The transfer of pollutants in porous media can be described [1] [3] by the following system of equations:

$\{\begin{array}{l}\frac{\partial \left(\epsilon d\right)}{\partial t}+\nabla \cdot \left(dq\right)=0\text{(2}\text{.1a)}\\ \frac{\partial \left(\epsilon d\mathcal{W}\right)}{\partial t}+\nabla \cdot \left(d\mathcal{W}q+J\right)=0\text{(2}\text{.1b)}\\ J=-dD\nabla \mathcal{W}\left(\text{Fick's law}\right)\text{(2}\text{.1c)}\\ d=d_0\exp \left({\beta }_T\left(T-T_0\right)+{\beta }_P\left(P-P_0\right)+\gamma \mathcal{W}\right),\left[1,6\right]\text{(2}\text{.1d)}\\ q=-\frac{k}{\mu }\left(\nabla P+dg{e}_3\right)\left(\text{Darcy's law}\right)\text{(2}\text{.1e)}\end{array}$

There were several important variables and parameters in the model. The flow density $d\left(\text{kg}/{\text{m}}^3\right)$ is a function of the temperature $\left(T\right)$, pressure $\left(P\right)$ and the mass pollutant ratio transported $\mathcal{W}$. The porosity of the porous medium, $\left(\epsilon \right)$, is a dimensionless value from 0 to 1, which is the ratio of the void volume to the medium volume. The fluid vector indicates $q\left(\text{m}/\text{s}\right)$ and indicates the Darcys’ velocity, which is a function of the pressure of the fluid $P\left(Pa\right)$. The pollutant mass transport is diffusive owing to the diffusion coefficient $D\left({\text{m}}^2/\text{s}\right)$ and is expressed in the diffusive flux $J\left(\text{kg}/\left({\text{m}}^2/\text{s}\right)\right)$, that follows Fick’s law. Time $t\left(\text{s}\right)$ as the execution of the system continues. Some reference values for temperature $T_0$ and pressure $P_0$ and sensitivity coefficients $\left({\beta }_T\right),\left({\beta }_P\right)$, and $\gamma$ for the temperature, pressure and $\left(\mathcal{W}\right)$, affect the fluid density $\left(d\right)$. The permeability of the porous medium $\left(\frac{k}{\mu }\right)$ represents the medium’s ability to allow fluid transmission, and ${}^{m}u$ fluid dynamic viscosity (it’s value is in (Pa∙s)) means the degree of the fluid flow resistance. Gravitational effects are introduced by means of gravitational acceleration $g\left({\text{m}}^2/\text{s}\right)$ and unit vector $e_3=\left[0;0;1\right]$ that indicates how gravity acts in the model.

In this approach, we avoid making assumptions about density d. Instead, which is treated as a variable influenced by external factors such as pressure, temperature, and pollutant concentration. This generalization allowed the model to remain applicable to a wider range of porous media and scenarios.

The system of Equations (2.1) can be rewritten according to [1] - [3] as:

$\begin{array}{c}\frac{\partial \left(\epsilon d\right)}{\partial t}-\nabla \cdot \left(\frac{k}{\mu }\nabla Pd\right)-\nabla \cdot \left(\frac{k}{\mu }{d}^{2}g{e}_3\right)=0,\\ \frac{\partial \left(\epsilon d\mathcal{W}\right)}{\partial t}-\nabla \cdot \left(d\mathcal{W}\frac{k}{\mu }\nabla P\right)-\nabla \cdot \left(d^2\mathcal{W}\frac{k}{\mu }g{e}_3\right)-\nabla \cdot \left(dD\nabla W\right)=0\end{array}$ (2.2)

REMARK 1 The porosity ε of the medium can be modeled using Garner’s law:

$\epsilon =\{\begin{array}{ll}\frac{{\epsilon }_s-{\epsilon }_r}{1+{\left(\alpha h\right)}^{\beta }},\hfill & h\le 0\hfill \\ {\epsilon }_s,\hfill & h>0\hfill \end{array}$

where $h$ represents the pressure relative to the atmospheric pressure.

