Mixed finite element method of a seawater intrusion problem in confined aquifers

We consider a model mixing sharp and diffuse interface approach for the seawater intrusion phenomenon in confined aquifers. The aim of this work is to introduce and analyze a new mixed formulation, obtained by writing the problem into a matrix form, and introducing a new variable representing σ = R(u)∇u the flux tensor of the primal variable . Here, h represents the depth of the salt/freshwater interface, u = (h, Φ f ) , the hydraulic head of freshwater, and R(u) a symmetric and positive definite diffusion matrix. We show that Φ f the continuous problem is well-posed. For the time discretization of this new mixed formulation, we use a semi-implicit scheme, and we show that the problem is well posed. Corresponding author: soumaiaslim@gmail.com © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). E3S Web of Conferences 314, 05004 (2021) https://doi.org/10.1051/e3sconf/202131405004 WMAD21


Introduction
In many countries and regions all over the globe, groundwater is considered the primary source of freshwater supply. Unfortunately, in coastal areas, a hydraulic exchange between groundwater and seawater may occur.
This exchange may arise for two main reasons: natural conditions such as the decline of the water table after a dry period or human impact such as intensive pumping. These factors lead to a decrease in the pressure of the water table, which, therefore, causes saltwater intrusion into coastal aquifers. Consequently, industrial and agricultural production may sustain significant damage. Thus, building a model, which simulates the movement of saltwater fronts in the coastal aquifer, is important for reasonable groundwater development and freshwater preservation. Within this context, several models based on numerical methods have been proposed and evaluated in the literature, see [1-3-6]. Considering the case of confined aquifers, we adopt a sharp/diffuse interface approach (see Fig. 1.). The domain is thus occupied by two immiscible fluids (freshwater and saltwater) separated by a sharp interface. For modeling the boundary conditions on the sharp interface, we use the Allen-Cahn model in the fluid/fluid context. We refer to [3], for more details on this approach .
The mathematical model associated with confined aquifers is given by a strongly coupled set of quasi-linear elliptic-parabolic equations. The considered unknowns are the depth freshwater/saltwater interface and , the ℎ Φ hydraulic head. It must be noted that the global in time existence result is demonstrated in [4]. The use of the mixed sharp/diffuse interface approach provides a result of solution regularity. Indeed, the gradient of the solution is contained in the space . This result gives the (0, , 1, (Ω)), > 2 uniqueness of the solution; we refer to [5] for more details. We propose, in this paper, a new mixed formulation of the problem. A time discretization of this new mixed formulation is based on a semi-implicit scheme. We show that the associated problem is well posed.

Modelling
We give, in this section, the model associated with a sharp/diffuse interface approach in a confined aquifer.

Assumptions and notations
The assumptions and results drawn in [4][5] are adopted. Moreover, we assume that the maximum principle is always verified and the hydraulic conductivity is a Homothety matrix. Therefore, the problem is formulated as follows: where is the parameter of density contrast, the α ϕ porosity of the medium and represents the , = , external source terms corresponding to the pumping or recharge of fresh or saltwater into the aquifer, respectively.
The matrix form of the system (1-4) is given by: where and is a symmetric positive It should be noted that the analysis of problem (5) in the stationary case has already been done [7]. Thus, the results will be helpful to prove those in the non-stationary case.

Mixed formulation
We consider homogeneous Dirichlet conditions, . without loss of generality. To derive a mixed = 0 formulation of the problem (5), we define the following spaces: Introducing the new variable , we may σ = ( )∇ rewrite the problem (5) as follows: similarly, for and .

Some auxiliary results
In this paragraph, we give beneficial results.
• is a symmetric and definite-positive ( ) matrix. We note its inverse matrix.   It's obvious to prove that problem (11) admits a solution , where and is a solution of (6). For ( , σ) σ = ( ) ∇ the uniqueness, we give the following result:  • Assuming that

Resolution of the continuous problem
, we obtain, with the same way as , that (

Results and conclusion
Using a sharp/diffuse interface approach, we propose a new formulation of the seawater intrusion problem in confined aquifers. The mathematical analysis of the problem is based on the regularity of the solution, A time discretization of this new mixed formulation is based on a semi-implicit scheme. We prove that the associated problem is well posed.
In a future work, we will show the convergence result of the semi-implicit scheme as well as an error estimation result for the case of the fully discretized problem.