydrogen sulfide[END_REF]. With sufficient sulfur source, the pH of concrete surface can be reduced to 1 by T.thiooxidans [START_REF] Mori | Interactions of nutrients, moisture and ph on microbial corrosion of concrete sewer pipes[END_REF]. That is destructive to concrete.

S 0 + H 2 O + 3 2 O 2 ASOM ----→ SO 2- 4 + 2H + (1.4)
Step 3: Chemical reaction between H 2 SO 4 and concrete.

During the final step of the biodeterioration process, the biogenic sulfuric acid penetrates into concrete and reacts with portlandite (CH) 1 and calcium silicate hydrate (C-S-H) which are the main Portland cement hydrates. This step is characterized by the production of a corroded layer on the surface of concrete.

XRD patterns [START_REF] Davis | Analysis of concrete from corroded sewer pipe[END_REF][START_REF] Kawai | Concrete deterioration caused by sulfuric acid attack[END_REF], and other experimental observations (as Fig. 1.5) of corroded layer revealed that corrosion products consist of gypsum (CaSO 4 •2H2O, noted as C SH2), silica gel (SiO 2 , noted as S) and moisture. Thus, during deterioration CH and C-S-H are dissolved, while gypsum and silica gel precipitate as reactions (1.5a) and (1.5b).

CH + H 2 SO 4 ⇋ C SH2 (1.5a)

C x S y H z + xH 2 SO 4 ⇋ xC SH2 + ySiO 2 + (z -x)H 2 O (1.5b)
Since gypsum is swelling while it has barely no strength, H 2 SO 4 attack could reduce the performance of concrete, and even lead to the eventual structural failure of the facility. Figure 1.5: (a) Concrete coupons exposed to the sewer atmosphere [START_REF] Okabe | Succession of sulfur-oxidizing bacteria in the microbial community on corroding concrete in sewer systems[END_REF]; (b) Scanning electron microphotograph of gypsum crystals in the cement paste [START_REF] Monteny | microbiological, and in situ test methods for biogenic sulfuric acid corrosion of concrete[END_REF].

Recent research works about biodeterioration of cementitious materials

To understand and to fight against such a biodeterioration process involving a succession of bacterial in biofilms and resulting in a change of concrete properties, many research works were conducted focusing on different aspects. Not only experimental (in-situ/laboratory) researches, but also modeling researches were conducted.

Samples of biofilm taken from sewer or cultured in laboratory, were monitored. The type, number and succession of SOB were measured by conventional culture-dependent techniques [START_REF] Harrison | The acidophilic thiobacilli and other acidophilic bacteria that share their habitat[END_REF], conventional cultivation techniques and molecular tools [START_REF] Vincke | Analysis of the microbial communities on corroded concrete sewer pipes-a case study[END_REF], and gene-cloning analysis [START_REF] Herisson | Toward an accelerated biodeterioration test to understand the behavior of Portland and calcium aluminate cementitious materials in sewer networks[END_REF]. The understanding of microbial community structures of SOB and their activities in biofilm has been improving continuously. It is generally believed that five species of T hiobacillus play important roles: T.thioparus, T.novellus, T.neapolitanus, T.intermedius and T.thiooxidans. The characteristics of these SOB are listed in Table 1.3. Since [START_REF] Kempner | Acid production by thiobacillus thiooxidans[END_REF] has found that the activity of T.thiooxidans is slowed at pH below 0.9, it is expected that if sulfur source is sufficient, the pH of biofilm would be limited around 1.

Table 1.3: Characteristics of T hiobacillus in biofilm [START_REF] Islander | Microbial ecology of crown corrosion in sewers[END_REF] .

Species pH range for growth Products

T.thioparus 4.5-10 Sulfur, polythionic acids T.novellus 5-9.2 Sulfur T.neapolitanus 4-9 Polythionic acids, sulfuric acid T.intermedius 1.7-9 Polythionic acids, sulfuric acid T.thiooxidans 0.5-4 Sulfur, sulfuric acid

To figure out the components of corrosion products, in the in-situ experiments carried out by Davis [START_REF] Davis | Analysis of concrete from corroded sewer pipe[END_REF], samples of concrete were drilled from the Houston's sewage collection pipes. Several tests revealed that the average compressive strength of the concrete undergoing degradation was reduced by more than 20%. The mineralogical analysis found that the corrosion layer of concrete contained large amounts of gypsum, which is expansive and has no strength, and yet ettringite was not present. Such observations were reported by other experimental laboratory researches [START_REF] Ehrich | Biogenic and chemical sulfuric acid corrosion of mortars[END_REF][START_REF] Vincke | Recent developments in research on biogenic sulfuric acid attack of concrete[END_REF] as well. It was explained that ettringite forms at pH ranging from 12.5 to 12, while it starts decomposing into gypsum when pH < 10.6 [START_REF] Allahverdi | Acidic corrosion of hydrated cement based materials[END_REF].

To study the consequences of gypsum precipitation, C 3 S mortar and paste specimens were exposed to Na 2 SO 4 solution [START_REF] Tian | Does gypsum formation during sulfate attack on concrete lead to expansion?[END_REF]. The physical properties and chemical composition of samples after corrosion were tested. The experimental results suggest that during gypsum formation tensile stresses may play a role in expansion and subsequent cracking.

To study the behaviour of cementitious materials corroded by H 2 SO 4 , researchers [START_REF] Jahani | Investigations of sulfuric acid corrosion of concrete. ii: electrochemical and visual observations[END_REF][START_REF] Bassuoni | Resistance of self-consolidating concrete to sulfuric acid attack with consecutive ph reduction[END_REF][START_REF] Kawai | Concrete deterioration caused by sulfuric acid attack[END_REF] conducted experiments to simulate the chemical corrosion process of concrete in contact with H 2 SO 4 solution.

In the experiment of [START_REF] Kawai | Concrete deterioration caused by sulfuric acid attack[END_REF] ordinary Portland cement cubes were immersed in H 2 SO 4 solution with pH=1 in order to simulate the chemical corrosion process in sewer pipe. To keep the concentration of H 2 SO 4 in the solution constant, H 2 SO 4 was added suitably to the solution, and the solution was statical (Method 1) or circulated with a pump and flew into the tank (Method 2). In method 2, the specimens were subjected to the shearing force of the solution flow. Therefore the reaction products were driven out by the flow of solution without precipitating on the sample surfaces. During the tests, the corrosion depth was measured. In Method 1, a layer of reaction products was observed on the surface of the mortar samples. The XRD patterns of this degraded layer showed that reaction products of sulfuric acid attack are mainly gypsum and silica gel. With method 2, few products can be seen on the sample surfaces since most of the reaction products were swept by the solution flow. The solution flow washed the new surface of reaction, thus the samples immersed with Method 2 degraded much deeper and faster than those with the Method 1. In Kawai's experiment, an immersion test using a solution with low H 2 SO 4 concentration solution (pH=2) was carried out as well. It turned out that corrosion process is very sensitive to H 2 SO 4 concentration: all of the samples immersed in H 2 SO 4 solution of pH=2 were almost uncorroded either with Method 1 or Method 2.

In order to investigate the effect of biofilm on the bio-degradation process, some experiments have been done to simulate the biodeterioration process of sewer pipe [START_REF] De Muynck | Effectiveness of admixtures, surface treatments and antimicrobial compounds against biogenic sulfuric acid corrosion of concrete[END_REF]Gutiérrez-Padilla et al., 2010;[START_REF] Alexander | Performance of sewer pipe concrete mixtures with portland and calcium aluminate cements subject to mineral and biogenic acid attack[END_REF].

In the experiments conducted by [START_REF] De Muynck | Effectiveness of admixtures, surface treatments and antimicrobial compounds against biogenic sulfuric acid corrosion of concrete[END_REF], CEM I cement cylinders were subjected to 8 cycles of accelerated test. Each cycle consisted in 4 steps: (1) incubation in H 2 S (200 ppm) for 2 days;

(2) submersion in 1.5 L of mixed cultures of SOB obtained from a sewer pipe biofilm (medium composition: 10 g/L element sulfur, 3 g/L KH 2 PO 4 , 0.1 g/L NH 4 Cl, 0.1 g/L MgCl•H 2 O) for 10 days;

(3) submersion in distilled water for 2 days; (4) drying at room temperature for 1 day. In the second and third steps, the SOB suspension and water were rotated. The degradation products were removed at the third step.

The last step could eliminate SOB remaining on the concrete surface. During the second step, the pH in the SOB suspensions was measured frequently. In the experiment, as a result of the conversion of element sulfur to sulfuric acid by cultures of SOB, a decrease of pH was observed 3-4 days after submersion. The pH continued to decrease to around 1 after 10 days of testing.

In the works of (Gutiérrez-Padilla et al., 2010), the biodeterioration of low and high quality concrete under conditions simulating sewer pipes were compared with laboratory experiments. With the same conditions of H 2 S and bacteria, the concrete with low W/C (0.33) showed higher resistance to biogenic sulfuric acid attack than the concrete with high W/C (0.42). While in another research work [START_REF] Ehrich | Investigation into the corrosion of cement-bound construction materials by biogenic sulfuric acid[END_REF], the weight loss of mortar samples with different W/C ratio of 0.35, 0.40, 0.45, 0.5, 0.55 were measured respectively. The results of that experiment stated that the lowest weight losses were determined on samples with a W/C ratio of 0.5 and the resistance of mortars was reduced for both high and low W/C ratios. This can be explained as the concrete with a high W/C has larger and more pores. The pores play the role of a capacity to absorb expansion caused by the production of gypsum [START_REF] Kawai | Concrete deterioration caused by sulfuric acid attack[END_REF]. The influence of W/C of concrete is discussed in Chapter 6 of this thesis as well.

It is necessary to improve the understanding of the specific influence of environmental variables, such as H 2 S level, temperature, and relative humidity. [START_REF] Joseph | Surface neutralization and h2s oxidation at early stages of sewer corrosion: Influence of temperature, relative humidity and h2s concentration[END_REF] examined changes in the surface chemistry of concrete during the early stages of corrosion by exposing concrete coupons to thirty-six independent conditions in well-controlled laboratory chambers that simulated conditions typically found in various sewer environments across Australia. The results indicated that the role of CO 2 on initial surface pH reduction is insignificant as compared to the influence of H 2 S. Within the first 12 months, a decrease in surface pH by 4.8 units was observed for coupons exposed to 30 • C and 50 ppm H 2 S while significantly lower pH reductions of 3.5 and 1.8 units were detected for coupons exposed to 25 • C and 18 • C respectively, and 50 ppm H 2 S. However, [START_REF] Ismail | Effect of carbonation on microbial corrosion of concrete[END_REF] claimed that under quite high CO 2 gas concentration, bio-corrosion rate would decrease due to formation of much calcite which could reduce the porosity of pipe surface. The role of carbonation layer is investigated in Chapter 6.

Furthermore, some works have been done to improve the biodeterioration resistance of sewer pipes. A set of in-situ experiments to determine the biochemical parameters influencing the behaviors of ordinary Portland cement (OPC) and calcium aluminate cement (CAC) materials was conducted by [START_REF] Herisson | Toward an accelerated biodeterioration test to understand the behavior of Portland and calcium aluminate cementitious materials in sewer networks[END_REF]. It has been found that CAC materials perform better than those with OPC, and the biomass that develops on the surface of the former is different and less abundant. The good behavior of CAC materials could be due to their high aluminum content, which could inhibit the growth of microorganisms.

( [START_REF] De Muynck | Effectiveness of admixtures, surface treatments and antimicrobial compounds against biogenic sulfuric acid corrosion of concrete[END_REF] investigated the effectiveness of different admixtures, and surface treatments towards biogenic sulfuric acid corrosion of concrete. The best protective method is epoxy coating, while the strongest degradation was observed for the cementitious coating. The addition of hydrous silicate failed to improve the resistance performance.

In addition to experiments, some researches have been done on modeling. Existing models mainly focus on the biochemical process or the chemical process separately.

For the chemical corrosion process, which directly results in the degradation of materials, [START_REF] Jahani | Investigations of sulfuric acid corrosion of concrete. ii: electrochemical and visual observations[END_REF] proposed a diffusion-reaction based model with a moving boundary to predict the corrosion rate. [START_REF] Chalupeckỳ | Macroscopic corrosion front computations of sulfate attack in sewer pipes based on a micro-macro reaction-diffusion model[END_REF] consider a two-scale reaction diffusion system containing one microscopic scale (pore scale) and one macroscopic scale (sewer pipe scale). In both of them, only dissolution of portlandite or calcite was considered. As another important solid component in cementitious materials, decalcification of C-S-H gel was absent. In addition, only the front of acid neutralization and gypsum formation were captured by the modeling.

In such a biogenic degradation process, the concentration of sulfuric acid is not constant or given, but imposed by the biochemical reactions in the SOB community. Thus the modeling of biochemistry reactions taking place in the biofilm is needed as well.

To simulate the biochemical process, models such as "Wastewater Aerobic/anaerobic Transformations in Sewer (WATS)" model [START_REF] Jensen | Modeling of hydrogen sulfide oxidation in concrete corrosion products from sewer pipes[END_REF] and sulfide oxidation modelling [START_REF] Nielsen | Kinetics and stoichiometry of sulfide oxidation by sewer biofilms[END_REF] were developed. Based on experiment data, kinetics and stoichiometry of sulfur oxidation by biofilm were investigated, while rate equations and stoichiometry constants were proposed. These simulations indicated that oxidation of accumulated slowly degradable element sulfur could cause a steady production of sulfuric acid, even when hydrogen sulfide is not present. However, the relationship between pH and H 2 SO 4 production in biofilm is not found in previous research results.

Few modeling focus on the coupling of biochemical process and chemical corrosion process. With HYTEC, (De Windt and Devillers, 2010) simulate a bioleaching test applied to ordinary Portland cement pastes during 15 months. By coupling of (bio)chemical alteration, porosity and diffusion evolutions, the long term performance of cementitious materials subjected to biodeterioration was assessed. Various species were considered in this work, and yet their diffusion coefficient were uniform. Moreover, a continuous approach of C-S-H dissolution is absent in this modeling and the effect of swelling of gypsum on the microstructure of concrete was not considered either.

Research objectives

Apparently, there is a lot of financial implications to change or reinforce the networks of pipes. Thus, it is necessary to develop a tool to address questions such as the service life of current pipes, or the durability of the new pipes made of other materials or protected by new methods. Since the present modeling are still imperfect, the objectives of this PhD study are to improve the understanding of the biogenic acid attack of concrete, and propose a method to predict the performance of concrete servicing in such extreme conditions.

The biochemistry process in biofilm and the chemical process in cement paste are investigated and coupled with each other. A reactive transport model to predict the behaviour of cement paste in contact with H 2 SO 4 solution or sulfur-oxidizing bacteria community is expected to be proposed. Therefore, the main parts of biogenical acid attack should be included in this model as follows:

1. Production variation of H 2 SO 4 with pH in biofilm.

2. Dissolution and precipitation reactions of portlandite (CH) and gypsum (C SH2).

3. Continuous variation (decrease) of the Ca/Si ratio during the dissolution reaction of C-S-H.

4. Diffusion of different species and thermodynamical equilibrium.

5. Changes of porosity and damage of materials induced by the production of gypsum.

6. Coupling of H 2 SO 4 production, transport equations and chemical reactions.

7. Effect of pore clogging resulting from carbonation on the biodeterioration rate during early time.

The model is implemented within a finite-volume code Bil (http://perso.lcpc.fr/dangla.patrick/bil).

Coupled with the H 2 SO 4 degradation modeling of concrete, the change of pH in biofilm and the composition of cement paste are expected to be calculated during the biodeterioration process. Furthermore, the degradation rate of concrete submitted to biogenic acid attack will be predicted. Simulations results of a set of chemical exposure tests (external sulfuric acid solution attack, conducted by [START_REF] Kawai | Concrete deterioration caused by sulfuric acid attack[END_REF][START_REF] Vincke | A new test procedure for biogenic sulfuric acid corrosion of concrete[END_REF]) and accelerated micro-biological tests conducted by [START_REF] De Muynck | Effectiveness of admixtures, surface treatments and antimicrobial compounds against biogenic sulfuric acid corrosion of concrete[END_REF] are presented and compared with experimental results. Due to the lack of bacteria activity data (such as the relationship between H 2 SO 4 production rate of various bacteria and pH in biofilm), this study only simulates the biological process of biofilm by fitting on the measured experimental data.

Outline of thesis

Since this thesis focuses on the biodeterioration of cementitious materials induced by an existing SOB community in an environment containing H 2 S gas, the incubation process of bacterial and the production process of H 2 S in wastewater is not addressed. products. An innovative approach for the C-S-H is applied to explain the continuous decalcification and facilitate the numerical modeling.

Resulting from the precipitation-dissolution of the various minerals, the porosity of cementitious materials will change. Thus the diffusion of species would be influenced as well. Regarding the swelling of gypsum, Chapter 4 introduces a simplified damage model of the microstructure of cement paste. Based on the balance of volume, the microstructure evolution is accounted for by adjusting the porosity using different methods during different damage stages.

Chapter 5 introduces the reactive transport modeling and numerical procedures. To both of the biofilm and cement paste, the field equations are presented. Different diffusion coefficients are employed for various ions, yet only transport of aqueous species are considered. Following the introduction of principle of the finite volume method, coupling of the bio-chemistry process in biofilm and chemistry process in cement paste is illustrated.

In Chapter 6, several calculations are carried out to simulate experimental works from [START_REF] Kawai | Concrete deterioration caused by sulfuric acid attack[END_REF][START_REF] Vincke | A new test procedure for biogenic sulfuric acid corrosion of concrete[END_REF] where cement samples are directly immersed in H 2 SO 4 solution with constant or cyclical variable concentrations. Simulation results of different conditions, e.g., statical solution and circulated solution, are compared with experiment observations. The sensitivity of H 2 SO 4 concentration, initial porosity, and damage criterion are analyzed. The effect of initial porosity and carbonation layer in early time are investigated as well.

After the modeling of chemical deterioration has been verified, in Chapter 7 the experiment conducted by [START_REF] De Muynck | Effectiveness of admixtures, surface treatments and antimicrobial compounds against biogenic sulfuric acid corrosion of concrete[END_REF] is simulated to predict the bio-degradation process. The simulation results of pH evolution in biofilm and the change of cement paste properties are compared with experimental results.

Effect of environmental factors (such as H 2 S content) is discussed and a long term prediction is presented.

Finally, the conclusion of this thesis and some future improvements are presented in Chapter 8.

Chapter 2

The production of H 2 SO 4 by micro-organisms in biofilm

Introduction

For biogenic sulfuric acid attack, H 2 SO 4 production is a fundamental process, which provides the strong corrosion medium causing the deterioration of concrete. Thus, the first step to simulate the bio-deterioration of sewer pipes is the calculation of the production of H 2 SO 4 in biofilms. As the major issue associated with the corrosion of concrete, the relationship between H 2 S gas level and the corrosion rate of the sewer pipes is still a major research question [START_REF] Apgar | Minimization of odors and corrosion in collection systems: Phase i[END_REF]. As introduced in Chapter 1, H 2 S absorption by biofilms and concrete surface provides the sulfur source of H 2 SO 4 production, whose rate is controlled by the oxidation of H 2 S in biofilms (see Fig. 2.1). In this paper, the generation and emission of H 2 S from wastewater is not considered and the level of H 2 S gas in sewer pipes is constant.

For sulfur-oxidizing bacteria (SOB), the initial pH of ordinary Portland cement concrete is too high to grow. [START_REF] Mori | Interactions of nutrients, moisture and ph on microbial corrosion of concrete sewer pipes[END_REF][START_REF] Rigdon | Corrosion of concrete by autotrophes[END_REF] proposed that the microbial activity on the concrete surface is initiated at pH around 9 and thereafter, a succession of microbial communities develops, which can utilize sulfide and/or its oxidized forms, such as element sulfur [START_REF] Joseph | Surface neutralization and h2s oxidation at early stages of sewer corrosion: Influence of temperature, relative humidity and h2s concentration[END_REF]. The abiotic pH reduction of concrete surface results to an initial lag period before the start of active corrosion phase [START_REF] Islander | Microbial ecology of crown corrosion in sewers[END_REF].

Therefore, the primary stage is the reduction of pH at the concrete surface, where pH decreases from about 12.5 to 9 by chemical acid reactions of CO 2 and H 2 S with concrete. The experimental researches of [START_REF] Joseph | Surface neutralization and h2s oxidation at early stages of sewer corrosion: Influence of temperature, relative humidity and h2s concentration[END_REF] suggested that H 2 S gas is the major factor for the surface pH reduction in sewers during early stages of exposure rather than carbonation.

Figure 2.1: Schematic outline of the H 2 S production process. [START_REF] Jensen | Hydrogen sulfide initiated corrosion in concrete sewers-a conceptual approach for prediction[END_REF] According to literatures [START_REF] Harrison | The acidophilic thiobacilli and other acidophilic bacteria that share their habitat[END_REF][START_REF] Islander | Microbial ecology of crown corrosion in sewers[END_REF][START_REF] Herisson | Toward an accelerated biodeterioration test to understand the behavior of Portland and calcium aluminate cementitious materials in sewer networks[END_REF], different types of SOB are involved in H 2 S oxidation process which is influenced by various factors, such as the amount of sulfur source, pH and temperature. For such a complex process, there is a lack of fundamental understanding of the quantitative relationship among these factors. However, the kinetics and stoichiometry of sulfide oxidation by biofilms can be described by Wastewater aerobic/anaerobic Transformations in Sewer (WATS) model [START_REF] Jensen | Modeling of hydrogen sulfide oxidation in concrete corrosion products from sewer pipes[END_REF][START_REF] Nielsen | Kinetics and stoichiometry of sulfide oxidation by sewer biofilms[END_REF] regardless the influence of pH. Yet, to our knowledge, no attempt has been made to model both the neutralization of concrete surface and H 2 SO 4 production in biofilms.

This chapter studies first with the absorption of H 2 S gas by concrete surface. Then, the chemical reactions of H 2 S and concrete with constant temperature and humidity is described. For the specific biofilms, we model the production of H 2 SO 4 by fitting the measurements of the pH evolution in biofilms.

For a more general condition with varying H 2 S content, we follow the WATS model to calculate the production rate based on H 2 S content. , 1974) suggested that all released H 2 S from wastewater is immediately absorbed by concrete surface, resulting in zero H 2 S in the gas phase. However, experimental results of [START_REF] Vollertsen | Corrosion of concrete sewers-the kinetics of hydrogen sulfide oxidation[END_REF][START_REF] Nielsen | Influence of pipe material and surfaces on sulfide related odor and corrosion in sewers[END_REF] revealed that transferring of H 2 S from the gas phase to the concrete surface is not an instantaneous process and absorption kinetics should be taken into account.

Neutralization of pipe surface during early stage

By experiments with pilot-scale sewer reactors as in Fig. 2.2, [START_REF] Vollertsen | Corrosion of concrete sewers-the kinetics of hydrogen sulfide oxidation[END_REF] found that the absorption rate (F H2S ) could be described as a power function in the gas phase H 2 S concentration (p H2S ).

Figure 2.2: Schematics of the pilot scale reactors. [START_REF] Vollertsen | Corrosion of concrete sewers-the kinetics of hydrogen sulfide oxidation[END_REF] Furthermore, [START_REF] Nielsen | Effect of sewer headspace air-flow on hydrogen sulfide removal by corroding concrete surfaces[END_REF] presented an empirical relationship between the Reynolds (Re) number of the gas flow and the absorption rate. For the sake of simplicity, the effect of gas flow regime is ignored in this paper. The n th order kinetics of H 2 S absorption of concrete surface and biofilms is described as follows:

F H2S = k abs p n H2S (2.1)
where, p H2S is the content of H 2 S gas (ppm). k abs is the surface specific H 2 S absorption rate constant, which is reported to vary between 6.25×10 -8 and 3.12×10 -7 mol S m -2 s -1 (ppm H 2 S) -n depending on temperature [START_REF] Vollertsen | Corrosion of concrete sewers-the kinetics of hydrogen sulfide oxidation[END_REF]. The reaction order n is correlated with k abs , with n approximately 0.5 for low k abs and 0.8 for high k abs . Since temperature is constant at 25 • C in this paper, k abs is determined as 2×10 -7 mol S m -2 s -1 (ppm H 2 S) -n and n = 0.75. Then the absorption rate of H 2 S versus content of H 2 S in gas phase is plotted in Fig. 2.3.

Chemical reaction between aqueous H 2 S and concrete

Although the acidification of aqueous hydrogen sulfide is much weaker than that of H 2 SO 4 , it dissociates in the same manner as sulfuric acid and releases H + as follows [START_REF] Tsonopoulos | Ionization constants of water pollutants[END_REF][START_REF] Su | Amplified potentiometric determination of pk(00), pk(0), pk(l), and pk(2) of hydrogen sulfides with ag(2)s ise e[END_REF]: where the equilibrium constant is at 298K which keeps constant. Thus calcium hydroxide could be dissolved and calcium sulfide forms when H 2 S is absorbed by concrete surface. These global reactions can be understood by combining three basic dissociation reactions Eq.(2.4)-Eq.(2.6), which involve the minerals and the aqueous species in pore solution.

H 2 S ⇋ HS -+ H + K H2S = 8.9 × 10 -8 (2.2) HS -⇋ S 2-+ H + K HS -= 1.2 × 10 -13 (2.3)
CH ⇋ Ca 2+ + 2OH - (2.4) C x S y H z ⇋ xCa 2+ + 2xOH -+ ySiO 0 2 + (z -x)H 2 O (2.5) CaS ⇋ Ca 2+ + S 2- (2.6)
Furthermore, CaS precipitation is not stable in acidic solution. When there is enough H 2 S, calcium hydrogen sulfide, which is soluble, would be produced as follows [START_REF] Idriss | Effect of hydrogen sulphide emissions on cement mortar specimens[END_REF]:

CaS + H 2 S ⇋ Ca(HS) 2 (2.7)
The equilibrium condition of these dissociation reactions are depended on the concentration of H 2 S.

