Model Analysis

Abstract

We employ the ThermoRichardsMechanics (TRM) process, Wang et al. (2010) implemented in OpenGeoSys (OGS), Bilke et al. (2019). The process captures a non-isothermal porous medium with a solid phase, a liquid phase and a gas phase which is considered isobaric Richards (1931).

4.1 THM-Modelling of the Mock-Up Test in OpenGeoSys

Aqeel Afzal Chaudhry, Thomas Nagel

4.1.1 Description of the TRM Model

We employ the ThermoRichardsMechanics (TRM) process, Wang et al. (2010) implemented in OpenGeoSys (OGS), Bilke et al. (2019). The process captures a non-isothermal porous medium with a solid phase, a liquid phase and a gas phase which is considered isobaric (Richards, 1931). Thus, the model is governed by three balance equations i.e., an energy balance, a mass balance of the liquid phase and the linear momentum balance of the mixture which are thus formulated based on three independent state variables i.e., temperature T, liquid pressure $p_\text {LR}$ and solid displacement $\textbf{u}_\text {S}$, respectively. In addition, the TRM process allows for the inclusion of liquid evaporation and a vapor diffusion model based on Philip and De Vries (1957a) is implemented. For the sake of brevity, we will not mention all the terms appearing in the equations in this section but focus on terms that are not standard in similar implementations or of particular importance for interpreting the results. For a detailed description of the TRM model summarized here as well as it’s comparison to a non-isothermal two-phase two-component flow with mechanics model ($\text {TH}^2 \text {M}$), published by Grunwald et al. (2022), the reader is referred to Pitz et al. (2022).

The heat balance equation is written as

$$\begin{aligned} \begin{aligned} & (\rho c_p)_{\text {eff}}\frac{\text {d} T}{\text {d} t} + L_\text {0}\frac{\text {d} \theta _\text {vap}}{\text {d} t} - \text {div}\left( \boldsymbol{\lambda }_{\text {eff}}\,\text {grad}\,T\right) \\ & +\text {div}\left( \dfrac{L_\text {0}\boldsymbol{J^\text {W}_\text {G}}}{\rho ^\text {W}_\text {GR}}\right) +\text {grad}\,T \cdot \left( c_{p\text {L}}\boldsymbol{A}_\text {L}+c_{p\text {,vap}}\,\boldsymbol{J^\text {W}_\text {G}} \right) = Q_T \end{aligned} \end{aligned}$$

(4.1)

where $(\rho c_p)_{\text {eff}}$ is the effective volumetric heat capacity of the medium and can be obtained using the respective phase properties as

$$\begin{aligned} (\rho c_p)_{\text {eff}} = \phi S_\text {L} \rho _\text {LR}c_{p\text {L}} + (1-\phi ) \rho _\text {SR}c_{p\text {S}} \end{aligned}$$

(4.2)

At this stage, to avoid confusion, it is appropriate to point out that $S_\text {L}$ represents the liquid saturation while “S” in the subscript indicates the corresponding property of the solid phase. The effective thermal conductivity $\boldsymbol{\lambda }_{\text {eff}}$ in Eq. 4.1 is considered to be saturation-dependent for bentonite in our case and can be written as a function of dry ($\boldsymbol{\lambda }_{\text {dry}}$) and wet / saturated ($\boldsymbol{\lambda }_{\text {wet}}$) thermal conductivity as

$$\begin{aligned} \boldsymbol{\lambda }_{\text {eff}} = \boldsymbol{\lambda }_{\text {dry}}+S_{\text {L}}\left( \boldsymbol{\lambda }_{\text {wet}}-\boldsymbol{\lambda }_{\text {dry}} \right) \end{aligned}$$

(4.3)

Furthermore, $L_\text {0}$ in Eq. 4.1 is the volumetric latent heat of vaporization and can be written for water following (Saito et al. 2006) as

$$\begin{aligned} L_\text {0} = \rho _{\text {LR}} \left( {2.501\cdot 10^6}\,\text {J kg}^{-1} - {2369.2}\,{\,\text {J kg}^{-1}} \left( T-{273.15\,\mathrm{\text {K}}} \right) \right) \end{aligned}$$

(4.4)

$\theta _\text {vap}$ in Eq. 4.1 is the water vapor content and is given following (De Vries 1958; Saito et al. 2006) as

$$\begin{aligned} \theta _\text {vap} = \phi (1-S_\text{ L})\dfrac{\rho ^{\text {W}}_{\text {GR}}}{\rho _{\text {LR}}} \end{aligned}$$

(4.5)

where the vapor density $\rho ^{\text {W}}_{\text {GR}}$ can be deduced from Kelvin-Laplace equation as

$$\begin{aligned} \rho ^{\text {W}}_{\text {GR}} = \rho ^{\text {W}}_{\text {vap}} \exp \left( -\dfrac{p_{\text {cap}}}{\rho _{\text {LR}} RT}\right) \end{aligned}$$

(4.6)

where $p_\text {cap}$ is the capillary pressure, R is specific gas constant for water vapor and $\rho ^{\text {W}}_{\text {vap}}$ is the saturated vapor density given as, (Harrison 1965)

$$\begin{aligned} \rho ^{\text {W}}_{\text {vap}} = {10^{-3}}\,\text {kg m}^{-3}\, \exp \left( 19.819-\dfrac{{4975.9\,\mathrm{\text {K}}}}{T}\right) \end{aligned}$$

(4.7)

The fourth term on the left hand side of Eq. 4.1 represents the transport of latent heat while the last term represents the sensible heat transported by water and vapor per unit area. The term on the right hand side of Eq. 4.1 represents the heat source (or sink).

The mass balance equation for the liquid phase is given as

$$\begin{aligned} \begin{aligned} & \rho _\text {LR}S_\text {L}(\alpha _\text {B} - \phi ) \beta _{p\text {,SR}}\dfrac{\text {d}p_\text {LR}}{\text {d}t}-\rho _\text {LR}S_\text {L}(\alpha _\text {B} - \phi )\, \text {tr}(\boldsymbol{\alpha }_{T\text {,SR}})\dfrac{\text {d}T}{\text {d}t} \\ & +\phi \left( (1-S_\text {L})\dfrac{\text {d}\rho ^\text {W}_\text {GR}}{\text {d}t}+S_\text {L}\dfrac{\text {d}\rho _\text {LR}}{\text {d}t} \right) + (\rho _\text {LR}-\rho ^\text {W}_\text {GR})\left[ \phi + p_\text {LR}S_\text {L}(\alpha _\text {B} - \phi )\right] \dfrac{\text {d}S_\text {L}}{\text {d}t} \\ & +\rho _\text {LR}S_\text {L}\alpha _\text {B}\text {div}\left( \dfrac{\text {d}\boldsymbol{u}_\text {S}}{\text {d}t}\right) + \text {div} \left( \boldsymbol{A^\text {W}_\text {L}} + \boldsymbol{J^\text {W}_\text {G}} \right) = Q_H \end{aligned} \end{aligned}$$

(4.8)

where $\alpha _\text {B}$, $\beta _{p\text {,SR}}$ and $\boldsymbol{\alpha }_{T\text {,SR}}$ represent the Biot-Willis coefficient, intrinsic solid compressibility and the solid’s linear thermal expansion tensor, respectively, whereas $Q_H$ on the right hand side represents the fluid source (or sink).

The linear momentum balance of the overall mixture is given as

$$\begin{aligned} \text {div}\, \left( \boldsymbol{\sigma }^{\text {eff}} -{\alpha }_\text {B} \chi (S_\text {L}) p_\text {LR}\, \textbf{I}\right) + \rho \textbf{g} = \textbf{0} \end{aligned}$$

(4.9)

with

$$\begin{aligned} \dot{\boldsymbol{\sigma }}^{\text {eff}} = \boldsymbol{\mathcal {C}}\, \mathbf {:}\, (\dot{\boldsymbol{\epsilon }}-\dot{\boldsymbol{\epsilon }}_\text {pl}-\dot{\boldsymbol{\epsilon }}_\text {th}-\dot{\boldsymbol{\epsilon }}_\text {sw}) \end{aligned}$$

(4.10)

where $\boldsymbol{\mathcal {C}}$ is the fourth order elastic stiffness tensor while ${\boldsymbol{\epsilon }}$, ${\boldsymbol{\epsilon }}_\text {pl}$, ${\boldsymbol{\epsilon }}_\text {th}$ and ${\boldsymbol{\epsilon }}_\text {sw}$ represent the total, plastic, thermal and swelling strains, respectively. Furthermore, $\chi (S_\text {L})$ in Eq. 4.9 is the Bishop coefficient. The implementation in OGS-6 allows different possibilities to account for Richard coefficient, e.g., BishopsPowerLaw and BishopsSaturationCutoff which can be written respectively as

$$\begin{aligned} \chi (S_\text {L})={S_\text {L}}^n, \qquad \qquad \chi (S_\text {L}) = {\left\{ \begin{array}{ll} 1 \,\, \text {for} \, S_\text {L} \ge S_\text {cutoff} \\ 0 \,\, \text {for} \, S_\text {L} < S_\text {cutoff}\,, \\ \end{array}\right. } \end{aligned}$$

(4.11)

where the exponent n and $S_\text {cutoff}$ are the controlling parameters in each case. The swelling strain rate in Eq. 4.10 is given as

$$\begin{aligned} \dot{\boldsymbol{\epsilon }}_\text {sw} = -\boldsymbol{\mathcal {C}}^{-1} \mathbf {:} \dot{\boldsymbol{\sigma }}_\text {sw} \end{aligned}$$

(4.12)

where the anisotropic swelling stress is implemented as

$$\begin{aligned} \boldsymbol{\sigma }_\text {sw} = - \sum \limits _{i=1}^3 \hat{p}_{\text {sw},i} \left( S_\text {eff,sw}^{\lambda _{\text {sw},i}} - S_\text {eff,sw\,0}^{\lambda _{\text {sw},i}} \right) \textbf{n}_i \otimes \textbf{n}_i \end{aligned}$$

(4.13)

where $\hat{p}_i$ is the maximum swelling pressure in direction $\textbf{n}_i$. Furthermore,

$$\begin{aligned} S_\text {eff,sw} = \frac{S_\text {L} - S_\text {min,sw}}{S_\text {max,sw} - S_\text {min,sw}} \quad \text {and} \quad S_\text {eff,sw} \in [0,1] \end{aligned}$$

(4.14)

Instead of using Eq. 4.13 directly, the model is implemented in the rate form as

$$\begin{aligned} \dot{\boldsymbol{\sigma }}_\text {sw} = - \sum \limits _{i=1}^3 \lambda _{\text {sw},i} \hat{p}_{\text {sw},i} S_\text {eff,sw}^{\lambda _{\text {sw},i}-1}\, \textbf{n}_i \otimes \textbf{n}_i\ \dot{S}_\text {eff,sw} \end{aligned}$$

(4.15)

Table 4.1 List of parameters to be specified for the anisotropic swelling model

The parameters required for this anisotropic swelling law implementation in OGS-6 are summarized in Table 4.1. For this study, we limited ourselves to the use of isotropic swelling model only.

