Two separate samples were analysed for this test phase with a view to determining the concentrations of nitrate ions ( ${\text{NO}}_3^{-}$) and manganese (Mn²⁺), considered as potential pollutants of the stored water contained in the test tanks. The concrete reserves under consideration are miniaturized to a laboratory scale.

Figure 3 and Figure 4 illustrate the experimental conditions in the laboratory.

The experimental simulation set-up consists of:

The pollutant solution passes through the concrete walls by the principle of diffusion, thus altering the quality of the stored water.

Figure 5 shows an explicit diagram of this experimental set-up.

The equipment used to determine concentrations is the WPA UV visible spectrophotometer ( Figure 6 ) for time and space estimation of concentrations. In practice, nitrate ions ${\text{NO}}_3^{-}$ (nitrave 5) and manganese ions Mn²⁺ (manganecole n 1) were used as pollutants.

On the basis of measurements obtained on the laboratory-scale phenomenon, the approach consists of approximating the parameters that will enable these measurements to be confirmed. This can be done by numerical simulation, if the model can approximate its physical observations, sometimes deviating from the real parameters. It is therefore necessary to add constraints or a priori factors to narrow down the range of possibilities in order to arrive at a unique solution.

Diffusion takes place according to the law of conservation of matter and thermodynamic equilibrium.

The diffusion equation is the relationship that governs the time and space evolution of particle density. It is the result of a material balance associated with Fick’s law.

According to this hypothesis, particle flow J, in the presence of a concentration gradient in a porous medium, can be modelled by equation 1:

$J=-\varnothing D\frac{\partial C}{\partial x}$ (1)

with D as the diffusion coefficient or diffusivity, and ∅ the porosity of the medium.

The concentration C of the solute being defined, as the mass of the solute per unit volume of the solution and ∅, the porosity of the medium; the one-dimensional particle balance in the transient regime (flux varying at each point), for an elementary volume dV of containing dN = CSdx particles and a closed surface S, gives:

At x, the total mass of solute entering the element per unit time is:

Me = j (x,t) Sdt (2)

At x + dx abscissa, the total outgoing mass during the elementary displacement is:

$Ms=j\left(x+dx,t\right)Sdt$ (3)

The difference between incoming and outgoing mass ∆M is:

$\Delta M=Me-Ms=j\left(x,t\right)Sdt-j\left(x+dx,t\right)Sdt=-\frac{\partial j\left(x,t\right)}{\partial x}Sdtdx$ (4)

Neglecting adsorption, the difference between the flux entering and flux leaving the element is equal to the amount of substance dissolved in the element.

The mass exchange rate in the infinitésimal element is given by the following expression:

$dN\left(t+dt\right)-dN\left(t\right)=\varnothing \left\{C\left(t+dt,x\right)-C\left(t,x\right)\right\}Sdt=\varnothing \frac{\partial C\left(x,t\right)}{\partial t}dtSdx$ (5)

Finally, if we assume that there is no process of creation or annihilation of particles in the elementary volume dV = Sdx, for the duration dt, the conservation of matter implies that the difference of mass per unit time must be equal to the mass exchanged per unit time, i.e.:

$\Delta M=dN\left(t+dt\right)-dN\left(t\right)$ (6)

Considering the following equation:

$\frac{\partial C\left(x,t\right)}{\partial t}Sdtdx=-\frac{\partial j\left(x,t\right)}{\partial x}Sdtdx$ (7)

which results in:

$\varnothing \frac{\partial C\left(x,t\right)}{\partial t}=-\frac{\partial j\left(x,t\right)}{\partial x}$ (8)

According to Fick’s law in porous media:

$j=-\varnothing D\frac{\partial C\left(x,t\right)}{\partial x}$ (9)

Hence:

$\varnothing \frac{\partial C\left(x,t\right)}{\partial t}=\frac{\partial }{\partial x}\left(\varnothing D\frac{\partial C\left(x,t\right)}{\partial x}\right)$ (10)

This equation will be completed with boundary and initial conditions corresponding to the experimental conditions we wish to model, for its resolution.

