Corps de l’article

1. Introduction

The reuse of treated wastewater is already practiced in many countries, both in dry regions (e.g. Sahel, Golf Persian, etc.) and more widely regions with high water stress index (e.g. around the Mediterranean), a frequent use being agricultural irrigation. Its use to refill reservoirs (lakes, ponds, aquifers, etc.) is not widely practiced or currently forbidden in countries like France. Indeed, adding treated wastewater may modify the composition of the water reservoir (chemically and biologically) and lead to health risks. A possible solution could be to purify the wastewater to a high quality (such as drinking quality), this approach being the safest but not economically viable due to advanced treatment costs. Seeking the best compromise between the exploitation of available treated water and the restrictions on its use is currently mobilizing researchers and policy makers to propose solutions in order to face water scarcity (ALCALDE-SANT and GAWLIK 2014; CONDOM et al., 2013).

In this work, we aim to propose optimal strategies of wastewater reuse for refilling a generic water reservoir and then apply the proposed methodology to a concrete case study suggested by Vendée Eau. Vendée Eau is a nonprofit public body in charge of the drinking water supply on the French western coast, which produces drinking water mainly from surface resources. Particularly, it is in charge of the water intake in Jaunay Lake (a fresh water reservoir of 3 700 000 m3), its purification and its distribution to neighboring populations. This water intake results in a reduction of the lake volume, which becomes alarming in dry seasons when its volume may decrease to half of its capacity. In order to preserve the lake water volume to a desired value, Vendée Eau proposes to refill the lake with reused water, coming from a coastal wastewater treatment plant, so that the volume of the lake stays roughly constant during the refill operation (see figure 1 for a detailed description of the water balance occurring in Jaunay Lake). The reused water is obtained after adding a tertiary treatment unit and it may still contain pollutant. Our objective is to find an optimal location of the refilling point such that the pollutant concentration is minimized at the region of the lake devoted to recreational activities, and at the same time maintained under a desired threshold near the removal point. On account of the absence of regulations and unprecedented cases of indirect potable reuse in France, Vendée Eau envisions the implementation of a 1:4 scale demonstrator during the 2018-2024 period including tertiary treatment unit, transfer pipe and discharge zone.

Figure 1

Schematic representation of the Jaunay Lake configuration

Représentation schématique de la configuration du lac de Jaunay

Schematic representation of the Jaunay Lake configuration

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To grasp the multiple issues to ascertain the appropriate location for the reused water discharge, mathematical modelling is particularly adapted for describing the water quality under different system and meteorological conditions (ALAVANI et al., 2010; BARBIER et al., 2016; GAJARDO et al., 2017; RAPAPORT et al., 2014). Our strategy resides in introducing a mathematical model which describes the evolution of the water quality in the reservoir, carrying out numerical simulations and solving the desired optimization problem with an appropriate optimization algorithm. In this work, we focus on modelling the distribution of a generic pollutant in the reservoir through the refilling process with treated water from a wastewater treatment plant. Following ALAVANI et al. (2010) and references thereinafter, we assume that the density of the pollutant is smaller than that of the lake water (so the pollutant remains at the top level of the water column). Moreover, we consider that the reservoir volume remains roughly constant (because the flow rates at the refilling and removal points are assumed quasi-identical). Furthermore, we consider that two main effects influence the pollutant distribution: wind and water currents, the latter resulting from the pumping processes and the discharge of Jaunay River into the reservoir. In order to tackle the proposed bi-objective optimization problem, which aims to control the water quality at two different lake sectors, we present a Pareto front showing how improving one objective is related to deteriorating the second one. This methodology has been broadly used when solving multi-objective problems for water management (AL-ZAHRANI et al., 2016; MORTAZAVI et al., 2012; VEMURI 1974; ZHANG et al., 2014), since it provides a decision-tool to a posteriori help in choosing the optimal strategy according to different water quality criteria.

The article is organized as follows: in section 2 we introduce the model describing the distribution of a generic pollutant in a large water reservoir through the refill process. In section 3 we state the optimization problem which aims to preserve the water quality at two specific regions by choosing a suitable refilling location. In section 4, we explain the numerical experiments carried out for the optimization problem and show the results obtained for the Jaunay case study.