Porosity evolution is dictated by the solid mass balance and drives a power-law permeability evolution:

$$\begin{aligned} \textbf{k} = \sum \limits _{i=1}^3 k_{0,i} \left( \dfrac{\phi }{\phi _0}\right) ^{\lambda ^{k,i}}\, \textbf{n}_i \otimes \textbf{n}_i \end{aligned}$$

(4.16)

4.1.2 Model Setup and Partial Assembly

The waste repository concept which is the design basis of the China mock-up experiment as well as the sketch showing the components of the lab experiment are depicted in Figs. 4 and 5 in Wang (2010b), respectively. The main components are a heating strip and heater body in the center, a temperature control system, highly-compacted GMZ bentonite blocks, crushed bentonite, a steel tank enclosing the bentonite blocks, an insulation layer, a fluid injection system, sensors to measure temperature, stresses and relative humidity, gas collection and measurement system as well as a data acquisition system (Wang 2010b). As from the numerical simulation perspective, not all of these components can or need to be modeled using the fully coupled THM process. It is common practice to either exclude many of these components like the heater body, steel tank or insulation layer from the numerical model or the thermal, hydraulic and mechanical processes are “switched off” by modifying material properties such as permeability or porosity, which is not without numerical or conceptual issues. To circumvent this problem, a new feature called partial domain deactivation was implemented in OGS-6 which, as the name indicates, allows to disable or exclude a subset of physical processes for selected sub-domains by making use of partial assembly of the finite element matrices. This features brings along several advantages: at the pre-processing stage, all the components can be included while creating the geometry or mesh without any concern to go through this whole process again if some component needs to be excluded at some later stage. Furthermore, this feature allows to reduce computational cost tremendously by reducing the number of degrees of freedom as the governing equation for the process to be deactivated is not solved for that particular sub-domain. The reduced computational cost becomes quite significant when the disabled process is related to a vector field like solid displacement. Defining the boundary conditions is also simplified as for example, if the hydraulic process is disabled for the outer most layer in a mesh, there is no need to define physically redundant boundary conditions on the outer boundary for fluid flow in such a case. Furthermore, there is no need to define artificial material parameters like a very small value of porosity or permeability for steel for example, which further helps to eliminate convergence problems or numerical instabilities caused by sharp material parameter jumps on the interfaces between adjoining domains. Overall, the physical description becomes more plausible and is closer to the actual problem while at the same time improving numerical behavior.

Fig. 4.1

Setup for the numerical experiment

Figure 4.1 shows the sketch of the geometry used for this work (left) and the corresponding mesh (right). The hydraulic process is deactivated using the newly implemented sub-domain deactivation feature in OGS-6 for the domains representing the heater body, steel tank and insulation layer, in which no distinct fluid phase needs to be modeled. This is also shown by labeling the corresponding domains in Fig. 4.1a using TM which stands for Thermal-Mechanical process. The material properties used for bentonite are mostly taken from BRIUG (2013, 2014), those for the steel tank and insulation layer from Table 1 in Liu et al. (2014). For the heater body, we used the same properties as steel, except a higher value of the thermal conductivity (${80\,\mathrm{\text {W}\text {m}^{-1}\text {K}^{-1}}}$). A higher value of permeability is used for the crushed bentonite layer, as well as a lowered water retention potential as compared to the bentonite blocks. We will come back to this later in the upcoming section. The initial condition for temperature is $T_0={20\,\mathrm{{{}^{\circ }\text {C}}}}$ and the fluid pressure is initially set to $p_{\text {LR},0}={-80\,\mathrm{\text {M}\text {Pa}}}$ which corresponds to an initial saturation of $S_\text {L} = 0.5346$, while the initial horizontal and vertical displacements are kept at zero. As a result of malfunctioning of the lower part of heating strip in the experiment, the heating is imposed via the upper left half of the heater boundary only as shown in Fig. 4.1a, where the temperature is raised from ${20\,\mathrm{{{}^{\circ }\text {C}}}}$ to ${30\,\mathrm{{{}^{\circ }\text {C}}}}$ in ${100\,\mathrm{\text {d}}}$, from ${30\,\mathrm{{{}^{\circ }\text {C}}}}$ to ${90\,\mathrm{{{}^{\circ }\text {C}}}}$ in ${60\,\mathrm{\text {d}}}$ and then kept constant. In the actual lab experiment, the room temperature varied significantly throughout the entire duration of the experiment. We, thus, used the temperature recorded as the room temperature boundary condition on the outer boundaries. The fluid injection is done in two stages; in the first stage, the fluid is injected by using a Neumann boundary condition for the first ${500\,\mathrm{\text {d}}}$ through the lower part of the interface between the steel and crushed bentonite layer as shown in Fig. 4.1a and in the second stage by keeping the fluid pressure constant at ${0.2\,\mathrm{\text {M}\text {Pa}}}$ on the whole vertical interface between the steel tank and the crushed bentonite layer. Furthermore, the horizontal displacements are fixed on the left boundary (axial symmetry) while the vertical displacement is fixed on the bottom edge.

4.1.3 Selected Results

Before proceeding further, it is important to mention that there are several sources of uncertainties related to the experiment, and we proceed with the discussion of the results in two stages. In the first stage, we briefly discuss selected results obtained using the material parameters and boundary conditions provided in the previous section, while in the second stage, we discuss the related uncertainties in more detail and how they affect the results, as well as, give our perspective for further research in this regard.

Fig. 4.2

Temperature profiles at times $t = [100, 115, 130, 145, 160]$ days (left to right). Only the domains representing the heater, bentonite pellets and crushed bentonite are shown here

Figure 4.2 shows the distribution of temperature during the earlier heating stage i.e., the time to reach the peak temperature. The effect of heating through the upper part of the heater only, is clearly visible here and can also be seen in BRIUG (2014), even when a high value of thermal conductivity is used for the heater body. Therefore, it can be inferred that it is better to include the heater body in the numerical model, instead of applying the heat source as a boundary condition to the bentonite blocks directly. The latter would create an assymmetry in the temperature field that would be too strong compared to the measurements. In other words, heat conduction to the lower part of the heater is a significant effect for capturing heat transfer into the bentonite. We do not show any comparison of the distribution results with BRIUG (2014) as firstly, this type of plot merely gives a qualitative picture of the results and secondly, the considerable fluctuation in the room temperature makes it even harder to serve the purpose of a quantitative comparison. Figure 4.3 shows the saturation distribution in the domains representing the bentonite blocks and the crushed bentonite section at different times. As the fluid injection during the first stage is done through the lower part of the crushed bentonite section, the bottom right part shows higher saturation (gravity effects are also included). The saturation front moves faster in the crushed bentonite layer due to higher permeability as compared to the bentonite blocks. As the fluid pressure is raised to ${0.2\,\mathrm{\text {M}\text {Pa}}}$ after ${500\,\mathrm{\text {d}}}$, the crushed bentonite section gets fully saturated soon after and the saturation front moves to the left. But even after ${1000\,\mathrm{\text {d}}}$, the bentonite blocks do not get fully saturated in the upper left region which can partly be attributed to the saturation starting from the bottom right as well as the drying effect of the heater in the upper left region.

Fig. 4.3

Saturation profiles at times $t = [200, 400, 600, 800, 1000]$ days (left to right). Only the domains representing the bentonite pellets and crushed bentonite are shown here

Fig. 4.4

Temperature evolution in time at points in the area near the heater. Dashed red and green lines show the heater and room temperature boundary condition, respectively. “BRIUG 2014” in figure legend refers to the experimental data

Fig. 4.5

Temperature evolution in time at points in the middle region between the heater and the outer boundary. Dashed red and green lines show the heater and room temperature boundary condition, respectively

Figure 4.4 shows the evolution of temperature with time at two different points which are at the same horizontal distance from the heat source but at different heights. For all the plots showing temperature evolution, we also show the heater temperature (dashed red line) and room temperature (dashed green line) to have a clear view of the effect of these boundary conditions. The OGS-6 results at the point in the top area match very well with the experimental results, especially during the later times, while we see that OGS-6 shows higher temperatures for the point near the bottom side. Furthermore, it is to be noted that the influence of room temperature is relatively more dominant than the heater. The different temperature spreads between simulation and experiment show that the experiment is even more dominated by the lab temperature. Figure 4.5 shows the temperature evolution at points which are at a somewhat similar horizontal distance from the heater as well as outer boundary. It can be observed here again that the OGS-6 results in the upper region are in good agreement with the experimental results as compared to the lower region. But, in contrast to Fig. 4.4, the current parameterization gives relatively lower temperatures than the experimental results. Figure 4.6 shows the temperature evolution in the crushed bentonite region which is not only relatively nearer to the room temperature boundary but also is the area where the fluid injection takes place. The trend observed earlier can also be seen here as the OGS-6 results are able to reproduce the experimental results at the upper point very well, especially in the later stage which is also the time at which the fluid pressure is raised to ${0.2\,\mathrm{\text {M}\text {Pa}}}$. At the position in the lower area inside the crushed bentonite region, the model again gives slightly high temperatures.