The problem of contamination of water stored in an underground reservoir located in a saturated and polluted environment can be modelled mathematically by a linear partial differential equation of the parabolic type developed in numerical methods used to solue elliptic of parabolic differential equation:

$\{\begin{array}{c}\varnothing \left(x\right)\frac{\partial C}{\partial t}\left(x,t\right)=\frac{\partial }{\partial x}\left(\varnothing \left(x\right)D\left(x\right)\frac{\partial C}{\partial x}\left(x,t\right)\right),\forall \left(x,t\right)\in \left]0,L_t\right[\times \left]0,T\right[\\ \frac{\partial C}{\partial x}\left(0,t\right)=\frac{\partial C}{\partial x}\left(L_t,t\right)=0,\forall t\in \left]0,T\right[\\ C\left(x,0\right)=C_0\left(x\right),\forall x\in \left]0,L_t\right[\end{array}$ (11)

in which:

$D\left(x\right)>0$ is the diffusion coefficient; $\varnothing \left(x\right)>0$ is the porosity of the medium; $C_0$ is the given initial pollutant concentration and C the unknown concentration function defined on $\left]0,L_t\right[\times \left]0,T\right[$ and with values in $ℝ$.

$L_t>0$ (in meters) is the total length of the integration domain of the equation and T (in seconds) is the final time of the experiment.

Modelling this control system on the basis of a dynamic parameter (control) involves the use of differential, integral, functional, finite-difference, partial-derivative equations, etc. The aim is to stabilize the control system and optimize its performance.

Controls consist of highlighting functions or parameters that are generally subject to constraints. Optimal control, therefore, is a variational method which consists in finding the state of a numerical forecasting model, in order to best fit the observed measurements over a given period of time and make a natural link with the model equations.

Variational analysis relies on the minimization of a “cost” functional J as an additional condition to ensure the uniqueness of the solution. This function is determined by physical or numerical considerations (stability of optimization algorithms) [23] - [25] .

For simplicity’s sake, we’ll use the finite-difference formulation on a rectangular grid with a regular mesh.

Equation (11) discretized on a grid is then written as:

$\frac{dX\left(u,t\right)}{dt}=F\left(X\left(u,t\right),u\right)$ (12)

with $uЄ{ℝ}^l$, a parameter to be identified (e.g. diffusion coefficient) and X (u, t), a vector of ${ℝ}^N$, the discretized concentration. The Cauchy problem with initial condition:

$\{\begin{array}{c}\frac{dX\left(u,t\right)}{dt}=F\left(X\left(u,t\right),u\right)\\ X\left(u,0\right)=X_0\end{array}$ (13)

then admits as a solution $X\left(u,t\right)$ over the interval [0, T].

We assume that the observations distributed in space and time have made it possible to define a time-dependent vector Xobs (t) of ${ℝ}^N$, which we’ll call the observation vector.

The cost function or objective function J(u) is defined by:

$J\left(u\right)=\frac{1}{2}\underset{0}{\overset{T}{{\displaystyle \int }}}<\left(X\left(u,t\right)-Xobs\left(t\right)\right),X\left(u,t\right)-Xobs\left(t\right)>dt$ (14)

where <· , ·> denotes the inner product in ${ℝ}^N$.

Since the set of possible states is theoretically all ${ℝ}^l$, one has an unconstrained optimization problem written as:

$\{\begin{array}{c}u\ast Є{ℝ}^l\\ J\left(\text{u}\ast \right)=\inf J\left(u\right)\\ uЄ{ℝ}^l\end{array}$ (15)

Variational methods solve the inverse problem by directly minimizing the cost function, usually using a gradient descent method. The gradient is computed by the adjoint method, which comes from the optimal control theory of PDEs [10] [11] .

Let J be a real-valued function defined on a Hilbert space ${\rm H}$ (with inner product <.,.>), of variable u.

To minimize the functional J, we must first perform the derivation operation on it. To do this, we introduce the Gateaux derivative (or directional derivative) $\hat{J}$ of J in direction h Є ${\rm H}$ defined by:

$\hat{J}\left(u,h\right)=\underset{\alpha \to 0}{\lim }\frac{J\left(u+\alpha h\right)-J\left(u\right)}{\alpha }$ (16)

In the case where $\hat{J}$ is linearly continuous in h, we can then note the following relationship:

$\hat{J}\left(u,h\right)=\langle \nabla J\left(u\right),h\rangle$ (17)

$\nabla J\left(u\right)$ is the gradient of J with respect to u.

When J admits continuous partial derivatives for all u in ${ℝ}^l$, a necessary condition for $u^{\ast }$ to be a minimum is that $\nabla j\left(u^{\ast }\right)=0$.