By developing the second equation in (2.2), we have

$\mathcal{W}\left[\frac{\partial \left(\epsilon d\right)}{\partial t}-\nabla \cdot \left(\frac{k}{\mu }\left(\nabla P\right)d_s\right)-\nabla \cdot \left(\frac{k}{\mu }{d}^{2}g{e}_3\right)\right]+\epsilon d\frac{\partial \mathcal{W}}{\partial t}-d\frac{k}{\mu }\nabla \mathcal{W}\cdot \nabla P-d^2\frac{k}{\mu }g\nabla \mathcal{W}\cdot e_3-\nabla \cdot \left(dD\nabla \mathcal{W}\right)=0$ (2.3)

According to the first equation of (2.2), we have:

$\epsilon d\frac{\partial \left(\mathcal{W}\right)}{\partial t}-d\frac{k}{\mu }\nabla \mathcal{W}\nabla P-d^2\frac{k}{\mu }g{\partial }_z\mathcal{W}-\nabla \cdot \left(dD\nabla \mathcal{W}\right)=0$

Due to (2.1d) and the temperature does not vary, one gets

$\begin{array}{c}\nabla P=-\frac{1}{\beta }\nabla W+\frac{1}{\gamma {\beta }_p}\nabla \left(\ln \left(\frac{d}{d_0}\right)\right)\end{array}$ (2.4)

$\epsilon d\frac{\partial \mathcal{W}}{\partial t}-d\frac{k}{\mu }\nabla \mathcal{W}\cdot \left[-\frac{1}{\beta }\nabla \mathcal{W}+\frac{1}{\gamma {\beta }_p}\nabla \left(\text{ln}\left(\frac{d}{d_0}\right)\right)\right]-d^2\frac{k}{\mu }g{\partial }_z\mathcal{W}-\nabla \cdot \left(dD\nabla \mathcal{W}\right)=0$

It follows that:

$\epsilon d\frac{\partial \mathcal{W}}{\partial t}+\frac{dk}{\mu \beta }{\left|\nabla \mathcal{W}\right|}^2-\frac{dk}{\mu \gamma {\beta }_p}\nabla \left(\text{ln}\left(\frac{d}{d_0}\right)\right)\cdot \nabla \mathcal{W}-d^2\frac{k}{\mu }g{\partial }_z\mathcal{W}-\nabla \cdot \left(dD\nabla \mathcal{W}\right)=0$

By rearranging, we have

$\frac{\partial \mathcal{W}}{\partial t}+\frac{k}{\epsilon \mu \beta }{\left|\nabla \mathcal{W}\right|}^2-\frac{k}{\mu \epsilon \gamma {\beta }_p}\nabla \left(\ln \left(\frac{d}{d_0}\right)\right)\nabla \mathcal{W}-\frac{kd}{\epsilon \mu }g{\partial }_z\mathcal{W}-\frac{\nabla \left(dD\right)}{d\epsilon }\nabla \mathcal{W}-\frac{D}{\epsilon }\text{Δ}\mathcal{W}=0$

we have

$\frac{\partial \mathcal{W}}{\partial t}+\frac{k}{\epsilon \mu \beta }{\left|\nabla \mathcal{W}\right|}^2-\left[\frac{k}{\mu \epsilon \gamma {\beta }_p}\nabla \left(\text{ln}\left(\frac{d}{d_0}\right)\right)+\frac{\nabla \left(dD\right)}{d\epsilon }\right]\nabla \mathcal{W}-\frac{kd}{\epsilon \mu }g{\partial }_z\mathcal{W}-\frac{D}{\epsilon }\text{Δ}\mathcal{W}=0$

Setting ${\alpha }_1=\frac{k}{\epsilon \mu \beta },\overline{{\alpha }_2}=\frac{k}{\mu \epsilon \gamma {\beta }_p}\nabla \left(\text{ln}\left(\frac{d}{d_0}\right)\right)+\frac{\nabla \left(dD\right)}{d\epsilon },{\alpha }_3=\frac{kd}{\epsilon \mu }g$, and ${\alpha }_4=\frac{D}{\epsilon }$,

one has

$\frac{\partial \mathcal{W}}{\partial t}+{\alpha }_1{\left|\nabla \mathcal{W}\right|}^2-\overline{{\alpha }_2}\cdot \nabla \mathcal{W}-{\alpha }_3{\partial }_z\mathcal{W}-{\alpha }_4\text{Δ}\mathcal{W}=0$

We aim to fix the boundary and initial conditions to guarantee the good posedness of the pollution model [11] [12] . Thus