2. Mathematical modelling

Let us denote by Ω ⊂ ℝ2 the spatial domain describing the surface of the water resource (see figure 2 for a physical description). The boundary of the domain, denoted by Ω, can be seen as ∂Ω = Γin ∪ Γout ∪ Γwall, where Γin and Γout are the parts of the boundary through which the water enters and leaves the reservoir due to natural flow, respectively, and Γwall = ∂Ω / ΓinΓout is the part of the boundary where null flux is considered. The refilling and removal locations are denoted by Γref and Γrem, respectively. We denote by ΩcritΩ the critical area of the domain usually devoted to recreational activities.

Figure 2

Domain representation of the Jaunay Lake surface geometry

Représentation de la géométrie de la surface du lac de Jaunay

Domain representation of the Jaunay Lake surface geometry

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Following ALAVANI et al. (2010), we assume that the density of the pollutant is smaller than that of the lake water, so that it remains at the top level of the water column. Additionally, we consider that the possible changes on the lake volume occurring along the process are negligible.

We denote by c(x, t) the pollutant superficial concentration, measured as the volume of pollutant per surface area at (x, t) ∈Ω (0, T), where T is the final time for which we want to model the process. We consider that the evolution of c is governed by four main effects, namely:

  • The diffusion of the pollutant.

  • The wind induced transport.

  • The water currents induced transport.

  • The spill and removal of pollutant resulting from the pumping processes.

Under these assumptions, the space-time distribution of c is governed by the following advection-diffusion type equation:

where D is the diffusion coefficient of the pollutant in the reservoir water, equation: equation pleine grandeur is the wind velocity vector and α is a drag factor measuring the percentage of the wind speed inducing the pollutant transport. The water currents velocity vector, denoted by equation: equation pleine grandeur, is computed by solving the well-known Navier-Stokes equations (GLOWINSKI, 2013) taking into account the lake geometry, the river velocity at its mouth and the pumping velocities at the removal and refilling locations. Furthermore, we denote by c0, cref and cin the pollutant concentration in the lake at the beginning of the process, the pollutant concentration at the refilling location Γref and the pollutant concentration at the river mouth Γin, respectively.

3. Optimization problem

We consider the optimization problem consisting in minimizing the amount of pollutant in the critical region Ωcrit, while the pollutant concentration is maintained under a desired threshold equation: equation pleine grandeur at the removal point Γrem, by choosing a suitable refilling location Γref. Given the final T > 0 for which we want to model the process, the optimization problem can be formulated as follows:

where JT is defined as the amount of pollutant in Ωcrit along the process,

and equation: equation pleine grandeur denotes the maximum pollutant concentration reached at Γrem,

4. Numerical experiments

In this section, we first introduce the numerical solver used for computing the solutions of the system (Equation 1) and describe the considered numerical experiments based on the optimization problem (Equation 2). Then, in section 4.2 we analyze the effect of the wind and water currents on the pollutant distribution for the Jaunay case study. Section 4.3 presents the optimization results and outline the influence of setting different water quality thresholds on the obtained optimal refilling location for the Jaunay case study.

4.1 Numerical implementation of the model

The solution of equation 1 was computed using the software COMSOL Multiphysics 5.3 (www.comsol.com) based on the Finite Element Method. Model variables (Equations 3 and 4) were estimated using the functions Domain Integration (based on a trapezoidal approximation of the integral) and Boundary Maximum of COMSOL, respectively. The numerical experiments were carried out in a 2.8 Ghz Intel i7-930 64 bits with 12 Gb of RAM. We used a triangular mesh with around 14 000 elements. Depending on the considered case (detailed below), each function evaluation in equation 2 may take from 20 min up to 4 h.