In general, problem (15) cannot be solved analytically. However, there are numerical solution algorithms that can be used to determine approximate solutions, including the conjugate gradient method and the quasi-Newton method.

Most are iterative descent procedures which, starting from an initial value $u_0$, seek the best approximation $u_1$ of the solution $u^{*}$ such that j ( $u_1$) ≤ j ( $u_0$) and iterate the process until a given convergence condition is satisfied.

All these methods use the gradient of the function to be minimized with respect to the control variable (in our case). Hence the importance of determining the gradient of J with respect to u.

A very efficient technique for obtaining the gradient of the cost function with respect to the control variable is the use of the adjoint system that we will present below.

To do this, we consider a vector h belonging to the same space as u and a real number α.

$X\left(u+\alpha h,t\right)$ and $X\left(u,t\right)$ are then respectively solutions of the ordinary differential equations with initial conditions:

$\{\begin{array}{c}\frac{dX\left(u+\alpha h,t\right)}{dt}=F\left(X\left(u+\alpha h\right),t\right)\\ X\left(u+\alpha h,0\right)=X_0\end{array}$ (18)

and:

$\{\begin{array}{c}\frac{dX\left(u,t\right)}{dt}=F\left(X\left(u,t\right),u\right)\\ X\left(u,0\right)=X_0\end{array}$ (19)

Subtracting (18) from (19) then dividing by α and taking the limit when the latter tends to, we obtain that the derivative $\hat{X}$ of X in the h direction is then solution of the system:

$\{\begin{array}{c}\frac{d\hat{X}\left(u,h,t\right)}{dt}=\left[\frac{\partial F}{\partial X}\right]\hat{X}\left(u,h,t\right)+\left[\frac{\partial F}{\partial u}\right]h\\ \hat{X}\left(u,h,0\right)=0\end{array}$ (20)

where:

$\frac{\partial F}{\partial X}$ is the Jacobian matrix of F with respect to X;

$\frac{\partial F}{\partial U}$ is the Jacobian matrix of F with respect to u.

The Gateaux derivative of J in the h direction is given by

$\hat{J}\left(u,h\right)=\langle X\left(u,t\right)-Xobs\left(t\right),\hat{X}\left(u,t\right)\rangle$ (21)

To find the linearity of $\hat{J}$ with respect to h, in equation (22), let’s multiply each by a variable P ${ℝ}^{\text{N}}$, and integrate the result over the interval $\left[0,T\right]$. The result is:

$\underset{0}{\overset{T}{{\displaystyle \int }}}\langle \frac{\partial \hat{X}}{\partial t},P\rangle dt=\underset{0}{\overset{T}{{\displaystyle \int }}}\langle \left[\frac{\partial F}{\partial X}\right]\hat{X},P\rangle dt+\underset{0}{\overset{T}{{\displaystyle \int }}}\langle \left[\frac{\partial F}{\partial u}\right]h,P\rangle dt$ (22)

or after integration by parts of the:

$\langle \hat{X}\left(T\right),P\left(T\right)\rangle -\langle \hat{X}\left(0\right),P\left(0\right)\rangle =\underset{0}{\overset{T}{{\displaystyle \int }}}\hat{X},\frac{dP}{dt}+{\left[\frac{\partial F}{\partial X}\right]}^{t}P>dt+\underset{0}{\overset{T}{{\displaystyle \int }}}\langle h,\left[\frac{\partial F}{\partial u}\right]P\rangle dt$ (23)

When P is chosen as the solution to the equation:

$\{\begin{array}{c}\frac{dP}{dt}+{\left[\frac{\partial F}{\partial X}\right]}^{t}P=X\left(u,t\right)-Xobs\left(t\right)\\ P\left(T\right)=0\end{array}$ (24)

finds:

$\underset{0}{\overset{T}{{\displaystyle \int }}}\langle \hat{X}\left(u,t\right),X\left(u,t\right)-Xobs\left(t\right)\rangle dt=-\underset{0}{\overset{T}{{\displaystyle \int }}}\langle h,{\left[\frac{\partial F}{\partial X}\right]}^{t}P\rangle dt$ (25)

$\hat{J}\left(u,h\right)=-\underset{0}{\overset{T}{{\displaystyle \int }}}\langle h,{\left[\frac{\partial F}{\partial u}\right]}^{t}P\rangle dt$ (26)

To obtain the gradient of J with respect to explicitly, starting from, need to h and P in the canonical bases of ${ℝ}^l$ and ${ℝ}^N$. This gives the components of $\nabla J\left(u\right)$ in the form:

${\left(\nabla J\left(u\right)\right)}_i=-\underset{0}{\overset{T}{{\displaystyle \int }}}{\left({\left[\frac{\partial F}{\partial u}\right]}^{t}P\right)}_{i}dt1\le i\le l$ (27)

Different numerical methods can be used for solving diffusion problems.