Figure 4.7 shows the comparison of RH (relative humidity) evolution results between OGS-6 and the experimental data in the bottom region at three different points. At the point near the fluid injection boundary (green), the chosen settings for the numerical test, result in a good match with the experimental data while in the area away from the fluid source (orange and blue), the differences are relatively more pronounced but still the overall trend is captured well by the numerical test. In Fig. 4.8, which shows the RH evolution at two points in the central region ($y={1.1\,\mathrm{\text {m}}}$), the effect of desaturation by drying effect caused by the heater and the subsequent resaturation caused by the fluid injection can be observed near the heater boundary. This drying effect is not observed in the numerical results near the fluid injection boundary which is plausible given the temperature evolution. In the upper region as shown in Fig. 4.9, the numerical results are again in a better agreement with the experimental results, and show that the bentonite structure is still not fully saturated in the area near the heater.

Fig. 4.6

Temperature evolution in time at points in the area near the outer boundary (crushed bentonite area). Dashed red and green lines show the heater and room temperature boundary condition, respectively

Fig. 4.7

Evolution of relative humidity in time in the lower region. Solid lines show the OGS-6 results while the experimental results are plotted using dots

Fig. 4.8

Evolution of relative humidity in time in the middle region. Solid lines show the OGS-6 results while the experimental results are plotted using dots

Fig. 4.9

Evolution of relative humidity in time in the upper region. Solid lines show the OGS-6 results while the experimental results are plotted using dots

4.1.4 Discussion in View of Uncertainties and Outlook

In previous work for the China-Mock-Up experiment, numerical simulations were performed in two stages: a thermal process is simulated and solely used to compare the temperature results with the experimental data while a thermal-hydraulic process is used to compare results for relative humidity only. Such a separation may not be appropriate due to the highly complex nature of the coupled THM problem at hand. The drying effect caused by the heater, for example, results in a change in the saturation which in turn changes the thermal conductivity, thus influencing the temperature evolution and in turn the evolution of relative humidity following from both the temperature as well as fluid pressure. Thus, to analyze the results properly, both temperature and RH evolution must be calculated from a coupled model. As a consequence, the choice of material parameters as well as boundary conditions relevant to the fluid phase will strongly affect the temperature evolution. Similarly, the choice of material parameters which may only seem to affect the temperature evolution at first glance, like thermal conductivity, will also affect the RH evolution. To compare the RH results, only two observation points are usually discussed in the literature, namely the point shown with blue color in Fig. 4.8 and the point shown with green color in Fig. 4.9. Moreover, the data is plotted only for times less than ${500\,\mathrm{\text {d}}}$ (cf. also Fig. 4.18). We performed several numerical tests with different sets of material parameters as well as different fluid injection boundary conditions and were able to reproduce the results for these two mentioned points for most of the tests during this early time period. As seen from our results, this early time period ends prior to the arrival of the saturation front, i.e., the phase during which most changes and non-linearities occur. It must thus be considered insufficient for a model validation. Based on available data, significant uncertainty about the sensor locations remains, especially when it comes to their radial/horizontal position, and makes an exact comparison difficult. In Figs. 10–16 showing the experimental results for RH in BRIUG (2014), the horizontal position of the sensors is not specified. The comparison of RH data shown in our work is based on the experimental data obtained from BRIUG (2014) and the visual guess of the sensor locations shown in Chen et al. (2014). This additional data for RH evolution helped us to further confirm that it is important to use different material parameters affecting fluid flow for the crushed bentonite and the bentonite blocks, especially for the intrinsic permeability which appeared to have a strong influence on the results. We will come back to the observations made regarding the significance of intrinsic permeability later. Based on this comparison of the RH results at more locations and over the entire duration of the experiment (which has not been published earlier to the best of our knowledge), we believe that future works should focus on material parameters relevant especially to the water transport process (both in the liquid and the gas phase) as well as the fluid flow boundary condition.

In our previous study (Chaudhry et al., 2021) on the sensitivity analysis of a coupled THM problem, we observed that temperature and fluid pressure are strongly sensitive to the choice of thermal conductivity and intrinsic permeability which is confirmed also for the present case. The thermal conductivity of the heater body, steel tank and insulation layer seem to strongly affect the temperature evolution and thus more information is needed about the actual materials used in the experiment. The only work to the best of our knowledge which accounts for the insulation layer in the numerical model is Liu et al. (2014) where the thermal conductivity of the insulation is chosen as ${0.04\,\mathrm{\text {W}\text {m}^{-1}\text {K}^{-1}}}$. Thermal conductivities for insulation materials can reach up to ${0.1\,\mathrm{\text {W}\text {m}^{-1}\text {K}^{-1}}}$ and we observed that the temperature evolution is influenced by even a small change within this range. Intrinsic permeability on the other hand, showed a very strong influence on the evolution of RH. For example, the drying effect observed near the heater shown in Fig. 4.8 in blue color, becomes very pronounced if the intrinsic permeability of the bentonite pellets is reduced even slightly from $7\cdot 10^{-21}\,\text {m}^{2}$ to $5\cdot 10^{-21}\,\text {m}^{2}$. Furthermore, the choice of intrinsic permeability of the crushed bentonite layer appears to strongly dictate the horizontal and vertical direction of the movement of the saturation front. Thus, although an exact estimate of the value of intrinsic permeability of crushed bentonite may be difficult from experimental perspective but an informed expert guess can help the numerical implementation significantly. This is not a trivial issue because porosity and permeability exists at different scales: there are inter-block gaps acting as potential path ways and one may distinguish macro- and micro porosity in the blocks themselves. The present results indicate that it may be worth extending the model description accordingly, provided meaningful information on material and structural characterization can be obtained.

When it comes to the intrinsic permeability of expansive porous media, not only is the reduction of uncertainties in the material parameters significant but also the choice and complexity of the numerical model in terms of secondary coupling. It is a well-known fact that the intrinsic permeability of swelling porous media can be influenced by changes in both the effective transport porosity which is the proportion of the pore space available for fluid flow and the micro porosity related to the swelling potential. The relationship between the intrinsic permeability and these porosity measures is non-linear and transient with an additional time scale. Considering the high sensitivity to intrinsic permeability mentioned earlier, the coupling of swelling to porosity to permeability is expected to contribute to the complex patterns observed in the China-Mock-Up test, in which the drying front and saturation front approach each other from opposite directions causing subsequent shrinkage and swelling of these regions locally with a simultaneous long-range interaction imparted by the mechanical process (e.g., compression of regions adjacent to a swelling region). As the TRM process in OGS-6 allows to account for dual porosity media, the dependence of intrinsic permeability on the transport porosity, and takes into account the mechanical coupling we plan to further investigate the China-Mock-Up test by considering such extended formulations.

Currently, three-dimensional simulations are employed to study the effect of different representations of the fluid boundary conditions in combination with heterogeneity in the bentonite fill. While current analyses were based on data published previously on the China Mock-Up experiment, more data has been acquired as part of this project. This data addresses some of the points discussed above in the context of uncertainty. In particular, information on micro and macroporosity as well as HM-coupled tests and data on the swelling behavior are available Sect. 3.1. Incorporating this data into the simulations outlined above and alleviating open issues related to the thermal and hydraulic boundary conditions is expected to further improve the results.

4.2 THM–Modelling of Mock-Up Test in FLAC3D-TOUGH2

Juan Zhao, Düsterloh Uwe

4.2.1 FLAC3D-TOUGH2-Simulator

The TUC-chair for Geomechanics and Multiphysics Systems used their self-developed FLAC3D-TOUGH2-simulator (FTK-simulator) for numerical simulations in this research project. The FTK-simulator is based on the TOUGH-FLAC-simulator from Lawrence Berkeley National Laboratory (LBNL) and has been updated with the ability to handle finite strains and time-dependent rheology, specifically for the study of underground disposal of nuclear waste in various host rock formations, such as rock salt, claystone, and crystalline.

Fig. 4.10

FLAC3D-TOUGH2-coupled simulation processes

The FTK-simulator is a numerical tool that sequentially combines the TOUGH2-simulator and the FLAC3D-simulator according to the undrained split method. At the FTK-simulator, the TOUGH2 component handles the non-isothermal, multicomponent and multiphase flow sub-problem, while the FLAC3D-component handles the geomechanics sub-problem. The detailed process of this coupling method is shown in Fig. 4.10. In this method, the geomechanics sub-problem is solved first, assuming fluid mass and temperature remain constant. Pore pressure changes due to volumetric deformation are computed and the pore pressure induced by mechanical deformation is transferred to TOUGH2 during each FLAC3D-TOUGH2-communication step. Changes in porosity and permeability and geometry (large strain mode) are also transferred in these steps. TOUGH2 then solves the flow sub-problem, computes new pore pressures and variables to reach thermodynamic equilibrium (temperature, saturations). These new variables are transferred back to FLAC3D in each TOUGH2-FLAC3D-communication step to compute the new state of mechanical equilibrium. This process is repeated sequentially (Blanco-Martín et al., 2016).

4.2.2 Validation of the Implementation of the Barcelona Basic Model (BBM) into the FTK-Simulator Via Retrospective Analysis of a Bentonite Cube Test from DECOVALEX Project

The Barcelona Basic Model is a constitutive model for simulating the swelling behavior of bentonite when exposed to water. This process can significantly impact the long-term performance of a repository in crystalline/claystone rock, making this constitutive model crucial for accurate simulations. It has been implemented into the FTK-simulator at the TUC-chair for Geomechanics and multiphysics Systems. The implemented model code was first validated by performing retrospective analyses of simulation results from a swelling pressure test on a cube-shaped bentonite test specimen (shown in Fig. 4.11). This test was part of the FEBEX-experiment (Full-scale Engineered Barriers Experiment in Crystalline Host Rock) and has been studied as part of the DECOVALEX-project (Rutqvist et al., 2011; Alonso et al., 2005b).