Model parameters were taken following the data provided by Vendée Eau. The period of time for which we have modeled the process is 1 June 2016 - 31 August 2016. In order to compute the velocity vector equation: equation pleine grandeur we assumed that the river enters in the lake with a flow rate of 1.96 x 10-3 (m3∙s-1), the flow rate of the pumping is 3.47 x 10-1 (m3∙s-1) and the physical pipe in charge of the pumping is a cylinder with a cross section of radius 3 m. Additionally, data regarding the wind velocity equation: equation pleine grandeur was extracted from the free source COPERNICUS (http://marine.copernicus.eu/services-portfolio/access-to-products/). The pollutant was assumed to diffuse with rate D = 1.31 x 10-8 m2∙s-1 (HAMAM, 1987) and show a drag factor α = 2 x 10-3 (ALAVANI et al., 2010). At initial time, the pollutant concentration in the lake was taken constant with value c0 = 0.05 kg∙m-2. We assumed that the water entering the lake through the refilling pipe was charged with a pollutant concentration cref = 0.19 kg∙m-2, while the water entering the lake through the river was clean, i.e. cin = 0 kg∙m-2. The solution of equation 2 was approximated by taking 1 400 possible refilling locations uniformly distributed through the lake surface, equation: equation pleine grandeur, and computing the objective values equation: equation pleine grandeur (kg) and equation: equation pleine grandeur (kg∙m-2) (see equations 3-4 associated to each prospective location equation: equation pleine grandeur, i = 1,..., 1400).

4.2 Analysis of the wind and water current induced transport

As explained in section 2, we assume that the pollutant transport is due by two main factors: water currents and wind. The water currents speed vector equation: equation pleine grandeur does not depend on time, since we consider that the river velocity and pumping flow rates are constant along the process. Figure 3a shows an example of the streamlines of vector equation: equation pleine grandeur (computed for a specific choice of refilling and removal locations) showing the direction in which a Lagrangian particle travels at any point in the lake surface. We observe that the trajectory of the particles follows the natural flow induced by the upstream river mouth. The removal pipe absorbs some of the particles while the rest leave the lake through the downstream lake boundary. On the other hand, real data seem to show that in the region of France where Jaunay Lake is located, the wind velocity vector equation: equation pleine grandeur usually has a direction from north-west to south-east, as depicted in figure 3b. When this occurs, the wind pushes the pollutant to the south eastern zones of the lake (as for instance the recreational activities area Ωcrit).

Figure 3

Analysis of the wind and water current velocity vectors for Jaunay Lake: a) water current streamlines associated to an specific choice of Γref and Γrem, b) usual wind directions registered in the lake region

Analyse des vecteurs de vitesse du courant d'eau et du vent pour le lac de Jaunay : a) lignes de courant d'eau associées à un choix spécifique de Γref and Γrem, b) directions habituelles du vent enregistrées dans la région du lac

Analysis of the wind and water current velocity vectors for Jaunay Lake: a) water current streamlines associated to an specific choice of Γref and Γrem, b) usual wind directions registered in the lake region

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4.3 Optimization results

The bi-objective optimization problem (Equation 2) may have multiple optimal solutions depending on how restrictive are the constraints in each of the objectives (in our case, reducing the pollutant concentration at areas Γrem and Ωcrit). In this case, a usual methodology to visualize the possible optimization results is to plot the Pareto front (AUBIN, 1984), a curve that informs the decision-maker how improving one objective is related to deteriorating the second one while moving along the curve. In figure 4a, the tested refilling locations equation: equation pleine grandeur are plotted such that the distance to the removal point decreases from blue to red. In figure 4b, each depicted point corresponds to the pair of objective values, equation: equation pleine grandeur, obtained for an specific refilling location equation: equation pleine grandeur in figure 4a, from which the color plot is inherited with a view to easily associate distinct ranges of the objective values with specific sectors in the lake. The Pareto front is depicted with a black curve. As expected, numerical simulations seem to show that refilling locations near the removal point Γrem reduce the amount of pollutant at the recreational activities area Ωcrit. On the contrary, refilling locations near the river mouth induce low pollutant concentration at the removal point. This representation may help the decision makers to balance their choice between the two criteria.

Figure 4

Graphical interpretation of a) tested refilling locations equation: equation pleine grandeur used to obtain b) the objective values equation: equation pleine grandeur, equation: equation pleine grandeur in equation 2. The black curve represents the Pareto front

Interprétation graphique a) des emplacements de recharge testés equation: equation pleine grandeur utilisés pour obtenir b) les valeurs des objectifs equation: equation pleine grandeur, equation: equation pleine grandeur dans l’équation 2. La courbe noire répresente le front de Pareto

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Figure 5a shows the optimal refilling locations, solution of equation 2, obtained when setting equation: equation pleine grandeur ∈ {0.06, 0.1, 0.14} kg∙m-2, that is, obtained when imposing three different water quality thresholds at the removal point. Figure 5b plots the points of the Pareto front corresponding to the optimal locations in figure 5a. By choosing the most restrictive threshold, equation: equation pleine grandeur = 0.06, we aim a high quality effluent at the removal point Γrem, while setting equation: equation pleine grandeur = 0.14, we priorize the water quality at the region of the lake devoted to recreational activities, Ωcrit. The intermediate constraint equation: equation pleine grandeur = 0.1 represents a trade-off for which the pollutant concentration is controlled at both areas Γrem and Ωcrit.