In this work, the finite volume method was used to discretize equation (11) or (12) in space, and for stability reasons, the implicit Euler method was used for its discretization in time. At each time step, the resulting linear system was solved using the GMRES method.

Indeed, for a regular one-dimensional mesh, with a flow conservation approximation, the finite volume method seems suitable. it adapts well to nonlinear conservation equations, as it ensures discrete conservation of extensive quantities [13] . It is a method of decomposition by subdomains (cell or control volumes) in which flows are approximated according to the principle of “finite differences”, but quite different from the partial differential equation (PDE) scheme.

The stability of this method also preserves the divergence problems that can arise with finite elements. Last but not least, this method fits in well with Fick’s law of particle flow diffusion.

Equation (20) was solved using a derivation technique for the discretized equation of (11), while (24) was solved using a transposition technique for the discretized equation of (22).

An essential point for the implementation of the method, is the calculation of the gradient as well as its verification. From the direct system, we obtain the tangent system by linearization around a perturbation, then the adjoint system by transposition. To obtain the adjoint, we write the code for the tangent system, then transpose the instructions line by line ( Table 1 ).

To check the accuracy of the gradient obtained, we consider Taylor’s formula:

$J\left(u+\alpha h\right)-j\left(u\right)=\alpha \nabla J\left(u\right).h+\frac{{\alpha }^2}{2}{\nabla }^{2}j\left(u\right)\left(h,h\right)+0\left({\alpha }^2\right)$ (28)

Where ${\nabla }^{2}j\left(u\right)$ denotes the matrix of J’s second derivatives with respect to u. We then have:

$\underset{\propto \to 0}{\lim }f\left(\alpha \right)=\underset{\alpha \to 0}{\lim }\left(\frac{J\left(u+\alpha h\right)-J\left(u\right)}{\alpha \nabla J\left(u\right)h}\right)=1$ (29)

Figure 9 and Figure 10 illustrate the gradient verification curves obtained from numerical simulations 1 and 2 using the optimal control technique. At the same time, they illustrate the instability of the gradient descent directions and show that, for α between ${10}^{-1}et{10}^{-6}$, the value of $f\left(\alpha \right)$ is equal to 1. Thus, gradient descent is an optimization algorithm that makes it possible to find the minimum of any function by gradually converging towards it. Similarly, at the current point, a shift is made in the opposite direction to the gradient, so that the cost function decreases.

Gradient descent, on the other hand, is an optimization algorithm that finds the minimum of any function by gradually converging towards it. At the current point, a shift is made in the opposite direction to the gradient, so that the cost function decreases.

Figure 11 and Figure 12 summarize the characteristics of numerical simulations 1 and 2, using the optimal control technique for the ammonia pollutant, after six (06) iterations ( Table 2 ).

The gradient optimization algorithm reduces the function at each iteration. The cost function is reduced respectively from 456.7 to 178 for numerical simulation 1 and from 450.6 to 21.3 for numerical simulation 2.

Table 3 shows the concentration values of numerical simulations 1 and 2 obtained by optimal control on the basis of experimental simulations 1 and 2 by Malanda (2014). Here the results of the numerical simulation tests in this control study are compared with those obtained by Mlalanda in 2014.

Figure 13 and Figure 14 show the concentration variation curves of numerical simulations obtained by the optimal control method from experimental measurements by Malanda (2014).

Interpretation of the figures below reveals the following:

With regard to the experimental simulation curves, the experimental results are close at the start of the experiment, and the pattern at the end of the first two weeks may be due to the influence of the granular structure of the cementitious matrix, or even its porosity. The phenomenon of water saturation of the concrete (experimental simulation) accelerates pollutant diffusion as all the pores become interconnected. On the other hand, the delay observed in the first measurements can be attributed to the difficulty for (polluted) water molecules to fill all the voids in the cement matrix during the imbibition phase [12] [14] [15] [17] [18] [26] .

However, in the numerical simulation, these parameters have not been taken into account, and the focus has been on diffusion, without taking into account the time required for the (polluted) water to be absorbed by the concrete, so that equilibrium of concentrations in the tank can be established.