The numerical simulation of the swelling behavior of a bentonite cube with an edge length of 20 mm, which has been clamped on all sides, during a 10-day gradual saturation process using the TOUGH-FLAC-simulator developed at Lawrence Berkeley National Laboratory (LBNL) is described in Rutqvist et al. (2011). The cube is assumed to have hydraulically tight side and top surfaces, corresponding to a bentonite cube installed in a cube-shaped steel tank with a permeable bottom surface, standing in a water reservoir. Note that wall friction effects are not considered in the simulation.

As described in Rutqvist et al. (2011), the water pressure in the reservoir is slightly above atmospheric pressure, at about p = 0.52 MPa. The initial pore gas pressure in the partially saturated bentonite cube is assumed to be the same as atmospheric pressure. The bentonite cubes initial water saturation is $S_l$ = 0.65, which leads to a negative pore water pressure caused by the capillary pressure according to the initial partial saturation in the pore space of the bentonite. This creates an additional hydraulic gradient between the water reservoir and the pore space in the bentonite cube, which accelerates the upward Darcy-flow of water and the gradual rise of water within the bentonite cube.

Fig. 4.11

Bentonite cube in water reservoir–FEBEX-experiment (Full-scale Engineered Barriers in Crystalline Host Rock)–DECOVALEX-project

Figure 4.12 illustrates the distribution of water saturation in the bentonite cube at time t = 4d, obtained using the TOUGH-FLAC-simulator (left) and the FTK-simulator (right). Both simulations show that the lower region of the cube is fully saturated at time t = 4d, while the upper region has a saturation of $S_l$ $\approx$ 0.85–0.9.

Fig. 4.12

Water saturation in bentonite cube–simulation results at t = 4d –TOUGH-FLAC-simulator (left) as presented in Rutqvist et al. (2011) and FTK-simulator (right)

Figure 4.13(left) shows the simulation results of the time-dependent development of water saturation and gas pressure in the center and upper part of the bentonite cube, obtained using the TOUGH-FLAC-simulator as documented in Rutqvist et al. (2011) and the FTK-simulator. Figure 4.13 (right) shows the simulation results of the time-dependent development of swelling pressure in the upper part of the bentonite cube, obtained using the TOUGH-FLAC-simulator as documented in Rutqvist et al. (2011) and the FTK-simulator. It should be noted that, in addition to the Barcelona Basic Model for modeling THM-coupled processes in the bentonite, for comparative consideration the Linear Elastic Swelling Model described in Nguyen et al. (2007) was also used in Rutqvist et al. (2011), but the swelling pressure development obtained with the Barcelona Basic Model is considered more realistic in Rutqvist et al. (2011).

Fig. 4.13

Time-dependent water saturation, gas pressure (left) and swelling pressure (right) in bentonite cube–simulation results–TOUGH-FLAC-simulator (Rutqvist et al., 2011) and FTK-simulator

From Fig. 4.13 (left), it is apparent that the bentonite cube reaches full water saturation at around t = 10d, with water saturation $S_l$ = 1 in the upper region. As expected, saturation occurs earlier in the middle of the cube (red solid and dashed curves) than at the top of the cube (green solid and dashed curves). The water saturation increases monotonically over time, while the gas pressure reaches a maximum of about $p_g$ $\approx$ 0.6–0.7 MPa at around t = 6.5d and then decreases. The reason for this is that the initially present air in the pore space is compressed by the rising water level in the upper region of the cube during the first days, but then gets gradually squeezed out passing the water molecules on the bottom of the cube due to the compression-induced rising gas pressure. This process leads to a gradual reduction of the gas pressure, which is not yet complete even at the end of the simulation, as the water pressure and gas pressure must reach the level of the water pressure in the reservoir at the bottom of the bentonite cube at complete saturation. Figure 4.13(right) shows that the swelling pressure reaches a level of about $p_\text {swelling}$ $\approx$ 5–5.5 MPa at the end of the saturation process, which is in good agreement with the observed swelling pressure in Alonso et al. (2005b), according to Rutqvist et al. (2011).

From Figs. 4.12 and 4.13, it is clear that the numerical results obtained using the TOUGH-FLAC-simulator and the FTK-simulator are very similar, thus the validation of the FTK-simulator in terms of the implementation of the Barcelona Basic Model for the physical modeling of the swelling behavior of bentonite appears to be successful.

4.2.3 Validation of the Implementation of the Barcelona Basic Model (BBM) into the FTK-Simulator Via Retrospective Analysis of the China-Mock-Up-Experiment

As part of a retrospective analysis of the measured test data for the Chinese Mock-up Test on GMZ, BRIUG has performed several numerical simulations, as documented in BRIUG (2014). These simulations included thermal, thermally-hydraulically coupled, and thermally-hydraulically-mechanically coupled simulations using the finite element code LAGAMINE. These simulations have been performed using simplified, rotationally symmetric 2D-models without considering the joints between the bentonite blocks. The diffusion of water vapor resulting from evaporation processes in the existing pore air was considered in all simulations, as it was necessary to reproduce the measured data on the time-dependent development of relative humidity in different measurement positions.

Fig. 4.14

Rotationally symmetric simulation model for retrospective analysis of Chinese Mock-up Test on GMZ: comparison of FTK-simulator a at Chair of Geomechanics and multiphysics Systems and LAGAMINE-simulator b and c at BRIUG

TUC has also done the retrospective analyses performed at BRIUG, but using the FTK-simulator with a similarly simplified rotationally symmetric 3D-segment model. The rotationally symmetric 3D-segment model used in the FTK-simulator, as shown in Fig. 4.14a, represents an 11.25$^\circ$ -segment of a cylindrical structure. This significantly reduces the needed computational effort compared to a simulation with a full 3D-cylinder model. This computational model is used at the Chair of Geomechanics and multiphysics Systems for all simulations performed with the FTK-simulator for the China Mock-up Test on GMZ (thermal / thermally-hydraulically coupled/thermally-hydraulically-mechanically coupled). In contrast, BRIUG (2014) used different models for thermal, thermally-hydraulically and thermally-hydraulically-mechanically coupled simulations, as shown in Fig. 4.14b, c.

Fig. 4.15

Time-dependent heater temperature and injected water quantity, as documented in BRIUG (2014)

As seen in Fig. 4.14, the heater core (gray zones), which represents the actual heat source, is not modeled with its planned height of 1.600 mm, but only with a height of 800 mm. Therefore, only the upper half of the heater is considered in the model. This is because the lower half of the heater was malfunctioning at the time of activation and could not be fixed after the test setup was completed. The radius of the heater core is 30 mm and it is surrounded by a steel jacket (brown zones) with a radius of 150 mm. A layer of bentonite pellets (magenta zones) with a thickness of 15 mm surrounds the heater and separates it from the compacted bentonite blocks (green zones), whose outer radius is 400 mm. To prevent direct contact between the compacted bentonite blocks and the surrounding steel cylinder, a layer of bentonite pellets with 50 mm thickness is placed around the bentonite blocks.

For the retrospective analysis of this TH2M-coupled large-scale experiment, various physical quantities must be taken into account. The most straightforward quantity to model in numerical simulations is the temperature at different positions of the experimental setup, as the temperature distribution primarily depends on the time-dependent heater temperature shown in Fig. 4.15 and the room temperature shown in Fig. 4.16. Although the temperature distribution is also influenced by the thermal conductivity and heat capacity of the bentonite sample, these factors are not significantly affected by other physical quantities, but mainly from water saturation of the bentonite pore space.

However, since the time-dependent heater temperature shown in Fig. 4.15 is relatively complex, the time-dependent heater temperature used in the numerical simulations documented in BRIUG (2014) is simplified as shown in Fig. 4.16.

• (a) Thermal Simulation

In accordance to the procedure documented in BRIUG (2014), the TUC-retrospective analysis of the large-scale bentonite test performed at BRIUG uses the FTK-simulator to initially consider only the thermal processes. Figure 4.17(left) shows the temperature distributions obtained with the FTK- simulator at three different times (top) and the simulation results obtained at BRIUG for the temperature distribution at the same times (bottom). It can be seen that the simulation results of the FTK-simulator are in good agreement with the simulation results of BRIUG. In both simulations, temperatures initially increase rapidly in the first few months after the start of the test, with larger temperature increases observed near the heater than farer away from the heater, as expected. With increasing distance from the heater, temperatures approach the outer boundary condition (room temperature outside of the test setup).

Fig. 4.16

Simplified time-dependent heater temperature and room temperature, as documented in BRIUG (2014)

As can be seen from Fig. 4.17 (right), the time-dependent temperature developments at selected positions of the specimen obtained with the FTK-simulator are in good agreement with the corresponding time-dependent temperature developments obtained at BRIUG. Additionally, the simulation results of both simulators are consistent with the measured data which are also shown in Fig. 4.17 (right). This validates the use of the FTK-simulator for the case of exclusively thermal simulation.

Fig. 4.17

Temperature distribution in bentonite block structure at $t_1$ = 30 d, $t_2$ = 1 a, and $t_3$ = 3 a after test start (left): comparison of FTK-simulator (top) and LAGAMINE-simulator (bottom) results with BRIUG (2014); time-dependent temperature in different areas of the bentonite block structure (right)

Fig. 4.18

Pore water pressure distribution in bentonite block structure at $t_1$ = 30 d, $t_2$ = 200 d, and $t_3$ = 430 d after test start (left): comparison of FTK-simulator (top) and LAGAMINE-simulator (bottom) results with BRIUG (2014); time-dependent relative humidity at chosen locations in the bentonite block structure (right)

• (b) Thermally-hydraulically coupled simulation

In BRIUG (2014), simulation results of thermally-hydraulically coupled simulations for the large-scale bentonite test are documented, in addition to the simulation results of the exclusively thermal simulations described above. These simulation results were also used to validate the FTK-simulator.