The dissolution of CH and the decalcification of C-S-H are the same as those with H 2 SO 4 attack, which are detailed in Chapter 3. Taking the equilibrium between CH and CaS for instance, we have

Q CaS K CaS = Q CH K CH ρ H2S ρ CH H2S (2.8)
where, K and Q are the equilibrium constant and ion activity product of CH and CaS respectively. K CH = 6.5 ×10 -6 and K CaS = 7.9 ×10 -7 . ρ CH H2S is the value of ρ H2S defined by the coexistence of CH , CaS and aqueous phase. It can be calculated by a given function of the solubility constants of other species as Eq.(2.9).

ρ CH H2S = K 2 H2O K CaS K H2S K HS -K CH (2.9)
With the help of the constants mentioned above, ρ CH H2S ≈ 8.90 × 10 -10 mol/L. Hence, CaS would not precipitate when ρ H2S < ρ CH H2S , while dissolution of CH takes place when ρ H2S > ρ CH H2S .

In the model, the CH dissolution process is governed by a simple kinetic law which is assumed to be governed only by the difference of chemical potentials of H 2 S between the current state (µ H2S(current) ) and the equilibrium state (µ H2S(eq) ). Details of the kinetic and decalcification of C-S-H is demonstrated in Chapter 3, as well as an example of H 2 S attack.

pH reduction of concrete surface

With the absorption of H 2 S in gas phase and the chemical reaction of H 2 S with concrete mentioned above, the pH reduction of concrete surface can be calculated. [START_REF] Roberts | Quantifying microbially induced deterioration of concrete: initial studies[END_REF] experimentally reproduced a reduction of pH in pore water by exposing fresh concrete samples to hydrogen sulfide gas concentrations of 50, 100, and 250 ppm. The pH reduction is given by 4.4 × 10 -5 pH unit/day/ppm of H 2 S • p H2S + 0.021 pH unit/day (2.10) [START_REF] Okabe | Succession of sulfur-oxidizing bacteria in the microbial community on corroding concrete in sewer systems[END_REF] measured the pH reduction rate for mortar coupons placed in sewers with H 2 S of 30 ppm ± 20 ppm for a period of 56 days. The result was 0.06 pH unit/day, which is approximately 3 times of the result of Eq.(2.10). However, the experiment was in-situ, where microbial processes could be involved in.

Thus, we simulate a set of experiments conducted and clearly described by [START_REF] Joseph | Surface neutralization and h2s oxidation at early stages of sewer corrosion: Influence of temperature, relative humidity and h2s concentration[END_REF]. In the experiments, concrete coupons were exposed to different H 2 S gas level (4.5, 7.5, 15.8, 26.5, 48.9 ppm) in well-controlled laboratory chambers, where the temperature is 25 • C, for 1 year. The surface pH of the samples before exposure was 10.6 ± 0.3, which is near the pH of a carbonated sample, indicating that the samples were carbonated during the manufacturing and/or transportion. Therefore, in our simulation calcite is considered as the solid phase of samples instead of portlandite. After exposure of 1 year, the simulation results and experimental results of the decrease of pH in surface are compared in Fig. 2.4.

The production of H 2 SO 4 by sulfur-oxidizing bacteria

When the environment of pipe surface is suitable for the microbial succession, biofilms starts to form.

With the supply of hydrogen sulfide, these microorganisms can produce biogenic sulfuric acid which attacks the concrete. Some experimental results [START_REF] Herisson | Toward an accelerated biodeterioration test to understand the behavior of Portland and calcium aluminate cementitious materials in sewer networks[END_REF] indicated that the oxidation rate of H 2 S is controlled by the activity and the amount of SOB, which is determined by the pH in SOB media, H 2 S content in sewer, temperature and the concentration of Al 3+ in the biofilm. However, the quantitative effect of these factors on H 2 SO 4 production is not clear for now.

Different types of SOB are responsible to different deterioration degree of concrete as listed in Table 1.2. They are active in different ranges of pH (see Fig. 2.5) and produce H 2 SO 4 with different rates. Two general categories of SOB can be differentiated based on their optimal pH for growth [START_REF] Hudon | Biodeterioration of concrete sewer pipes: State of the art and research needs[END_REF]:

Neutrophilic Sulfur-Oxidizing Bacteria (NSOB), which develops at pH near neutral and are found at the beginning of biodeterioration; Acidophilic Sulfur-Oxidizing Bacteria (ASOB), which prefer acidic media.

Normally, the H 2 SO 4 production rate of NSOB is much slower than that of ASOB. Thus, NSOB only can cause slight deterioration.

Most types of SOB are not able to oxidize sulfide directly. Fig. 2.6 summarizes the possible oxidation pathways in biofilms, which can occur concurrently [START_REF] Hudon | Biodeterioration of concrete sewer pipes: State of the art and research needs[END_REF][START_REF] Islander | Microbial ecology of crown corrosion in sewers[END_REF].

Figure 2.5: Succession of SOB in different pH range [START_REF] Herisson | Biodétérioration des matériaux cimentaires dans les ouvrages d'assainissement: étude comparative du ciment d'aluminate de calcium et du ciment Portland[END_REF] Figure 2.6: Sulfur oxidation states typical in sewer. (Biologic pathway is shown with full lines; chemical, with dashed lines.) [START_REF] Islander | Microbial ecology of crown corrosion in sewers[END_REF] For heavily deteriorated sewer pipes, Acidithiobacillus thiooxidans bacteria is normally detected in biofilms [START_REF] Okabe | Succession of sulfur-oxidizing bacteria in the microbial community on corroding concrete in sewer systems[END_REF][START_REF] Cho | A newly isolated fungus participates in the corrosion of concrete sewer pipes[END_REF][START_REF] Nica | Isolation and characterization of microorganisms involved in the biodeterioration of concrete in sewers[END_REF]. Most of the present laboratory researches are carried out based on the presence of Acidithiobacillus thiooxidans, whose oxidation process is modeled by WATS model [START_REF] Hvitved-Jacobsen | Wastewater quality changes during transport in sewers-an integrated aerobic and anaerobic model concept for carbon and sulfur microbial transformations[END_REF][START_REF] Nielsen | Kinetics and stoichiometry of sulfide oxidation by sewer biofilms[END_REF][START_REF] Jensen | Modeling of hydrogen sulfide oxidation in concrete corrosion products from sewer pipes[END_REF]. [START_REF] Jensen | Modeling of hydrogen sulfide oxidation in concrete corrosion products from sewer pipes[END_REF] showed that in the corrosion products of concrete, H 2 S is rapidly oxidized to a mix of elemental sulfur and H 2 SO 4 . They observed two fractions of elemental sulfur: a readily degradable fraction which was oxidized almost as fast as the oxidation of H 2 S, and a slowly degradable fraction which was oxidized much slower.

If H 2 S is continuously present during the early time, slowly degradable elemental sulfur would accumulate and cause a delayed formation of sulfuric acid even if H 2 S is no longer absorbed later.

There is still a lack of the quantitative understanding of the H 2 SO 4 production, the succession of different types of SOB and the influence of environmental factors. Among them, Al 3+ ion has bacteriostatic effect [START_REF] Herisson | Biodétérioration des matériaux cimentaires dans les ouvrages d'assainissement: étude comparative du ciment d'aluminate de calcium et du ciment Portland[END_REF]. Since we deal with the Portland cement based sewer pipes in this paper, Al 3+ is neglected. For sewer pipes degraded heavily, the temperature is normally 25 • C to 30 • C [START_REF] Yongsiri | Effect of temperature on air-water transfer of hydrogen sulfide[END_REF]. Therefore, in our modeling the temperature and humidity is constant. In this paper, dealing with different experimental conditions, H 2 SO 4 production controlled by pH and H 2 S content are considered respectively.

Modeling of H 2 SO 4 production with given pH evolution in biofilm

In some experimental researches, the pH evolution in biofilms or SOB suspension with time were measured [START_REF] Vincke | A new test procedure for biogenic sulfuric acid corrosion of concrete[END_REF][START_REF] De Muynck | Effectiveness of admixtures, surface treatments and antimicrobial compounds against biogenic sulfuric acid corrosion of concrete[END_REF]. Thus, it is possible to model the H 2 SO 4 production process by fitting the pH evolution.

In the sewer pipes degraded heavily, the content of H 2 S gas in sewer could reach 400 ppm [START_REF] Mori | Interactions of nutrients, moisture and ph on microbial corrosion of concrete sewer pipes[END_REF]. In some laboratory researches focusing on the deterioration process, element sulfur content in SOB culture was quite high and the concrete samples were incubated by H 2 S gas or biological sulfur solution [START_REF] Vincke | A new test procedure for biogenic sulfuric acid corrosion of concrete[END_REF][START_REF] De Muynck | Effectiveness of admixtures, surface treatments and antimicrobial compounds against biogenic sulfuric acid corrosion of concrete[END_REF]. For such conditions, we assume that element S source is always enough for SOB. Thus, in this simplified modeling the production of H 2 SO 4 is controlled by pH which itself is determined by the amount of H 2 SO 4 . According to the experimental observations [START_REF] Ehrich | Biogenic and chemical sulfuric acid corrosion of mortars[END_REF], the activity of SOB increases until pH is reduced to a critical value, depending on the type of bacteria species. When pH is lower than the critical value, H 2 SO 4 production rate drops rapidly. In this paper, the production process of sulfuric acid in SOB media is divided into two step: NSOB dominating step: When pH in SOB media is higher than the value of pH Under in-situ conditions, H 2 S is not always at high level and it is easier to monitor H 2 S level in gas phase than pH in SOB media. According to experimental researches [START_REF] Jensen | Hydrogen sulfide initiated corrosion in concrete sewers-a conceptual approach for prediction[END_REF][START_REF] Vollertsen | Corrosion of concrete sewers-the kinetics of hydrogen sulfide oxidation[END_REF][START_REF] Vollertsen | Modeling the corrosion of concrete sewers[END_REF] where H 2 S content was measured, content of H 2 S is an important factor which can control the corrosion rate of sewer pipes.

With given H 2 S content in gas phase, WATS model is employed to calculate the H 2 SO 4 production by considering the kinetics of H 2 S oxidation. WATS concept was originally presented in 1998 and continuously developed [START_REF] Hvitved-Jacobsen | Wastewater quality changes during transport in sewers-an integrated aerobic and anaerobic model concept for carbon and sulfur microbial transformations[END_REF][START_REF] Vollertsen | Corrosion of concrete sewers-the kinetics of hydrogen sulfide oxidation[END_REF][START_REF] Jensen | Hydrogen sulfide induced concrete corrosion of sewer networks[END_REF]. Based on experimental observation, WATS model suggests that hydrogen sulfide is initially oxidized to readily biodegradable elemental sulfur (Sf). This elemental sulfur is partly aged to slowly biodegradable elemental sulfur (Ss) and partly oxidized to sulfuric acid. The slowly biodegradable elemental sulfur is also oxidized to sulfuric acid, only at a lower rate (see Fig. 2.8).

Figure 2.8: The pathway used to describe the hydrogen sulfide and oxygen consumption pattern observed in the experiments on biotic hydrogen sulfide oxidation [START_REF] Jensen | Hydrogen sulfide induced concrete corrosion of sewer networks[END_REF].

However, biofilms are considered as containing a general mixture of SOB of A. thiooxidans, i.e., the difference between NSOB and ASOB is neglected in WATS model. Thus, the effect of pH on H 2 SO 4 production is neglected. The oxidation rate is governed by the concentration of biomass, the amount of aqueous H 2 S and O 2 . In WATS model, the oxidation rate of H 2 S to Sf (R H2S,WATS ), Sf to H 2 SO 4 (R Sf,WATS ), Ss to H 2 SO 4 (R Ss,WATS ) and the ageing rate of Sf to Ss (R Sf-Ss,WATS ) are described by Eq.(2.13) -Eq.(2.16).

R H2S,WATS = k oxi-H2S X SOB ρ H2S ρ H2S + H H2S ρ O2 ρ O2 + H O2,H2S
(2.13)

R Sf,WATS = k oxi-Sf X SOB ρ Sf ρ Sf + H Sf ρ O2 ρ O2 + H O2,Sf (2.14) R Ss,WATS = k oxi-Sf X SOB ρ c O2 (2.15) R Sf-Ss,WATS = k Sf-Ss ρ Sf (2.16)
where all parameters are defined in Table 2.1. Obviously, the production rate of H 2 SO 4 equals to the sum of oxidation rate of Sf and Ss.

In our modeling, the oxidation pathways are simplified as H 2 S -element S -H 2 SO 4 , i.e., the slowly biodegradable elemental sulfur is neglected. Oxygen is always sufficient for the sake of simplification. Thus, To simulate the corrosion of sewer pipes in the experiments of [START_REF] Jensen | Hydrogen sulfide initiated corrosion in concrete sewers-a conceptual approach for prediction[END_REF][START_REF] Vollertsen | Modeling the corrosion of concrete sewers[END_REF] , which is detailed in Chapter 7, the value of parameters in Eq.(2.17) and Eq.(2.18) are listed in

Conclusions

This chapter starts with the introduction of the mechanisms of H 2 SO 4 production by sulfide-oxidizing bacteria (SOB). Before SOB starts to grow on the surface of sewer pipes, the concrete is neutralized by H 2 S in gas phase. The absorption of H 2 S gas in pore solution of concrete or SOB media is considered.

The chemical reaction of H 2 S and concrete is introduced into our modeling as well. Thus, the surface neutralization process of biodeterioration of sewer pipes can be simulated. The lag time of the start of active corrosion can be predicted.

By simulating the experiment conducted by [START_REF] Joseph | Surface neutralization and h2s oxidation at early stages of sewer corrosion: Influence of temperature, relative humidity and h2s concentration[END_REF] When the values of parameters in both methods are determined, the production rate of H 2 SO 4 during the experiments of [START_REF] De Muynck | Effectiveness of admixtures, surface treatments and antimicrobial compounds against biogenic sulfuric acid corrosion of concrete[END_REF] and [START_REF] Vollertsen | Modeling the corrosion of concrete sewers[END_REF] are plotted. When biogenic sulfuric acid is produced, it starts to corrode concrete. The chemical reaction of H 2 SO 4 attack of concrete will be introduced in the next chapter.

Chapter 3

Degradation of cementitious materials subjected to sulfuric acid attack

Introduction

Once H 2 SO 4 is produced by biofilms, cement-based materials are directly in contact with H 2 SO 4 which diffuses into the pores of concrete. As a strong acid, a series of homogeneous chemical reactions between H 2 SO 4 and concrete could result in the reduction of the concrete alkalinity and the dissolution of the calcium hydroxide. In pore solution of concrete, the ionization of H 2 SO 4 is influenced by other ions. Unlike pure solution, it is difficult to determine the concentration of H 2 SO 4 by pH in pore solution. Thus, the concentration of unionized molecules of H 2 SO 4 (noted as ρ 0 H2SO4 ) is employed to indicate the amount of H 2 SO 4 in pore solution of equilibrium state in this study. When ρ 0 H2SO4 is high enough (≥ 3.9 × 10 -32 mol/L at 298 K, i.e., pH≤ 12.4), portlandite (Ca(OH) 2 , noted as CH in this study) will start to dissolve. As the concentration of H 2 SO 4 getting higher, decalcification process of calcium silicate hydrate (xCaO•ySiO 2 • zH 2 O, noted as C-S-H) will take place.

The sulfuric acid corrosion product is a layer of white precipitate that accumulates on the concrete surface. This corroded layer consists of a combination of sand particles, unreacted cement particles and calcium sulfate mineral. By electron microscopy, [START_REF] Bock | Applied electron microscopy on the biogenic destruction of concrete and blocks. use of the transmission electron microscope for identification of mineral acid producing bacteria[END_REF] identified the main part of the corrosion products as gypsum (CaSO 4 •2H 2 O, noted as C SH2 in this paper) with no structural strength. Some publications [START_REF] Mori | Interactions of nutrients, moisture and ph on microbial corrosion of concrete sewer pipes[END_REF][START_REF] Redner | Evaluating coatings for concrete in wastewater facilities: update[END_REF] claimed that the ample supply of sulfate and the intrusion of the acid may result in the formation of ettringite (3CaO•Al 2 O 3 •3CaSO 4 •32H 2 O) which is expansive. However, in other experimental observations [START_REF] Ehrich | Biogenic and chemical sulfuric acid corrosion of mortars[END_REF][START_REF] Vincke | Recent developments in research on biogenic sulfuric acid attack of concrete[END_REF] ettringite was not present. According to the investigations conducted by [START_REF] Gabrisova | Stability of calcium sulphoaluminate hydrates in water solutions with various ph values[END_REF][START_REF] Allahverdi | Acidic corrosion of hydrated cement based materials[END_REF], ettringite starts to form at pH ranging from 12.5 to 12. When pH decreases below 10.7, ettringite starts to decompose into gypsum. Since normally pH in biofilms is less than 3, ettringite exists temporarily and occasionally appears in the corrosion products during the biodeterioration of sewer pipes. Therefore, in this paper only the precipitation of gypsum is considered during the sulfuric acid attack process.

Not much modeling researches about the sulfuric acid attack of cementitious materials were reported for now. Based on a moving-boundary system, [START_REF] Böhm | On a moving-boundary system modeling corrosion in sewer pipes[END_REF] firstly proposed a modeling of corrosion of sewer pipes. In this modelling, the chemical reactions are simplified to the dissolution of calcite (CaCO 3 ).

Such simplification was still employed in recent research [START_REF] Fatima | Sulfate attack in sewer pipes: derivation of a concrete corrosion model via two-scale convergence[END_REF]. Dissolution of portlandite and C-S-H gel were mentioned in the modeling research of [START_REF] Beddoe | Modelling acid attack on concrete: Part i. the essential mechanisms[END_REF]. However, the quantities of portlandite and C-S-H gel were given by the total calcium content rather than taken into account individually. In addition, the equilibrium between solid phases was not considered.

In the modeling of biogenic acid attack based on HYTEC, (De Windt and Devillers, 2010) studied the thermodynamic equilibrium of the chemical reactions. The dissolution of portlandite and C-S-H gel were described separately. However, a discrete approach of C-S-H dissolution was used by considering three end members with Ca/Si ratio of 1.8, 1.1 and 0.8.

In this chapter, the approach of chemical reactions between H 2 SO 4 and cement-based materials are detailed. Chemical reactions taking place in aqueous phase and solid phases are profiled in the first section.

The effects of the chemical activity are disregarded as a first approximation (the activity coefficient is assumed to be one throughout the paper). The dissolution of CH and C-S-H, and the precipitation are thereafter described in the next two sections. A general approach to describe the C-S-H dissolution is employed. The proposed approach provides a method to characterize the continuous decalcification of the C-S-H during H 2 SO 4 attack and facilitates the numerical modeling. With the help of chemical equilibrium between portlandite and calcium sulfide (CaS) introduced in Chapter 2, an example of H 2 S attack is conducted on representative elementary volume (REV). The evolution of pH and solid composition during H 2 SO 4 attack are illustrated via a REV example as well.

Chemical reactions in pore solution

For Portland cementitious materials, portlandite and calcium silicate hydrates are the solid components which react with sulfuric acid to form gypsum (C SH2). These global reactions can be understood by com-bining three basic dissociation reactions Eq.(3.1)-(3.3), which involve the minerals and the aqueous species in pore solution. The equilibrium condition of these dissociation reactions depends on the concentration of

H 2 SO 4 . CH ⇋ Ca 2+ + 2OH - (3.1) C SH2 ⇋ Ca 2+ + SO 2- 4 + 2H 2 O (3.2) C x S y H z ⇋ xCa 2+ + 2xOH -+ ySiO 0 2 + (z -x)H 2 O (3.3)
Therefore the ingress of H 2 SO 4 solution will cause a series of homogeneous chemical reactions gathered in the Table 3.1, where the equilibrium constant are given at 298K. Throughout this paper, two points should be noticed: 1) We use a superscript 0 in the chemical formula of any species (e.g. in SiO 0 2 ) as a convention to denote the dissolved form of this species; 2) We assume infinite dilution approximation for aqueous species. Thus the activity of each aqueous species is equal to the molar concentration expressed in mol/L, and that of each solid component equals to 1 (CH and C SH2) except the different poles of C-S-H.

Table 3.1: Chemical reactions taking place in the aqueous solution

Aqueous reactions

Equilibrium constant=

H 2 O ⇋ H + + OH - K H2O = 1.0 × 10 -14 H 2 SO 4 ⇋ HSO - 4 + H + K H2SO4 = 1.0 × 10 6 HSO - 4 ⇋ SO 2- 4 + H + K HSO - 4 = 1.0 × 10 -2 CaOH + ⇋ Ca 2+ + OH - K CaOH + = 1.66 × 10 1 Ca 2+ + SO 2- 4 ⇋ CaSO 0 4 K CaSO 0 4 = 1.4 × 10 3 Ca 2+ + HSO - 4 ⇋ CaHSO + 4 K CaHSO + 4 = 1.276 × 10 1 Ca 2+ + H 2 SiO 2- 4 ⇋ CaH 2 SiO 0 4 K CaH2SiO 0 4 = 3.89 × 10 4 Ca 2+ + H 3 SiO - 4 ⇋ CaH 3 SiO + 4 K CaH3SiO + 4 = 1.58 × 10 1 SiO 0 2 + 2H 2 O ⇋ H 4 SiO 0 4 K SiO 0 2 = 1.94 × 10 -3 H 4 SiO 0 4 ⇋ H 3 SiO - 4 + H + K H4SiO 0 4 = 1.55 × 10 -10 H 3 SiO - 4 ⇋ H 2 SiO 2- 4 + H + K H3SiO - 4 = 4.68 × 10 -14
* The equilibrium constants are reported by [START_REF] Galíndez | Assessment of the long-term stability of cementitious barriers of radioactive waste repositories by using digital-image-based microstructure generation and reactive transport modelling[END_REF][START_REF] Stronach | Modelling the impact of abundant geochemical components on phase stability and solubility of the cao-sio2-h2o system at 25°c: Na+, k+, so42-, cl-and co32[END_REF][START_REF] Westall | A computer program for the calculation of chemical equilibrium composition of aqueous systems[END_REF] In Table 3.1, each ion, i, takes a valence number, z i , which depends on the charge carried by the ion.

Since electroneutrality must be kept in the medium and the solid phase is not charged, the total charge of the aqueous solution is 0 as Eq.3.4.

i z i ρ i = 0 (3.4)
where, ρ i represents the concentration of ion i (mol/L). From Table 3.1, we can found that the concentration of H 2 SO 4 , Ca 2+ and H 4 SiO 0 4 are key variables. With known value of ρ H2SO4 , ρ Ca 2+ and ρ H4SiO 0 4 , ρ H + can be derived via Eq.3.4. Consequently, the concentration of the other species can be calculated from Table 3.1.

In addition, alkali K + /Na + are included in the model. For the electroneutrality, the same amount of anion (such as Cl -and OH -) should introduced as well. Since those species are not involved in any chemical reaction in this study, an unified form as KCl is considered.

Dissolution of portlandite

The stability of the thermodynamical equilibrium of reactions (3.1) and (3.2) can be characterized by a threshold of the ion activity products as follows:

Q CH = ρ Ca 2+ ρ 2 OH - ≤ K CH (3.5) Q C SH2 = ρ Ca 2+ ρ SO 2- 4 ρ 2 H2O ≤ K C SH2 (3.6)
where, K and Q are the equilibrium constant and ion activity products of CH and C SH2 respectively.

K CH = 6.5 × 10 -6 and K C SH2 = 2.5 × 10 -5 . According to Table 3.1, ρ SO 2- 4 is governed by three chemical equilibriums:

K H2O = ρ H + ρ OH - (3.7) K H2SO4 = ρ H + ρ HSO - 4 ρ H2SO4 (3.8) K HSO - 4 = ρ H + ρ SO 2- 4 ρ HSO - 4 (3.9)
Combining Eq.(3.7), Eq.(3.8) and Eq.(3.9), ρ SO 2- 4 can be derived as follows:

ρ SO 2- 4 = K H2SO4 K HSO - 4 K 2 H2O ρ H2SO4 ρ 2 OH - (3.10)
Substituting Eq.(3.10) into Eq.(3.6), the ion activity of gypsum can be expressed by the key variables as Eq.(3.11).

Q C SH2 = ρ Ca 2+ K H2SO4 K HSO - 4 K 2 H2O ρ H2SO4 ρ 2 OH - (3.11)
The relationship between the ion activity of portlandite and that of gypsum can be derived from Eq.(3.5) and Eq.(3.11) as follows:

Q C SH2 = Q CH K H2SO4 K HSO - 4 K 2 H2O ρ H2SO4 (3.12)
Thus, the equilibrium relationship between CH and C SH2 can be written as:

Q C SH2 K C SH2 = Q CH K CH ρ H2SO4 ρ CH H2SO4 (3.13)
where, ρ CH H2SO4 is the value of ρ H2SO4 defined by the coexistence of CH , C SH2 and aqueous phase. It can be calculated by a given function of the solubility constants of other species as Eq.(3.14).

ρ CH H2SO4 = K 2 H2O K C SH2 K H2SO4 K HSO - 4 K CH (3.14)
With the help of the constants listed in Table 3.1, ρ CH H2SO4 ≈ 3.90 × 10 -32 mol/L. Hence, C SH2 would not precipitate when ρ H2SO4 < ρ CH H2SO4 , while dissolution of CH takes place when ρ H2SO4 > ρ CH H2SO4 .

These stability domains are summarized in Fig. 3.1. For the fresh ordinary Portland cement paste, pH is approximately 12.4, which means ρ H2SO4 < ρ CH H2SO4 . Thus portlandite is stable and no gypsum exists. When H 2 SO 4 penetrates into the pores of cement paste and further exceeds ρ CH H2SO4 , portlandite starts to dissolve and C SH2 forms.

dn CH dt = - n CH τ CH ( µ H2SO4(current) -µ H2SO4(eq) RT ) = - n CH τ CH ln( ρ H2SO4 ρ CH H2SO4 ) (3.15)
The kinetics is simplified to be governed only by the difference of chemical potentials of H 2 SO 4 between the current state (µ H2SO4(current) ) and the equilibrium state (µ H2SO4(eq) ). Since the pH of the biofilms may be as low as 1, τ CH should be set in practice small enough to oblige the reaction to be close to equilibrium.

