Insight into the operator splitting error and its combination with numerical methods 

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Repository Safety

A set of basic safety rules have been defined by the « Autorité de Sûreté Nucléaire » (ASN) (ANDRA, 2005a; ASN, 2008) setting the main objectives for the repository such as the absence of seismic risks in the long term, confinement properties for radioactive substances, and rock suitable to underground excavations. The target is to preserve the environment and human beings from risks associated with nuclear waste. Consequently, the following functions must be fulfilled:
• Preventing water circulation because it can degrade waste packages and migration of radionuclides into the environment;
• Limiting the release of radioactive substances by the package and immobilizing them in the repository as long as possible;
• Delaying and reducing the migration of radioactive substances beyond the repository or geological layer.
In order to complete such functions a passive engineered barrier system is designed comprising a variety of sub-systems: canister, buffer, backfill, and so on. The main purpose of such systems is to delay as much as possible the release of radionuclides from the waste to the host rock. Consequently, ANDRA and homologues institution of other countries (e.g. SKB) facing similar problems have developed R&D programs to study the behavior of rocks and radionuclides in order to assess the design of future repositories. Among these studies is possible to find problems related to the migration of radionuclides such as uranium through the host rock using experiments or numerical simulations (Dittrich and Reimus, 2015; Pfingsten, 2014; Xiong et al., 2015). Studies focused on HLW which will be confined in a vitrified glass either in contact with a bentonite buffer or with the host rock. Consequently, the evolution of the dissolution of the vitrified glass has been estimated through numerical and experimental simulations (Debure et al., 2013), and also the interaction between the glass and bentonite, and between the glass and the host rock through numerical simulations (Ngo et al., 2014). Numerical experiences have also contributed to give insights on the geochemical evolution of the HLW, engineered barriers and host rock through the several thousand of years (Trotignon et al., 2007; Yang et al., 2008), in some cases taking into account possible climate change scenarios (Nasir et al., 2014; Spycher et al., 2003), or comparing with analogous natural sites (Chen et al., 2015; Martin et al., 2016). Since many of these studies have been carried out using or relying on numerical simulations and no analytical solutions exist, code intercomparison work has also been performed in order to compare codes results (Marty et al., 2015; Xie et al., 2015).
Here we aim on solving numerical simulations about the effects of atmospheric carbonation process over concrete in a nuclear waste context.

Description of the problem: Atmospheric carbonation

ILW-LL is proposed to be conditioned in cylinders of bitumen or concrete according to the type of waste which includes metals (fuel claddings), effluent treatment sludges and nuclear plant operating equipment. The primary ILW-LL package will be placed in a high-performance reinforced concrete container (Figure I.2), containing from 1 to 4 primary packages (ANDRA, 2005a; ANDRA, 2005b).
Figure I.2: Disposal container for intermediate-level long-lived waste (ILW-LL) containing four primary waste packages (ANDRA, 2005b).
The disposal containers of ILW-LL are planned to be placed in vaults which will be ventilated during the operation period (up to 150 years). Ventilation is required to guarantee operating safety, evacuate radioactive gas such as hydrogen produced by radiolysis, and residual heat from the waste. One of the consequences of the vault ventilation is that it will desaturate the disposal container, leading to a physico-chemical process known as concrete atmospheric carbonation (Thouvenot et al., 2013).
The atmospheric carbonation process is summarized as follows:
1. The carbon dioxide (CO2) diffuses into the concrete and dissolves into the pore solution:
2. The water molecules react with CO2 to form carbonic acid (H2CO3):
3. H2CO3 dissociates as bicarbonate (HCO3-), also called hydrogen carbonate, and carbonate (CO32-) ions according to the pH of the solution (Figure I.3). The dissociation releases H+ ions, leading to a pH drop:
4. The principal hydration products (Table I-1) of the concrete, particularly portlandite (Ca(OH)2), dissolve in order to buffer the decrease of the pH level and maintain the equilibrium of the solution. Furthermore, the dissolution of portlandite releases Ca2+ ions which reacts with CO32- in the pore solution, precipitating as calcite (CaCO3).
Figure I.3: Molar fraction of the chemical species H2CO3, HCO3-, and CO32- respect pH at 20°C and equilibrium (Thiery, 2006).
At first glance, the carbonation process might not seem harmful for the concrete. Generally, even a decrease in porosity can be expected because the carbonation products usually have higher molar volume than their reactants (Glasser et al., 2008). Nevertheless, the decrease in alkalinity turns out to be an issue for the reinforcing steel bars of the concrete, and thus for the concrete structure. Normally, the pore solution in concrete has an alkaline environment with a pH between 12.5 and 13.5 in order to maintain the corrosion of the reinforcing steel bars in a range of very low rates. At such high pH a thin passive oxide layer forms on the steel and slows down the corrosion. If the passive layer is destroyed, for example due to the decrease of pH owing to atmospheric carbonation, corrosion occurs and might result in a failure of the structure (Zhang, 2016). Therefore, assessing the depth of the carbonation front is a main mean for evaluating the safety of the concrete package for ILW-LL (Figure I.4).
Figure I.4: Carbonation front in a simplified 1D model sketch. Three zones from left to right can be observed: a fully carbonated concrete with a pH around 9, a transition area where the carbonation process is occurring and an uncarbonated area with a pH around 13 (Ta et al., 2016).
Experimental studies of the carbonation process give an insight of the phenomenon in the concrete package (Duprat et al., 2014; Ekolu, 2016; Shi et al., 2016; Thiery et al., 2007), but due to the long time scale of the waste confinement, detailed numerical studies of the physico-chemical processes are necessary in order to assess the safety of the disposal containers.

