Corresponding author: A. Younes, Laboratoire d'Hydrologie et de Geochimie de Strasbourg, CNRS, UMR 7517, University of Strasbourg, 1 rue Blessig, 67084, Strasbourg Cedex, France. (email@example.com)
 The Henry semianalytical solution for salt water intrusion is widely used for benchmarking density dependent flow codes. The method consists of replacing the stream function and the concentration by a double set of Fourier series. These series are truncated at a given order and the remaining coefficients are calculated by solving a highly nonlinear system of algebraic equations. The solution of this system is often subject to substantial numerical difficulties. Previous works succeeded to provide semianalytical solutions only for saltwater intrusion problems with unrealistic large amount of dispersion. In this work, different truncations for the Fourier series are tested and the Levenberg-Marquardt algorithm, which has a quadratic rate of convergence, is applied to calculate their coefficients. The obtained results provide semianalytical solutions for the Henry problem in the case of reduced dispersion coefficients and for two freshwater recharge values: the initial value suggested by Henry (1964) and the reduced one suggested by Simpson and Clement (2004). The developed semianalytical solutions are compared against numerical results obtained by using the method of lines and advanced spatial discretization schemes. The obtained semianalytical solutions improve considerably the worthiness of the Henry problem and therefore, they are more suitable for testing density dependent flow codes.
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 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  used only 78 terms in these series and calculated the coefficients using a Gauss elimination procedure with full pivoting. Pinder and Cooper  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  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. , 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 , Huyakorn et al. , and Voss and Souza  was not consistent with the original Henry problem. Croucher and O'Sullivan  presented a grid convergence study to evaluate the truncation error due to the spatial discretization. Kolditz et al.  claimed that the discrepancies may be due to the inaccuracy of the Boussinesq approximation assumed by Henry.
 In 2003, Simpson and Clement  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].
Henry , Segol , and Simpson and Clement  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 , 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.  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.”
 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 . 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].
2. Semianalytical Method
 To obtain the semianalytical solution, Henry  used a constant dispersion coefficient and assumed the Boussinesq approximation valid which implies the existence of stream function. Using these assumptions, the steady state flow and transport can be written in the following nondimensional form [Henry, 1964, Segol, 1994]:
where is the dimensionless stream function, is the dimensionless concentration, is the aspect ratio of the domain with and , which are the length and the depth of the aquifer, respectively.
 The nondimensional parameters and in the previous equations are given by
where Q [L2T−1] is the freshwater recharge, [L2T−1] is the coefficient of dispersion, with [LT−1] the saturated hydraulic conductivity, [ML−3] and [ML−3] are the freshwater and saltwater densities, respectively.
 The solution technique, known as Galerkin or Fourier-Galerkin solution [Forbes, 1988], is obtained by replacing the stream function and the salt concentration by double Fourier series of the form:
 Substituting these relations into equations (1) and (2), multiplying equation (1) by and equation (2) by , and integrating over the rectangular domain gives an infinite set of algebraic equations for and namely,
 The functions and are detailed in Appendix A.
Segol [1994, p. 272] wrote about how Henry  described his solution of the set of algebraic equations (6) and (7): “An iterative solution of equation (6) was used in which the were computed by several sub iterations and vice versa. The subiterative cycle consisted of recomputing the values of the quadratic terms ( ) using the revised values of (g > 0) while continuing to hold the constant and then using these values to recompute the (g > 0).”
 This procedure for computing the coefficients of the Fourier series was also used by Segol  and Simpson and Clement . The convergence rate of the method depends upon the values of the parameters and . To overcome the convergence difficulties, Segol  and Simpson and Clement  used the solution with the parameters and as an initial guess for the other parameterizations. Then, the parameters and , are reduced with small stepwise changes until the desired solution is obtained. Segol  stated that the value was the lower limit of the range for which a stable and convergent solution can be obtained.
3. New Semianalytical Strategy and Numerical Code
 In the first part of this section, we describe the new strategy used for solving the nonlinear system of algebraic equations to calculate the Fourier series coefficients of the semianalytical solution. In the second part, we briefly describe the numerical code used to compare numerical and semianalytical results.
3.1. Semianalytical Strategy
 The procedure used by Henry , Segol  and Simpson and Clement  encounters substantial numerical difficulties because of its low convergence rate when lowering the values of the parameters and/or . Indeed, Segol  stated that when lowering the value of , more coefficients are required to obtain a stable solution and the convergence of the scheme becomes difficult. To obtain a stable solution, Segol  decreased the value of by considering small stepwise changes and iterating at intermediate steps. The value was the lower limit of the range for which a stable and convergent solution can be obtained.
