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.

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.

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 = Γ_in ∪ Γ_out ∪ Γ_wall, where Γ_in and Γ_out are the parts of the boundary through which the water enters and leaves the reservoir due to natural flow, respectively, and Γ_wall = ∂Ω / Γ_in ∪ Γ_out is the part of the boundary where null flux is considered. The refilling and removal locations are denoted by Γ_ref and Γ_rem, respectively. We denote by Ω_crit ⊂ Ω the critical area of the domain usually devoted to recreational activities.

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:

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, 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 , 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, c_ref and c_in the pollutant concentration in the lake at the beginning of the process, the pollutant concentration at the refilling location Γ_ref and the pollutant concentration at the river mouth Γ_in, respectively.

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

where J_T is defined as the amount of pollutant in Ω_crit along the process,

and denotes the maximum pollutant concentration reached at Γ_rem,

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.

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.