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State-of-the-art: Reactive transport modeling

Mathematical model

Before any simulation, a conceptualization of the reality must be carried out. Mathematical models are tools that can help to conceptualize such reality. According to the hypothesis and assumptions that are taken, different models with their own intrinsic difficulties and simplicities arise. Comparisons between the results of the mathematical model and reality will determine the validity of the model (Hassan, 2004). Two main processes have to be modeled in terms of reactive transport: species transport and chemical reactions. The selection of laws, therefore the system of equations, are subject to the working scale.

Spatial scale

Here we work at a mesoscopic scale, where transport and reactions are described by macroscale equations based on a continuum formulation. The properties of the porous media such as porosity and density, are averaged over a control volume known as Representative Elementary Volume (REV) (Figure I.5) (Bear, 1972). REV works under the following assumptions (Steefel et al., 2005):
• REV is large enough to have a meaningful average but small enough to assume that the volume of the REV is infinitesimal.

Table of contents :

I Introduction
I.1 Context of the thesis
I.1.1 Repository Safety
I.1.2 Description of the problem: Atmospheric carbonation
I.2 State-of-the-art: Reactive transport modeling
I.2.1 Mathematical model
I.2.1.1 Spatial scale
I.2.1.2 Transport and reaction operators
I.2.2 Numerical approaches
I.2.3 Codes
I.3 Objectives and issues
II Development of TReacLab
II.1 Article
II.2 Additional benchmarks
II.2.1 Benchmark 1: Transport validation
II.2.2 Benchmark 2: Cation exchange
II.2.3 Benchmark 3: Multispecies sorption and decay
II.3 External transport and geochemical plugged codes
II.3.1 Transport codes
II.3.1.2 FVTool
II.3.1.3 pdepe MATLAB
II.3.1.4 FD script
II.3.2 Geochemical codes
II.3.2.1 PHREEQC, iPhreeqc, and PhreeqcRM
II.4 Insight into the operator splitting error and its combination with numerical methods
II.4.1 Error of the operator splitting methods
II.4.2 Operator splitting methods and numerical methods
III Atmospheric Carbonation
III.1 Concrete conceptualization
III.1.1 Geometry
III.1.2 Concrete composition
III.1.3 Decoupling atmospheric carbonation processes
III.1.3.1 Fluid flow
III.1.3.2 Multicomponent Transport
III.1.3.3 Geochemical reactions
III.2 First modeling approach to the atmospheric carbonation problem
III.2.1 Constant Saturation Test
III.2.1.1 Coupling procedure and hydraulic properties
III.2.1.2 Initial values and boundary conditions
III.2.1.3 Discretization and Von Neumann number
III.2.1.4 Preliminary results for the constant saturation test, case Sl=0.802
III.2.1.5 Preliminary results for the constant saturation test, case Sl=0.602
III.3 Discussion and perspectives
IV Conclusion


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