In the thermally-hydraulically coupled simulations of the test, the heat output of the heaters is specified the same as in the thermal-only simulations. The water injection rate applied to the outer edge of the test structure is based on Fig. 4.6. Similar to the procedure documented in BRIUG (2014), a non-uniform water supply is assumed in the lower area of the test structure than the rest of the bentonite block structure, taking into account the effect of gravity within the water injection area.

The simulation is performed here only up to time t = 430 d after the start of the experiment, since no further simulation results are given in BRIUG (2014) for later times of the experiment that can be used to validate the FTK-simulator.

Figure 4.18(left) shows a comparison of the pore water pressure distribution obtained with the FTK-simulator at times $t_1$ = 30 d, $t_2$ = 200 d and $t_3$ = 430 d, with corresponding results shown in BRIUG (2014). It can be seen that the pore water pressure near the heater increases over time, from an initial value of $p_1$ $\approx$ –93 MPa to $p_1$ < –283 MPa, indicating a decrease in water saturation and therefore an increase in capillary pressure. Analysis of the simulation results reveals that this decrease in water saturation is caused by evaporation of pore water in the pore space of the bentonite block structure, which may also be considered in the FTK-simulator. The evaporation process is dominated by the simultaneous thermally-induced expansion of liquid water, resulting in the superposition of these two processes not causing a decrease in saturation. However, the decrease in saturation can be explained by analyzing the influence of water vapor diffusion resulting from the evaporation of pore water in the gas phase. The diffusion direction is oriented to the concentration gradient, so the water vapor moves from areas of higher concentration to areas of lower concentration, from the heater near-field to cooler areas of the bentonite block structure. This causes further pore water to evaporate in the heater near-field, resulting in a gradual decrease in saturation.

The simulation results obtained with the FTK-simulator and those documented in BRIUG (2014) are very similar, but not identical, at the 3 points in time. The discrepancy is attributed to the lack of a specified diffusion coefficient for the diffusion of water vapor in the gas phase in the simulation documented in BRIUG (2014). To achieve a better match, the diffusion coefficient was adjusted in the FTK-simulation by a sensitivity analysis.

In BRIUG (2014), the simulation results for selected positions of the bentonite block structure are compared with the corresponding measured values for the time-dependent relative humidity in addition to the extensive representation of the pore water pressure distribution shown in Fig. 4.18(left). As per Fredlund and Rahardjo (1993), the relative humidity RH can be calculated from the capillary pressure $p_c$ and suction $s = p_c$ using Eq. (4.17):

$$\begin{aligned} RH = \exp \left( \frac{-s\cdot w_v}{R\cdot T \cdot \rho _w}\right) \end{aligned}$$

(4.17)

 

s:

– suction kPa,

$w_v$:

– molar mass of water vapor with $w_v$ $\approx$ 18.016 g/mol,

R:

– general gas constant with R $\approx$ 8.31432 J/(mol K),

$\rho _w$:

– water density with $\rho _w$ $\approx$ 998 kg/m$^3$ at 20 $^\circ$C, and

T:

– absolute temperature with $T = (273.16 + \frac{\theta }{1{^\circ }C}) K$, where $\theta$ is temperature in $^\circ$C.

 

The results of the FTK-simulator are in good agreement with the measured values as well as with the simulation results from BRIUG (2014) for the time-dependent development of relative humidity at two selected positions within the bentonite block structure, as shown in Fig. 4.18(right). The relative humidity at one position decreases due to the diffusion of water vapor resulting from evaporation into cooler areas, while at the other position it increases continuously due to distance from the heater and proximity to the external water supply. The validation of the FTK-simulator has been successful for the case of thermally-hydraulically coupled simulations.

• (c) Thermally-hydraulically-mechanically coupled simulation

In BRIUG (2013), a thermally-hydraulically-mechanically coupled simulation for the large-scale bentonite test is documented, along with some related simulation results. The Barcelona Basic Model (BBM) was used to model the mechanical behavior of the bentonite material, as it also accounts for the swelling pressure development. However, it should be noted that different material parameters and initial and boundary conditions were applied in this simulation, as it was not a retrospective analysis but a prognostic numerical simulation.

The thermally-hydraulically-mechanically coupled simulation using the FTK-simulator was able to replicate the results from BRIUG (2013) in terms of temperature, degree of saturation and suction development for a horizontal section at half height of the bentonite block structure, as shown in Fig. 4.19. However, some material parameters had to be estimated due to a lack of sufficient information in BRIUG (2013).

Fig. 4.19

Time-dependent development of temperature, water saturation, suction, and swelling pressure in a horizontal section nearly at half height of the bentonite block structure (left: LAGAMINE-simulator BRIUG (2013), right: FTK-simulator)

In BRIUG (2013), the time-dependent development of the swelling pressure calculated in the simulation is also documented for a selected observation point ‘A’, which is located nearly at half height of the bentonite block structure in direct contact with the heater surface, as shown in Fig. 4.19(left). It can be seen that a swelling pressure of about $p_\text {swelling}$ = 0.55 MPa builds up relatively quickly at observation point ‘A’ within the first 6 months after the start of the test, which then continues to rise at a lower rate up to about $p_\text {swelling}$ = 1.45 MPa until the end of the simulation. As can be seen from Fig. 4.19(right), the shape of the swelling pressure development curve shown in BRIUG (2013) could unfortunately not be reproduced with the FTK-simulator, since at the FTK-simulation the swelling pressure increases only slowly at first and then more and more rapidly with increasing saturation. Nevertheless, the validation of the FTK-simulator for the case of the thermally-hydraulically-mechanically coupled simulation appears to be largely successful.

4.3 THM–Modelling of Mock-Up Test in LAGAMINE

Jingbo Zhao, Liang Chen, Ju Wang, Yuemiao Liu, Shengfei Cao, Qi Zhang

4.3.1 Description of the Coupled THM

Based on previous experiments, we have introduced a constitutive model (Chen et al., 2012b; Zhao et al., 2016) to replicate the principal THM coupling behavior of GMZ bentonite. Our model incorporates the transfer of heat, moisture (in liquid water and water vapor form), and air in a deformable, unsaturated soil, while also utilizing the BBM model to simulate mechanical behavior. This section provides a brief introduction to our proposed model.

4.3.1.1 Mass Conservation of the Phases

For water, the mass conservation equation is derived by adding the balance equation of liquid water and water vapor. This equation considers changes in water storage and the divergence of water flow in each phase, as Collin et al. (2002) explains:

$$\begin{aligned} \frac{\partial { \rho }_{\textrm{w}} nS_{\textrm{r,w}} }{\partial t} +div(\rho _{\textrm{w}} \underline{f}_{\textrm{w}} )+\frac{\partial \rho _{\textrm{v}} nS_{\textrm{r,g}} }{\partial t} +div(\underline{i}_{{} {} {} \textrm{v}} \rho _{\textrm{v}} \underline{f}_{\textrm{g}} )=0 \end{aligned}$$

(4.18)

where $\rho$_w is the liquid water density, n is the medium porosity, S_r,w is water saturation degree in volume, f $_\alpha$ is the macroscopic velocity of the phase $\alpha$ ($\alpha$ = liquid water, gas), $\rho$_v is the water vapor density, S_r,g is water vapor saturation degree in volume, i_v is the non-advective flux of water vapor.

Generally, the medium consists of a mixture of water vapor and dry air. To conserve mass of the dry air, the equation must consider the contributions of both the dry air and the dissolved air in water, which can be mathematically expressed as:

$$\begin{aligned} \frac{\partial \rho _{a} nS_{\textrm{r,g}} }{\partial t} +div(\underline{i}_{{} {} {} a} +\rho _{a} \underline{f}_{\textrm{g}} )+\frac{\partial H\rho _{a} nS_{\textrm{r,w}} }{\partial t} +div(H\rho _{a} \underline{f}_{w} )=0 \end{aligned}$$

(4.19)

where i_a is the non-advective flux of dry air, $\rho$_a is dry air density and H is Henry’s coefficient.

4.3.1.2 Momentum Conservation of the Phases

The generalized Darcy’s law for multiphase porous medium gives the velocity:

$$\begin{aligned} \underline{f}_{\alpha } =-\frac{k_{int} k_{\textrm{r,}\alpha } }{\mu _{\alpha } } [\underline{\nabla }p_{\alpha } +\rho _{\alpha } g\underline{\nabla }y] \end{aligned}$$

(4.20)

where p $_\alpha$ is the pressure of the phase $\alpha$, g is the gravitational acceleration, u $_\alpha$ is the dynamic viscosity of the phase $\alpha$, k_int is the intrinsic permeability, and k $_{\text {r},\alpha }$ is the relative permeability of the phase $\alpha$, $\alpha$=liquid water, gas.

In a convoluted medium, both water vapor and dry air are assumed to adhere to Fick’s diffusion law, as formulated by Philip and De Vries (1957b):

$$\begin{aligned} \underline{i}_{{} {} {} \alpha } =-D_{\textrm{atm}} \tau _{\textrm{v}} nS_{\textrm{r,g}} \underline{\nabla }\rho _{\alpha } \end{aligned}$$

(4.21)

where D_atm is the molecular diffusion coefficient, and $\tau$_v is the tortuosity, $\alpha$ = water vapor, dry air.