 To avoid these difficulties, we use in this work the Levenberg-Marquardt algorithm [Levenberg, 1944; Marquardt, 1963], which has a quadratic rate of convergence to solve the set of nonlinear algebraic equations [Yamashita and Fukushima, 2001]. The Levenberg-Marquardt method is considered as one of the most efficient algorithms for solving systems of nonlinear equations. The nonlinear algebraic system of equations (6)–(7) is written in the form where is a vector formed by the coefficients and . The algorithm attempts to minimize the sum of the squares of the function. The method is a combination of two minimization methods: the gradient descent method and the Gauss-Newton method. Far from the optimum, the Levenberg-Marquard method behaves like a gradient descent method, whereas, it acts like the Gauss-Newton method nearby the optimum.
 The Levenberg-Marquardt iterates starting from an initial solution . At each iteration , the new solution is obtained from the solution of the following linear system
where is the Jacobian.
 Small values of the parameter correspond to the Gauss-Newton update while the large ones correspond to the gradient descent update. When the solution approaches the minimum, the parameter is decreased, what makes the algorithm tends to the Gauss-Newton method.
 In this study the Jacobian is approximated numerically using finite differences and the Levenberg-Marquardt parameter is initially fixed to . During iterations, if the new estimate is sufficiently better than the old one, the parameter is reduced by ten. Otherwise it is increased by a factor of two. The tolerance is fixed to for the sum of the squares of the function.
3.2. Numerical Code
 The semianalytical results will be compared against accurate numerical results obtained using a combination of the method of lines and advanced spatial discretization schemes.
where is the fluid density [ML−3], the specific mass storativity related to head changes [L−1], the equivalent freshwater head [L], t the time [T], the porosity [-], the solute mass fraction [M. salt/M. fluid], the Darcy's velocity [LT−1], the density of the displaced fluid [ML−3], the gravity acceleration [LT−2], the fluid dynamic viscosity [ML−1T−1], the permeability tensor [L2] and z the depth [L].
 The solute mass conservation is written in term of mass fraction:
 For the Henry problem, the dispersion tensor is assumed constant ( is the identity matrix) and the density is assumed to vary linearly with respect to the mass fraction. Due to the form of the governing equation used by Henry, the value of the dispersion coefficient used in the numerical code is equal to the value used in the semianalytical solution divided by porosity .
 To achieve high accuracy for the spatial discretization, each equation within the flow-transport system (9)–(11) is modeled with a specific numerical method. The flow equation is discretized with the locally mass conservative Mixed Finite Element method (MFE), that produces accurate and consistent velocity field [Ackerer and Younes, 2008; Younes et al., 2009; Younes et al., 2010]. The advection part of the transport equation is discretized with the Discontinuous Galerkin (DG) method which produces accurate solution even for problems involving sharp fronts [Siegel et al., 1997]. Finally, the multi-point flux approximation (MPFA) is used to discretize the dispersion equation [Younes and Ackerer, 2008] since it is locally conservative and can treat general irregular grids on anisotropic heterogeneous domains [Aavatsmark, 2002; Younes and Fontaine, 2008a, 2008b]. The combination of the three spatial discretization methods MPFA, DG and MFE, has proven to be accurate and robust for modeling density driven flow problems [Younes and Ackerer, 2008; Konz et al., 2009: Zechner et al., 2011].
 The Differential Algebraic Solver with Preconditioned Krylov (DASPK) time solver is used to solve the MFE_DG_MPFA spatial discretization with the method of lines. DASPK is a mature and sophisticated time integration package for large-scale Ordinary Differential Equations (ODE) or Differential Algebraic Equations (DAE) systems. DASPK uses the Fixed Leading Coefficient Backward Difference Formulas (FLCBDF). The BDF approach works well on stiff problems and has good stability properties. DASPK solves systems of the general form:
 Predictor and corrector polynomials are constructed in order to estimate the truncation error. The time step length and choice of order of integration (up to fifth order) are adapted to minimize the computational effort while keeping the local temporal truncation error small, where the relative and absolute local error tolerances are specified by the user.
 In this study, the Henry problem is discretized with a uniform triangular mesh of 3200 elements (same results were obtained with 5000 and 8000 elements).
