## 1. Introduction

[2] Saltwater intrusion into unconfined coastal aquifers has been largely investigated using laboratory experiments [e.g., *Goswami and Clement*, 2007; *Thorenz et al.*, 2002] and/or numerical simulations [e.g., *Park and Aral*, 2008]. However, the existence of a semianalytical solution made the synthetic Henry saltwater intrusion problem [*Henry*, 1964] as one of the most widely tests used for verification of density driven flow codes. The problem describes steady state saltwater intrusion through an isotropic confined aquifer. Freshwater enters the idealized rectangular aquifer (Figure 1 ) with a constant flux rate from the inland (left) boundary. A hydrostatic pressure is prescribed along the coast (right) boundary where the concentration corresponds to seawater concentration. The top and the bottom of the domain are impermeable boundaries. The saltwater intrudes from the right until an equilibrium with the injected freshwater is reached. The semianalytical solution of Henry [*Henry*, 1964] provides the steady state isochlors positions by expanding the salt concentration and the stream function in double Fourier series. *Henry* [1964] used only 78 terms in these series and calculated the coefficients using a Gauss elimination procedure with full pivoting. *Pinder and Cooper* [1970] were the first to simulate the Henry problem using a transient numerical code with two different initial conditions to ensure convergence to the steady state solution. The obtained results as well as those obtained later by [*Segol et al.*, 1975; *Frind*, 1982; *Huyakorn et al.*, 1987] were not in agreement with Henry's solution. In 1987, *Voss and Souza* [1987] showed that the discrepancies in the published papers were due to the use of different dispersion coefficients in numerical and semianalytical calculations. However, solving this problem did not lead to a satisfactory matching. Many possible reasons for the discrepancies have been invoked in the literature: for *Huyakorn et al.* [1987], the discrepancies may be due to the discretization errors within the numerical codes and/or to the use of different boundary conditions at the seaward side between semianalytical and numerical codes. Indeed, the sea boundary condition used in the work of *Frind* [1982], *Huyakorn et al.* [1987], and *Voss and Souza* [1987] was not consistent with the original Henry problem. *Croucher and O'Sullivan* [1995] presented a grid convergence study to evaluate the truncation error due to the spatial discretization. *Kolditz et al.* [1998] claimed that the discrepancies may be due to the inaccuracy of the Boussinesq approximation assumed by Henry.

[3] The most important reason for discrepancies has been invoked by *Voss and Souza* [1987] who claimed that, due to the lack of computing resources, Henry's truncation may not contain enough terms in the Fourier series to represent accurately the solution. In 1994, *Segol* [1994] revaluated the semianalytical solution of Henry by using a new truncation of the Fourier series with 138 terms instead of the 78 used by Henry. The revaluated solution shows a good agreement with the numerical results [e.g., *Oldenbourg and Prues*, 1995; *Herbert et al.*, 1988; *Ackerer et al.*, 1999; *Buès and Oltéan*, 2000; *Abarca et al.*, 2007, *Younes et al.*, 2009].

[4] In 2003, *Simpson and Clement* [2003] performed a coupled versus uncoupled analysis to show that the true profile in the Henry problem is largely determined by boundary forcing and much less by the density-dependent effects. In the uncoupled flow, the solute transport acts as a tracer and has no influence on the flow equation. To improve the worthiness of the Henry problem, they suggested a decreasing of the fresh water recharge by half [*Simpson and Clement*, 2004]. The semianalytical solution is revaluated in this case by using 203 terms in the Fourier series [*Simpson and Clement*, 2004].

[5] *Henry* [1964], *Segol* [1994], and *Simpson and Clement* [2004] used the same iterative technique to calculate the coefficients of the Fourier series. They solved the nonlinear system as a system of linear equations where the expansion coefficients are considered as unknowns. The nonlinear right hand side is treated as a known quantity, updated iteratively until convergence. As stated by *Segol* [1994], this technique encountered substantial convergence difficulties for small values of the dispersion coefficient. Note that all published works succeeded to develop semianalytical solutions only when an unrealistically large amount of dispersion is introduced in the solution. This deficiency was pointed out by *Kolditz et al.* [1998] and by *Voss and Souza* [1987, p. 1857], who stated that due to the large amount of dispersion, “this test does not check whether a model is consistent or whether it accurately represents density driven flows, nor does it check whether a model can represent field situation with relatively narrow transition zones.”

[6] In this work, we calculate the coefficients of the Fourier series by using the Levenberg-Marquardt algorithm, which has a quadratic rate of convergence, to solve the nonlinear algebraic system of equations. Different truncations of the infinite Fourier series are tested. Semianalytical solutions for the Henry problem are developed in the case of reduced dispersion coefficients and for two freshwater recharge values: the initial value suggested by Henry and the reduced one suggested by *Simpson and Clement* [2004]. The semianalytical solutions are compared against numerical results obtained using a robust numerical model based on the method of lines and advanced spatial discretization schemes [*Younes et al.*, 2009].