4.3.1.3 Energy Conservation of the Phases

Regarding the thermal problem, we assume that there is equilibrium between the phases, resulting in the same temperature for all phases. The total enthalpy of the system can then be expressed as the sum of each component’s enthalpy:

$$\begin{aligned} \begin{array}{l} {\phi =nS_{\textrm{r,w}} \rho _{\textrm{w}} c_{\textrm{p,w}} (T-T_{0} )+nS_{\textrm{r,g}} \rho _{\textrm{a}} c_{\textrm{p,a}} (T-T_{0} )+(1-n)\rho _{\textrm{s}} c_{\textrm{p,s}} (T-T_{0} )+} \\ {nS_{\textrm{r,g}} \rho _{\textrm{v}} c_{\textrm{p,v}} (T-T_{0} )+nS_{\textrm{r,g}} \rho _{\textrm{v}} L} \end{array} \end{aligned}$$

(4.22)

where T is the temperature, T₀ is definition temperature, c $_{\text {p},\alpha }$ is the specific heat of the phase $\alpha$ ($\alpha$= solid, liquid water, gas), $\rho$_s is the solid density, The last enthalpy term corresponds to the heat stored during the water vaporization.

In this medium, heat is transported through conduction, convection, and vaporization, as shown in the following equation:

$$\begin{aligned} \begin{array}{l} {q=-\Gamma \underline{\nabla }T+(\underline{i}_{{} {} {} {} \textrm{v}} +\rho _{\textrm{v}} \underline{f}_{\textrm{g}} )L+} \\ {[c_{\textrm{p,w}} \rho _{\textrm{w}} \underline{f}_{\textrm{w}} +c_{\textrm{p,a}} (\underline{i}_{{} {} {} {} \textrm{a}} +\rho _{\textrm{a}} \underline{f}_{\textrm{g}} )+c_{\textrm{p,v}} (\underline{i}_{{} {} {} {} \textrm{v}} +\rho _{\textrm{v}} \underline{f}_{\textrm{g}} )](T-T_{0} )} \end{array} \end{aligned}$$

(4.23)

where $\Gamma$ is the medium conductivity.

Neglecting the terms for kinetic energy and pressure energy, we can ultimately obtain the energy balance:

$$\begin{aligned} \frac{\partial \phi }{\partial t} +div(q)-Q=0 \end{aligned}$$

(4.24)

Q is a volume heat source.

4.3.1.4 Mechanical Model

For the BBM (Alonso et al., 1999), the yield surfaces are composed of three parts. In the (p, q) space, for a given suction, the yield surface can be written as:

$$\begin{aligned} q^{2} -M^{2} (p+p_{\textrm{s}} )(p_{0} -p)=0 \end{aligned}$$

(4.25)

where p is the mean stress, q is the deviatoric stress, M is the slop of the critical line, p_s is the soil strength in extension and p₀ is the pre-consolidation pressure.

In the (p, s) space, pre-consolidation pressure p₀ varies with the suction s, which is well known as the LC curve:

$$\begin{aligned} p_{0} =p_{c} \left( \frac{p_{0}^{*} }{p_{c} } \right) ^{\frac{\lambda (0)-k}{\lambda (s)-{ k}} } \end{aligned}$$

(4.26)

where p_c is a reference pressure, p₀^* is the pre-consolidation in saturated condition, k is the elastic slope of the compressibility curve against the net mean stress, $\lambda$(s) refers to the plastic slope of the compressibility curve against the net mean stress and $\lambda$(0) is the plastic slope in saturated condition. The relationship between $\lambda$(s) and $\lambda$(0) can be expressed as:

$$\begin{aligned} \lambda (s)=\lambda (0)[(1-\gamma )\exp (-\beta s)+\gamma ] \end{aligned}$$

(4.27)

where $\gamma$ and $\beta$ are the parameters describing the changes in soil stiffness with suction. When the soil is saturated, $\lambda$(s) is equal to $\lambda$(0).

The confirmed phenomenon of suction variations causing irreversible volumetric deformations has led to the adoption Thus the following yield locus named SI is adopted:

$$\begin{aligned} F_{2} =s-s_{0} =0 \end{aligned}$$

(4.28)

where s is the suction and s₀ is the maximum historic suction of the soil.

The evolution of the yield surface is controlled by the hardening parameters p₀^* and s_0. They depend on the total plastic volumetric strain increment d $\varepsilon$_v^p as follows:

$$\begin{aligned} dp_{0}^{*} =\frac{(1+e)p_{0}^{*} }{\lambda (0)-\kappa } d\varepsilon _{v}^{p} \end{aligned}$$

(4.29)

$$\begin{aligned} ds_{0} =\frac{(1+e)(s_{0} +P_{at} )}{\lambda _{\textrm{s}} -k_{\textrm{s}} } d\varepsilon _{v}^{p} \end{aligned}$$

(4.30)

where e is the porosity of the soil, p_at is the atmosphere pressure, k_s and $\lambda$_s are the elastic and plastic stiffness parameter for suction variation, respectively.

4.3.2 Model Setup

4.3.2.1 Geometry and Boundary Conditions

A two-dimensional axisymmetric numerical model is proposed due to the symmetrical nature of the infrastructure of the China mock-up. The program LAGAMINE was utilized to discretize the model domain using rectangular grids which consisted of 4244 nodes and 1482 elements, as shown in Fig. 4.20. The mesh clearly highlighted the five different material zones. The thermal and hydraulic boundary conditions were controlled by heating and hydration systems, while the nodes were subjected to a fixed horizontal/vertical displacement at x=0 and y=0 respectively. In this section, we present numerical simulations conducted over a period of 1800 days using experimental data between April 1, 2011, and December 31, 2015. It is important to note that April 1, 2011, was designated as “day 0” on the time scale in the model.

Fig. 4.20

Boundary conditions and meshes (unit: mm)

The numerical simulation of the heater strip entails a heating process that occurs in three stages. Initially, for the first 135 days, the temperature was maintained at 30 $^\circ$C. Next, from day 136 to day 1800, the temperature gradually increased to 90 $^\circ$C. Finally, the temperature remained constant for the remainder of the test, as depicted in Fig. 4.21. Although the thermal insulation materials were installed on the outer surface of the China-mock-up facility, the interior temperature was significantly influenced by the variation of the room temperature. To account for this, we considered the room temperature as the outer thermal boundary condition and defined it as a piecewise function in the model using recorded data, as shown in Fig. 4.22.

Fig. 4.21

Heating process of the heater strip

Fig. 4.22

Variation in room temperature with time

In the China-mock-up test, the hydration process is regulated by a rate of water injection. The total mass variation of water injection is displayed in Fig. 4.23a. The average water injection rate was used in the numerical model to depict the water injection process on the right boundary, as illustrated in Fig. 4.23b. The remaining borders were assigned a default zero-flow boundary condition. In fact, the hydration process on the right boundary in the vertical direction was inhomogeneous due to the gravity effect. As there was no experimental data available on water injection rate, the right boundary was divided into two parts, namely upper and lower boundary (see Fig. 4.20). During the initial hydration stage, the majority of the injected water accumulated at the bottom, resulting in a higher injection rate at the lower boundary than the upper boundary. As the bentonite at the bottom became saturated, the injection of water in this area became less impactful.

Fig. 4.23

Water injection process of China-mock-up

In the numerical model, the GMZ bentonite was initially saturated with 31 % water and had a suction of 80 MPa. To achieve better numerical convergence, the gas pressure was held constant and dissolved air in water was not taken into account. In the hydro-mechanical simulation, the study focused on compacted bentonite blocks and crushed pellets. The water content and stresses in the heater strip, high thermal conductivity material, and steel tank remained constant throughout the heating and hydration process.

4.3.2.2 Model Parameters Setting

Among the hydrothermal properties, the soil water characteristic curve plays a crucial role in determining water intake volume and the degree of saturation variation. Chen et al. (2015) proposed the following correlation between suction and degree of saturation, based on their experimental investigation of GMZ bentonite:

$$\begin{aligned} S_{\textrm{r,w}} =S_{\textrm{r,res}} +a_{3} \frac{S_{\textrm{r,u}} -S_{\textrm{r,res}} }{a_{3} +(a_{1} s)^{a_{2} } } \end{aligned}$$

(4.31)

Where S_r,u is the maximum degree of saturation in the soil, S_r,res is the residual degree of saturation for a very high value of suction, and a_i is the parameters (i=1,2,3).

For the compacted bentonite, values of S_r,u and S_r,res are 1.0 and 0.1, According to the numerical results of soil water retention curve shown in Fig. 4.24, the values of the parameters a_i can be obtained: a₁ = 7.0$\times$10^–6 Pa^–1, a₂ = 0.9, a₃ = 70.

Fig. 4.24

Water retention curve of the GMZ bentonite

The unsaturated hydraulic conductivity of the bentonite is highly dependent on the degree of saturation S_r,u. According to the unsaturated permeability test (Chen et al., 2015), the intrinsic permeability of saturated GMZ bentonite is about k_int = 2.5$\times$10^–20 m². An empirical relationship for the relative permeability of the bentonite in terms of the degree of saturation is given as follows in Fig. 4.25:

$$\begin{aligned} k_{\textrm{r,w}} =\frac{(S_{\textrm{r,w}} -S_{\textrm{r,res}} )^{3} }{(S_{\textrm{r,u}} -S_{\textrm{r,res}} )^{3} } \end{aligned}$$

(4.32)

Fig. 4.25

Relative permeability of the GMZ bentonite as a function of degree of saturation

The relationship between thermal conductivity and water content is linear, as noted in the study by Zhao et al. (2016), which also presents other parameters describing the hydrothermal and mechanical properties.

4.3.3 Result Analysis

4.3.3.1 Thermal Response

The temperature at various locations was compared between the predicted and experimental results. These comparisons are shown in Fig. 4.26. Initially, the temperature increased due to the heating process during the first 255 days. However, the subsequent temperature fluctuations were primarily caused by the room temperature variations and the heating process, resulting in minor thermal fluctuations. For example, Figs. 4.21 and 4.22 illustrate that the temperature of the heater strip remained at 60$^\circ$C from the 180th to the 212th day. The room temperature was the dominant factor during this period and caused a more pronounced decrease in temperature. Subsequently, on the 212th day, the temperature of the heater strip was increased to 70$^\circ$C, resulting in a corresponding temperature increase due to heating. As the temperature field approached steady state, the room temperature once again became a significant factor and caused a noticeable decrease in overall temperature.