4. Results and Discussion
 As stated, by Abarca et al. , the drawbacks of the Henry problem arise from the high values of the parameters a and b that Henry used to obtain the semianalytical solution. Indeed, Henry chose the dimensionless parameters , and . Segol  stated that the value was the lower limit of the range for which a stable and convergent solution can be obtained. Simpson and Clement  proposed a semianalytical solution where the parameter is reduced by half and the parameter is increased by a factor of 2 ( , and ). In this section, the semianalytical solution for the Henry problem is calculated using the Levenberg-Marquardt algorithm and compared to the numerical solution for different test cases. Table 1 provides the freshwater flux, the saltwater density and the diffusion coefficients for each test case. Two values for the parameter are tested: the initial one and the halved one. The values of the parameter are reduced until (Table 1). Different truncations of the Fourier series are tested: the 78 coefficients of Henry , the 138 coefficients of Segol  and the 203 coefficients of Simpson and Clement . A new truncation using 424 terms is introduced to calculate the semianalytical solution in the case of small dispersion values. Two domains are used for the calculation of the semianalytical solution: the initial domain with an aspect ratio and a larger domain with (Table 1). Three concentration isochlors (0.25, 0.5 and 0.75) are used for the comparison between the semianalytical and the numerical solutions. Table 2 provides the position of the three isochlors for the new provided test cases. The transient numerical simulations are performed until a long time to ensure a steady state solution. Figure 2 shows the temporal variation of the concentration in the element corresponding to the intersection of the 0.25 isochlor with the base at t = 800 min for the studied test cases. The results show that the time required to reach a steady state depends on the parameters a and b. For all test cases, 500 min of time seems to be sufficient to obtain a steady position for the isochlors.
Table 1. Aspect Ratio, Freshwater Reacharge, Saltwater Density, Diffusion Coefficient and the Parameters (a;b) for the Different Test Cases
Table 2. Position of the 0.25, 0.5 and 0.75 Isochlors for the Test Cases 3, 4, 5 and 6b
Case 3 ( , , )
Case 4 ( , , )
Case 5 ( , , )
Case 6b ( , , )
(0.25) x =
(0.5) x =
(0.75) x =
(0.25) x =
(0.5) x =
(0.75) x =
(0.25) x =
(0.5) x =
(0.75) x =
(0.25) x =
(0.5) x =
(0.75) x =
4.1. Case 1: The Standard Henry Problem With a = 0.263, b = 0.1, and ξ = 2.0
 This case is the standard test case solved by Henry  on an IBM 650 digital computer using 38 coefficients for the expansion of the stream function and 40 coefficients for the expansion of the concentration. Segol  revaluated the semianalytical solution using an expansion with 38 coefficients for the stream function and 100 coefficients for the concentration. Simpson and Clement  calculated the solution using 103 and 100 coefficients for the expansion of the stream function and concentration, respectively. The same iteration procedure, including a number of subiterations, was used by all the authors.
 We recalculate this standard test case with the Levenberg-Marquardt algorithm using the 78 coefficients of Henry, the 138 coefficients of Segol and the 203 coefficients of Simpson and Clement. Figure 3 compares the new calculated isochlors and the semianalytical solution of Segol  to the numerical solution.
 The semianalytical solution obtained using the Henry truncation (78 terms) presents strong oscillations along the three isochlors (0.25, 0.5 and 0.75). This shows that the Henry solution was in error not because the full equilibrium solution was not reached, but rather because of the limited number of coefficients used in the Fourier series. Note that Segol  stated that the values obtained by Henry are interpolated and “are drawn to eliminate local variations that appear due to truncation of Fourier series”.
 The Simpson and Clement truncation gives the same semianalytical solution (not plotted in Figure 3) than the Segol truncation (138 terms). Contrarily to the solution of Segol , which is slightly shifted to the right (especially the 0.75 isochlor), the calculated semianalytical solution has an excellent agreement with the numerical solution (Figure 3).
4.2. Case 2: The Modified Henry Problem With a = 0.1315, b = 0.2, and ξ = 2.0
Simpson and Clement  discussed the worthiness of the Henry problem by comparing the isochlors obtained with the coupled and uncoupled flow transport models. To increase the density-dependent effects compared to boundary forces, they proposed to decrease the freshwater recharge by half. Compared to the previous test case, the parameter is reduced by 50% and the parameter is increased by a factor of 2.