$\begin{array}{c}\{\begin{array}{l}\frac{\partial \mathcal{W}}{\partial t}+{\alpha }_1{\left|\nabla \mathcal{W}\right|}^2-\overline{{\alpha }_2}\cdot \nabla \mathcal{W}-{\alpha }_3{\partial }_z\mathcal{W}-{\alpha }_4\text{Δ}\mathcal{W}=0,\text{in}ℬ\times \left(0,T_1\right),\hfill \\ \mathcal{W}={\mathcal{W}}_1{\text{on Γ}}_1\times \left(0,T_1\right),\hfill \\ \frac{\partial \mathcal{W}}{\partial n}=0\text{on}\partial ℬ\setminus {\text{Γ}}_1\times \left(0,T_1\right),\hfill \\ \mathcal{W}\left(x,0\right)={\mathcal{W}}_i,x\in ℬ\times \left\{0\right\}.\hfill \end{array}\end{array}$ (2.5)

REMARK 2 The pollution model we obtain is a generalization of the models obtained in [1] - [3] . In fact, assuming the same hypotheses used in [3] of our generalized model, we obtain the same pollution model.

More details of the generalized adjoint method can be found in [22] [23] .

Let $\mathscr{H}$ be a fixed Hilbert space and $ℒ\left(\mathscr{H}\right)\left(\text{resp}{ℒ}_2\left(\mathscr{H}\right)\right)$ denotes the spaces of linear (resp bilinear) forms on $\mathscr{H}$. Let us assume the following hypotheses.

HYPOTHESIS 1: There exists a real function $f$, a bilinear form ${\delta }_a\in {ℒ}_2\left(\mathscr{H}\right)$ and a linear form ${\delta }_l$ such that

$f\left(r\right)\to 0,r\to 0$ (3.1)

${\Vert a_r-a_0-f\left(r\right){\delta }_a\Vert }_{L_2\left(\mathscr{H}\right)}=o\left(f\left(r\right)\right)$ (3.2)

${\Vert l_r-l_0-f\left(r\right){\delta }_l\Vert }_{L\left(\mathscr{H}\right)}=o\left(f\left(r\right)\right)$ (3.3)

where $a_r,a_0$ resp ( $l_r,l_0$) denotes the bilinear resp (linear) form.

HYPOTHESIS 2: The bilinear form $a_0$ is coercive. There exists a constant $\alpha >0$ such that

$\begin{array}{c}{a}_0\left(u,u\right)\ge \alpha {\Vert u\Vert }^2,\forall u\in \mathscr{H}\end{array}$ (3.4)

According to (3.2) the bilinear form $a_r$ depends continuously on $r$, hence there exists $r_0$ and $\beta >0$ such that for $r\in \left[0,r_0\right]$ the following uniform coercive holds $a_r\left(u,u\right)\ge \beta {\Vert u\Vert }^2,\forall u\in \mathscr{H}$. Moreover, according to Lax-Milligram’s theorem, for $r\in \left[0,r_0\right]$, the problem finds $u_r\in \mathscr{H}$, such that

$\begin{array}{c}{a}_r\left(u_r,v\right)=l_r\left(v\right),\forall v\in \mathscr{H}\end{array}$ (3.5)

has a unique solution and the following result holds.

LEMMA 1 If Hypotheses 1 and 2 are satisfied, there exists a unique solution $u_r$ to (3.5). Furthermore, the following estimate holds:

${\Vert u_r-u\Vert }_{{\mathscr{H}}_r}\begin{array}{c}=O\left(f\left(r\right)\right)\end{array}$ (3.6)

where $u$ is the solution of: $a_0\left(u,v\right)=l_0\left(v\right),\forall v\in \mathscr{H}$.

Proof. Because of the ellipticity of $a_r$ we obtain

$\alpha {\Vert u_r-u_0\Vert }^2\le a_r\left(u_r-u_0,u_r-u_0\right)$

From

$\begin{array}{l}{a}_r\left(u_r-u_0,u_r-u_0\right)=\left|a_r\left(u_r,u_r-u_0\right)-a_r\left(u_0,u_r-u_0\right)\right|\\ =\left|l_r\left(u_r-u_0\right)-a_r\left(u_0,u_r-u_0\right)\right|\\ =\left|l_0\left(u_r-u_0\right)+\left(l_r-l_0\right)\left(u_r-u_0\right)-a_r\left(u_0,u_r-u_0\right)\right|\\ \le \Vert a_0\left(u_0,u_r-u_0\right)-a_r\left(u_0,u_r-u_0\right)\Vert +\Vert \left(l_r-l_0\right)\left(u_r-u_0\right)\Vert \end{array}$