Combined with sulfate ions, the dissolved calcium ions will precipitate as C SH2 with almost at the same rate as that of the portlandite dissolution.

Decalcification of calcium silicate hydrates (C-S-H)

As a very important solid phase, C-S-H contributes to the strength of cement-based materials. The dissolution of C-S-H gel during H 2 SO 4 attack, is presented by reactions Eq.(3.3). Dissolution modeling of C-S-H is important to understand the degradation process of cement hydrates.

Compared to CH, C-S-H is a very complex hydration product whose composition is not specific. Thus the same approach as the one adopted for the CH dissolution is inappropriate for C-S-H. According to the researches about the structure and stoichiometry of C-S-H [START_REF] Greenberg | Investigation of Colloidal Hydrated Calcium Silicates: Pt. II. Solubility Relationships in the Calcium Oxide-silica-water System at 259[END_REF][START_REF] Chen | Solubility and structure of calcium silicate hydrate[END_REF][START_REF] Constantinides | The effect of two types of csh on the elasticity of cement-based materials: Results from nanoindentation and micromechanical modeling[END_REF], the ratio of calcium to silica (Ca/Si ratio), which is generally used to characterize the C-S-H, is about 1.7 for fresh hydrated Portland cement and tends to decrease during the dissolution process.

Literatures provide several methods to model the thermodynamic properties of C-S-H and its incongruent behaviour as well. Comparing with the empirical or semi-empirical models of the early method [START_REF] Reardon | An ion interaction model for the determination of chemical equilibria in cement/water systems[END_REF] and solid solution method [START_REF] Kulik | Aqueous solubility diagrams for cementitious waste stabilization systems: Ii, end-member stoichiometries of ideal calcium silicate hydrate solid solutions[END_REF], a more general characterization method is proposed to address the continuous change of the stoichiometric coefficients of C-S-H during decalcification. For a better understanding of the new approach, the discrete model and solid solution model are introduced in Appendix A.

Based on thermodynamics, the new approach encompasses the solid solution model. There is no explicit mention of end-members in this approach. The Gibbs energy of C-S-H gel is eventually assessed directly by experimental measurements as a function of the Ca/Si ratio rather than being assumed to fit a mixture of given members. Thus, this approach is expected to be more general than solid solution model. The Ca/Si ratio of C-S-H is taken as a variable in the model. Defining the unit mole content of C-S-H with a unit mole content of Si (i.e., y = 1), the dissociation reaction of an infinitesimal small amount of C-S-H can be written as:

C x S 1 H z ⇋ xCa 2+ + 2xOH -+ SiO 0 2 + (z -x)H 2 O (3.16)
where, x stands for the current Ca/Si ratio and z for the Water/Si ratio. Obviously, the solid content of Si (n Si ) is equal to the content of C-S-H (n C-S-H ). Thus, the molar Gibbs energy of C-S-H depends on x and z which is assumed as a function of x. We write the molar Gibbs energy of C-S-H as g(x). During an infinitesimal small variation of C-S-H content, dn Si , the reaction Gibbs free energy, G, must decrease spontaneously:

dG = -µ Ca 2+ d(nx) -µ OH -d(2nx) -µ SiO 0 2 dn -µ H2O d(n(z -x)) + d(ng(x)) ≤ 0 (3.17)
where, µ i is the chemical potential of each specie. At equilibrium condition, G is minimum. Therefore, for arbitrary increment dn and C-S-H mole content n, Eq.(3.17) can be changed to Eq.(3.18).

[-x(µ

Ca 2+ + 2µ OH -) -µ SiO 0 2 -(z -x)µ H2O + g(x)]dn +[-µ Ca 2+ -2µ OH --(z -1)µ H2O + ∂g ∂x ]ndx = 0 (3.18)
Since the chemical potential is independent on amount, both of the two terms of Eq.(3.18) are equal to 0. Thus,

A = x(µ Ca 2+ + 2µ OH -) + µ SiO 0 2 + (z -x)µ H2O -g(x) = 0 (3.19) ∂A ∂x = (µ Ca 2+ + 2µ OH -) + (z ′ -1)µ H2O - ∂g ∂x = 0 (3.20)
where, z ′ = dz/dx.

Eq. (3.19) indicates that the chemical affinity of reaction (3.16), A, vanishes at equilibrium. Eq. (3.20)

indicates that the chemical affinity keeps at 0. Thus, equilibrium must hold during the decalcification of the C-S-H. Note that we have the expression of the chemical potential of CH and SH t as follows:

µ Ca 2+ + 2µ OH -= µ 0 CH + RT ln Q CH K CH (3.21) µ SiO 0 2 + tµ H2O = µ 0 SHt + RT ln Q SHt K SHt (3.22)
where, t is the hydration level of the amorpheous silica SH t which can be obtained after a complete decalcification of the C-S-H. K CH and K SHt (Q CH and Q SHt ) are the equilibrium constants (ion activity products) of portlandite and amorpheous silica. Combining Eq. (3.19), Eq.(3.21) and (Eq.3.22), a basic relationship between the ion activity products of portlandite and that of amorphous silica is derived as (Eq.(3.23) and Eq.(3.24).

x ln

Q CH K CH + ln Q SHt K SHt = ln a(x) (3.23) ln Q CH K CH = ∂ ln a ∂x (3.24)
where, ln

a(x) = (g(x) -xµ 0 CH -µ 0 SHt -(z -t -x)µ H2O )/RT .
Given that the molar Gibbs energy of the amorphous silica in the form of g(0) equals to µ 0 SHt , ln a(0) = 0. Eq. (3.24) indicates a constitutive-like equation as a one to one relationship between the Ca/Si ratio and

Q CH K CH : x = χ( Q CH K CH ).
Consequently, a general mass action law in the form of the differential equation is then derived from Eq. (3.23) and Eq. (3.24) as follows:

d(ln Q SHt K SHt ) + xd(ln Q CH K CH ) = 0 (3.25) entailing ln Q SHt K SHt = - Q CH K CH 0 χ(q) q dq (3.26)
According to Eq.(3.26) , the relationship between

Q SHt K SHt and Q CH K CH is shown in Fig. 3.2.
In the following parts of this study, note q CH = Q CH K CH as the saturation degree of dissolved CH.

To confirm this relationship, we employ the results of an experiment conducted by Greenberg and Chang [START_REF] Greenberg | Investigation of Colloidal Hydrated Calcium Silicates: Pt. II. Solubility Relationships in the Calcium Oxide-silica-water System at 259[END_REF]. In the experiment the solubilities of reaction mixtures of calcium oxide, silica and water were investigated (0.805 g of SiO 2 was poured in 100 mL of water with varying additions of CaO). The calcium ion concentration, the silicic acid concentration (ρ H4SiO4 + ρ H3SiO -

4 + ρ H2SiO 2- 4
) and the pH were measured. In 1.8

Q SH t /K SHt χ Q CH /K CH Q SH t /K SHt χ (Ca/Si ratio) Figure 3.2: A typical pattern of the relationship between Q SHt K SHt and Q CH K CH 1e-04 1e-02 1e+00 1e-06 1e-04 1e-02 1e+00 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Q SH t /K SHt Ca/Si ratio Q CH /K CH Q SH t /K SHt -model Q SH t /K SHt -experiment Ca/Si-model Ca/Si-experiment Figure 3
.3: Composition of a calcium-silicon solution in equilibrium with its solid phase vs. Ca/Si ratio.

Examples of the evolution of pH and solid compositions

To illustrate the evolution of pH and solid compositions, the H 2 S attack and H 2 SO 4 attack of homogeneous representative volume element (RVE) of hydrated cement paste are presented separately. The cement paste contains initially 5.2 mol/L of portlandite (CH) and 3.5 mol/L of jennite (i.e. C 1.66 SH 2.6 ), the same value as a well hydrated Class H cement with water/cement ratio (W/C) of 0.38 [START_REF] Huet | Quantitative reactive transport modeling of portland cement in co2-saturated water[END_REF]. There is no mass transport at the boundary due to diffusion. In the simulation, the model given by Eq. (3.26) is employed to describe the dissolution of C-S-H and the Ca/Si ratio evolution is shown in Fig. 3.2.

For H 2 S attack, a small value (about 1×10 -9 mol/L) is assigned as the initial aqueous H 2 S concentration in the pore solution to keep the portlandite stable. The aqueous H 2 S concentration increases to 0.19 mol/L (the pH reaches about 9) in 86400 s (i.e., 1 day). There is no H 2 S absorption through the concrete surface.

Concentration of aqueous H 2 S, HS -and S 2-, and pH evolution are shown in Fig. 3.4. Since the ionization constant of H 2 S is very low (8.9×10 -8 ), the pH drops from 12.4 to 9 even if ρ H2S reaches up to 0.19 mol/L. Due to the lower ionization constant of HS -(1.2×10 -13 ), ρ HS -is always higher than ρ H2S .

Conclusion

This chapter introduces the chemical reaction part of biogenic sulfuric acid attack modelling. Homogeneous chemical reactions are considered in the pore solution of cementitious materials. It is assumed that all the aqueous reactions are in thermodynamical equilibrium governed by equilibrium constant.

The threshold of ion activity products are employed to determine the stability of CH and C SH2. With mass action laws, the equilibrium between CH and C SH2 is calculated. A simplified chemical kinetics for the dissolution of CH is introduced to facilitate numerical convergence. In heavily corroded sewer pipe, the pH of biofilm is normally below 2. Thus the characteristic time of CH dissolution is chosen as small as possible during simulation to achieve a quick dissolution.

Rather than discrete or solid solution methods, a general approach is applied to characterize the continuous decalcification of C-S-H gel. A generalization of the mass action law is employed for C-S-H dissolution.

Comparing with discrete or solid solution methods, this approach encompasses the solid solution model involving any number of poles of coefficient, and is easy to implement in a transport-reaction modelling.

The evolution of pH and solid composition during H 2 S and H 2 SO 4 attack are illustrated by the study of simple cases of representative volume element. Qualitatively, the profiles of simulation results coincide with experimental observations. However, due to the lack of transport terms, the simulation results can not be compared with the experimental results quantitatively. The influence of the decomposition of calcium sulfide and the deposition of gypsum on the porosity of materials will be discussed in the next chapter.

Chapter 4

Porosity evolution and damage of cementitious materials

Introduction

As a porous media, the physical/chemical performance of cementitious materials depends not only on solid compositions but also on pore structure, which plays important role in this model as well. Diffusion of aqueous species is considered in this paper, then the diffusivity of all ionic species is governed by a function of pore structure (given in Section 5.3). The corrosion depth is determined by the damage of pore structure as well. The characteristic parameters of pore structure are various (such as porosity, pore size distribution, specific internal area etc. [START_REF] Hall | Transport of fluids in homogeneous isotropic cementitious materials[END_REF]). However, the purpose of our study is to propose a method to model the whole process of biodeterioration of cementitious materials rather than the specific effect of pore structure. In this paper, our model accounts for the microstructure evolution by adjusting the porosity.

During the abiotic pH reduction process of concrete surface and the sulfuric acid attack process, solid phases such as portlandite and C-S-H gel are dissolved. For H 2 S attack, CaS, whose molar volume is less than CH and C-S-H, precipitates in pores. Furthermore, when H 2 S is enough, CaS could be dissolved.

Obviously, when the phases which occupied the initial solid volume of materials disappear, the pore volume would increase.

While for H 2 SO 4 attack process, corrosion product is gypsum, which is expansive and insoluble. Observations revealed that the volume of solid substances increases largely [START_REF] Okabe | Succession of sulfur-oxidizing bacteria in the microbial community on corroding concrete in sewer systems[END_REF][START_REF] Davis | Analysis of concrete from corroded sewer pipe[END_REF]. Compared to H 2 S attack, the evolution of porosity during H 2 SO 4 attack process is much more complex.

Firstly, the molar volume of C-S-H and the porosity evolution during H 2 S attack will be introduced in this chapter. The next section introduces the swelling of gypsum precipitation. Using different methods for different conditions, the porosity evolution during H 2 SO 4 attack process is calculated. In the last section, two simple examples without transport are presented to illustrate the evolution of the solid volume assemblage of H 2 S and H 2 SO 4 attack separately.

Porosity change during H 2 S attack

Let us note the initial porosity of the sample materials and the porosity after deterioration as φ 0 and φ H2S respectively. Induced by the precipitation/dissolution of the various solid compounds, the porosity change can be easily addressed by a simple balance of volume as Eq.(4.1).

φ H2S -φ 0 = -V CH (n CH -n 0 CH ) -V CaS (n CaS -n 0 CaS ) -V C-S-H n Si + V 0 C-S-H n 0 Si (4.1)
where, V i is the molar volume of the solid compound i (L/mol). n i and n 0 i represent the current and initial solid content of i (mol/L). V CH = 33 cm 3 /mol is used for the molar volume of portlandite. For CaS, the molar volume V CaS is chosen as 28 cm 3 /mol [START_REF] Wiberg | Holleman-wiberg's inorganic chemistry[END_REF][START_REF] Zekker | Dissolution modeling and experimental measurement of cas-h2o binary system[END_REF].

However there is a lack of data regarding the molar volume of C-S-H, which is theoretically a function of the Ca/Si ratio, x, and the H 2 O/Si ratio, z. According to [START_REF] Jennings | Colloid model of c-s-h and implications to the problem of creep and shrinkage[END_REF], the molar volume of C 1.7 SH 1.4 is approximately 64.5 cm 3 /mol, that of C 1.7 SH 2.1 is 78.8 cm 3 /mol and that of C 1.7 SH 4 is 113.6 cm 3 /mol.

On the other hand, the molar volume of amorpheous silica is approximately 29 cm 3 /mol [START_REF] Lothenbach | Thermodynamic modelling of the effect of temperature on the hydration and porosity of portland cement[END_REF]. In this paper, we choose the values provided by [START_REF] Thiéry | Effect of carbonation on the microstructure and the moisture properties of cement-based materials[END_REF][START_REF] Morandeau | An original use of carbonation profiles to investigate kinetics, microstructurale changes and released water for type-i cement-based materials[END_REF], i.e., V C1.67SH2.6 = 81.3 cm 3 /mol, V C0.83SH1.83 = 54.8 cm 3 /mol, V SH2 = 43 cm 3 /mol.

In absence of knowledge, assuming volume and H 2 O/Si ratio of three end members, linear functions for V C-S-H (x) are used as follows:

V C-S-H (x) =        x -x Tob x Jen -x Tob V Jen C-S-H + (1 - x -x Tob x Jen -x Tob )V Tob C-S-H ,x ≥ x Tob x x Tob V Tob C-S-H + (1 - x x Tob )V SHt ,x ≤ x Tob (4.2)
where, x Jen = 1.667 and x Tob = 0.8333 represents the Ca/Si of Jennite and Tobermorite respectively.

V Jen C-S-H = 81.3 cm 3 /mol and V Tob C-S-H = 54.8 cm 3 /mol are the molar volume of Jennite and Tobermorite. Similarly, z(x) can be expressed as follows:

z(x) =        x -x Tob x Jen -x Tob z Jen + (1 - x -x Tob x Jen -x Tob )z Tob ,x ≥ x Tob x x Tob z Tob + (1 - x x Tob )t ,x ≤ x Tob

Expansion of gypsum precipitation

Lots of observations [START_REF] Monteny | microbiological, and in situ test methods for biogenic sulfuric acid corrosion of concrete[END_REF][START_REF] Ehrich | Biogenic and chemical sulfuric acid corrosion of mortars[END_REF] found that biodeterioration forms gypsum with an increase in volume that results in significant expansion of concrete (as shown in Fig. 4.2a). The molar volume of gypsum is 75 cm 3 /mol, which is more than 2 times of that of portlandite. The volume of the dissolution products of 1 mole Jennite (i.e., 1.67 mole C SH2 and 1 mole SH 2 ) is about 168 cm 3 , while the molar volume of Jennite is 81.3 cm 3 . Therefore, unlike H 2 S attack, H 2 SO 4 attack leads to volume expansion.

Furthermore, the gypsum forming during H 2 SO 4 attack process is a kind of softy gel which provides no structural strength [START_REF] Vincke | Recent developments in research on biogenic sulfuric acid attack of concrete[END_REF]. Normally, it is a white-powder like deposition saturated with water (having the consistency of "cottage cheese") as shown in Fig. 4.2b and loosely connected with uncorroded concrete core [START_REF] Israel | Acid attack on pore-reduced cements[END_REF]. The porosity of the gypsum gel is about 0.85 -0.92 [START_REF] Jahani | Investigation of sulfuric acid corrosion of concrete: Part i. modeling and chemical observations[END_REF].

Although the formation of gypsum as a result of sulfuric acid attack of concrete has been reported frequently, there is no agreement on its mechanism.

Since gypsum layer loosely connects with uncorroded concrete core and very few sulfur was detected beyond this zone, [START_REF] Beddoe | Modelling acid attack on concrete: Part i. the essential mechanisms[END_REF][START_REF] Cohen | Sulfate attack on concrete: research needs[END_REF] claimed that gypsum forms via a through-solution mechanism. In this way, Ca 2+ is depleted by an ion exchange reaction with H + , and Samples on left side is reference sample,whereas on right side sample exposed to corrosion condition. [START_REF] Ehrich | Biogenic and chemical sulfuric acid corrosion of mortars[END_REF]; (b) The appearance of concrete coupons exposed to the sewer atmosphere for 1 year. [START_REF] Okabe | Succession of sulfur-oxidizing bacteria in the microbial community on corroding concrete in sewer systems[END_REF] thereafter diffuses towards the acid solution. By a reaction with sulfate anion from the acid solution, the deposition of gypsum forms the corroding layer (illustrated by Fig. 4.3a). Thus, the expansion and damage process of pore structure can be dismissed.

On the other hand, [START_REF] Lawrence | Sulphate attack on concrete[END_REF] had found that substantial deposits of gypsum forms in concrete and cause expansion. Similarly, [START_REF] Gollop | Microstructural and microanalytical studies of sulfate attack iii. sulfate-resisting portland cement: reactions with sodium and magnesium sulfate solutions[END_REF] reported cracking and expansion in the gypsum layer, and the spontaneous spalling of gypsum layer was observed by [START_REF] Torii | Effects of fly ash and silica fume on the resistance of mortar to sulfuric acid and sulfate attack[END_REF]. Therefore, sulfuric acid penetrates into mortar or concrete and reacts with calcium hydroxide of cement hydrates to produce gypsum. The volume of solid substances increases largely, which causes expansion of reaction products resulting in the damage of pore structure. Moreover, some investigations mentioned the effect of w/c ratios of cement paste on the resistance to H 2 SO 4 attack. It is well known that, a low w/c brings low porosity which normally leads to more durability.

However, researches revealed that H 2 SO 4 solution showed greater degradation for low w/c ratio than for high w/c ratio samples [START_REF] Fattuhi | The performance of cement paste and concrete subjected to sulphuric acid attack[END_REF][START_REF] Hughes | Limestone and siliceous aggregate concretes subjected to sulphuric acid attack[END_REF][START_REF] Kawai | Concrete deterioration caused by sulfuric acid attack[END_REF].

The possible reason is that samples with a high water cement ratio has larger and more pores than the samples with a low water cement ratio. These pores play the role of a capacity to absorb expansion caused by the production of gypsum. Hence samples with a high w/c ratio has a higher capacity to absorb the expansion gypsum than samples with a low water cement ratio. Moreover, lower w/c ratio samples contain more calcium hydroxide than that with high w/c samples. That is to say more gypsum is precipitated in pores with low w/c samples. This mechanism is schematically illustrated in Fig. 4.3.

Therefore, in this paper we consider that the pores will be filled with gypsum during the early time of H 2 SO 4 attack. Since gypsum gel is compressible [START_REF] Azam | Study on the geological and engineering aspects of anhydrite/gypsum transition in the arabian gulf coastal deposits[END_REF], it is assumed that gypsum will be compressed after a complete filling of pores. Under this condition, the sample can be treated as cementitious pore structure with gypsum inclusion. The gypsum accumulation leads to pressure rising until the inner pressure exceeds the ultimate bearing capacity of the materials. If so, the pore structure will be damaged.

The zone, where the pore structure is damaged by inner pressure, should be treated as uncompressed gypsum with unreacted solid compounds inclusion (such as silica gel, remaining cement hydrates, etc.).

Our model accounts for the microstructure evolution by adjusting the porosity using different methods

Porosity of cementitous materials with uncompressed gypsum inclusion

The initial porosity of the sample materials is noted as φ 0 , and the porosity of uncompressed gypsum is φ 0 C SH2 . Resulting from the dissolution of the portlandite and C-S-H gel, the porosity of cementitious materials will increase to φ 0 C as follows:

φ 0 C = φ 0 -V CH (n CH -n 0 CH ) -V C-S-H n Si + V 0 C-S-H n 0 Si (4.4) If V s C SH2 = 75 cm 3 /mol is the molar volume of the solid phase of gypsum, the molar volume of un- compressed gypsum ,V 0 C SH2 , is V s C SH2 /(1 -φ 0 C SH2 ).
Before the saturation degree of gypsum has reached 1, neither gypsum nor the pore structure are subjected to pressure. Therefore, during this phase V C SH2 and φ C is equal to V 0 C SH2 and φ 0 C respectively. The porosity of the sample is therefore given by Eq.(4.5).

φ = φ 0 C -V s C SH2 (n C SH2 -n 0 C SH2 ) (4.5)

Porosity of cementitous materials with compressed gypsum inclusion

After gypsum has filled the pore volume, the gypsum gel starts to exert pressure on the pore wall.

Subjected to the pressure p, the gypsum will be compressed while the pore volume will be expanded. As gypsum precipitates in the pores, the saturation degree of gypsum can be written as:

S C SH2 = n C SH2 V C SH2 φ C (4.6)
where, V C SH2 and φ C are the molar volume of gypsum gel and the porosity of cementitious material respectively. The inner pressure is 0 until the saturation degree of gypsum reaches 1 (see Figure.4.6 ).

In the condition of gypsum saturated, the porosity of cementitious material and gypsum can be described as

φ C = φ 0 C + ϕ C (4.7a) φ C SH2 = φ 0 C SH2 + ϕ C SH2 (4.7b)
where, ϕ C and ϕ C SH2 represent the porosity changes of cementitious materials and gypsum caused by pressure. It is assumed that the behaviour of the porous material is elastic linear and brittle as shown in Fig. 4.7. The relationship between pressure and porosity change is:

ϕ C = φ 0 C p K C (4.8a) ϕ C SH2 = -φ 0 C SH2 p K C SH2 (4.8b)
where, K C and K C SH2 are the compression coefficients of cementitious material and gypsum respectively.

The elastic modulus of mortar, which is approximately 10 GPa, is employed as K C in the model. Regarding the compression property of gypsum, few data can be found. However, the similar gypsum was found in soil [START_REF] Azam | Study on the geological and engineering aspects of anhydrite/gypsum transition in the arabian gulf coastal deposits[END_REF]. Therefore, the compression coefficient of soil, which is approximately 5 MPa, is applied as a first approximation of K C SH2 .

Since a change of molar volume of gypsum gel is induced by the decreasing of the pore volume of gypsum, according to Eq.(4.8b) it can be derived that:

V C SH2 -V 0 C SH2 V 0 C SH2 = ϕ C SH2 φ 0 C SH2 = - p K C SH2
(4.9) Thus, the molar volume of gypsum gel under pressure p is given by:

V C SH2 = V 0 C SH2 (1 - p K C SH2 ) (4.10)
Note that the saturation degree of gypsum, S C SH2 , is equal to 1 during this stage. Substituting Eq.(4.10) into Eq.(4.6), pressure p can be obtained as:

p = K C SH2 (1 - φ C n C SH2 φ 0 C SH2 ) (4.11)
Substituting Eq.(4.8a) into Eq.(4.11), the pressure can be further calculated by:

p = (V 0 C SH2 n C SH2 -φ 0 C )K C SH2 K C K C SH2 φ 0 C + V 0 C SH2 n C SH2 K C (4.12)
Consequently, the porosity of gypsum and cementitious materials can be derived from the assembly of Eq.(4.12), Eq.(4.4), Eq.(4.8) and Eq.(4.7).

For the whole sample, the porosity is the ratio of pore volume to the total volume. For S C SH2 = 1, the pore volume of the whole sample is the pore volume in gypsum gel. Thus the porosity of the sample is:

φ = φ C SH2 φ C (4.13)

Porosity of damaged cementitous materials

Along with the accumulation of gypsum in cementitious material pores, the inner pressure could exceed the tensile strength of the cementitious material (p cr , which is 2-5 MPa for normal concrete (CEB-FIP, 1990) and set as 3.5 MPa in this paper). In that case, the pore structure is damaged. For now, the damage process is simplified in Fig. 4.7 with abrupt softening. Once the failure criterion (p > p cr ) is reached, the pore structure is fully damaged and characterized by a compression modulus equal to 0. For the damaged materials, the inner pressure will be released (p = 0)

and gypsum gel will swell freely. The microstructure of such sample can be considered as uncompressed gypsum gel with a porosity of φ 0 C SH2 , and remained solid particles compounds such as unreacted portlandite and C-S-H, quartz and other non-reactive phases. The volume of remained solid particle, V remainedsolid , can be calculated as follows:

V remained-solid = (1 -φ 0 ) -V CH (n 0 CH -n CH ) -(V 0 C-S-H n 0 Si -V C-S-H n Si ) (4.14)
Therefore, the pore volume of the whole sample equals to pore volume of the gypsum fraction of sample and the porosity can be expressed as follows:

φ = (1 -V remained-solid )φ 0 C SH2 (4.15)

Examples of the evolution of porosity and solid volume

To illustrate the evolution of the porosity and solid volume, the H 2 S attack and H 2 SO 4 attack are simulated in a REV similar to the case in Section 3.5.