 The semianalytical solution is revaluated using a scheme of 203 terms. To facilitate the convergence, they used the solution with the parameters and as an initial guess and then lowered the parameter using 10 nonuniform steps.
 The semianalytical solution calculated with the Levenberg-Marquardt algorithm in the case of Henry truncation (78 terms) and the case of Segol truncation (138 terms) are shown in Figure 4. For both calculations, the initial solution for the Levenberg-Marquardt algorithm was set to zero.
Figure 4 shows that the truncation of Henry (78 terms) is not sufficient to obtain accurate results. The Simpson truncation (203 terms) gives the same results (not plotted in Figure 4) than the Segol truncation (138 terms). Indeed, Simpson and Clement  used the same expansion for the concentration (100 terms) than Segol . This confirms the observation of Segol  who stated that the system is much less sensitive to the number of coefficients for the expansion of the stream function than the number of coefficients for the expansion of the concentration . As in the previous case, the 38 coefficients proposed by Henry and Segol for the expansion of the stream function are adequate. Comparison with numerical solution shows a very good agreement with numerical results. Note that the steady state position of the 0.25 isochlor is obtained after around 350 min whereas for the previous case it was around 150 min (Figure 2).
Figure 4 presents also the numerical results of the uncoupled flow model where the solute transport acts as a tracer. In this case, the solute intrusion is due to boundary forcing. Indeed, the right hand side boundary condition corresponds to hydrostatic pressure with heavier saline water. This induces solute intrusion even in the case of uncoupled flow where the solute acts as a tracer inside the domain (for more details see Simpson and Clement ).
4.3. Case 3: The Modified Henry Problem With a = 0.1315, b = 0.1, and ξ = 2.0
 This test case is obtained from the modified Henry problem of Simpson and Clement  and by lowering the dispersion coefficient by half (Table 1). This increases the density-dependent effects compared to boundary forcing. Figure 5 shows that the semianalytical isochlors move inland significantly farther then the uncoupled numerical isochlors. Therefore, this case is more sensitive to density-dependent effects than the Simpson and Clement  test case. A very good agreement is observed between semianalytical and numerical results, except the unphysical oscillations (local over- and undershoots) observed in the 0.25 semianalytical isochlor at the upper right corner of the domain (Figure 5). Note that these unphysical oscillations, due to truncation of the Fourier series, remain in the solution with the Simpson and Clement truncation of 203 terms. Therefore, a new truncation is performed by using 424 terms of the Fourier series with 214 terms for the expansion of the stream function and 210 terms for the expansion of the concentration. The results of this new truncation avoid the unphysical oscillations at the right top corner of the domain, although the position of the isochlors does not change. Note that the new truncation requires significantly more computational time (around 5 days) than the Segol truncation (around 10 h). On the other hand, the numerical simulation with the DASPK numerical code requires less than 3 min of CPU time.
 Finally, as shown in Table 1, this test case ( and ) can also be obtained using the initial freshwater recharge (instead of the halved one), the initial dispersion coefficient (6.610−6 m2 s−1) and a salt water density of (instead of ). A new numerical simulation is performed with these values. The obtained isochlors are in excellent agreement with the semianalytical solution. This shows the validity of the Boussinesq approximation for the Henry problem even for a saltwater density of . Indeed, the Boussinesq approximation is used in the semianalytical calculation but not in the numerical calculation and both give the same results.
4.4. Case 4: The Modified Henry Problem With a = 0.263, b = 0.04, and ξ = 2.0
 As stated by Voss and Souza , Kolditz et al. , and Abarca et al. , the major drawback of the Henry problem arises from the high dispersion value used by Henry to obtain the semianalytical solution. Therefore the last three test cases are devoted to the semianalytical solution with a strong reduction of the dispersion coefficient which implies strong reduction of the parameter .
 The results for , and presented in Figure 6 show a reduced transition zone and a significant difference between coupled and uncoupled results. This makes this test case suitable for testing seawater intrusion codes. The semianalytical solution obtained with the Segol truncation (138 terms) is in very good agreement with the coupled numerical solution. Note, however, that small unphysical oscillations appear in the right and left corners at the top of the domain. These unphysical oscillations remain with the Simpson and Clement truncation (203 terms). As previously mentioned, the results of the new truncation with 424 terms show that the unphysical oscillations at the top of the domain are completely avoided, although the position of the isochlors does not change.