In reality, as the phenomenon of tortuosity has not been taken into account, the proposed modelling concerns effective diffusion rather than pollutant diffusion.

Using experimental measurements 1 and 2 by Malanda (2014), numerical tests were carried out on the basis of the calculation code developed with the Fortan 90 language and the curves were plotted with Origin Pro software.

Table 4 shows the results obtained following the numerical simulation tests.

Similarly, the comparative analysis of the two simulations carried out (experimental and numerical) is shown by assembling the curves ( Figures 15-18 ) and from Table 4 .

Table 5. Results of identified ammonia diffusion parameters.

Interpretation of the figures shown above reveals the following two phases:

Nevertheless, the physical convergence of the results shows that the numerical method chosen better simulates the diffusion of ions in solution in the cementitious matrix (concrete) and, consequently, the effective migration of pollutants through it. On the basis of the above, it follows that the proposed model approaches experimental measurements better than that proposed by Malanda (2014). Numerically, this model better predicts the evolution of particle diffusion flow through a cemented granular material (CGM), and ultimately the transfer of dissolved pollutants from one point to another.

The ammonia diffusion parameters identified are given in Table 5 .

Generally speaking, this experiment shows that the diffusion coefficient in concrete is higher than in water and sand. This implies ipso facto the influence of the nature of the material, which constitutes an explanatory variable for the differences encountered in the value of the diffusion coefficient. Indeed, the greater the porosity, the greater the proportion of pollutants in the concrete tank.

This observation can also be explained by the fact that the diffusion coefficient in the porous medium increases as the porosity decreases. Consequently, the diffusion coefficient is greater for lower porosity, and therefore inversely proportional.

In accordance with the experimental set-up, these two tanks respectively contained two pollutants (nitrates and manganese), diluted in water and then discharged into fine sand (see Figure 2 and Figure 4 ). The concentrations of the selected pollutants were determined over time using the above-mentioned spectrophotometer.

The results of the eleven (11) concentration measurements obtained over 32 days are given in Table 6 .

Overall, there is a temporal variation in the concentration results measured in the two samples. In fact, both samples show gradual changes in the concentrations measured as they diffuse through the concrete walls into drinking water. We also notice :

So, although the diffusion of the selected pollutants is clearly established, we note that the measured values are differentiated for each measurement carried out for the two samples.

The results of numerical simulation tests on nitrate ion diffusion are summarized in Table 7 .

This produces a curve with some variability in nitrate ion concentrations for the numerical test performed. Figure 19 shows the concentration of nitrate ions in the cement matrix and inside the tank containing stored water after 32 days. Figure 20 , on the other hand, illustrates the temporal variations in concentrations obtained from the experimental and numerical simulations.

In addition, Figure 19 above shows the spatial variation in nitrate ion concentration at each point of the chosen one-dimensional mesh at a given time, based on the principle of the calculation code developed.

In addition, as the laboratory experiment has not yet reached its stationary state, the values of the effective diffusion coefficients obtained D1 and D2 for nitrate and manganese ions could evolve.

Similarly, the comparative analysis of the two types of simulations (experimental and numerical) is shown by assembling the curves ( Figure 20 ) and from Table 7 .

It should be noted that concentrations increase continuously as a function of time only, and the concentration variations observed in these evolutions confirm the random nature of diffusion: it is difficult to predict future concentrations, and the speed with which pollutant particles move is uniformly varied.

Furthermore, Figure 21 shows the temporal variations in nitrate ion concentrations (numerical simulation and experimental simulation), and highlights the following sentences:

These results show that the measurements taken in the laboratory experiments and the numerical simulation confirm the temporal evolution of pollutant diffusion. The process is increasing and the equilibrium state has not yet been reached, but the phenomenon seems to be stabilizing at the end of the experiment (11th measurement).

On the basis of these results, we can see that in numerical modelling of this type, the longer the duration of the experimental measurements, the less stringent the stability requirement becomes.

We thus obtain a curve with a certain variability of manganese ion concentrations for the numerical test carried out. Figure 21 shows the concentration of nitrate ions in the cement matrix and inside the tank containing stored water after 32 days ( Table 8 ).