For H 2 S attack, solid volume assembling for calcium sulfide, CH and C-S-H is profiled in Fig. 4.9. The volume occupied by portlandite decreases with the fast dissolution of portlandite. On the other hand, the space occupied by calcium sulfide, which starts to precipitate, increases. Since the molar volume of calcium sulfide is slightly less than that of portlandite, porosity increases from 0.33 to 0.36 when portlandite is completely dissolved. Compared to portlandite, the dissolution product of C-S-H (i.e., amorphous silica) occupies some space rather than decomposing in pore solution. Therefore, during the decrease of Ca/Si ratio of C-S-H, porosity increases very little. However, after ρ H2S exceeds 0.0025 mol/L, a fast rising of porosity results from the dissolution of calcium sulfide. When calcium sulfide is fully dissolved, the only solid phase occupying space is amorphous silica and porosity reaches 0.64. After C-S-H is fully decalcified, 95% of initial volume of REV is occupied by gypsum and amorphous silica, while porosity increases to 0.62. Hence, the total volume of corrosion products is 2.5 times larger of the volume of initial sample. In the experiment conducted by [START_REF] Okabe | Succession of sulfur-oxidizing bacteria in the microbial community on corroding concrete in sewer systems[END_REF], a mortar sample with thickness of 8 mm was exposed to the sewer atmosphere for 1 year. According to the measurement of mortar surface, about 4 mm of mortar was corroded and the thickness of gypsum layer is 10 mm (as shown in Fig. 4.12). Thus, the simulation result agrees with experiment observation.

Conclusion

This chapter introduces the porosity evolution induced by the precipitation and dissolution reactions during H 2 S attack and H 2 SO 4 attack. The molar volume of C-S-H depending on Ca/Si ratio is assumed by a bilinear function. When reacting with H 2 S, the volume of sample does not change and the change in porosity is calculated by the volume balance equation. Then the mechanism of the formation of gypsum is discussed. Thereafter, the failure of pore structure is taken into account, and the change in porosity is addressed by different methods for different conditions.

By simple cases of RVE without transport, the evolution of porosity and solid volume are illustrated.

Both of H 2 S and H 2 SO 4 leads to the increase of porosity. During H 2 S attack, the smaller molar volume and dissolution of CaS cause the reduction of solid volume which is responsible for the increase of porosity.

For H 2 SO 4 attack, the expansion of gypsum fills pores to reduce the porosity during early time. When the material is damaged, porous gypsum occupies most of the space and the sample expands to about 2.5 times of the initial volume.

Since no publication reports the porosity of the corrosion layer of H 2 S or H 2 SO 4 attack, the simulation results of porosity evolution can not be verified quantitatively and directly. On the other hand, the volume of initial and corroded sample is much easier to measure. For the expansion of gypsum layer, the simulation results are in good agreement with experimental observations. In our study, we just applied a simplified damage model to determine the porosity change. To make a more precise prediction, a more detailed description of the physical behaviour of gypsum precipitation may be implemented. For example, according to [START_REF] Azam | Study on the geological and engineering aspects of anhydrite/gypsum transition in the arabian gulf coastal deposits[END_REF], the relationship between log of pressure and reduction of porosity of gypsum gel is linear, and the compression index is 0.12. Furthermore, as more and more cement hydrates dissolve, the cementitious materials is soften gradually, while in our study the material is soften abruptly.

Chapter 5

Reactive transport modeling and numerical procedures

Introduction

The deterioration process is not only controlled by the chemical equilibrium but also by the transport of different species. In this paper, only diffusion of various species in saturated porous media is considered.

Furthermore, the change of porosity, which is the result of dissolution/precipitation of solid phases, leads to the change of the diffusion rate of aqueous species. The coupling of transport and chemical equilibrium could be approached by reactive transport modeling.

To simulate the whole process of biodeterioration of sewer pipe, two types of porous materials are considered. One is cementitious materials which initially contains portlandite and C-S-H gel as hydration products and some non-reactive solid parts. The other one is biofilm which contains bacteria producing H 2 SO 4 . The chemical reactions and bio-chemical reactions are considered in cementitious materials part and biofilm part separately. Yet the various species diffuse not only in the cement part and biofilm but also across the interface between the two materials. This work is implemented within the modeling platform, Bil1 , based on the finite volume methods.

Comparing with other reactive transport codes [START_REF] Xu | Toughreact user's guide: A simulation program for non-isothermal multiphase reactive geochemical transport in variably saturated geologic media[END_REF][START_REF] Steefel | version 1.2: Software for modeling multicomponent, multidimensional reactive transport. User's Guide[END_REF][START_REF] Lichtner | Flotran user's manual[END_REF], Bil can solve the couplings between the chemical reactions and the transport equations in one step.

In this chapter, the explanation of the field equations governing the coupling of transport and reactions is followed by the introduction of electro-diffusion equation of aqueous species. Then, the numerical tool Bil is introduced briefly. Numerical procedures and the coupling of transport equations and reaction equations are presented. By a simple biodeterioration example (a hydrated cement paste immersed in mixed culture of SOB which produces H 2 SO 4 at a constant rate), the coupling of biofilm and cementitious materials is illustrated.

Field equations

Mass balance equation of element

The coupling of the transport and the chemical reactions is treated with the help of a set of mass balance equations. In this paper, the mass balance equations are applied to atoms such as sulfur (S), calcium (Ca), silicon (Si), potassium (K) and chlorine (Cl). The amount of each element A (A = S, Ca, Si, K, Cl) keeps balance in the system as follows:

∂n A ∂t = -div w A (5.1)
where n A represents the total molar content of element A per unit volume of porous medium (mol/L).

There are two contributions to n A associated to the liquid and solid (or gel) phases:

n A = φρ L A + n S A , where ρ L
A is the concentration of element A in pore solution (mol/L) and n S A is the amount of element A in solid phase per litre of porous medium (mol/L). The concentration of each element can be found in Table 5.1, where n C-S-H represents the amount of element Si in all types of C-S-H gel with different Ca/Si ratio.

Obviously, the total molar flow of A, w A (mol/m 2 •s), can be decomposed in the same manner as n A . The transport of species will be given in the following sections.

+ ρ CaH2SiO4 + ρ SiO 0 2 n C-S-H

Balance equation of charge

Each molecule, i, takes a fixed valence number, z i , hence carrying a constant charge. Beside mass balance of elements, for each chemical reaction the charge keeps balanced (there is no source of charge in the chemical reactions). Such hypothesis of null current is commonly used in reactive transport models to take into account the ionic interactions [START_REF] Xu | Toughreact user's guide: A simulation program for non-isothermal multiphase reactive geochemical transport in variably saturated geologic media[END_REF][START_REF] Steefel | version 1.2: Software for modeling multicomponent, multidimensional reactive transport. User's Guide[END_REF][START_REF] Lichtner | Flotran user's manual[END_REF]. As a consequence, the charge is governed by a global balance equation as follows:

div i = 0 (5.2)
where i stands for the ionic current written as:

i = i z i w i (5.3)
where the summation applies on the set of electrolyte ions. The ionic flow w i (mol/m 2 •s), is given by a Nernst-Planck equation (Eq.(5.5)). Therefore an electric potential is generated in the medium providing electrostatic force on each ion so as to form an electroneutral pore solution. It follows that electroneutrality must be held in the medium:

i z i ρ i = 0 (5.4)
Eqs.

(5.1-5.5) are the set of the field equations governing the coupling of transport and chemistry.

Transport of aqueous species

Concerning the transport equations, we only consider the diffusion of aqueous species in saturated porous materials. The electro-diffusion (Nernst-Plank equation) as Eq.(5.5) is employed.

w i = -D i ∇ρ i + ρ i F z i RT ∇ψ (5.5)
where, D i represents the effective diffusion coefficient (m 2 •s -1 ) of species i. F is the Faraday's constant (96485 C•mol -1 ). R is the gas constant (8.31 J•mol -1 K -1 ). T is the absolute temperature in K. z i and ρ i stand for the ionic valence and the concentration species i. The first term in Eq.(5.5) refers to the diffusion effect due to the concentration gradient (in agreement with the Fick's law), and the second part accounts for the electrostatic effect.

For a saturated cement paste, D i of each species i depends on the porosity. According to an empir-ical relationship between the effective diffusion coefficient and the porosity of sound materials (Eq.(5.6)) [START_REF] Mainguy | Modelling of leaching in pure cement paste and mortar[END_REF], when φ < 0.92, D i can be calculated as follows:

D i = D 0 i 2.9 × 10 -4 e 9.95φ
(5.6)

The diffusion coefficient in pure water, D 0 i could be obtained from experimental measurement. When experimental data is not available, D 0 i is obtained from Stokes-Einstein equation (Eq.5.7) as follow:

D 0 i = k B T 6πµ L r i (5.7)
where k B =1.3806488 10 -23 J/K is Boltzmann's constant. µ L is liquid water viscosity (Pa•s). And r i is the radius of the aqueous species i, which is taken from [START_REF] Lide | CRC handbook of chemistry and physics: a ready-reference book of chemical and physical data[END_REF][START_REF] Conway | Ionic hydration in chemistry and biophysics[END_REF]. The collected diffusion coefficients of different aqueous species at T=298 K are listed in Table 5.2. Note that the diffusion coefficients of Na + and K + are quite similar, it is thus acceptable to assimilate these two species. r=2.×10 -10 m

D 0 CaHS + 1.07×10 -7 r=2.×10 -10 m D 0 CaOH + 1.07×10 -7 r=2.×10 -10 m D 0 CaSO 4(aq) 1.43×10 -7 r=1.5×10 -10 m D 0 CaS (aq) 1.43×10 -7 r=1.5×110 -10 m D 0 H4SiO 0 4 1.07×10 -7 r=2.×10 -10 m D 0 H3SiO - 4 1.07×10 -7 r=2.×110 -10 m D 0 H2SiO 2- 4 1.07×10 -7 r=2.×10 -10 m D 0 CaH2SiO 0 4 1.07×10 -7 r=2.×10 -10 m D 0 CaH3SiO + 4 1.07×10 -7
r=2.×10 -10 m

D 0 K + 1.957×10 -7 (Weast et al., 1988) D 0 Cl - 2.032×10 -7 (Weast et al., 1988) D 0 Na + 1.334×10 -7
( [START_REF] Weast | CRC handbook of chemistry and physics[END_REF] 5.4 Introduction of numerical procedures

Introduction of finite volume method and Bil

To solve the coupling of chemical equations and transport equations, a modeling platform, Bil, based on finite volume method is employed. The finite volume method is a discretization method which is well suited for the numerical simulation of various types of conservation laws. Similar to the finite difference method or finite element method, values are calculated at the nodes of a mesh. "Finite volume" refers to the small volume surrounding each node point of a mesh. In the finite volume method, volume integrals in a partial differential equation that contains a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume.

This method is widely used in computational fluid dynamics packages. In Appendix B, a more precisely introduction of the finite volume method is conducted through a simple example.

The modeling platform Bil, which is based on finite volume method, is dedicated to couple problems involved in environmental engineering, geomechanical engineering, material sciences, etc.(it can be downloaded on http://perso.lcpc.fr/dangla.patrick/bil). The structure of the code facilitates the development of new models. Bil is written in C language and runs on Linux-based OS. This platform is developed for 1D, 2D and 3D problems. It doesn't include mesh generator for 2D and 3D problems or post-processing treatment of outputs. However, it can read mesh files created by open-source software like Gmsh [START_REF] Geuzaine | Gmsh: A 3-d finite element mesh generator with built-in pre-and postprocessing facilities[END_REF]. The output files created by Bil can easily be used by some plotting programs such as Gnuplot [START_REF] Racine | gnuplot 4.0: a portable interactive plotting utility[END_REF]). Compared to other numerical simulation tools used for the (biogenic) sulfuric acid attack [START_REF] Jahani | Investigations of sulfuric acid corrosion of concrete. ii: electrochemical and visual observations[END_REF][START_REF] Chalupeckỳ | Macroscopic corrosion front computations of sulfate attack in sewer pipes based on a micro-macro reaction-diffusion model[END_REF][START_REF] De Windt | Modeling the degradation of portland cement pastes by biogenic organic acids[END_REF], Bil has two major advantages: first it solves the coupling of the chemical reactions and the transport equations in one step; second it uses different diffusion coefficients for each aqueous species. This is essential since the change in composition of cement-based material is generally diffusion controlled under saturated conditions. In addition, it handles the influence of the porosity changing on diffusion.

Numerical procedures of dissolution and precipitation of solid phases

As described in Chapter 3, dissolution and precipitation of any solid phases are controlled by a thermodynamical equilibrium. Out of the stability domain, solid component would dissolve completely. Thus for the stable solid component, i, the ion activity products (Q i ) is equal to the equilibrium constant (K i ).

While at Q i < K i , the content of solid phase, n i (mol/L), is equal to 0. Thus Eq.5.8 can be obtained.

n i • log( Q i K i ) = 0 (5.8)
Obviously, n i is no less than 0, while log( Qi Ki ) ≤ 0 at stable state. Then, a variable, ζ i , can be introduced to determine if a solid component i exists is dissolved as follows:

ζ i = n i n 0 i + log( Q i K i ) (5.9)
where, n 0 i is the initial content of solid component i (mol/L). At stable state, Q i is no bigger than K i . Thus, ζ i > 0 means that i exists stable and ζ i is noted as

ζ + i = ni n 0 i . ζ i <0 represents the dissolved i and ζ i is noted as ζ - i = log( Qi Ki ).
In this section, we take H 2 SO 4 attack for instance to describe the numerical procedures of dissolution/precipitation progress. For the equilibrium between CH and C SH2, according to the equilibrium equation (Eq.3.13), Table 5.3 can be obtained easily.

Q CH K CH Q C SH2 K C SH2 Only CH < 1 = 1 < 1 n CH ≥ 0 Only C SH2 > 1 < 1 = 1 n C SH2 ≥ 0 CH & C SH2 = 1 = 1 = 1 n CH ≥ 0, n C SH2 ≥ 0
Thus, it is possible to illustrate the dissolution of both CH and C SH2 with one variable ζ Ca , defined as follows:

ζ Ca = n CH + n C SH2 n 0 Ca + log[max( Q CH K CH , Q C SH2 K C SH2 )] (5.10)
where n 0 Ca is the initial CH content. Therefore, when ρ H2SO4 > ρ CH H2SO4 , CH is not stable. ζ Ca is in the form as:

ζ Ca = n CH + n C SH2 n 0 Ca + log( Q C SH2 K C SH2 ) (5.11)
For ζ Ca > 0, gypsum can precipitate. If ζ Ca < 0, gypsum is not stable either.

If ρ H2SO4 ≤ ρ CH H2SO4 , C SH2 is not stable. We have:

ζ Ca = n CH + n C SH2 n 0 Ca + log( Q CH K CH ) (5.12)
For ζ Ca > 0, portlandite can precipitate and for ζ Ca < 0, portlandite is not stable.

Let's assume that at time t, the system of CH/C SH2 contains n t CH and n t C SH2 . After a time difference,

Q C SH2 = 10 (ζ - Ca ) min(z H2SO4 , 1)K C SH2 (5.18)
where, z H2SO4 = ρ H2SO4 ρ CH

H2SO4

. Thus, ρ Ca 2+ can be calculated by the following equation:

ρ Ca 2+ = Q C SH2 ρ SO 2- 4 (5.19)
where, ρ SO 2- 4 can be easily obtained with ρ H2SO4 and Eq. (3.10).

Similarly, ρ H4SiO 0 4 can be attained as follows:

ρ H4SiO 0 4 = Q SHt = 10 ζ - Si Q SHt(eq) (see Eq. (5.16)) (5.20)
where Q SHt(eq) is defined in Eq. (5.17).

Examples of bio-deterioration modeling

Coupling of transport and chemical reactions

For a better understanding of the coupling of bio-chemistry process and chemistry process, it is necessary to demonstrate the chemistry process firstly. A simple H 2 SO 4 attack example is employed to illustrate the coupling of transport and chemical reactions. A cement paste cube with side length of 1 cm is considered and divided into 100 nodes. The cement paste contains initially 5.2 mol/L of portlandite and 3.9 mol/ of jennite (i.e., C 1.67 SH 2.6 ), with porosity 33%. The material is saturated with water and immersed in H 2 SO 4 solution. To simulate a severe acid environment, the concentration of unionized molecules of H 2 SO 4 , ρ 0 H2SO4 , at the sample surface is constant at 1.0×10 -8 mol/L (i.e., the concentration of total sulfate ions is 8.0×10 -2 mol/L is and the approximate pH keeps at 1). Due to the high concentration of hydrogen ion, the characteristic time τ CH and τ C SH2 is 2000 s. For the sake of simplicity, no alkali is added in the example.

Given the very small solubility of C SH2 and SH t in low pH condition, the leaching of calcium and silicon should be prevented at the sample surface. Thus, no diffusion is allowed at boundary. Considering that the surface of sample is in contact with H 2 SO 4 solution, the schematic of sample is summarized in Table 5.5 as well as the initial and boundary conditions. Due to symmetry, only half of the sample is discretized.

+ K =0 Chlorine (Cl) no diffusion ρ - Cl =0 Charge ψ=0 ψ=0
The distribution of H 2 SO 4 concentration and the evolution of pH in the material from 0 to 60 days are profiled in Fig. 5.1 and Fig. Damage front, indicated by dash line: The place where the material is fully damaged.

With the help of these fronts, several zones can be distinguished:

Uncorroded zone: The zone between the core of the sample and portlandite dissolution front, where the material remains intact.

C-S-H decalcification zone:

The zone between C-S-H decalcification front and SH t precipitation front, where the Ca/Si ratio decreases gradually.

Gypsum layer:

The zone between gypsum precipitation front and the material surface in contact with H 2 SO 4 , where gypsum accumulates. The volume of this layer is larger than the volume of uncorroded material.

Damaged zone:

The zone between damage front and the initial surface of material, where pore structure is damaged and the material has no strength. In this study, we take the thickness of this layer as the corrosion depth which is commonly measured to evaluate the deterioration in experiments.

Fully degraded zone:

The zone between SH t precipitation front and the material surface in contact with H 2 SO 4 , where only gypsum, amorphous silica gel and unreactable solid phased exists.

With the change in the solid compositions, the porosity changes as well. From From the CH dissolution front to the damage front, the porosity increases from 0.33 to 0.47 due to CH dissolution as well as C-S-H decalcification. Along with gypsum forming, pores are filled up. Given that gypsum gel is compressed by the inner pressure, the porosity decreases then sharply to 0.43. The inner pressure exceeds the strength of the material very fast. Thereafter, the pore structure is damaged and the porosity turns back to rise until C-S-H is fully decalcificated indicated by Sf. In the fully degraded zone, porosity reaches to 0.556.

As introduced before, the equilibrium between CH and C SH2 is controlled by the concentration of Example of H 2 SO 4 attack Ca/Si ratio log(q CH ) Figure 5.8: Ca/Si ratio of C-S-H and log(q CH ) after 15 days immersion

Coupling of bio-chemical process and chemical process

To illustrate the coupling of bio-chemistry process and chemistry process, we simply consider a cement paste cube immersed in 500 mL of the solution containing idealized sulfur-oxidizing bacteria (SOB). The cement paste sample is the same as in the previous section. And the idealized SOB solution is considered as in contact with cement paste cube. Assuming an enough source of sulfur, the idealized SOB solution produces H 2 SO 4 at a constant volumetric rate. Initially, H 2 SO 4 , HSO - 4 and SO 2- 4 are the only aqueous species in SOB solution, where the initial concentration of H 2 SO 4 0 is 1×10 -25.5 mol/L (i.e., the approximate pH is 7). Since the income of sulfur is neglected, there is no mass transfer at the boundary of the solution, while aqueous species can diffuse across the interface between cement paste and SOB solution.

The schematic of the sample and the initial/boundary conditions are summarized in Table 5.6. Almost the same fronts and zones can be observed in SOB solution attack as those in H 2 SO 4 attack (see Fig. 5.3). However, a gypsum dissolution front (noted as Gdf) appears close to the cement surface. To make sure neither CH nor C-S-H exist in SOB solution part, the initial ζ Ca in SOB solution is set to < 0. Thus, a few of gypsum leaches out. The flux of aqueous species related with calcium from cement paste to SOB solution can be found in Fig. 5.12.

-

Conclusion

In this chapter, the field equations which govern the mass and charge balance are introduced. The transport of aqueous species are considered as diffusion in saturated porous media. The different diffusion coefficients are employed for various species. Moreover, the effect of porosity on the diffusion coefficients are taken into account.

Afterwards, the numerical procedures for the couplings between transport and reaction equations have been introduced. Brief introductions of the principle of the finite volume method and the modeling platform, Bil are given. The method to calculate the dissolution and precipitation of solid components, as well as the kinetics effects, have been presented.

For a better understanding of our modeling, two simulation examples of cement paste in contact with constant H 2 SO 4 solution and idealized SOB solution are illustrated respectively. The evolution of pH in cement paste and SOB solution is predicted. Suffering from H 2 SO 4 attack, regardless pure H 2 SO 4 solution or biogenic H 2 SO 4 , similar solid dissolution/precipitation fronts are observed in cement paste via the profile of solid compositions. The porosity evolution caused by the special properties of gypsum precipitation is predicted. Furthermore, the damage position is identified by the method introduced in Chapter 4. With the help of these fronts, several zones (e.g., uncorroded zone, C-S-H decalcification zone, gypsum layer, damaged zone) can be distinguished.

Besides simple conditions, such as constant H 2 SO 4 solution and idealized SOB solution, our modeling can work with complex boundary condition and real biofilms. In the following chapters, laboratory experiments with various conditions are simulated and long time predictions are conducted.

Chapter 6

Simulation in the case of constant H 2 SO 4 solution

Introduction

In order to predict the deterioration rate of cementitious materials, some experiments have been done to simulate the corrosion process of sewer pipe (Gutiérrez-Padilla et al., 2010;[START_REF] De Muynck | Effectiveness of admixtures, surface treatments and antimicrobial compounds against biogenic sulfuric acid corrosion of concrete[END_REF][START_REF] Alexander | Performance of sewer pipe concrete mixtures with portland and calcium aluminate cements subject to mineral and biogenic acid attack[END_REF].

Based on laboratory tests where specimens are continuously immersed in acid solutions over a specified period of time, it is understood that the depth of deterioration of concrete due to sulfuric acid attack is proportional to the square root of the exposure time. But in actual structures, concrete is often exposed to flowing water that contains an acid solution such as sewage. In other words, concrete is subjected to the shearing force of fluid that erodes the surface areas of deteriorated concrete.

For cementitious materials in static solution environment and flowing solution environment, two calculations are carried out to simulate experimental works of [START_REF] Kawai | Concrete deterioration caused by sulfuric acid attack[END_REF]. The influence of different H 2 SO 4 concentrations of the solution on the corrosion is studied. Furthermore, in order to understand the difference between the experimental results and simulation results, the effect of carbonated layer and initial porosity of materials are analysed. At last, a long term prediction of H 2 SO 4 attack is conducted.

High H 2 SO 4 concentration condition (pH = 1)

In the experiment of [START_REF] Kawai | Concrete deterioration caused by sulfuric acid attack[END_REF], mortar samples in size of 40×40×160 mm were made of ordinary Portland cement (OPC) of various water/cement ratios. After cured for 28 days, the samples were immersed in sulfuric acid of various concentrations. The sulfuric acid solution was static (noted as Method 1) or circulated onto specimens (noted as Method 2). To keep the concentration of sulfuric acid solution constant, sulfuric acid was added suitably to the solution. A pH meter was used to measure the value of pH and during the experiment the concentration of sulfuric acid solution was adjusted to the initial value of pH. In both instances, the depths of corroded zones were measured with a vernier micrometer every seven days. The corrosion depth was defined as a distance between the initial surface and current surface.

Before every measurement, intentional removal of deteriorated zones on the surfaces was not carried out.

The zones of deteriorated concrete were analysed with XRD and ion chromatoanalyzer.

In this section, we simulate the experiments where mortar samples with W/C = 0.35 were immersed in static and circulated H 2 SO 4 solution whose pH is kept at 1 for 90 days.

To simulate the experiments, mortar samples with the same size as that in experiments are considered.

The samples are divided in 720 nodes. For numerical stability, the initial H 2 SO 4 concentration in the pore solution of sample is 1×10 -31.5 mol/L, while at the right boundary the concentration of unionized molecules of H 2 SO 4 , ρ 0 H2SO4 , is set to 1×10 -8 mol/L (i.e., pH = 1).

The content of sand and other composition information (such as the content of CH and C-S-H and the initial porosity) were not given by the authors. By collecting and adjusting the data of Portland cement hydration [START_REF] Garboczi | An electronic monograph: Modeling and measuring the structure and properties of cement-based materials[END_REF][START_REF] Voigt | Properties of early-age portland cement mortar monitored with shear wave reflection method[END_REF][START_REF] Maruyama | Hydration model of portland cement for structural integrity analysis[END_REF], which can be simulated by hydration models (e.g., CEMHYD3D [START_REF] Bentz | Cemhyd3d: A three-dimensional cement hydration and microstructure development modeling package[END_REF] and HYMOSTRUC3D [START_REF] Ye | A micromechanic model for characterization of cement paste at early age validated with experiments[END_REF])), the solid compositions of samples can reasonably be assumed. For the OPC mortar with W/C = 0.35 of 28 days the cement hydrates contains initially 5.2 mol/L of portlandite and 5 mol/L of C-S-H as jennite.

The initial porosity is 30%.

From the measurement of experiments [START_REF] Brouwers | Alkali concentrations of pore solution in hydrating opc[END_REF], the alkali concentrations (i.e., K + and Na + ) in the pore solution of cement paste cured for 28 days is about 0.5 mol/L. Since these two ions have almost the same diffusion coefficient, K + and Na + are combined as one ion for the sake of simplicity.

