Introduction
===========

Radioactive waste disposal systems typically comprise a series of barriers that aim at protecting the environment and human health. The presence of several barriers enhances confidence that the waste will be adequately contained. In deep geological disposal systems, the barriers include the natural geological barrier and the engineered barrier system (EBS).

The efficiency of the EBS and the geological barrier is often tested through complex numerical models. These models serve to evaluate the response of the barrier under different sensitivity scenarios. One important factor to be evaluated is the groundwater flow crossing the barriers under different scenarios, which often consist in changes in the hydraulic properties of the different materials or changes in the waste compartment location. 

The present article shows an example of a COMSOL App set up to allow such sensitivity analyses. The model behind the app contains a hydrogeological groundwater flow model of a specific vault (1BMA) for intermediate level radioactive waste disposal operated by the Swedish nuclear waste management company (SKB).

Model Set-up
===========

The Multiphysics model underlying the application simulates groundwater flow through a rock vault in the Swedish final repository for short-lived radioactive waste [1]. The vault geometry is shown in Figure 1.

<figure className="image">
 <StrapiImage src="/uploads/BMA_app_1_d971a095ec.png" alt=""/>
 <figcaption>Figure 1. On the Geometry page the user can visualize the different structures and material domains available in the application.</figcaption>
</figure>

The Darcy’s law and Solute Transport interfaces have been applied to calculate groundwater flow rates and residence times in the vault. The residence time is defined here as the spatial distribution of the mean groundwater age, taking into account advection, diffusion and dispersion processes. It is calculated by solving a partial differential equation proposed by Goode et al [2].

$$
\frac{\partial A}{\partial{t}} = 1 + \nabla \cdot \left ( \mathbf{D} \cdot \nabla A \right ) - \nabla \cdot \left ( \frac{\mathbf{q}}{\theta} A \right ) + \frac{F}{\theta \rho}
$$

Above, $A$ is the water age ($s$), $\mathbf{q}$ is the Darcy velocity field (m/s), $\mathbf{D}$ is the hydrodynamic dispersion tensor (m2/s), $θ$ is the porosity, $\rho$ is the water density (kg/m3) and $F$ is a source term. 

Application description
===========

Figure 2 shows the ribbon at the top of the application graphical user interface. Clicking the buttons, from left to right, provides a workflow that guides the user through the application.


<figure className="image">
 <StrapiImage src="/uploads/BMA_app_2_19c85c0cd3.png" alt=""/>
 <figcaption>Figure 2. The application ribbon provides the workflow to guide the user.</figcaption>
</figure>

The user can change model inputs on the Setup page. This involves assigning values to the porosity, hydraulic conductivity and effective diffusivity of different materials and engineered structures.

The user can also select among predefined pressure fields to prescribe the boundary conditions for the groundwater flow (Figure 3). As an option, pressure boundary conditions can be imported from file.

<figure className="image">
 <StrapiImage src="/uploads/BMA_app_3_7394511a60.png" alt=""/>
 <figcaption>Figure 3. On the Setup page the user can modify the material transport properties and the boundary conditions (1). Selecting from the list of shoreline positions (2) updates the information graphics and text (3) as well as the displayed pressure field (4).</figcaption>
</figure>

After model setup the user can compute the groundwater flow and the residence time in the rock vault. Results can be analyzed through a set of predefined plots in 2D and 3D (Figure 4). Numerical values can be output in table exports. Formats are standardized such that quantitative simulation results can be used directly in connected modelling activities.

<figure className="image">
 <StrapiImage src="/uploads/BMA_app_4_8a6f9131a5.png" alt=""/>
 <figcaption>Figure 4. On the Plots page the user can select between flow or residence time results (1) displayed in 2D and 3D. Flow results presented in 2D use predefined cutplanes (2). Selecting from the list updates the information graphics and text (3 and 4) as well as the displayed results (5).</figcaption>
</figure>

After simulation, modelling reports can be generated automatically, summarizing the model setup and results.

Perspective
===========
The COMSOL Application Builder has been used to rework a coupled Multiphysics model in 3D into a ready-to-use simulation application. The application provides the opportunity for non-modelling experts and engineers to perform extended simulation trials and production runs. The accessibility of the application format furthermore facilitates review and quality control work associated with the model development process itself.