4.5. Case 5: The Modified Henry Problem With a = 0.263, b = 0.02, and ξ = 2.0
 In this test case, we lowered the dispersion by half compared to the previous case. The truncations proposed by Segol (138 terms) and Simpson and Clement (203 terms) are not sufficient to obtain a stable solution in this case. Both truncations lead to significant unphysical oscillations not only in the top of the domain but also in the isochlors curves. The results of the new truncation (424 terms) are compared to the numerical solution in Figure 7. Despite the unphysical oscillations observed at the top of the domain, a good agreement is observed between the semianalytical and the numerical isochlors. These unphysical oscillations could be avoided if one uses a truncation with much more coefficients. This requires very long computational time (several weeks) and as previously this will have no significant effects on the three isochlors positions. The results of Figure 7 show a very narrow transition zone due the low dispersion coefficient, thus, saltwater invades more the domain than the previous case. Coupled and uncoupled numerical results are significantly different (Figure 7) which reflects the importance of the density-dependent effects compared to boundary forcing.
4.6. Case 6a: The Modified Henry Problem With a = 0.1315, b = 0.04, and ξ = 2.0
 This case has a small dispersion and a freshwater recharge reduced by half as suggested by Simpson and Clement . As mentioned before, the solutions obtained using the Segol (138 terms) and Simpson and Clement (203 terms) truncations are not accurate and lead to significant unphysical oscillations not only at top of the domain but also in the isochlor curves. Figure 8 shows that the new truncation (424 terms) gives a stable solution which is in good agreement with the numerical results except in the lower left corner of the domain where the semianalytical 0.25 isochlor is slightly more advanced than the numerical one. This discrepancy is observed because the semianalytical 0.25 isochlor is very close to the inland boundary with zero concentration. Note that the 0.5 and 0.75 isochlors are well reproduced by the numerical code.
4.7. Case 6b: The Modified Henry Problem With a = 0.1315, b = 0.04, and ξ = 3.0
 To reduce the influence of the zero-concentration Dirichlet left boundary condition on the 0.25 isochlor for the previous case, a new semi analytical solution is performed on a larger rectangular domain with an aspect ration . The obtained semianalytical results are in very good agreement with the numerical solution as shown in Figure 9. Note that in this case, the steady state position of the isochors is reached within 500 min. In Figure 9, the saltwater intrudes much more in the domain than all previous cases. Large differences can be observed between coupled and uncoupled results. Therefore, this test case is the most sensitive one upon all the previous to density-dependent effects and should be preferred for benchmarking density driven flow codes.
 The Henry saltwater intrusion problem is considered as one of the most popular test cases of density-dependent groundwater flow models, since Henry  provided a semianalytical solution of the problem by expanding the stream function and the salt concentration in double Fourier series. These series are truncated at a given order and the remaining coefficients are calculated by solving a highly nonlinear system of algebraic equations. Henry  used a truncation of 78 terms, Segol  used a truncation of 138 terms and Simpson and Clement  used a truncation of 203 terms. These authors solved the nonlinear system as a system of linear equations treating the nonlinear right hand side as a known quantity, updated iteratively until convergence. This procedure encountered substantial convergence difficulties especially for reduced dispersion values. Consequently, all published works succeeded to develop semianalytical solutions only when an unrealistically large amount of dispersion is introduced in the solution.
 In this work, this deficiency of the Henry problem was avoided by using the Levenberg-Marquardt algorithm, which has a quadratic rate of convergence, to calculate the coefficients of the Fourier series. Different truncations are studied and a new truncation based on 424 terms is proposed to develop the semianalytical solutions of the Henry problem with reduced dispersion coefficients.
 The obtained semianalytical solutions are in a very good agreement with the numerical results obtained using the method of lines and advanced spatial discretization schemes. These solutions improve the applicability of the semianalytical solution of the Henry problem to saltwater intrusion problems with reduced diffusion and are therefore more suitable to benchmark density-driven flow codes.
 The non linear algebraic equations are as follows:
and is the Kronecker delta such that
 The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper. This study was partially supported by the GnR MoMaS (PACEN/CNRS ANDRA, BRGM CEA EDF IRSN) France. And in part, was made possible by the support of SNF (Swiss National Foundation, grant 200020_125167), whose support is gratefully acknowledged. And a special thanks to Stefan Wiesmeier and Silvia Leupin for their appreciated support.