For the electroneutrality, the same amount of anions which are not involved in any chemical reaction, is introduced. Thus, an unified form as KCl is considered. Thus, 0.5 mol/L KCl is added in the simulated pore solution, while no KCl exists in the external H 2 SO 4 solution.

Static solution environment (Method 1)

With Method 1, corrosion of specimens was visible after immersion of dozens of days. A white layer of reaction products was observed on the surface of samples as Fig. 6.1a. The XRD patterns of this degraded layer (see Fig. 6.1b) indicated that reaction products of sulfuric acid attack are mainly gypsum, portlandite and quartz. To simulation the precipitating of the reaction products at the surface of specimen, no Ca 2+ and Si 4+ flow across the boundary. Boundary and initial conditions of our modeling are summarized in Table 6.1.

= 10 -8 mol/L Calcium (Ca) no diffusion ζ Ca = 1 Silicon (Si) no diffusion ζ Si = 1 Potassium (K) diffusion allowed ρ K + =0 mol/L ρ K + = 0.5 mol/L Chlorine (Cl) diffusion allowed ρ Cl -=0 mol/L ρ Cl -= 0.5 mol/L Charge no current ψ=0
After immersion of 90 days, the calculated penetration of H 2 SO 4 and pH evolution are shown in Fig. 6.2.

At the surface, the H 2 SO 4 0 concentration is 1×10 -8 mol/L as set in boundary conditions. The surface pH keeps at 1 which was measured during experiments as well. As expected, H 2 SO 4 penetrates into sample and leads to the acidification of cementitious materials, whose pH decreases form 12.35 to 1. Due to the low concentration and high ionization constant of H 2 SO 4 , the transition zone between the H 2 SO 4 penetration front (where pH is less than 2), and the unneutralized zone (where pH is close to 12.4) is not very remarkable.

The calculated profiles of the concentration and volume of solid compounds, as well as porosity profiles The corroded sample starts with a non-uniform gypsum-rich layer, which corresponds to the white part in Fig. 6.1a. At the surface, gypsum accumulates resulting from the diffusion of Ca 2+ as described in Section 5.5.1. Consequently, the volume of this layer expands more than the inner gypsum-rich zone. At the place just below the surface, the content of gypsum decreases slightly due to deplete of Ca 2+ which has transported to the surface. In the gypsum layer, Ca/Si ratio of C-S-H is zero (Fig. 6.3b). Thus, only silica gel and unreactive phases are contained in the gypsum layer, which is defined as the fully degraded zone in Section 5.5.1. The porosity of this zone reaches nearly 0.6. The evolution of porosity is plotted in Fig. 6.7a.

Following this layer, a region containing C SH2, CH and C-S-H is observed. The H 2 SO 4 attack is taking place in this region which represents the C-S-H decalcification zone, where Ca/Si ratio drops from 1.67 to 0, as well as the C SH2 precipitation front and CH dissolution front. With the change in solid phases, porosity of this region increases from 0.3 to 0.56, and the sample starts to expands. The damage of pores happens in this region as well. In this paper, the corrosion depth is defined as the distance between the initial surface and the location where the pore structure is damaged following the simplified damage model introduced in Chapter 4.

To illustrate the damage process of pore structure, the evolution of solid compositions and inner pressure at 5 mm depth of sample are plotted in Fig. 6.6. From 105 days, the porosity starts to increase due to the dissolution of CH and C-S-H. From about 120 days, C-S-H and CH are dissolved completely, while C SH2 continues to form. At that time, the porosity decrease a little temporally until 120 days (see Fig. 6.6a). Indicated by Fig. 6.6b, inner pressure rises sharply during this period due to the compression of gypsum. are not mentioned by [START_REF] Kawai | Concrete deterioration caused by sulfuric acid attack[END_REF]. To simulate the removing of corrosion products, we set the saturation degree of dissolved gypsum (Q C SH2 /K C SH2 ) and that of dissolved silica gel (Q SHt /K SHt ) to a small value 0.01 at the boundary of sample. Thus, calcium and silicon will leach out and corrosion products will be dissolved rather than accumulated at the surface. The rest of the conditions are the same as those in the simulation of Method 1. Boundary and initial conditions are summarized in Table 6.2.

ρ H2SO4 0 = 10 -31.5 mol/L ρ H2SO4 0 = 10 -8 mol/L Calcium (Ca) diffusion allowed ζ Ca = -2 ζ Ca = 1 Silicon (Si) diffusion allowed ζ Si = -2 ζ Si = 1 Potassium (K) diffusion allowed ρ K + =0 mol/L ρ K + = 0.5 mol/L Chlorine (Cl) diffusion allowed ρ Cl -=0 mol/L ρ Cl -= 0.5 mol/L Charge ψ=0 ψ=0
Since reaction products are swept from the surface, the gypsum layer which could obstruct transport is removed and more reactive surface is exposed. Hence, the penetration rate of H 2 SO 4 and the reduction rate of pH should increase. Our simulation results agree with such expectation (see Fig. 6.9). From the boundary to the core of sample, successive zones different from those of Method 1 are observed (see Fig.6.10,Fig.6.11 and Fig.6.12). A leaching effect is illustrated since there is neither gypsum nor silica gel at the surface. Such result coincides with experimental observations. Moreover, the remaining unreacted solid components result in a porosity of 0.88. This region is considered as the fully decomposed zone in the simulation of Method 2.

Due to the difference between the leaching effect of gypsum and that of silica gel, a zone containing only C-S-H, whose Ca/Si ratio is zero (as indicated by the blue line in the following plots), is observed following the fully decomposed zone. Consequently, the porosity reaches 0.68. Following this region, a gypsum-rich layer is observed, where gypsum accumulates and the porosity is 0.56 which equals to the porosity of gypsum layer in the simulation of Method 1. The Ca/Si ratio of C-S-H is still 0 in this zone.

Gypsum is formed at the gypsum precipitation front (the inner side of this layer) and is dissolved at gypsum dissolution front (the outside of this layer) due to the leaching simulated at the boundary condition.

The subsequent zones such as damage zone and uncorroded zone are similar to that observed in the simulation of Method 1. and Method 2, there are positive concentration gradient of Ca 2+ between the gypsum precipitation front and the core of sample. Thus the diffusion of Ca 2+ toward the gypsum precipitation front results in the accumulation of gypsum. However, for Method 2, between the surface and dissolution front of gypsum there is another positive concentration gradient of Ca 2+ , which is not observed in simulation of Method 1.

Ca 2+ concentration in this region decreases from 0.003 mol/L, which is equal to Ca 2+ concentration at the surface of Method 1, to 0. The diffusion of Ca 2+ toward the surface (see Fig. 6.14) causes a leaching effect. which, in our model, was simulated by leaching out of gypsum and silica gel. Thus, some unreactive solid phases still remain in the fully decomposed zone which were completely washed away in experiment.

Therefore, the porosity of fully decomposed zone is 0.88 rather than 1 (see Fig. 6.15a). According to the experimental measurements, the sulphate ions concentration around the uncorroded surface was equal to that in the solution. But in our simulation, the sulphate concentration decreased gradually in the fully decomposed zone as shown in Fig. 6.16. We simulated this experiment by decreasing the H 2 SO 4 concentration at the right boundary to achieve pH = 2. Boundary and initial conditions are summarized in Table 6.3. Since the concentration of H 2 SO 4 is low, the characteristic time (i.e., τ CH and τ C SH2 ) are extended to 200000 s.

ρ H2SO4 0 = 10 -31.5 mol/L ρ H2SO4 0 = 10 -10 mol/L Calcium (Ca) Method 1:no diffusion ζ Ca = 1 Method 2: diffusion allowed ζ Ca = -2 Silicon (Si) Method 1:no diffusion ζ Si = 1 Method 2: diffusion allowed ζ Si = -2 Potassium (K) diffusion allowed ρ K + =0 mol/L ρ K + = 0.5 mol/L Chlorine (Cl) diffusion allowed ρ Cl -=0 mol/L ρ Cl -= 0.5 mol/L Charge Method 1: no current ψ=0 Method 2: ψ=0
The concentration of unionized molecules of H 2 SO 4 , ρ 0 H2SO4 , is 1×10 -10 mol/L in the solution with pH = 2. Comparing with the solution with pH = 1, the penetration of H 2 SO 4 0 with the solution with pH = 2 is much slower (see Fig. 6.17). Therefore the acidification of cement paste is slow as shown in Fig. 6.18.

For both Method 1 and Method 2, the acidification front of the solution with pH = 2 at 90 days does not even reach that of solution with pH = 1 at 30 days. For Method 2, the difference between the attack of pH =1 and that of pH =2 is even bigger than that simulated with Method 1 due to the removing of reaction products which obstructs the transport of H 2 SO 4 . For pH = 2, the same zones as those observed in the simulation of pH = 1 are presented in Fig. 6.19 to Fig. 6.21. Due to the low concentration of H 2 SO 4 , the corrosion rate are slow for both Method 1 and Method 2. In the experiment, corrosion is almost undetectable with Method 1 and very little corrosion was observed after 90 days of immersion of Method 2. The predicted corrosion rate is overestimated, yet much slower than that of the simulation of pH = 1.

Due to the high first order ionization constant of H 2 SO 4 , the concentrations of HSO - 4 and SO 2- 4 are much higher than H 2 SO 4 concentration. Hence the source of sulfur in gypsum is provided mainly by the diffusion of HSO - 4 and SO 2- 4 . Taking the immersion after 90 days for example, the concentration of HSO - and SO 2- 4 of the immersion of pH = 2 are much less than that of the immersion with pH = 1 (see Fig. 6.22).

For pH = 2, due to the smaller concentration gradients of both HSO - 4 and SO 2- 4 , the flux of sulfur is much lower than that under high H 2 SO 4 concentration condition as shown in Fig. 6.23.

For Method 2, in the zone where gypsum accumulates, the concentration gradients of both HSO - 4 and SO 2- 4 are negative. Thus sulfur diffuses from the gypsum dissolved front to the gypsum precipitate front.

For the condition of pH = 1, concentration gradient of SO 2- 4 is positive in the fully decomposed zone, while for pH = 2 condition it is negative. Therefore, when H 2 SO 4 is few the SO 2- 4 released from the gypsum dissolution front is not sufficient for gypsum precipitation. The reason could be that the difference of Ca 2+ concentration at pH = 2 condition and that at pH = 1 condition is smaller than the difference of SO 2- 4 concentration. 6.4 Analysis of the discrepancy between simulation results and experimental results

Effect of carbonated layer

The degradation depth measured in the experiment was about 4.2 mm and 7.2 mm with Method 1 and Method 2 respectively after 90 days immersion test which is close to our predictions (indicated by vertical lines in Fig. 6.5b and Fig. 6.12b). However, predicted degradation depth is not always consistent with experimental results as shown in in Fig. 6.7b and Fig. 6.15b. The corrosion rate in the experiments increased during immersion time, while in the early time the predicted corrosion rate is much higher than experimental results.

This may be justified by several reasons, such as inaccurate parameters in model, different conditions between simulation and experiment and carbonation occurring during the initiation stage of biodeterioration [START_REF] Hudon | Biodeterioration of concrete sewer pipes: State of the art and research needs[END_REF][START_REF] Magniont | A new test method to assess the bacterial deterioration of cementitious materials[END_REF][START_REF] Joseph | Surface neutralization and h2s oxidation at early stages of sewer corrosion: Influence of temperature, relative humidity and h2s concentration[END_REF]. Normally, overestimated/ underestimated diffusion coefficients would affect the corrosion rate during the whole process. However, in Fig. 6.7b and Fig. 6.15b the corrosion rate of simulation is higher than that of experiment only during early time. This suggests that carbonation must be taken into account in our model.

During the curing time the samples were subjected to carbonation, which could clog the pores of the surface of sample. According to the experiments on the effect of carbonation on porosity [START_REF] Thiéry | Effect of carbonation on the microstructure and the moisture properties of cement-based materials[END_REF][START_REF] Van Gerven | Effects of carbonation and leaching on porosity in cement-bound waste[END_REF], carbonation could decrease the average porosity of mortar by more than 15%. In some extreme cases, the pores of surface could be clogged [START_REF] Shen | Reactive transport modeling of CO2 through cementitious materials under CO2 geological storage conditions[END_REF]. In that case, it would have taken long time for H 2 SO 4 to penetrate into pores at the early time. After the surface area was corroded, more reactive surface was exposed, thus the degradation process accelerated. Therefore, we take carbonation as the most suspected reason leading such discrepancy.

To verify this explanation, we conducted a preliminary investigation. From the experiment results shown in Fig. 6.7b, the degradation rate accelerated after corroded depth reached to 1 mm. Thus, a simulation of the sample with a simplified carbonated layer, whose thickness is 1 mm, is carried out. During carbonation, portlandite and C-S-H gel dissolves and calcite (CaCO 3 , noted as C C) precipitates as the reactions Eq.(6.1) and Eq.(6.2). We assume that this layer is homogeneous and fully carbonated. Since the initial material contains portlandite (5.2 mol/L) and C 1.66 SH 2.6 (5 mol/L), the solid compositions of the fully carbonated layer are calcite (13.3 mol/L) and silica gel (5 mol/L). Given that the molar volume of calcite is 37 cm 3 /mol [START_REF] Shen | Reactive transport modeling of CO2 through cementitious materials under CO2 geological storage conditions[END_REF], the porosity of carbonated layer decreases from 0.3 to 0.166 by a simple balance of volume.

It is known that a reduction of porosity in cementitious materials increases its strength in general [START_REF] Yudenfreund | Hardened portland cement pastes of low porosity i. materials and experimental methods[END_REF][START_REF] Pantazopoulou | Microstructural aspects of the mechanical response of plain concrete[END_REF]. The physical properties of carbonated layer is considered to be similar with that of normal cement paste for simplification. According to the study of [START_REF] Chen | Influence of porosity on compressive and tensile strength of cement mortar[END_REF], the tensile strength of cement mortar whose porosity is 0.166 is about 1.4 times of that of the cement mortar with porosity = 0.3. Therefore the damage criterion is set as 5 MPa.

CH + CO

2 ⇋ C C + H 2 O (6.1) C x S y H z + xCO 2 ⇋ xC C + yH 4 SiO 0 4 + (z -2y)H 2 O (6.2)
Since the materials is saturated with water in our modeling, the H 2 SO 4 attack product of the carbonated layer is only gypsum.

Since we didn't take the calcite into our modeling yet, we consider the reaction of H 2 SO 4 and C C as same as that of CH in carbonated layer. Thus, the equilibrium concentration of H 2 SO 4 in carbonated layer is equals to that of initial cement paste. Boundary and initial conditions are summarized in Table 6.4.

ρ H2SO4 0 = 10 -31.5 mol/L ρ H2SO4 0 = 10 -31.5 mol/L ρ H2SO4 0 = 10 -8 mol/L Calcium (Ca) Method 1:no diffusion ζ Ca = 1 ζ Ca = 1 Method 2: diffusion allowed C C 13.3 mol/L CH 5.2 mol/L Silicon (Si) Method 1:no diffusion ζ Si = 1 ζ Si = 1 Method 2: diffusion allowed SH t 5 mol/L Jennite 5 mol/L Potassium (K) diffusion allowed ρ K + =0 mol/L ρ K + = 0.5 mol/L ρ K + = 0.5 mol/L Chlorine (Cl) diffusion allowed ρ Cl -=0 mol/L ρ Cl -= 0.5 mol/L ρ Cl -= 0.5 mol/L Charge Method 1: no current ψ=0 ψ=0 Method 2: ψ=0
The diffusion rate of species was expected to decrease during early time due to lower porosity. Therefore the pH reduction rate of both Method 1 and Method 2 are slowed down as in Fig. 6.25. Before 30 days, the acidification front of carbonated sample is behind that of fresh sample. However, after 60 days the diffusion of H 2 SO 4 in carbonated sample catches up that of fresh sample due to the fully degradation of carbonated layer.

The solid compositions of samples with carbonated layer at the early time of immersion are plotted in Fig. 6.26. At 10 days of Method 1, some Calcite still remains in the carbonated layer, where porosity keeps at 0.166. For Method 2, all of the calcite is gone due to a faster corrosion rate. At 90 days, the solid compositions are almost the same with that of normal samples.

The predicted degradation depth of the sample with carbonated layer (as shown in Fig. 6.27) indicates that the carbonated layer could reduce the degradation rate at early time. However, the carbonated layer

Effect of water-cement ratio of cemment paste

In Kawai's experiments, the degradation rate of mortar samples with different W/C ratio were compared.

It is remarkable that the degradation depth of samples with low W/C ratios was deeper than those with high W/C ratios. This phenomenon was explained as follows [START_REF] Kawai | Concrete deterioration caused by sulfuric acid attack[END_REF]:

Concrete with a high water cement ratio has larger and more pores than that with a low water cement ratio. These pores play the role of a capacity to absorb expansion caused by the production of gypsum.

Therefore concrete with a high water cement ratio has a higher capacity to absorb the expansion of production reaction of gypsum than that with a low water cement ratio, that is to say, concrete with a low water cement ratio erodes earlier than that with a high water cement ratio. Such results were found by other researchers as well: Attacks with 2% H 2 SO 4 solution showed no differences between degradation of high and low porosity mortar paste sample [START_REF] Israel | Acid attack on pore-reduced cements[END_REF]; 1% H 2 SO 4 solution showed greater degradation for low W/C ratios than high W/C ratios samples [START_REF] Fattuhi | The performance of cement paste and concrete subjected to sulphuric acid attack[END_REF]; low H 2 SO 4 concentration (0.0016 and 0.02%) showed greater deterioration depth for samples with low W/C ratios than with high W/C ratios [START_REF] Hughes | Limestone and siliceous aggregate concretes subjected to sulphuric acid attack[END_REF].

For high H 2 SO 4 concentration condition, we conducted two simulations of immersion test (Method 1 and pH = 0.5) of cement mortars whose W/C ratios are 0.35 and 0.5 respectively. For the sample with W/C ratio of 0.5, we consider that the cement hydration contains initially 3.9 mol/L of portlandite and 3.7 mol/L of C-S-H as jennite. The initial porosity is 0.45. The initial condition and boundary condition is same with that in Table 6.1. The profiles of solid compositions after 30 days (see Fig. 6.28) shows that the predicted degradation rate for samples with high W/C ratio is greater than that with W/C ratio. For low H 2 SO 4 concentration condition, similar simulations (Method 1 and pH = 2) were conducted to cement mortar with W/C ratios of 0.35 and 0.5. Fig. 6.29 shows that under low H 2 SO 4 concentration condition the predicted degradation rate for samples with high W/C ratio is still greater than that with low W/C ratio. When H 2 SO 4 is few, the corrosion rate is slow. Thus the difference between low W/C and high W/C is not as significant as that for high H 2 SO 4 concentration condition. The simulation results could be explained because the diffusion of sulfate species (H 2 SO 4 , HSO - 4 , SO 2- 4 )

is very fast and sensitive to the porosity. Thus the degradation process is diffusion controlled. However, [START_REF] Ueda | Deterioration mechanism of hardened cement paste by various acids[END_REF] claimed that sulfuric acid hardly penetrates into hardened cement. The reactions of cement hydrates and sulfuric acid should occur just beneath the surface portion of specimens. This may be caused by the production of gypsum in the surface portion which is faster than the penetration rate of sulfate ions into the specimen. In other words, the corrosion process is governed by reaction rather than diffusion. And in the simulation study [START_REF] Jahani | Investigation of sulfuric acid corrosion of concrete: Part i. modeling and chemical observations[END_REF], it was indicated that the diffusion coefficient of sulphate ions in the corrosion products layer is 7.70×10 -4 cm 2 /day, while the reaction rate constant of precipitation of gypsum is about 112 cm 2 /day according to the simulation of Bouchelaghem [START_REF] Bouchelaghem | A numerical and analytical study on calcite dissolution and gypsum precipitation[END_REF]. Therefore, further investigations about the damage mechanism and diffusion of sulfate ion in cementitious materials should be carried out.

Long-term prediction

Besides short-term predictions, long-term simulation should be conducted to assess the relevance of the model. Kawai's experiments only lasted 90 days, yet it is one of the longest experiments where enough information is available for simulation. It is generally believed that the degradation depth of concrete due to acid attack including sulfuric acid attack is proportional to the square root of the exposure time [START_REF] Pavlik | Corrosion of hardened cement paste by acetic and nitric acids part i: Calculation of corrosion depth[END_REF] as follows:

y = b √ t (6.3)
where, y is the degradation depth. t represents the exposure time in acid solution and b is constant.

According to this equation, the rate of degradation is dominated by the diffusion rate of acid under the assumption that corrosion products remain on surface.

Two simulations of sulfuric acid attack (pH = 1 and pH = 2) of Method 1 was conducted for 2 years. The carbonated layer is taken into account. The predicted corrosion depth versus time are shown in Fig. 6.30, where the relationship between exposure time and degradation coincides with Eq.( 6.3). For the condition of pH = 1, the constant b is about 2.5 mm/ √ month. For the condition of pH = 2, the constant b is 1.2 mm/ √ month, which is about 2 times less than that for pH = 1.

In sewer pipes severely corroded, the pH at the concrete surface is normally less than 2 [START_REF] Okabe | Succession of sulfur-oxidizing bacteria in the microbial community on corroding concrete in sewer systems[END_REF][START_REF] Islander | Microbial ecology of crown corrosion in sewers[END_REF]. According to the long term prediction, under such condition the corrosion rate is more than 4 mm/year, which is agreement with in-situ and laboratory observations [START_REF] Mori | Interactions of nutrients, moisture and ph on microbial corrosion of concrete sewer pipes[END_REF][START_REF] Vincke | A new test procedure for biogenic sulfuric acid corrosion of concrete[END_REF][START_REF] Okabe | Succession of sulfur-oxidizing bacteria in the microbial community on corroding concrete in sewer systems[END_REF].

Conclusion

In this chapter, the experiments of sulfuric acid attack of cement mortar conducted by [START_REF] Kawai | Concrete deterioration caused by sulfuric acid attack[END_REF] was simulated. The evolution of solid compositions and porosity of materials were addressed. Moreover, the model has provided a prediction of the deterioration depth.

Exposed to high concentration of sulfuric acid (pH = 1), cementitious materials would be corroded heavily. In statical solution condition, a porous layer containing gypsum and silica gel can be found in both of experiments and simulations. In the flowing solution condition, no corrosion products layer exists due to the removing of corrosion products by the shear force of flowing solution. However, low concentration of sulfuric acid (pH = 2) cannot cause significant corrosion. Indeed the solubility product of H 2 SO 4 is so high that when pH is 2, too few acid penetrating into concrete is available to form gypsum. Therefore, sewer pipes will not be corroded unless enough H 2 SO 4 is produced by bacteria.

The degradation depth measured in the experiments after 90 days immersion test are close to our predictions. However, the modeled corrosion was faster than that measured in experiments during the early time. According to experimental results, the corrosion rate is quite slow during the early time. Such phenomenon draws attention to the effect of carbonation. We believe that carbonated layer slowed down the corrosion. By adjusting the porosity and initial solid compositions in the surface region of the sample, the effect of carbonated layer can be handled. Besides porosity, pH in carbonated layer is lower than that in the core of sample, which can affect the H 2 SO 4 flux into materials. Such influence should be investigated as well as the influence of porosity.

Furthermore, according to the simulation results, the degradation rate of samples with high porosity is greater than that with low porosity, thereby disagreeing with some experiment observations. Thus, more accurate estimation of some parameters (e.g., the precipitation kinetics of gypsum, the diffusion coefficient of sulfate ions) should be employed to investigate the possible reasons.

For the long-term prediction of corrosion depth, the thickness of the corroded layer increases linearly with the square root of time, which is typical of a diffusion-controlled acid attack profile. The long-term predicted corrosion rate is roughly in agreement with experimental observations. In the next chapter, a set of biogenic H 2 SO 4 attack experiments are simulated in order to take into account the bacterial activity producing sulfuric acid on the pipe surface.

Chapter 7

Simulation in the case of biofilms condition

Introduction

In order to predict the deterioration rate of concrete subjected to bio-degradation, laboratory experiments have been done to simulate the corrosion process of concrete in different environments [START_REF] Ehrich | Biogenic and chemical sulfuric acid corrosion of mortars[END_REF][START_REF] Vincke | Recent developments in research on biogenic sulfuric acid attack of concrete[END_REF]Gutiérrez-Padilla et al., 2010), as well as in-situ experiments [START_REF] Monteny | microbiological, and in situ test methods for biogenic sulfuric acid corrosion of concrete[END_REF][START_REF] Herisson | Toward an accelerated biodeterioration test to understand the behavior of Portland and calcium aluminate cementitious materials in sewer networks[END_REF].

Comparing with chemical corrosion by constant H 2 SO 4 solution, the supply of H 2 SO 4 during the biodeterioration process is neither given nor constant, but governed by SOB activity. Some laboratory experiments were carried out with mixed culture of several types of SOB containing elemental sulfur, biomass and other nutrient components [START_REF] Tichỳ | Possibilities for using biologicallyproduced sulphur for cultivation of thiobacilli with respect to bioleaching processes[END_REF]. The mixed cultures of SOB were produced from laboratory or directly obtained from the biofilms of corroded sewer pipes [START_REF] Vincke | Recent developments in research on biogenic sulfuric acid attack of concrete[END_REF][START_REF] De Muynck | Effectiveness of admixtures, surface treatments and antimicrobial compounds against biogenic sulfuric acid corrosion of concrete[END_REF].

After incubation in H 2 S atmosphere, the cementitious samples were immersed in the aqueous solution of the SOB suspension (see Fig. 7.1). The pH evolution in SOB suspension was measured during immersion.

In some other laboratory experiments, simulation reactors were built to simulate the environment in sewer pipes. In the simulation reactors, H 2 S gas and wastewater, which provides the strain of SOB, were circulated (see Fig. 7.2). The content of H 2 S gas was adjusted to different levels. The cementitious samples were exposed to H 2 S atmosphere rather than immersed in SOB suspension. Thus, H 2 SO 4 was produced from biofilms which formed on the sample surface. After a certain time, the corrosion state was illustrated by measurement of corrosion depth or weight loss.