References
===========

*COMSOL, 2015*. **COMSOL Multiphysics 5.1.** COMSOL AB, 2015.

*Goode D J,* 1996. **Direct simulation of groundwater age.** Water resources research, vol 32. N.12, pages 289-296.

medium_BMA_app_1.png

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BMA_app_1.png

Radioactive waste disposal systems typically comprise a series of barriers that aim at protecting the environment and human health. 

Comsol App for the evaluation of an engineered barrier system

Motivation
=========

A correct design and control of the irrigation cycles is very important for the correct and efficient grow of crops. It is well known that a correct design of irrigation cycles increases its efficiency and reduces the amount of water required. An important objective in the agricultural industry is to design an optimal irrigation plan depending on crop requirements. The irrigation cycles have an effect on the quality and quantity of water. It becomes more relevant in areas of the world where water is scarce. Nowadays, several scientists are involved in different research projects that focus on finding alternative water sources for irrigation. One idea is to reuse human waste water [1]. This concept is based on the pollutant removal effect of the sun, the soil and the plants itself. However, it requires an important control of the polluted water evolution in the subsurface. 

Numerical modelling is an important tool to help engineers and scientists design structures, simulate experiments, etc. In the context of this study, it can be used to design irrigation structures. It serves also to evaluate the propagation of pollutants or nutrients available in the irrigation water through the subsurface.

We present a useful app designed in Comsol that simulates the effects of different irrigation cycles on the soil saturation. It also can simulate the evolution of the pollutants dissolved in the water through the subsurface attending to its degradation. The application solves the Richard’s equation (eq 1) to obtain the groundwater flow field and its saturation. 

$$
\rho(\frac{C_m}{\rho g}+S_eS)\frac{\partial p}{\partial t}+\nabla \cdot \rho (-\frac{k_s}{\mu}\mathbf{k_r}(\nabla p+\rho g \nabla \mathbf{D}))=Q_m
$$

The advection diffusion equation (eq 2) is used to simulate the transport of pollutants and nutrients. 

$$
\phi \frac{\partial c_i}{\partial t}+\mathbf{u}c_i=\nabla \cdot [(\mathbf{D_{D,i}}+\mathbf{D_e})\nabla c_i]+R_i
$$

The model is also prepared to evaluate the evolution of the dissolved elements involved in the organic matter degradation chain (figure 1). 


<figure className="image">
 <StrapiImage
 src="/uploads/com_irri_app_fig_1_98f8084b34.png" className="mx-auto" alt=""
 />
 <figcaption>Figure 1. Sequence of important redox processes at pH 7. Figure taken from: Appelo and Postma, 2014 [2].</figcaption>
</figure>


The chemical reactions considered in the model are the following:

$$
O_2+CH_2O \leftrightarrow CO_2+H_2O
$$
$$
MnO_2 + 0.5CH_2O + 1.5CO_2 + 0.5H_2O \leftrightarrow Mn^{2+}+2HCO_3^-
$$
$$
Fe(OH)_3 + 0.25CH_2O + 1.75CO_2 \leftrightarrow Fe^{2+}+2HCO_3^-+0.75H_2O
$$
$$
SO_4 + 2CH_2O \leftrightarrow H_2S + 2HCO_3^-
$$
$$

This sequence is progressive which means that the denitrification, e.g., does not begin until all the oxigen is depleted. The reactions are included in the model as second order kinetics (Eq 4) where the rate is defined by the user.


\frac{\partial O_2}{\partial t}=-k_{O_2}[O_2][CH_2O]
$$
$$
\frac{\partial NO_3}{\partial t}=-k_{NO_3}[NO_3][CH_2O]
$$
$$
\frac{\partial MnO_2}{\partial t}=-k_{MnO_2}[MnO_2][CH_2O]
$$
$$
\frac{\partial SO_4}{\partial t}=-k_{SO_4}[SO_4][CH_2O]
$$
$$
\frac{\partial CH_2O}{\partial t}=\frac{\partial O_2}{\partial t}+0.5\frac{\partial NO_3}{\partial t}+0.5\frac{\partial MnO_2}{\partial t}+0.25\frac{\partial Fe(OH)_3}{\partial t}+2\frac{\partial SO_4}{\partial t}
$$

The app is fully parametrized. Regarding the geometry, it is up to the user to select the extension of the modelled area (Figure 2), the extension and location of the irrigation areas, etc. The app also considers different materials for the subsurface. Once the geometry is generated, the app allows the user to assign the properties of the materials such as the permeability or the porosity and introduce the irrigation planning which is read from an external file. Finally, the initial concentration of the different species is assigned together with the composition of the irrigation water. 