Similarly, the comparative analysis of the two types of simulation (experimental and numerical) is reflected in the assembly of curves ( Figure 21 ) from Table 8 . Figure 22 illustrates the temporal variations in concentrations between the numerical and experimental simulations. Here, the divergence in the shape of 02 curves (experimental and numerical simulations) does not call into question the approved theory, but rather provides evidence that the cementitious matrix (concrete) is not watertight to preserve the quality of the stored water. The results shown in Figure 22 above, for the pollutant used (manganese ions), seem to show that at the start of the experiment, the concentrations for the two types of simulation appear close together, and then the behaviour for the two simulations becomes almost fairly parallel until the end of the experiment (at t = 32 days). This may be explained by the fact that the diffusion time was not optimal, and by the low initial concentration of manganese ions.

The divergence of the curves of the two (02) methods (experimental measurements and numerical simulation) does not call into question the proven theory, but demonstrates that the cement matrix is not watertight enough to preserve the quality of the water in buried receivers in a polluted environment.

In light of these results, we can appreciate the temporal variation in concentration according to the mathematical model adopted and as a function of concrete porosity. This is because concentrations depend on porosity size. If we try to reduce the porosity to a value between 0.04 and 0.07, it becomes clear that the model has its limits. In other words, when the concrete becomes practically watertight, mathematically, pollutant diffusion stops. This is undoubtedly due to the degeneracy of the diffusion equation, which becomes stationary and changes from parabolic to elliptical ( Figure 23 ).

It may be noted that porosity and diffusion coefficient are inversely proportional, and that the variation in major ion concentrations observed depends fundamentally on the porosity of the concrete ( Table 9 ).

Table 10 , which groups together the results of the last three (03) numerical simulations for concrete porosities of 14%, 16% and 18%, shows us that the temporal variation of the diffusion coefficient through the concrete obtained in the tank depends on the porosity of the concrete. These results show that porosity is an explanatory variable in the variation of the diffusion coefficient in the concrete obtained.

Looking at the figure, we can see that the effective porosity and diffusion coefficient are inversely proportional, and that the variation in concentration observed depends fundamentally on the porosity of the concrete.

In addition, the diffusion coefficients identified for the two selected pollutants are given in the table ( Table 11 ).

Table 10. Summary of diffusion coefficient values obtained for concrete porosities equal to 14%, 16% and 18%.

Table 11. Results of identified diffusion coefficients for selected pollutants.

In addition, several other authors have already worked on the problems of transfer and diffusion through the cementitious matrix that is concrete. The focus is on the porosity parameter, the microstructural analysis of concrete and the corrective aspects of pores. We can point to the work of Hailong Ye et al. (2013), Chen (2011), Tetsuya Tshida et al., (2009), Care er Have (1999), Mainguy (1999), Mchirgui (2013), and Dana (1999) [12] [13] [20] . All these works demonstrate that concrete porosity is its weakest link. Indeed, porosity facilitates ion transfer and migration via the pore network, thanks to its connectivity within the cementitious matrix.

All in all, it has been proven that ${\text{NO}}_3^{-}$ and Mn²⁺ ions pass through the porous concrete matrix contained in the material zones. Thus, the rate of transfer depends on the volume fraction and connectivity of the soils according to Janz (1997), Ye et al. (2013), and other anthors Hailong ye et al. (2013) [26] - [30] . By the way, the work carried out by Chen (2011) [15] , on the experimental study of the permeability of a sound concrete, under variable thermal and hydric conditions, made it possible to formulate a concrete intended for storage structures for various materials and installed in any type of environment. The results obtained are consistent with those of our research work. Nevertheless, for concrete porosities below 14%, the results can be considered optimum [13] .

Also, the cases studied with the nitrate ion pollutant, for porosity values of 14%, 16% and 18%, give us results that appear reasonable. It can therefore be noted that pollutant transport in cementitious materials under the conditions mentioned is an adventitious-diffusive phenomenon. However, in the modelling, advective transfer due to bulk movement or interstitial solution phase is taken into account, as well as ionic diffusion due to concentration gradients according to [Tetsuya Tshida et al., 2009] [28] . This justifies the fact that the transfer or diffusion coefficients depend on the characteristics of the material, and therefore on the formation of the concrete. However, the porosity of the cementitious matrix subjected to a concentration gradient during the transfer phase also depends on its microstructure. Larger porosity zones, known as “transition halos”, appear around the aggregates and promote transfer [28] - [31] . It should be noted, however, that the porosity of the material has a particularly strong influence on concrete when it comes to the transfer and diffusion of pollutants.