Thus

$\left|\alpha \right|{\Vert u_r-u_0\Vert }^2\le \Vert \delta a\left(u_0,u_r-u_0\right)\Vert f\left(r\right)+\Vert \delta l\left(u_r-u_0\right)\Vert f\left(r\right)+\Vert u_0\Vert \Vert u_r-u_0\Vert o\left(f\left(r\right)\right)+\Vert u_r-u_0\Vert o\left(f\left(r\right)\right)$.

Thus,

$\alpha {\Vert u_r-u_0\Vert }^2\le \left(\Vert \delta a\Vert \Vert u_0\Vert +\Vert \delta l\Vert \right)\Vert u_r-u_0\Vert f\left(r\right)+\left(1+\Vert u_0\Vert \right)\Vert u_r-u_0\Vert o\left(f\left(r\right)\right)$.

which implies that

$\alpha \Vert u_r-u_0\Vert \le \left(\Vert \delta a\Vert \Vert u_0\Vert +\Vert \delta l\Vert \right)f\left(r\right)+\left(1+\Vert u_0\Vert \right)o\left(f\left(r\right)\right)$,

this confirms the proof. ■ We also assume the following hypothesis.

HYPOTHESIS 3: Let $j$ be a functional such that $j\left(r\right)=J\left(u_r\right)$ where $J$ is differentiable. For any $u\in \mathscr{H}$, there exists a linear and continuous form $DJ\left(u\right)\in ℒ\left(\mathscr{H}\right)$ and ${\delta }_J$ such that:

$\begin{array}{c}J\left(u\right)-J\left(v\right)=DJ\left(u\right)\left(v-u\right)+f\left(r\right){\delta }_J\left(u\right)+o\left({\Vert u-v\Vert }_{\mathscr{H}}\right)\end{array}$ (3.7)

The Lagrangian $ℒ$ is given by

$\begin{array}{c}L\left(u,v\right)=a\left(u,v\right)-l\left(v\right)+J\left(u\right),\forall u,v\in \mathscr{H}\end{array}$ (3.8)

Its variation is given by

$\begin{array}{c}{ℒ}_r\left(u,v\right)=a_r\left(u,v\right)-l_r\left(v\right)+J\left(u\right),\forall u,v\in \mathscr{H}\end{array}$ (3.9)

We also have the following generic theorem.

THEOREM 1 Under hypotheses (1), (2) and (3), we have the following asymptotic expansion of $j$:

$\begin{array}{c}j\left(r\right)-j\left(0\right)=f\left(r\right)\delta L\left(u_0,v_0\right)+o\left(f\left(r\right)\right)\end{array}$ (3.10)

where $u_0$ is the solution of (3.5) with $r=0,v_0$ is the solution to the adjoint problem, and $v_0$ is such that

$\begin{array}{c}{a}_0\left(ℬ,v_0\right)=-DJ\left(u_0\right)\int ,\forall f\in \mathscr{H}\end{array}$ (3.11)

and $\delta ℒ\left(u,v\right)={\delta }_a\left(u,v\right)-{\delta }_l\left(v\right)+{\delta }_J\left(u\right)$, denotes the topological derivative.

Proof. For the proof of this theorem, see [23] . ■

To optimize the configuration of the porous media for pollution control, we employed topological optimization methods. This approach aims to minimize the function representing the pollutant mass ratio while satisfying physical constraints.

For a real number $r\ge 0$, we define the perturbed domain ${\text{Ω}}_r=\text{Ω}\setminus \overline{{ℬ}_r}$, where ${ℬ}_r=x_0+rℬ$ and $ℬ\subset {ℝ}^n$, with $n=2$ or 3.

We consider the sets $Q=\left(0,T\right)\times \text{Ω},{\displaystyle {\sum }_{{}_r}=}\left(0,T\right)\times {ℬ}_r$, and $Q_r=\left(0,T\right)\times {\text{Ω}}_r$.