Good agreement is evident between the experimental data and the numerical results both at the bottom (Fig. 4.26a, b) and in the middle section of the facility (Fig. 4.26c, d). Notably, minor temperature fluctuations are accurately reproduced in the numerical modeling. For instance, a significant temperature fluctuation is observed from the 700th to the 720th day, and the numerical analysis replicates this observation effectively. However, a relatively significant difference between the predicted and experimental results is observed in the upper zone of the facility (Fig. 4.26e, f). This disparity can be attributed to various factors, such as non-uniform hydration rates and improved thermal insulation conditions at the top of the facility. The authors suggest that simplifications in gas transportation may be a key factor. In reality, as the heating process progresses, heat energy gradually accumulates in the upper zone, leading to a temperature increase. In numerical simulations, gas pressure is assumed to be constant, so the gas velocity caused by pressure differences is not considered, and the heat transfer induced by gas pressure differences is not represented in the numerical study. Additionally, it’s observed that room temperature remains stable from the 1510th day in Fig. 4.22, and the simulated temperature effectively replicates this phenomenon.

Fig. 4.26

Comparison of the predictive temperature with the experimental results at different locations of the China-mock-up facility

Figure 4.27 presents that the distribution of temperature with time during the heating process. The heater strip and room temperature have an obvious influence on the temperature distribution. In the first 135 days the temperature of the heat strip was kept at 30 $^\circ$C that was close to room temperature. Thus, there does not exist the difference of the temperature distribution in the model domain at time 100days. Then with the increase of the temperature of the heat strip from 136 days to 255 days, the influence of the heat stirp on the temperature distribution became a dominant factor. This phenomenon could be found at times of 100 days, 200 days, 300 days in Fig. 4.27. When the temperature field was close to the steady state, the room temperature has a significant impact on the temperature distribution. For instance, the room temperature was relatively low at time 300 days. The corresponding temperature close to the boundary was also low and changed with the increase of the room temperature. Until the room temperature gradually reached to a stable condition after 1510 days, the temperature distribution in the domain almost keep the same at times of 1600 days and 1800 days.

Fig. 4.27

Temperature distribution at different times

4.3.3.2 Hydraulic Response

Figure 4.28 presents the predicted relative humidity distributions at various locations within the facility. Initially, relative humidity remained relatively constant before the hydration process, and the simulated relative humidity closely matched experimental data for this period. However, as the hydration process commenced, relative humidity variations became more significant. Notably, in areas near the electrical heater (Fig. 4.28b, c, d, e), a desaturation-saturation process occurred. This was attributed to the interplay between saturation due to water penetration and drying caused by the heater. The low permeability of GMZ bentonite meant that the drying effect initially dominated. While the numerical study showed good overall agreement with experimental data, the variation in relative humidity was slower than observed in the experiments. This difference is largely attributed to the presence of gaps between the compacted bentonite blocks. In the experiments, water initially penetrated into the blocks through these gaps or along sensor cables. In the numerical study, the compacted bentonite blocks were assumed to be homogeneous, neglecting the existence of gaps. As a result, the predicted saturation and desaturation were less pronounced than the experimental measurements, as also evident in Fig. 4.29.

Fig. 4.28

Comparison of the predictive relative humidity with the experimental results at different locations of the China-mock-up facility

Figure 4.29 presents porewater pressure distribution at different times. The simulated results shows that the compacted bentonite has not yet reached saturation after 1800 days.The maximum of porewater pressure is –9.2 MPa. The saturated process was much slower than the experimental data due to the neglect of the gaps in the model. Besides, a desaturation-saturation process could be clearly presented due to the heating process at times of 200 days, 600 days, 1200 days.The minimum of porewater pressure is around –230 MPa that is much lower than the initial suction. These phenomena keep consistent with the simulated results in Fig. 4.28.

Fig. 4.29

Porewater pressure distribution at different times. Only the domains representing compacted bentonite blocks and crushed bentonite pellets are presented here

4.3.3.3 Mechanical Response

Figure 4.30 provides a comparison between the predicted swelling pressure and experimental values at different locations. The maximum measured swelling pressure ranged from 0.5 MPa to 2.5 MPa, depending on the location. The proposed model successfully replicated the overall trend of swelling pressure variation, capturing some key characteristics present in the experimental data and predicted results. As shown in Figs. 4.23 and 4.30b, c, e, a pronounced decrease in swelling pressure is observed due to the temporal cessation of water injection from the 972nd to the 1048th day and the drying effect. However, noticeable differences are also observed at certain locations, particularly in points near the hydration boundary. This discrepancy can be partly attributed to the fact that the expansion strain induced by changes in the microstructure of bentonite during the wetting process was not considered in the BBM model (Alonso et al., 1999). Additionally, the omission of stress release resulting from initial gaps between sensors and blocks may be another significant contributing factor.

Fig. 4.30

Comparison of the predictive swelling pressure with the experimental results at different locations of the China-mock-up facility

4.4 Diffusion of Radionuclides in Heterogeneous Sediment Formations

Haibing Shao, Chaofan Chen, Tao Yuan, Renchao Lu, Cornelius Fischer, Olaf Kolditz

This chapter is based on the publication by Chen et al. (2022) in a generalized form.

4.4.1 Upscaling Workflow for Effective Diffusion Coefficient Determination

In this study, the effective diffusion coefficients are derived through a comprehensive upscaling procedure founded on multiscale digital rock models. These models encompass three key scales: the nanometer scale, micrometer scale, and centimeter scale. At the nanometer scale, digital representations of pore network geometries are meticulously reconstructed from image data acquired using a focused ion beam scanning electron microscope (FIB-SEM). At this level, we take into account the electrostatic diffusion of cations and anions within the diffuse layer, and employ the Donnan approach to calculate the surface electrostatic potential (Appelo and Wersin, 2007; Parkhurst et al., 2013; Yuan and Fischer, 2021). These calculated effective diffusion coefficients at the nanometer scale serve as crucial input parameters for the diffusive transport calculations in the micrometer-scale model.

At the micrometer scale, the digital rock models introduce variability in mineral composition. Transport calculations conducted at the nanometer scale reveal that the effective diffusivity in matrix-supported (MS) texture pores is notably higher, up to two orders of magnitude, in comparison to grain-supported texture pores (GS) and intragranular pores (IG). Consequently, the microstructure of the host rock consists of a combination of highly permeable MS pores and less permeable non-MS pores (GS and IG) (for definition of MS, GS, and IG see Bollermann et al. (2022)). According to the experimental analysis (Bollermann et al., 2022), the clay laminae are comprised of 70% clay mineral aggregates, with 85% being matrix-supported (MS). The mixture of clay and sand layers consists of 50% clay mineral aggregates, with 20% being MS. The carbonate lenses are composed of 54% calcite and 10% clay mineral aggregates, resulting in 5% grain-supported (GS) content and 65% intragranular (IG) content. The effective diffusion coefficient at the micrometer scale is subsequently determined by solving Eqs. (4.33) and (4.34). For a more comprehensive understanding of the upscaling workflow from the nanometer (nm) to the micrometer ($\upmu$m) scale and the reconstruction of digital rock models, readers are encouraged to refer to the works of Yuan and Fischer (2021) and Bollermann et al. (2022).

The effective diffusivities calculated for the clay laminae, the mixture of clay and sand laminae, and carbonate lenses are employed to estimate the effective diffusivity of the sandy facies of Opalinus Clay (SF-OPA) at the centimeter scale. This estimation takes into account the subfacies compositions, which are quantified through $\mu$-CT segmentation. Subsequently, these effective diffusion coefficients for the sedimentary layers of SF-OPA are integrated into the large-scale structural model to facilitate diffusive transport simulations using the Finite Element Method (FEM)-based OpenGeoSys (OGS) simulator (Kolditz et al., 2012; Bilke et al., 2019; Lu et al., 2022).

4.4.2 Methodology

Governing equation of the component transport

In the study, the Component Transport module within the OGS software is employed for the computation of radionuclide migration in SF-OPA. The mass balance equation is formulated as per previous work by Van Loon et al. (2004); Appelo and Wersin (2007); Bear and Bachmat (2012).

$$\begin{aligned} \frac{\partial \left( \alpha C \right) }{\partial t} - \nabla \cdot ( \mathbf {D_{e}} \nabla C) = 0, \end{aligned}$$

(4.33)

where C (mol m$^{-3}$) is the concentration of aqueous solute and $\mathbf {D_e}$ (m$^2$ s$^{-1}$) is the effective diffusion coefficient tensor. Assuming linear isotherm sorption (Wersin et al., 2008), the constant rock capacity $\alpha$ can be calculated by $\alpha = \phi + \rho _{bd}K_d$ with the porosity $\phi$ (–), the bulk dry density $\rho _{bd}$ (kg/m$^3$), and distribution coefficient $K_d$ (m$^3$/kg).

To determine the effective diffusion coefficient at the field scale, the diffusion equation (Eq. (4.33)) is numerically solved at the pore scale (Yuan et al., 2019; Yuan and Qin, 2020). By imposing constant concentrations at both the inlet and outlet, and no-flux conditions at the remaining boundaries, the effective diffusion coefficient is computed through the total mass flux, denoted as J, per unit cross-sectional area under steady-state conditions, as outlined in Yuan and Fischer (2021):

$$\begin{aligned} D_e = \frac{J\times L}{C_{in} - C_{out}}, \end{aligned}$$

(4.34)

where $C_{in}$ and $C_{out}$ are the prescribed concentrations at the inlet and outlet, respectively. L is the length of the domain.