Figure 5

Graphical interpretation of the optimization results solution of equation 2 when equation: equation pleine grandeur ∈ {0.06, 0.1 0.14} kg∙m-2: a) optimal refilling locations equation: equation pleine grandeur, b) objective values equation: equation pleine grandeur, equation: equation pleine grandeur associated to these optimal refilling locations. The black curve represents the Pareto front

Interprétation graphique des résultats d'optimisation solution de l’équation 2 lorsque equation: equation pleine grandeur ∈ { 0,06, 0,1 0,14} kg∙m-2 : a) emplacements optimaux de recharge equation: equation pleine grandeur et b) valeurs des objectifs equation: equation pleine grandeur, equation: equation pleine grandeur associées à ces emplacements optimaux. La courbe noire représente le front de Pareto

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Finally, figure 6 represents the pollutant distribution at final time (31 August 2016) obtained with the optimal refilling locations in figure 4a. As explained in section 4.2, one can observe that due to the wind effect, the pollutant concentration is notably accumulated at the southern areas of the lake. Indeed, high pollutant concentrations are reported at Ωcrit whenever the refilling pipe is placed at the right hand side of this region. As a result, one can conclude that in order to reduce the pollutant concentration at areas Γrem and Ωcrit, the refilling pipe must be placed as far as possible from Γrem and at the left hand side of Ωcrit.

Figure 6

Pollutant concentration c (kg∙m-2) at the simulated final time (31 August 2016) associated to the optimal refilling locations equation: equation pleine grandeur solution of equation 2 when equation: equation pleine grandeur = a) 0.06, b) 0.1, c) 0.14 kg∙m-2

Concentration du polluant c (kg∙m-2) à la date finale simulée (31 août 2016) associée aux emplacements optimaux de recharge equation: equation pleine grandeur solution de l’équation 2 lorsque equation: equation pleine grandeur = a) 0,06, b) 0,1, c) 0,14 kg∙m-2

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5. Conclusions

We have proposed a methodology to determine optimal strategies for refilling water resources with reused water still containing some pollutant. The methodology has been applied in the case of Jaunay Lake, a water reservoir located on the French western coast, which shows an alarming volume reduction due to the human water intake. The main objective was to find optimal refilling locations ensuring that a water quality threshold was maintained at the region of the lake devoted to recreational activities but also at the intake location, while maintaining the volume of the lake almost constant.

We have used a mathematical model, based on a partial differential equation of advection-diffusion type, which describes the distribution of a generic pollutant through the lake. The model assumes that pollutant remains at the surface of the water reservoir and the evolution of its distribution is mainly influenced by wind and lake water currents. Using the Finite Element Method, we have numerically computed the transient pollutant distribution associated to a particular refilling location. A total of 1 400 prospective refilling positions have been computationally tested and a Pareto front has been obtained, informing the policy-maker about the trade-offs among the water quality at both lake regions. Besides, real data seem to show that, in the area of France where Jaunay Lake is located, the wind velocity usually has a direction from north-west to south-east, which results in an accumulation of pollutant at the south-eastern zones of the lake (as for instance the recreational activities area). One concludes that, in order to achieve a reasonable trade-off among the two water quality objectives, the refilling pipe must be positioned as far as possible from the intake location and downstream the leisure region. Vendée Eau, the nonprofit public body in charge of the water management in Jaunay Lake, envisions the implementation of a 1:4 scale demonstrator during the 2018-2024 period based on the optimization results presented here.

In this work, we have considered that the volume of the lake remains roughly constant through the refilling process. Dropping this assumption is a matter of future work and the ultimate goal of this project, since we aim at decreasing the speed of the volume reduction more than at maintaining the volume to a constant value.