The optimization process involves minimizing the objective functional $J\left({\mathcal{W}}_r\right)$, defined as

$\begin{array}{c}\{\begin{array}{ll}\frac{\partial u_{\text{Ω}}}{\partial t}-\overline{{\alpha }_2}\nabla u_{\text{Ω}}-{\alpha }_3\frac{\partial u_{\text{Ω}}}{\partial z}-{\alpha }_4\text{Δ}{u}_{\text{Ω}}=0,\hfill & \text{in Ω}\times \left(0,T\right)\hfill \\ u_{\text{Ω}}=\psi \left({\mathcal{W}}_1\right),\hfill & {\text{on Γ}}_1\times \left(0,T\right)\hfill \\ \frac{\partial u_{\text{Ω}}}{\partial n}=\psi \left({\mathcal{W}}_2\right),\hfill & \text{on}\left(\partial \text{Ω}\setminus {\text{Γ}}_1\right)\times \left(0,T\right)\hfill \\ u_{\text{Ω}}\left(x,0\right)=\psi \left({\mathcal{W}}_0\right),\hfill & \text{for}x\in \text{Ω}\times \left\{0\right\}\hfill \end{array}\end{array}$ (3.15)

The weak formulation of (3.15) is to find $u\in ℱ\left(\text{Ω}\right)$ such that:

Thus, we have the following result:

THEOREM 2 (Main Result) Under the conditions of Lemma 1, the topological derivative of functional (3.2), where $ℬ$ denotes the unit ball centered at $x_0$, is given by

(3.18)

for Neumann boundary conditions

, for Dirichlet boundary conditions.

State $u_{\Omega }$ solves the direct problem (3.13) and $v_{\Omega }\left(x_0,t\right)$ is the solution of the corresponding adjoint problem.

Proof. To prove this theorem we determine the variation of the bilinear form, objective function and linear form. Subsequently Theorem 2 is applied.

Variation of the bilinear form

The variation of the bilinear form $a_r-a_0$ associated with problem (3.14) and (3.16) is given by

,

it follows that

.

In the case of a Dirichlet condition at the boundary of the hole, a similar reasoning applies.

Variation of the functional form

We have

since $\psi$ is a bijective and continuous function.

Thus:

(3.24)

We therefore have:

(3.25)

To complete the proof, it suffices to show that:

(3.26)

$\begin{array}{c}C:B\subset {ℝ}^n\to {ℬ}_r\subset {ℝ}^n,\\ y\mapsto C\left(y\right)=x=x_0+ry,\end{array}$ (3.27)

we get:

(3.28)

Rearranging gives:

$\delta J\left(u_{\text{Ω}}\left(x,t\right)\right)=0.$

According to Theorem 2, the expression for the topological derivative is:

(3.29)

for Neumann boundary conditions, or:

where $\mathcal{A}\left(u,v\right)$ is the bilinear operator given by:

(4.2)

The weak formulation at the time step $n+1$ becomes:

(4.3)

The problem is solved iteratively over the time steps, where $u^n$ represents the solution at step n.

The topological gradient ${\mathcal{G}}_{\tau }\left(x_0\right)$ is given by:

(4.4)

in the Neumann condition and

In 3-dimensional cases, we consider the initial domain ${\text{Ω}}_0=\left(-5,5\right)\times \left(-5,5\right)\times$ $\left(0,6\right)$. The simulations were performed under the following parameter values: the permeability is given by $\kappa =7x^2+3y^2+4z^2+1$, while the density is $d_s=\frac{100}{7x^2+3y^2+4z^2+1}$. The diffusion coefficient is given as $D=0.0003\left(7x^2+3y^2+4z^2+1\right)$, and the desired state is $u_d=0.5\text{cos}\left(x\right)+0.5\text{sin}\left(y\right)+0.5\text{sin}\left(z\right)$. In addition, we set the initial condition as $u_i=0.5\text{cos}\left(x\right)+0.5\text{sin}\left(y\right)+0.5\text{sin}\left(z\right)$, and the homogen boundary condition as $u_1=x^2{(x-1)}^2+2+t\left(-2+12x-11x^2-2x^3+x^4\right)$.

Based on these data, Figure 3 shows the distributions of pollutant mass fraction and topological gradient in the perforated domain by inserting the spherical hole $ℬ$ centered at the minimum of ${\mathcal{G}}_{\tau }$. We notice that creating a spherical hole, at the negative minimum of the topological derivative, decreases the objective function.

Thus, after 25 iterations, the topological gradient is almost null throughout the perforated domain (see Figure 4 ). The gradient descent algorithm converges, achieving the optimal domain.