Molecular diffusion

In the context of the diffusion process, the movement of molecules is depicted as a result of molecular random migration when viewed at a microscopic level, and this behavior can be effectively characterized by Brownian motion, as discussed in previous studies (Grathwohl, 1998; Berg, 2018). Since the diffusive spread of molecules follows a Gaussian normal distribution, the effective diffusion coefficient displays a linear correlation with the mean square displacement. This displacement can be calculated using the standard deviation of the molecular distribution, denoted as $\sigma$, as detailed in Grathwohl (1998).

$$\begin{aligned} N(x,t) = \frac{N_0}{2\sqrt{Dt\pi }}\textrm{exp}\left[ -\frac{x^2}{4Dt}\right] \end{aligned}$$

(4.35)

where, N(x, t) is the number density of molecules at x after time t, $N_0$ is the total number of molecules.

According to the Gaussian normal distribution, the mean square displacement in a 2D system can be described as a measure for the diffusion distance,

$$\begin{aligned} \sigma ^2 = 2Dt \end{aligned}$$

(4.36)

In this study, we account for molecules falling within the range of $\pm \sigma$ from the mean position, which encompasses approximately 68% of the total molecules at each diffusion time, as discussed in Yuan and Fischer (2022). Additionally, at the standard deviation $\sigma$, the concentration of molecules amounts to 61% of the maximum concentration at each diffusion time, in line with the findings of Grathwohl (1998).

4.4.3 Numerical Simulation Setup

For the field-scale simulation, we have established four representative structural models of SF-OPA, each featuring different subfacies compositions. These conceptual models are visually depicted in Fig. 4.31. Structure 1 (S1) is characterized by a low concentration of carbonate lenses distributed separately within the clay laminae of the shale. In Structure 2 (S2), there is a continuous thin layer of carbonate lenses. Structures 3 (S3) and 4 (S4) exhibit large patches of isolated carbonate lenses, resulting in a higher concentration of carbonate lenses. A detailed breakdown of the subfacies compositions for S1–S4, along with the corresponding effective diffusivities derived from the upscaling workflow, is provided in Table 4.2.

Fig. 4.31

Four SF-OPA structures with distinct subfacies compositions, including cases with low carbonate lens concentration (S1), a single layer (S2), and high carbonate lens concentration (S3 and S4)

Table 4.2 Subfacies compositions and effective diffusivities of four SF-OPA structures

Based on the four distinct types of SF-OPA and the corresponding effective diffusion coefficients obtained at the pore scale, we have developed a simplified 2D geological formation model, as depicted in Fig. 4.32. In this 2D model, we assume the diffusive migration of the $\textrm{Na}^+$ tracer from a canister (with dimensions: length = 1.2 m, width = 1.2 m) into the structured SF-OPA formation (with dimensions: length = 40 m, thickness = 20 m).

Each layer within the formation exhibits a homogeneously distributed structure, mirroring the four structures illustrated in Fig. 4.31. Consequently, every layer features a uniform diffusivity coefficient calculated at the pore scale. In the base case, the canister is placed within layer S3, positioned within a bentonite buffer (with dimensions: length = 2.4 m, width = 2.4 m). This buffer has a known porosity of 0.36 and an effective diffusion coefficient of $2.01\times 10^{-11}$ $\mathrm {m^2/s}$, as documented in Nagra (2002). To initiate the simulation, the initial concentration of $\textrm{Na}^+$ is set at 0.3 $\mathrm {mol/L}$. All boundaries are treated as no-flux conditions, and the total simulation duration spans 2000 years.

Fig. 4.32

2D structured model of SF-OPA formation with four sedimentary layers (S1–S4) and canister placement in layer S3

In the subsequent sensitivity analysis, we have devised five additional conceptual models, which are enumerated in Table 4.3.

First, in practice, the bedding planes of sedimentary SF-OPA formations at the field scale are typically not oriented perpendicular to the z direction, as established by Leupin et al. (2018). Due to the anisotropic diffusivity properties of SF-OPA, varying bedding angles can exert a considerable influence on the encapsulated effect of radionuclides in the z+ direction, as observed from a ground surface perspective. To explore the impact of these different bedding angles, we have undertaken simulations in Conceptual Model Series #1 (Fig. 4.33).

Second, in the context of radioactive waste disposal within SF-OPA, each sedimentary layer can potentially serve as the site for canister emplacement. The heterogeneity within these layers, where the canister is positioned, leads to distinct effects on radionuclide diffusion. In an effort to investigate and assess the implications of these different heterogeneities, we have emplaced the canister in various layers, as depicted in Fig. 4.34, within Conceptual Model Series #2.

Table 4.3 Conceptual model overview

Fig. 4.33

Distribution of SF-OPA layers in conceptual models 2A, 2B, and 2C with canister placement in S2, S1, and S4 layers

4.4.4 Results and Discussion

Temporal and spatial evolution of heterogeneous diffusion

In Conceptual Model 0 (base case), the highest concentration of sodium ($\textrm{Na}^+$) in the canister experiences a decrease from 0.071 to 0.055 mol/L between the 1000- and 2000-year marks. The sodium distribution profile along the z direction is visually represented in Fig. 4.34. After 1000 years, sodium has diffused to approximately 5.80 m in the z- direction and about 4.76 m in the z+ direction. This disparity arises because the neighboring layer in the z- direction (S2) possesses a higher diffusivity coefficient compared to the layer in the z+ direction (S4). The difference in sodium concentration distribution is attributed to the heterogeneous layer surrounding the canister.

Examining the diffusion front, which corresponds to the location where sodium concentration equals the standard deviation, it remains within the S3 layer even after 2000 years. Interestingly, the movement of the diffusion front extends further in the z+ direction, in contrast to the sodium diffusion’s path. Specifically, sodium can migrate an additional 1.04 m in the z- direction. In summary, the movement of the diffusion front is more distant from the canister center in the direction where the neighboring layer possesses a lower diffusivity coefficient. A layer with lower diffusivity coefficient effectively retards the tracer migration in that direction.

Fig. 4.34

The concentration profiles of $\textrm{Na}^+$ along the z direction after 1000 years (red line) and 2000 years (blue line) of diffusion. Subsets a and b provide a detailed view of the concentration distribution across the entire domain after 1000 years and 2000 years of diffusion, respectively. The red line in subset a and the blue line in subset b are used to facilitate a comparison of concentration profiles along the z direction. Notably, the canister is positioned at the center, located at $z = 0$ m. The white dashed lines delineate the boundaries between different layers, while the black line represents the diffusion front

The effect of bedding angles

For Conceptual Models 0, 1A, and 1B, where the bedding angle of SF-OPA is set at 0, 30, and 50 degrees, the progression of the diffusion front line in the z+ direction is depicted in Fig. 4.35.

Fig. 4.35

The displacements of diffusion front along $z+$ direction versus diffusion time under bedding angle of 0, 30 and 50 degrees

It is evident that the diffusion front advances further in the z+ direction as the bedding angle increases. After 100 years, with a bedding angle of 0 degrees, the diffusion front is situated at a distance of 1.39 m from the canister center in the z+ direction. This distance expands to 1.73 m and 1.81 m when the bedding angle is raised to 30 and 50 degrees, respectively. Remarkably, the locations of the diffusion front line are closely aligned between the two conceptual models with bedding angles of 30 and 50 degrees, with a mere 0.08-meter discrepancy.

Furthermore, it’s noteworthy that the diffusion front locations with non-zero bedding angles exceed the distance compared to the model with a bedding angle of 0 degrees. This phenomenon can be attributed to the fact that the diffusion plumes are situated in close proximity to the canister after 100 years, and the influence of heterogeneity caused by the neighboring layer is limited. Consequently, the variation in diffusion front displacement is primarily attributed to the anisotropic diffusivity property arising from different bedding angles within the SF-OPA layer.

As the simulation time progresses to 2000 years, a noteworthy change occurs in the sodium diffusion behavior. After 2000 years, the diffusion of sodium notably decelerates due to the influence of the upper neighboring layer S4, especially in the direction perpendicular to the bedding angle. Under these conditions, all diffusion fronts gravitate toward the layer interface in that specific direction.

In the z+ direction, the diffusion front is observed to be situated at distances of 2.37, 2.58, and 2.85 m away from the canister center after 2000 years for the conceptual models with bedding angles of 0, 30, and 50 degrees, respectively. The discrepancy between the conceptual models with bedding angles of 30 and 50 degrees amounts to 0.27 m, which is slightly larger (by 0.06 m) compared to the difference between the models with bedding angles of 30 and 0 degrees. This is a consistent trend observed in all three conceptual models, emphasizing the substantial retardation effect exerted by the upper neighboring layer S4 in this specific direction.

The effect of canister emplacement in different layers

The conceptual models 2A, 2B, and 2C have been simulated, and their respective diffusion front displacements are illustrated in Fig. 4.36. After 2000 years, it’s noteworthy that in the conceptual models 2A and 2C, the diffusion front remains within the layer where the canister was originally placed. The movement distance is notably smaller than 2.5 m, even falling below 2 m in the case of conceptual model 2C.

Fig. 4.36

The diffusion front movement in z direction after 1000 and 2000 years, with the canister placement in layers of S2, S1, and S4, respectively

In contrast, for conceptual model 2B, where the canister is located in the S1 layer with the highest diffusivity coefficient, the diffusion front extends to approximately 2.60 m in the z+ direction and 2.79 m in the z- direction. The movement in the z- direction is more pronounced, primarily due to the presence of the neighboring S4 layer, which exhibits the lowest diffusivity coefficient and thereby offers the most effective retardation effect on sodium migration within the 2000-year timespan.

The neighboring layers indeed exert a modest influence on the reshaping of the tracer’s diffusion front within a 2000-year timeframe. In the case of conceptual model 2B, where the canister is positioned in the layer featuring the highest diffusivity coefficient, the diffusion front extends an additional 0.19 m further in the vertical downward direction over the course of 2000 years, primarily due to the impact of the neighboring layer. However, it is important to note that the impact of m-scale heterogeneity, despite this effect, remains relatively limited in larger model geometries and longer timespans.