In this chapter, the experiments of the immersion method [START_REF] De Muynck | Effectiveness of admixtures, surface treatments and antimicrobial compounds against biogenic sulfuric acid corrosion of concrete[END_REF] and the simulation [START_REF] Mori | Interactions of nutrients, moisture and ph on microbial corrosion of concrete sewer pipes[END_REF]; (b) [START_REF] Herisson | Biodétérioration des matériaux cimentaires dans les ouvrages d'assainissement: étude comparative du ciment d'aluminate de calcium et du ciment Portland[END_REF].

reactor method [START_REF] Mori | Interactions of nutrients, moisture and ph on microbial corrosion of concrete sewer pipes[END_REF]

Immersion test condition

To study the effectiveness of different admixtures and surface treatments against biogenic sulfuric acid attack of concrete, the biodeterioration of different types of mortar were measured by means of microbiological tests in lab [START_REF] De Muynck | Effectiveness of admixtures, surface treatments and antimicrobial compounds against biogenic sulfuric acid corrosion of concrete[END_REF], where the specimens were drilled from sewer pipe made with CEM I cement, whose W/C = 0.37. The specimens with polished surfaces were cylinders with H = 15 mm, D = 80 mm. In the experiments, the specimens with or without protection (e.g., epoxy coating, polyurea lining) were subjected to 8 cycles of accelerate test. Each cycle consisted of 4 steps:

(1) As pH of fresh concrete specimen was too high to incubate SOB, specimens were subjected to H 2 S-incubation in a vessel containing 200 ppm of H 2 S gas for 2 days at beginning;

(2) Immersion in a vessel containing 1.5 L of mixed cultures of SOB obtained from biofilms of insitu sewer pipe (medium composition: 10 g/L elemental sulfur, 3g/L KH 2 PO 4 , 0.1 g/L NH 4 Cl, 0.1 g/L MgCl•H2O) for 10 days;

(3) Immersion in a vessel containing distilled water for 2 days;

(4) Drying at room temperature for 1 day.

During the second step of every cycle, the pH in SOB suspensions was measured at a regular basis.

Since water was rotated by a rotary shaker at 90 rpm, the degradation products were removed in the third step. The last step could eliminate SOB remaining on the samples surfaces. The corrosion depth of the specimens at the end of each cycle was measured using the automated laser measurement system.

Since the cycles are equal to each other, we simulate the first and second step in this section separately, while the removing of corrosion products and SOB are neglected. To simulate the test, concrete samples with the same size as that in experiments are considered. Comparing with the mortar specimens in Chapter 6, the concrete specimens in this experiment contains coarse aggregates. However, the amount of coarse aggregates are unknown. From the visual observations (see Fig. 7.3) of experiments, it can be seen that the material loss was mainly attributable to the attack of the cement paste part. Thus, we neglect the corrosion of aggregate and assume that the hydration products in the samples are 70% of those in the samples of Chapter 6, i.e., 3.7 mol/L of portlandite and 3.5 mol/L of C-S-H as jennite. The initial porosity is 30%.

Neutralization process

To simulate the H 2 S attack, the samples are divided in 300 nodes. We consider the boundary conditions of constant H 2 S gas level (200 ppm), while no H 2 SO 4 is present. For numerical stability, the initial aqueous H 2 S concentration in concrete sample is 1×10 -9 mol/L, i.e., pH = 12.4. Boundary and initial conditions of our modeling are summarized in Table .7.1. The change in the concentration and volume of solid compounds, as well as porosity profiles after 1 days and 2 days of exposure are plotted in Fig. 7.6 and Fig. 7.7. After 1 day, the sample surface turns to a CaS-rich layer, where parts of CH and C-S-H are dissolved. Even if the molar volume of CaS is smaller than that of CH and C-S-H, the porosity decreases due to CaS accumulation in pores, which results from the diffusion of Ca 2+ as described in Section 5.5.1. Thus, the solid volume at the surface increases. For the position just under the CaS-rich layer, the porosity slightly increases due to the small molar volume of CaS and the solid volume of concrete sample is slightly less than initial value. After 2 days, the CaS-rich layer is thicker than that of 1 day exposure (see Fig. 7.7). More CaS precipitating deeper, causes the further reduction of porosity and the rising of solid volume of the CaS-rich layer.

Thus, about 0.1 mm of sample is neutralized by H 2 S. CH at surface is completely dissolved while C-S-H is just partly decalcified because of the lack of aqueous H 2 S. However, the amount of aqueous H 2 S at surface is enough to dissolve the CaS precipitation after 2 days of gas H 2 Sabsorption. Therefore, CaS at surface starts to dissolve.

To illustrate the dissolution of CaS, the concentration of CaS and aqueous H 2 S at surface versus exposure time are plotted in Fig. 7.8. When aqueous H 2 S is absorbed into concrete, CaS starts to precipitate until ρ H2S 0 increases to 7×10 -4.9 mol/L. When aqueous H 2 S is more than that critical value, CaS starts to dissolve.

Furthermore, a simulation of 10 days of exposure is conducted. According to the solid profiles in Fig. 7.9, the corroded sample starts with a completely degraded layer, where CH , CaS and C-S-H are fully dissolved.

Consequently, the porosity of the completely degraded layer is high, while that in the CaS-rich layer is below 0.1.

Evolution of pH in SOB suspensions

In the set of experiments conducted by [START_REF] De Muynck | Effectiveness of admixtures, surface treatments and antimicrobial compounds against biogenic sulfuric acid corrosion of concrete[END_REF], concrete samples with polyurethane lining were immersed into SOB suspension. However, no degradation could be observed for the specimens after 8 test cycles. Therefore, we consider that no mass exchange takes place between samples with polyurethane lining and SOB suspension. In such SOB suspension, the pH evolution is equal to that in pure SOB suspensions. Thus we simulate the SOB activity in a pure SOB suspension sample by the model of H 2 SO 4 production introduced in Section 2.3.1 (see Eq.(2.11) and (2.12)). The initial pH of SOB suspension is 7. H 2 SO 4 is produced in the SOB suspension is governed by Eq.(7.1) which is determined by fitting measurements as follows:

R H2SO4 =                          8 × 10 -7 mol/L • s 0.48 √ 2π × e - 1 0.96 × 10 2.5-pH pH > 2.5 8 × 10 -7 mol/L • s 0.48 √ 2π × e - (pH -1.5) 2 0.96 1.5 ≤ pH ≥ 2.5 8 × 10 -7 mol/L • s 0.48 √ 2π × e - (pH -1.5) 2 0.48 pH < 1.5 (7.1)
The pH evolution obtained from both simulation and experiment are plotted in Fig. 7.10. In the experiment, as a result of the conversion of elemental sulfur to sulfuric acid by cultures of NSOB, pH decreases slightly during early time. However, a sharp reduction of pH was observed after 4 days of submersion due to the activity of ASOB. With H 2 SO 4 production by ASOB whose activity is suppressed when pH is below 1.5, the pH continues to decrease to around 1 after 10 days of testing. The comparison between the results of model and the experiment results is fairly good.

Biodeterioration of concrete immersed in SOB suspension

To simulate the second step of the experiment, we consider the concrete sample with neutralized surface in contact with the SOB suspension where H 2 SO 4 is produced and penetrates into concrete. For now, we are not able to merge the process of H 2 S attack and H 2 SO 4 attack into one modeling. Thus, neutralized layer is considered as another material whose thickness and solid compositions are determined by the modeling of H 2 S attack. According to the simulation results of the fist step in the previous section, after 2 days of exposure to H 2 S gas of 200 ppm, the thickness of neutralized layer is 0.1 mm. For numerical stability, the initial porosity of neutralized layer is equalized to 0.15, and the pH value and CaS concentration in neutralized layer are simplified to vary linearly.

Therefore, the fresh concrete core, neutralized surface and SOB suspension are combined together as ρ H2SO4 = 10 -20.3 ρ H2SO4 = 10 -31.5 -10 -20.3 mol/L ρ H2SO4 = 10 -31.5 mol/L Calcium (Ca)

ζ Ca = 0 ζ Ca = 1 ζ Ca = 1 Silicon (Si) ζ Si = 0 ζ Si = 1 ζ Si = 1 Potassium (K) ρ K + =

Simulation reactor condition

In the real sewer pipes, heavy degradation was not observed in the part immersed in waste water but in the part above the water level [START_REF] Davis | Analysis of concrete from corroded sewer pipe[END_REF]. As the deterioration of concrete is a slow process with corrosion rate typically ranging from 1 mm/year to 5 mm/year [START_REF] Mori | Microbial corrosion of concrete sewer pipes, h2s production from sediments and determination of corrosion rate[END_REF], long time test which simulate the real deterioration conditions are required to investigate the durability of concrete in sewer environment. Furthermore, the biogenic H 2 SO 4 attack is a very complex process, involving microbiological factors (such as biomass and substrate available) and chemical factors (such as H 2 S gas level and solid compositions of concrete). Therefore, [START_REF] Mori | Interactions of nutrients, moisture and ph on microbial corrosion of concrete sewer pipes[END_REF]) designed a simulation chamber, where optimal conditions for the bacterial growth are controlled, to investigate the biodeterioration of concrete in sewer pipes. The schematic diagram of the simulation chamber is shown in Fig. 7.2a.

In the experiment conducted by [START_REF] Mori | Interactions of nutrients, moisture and ph on microbial corrosion of concrete sewer pipes[END_REF], the chamber consisted of a plastic box with size of 44 cm × 30 cm × 30 cm. The H 2 S concentration was 400 ppm by regulating the flow rate of 4% of Na 2 S solution and 1.5 N HCl solution. The mortar specimens with size of 4 cm × 4 cm × 16 cm were placed in sewage of 6 cm depth and distilled water as reference. During the corrosion test of 6 months, T hiobacillus thiooxidans isolated from corroded materials in the concrete sewer pipes were inoculated on the surface of the mortar specimen every 2 weeks for the first 2 months.

After about 6 months, the pH of the corrosion products were determined using a pH meter after 1 g of sample was suspended in 1 mL of distilled water. The corrosion depth at different places of mortar samples were measured. The place under the sewage level was not corroded, while the portion just above the sewage level was more corroded than portion further above the sewage level. No corrosion occurred when mortar specimens were placed in distilled water (see Fig. 7.16). [START_REF] Mori | Interactions of nutrients, moisture and ph on microbial corrosion of concrete sewer pipes[END_REF].

In this study, the corrosion process of the mortar near sewage level is simulated by our modeling.

According to the analysis of [START_REF] Mori | Interactions of nutrients, moisture and ph on microbial corrosion of concrete sewer pipes[END_REF], this location provided a constant supply of moisture and nutrients by splashing, immersion and/or capillary action from the water. In addition, dissolved oxygen from the atmosphere is provided at this location. Thus, we assume that the biofilm and mortar are both saturated and nutrients and oxygen are both sufficient for the growth of SOB. Therefore, modeling of H 2 SO 4 production with given H 2 S content in gas phase introduced in Section 2.3.2 is employed. Since the W/C ratio of mortar samples are 0.6, the solid compositions are assumed to be 3.7 mol/L of Portlandite and 3.5 mol/L of C-S-H as jennite. The initial porosity is 0.35. Moreover, SOB was incubated on the mortar surface during experiments. Thus, the succession of SOB is neglected and we assumed that SOB species in biofilm starts to produce H 2 SO 4 as long as the pH in mortar surface is reduced to 9.5 by H 2 S gas.

The surface neutralization process is simulated firstly as described in Section 7. Hereafter, H 2 S oxidation process is started by SOB in biofilms on the mortar surface. In this model, biofilms are considered covering on the neutralized surface. In the initial biofilms, we assume a presence of 0.005 mol/L of element sulfur, which is formed by the chemical oxidation of H 2 S during the neutralization process [START_REF] Joseph | Surface neutralization and h2s oxidation at early stages of sewer corrosion: Influence of temperature, relative humidity and h2s concentration[END_REF]. The initial and boundary conditions of the core part of mortar, neutralized surface and biofilm are considered as accumulation peak is present at 0.1 mm from the surface rather than at the top surface. Such difference is caused by the surface neutralization process. With a higher H 2 S gas level, more CaS dissolves. Therefore, very few calcium solid phase is present at the top surface at the initial time of H 2 SO 4 attack. In our modeling the material is initially at equilibrium condition. Thus less calcium solid means less concentration of Ca 2+ , which results in less gypsum accumulation. The experiments lasted for 180 days. Thus the maximum corrosion rate of the specimens exposed to sewage was calculated to be 6.1 mm/year. By our modeling long term prediction is possible. After 1 year

Effect of H 2 S gas level

The difference of H 2 S gas level in sewer pipes could influence both of surface neutralization process and H 2 SO 4 production in biofilms.

However, in our modeling we consider constant biomass in biofilms where SOB produce H 2 SO 4 at high rate. In fact, the activity and the amount of SOB is controlled by the content of H 2 S in biofilms. Due to the lack of quantitative relationship between the content of H 2 S and the activity of SOB, the effect of H 2 S gas level on the H 2 SO 4 production process is difficult to illustrate by our modeling.

For the surface neutralization process, the decrease of pH at surface is directly governed by H 2 S level.

Lower H 2 S level leads to longer lag time of biogenic H 2 SO 4 attack. For most of the in-situ conditions, corrosion materials are removed by flow of sewage periodically and new surface where pH is too high for SOB growth is exposed cyclically. Thus, the lag time is crucial for biodeterioration of sewer pipes. We conduct simulations of surface neutralization with different H 2 S gas level. We consider the lag time as the time of the decrease of surface pH from 12.4 to 9. The lag time of biogenic H 2 SO 4 attack with different H 2 S gas level is plotted in Fig. 7.24. From Fig. 7.24, it is clear that more time is needed to prepare the proper environment for SOB with less H 2 S gas. For the condition of 10 ppm, biofilms even could not form if the period of flow is less than 80 days. When the H 2 S gas reaches more than 100 ppm, SOB could start to grow on the surface in few days.

Conclusion

In the cases where the activity of SOB is involved, our modelling coupled the process of surface neutralization, the production of biogenic H 2 SO 4 and the deterioration of concrete. We modeled the laboratory experiments of biodeterioration of cementitous materials conducted by [START_REF] De Muynck | Effectiveness of admixtures, surface treatments and antimicrobial compounds against biogenic sulfuric acid corrosion of concrete[END_REF] and [START_REF] Mori | Interactions of nutrients, moisture and ph on microbial corrosion of concrete sewer pipes[END_REF]. The decrease of surface pH and the depth of neutralized layer during the stage of H 2 Sincubation were determined. For different test conditions, the production of H 2 SO 4 in SOB suspension and biofilms were calculated by different methods. With the amount of biogenic H 2 SO 4 , the evolution of solid compositions and porosity of materials were addressed. Like the modeling of chemical H 2 SO 4 attack, the modeling of biogenic H 2 SO 4 attack provides a prediction of the deterioration depth of materials.

Exposed to the gas with high H 2 S level, concrete surface would absorb H 2 S gas into pore solution very fast. Consequently, Portlandite and C-S-H are dissolved by aqueous H 2 S and CaS precipitates. Thus, the alkalinity of concrete surface is neutralized in few days, which is called lag time in this paper. For the same type of concrete, the lag time is controlled by the H 2 S gas level and temperature, which vary in sewer pipes. The effect of H 2 S level on the surface neutralization process is studied by our modeling as well.

For the immersion test condition, H 2 SO 4 is produced in SOB suspension where the activity of SOB is governed by pH. pH in SOB suspension decreases very few until 4 days. However, during the later time of immersion, pH in SOB suspension is reduced to 1 which could result in heavily corrosion of concrete. After immersion of 10 days, the deterioration depth is about 0.22 mm. The predicted corrosion rate of each cycle coincides with that of experimental measurements.

For the simulation reactor condition, the production of H 2 SO 4 is calculated by the oxidation of H 2 S in biofilms. Due to the continuously supply of SOB and high H 2 S gas level, H 2 SO 4 is produced quite fast from the beginning. Since the oxidation rate of element S is lower than that of H 2 S, element S accumulates in biofilms. After simulation of 1 year, the predicted corrosion depth reaches about 6.8 mm which agrees the experimental results.

However, the SOB activity in our modeling is still relatively crude. Further research is needed to get the knowledge of the exact SOB ecology supporting long-term H 2 SO 4 production and the growth of the different types of SOB.

Chapter 8

Conclusions and further discussion

Summary of the thesis

In order to assess the deterioration process of sewer pipes, a reactive transport model is built. This model could be used to predict the deterioration depth and solid compounds of cementitious materials in contact with sulfuric acid solution or sulfur-oxidizing bacteria (SOB). In this study, the biodeterioration of concrete are divided in three parts: the neutralization of concrete surface, the production of sulfuric acid by SOB and H 2 SO 4 attack of concrete.

-The neutralization of concrete surface is caused by H 2 S attack of concrete. The model of this attack developed in the thesis considers the absorption of H 2 S gas in pore solution of concrete and the chemical reactions between aqueous H 2 S and concrete. Thus, the times of neutralization and the thickness of neutralized layer of concrete under different H 2 S levels can be calculated.

-For the production of H 2 SO 4 by SOB, the quantitative understanding is still very poor. In this study, the production rate of H 2 SO 4 under different conditions are calculated by fitting experimental measurements or WATS model. Thus, the amount of H 2 SO 4 which could deteriorate concrete can be determined.

-Dealing with H 2 SO 4 attack process, the dissolution of portlandite and C-S-H gel, and the precipitation of gypsum are considered. Due to the swelling of gypsum, a simplified damage model is employed to calculate the porosity change during degradation. Thus, the solid composites evolution and deterioration depth of concrete can be predicted.

The three parts of biodeterioration of concrete were programmed as a coupled process thanks to Bil platform. Thus this platform is able to simulate the whole process of biodeterioration of concrete. Some experiments were simulated by this model and the simulation results have been compared with experimental observations.

In the modelling, we consider homogeneous chemical reactions in the pore solution. The equilibrium of different solid phases (i.e., portlandite, C-S-H gel, gypsum and calcium sulfide) and aqueous species in concrete are considered as well. A generalization of the mass action law is used to describe the decalcification of C-S-H. This theory encompasses the solid solution model and is easy to implement in transport-reaction modeling. The diffusion of aqueous species is introduced into the model by Nernst-Planck equation, and the numerical method couples the transport equations and the chemical reactions.

Precipitation and dissolution reactions could result in a porosity increase or drop, due to the volume difference between cement hydrates and corrosion products. Thus, the changes in porosity have also been taken into account by the balance of volume on the solid phases and a model was developed to handle the swelling of gypsum induced by the damage of concrete. During H 2 S attack, the smaller molar volume and the dissolution of CaS cause the reduction of solid volume which is responsible for the increase of porosity. During the early time of H 2 SO 4 attack, the porosity will decrease as the formation of gypsum, whose molar volume is bigger than that of CH. Then the accumulation of gypsum could fill pores and increase the inner pressure. When the inner pressure exceeds the strength of concrete, pore structure fails and porosity increases. The formed gypsum could dissolve under peculiar boundary conditions and cause a further increase in porosity.

Thus, the model can predict the chemical composites and corrosion depth of cementitious materials immersed in static or flowing H 2 SO 4 solution with different pH. By changes in the porosity of surface layer, some differences between experimental observations and simulation results can be explained.

Not only chemical H 2 SO 4 attack, but also biogenic H 2 SO 4 attack can be simulated by this modelling.

We can deal with the biodeterioration of concrete in contact with two different SOB media: SOB suspension where the evolution of pH is given and biofilm where the type of SOB and H 2 S gas level are given. For the SOB suspension condition, H 2 SO 4 production rate is modeled by a function of pH to fit the measurement of pH evolution in suspension. The succession and activity of SOB in different pH range are considered.

For the biofilm, the amount and activity of SOB are constant. By WAST model [START_REF] Jensen | Modeling of hydrogen sulfide oxidation in concrete corrosion products from sewer pipes[END_REF], the oxidation rate of H 2 S and that of element S are governed by functions of concentration of H 2 S and S respectively.

In this study, this model is applied to simulate the experimental works on the neutralization of concrete surface [START_REF] Joseph | Surface neutralization and h2s oxidation at early stages of sewer corrosion: Influence of temperature, relative humidity and h2s concentration[END_REF], chemical H 2 SO 4 attack [START_REF] Kawai | Concrete deterioration caused by sulfuric acid attack[END_REF] and biogenic H 2 SO 4 attack [START_REF] De Muynck | Effectiveness of admixtures, surface treatments and antimicrobial compounds against biogenic sulfuric acid corrosion of concrete[END_REF][START_REF] Mori | Interactions of nutrients, moisture and ph on microbial corrosion of concrete sewer pipes[END_REF].

For H 2 S attack, a CaS-rich layer where CH and C-S-H are dissolved can be observed. When pH at the surface decreases to 9, SOB could start to grow. Then the width of neutralized layer is about 0.1 mm. The accumulation of CaS could reduce the porosity at surface to protect inner part of the sample from aqueous H 2 S penetration. However, if there is enough H 2 S the accumulated CaS would be dissolved. We consider the time for pH at surface to decrease to 9 as the neutralization time which is governed by the H 2 S gas level. When the amount of H 2 S gas is higher than 100 ppm, H 2 S is absorbed into pore solution of concrete quite fast and pH at concrete surface could be reduced from about 12.4 to 9 in 2 days. While for H 2 S gas less than 50 ppm, this time could be more than 1 month. Thus, the neutralization time is crucial for biodeterioration of sewer pipes. By our model, the proper time of H 2 S incubation step of accelerate test can be determined.

When immersed in a static H 2 SO 4 solution, a porous layer containing gypsum and silica gel can by observed at the surface of cement paste sample. Meanwhile, the C-S-H decalcification zone and CH dissolution zone are observed between the surface and the uncorroded zone. The diffusion of Ca 2+ from the CH dissolution front to the C SH2 precipitation front results in the accumulation of C SH2 at the surface. Hence the volume of gypsum-rich layer expands a lot due to the accumulation of gypsum whose molar volume is large. Due to the inner pressure caused by the precipitation of gypsum, damage of materials happens in the overlap region of gypsum-rich zone and dissolution zone. Thus, the deterioration depth is defined as the distance between the initial surface and the damage place of concrete sample.

In the flowing solution condition, the effect of flowing solution is modeled by the leaching effect which removes the corrosion products. We consider that the external solution is free of calcium and silicon.

Consequently, calcium and silicon released from the dissolution front leave the material as a result of concentration gradient. Thus, no significant gypsum layer can be observed at the surface where cement paste is fully degraded. Compared to static solution condition, sample is corroded much heavier when it is immersed in flowing solution which exposes more reactive surface for H 2 SO 4 attack. Therefore, during accelerate test the intensity of brush or flush is an important factor which should be controlled.

H 2 SO 4 attack of concrete is very sensitive to the concentration of H 2 SO 4 . For the solution with lower H 2 SO 4 concentration, less acid penetrating into concrete is available to form gypsum. Thus the cement paste is corroded slower. Long-term simulation showed that the corrosion rate of concrete under the condition of pH = 2 is about 2 time less than that of pH = 1. The corrosion depth increases linearly with the square root of time. Hence the environmental acidity in sewer pipes is a key factor which influence the service life.

For biogenic H 2 SO 4 attack, the activity of SOB and the cementitious material samples with neutralized layer are involved. For the immersion test condition, pH in SOB suspension decreases very few during early time due to the slow H 2 SO 4 production rate of NSOB (Neutrophilic Sulfur-Oxidizing Bacteria). Due to the alkalinity of concrete, pH in the SOB suspension near the concrete surface even increases at first.

During the later time of immersion, ASOB (Acidophilic Sulfur-Oxidizing Bacteria) rather than NSOB is producing H 2 SO 4 with a much higher rate. For the simulation reactor condition, H 2 S is absorbed into biofilm where oxidation of H 2 S and element S are conducted by SOB. Since the oxidation rate of element S is lower than that of H 2 S, element S accumulates in the biofilm. Due to the continuous supply of SOB and H 2 S, H 2 SO 4 is produced fast from the beginning. Hence pH in SOB suspensions or biofilms could be reduced to 1-2 which results in heavy corrosion of concrete. Therefore, applying antibacterial materials in sewer pipes (e.g., calcium aluminate cement) is a method to restrict the activity and amount of SOB in biofilm [START_REF] Alexander | Performance of sewer pipe concrete mixtures with portland and calcium aluminate cements subject to mineral and biogenic acid attack[END_REF][START_REF] Herisson | Biodétérioration des matériaux cimentaires dans les ouvrages d'assainissement: étude comparative du ciment d'aluminate de calcium et du ciment Portland[END_REF].

Under different conditions of biogenic H 2 SO 4 attack, similar zones as those in chemical H 2 SO 4 attack condition are observed. Not only neutralized layer, but also concrete core is degraded by biogenic H 2 SO 4 .

Due to the diffusion of calcium towards the gypsum precipitation front, lots of gypsum accumulates at concrete surface.

By adjusting the porosity and initial solid compositions in the surface region of the sample, the effect of carbonated layer is investigated. Due to calcite accumulation at surface, the carbonated layer where porosity is smaller than that of concrete core [START_REF] Shen | Reactive transport modeling of CO2 through cementitious materials under CO2 geological storage conditions[END_REF] protects the concrete core from H 2 SO 4 penetration. In this high density layer, the H 2 SO 4 penetration is slowed down and the reactions are restricted. Thus the corrosion rate is quite slow during the early time.