<figure className="image">
 <StrapiImage className="mx-auto" src="/uploads/com_irri_app_fig_2_4d49ba9199.png" alt=""/>
 <figcaption>Figure 2. Example of a geometry generated in the app with one irrigation zone and three different materials. Red lines and points show parameters such as thickness or position that are set by the user.</figcaption>
</figure>

The app is prepared to show several types of plots such as 2D slices that show the evolution of the saturation, the lead concentration, etc. It also shows the evolution of the groundwater level and water saturation at given observation points.

Example of application
==================

This example considers two permeable materials separated by a small aquitard. The hydraulic conductivity of the materials can be observed in Figure 2. Two irrigation zones are included (Figure 3). The model considers the irrigation of the small crop only. It considers a regional groundwater flow from south to north. Under these boundary conditions, the computed groundwater flow streams from the irrigation zone located south-west to the north north-east (Figure 4 and Figure 5). The groundwater level is higher in the upper aquifer than in the lower. There is a general flow from top to bottom through the aquitard. However, this flow between aquifers is small compared to the horizontal flow. This lower flow retards the movement of the concentration through the aquitard ( Figure 6). Figure 3 to 8 show different types of plots that can be used to visualize the results in the application. In addition, the app also contains different tables to export the results in a quantitative way.


<figure className="image">
 <StrapiImage className="mx-auto" src="/uploads/com_irri_app_fig_3_062efd8df6.png" alt=""/>
 <figcaption>Figure 3. Finite element grid formed by tetrahedra and prisms.</figcaption>
</figure>

<figure className="image">
 <StrapiImage className="mx-auto" src="/uploads/com_irri_app_fig_4_5ae687cd9e.png" alt=""/>
 <figcaption>Figure 4. Computed hydraulic head.</figcaption>
</figure>

<figure className="image">
 <StrapiImage className="mx-auto" src="/uploads/com_irri_app_fig_5_3b9723faa3.png" alt=""/>
 <figcaption>Figure 5. Cut plane showing the hydraulic head and the flow direction.</figcaption>
</figure>

<figure className="image">
 <StrapiImage className="mx-auto" src="/uploads/com_irri_app_fig_6_95c22195a8.png" alt=""/>
 <figcaption>Figure 6. Chloride concentration in a cut plane. Arrows show the flow direction.</figcaption>
</figure>

<figure className="image">
 <StrapiImage className="mx-auto" src="/uploads/com_irri_app_fig_7_68be4bf646.png" alt=""/>
 <figcaption>Figure 7. Cl mol/m3.</figcaption>
</figure>

<figure className="image">
 <StrapiImage className="mx-auto" src="/uploads/com_irri_app_fig_8_94dff2d315.png" alt=""/>
 <figcaption>Figure 8. Isosurface Cl 1.2 mol/m3.</figcaption>
</figure>

The possibility of generated custom apps with Comsol allows the user to generate general models such as the one presented here that can be applied for different crops and areas. 

<figure className="image">
 <StrapiVideo width="725" height="395" src="/uploads/comsol_app_irrigation_video_ddd59299b5.mp4" type="video/mp4" />
 <figcaption></figcaption>
</figure>



References
==========

[1] *Hussain I.; L. Raschid; M. A. Hanjra; F. Marikar; W. van der Hoek*. 2002. **Wastewater use in agriculture: Review of impacts and methodological issues in valuing impacts**. Working Paper 37. Colombo, Sri Lanka: International Water Management Institute.

[2] *Appelo, C.A.J and Postma.D,* 2014. **Geochemistry, groundwater and pollution**. 2nd edition.

large_com_irri_app_fig_0.png

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small_com_irri_app_fig_0.png

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com_irri_app_fig_0.png

We present a useful app designed in Comsol that simulates the effects of different irrigation cycles

Comsol-app

Comsol App for the evaluation of an engineered barrier system

A computational app for a proper evaluation of the irrigation effect over the aquifers