Further discussion and perspective

This model is able to predict the corrosion depth and service life of OPC concrete sewer pipes subjected to biogenic H 2 SO 4 attack. The change of solid compositions of concrete during biodeterioration can be calculated. The influence of environmental and chemical factors can be indicated by this model. However, there are still some developments could be conducted to improve the understanding of biodeterioration of concrete. Some modifies of modeling should be given priority for more accurate estimation:

-In this study, we assume infinite dilution approximation for aqueous species. Thus, the activity of each aqueous species is equal to the molar concentration and that of solid component equals 1. Due to the high ionization constant, this hypothesis is acceptable for H 2 SO 4 . However, for high H 2 S gas level such approximation may be criticized since the concentrations of the aqueous species could become quite high.

An improved model considering the effects of the chemical activity could be developed in a future work.

-For now, we cannot address the coupling of H 2 S attack and H 2 SO 4 attack in one step. Thus, the effect of neutralization process is taken into account by considering the sample as fresh concrete with a neutralized layer whose properties are determined by a separate simulation of H 2 S attack. Continuously simulation of the whole process of deterioration should be achieved in future work.

Due to the complex of biodeterioration process, we make some simplifying assumptions which should be reconsidered in future works:

-For the neutralization process, the absorption of H 2 S gas is calculated by the empirical equation of [START_REF] Nielsen | Effect of sewer headspace air-flow on hydrogen sulfide removal by corroding concrete surfaces[END_REF] rather than Henry's law which is commonly used to describe the solubility of a gas in a liquid. The exact mechanism of H 2 S absorption in sewer pipes should be studied. The temperature dependence is neglected in the absorption of H 2 S gas in this study. According to [START_REF] Vollertsen | Corrosion of concrete sewers-the kinetics of hydrogen sulfide oxidation[END_REF], higher temperature accelerates H 2 S absorption. Such phenomena was tested by the experimental works of [START_REF] Joseph | Surface neutralization and h2s oxidation at early stages of sewer corrosion: Influence of temperature, relative humidity and h2s concentration[END_REF]. Moreover, chemical oxidation of sulfide to element S may occur during the neutralization process [START_REF] Bagreev | On the mechanism of hydrogen sulfide removal from moist air on catalytic carbonaceous adsorbents[END_REF]. Such reaction can provide the source of S for biofilm formation during early time of biodeterioration. Thus, it is necessary to consider chemical oxidation of sulfide in future modelling.

-For the failure of pore structure, we applied a simplified damage model. To make a more precise prediction, a more detailed description of the physical behaviour of gypsum precipitation may also be implemented. For example, according to [START_REF] Azam | Study on the geological and engineering aspects of anhydrite/gypsum transition in the arabian gulf coastal deposits[END_REF], for gypsum gel the relationship between log of pressure and reduction of porosity is linear. Furthermore, as more and more cement hydrates dissolve, the cementitious materials is soften gradually, while in this study the materials is soften abruptly. Thus the damage model part should be modified.

Due to the lack of understanding about the mechanism of H 2 SO 4 attack of concrete, some experimental observation can not be explained by this model:

-According to some experimental observations of H 2 SO 4 attack [START_REF] Kawai | Concrete deterioration caused by sulfuric acid attack[END_REF][START_REF] Israel | Acid attack on pore-reduced cements[END_REF][START_REF] Fattuhi | The performance of cement paste and concrete subjected to sulphuric acid attack[END_REF], the corrosion rate of concrete with low W/C ratio is greater than that with high W/C ratio, thereby disagreeing with our simulation results. In our model, the corrosion process is diffusion controlled rather than reaction controlled which is claimed by [START_REF] Ueda | Deterioration mechanism of hardened cement paste by various acids[END_REF]. Furthermore, diffusion coefficient of some species (e.g., SO 2- 4 ) in concrete is not determined by measurement. Therefore, the mechanism of H 2 SO 4 attack should be investigated further. And more accurate parameters should be collected.

To refine the model, more experimental results should be collected:

-To calculate the production of H 2 SO 4 by SOB media, we employ a regression equation which is only suitable for specific experiment condition and empirical model where the activity of SOB is constant.

Dealing with such a complex process, lots of work could be conducted to dispose of a more accurate description of biodeterioration of sewer pipes. Further experimental study could be done to improve the understanding of the biogenic oxidation of sulfide and get more quantitative information about the effect of microbiological/chemical factors (such as biomass, pH, temperature) on the production of H 2 SO 4 .

-To validate the modelling, the corrosion depth of simulation results is compared to experimental results. Since change of solid compositions can be calculated by the model, some micro measurements of the corroded layer, such as the solid phases profile and the porosity evolution of the samples, should be conducted and compared with simulation results.

Moreover, the model could be used to describe the biodeterioration process of other types of concrete:

-The chemistry process of the biodeterioration of OPC concrete and that of concrete made by other blended cement with additions (like slag, fly ash and silica fume) is similar. Thus modelling of the biodeterioration of other type of cementitious materials can be achieved by changing the content of CH and C-S-H and the initial porosity. To provide assist in rational choice of adapted materials for sewage environment, such study will be worth in future.

K i = Q i a i (A.5)
where

a i = λ i X i (A.6)
In Eqs. (A.5) and (A.6), K i denotes the equilibrium constant. Q i is the activity product. a i is the chemical activity. X i is the mole fraction of the component i within the solid solution and λ i is the activity coefficient. For a pure end-member, a i equals to 1, while in the framework of solid solution, a i follows Eq.

(A.6). Thus, for each end-member, the equilibrium between a solid solution and an aqueous solution can be written as

Q i K i = λ i X i (A.7)
For an ideal solid solution, the chemical activity a i equals to the mole fraction X i , i.e., λ i =1. While in the non-ideal solid solution, this two values are different. To accurately represent the C-S-H as a solid solution, the end-members and the stoichiometric coefficients should be carefully chosen. Several possible values are reported in the literature. [START_REF] Atkinson | Aqueous chemistry and thermodynamic modelling of cao-sio2-h2o gels[END_REF] used two non-ideal solid solutions to describe the behavior of C-S-H, based on experimental data from [START_REF] Greenberg | Investigation of Colloidal Hydrated Calcium Silicates: Pt. II. Solubility Relationships in the Calcium Oxide-silica-water System at 259[END_REF]. For Ca/Si≤0.8, end-members were chosen as S -nC•S•mH (amorphous silica -"tobermorite") and for Ca/Si>0.8, endmembers were chosen as nC•S•mH -CH ("tobermorite" -portlandite), where n = 0.833 and m = 0.917. For

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 A sample of the concrete before (A) and after (B) eight cycles of the microbiological test. A loss of cement paste is clearly visible, while the aggregates remain relatively unaffected. (De Muynck et al., 2009). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 The calculated absorption of H 2 S and pH evolution profiles from 0 days to 2 days: (a) Aqueous H 2 S concentration; (b) pH value. (H 2 S level = 200 ppm) . . . . . . . . . . . . . . 7.5 The calculated absorption of H 2 S profiles from 0 days to 2 days: (a) Aqueous H 2 S concentration and pH evolution at surface; (b) Diffusion of sulfur. (H 2 S level = 200 ppm) . . . . . 7.6 After 1 day of exposure: (a) Solid concentration and porosity ; (b) Solid volume and Ca/Si ratio of C-S-H. (H 2 S level = 200 ppm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 After 2 days of exposure: (a) Solid concentration and porosity ; (b) Solid volume and Ca/Si ratio of C-S-H. (H 2 S level = 200 ppm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 The concentration of CaS and aqueous H 2 S at surface from 0 days to 2 days (H 2 S level =200 ppm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 After 10 days of exposure: (a) Solid concentration and porosity ; (b) Solid volume and Ca/Si ratio of C-S-H. (H 2 S level = 200 ppm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10 The evolution of pH in pure SOB suspension from 0 days to 10 days . . . . . . . . . . . . . 7.11 Initial concrete sample with neutralized surface: (a) pH and porosity ; (b) Solid volume and Ca/Si ratio of C-S-H. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.12 The calculated H 2 SO 4 and pH evolution profiles in SOB suspension and concrete from 0 days to 10 days: (a) H 2 SO 4 concentration; (b) pH value. . . . . . . . . . . . . . . . . . . . . 7.13 The pH evolution and production rate of H 2 SO 4 from 0 to 10 days: (a) bulk SOB suspension; (b) SOB closed to concrete surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.14 After 10 days of immersion in SOB suspension: (a) Solid concentration and porosity ; (b) Solid volume and Ca/Si ratio of C-S-H. D Simulation represents the degradation depth of the sample in simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.15 The predicted and measured corrosion depth. . . . . . . . . . . . . . . . . . . . . . . . . . . 7.16 Mortar specimens exposed to H 2 S gas: (a) Water; (b) Sewage.

  When surface pH decreases to 9.5: (a) Aqueous H 2 S concentration and pH evolution at surface; (b) Solid concentration and porosity of mortar sample. (H 2 S level = 400 ppm) . . . 7.18 The oxidation of H 2 S and S versus their concentration in the biofilms: (a) Aqueous H 2 S ; (b) Element S. (H 2 S level = 400 ppm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.19 The calculated pH evolution profiles (a) pH evolution in biofilms ; (b) pH change in biofilms and mortar. (H 2 S level = 400 ppm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.20 After 40 days of test in simulation reactor: (a) Solid concentration and porosity ; (b) Solid volume and Ca/Si ratio of C-S-H. (H 2 S level = 400 ppm) . . . . . . . . . . . . . . . . . . . 7.21 After 80 days of test in simulation reactor: (a) Solid concentration and porosity ; (b) Solid volume and Ca/Si ratio of C-S-H. (H 2 S level = 400 ppm) . . . . . . . . . . . . . . . . . . . 7.22 After 180 days of test in simulation reactor: (a) Solid concentration and porosity ; (b) Solid volume and Ca/Si ratio of C-S-H. (H 2 S level = 400 ppm) . . . . . . . . . . . . . . . . . . . 7.23 After 1 year of test in simulation reactor: (a) Solid concentration and porosity ; (b) Solid volume and Ca/Si ratio of C-S-H. (H 2 S level = 400 ppm) . . . . . . . . . . . . . . . . . . . 7.24 The lag time of surface neutralization process with different H 2 S level. . . . . . . . . . . . . A.1 Stability domains of jennite and amorphous silica gel. . . . . . . . . . . . . . . . . . . . . . A.2 Stability domains of jennite, tobermorite and amorphous silica gel. . . . . . . . . . . . . . . A.3 Mole fraction of different poles and Ca/Si evolution vs. q CH . . . . . . . . . . . . . . . . . . .

Figure

  Figure 1.1: Examples of biodeterioration

1.

  Cement chemistry notation is used throughout the paper: C = CaO, S = SiO 2 , H = H 2 O, S = SO 3 .

Figure 2

 2 Figure 2.4: Decrease in surface pH (pH (initial) -pH (exposed) ) after 1 year.

Figure 2

 2 Figure 2.9: Oxidation rate : (a) H 2 S ; (b) element S.

Figure 2

 2 Figure 2.10: The pH evolution in pure SOB media with different H 2 S level.

Figure 3 . 1 :

 31 Figure 3.1: Stability domains of portlandite and gypsum

  Fig.3.3, we plot Q SHt K SHt obtained from Eq.(3.26), and that from the Greenberg's results. To perform the latter calculation, the ion activity product of silica gel, Q SHt , is approximated by the concentration of the monosilicic acid H 4 SiO 0 4 . The solubility constant, K SHt , is 1.93 mM. The model results fit the the experiments fairly well. Furthermore, this theory encompasses the solid solution model involving any number of poles (detailed in Appendix A).

Figure 3 . 4 :Figure 3

 343 Figure 3.4: Concentration of H 2 S species and pH vs. time in REV of cement paste

Fig. 3 .Figure 3 . 6 :Figure 3

 3363 Fig.3.7 profiles the evolution of solid compositions versus H 2 SO 4 concentration. Due to the sufficient supply of H + , Ca/Si ratio of C-S-H decreases faster than that during H 2 S attack. Thanks to the stability of gypsum, at normal atmospheric temperature gypsum keeps depositing in pores rather than dissolving in H 2 SO 4 solution.

  (4.3) where, z Jen = 2.6 and z Tob = 1.83 represent the H 2 O/Si of Jennite and Tobermorite respectively, and t = 2. The change of V C-S-H and H 2 O/Si ratio with the change of Ca/Si ratio is shown in Fig.4.1.

Figure 4

 4 Figure 4.1: V C-S-H and H 2 O/Si ratio z vs. Ca/Si ratio x.

Figure 4

 4 Figure 4.2: (a) OPC Mortar after 350 Days of biogenic sulfuric acid corrosion in simulation chamber.Samples on left side is reference sample,whereas on right side sample exposed to corrosion condition.[START_REF] Ehrich | Biogenic and chemical sulfuric acid corrosion of mortars[END_REF]; (b) The appearance of concrete coupons exposed to the sewer atmosphere for 1 year.[START_REF] Okabe | Succession of sulfur-oxidizing bacteria in the microbial community on corroding concrete in sewer systems[END_REF]

Figure 4

 4 Figure 4.3: Schematics of the microstructure of cementitous materials with uncompressed gypsum inclusion: (a) Through-solution mechanism (Beddoe and Dorner, 2005); (b) Deposition in pores mechanism[START_REF] Kawai | Concrete deterioration caused by sulfuric acid attack[END_REF]

  for different conditions. The schematic figures of three conditions are shown in Fig.4.4,Fig.4.5 and Fig.4.8. 4.4 Porosity change during H 2 SO 4 attack 4.4.

Figure 4

 4 Figure 4.4: Schematics of the microstructure of cementitous materials with uncompressed gypsum inclusion

Figure 4

 4 Figure 4.5: Schematics of the microstructure of cementitous materials with compressed gypsum inclusion

  Figure 4.6: Schematics of the relationship between the saturation degree of gypsum and inner pressure.

Figure 4

 4 Figure 4.7: Elastic-Brittle behaviour of cementitious materials

Figure 4

 4 Figure 4.8: Schematics of the microstructure of damaged cementitous materials

Figure 4

 4 Figure 4.9: Evolution of the solid volume and porosity vs. H 2 S concentration in REV of cement paste

  Figure 4.10: Evolution of the solid volume and porosity vs. H 2 SO 4 concentration in REV of cement paste

Figure 4 .

 4 Figure 4.11: Evolution of inner pressure vs. content of CH and C SH2 in REV of cement paste

Figure 5 Figure 5

 55 Figure 5.1: H 2 SO 4 concentration profiles from 0 day to 60 days.

  Figure 5.3: Solid profiles after 15 days immersion

Figure 5 . 4 :

 54 Figure 5.4: Evolution of porosity and solid compositions of material versus immersion time.

H 2 Figure 5 Figure 5

 255 Figure 5.5: CH and C SH2 content, ζ Ca and log(z H2SO4 ) after 15 days immersion

Figure 5 Figure 5 4Figure 5

 555 Figure5.9: V C-S-H of C-S-H and log(q CH ) after 15 days immersion

FigureFigure 5

 5 Figure 5.12: Flow of aqueous species across the interface at 1 day

Figure 6

 6 Figure 6.1: After 90 days immersion test: (a) Observation of deterioration of sample; (b) XRD patterns of surface area of sample. (Method 1, pH = 1)[START_REF] Kawai | Concrete deterioration caused by sulfuric acid attack[END_REF]

Figure 6

 6 Figure 6.2: The calculated penetration of H 2 SO 4 and pH evolution profiles from 0 days to 90 days: (a) H 2 SO 4 concentration; (b) pH value. (Method 1, pH = 1)

Figure 6 Figure 6 Figure 6 Figure 6 Figure 6

 66666 Figure 6.3: After 10 days of immersion: (a) Solid concentration and porosity ; (b) Solid volume and Ca/Si ratio of C-S-H . D Simulation and D Experiment represent the degradation depth of the sample in simulation and experiment individually. (Method 1, pH = 1)

Figure 6

 6 Figure 6.8: After 90 days immersion test: (a) Observation of deterioration of sample; (b) XRD patterns of surface area of sample. (Method 2, pH = 1)[START_REF] Kawai | Concrete deterioration caused by sulfuric acid attack[END_REF]

Figure 6

 6 Figure 6.9: The calculated penetration of H 2 SO 4 and pH evolution profiles from 0 days to 90 days: (a) H 2 SO 4 concentration; (b) pH value. (Method 2, pH = 1)

Figure 6 Figure 6 Figure 6

 666 Figure 6.10: After 10 days of immersion: (a) Solid concentration and porosity ; (b) Solid volume and Ca/Si ratio of C-S-H . D Simulation and D Experiment represent the degradation depth of the sample in simulation and experiment individually. (Method 2, pH = 1)

Figure 6 Figure 6

 66 Figure 6.13: The calculated Ca 2+ concentration profiles during immersion: (a) Method 1 ; (b) Method 2 . (pH = 1)

Figure 6 Figure 6

 66 Figure 6.15: The predicted evolution of porosity and corrosion depth profiles from 0 days to 90 days: (a) Porosity; (b) Corrosion depth. (Method 2, pH = 1)

Figure 6 Figure 6

 66 Figure 6.17: The calculated penetration of H 2 SO 4 with different pH: (a) Method 1; (b) Method 2.

Figure 6 Figure 6 Figure 6

 666 Figure 6.19: The evolution of solid volume, Ca/Si ration of C-S-H and porosity after 10 days immersion : (a) Method 1; (b) Method 2. (pH = 2)

Figure 6

 6 Figure 6.22: The concentration of HSO - 4 and SO 2- 4 after 90 days immersion with different pH: (a) Method 1; (b) Method 2.

Figure 6 Figure 6

 66 Figure 6.23: The flux of sulfur after 90 days immersion with different pH: (a) Method 1; (b) Method 2.

Figure 6 Figure 6 Figure 6

 666 Figure 6.25: The evolution of pH from 10 days to 90 days for the sample with carbonated layer: (a) Method 1; (b) Method 2. (pH = 1)

Figure 6

 6 Figure 6.28: Solid compositions, Ca/Si ratio of C-S-H and porosity profiles after 30 days of immersion under high H 2 SO 4 concentration condition (pH = 0.5): (a) W/C = 0.35; (b) W/C = 0.5

Figure 6

 6 Figure 6.29: Solid compositions, Ca/Si ratio of C-S-H and porosity profiles after 30 days of immersion high H 2 SO 4 concentration condition (pH = 2): (a) initial porosity = 0.3; (b) initial porosity = 0.4

Figure 6

 6 Figure 6.30: Prediction of degradation depth of long exposure time.

Figure 7

 7 Figure 7.1: Experiment setup of immersion test of biodeterioration: (a) (Vincke et al., 2000); (b) (De Belie et al., 2004).

Figure 7

 7 Figure 7.2: Experiment setup of biodeterioration by simulation reactors: (a)[START_REF] Mori | Interactions of nutrients, moisture and ph on microbial corrosion of concrete sewer pipes[END_REF]; (b)[START_REF] Herisson | Biodétérioration des matériaux cimentaires dans les ouvrages d'assainissement: étude comparative du ciment d'aluminate de calcium et du ciment Portland[END_REF].

Figure 7

 7 Figure 7.3: A sample of the concrete before (A) and after (B) eight cycles of the microbiological test. A loss of cement paste is clearly visible, while the aggregates remain relatively unaffected. (De Muynck et al., 2009).

Figure 7 Figure 7 . 5 :

 775 Figure 7.4: The calculated absorption of H 2 S and pH evolution profiles from 0 days to 2 days: (a) Aqueous H 2 S concentration; (b) pH value. (H 2 S level = 200 ppm)

Figure 7

 7 Figure 7.6: After 1 day of exposure: (a) Solid concentration and porosity ; (b) Solid volume and Ca/Si ratio of C-S-H. (H 2 S level = 200 ppm)

Figure 7 Figure 7 . 8 :Figure 7

 7787 Figure 7.7: After 2 days of exposure: (a) Solid concentration and porosity ; (b) Solid volume and Ca/Si ratio of C-S-H. (H 2 S level = 200 ppm)

Figure 7

 7 Figure 7.10: The evolution of pH in pure SOB suspension from 0 days to 10 days

Figure 7

 7 Figure 7.11: Initial concrete sample with neutralized surface: (a) pH and porosity ; (b) Solid volume and Ca/Si ratio of C-S-H.

Figure 7 Figure 7

 77 Figure 7.14: After 10 days of immersion in SOB suspension: (a) Solid concentration and porosity ; (b) Solid volume and Ca/Si ratio of C-S-H. D Simulation represents the degradation depth of the sample in simulation

Figure 7

 7 Figure 7.16: Mortar specimens exposed to H 2 S gas: (a) Water; (b) Sewage.[START_REF] Mori | Interactions of nutrients, moisture and ph on microbial corrosion of concrete sewer pipes[END_REF].

pHFigure 7

 7 Figure 7.17: When surface pH decreases to 9.5: (a) Aqueous H 2 S concentration and pH evolution at surface; (b) Solid concentration and porosity of mortar sample. (H 2 S level = 400 ppm)

Figure 7 .Figure 7 Figure 7

 777 Figure 7.18: The oxidation of H 2 S and S versus their concentration in the biofilms: (a) Aqueous H 2 S ; (b) Element S. (H 2 S level = 400 ppm)

Figure 7

 7 Figure 7.21: After 80 days of test in simulation reactor: (a) Solid concentration and porosity ; (b) Solid volume and Ca/Si ratio of C-S-H. (H 2 S level = 400 ppm)

Figure 7 Figure 7

 77 Figure 7.22: After 180 days of test in simulation reactor: (a) Solid concentration and porosity ; (b) Solid volume and Ca/Si ratio of C-S-H. (H 2 S level = 400 ppm)

  Figure 7.24: The lag time of surface neutralization process with different H 2 S level.

Figure

  Figure A.2: Stability domains of jennite, tobermorite and amorphous silica gel.

  2 S in atmosphere, pH are considered. The modeling of H 2 SO 4 production in biofilm is described in detail. The inhibition/promotion of H 2 SO 4 production by the change of pH during the microbial metabolic process is introduced in the modeling. The H 2 SO 4 production modeling is verified at the end of Chapter 2.After H 2 SO 4 production is described, the deterioration of cementitious materials subjected to H 2 SO 4 attack is taken into account. Chapter 3 introduces the major chemical reactions taking place during H 2 SO 4 attack. Since the cementitious minerals are expected to completely dissolve, dissolution and precipitation reactions of CH and gypsum (C SH2) are described by mass action laws and threshold of ion activity

In Chapter 2, some details of the mechanisms of H 2 SO 4 production in the biofilm are presented. The influence of environmental factors, such as temperature, concentration of H

SO 4 production rate and pH 2.3.2 Modeling of H 2 SO 4 production with given H 2 S content in gas phase

  NSOB , H 2 SO 4 is produced by NSOB, whose production rate is quite low. ASOB dominating step: When pH in SOB media is lower than the value of pH NSOMB , H 2 SO 4 is produced by ASOB rather than NSOB which can't survive in acidic environment. The production rate of H 2 SO 4 keeps increasing until pH reaches pH ASOB , where the metabolic activity of ASOB reaches peak value. With pH below pH ASOB , the metabolic activity of ASOB is limited. Consequently, H 2 SO 4 production rate is

	NSOB : R H2SO4-NSOB =	σ	α 0 √ 2π	e	-	(pH NSOB -pH ASOB ) 2 2σ	10 pH NSOB -pH	(2.11)
	ASOB : R H2SO4-ASOB =	σ	α 0 √ 2π	e	-	(pH -pH ASOB ) 2 2σ	(2.12)
	where, α 0 and σ are constants depending on SOB numbers and activity. In this paper, the value of α 0
	and σ are determined by fitting experimental measurement. To simulate the experiments conducted by
	(De Muynck et al., 2009), which will be introduced in Chapter 7, we set pH NSOB = 2.5 and pH ASOB = 1.5.
	When pH > pH ASOB , α 0 = 8 ×10 -7 mol/L•s, σ = 0.48; while α 0 = 4 ×10 -7 mol/L•s, σ = 0.24, when pH <
	pH ASOB .
	The relationship between H 2 SO 4 production rate and pH is plotted in Fig.2.7, which indicates that few
	H 2 SO 4 is produced until pH decreases to 4. With further decreasing of pH, lots of H 2 SO 4 is produced by
	ASOB and the production rate rises sharply. Once pH was decreased below 1.5, the H 2 S production rate
	drops rapidly. The verification of this relationship could be found in Chapter 7, where the experiment of
	(De Muynck et al., 2009) is simulated and analysed.
		7e-07	Production rate of H 2 SO 4
	Production rate (mol/L s)	1e-07 2e-07 3e-07 4e-07 5e-07 6e-07
		0
		9	8	7		6			5	4	3	2	1	0
			Figure 2.7: H 2				pH
	reduced.
	Such process can be expressed by a function of pH as follows:

Table 2

 2

		.1: Definition of the parameters in WATS model
	Parameter	Definition		Unit
	R H2S,WATS /R Sf,WATS /R Ss,WATS Oxidation rate of H 2 S/Sf/Ss in SOB media	mol m -3 s -1
	k oxi-H2S /k oxi-Sf /k oxi-Ss	Rate constant for biotic oxidation of H 2 S/Sf/Ss	s -1
	R Sf-Ss,WATS	Ageing rate of Sf		mol m -3 s -1
	k Sf-Ss	Rate constant for the ageing rate of Sf	s -1
	X SOB	SOB concentration in SOB media	g m -3
	ρ H2S /ρ Sf /ρ Ss /ρ O2	Concentration of aqueous H 2 S/Sf/Ss in SOB media	mol m -3
	H H2S / H Sf	Half saturation constant for aqueous H 2 S/Sf	mol m -3
	H O2,H2S / H O2,Sf	Half saturation constant for O 2 in the oxidation of H 2 S/Sf mol m -3
	c	Reaction order for the oxidation of Ss	/
	the oxidation rate of H 2 S and S are expressed as follows:
	H 2 S oxidation : R H2S-oxi = k H2S-oxi X SOB	ρ H2S ρ H2S + H H2S	(2.17)
		S oxidation: R S-oxi = k S-oxi X SOB	ρ S ρ S + H S	(2.18)

Table 2

 2

	.2. The oxidation rate of H 2 S and S are plotted in Fig.2.9.
	Table 2.2: Value of the parameters in H 2 SO 4 production model. (Jensen et al., 2008)
	Parameter Value
	k H2S-oxi	1.1×10 -8 s -1
	k S-oxi	4.1×10 -9 s -1
	X SOB	1941 g/L
	H H2S	8.0×10 -4 mol/L
	H S	2.8×10 -5 mol/L
	Combining the absorption of H 2 S and the oxidation of H 2 S, the pH evolution in pure SOB media with
	different H 2 S level are plotted in Fig.2.10, which indicates that H 2 SO 4 is produced faster with high H 2 S
	level than that with low H 2 S level.

  , the pH reduction of carbonated concrete surface with different H 2 S gas level are calculated. The simulation results coincide with experimental observations. Dealing with different given conditions, the production rate of H 2 SO 4 by SOB media is calculated by fitting measured pH evolution or WATS model. With given pH evolution in SOB media with sufficient sulfur source, H 2 SO 4 production rate is governed by a function of pH. The succession and activity of ASOB and NSOB with different pH range are considered. With given H 2 S level in gas phase, SOB is simplified as a general type of bacteria with constant biomass and activity. H 2 SO 4 is produced by oxidation of element sulfur which accumulates in SOB media by oxidation of aqueous H 2 S. By WAST model, the oxidation rate of H 2 S and that of element S are governed by functions of concentration of H 2 S and S respectively.

Table 5

 5 H2S + ρ HS -+ ρ S 2- +ρ CaHS + + ρ CaS 0 n CaS Ca 2+ + ρ CaHS + + ρ CaOH + +ρ CaS 0 + ρ CaH3SiO +

		.1: Concentration of each element
	Element	In the liquid phase(ρ L A )	In the solid phase(n S A )
	H2SO4 attack H2S attack H2SO4 attack ρ Ca ρ H2SO4 + ρ HSO + 4 +ρ CaHSO + 4 + ρ CaSO 0 + ρ SO 2+ 4 4 S ρ Ca 2+ + ρ CaHSO + 4 + ρ CaOH + +ρ CaSO 0 4 + ρ CaH3SiO + 4 + ρ CaH2SiO4 H2S attack ρ Si H2SO4/H2S attack ρ H2SiO 2-4 + ρ H3SiO -+ ρ H4SiO 0 4 4 +ρ CaH3SiO + 4	n C SH2 n CH + n C SH2 + xn C-S-H

4

+ ρ CaH2SiO4 n CH + n CaS + xn C-S-H

Table 5

 5

	.2: Diffusion coefficients of different aqueous species at T=298 K
	Species	Value (dm 2 /s)	Reference
	D 0 OH -D 0 H + D 0 H2S (aq)	5.273× 10 -7 9.311×10 -7	(Weast et al., 1988) (Weast et al., 1988)

Table 5 .

 5 3:The key parameters of CH-C SH2 equilibrium at different stable states

	Stable phase	ρ H2SO4 ρ CH H2SO4

Table 5 .

 5 4: Governing equations and primary variables in the model Taking H 2 SO 4 attack for instance, according to section 5.4.2 Q C SH2 can be expressed as follows:

	Governing Equations Primary Variables
	Sulfur (S)	ρ H2SO4 / Gas pressure of H 2 S P H2S or ρ 0 H2S
	Calcium (Ca)	ζ Ca
	Silicon (Si) Potassium (K) Chlorine (Cl) Charge	ζ Si ρ + K ρ -Cl Electric potential ψ

Table 5 .

 5 5: Boundary and initial conditions of the H 2 SO 4 attack example

	Balance	Boundary conditions	Initial conditions
	Equation	Surface	Cement paste
	Sulfur (S)	diffusion allowed ρ H2SO4 = 10 -8 mol/L	ρ H2SO4 = 10 -31.5 mol/L
	Calcium (Ca)	no diffusion	ζ Ca =1
	Silicon (Si)	no diffusion	ζ Si =1
	Sodium/Potassium	no diffusion	ρ

  5.2 Note that the direction of H 2 SO 4 attack is from right to left. With H 2 SO 4 penetrating into sample, pH of the materials decreases form 12.4, which is the value of pH in the uncorroded zone, to 1 gradually.

	1e-05	Attack direction
	1e-25 1e-20 1e-15 1e-10 4 concentration (mol/L)		Example of H 2 SO 4 attack 0 days 15 days 30 days 45 days 60 days
	H 2 SO 1e-30
	1e-35
	0	2	4	6	8	10
		Distance to the surface (mm)

Table 5 .

 5 6: Boundary and initial conditions of the biogenic H 2 SO 4 attack example

	Balance	Boundary conditions	Initial conditions
	Equation	Surface	SOB solution Cement paste
	Sulfur (S)	no flow	ρ H2SO4 = 10 -20 mol/L 10 -31.5 mol/L ρ H2SO4 =
	Calcium (Ca)	no flow	ζ Ca =0	ζ Ca =1
	Silicon (Si) Sodium/Potassium Chlorine (Cl)	no flow no flow no flow	ζ Si =0 ρ + K =0 ρ -Cl =0	ζ Si =1 ρ + K =0 ρ -Cl =0
	Charge	no current	ψ=0	ψ=0

Table 6 .

 6 1: Boundary and initial conditions of the H 2 SO 4 attack with Method 1

	Balance	Boundary conditions	Initial conditions
	Equation	Left surface	Mortar
	Sulfur (S)	diffusion allowed ρ H2SO4 0	ρ H2SO4 0 = 10 -31.5 mol/L

Table 6 .

 6 2: Boundary and initial conditions of the H 2 SO 4 attack with Method 2

	Balance	Boundary conditions	Initial conditions
	Equation	Left surface	Mortar
	Sulfur (S)	diffusion allowed

Table 6 .

 6 3: Boundary and initial conditions of the H 2 SO 4 attack with (pH = 2)

	Balance	Boundary conditions	Initial conditions
	Equation	Left surface	Mortar
	Sulfur (S)	diffusion allowed

Table 6 .

 6 4: Boundary and initial conditions of the H 2 SO 4 attack of the sample with carbonated layer

	Balance	Boundary conditions	Initial conditions
	Equation	Left surface	Carbonated layer	Mortar
	Sulfur (S)	diffusion allowed

  are simulated. The neutralization of sample surface by H 2 S gas is considered. The H 2 SO 4 production from biological activity is calculated by the methods introduced in Chapter 2. Similar with the simulation of H 2 SO 4 chemical solution corrosion in Chapter 6, the evolution of solid compositions, corrosion depth and porosity of samples are illustrated. Furthermore, the effect of different H 2 S gas level is studied.

Table 7 .

 7 1: Boundary and initial conditions of the H 2 S attack (200 ppm) S gas is absorbed into pore solution of concrete from the left surface, and the absorption rate is governed by Eq.(2.1). Since no H 2 SO 4 is present during the neutralization process, CaS is considered as the only corrosion products. The dissolution of CH and C-S-H, and the precipitation of CaS are described by the model introduced in Chapter 2 and Chapter 3. After exposure of 2 days, the calculated absorption of H 2 S and pH evolution of concrete sample are shown in Fig.7.4.As expected, H 2 S in gas phase is absorbed into pore solution of concrete to neutralize the sample surface, where pH decreases form 12.35 to 9.5 gradually. The change of aqueous H 2 S concentration and pH in the pore solution of surface part are plotted in Fig.7.5a. Due to the low ionization constant of H 2 S, the neutralization rate of surface is low even if the H 2 S gas level is quite high at surface. Since H 2 S gas level keeps constant, the absorption of H 2 S at surface is constant (seeFig.7.5b).

	Balance	Boundary conditions Initial conditions
	Equation	Left surface	Concrete
	Sulfur (S)	Absorption allow p H2S,gas = 200 ppm	ρ H2S 0 = 10 -9 mol/L
	Calcium (Ca)	no diffusion	ζ Ca = 1
	Silicon (Si)	no diffusion	ζ Si = 1
	Potassium (K)	no diffusion	ρ K + = 0.5 mol/L
	Chlorine (Cl)	no diffusion	ρ Cl -= 0.5 mol/L
	Charge	no current	ψ=0
	H 2

Table .

 .

		Initial conditions
	Balance Equation
	SOB suspension	Neutralized layer	Concrete
	Sulfur (S)

7.2. The solid compositions, pH and porosity of the initial concrete sample with neutralized layer are indicated in Fig.7.11 Table 7.2: Boundary and initial conditions of biodeterioration by SOB suspension

  Figure 7.12: The calculated H 2 SO 4 and pH evolution profiles in SOB suspension and concrete from 0 days to 10 days: (a) H 2 SO 4 concentration; (b) pH value.The pH evolution in bulk SOB suspension, which refers to that part of suspension not influenced by is almost uniform in bulk SOB suspension where pH decreases from the very beginning. However, during the early time pH in SOB suspension close to concrete surface increases from 7 to about 9 due to the alkalinity of concrete and the low production rate of H 2 SO 4 (see Fig.7.13b), i.e., the acid production is slower than the acid consumption. A decrease of bacterial activity is expected. Thus, the production rate of H 2 SO 4 is too low to cause further deterioration of concrete. However, after about 8 days, pH starts to decrease by the H 2 SO 4 produced from bulk suspension.After immersion of 10 days, similar zones as those in the simulation of H 2 SO 4 solution condition are observed in SOB immersion test (seeFig.7.14). Not only neutralized layer, but also concrete core is degraded by biogenic H 2 SO 4 . Due to the diffusion of Ca 2+ towards the gypsum precipitation front, lots of gypsum accumulates at concrete surface. At the interface between neutralized layer and concrete core, the amount of C-S-H is discontinuous. That is probably caused by the discontinuity of initial porosity in the initial conditions.In the experiments[START_REF] De Muynck | Effectiveness of admixtures, surface treatments and antimicrobial compounds against biogenic sulfuric acid corrosion of concrete[END_REF], shaking in the distilled water resulted in both the removal of aggregate and reaction products. Drying removes all the remaining SOB at the concrete surface. Thus we consider that all the cycles are identical. Consequently, the deterioration depth of all cycles are the same. The predicted corrosion depth and that measured during experiments are compared in Fig.7.15. The predicted corrosion rate is 0.22 mm/cycle, which coincide with experimental results. However, no corrosion

			10						7e-07		10		7e-07
		pH	1 2 3 4 5 6 7 8 9	Bulk SOB suspension pH (Simulation) pH (Experiment) αH 2 SO 4			1e-07 2e-07 3e-07 4e-07 5e-07 6e-07	H 2 SO 4 production rate (mol/L s) pH	2 4 6 8	SOB suspension close to concrete surface pH (Simulation) αH 2 SO 4	1e-07 2e-07 3e-07 4e-07 5e-07 6e-07	H 2 SO 4 production rate (mol/L s)
			0						0		0		0
			0		2	4	6	8	10		0		2	4	6	8	10
						Time (Days)						Time (Days)
							(a)						(b)
	Concentration of H 2 SO 4 (mol/L)	1e-35 1e-30 1e-25 1e-20 1e-15 1e-10 1e-05 Figure 7.13: The pH evolution and production rate of H 2 SO 4 from 0 to 10 days: (a) bulk SOB suspension; 1 days 5 days 8 days 10 days 2 4 6 8 pH 1 days 5 days 8 days 10 days Immersion condition 0 days 0 days 10 12 Immersion condition SOB suspension Concrete 14 SOB suspension Concrete (b) SOB closed to concrete surface.
				0	0.2	0.4	0.6	0.8	1			0	0.2	0.4	0.6	0.8	1
					Distance to the surface (mm)						Distance to the surface (mm)
						(a)							(b)

concrete, is plotted in Fig.

7

.13a. Due to that NSOB produce H 2 SO 4 very slow, pH in SOB suspension decreases very few during early time. Since lower pH causes higher in-situ H 2 SO 4 production rate, pH starts to decrease sharply after several days. When pH reaches 1.5, H 2 SO 4 production starts to slow down.

The change in pH of SOB suspension was measured during the experiments (see Fig.

7

.13). The simulation results coincide with experimental results. pH depth was measured out in the first cycle of the experiments. Carbonated layer is probably responsible for that as described in Chapter 6.

  2.1. After 40 hours, the pH in mortar surface decreases to 9.5 (seeFig.7.17a). As shown in previous section, with H 2 S of 200 ppm the surface pH decreases to 9.5 after 48 hours. Higher H 2 S level results in faster surface neutralization rate.However, the difference between 200 ppm and 400 ppm is not significant. The solid compositions of mortar sample is plotted in Fig.7.17b which indicates a neutralized layer of approximate 0.1 mm.

	2 S (mol/L)	0.0001 0.001						12 13
	Concentration of aqueous H	1e-09 1e-08 1e-07 1e-06 1e-05	0	8	16	24 Surface neutralization 32 40 ρ H 2 S aq	48	8 9 10 11 pH
						Time (h)

  Table.7.3. According to the modeling of H 2 SO 4 production with a given H 2 S content in gas phase as Eq.7.2

		5e-07							0.0018			2.528e-07					0.09
		4.5e-07							0.0016			2.526e-07					0.08
	Oxidation rate (mol/L s)	0 5e-08 1e-07 1.5e-07 2e-07 2.5e-07 3e-07 3.5e-07 4e-07		Biofilms Oxidation rate of H 2 S Concentration of H 2 S		0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014	Concentration (mol/L)	Oxidation rate (mol/L s)	2.508e-07 2.51e-07 2.512e-07 2.514e-07 2.516e-07 2.518e-07 2.52e-07 2.522e-07 2.524e-07		Biofilms Oxidation rate of S Concentration of S	0 0.02 0.03 0.04 0.05 0.06 0.07 0.01	Concentration (mol/L)
		0	1	2	3	4	5	6	7			0	1	2	3	4	5	6	7
					Time (Days)									Time (Days)
					(a)										(b)

Afin de fournir un environnement approprié pour la croissance des BSO, la neutralisation de la surface du béton (i.e., l'absorption de H 2 S et la corrosion aqueuse de H 2 SO 4 ) est considérée. Pour obtenir la quantité de H 2 SO 4 biogénique, la bio-oxydation du H 2 S par l'activation des bactéries est simulée par un modèle simplifié. Puis, pour alimenter un environnement convenable pour la croissance des BSO, la réduction abiotique du pH du béton est introduite. Le taux de production de H 2 SO 4 est régi par la valeur du pH dans les biofilms et la quantité de H 2 S dans le gaz.On fait l'hypothèse que tous les processus chimiques sont en équilibre thermodynamique. La dissolution de la portlandite (CH) et du silicate de calcium hydratés (C-S-H), ainsi que la précipitation de gypse (C SH2) et du sulfure de calcium sont décrites par la loi d'action de masse et le seuil des produits d'activité ionique.Pour prendre en compte la décroissante continue du rapport Ca/Si lors de la dissolution de la C-S-H, une généralisation de la loi d'action de masse est appliquée.En simplifiant le processus de précipitation du gypse, un modèle d'endommagement est introduit pour caractériser la détérioration du béton due au gonflement du gypse. Ainsi, l'évolution de la porosité et de la profondeur de la détérioration pendant le processus de dégradation sont pris en compte.

Bil is developed by Patrick Dangla. The source code can be downloaded at http://perso.lcpc.fr/dangla.patrick/bil/

dt, the concentration of H 2 SO 4 changes to ρ H2SO4 . Using the simple kinetic law of the dissolution of CH given by Eq.(3.15), as indicated in Chapter 3, the change in the amount of CH is dn CH as follows:

(5.13) Thus, after dt the amount of CH can be expressed as follows:

With the help of ζ Ca , the amount of C SH2 is changed to n t+dt C SH2 as follows:

where, ζ + Ca = max(ζ Ca , 0) is the positive part of variable ζ Ca .

For the dissolution of C-S-H we can employ a similar variable ζ Si , defined as:

where, n C-S-H is the molar content of C-S-H in the form of C x S 1 H z including SH t . n 0 C-S-H is the initial molar content of C-S-H. Q SHt is the ion activity product of the dissociation reaction of SH t . Q SHt(eq) is the equilibrium ion activity product of SH t at given Ca/Si ratio. Q SHt(eq) can be calculated by Eq. (5.17) discussed in Section 3.4:

(5.17)

In Eq.(5.16),

represents the molar content of Si in solid phase. When

Governing equations and primary variables

As introduced in Chapter 3, with known value of ρ H2SO4 /ρ H2S , ρ Ca 2+ , and ρ H4SiO 0 4 , the concentration of all aqueous species (except ρ K + and ρ Cl -) can be calculated from Table .3.1. Through the primary variables (p H2S , ρ H2SO4 , ζ Ca and ζ Si ), the secondary variables such as ρ 0 H2S , ρ Ca 2+ and ρ H4SiO 0 4 can be computed. ρ K + and ρ Cl -are considered as primary variables as well.

Governing equations versus primary variables in the model are listed in Table 5.4. ρ H2SO4 = 10 -25.3 mol/L ρ H2SO4 = 10 -31.5 -10 -25.3 mol/L ρ H2SO4 = 10 -31.5 ρ H2S 0 = 0 mol/L / / ρ S = 0.005 mol/L / / Calcium (Ca)

Introduction of discrete and solid solution models of C-S-H dissolution

There are different types of C-S-H, e.g. jennite with Ca/Si ratio of approximately 1.7, tobermorite with Ca/Si ratio of approximately 0.83 and amorphous silica gel with Ca/Si ratio of 0. Table A.1 lists different forms of C-S-H and their thermodynamical properties proposed by [START_REF] Lothenbach | Thermodynamic modelling of the effect of temperature on the hydration and porosity of portland cement[END_REF]. In discrete method, to describe the dissolution of C-S-H, we can use the same approach as the one adopted for the CH dissolution (see Eq. (3.3) and Section 3.3),

where Q SHt = ρ H4SiO 0 4 is the activity product of amorphous silica. From Eq. (A.1), we can deduce the stability domains for the amorphous silica SH t and the different types of C-S-H. Taking jennite as example (see Table A.1), the equilibrium equation for jennite and amorphous silica is in the following form:

The stability domains are shown in Similarly we can calculate the stability domains with the system of jennite-tobermorite-silica gel, as

The threshold H 2 SO 4 concentration for tobermorite ρ Tob H2SO4 ≈ 6.2 × 10 -20 mol/L and for jennite ρ Jen H2SO4 ≈ 3.9 × 10 -25 mol/L. When the H 2 SO 4 concentration is lower than ρ Jen H2SO4 , jennite is stable, when it lays between ρ Jen H2SO4 and ρ Tob H2SO4 , tobermorite is stable, and when H 2 SO 4 concentration exceeds ρ Tob H2SO4 , amorphous silica gel is stable. Rather than the discrete model discussed above, a solid solution method which considers different solid poles in equilibrium, is more appropriate to describe the continuous decalcification of C-S-H. Let's consider a solid solution composed of N end-members, M i respectively (i = 1, ..., N ). In the framework of the solid solution theory, a reaction between the end-members (poles) to form a solid solution can be written as:

Each end-member is assumed to be in thermodynamical equilibrium with the aqueous solution. The chemical potential of the end-member i in the solid solution is given by:

where µ 0 i is the chemical potential of the pure end-member. The equilibrium between the end-member i and the aqueous solution results in: Ca/Si≥1.8, equilibrium with portlandite is added. [START_REF] Börjesson | A thermodynamic model for the calcium silicate hydrate gel, modelled as a non-ideal binary solid solution[END_REF] used an non-ideal solid solution of CH -CaH 2 SiO 4 for 1<Ca/Si≤1.43, based on data from [START_REF] Kalousek | Application of differential thermal analysis in a study of the system lime-silica-water[END_REF]. [START_REF] Rahman | A model for dissolution of cao-sio2-h2o gel at ca/si> 1[END_REF] proposed a similar model as [START_REF] Börjesson | A thermodynamic model for the calcium silicate hydrate gel, modelled as a non-ideal binary solid solution[END_REF], within the Ca/Si range from 1. to 1.5. [START_REF] Kersten | Aqueous solubility diagrams for cementitious waste stabilization systems. 1. the csh solidsolution system[END_REF] proposed a non-ideal binary solid solution CH -CaHSiO 3.5 •H2O based on the experimental data from [START_REF] Greenberg | Investigation of Colloidal Hydrated Calcium Silicates: Pt. II. Solubility Relationships in the Calcium Oxide-silica-water System at 259[END_REF] and [START_REF] Fujii | Heterogeneous equilibrium of calcium silicate hydrate in water at 30 c[END_REF]. [START_REF] Kulik | Aqueous solubility diagrams for cementitious waste stabilization systems: Ii, end-member stoichiometries of ideal calcium silicate hydrate solid solutions[END_REF] described the solubility of C-S-H with a system of two ideal solid solutions: C-S-H-(I): S -tobermorite and C-S-H-(II): tobermorite -jennite. The experimental data of [START_REF] Greenberg | Investigation of Colloidal Hydrated Calcium Silicates: Pt. II. Solubility Relationships in the Calcium Oxide-silica-water System at 259[END_REF] was used to calibrate the model, and the applicable range of Ca/Si ratio is between 0 to 1.7. [START_REF] Carey | Calcium silicate hydrate (csh) solid solution model applied to cement degradation using the continuum reactive transport model flotran[END_REF] proposed a non-ideal binary solid solution (S -CH) calibrated with the experimental data from [START_REF] Chen | Solubility and structure of calcium silicate hydrate[END_REF].

All the models above are able to reproduce the experimental results within the dispersion of experimental data [START_REF] Soler | Thermodynamic description of the solubility of c-s-h gels in hydrated portland cementliterature review[END_REF]. Therefore it is hard to say which one is better. However, the models proposed by [START_REF] Atkinson | Aqueous chemistry and thermodynamic modelling of cao-sio2-h2o gels[END_REF], [START_REF] Kersten | Aqueous solubility diagrams for cementitious waste stabilization systems. 1. the csh solidsolution system[END_REF], [START_REF] Kulik | Aqueous solubility diagrams for cementitious waste stabilization systems: Ii, end-member stoichiometries of ideal calcium silicate hydrate solid solutions[END_REF] and [START_REF] Carey | Calcium silicate hydrate (csh) solid solution model applied to cement degradation using the continuum reactive transport model flotran[END_REF] are applicable over a wider range of Ca/Si ratio than those of others.

Here we use an ideal solid solution with four poles, similar as that proposed by [START_REF] Kulik | Aqueous solubility diagrams for cementitious waste stabilization systems: Ii, end-member stoichiometries of ideal calcium silicate hydrate solid solutions[END_REF] with thermodynamical data from [START_REF] Lothenbach | Thermodynamic modelling of the effect of temperature on the hydration and porosity of portland cement[END_REF]. The stoichiometric and equilibrium constants for the different poles are listed in Table A.2.

Combing Eq. (A.1) and Eq.(A.7), for each pole, we get

(2,2.4,4) -28.26 Tobermorite II

(1.5,1.8,3) -21.19 Jennite (1.5,0.9,2.4) -15.62

1 data from [START_REF] Lothenbach | Thermodynamic modelling of the effect of temperature on the hydration and porosity of portland cement[END_REF] With the hypothesis of ideal solid solution, λ i = 1 and X i = 1, we have the following equation:

where,

Eq. (A.9) indicates that q S is a function of q CH , q S (q CH ), that we can solve. Thus, the mole fraction of each end-member can be calculated by Eq. (A.8) as a function of q CH . The Ca/Si ratio can be attained as a function of q CH by using Eq. (A.11).

The calculated mole fraction for each pole and the Ca/Si ratio as a function of q CH are illustrated in Fig.

A.3. We can see that for q CH =1, which means CH is in equilibrium with the aqueous solution, the C-S-H is in the form of jennite. When q CH decreases, C-S-H will first decalcified to the form of tobermorite, and finally to amorphous silica gel. Thus, Ca/Si ratio drops continuously as q CH decreases. It is noteworthy that our new approach, which is introduced in Section 3.4, encompasses the solid solution model involving any number of poles with coefficient (x i ,y i ,z i ). Indeed, the solid solution theory shows that X i = A i q xi (F ) yi , where

It is then easy to show that the average stoichiometric coefficients are given by x = X(q) and y = Y (q) with

Since the solid fraction has to satisfy X i = 1, we obtain:

Hence deriving this last equation with respect to q leads to: .15) which is akin to Eq. (3.26). Eq. (3.26) is a generalization of the mass action law for C-S-H with variable Ca/Si ratio.

Appendix B

The finite volume method

The finite volume method is a discretization method well suited for the numerical simulation of various types of conservation laws such as mass balance equations [START_REF] Robert | Finite volume methods[END_REF]. Some of the important features of the finite volume method are similar to those of the finite element method (arbitrary geometries, structured/unstructured meshes). An additional feature is the local conservativity of the numerical fluxes at the interface of two neighoring cells. conserved from one discretization cell to its neighbor. it is based on a balance approach: a local balance is written on each discretization cell which is often called "control volume" the boundary of the control volume is then obtained.

Let us introduce the method on a simple example. Consider a mass balance equation of the form

where m(ρ) is a mass content of a substance per unit volume of porous material and considered as a function of a primary variable ρ which is typically of a concentration. The flow vector of that substance, w, is assumed to be given by a transport's law, typically a the Fick's law:

The finite volume method is obtained by a discrete balance in the elements of a mesh called "control volumes". Integrating the previous equation over a control volume K yields the following balance equation over K:

where w KK ′ expresses an approximation of ∂Ω KK ′ w • n ds for any K and any neighor K ′ of K:

where |∂Ω KK ′ | is the surface area between K and K ′ and d KK ′ is the distance between the "centers" of K and K ′ . The surface ∂Ω KK ′ is also assumed to be perpendicular to the line joining the "centers" of K and K ′ in order to be consistent with the Fick's law. The resolution of the set of equations (B.3) is performed by a Newton's iteration method until the variation of ρ n+1 K between two successive iterations, δρ n+1 K , is small enough and meets a convergence criterion of the form

where ∆ρ obj is a given objective variation of ρ and ǫ a small positive number compared to unity, typically 10 -4 .