The mixed finite element (MFE) method is well adapted for the simulation of fluid flow in heterogeneous porous media. However, when employed for the transport equation, it can generate solutions with strong unphysical oscillations because of the hyperbolic nature of advection. In this work, a robust upwind MFE scheme is proposed to avoid such unphysical oscillations. The new scheme is a combination of the upwind edge/face centered finite volume method with the hybrid formulation of the MFE method. The scheme ensures continuity of both advective and dispersive fluxes between adjacent elements and allows to maintain the time derivative continuous, which permits employment of high-order time integration methods via the method of lines (MOL).

Numerical simulations are performed in both saturated and unsaturated porous media to investigate the robustness of the new upwind MFE scheme. Results show that, contrarily to the standard scheme, the upwind MFE method generates stable solutions without under and overshoots. The simulation of contaminant transport into a variably saturated porous medium highlights the robustness of the proposed upwind scheme when combined with the MOL for solving nonlinear problems.

The mixed finite element (MFE) method (Raviart and Thomas, 1977; Brezzi et al., 1985; Chavent and Jaffré, 1986; Brezzi and Fortin, 1991; Younes et al., 2010) is known to be a robust numerical scheme for solving elliptic diffusion problems such as the fluid flow in heterogeneous porous media. The method combines advantages of the finite volumes, by ensuring local mass conservation and continuity of fluxes between adjacent cells, and advantages of finite elements by easily handling heterogeneous domains with discontinuous parameter distributions and unstructured meshes. As a consequence, the MFE method has been largely used for flow in porous media (see, for instance, the review of Younes et al. (2010) and references therein). The hybridization technique has been largely used with the MFE method to improve its efficiency (Chavent and Roberts, 1991; Traverso et al., 2013). This technique allows to reduce the total number of unknowns and produces a final system with a symmetric positive definite matrix. The unknowns with the hybrid MFE method are the Lagrange multipliers which correspond to the traces of the scalar variable at edges/faces (Chavent and Jaffré, 1986).

When applied to transient diffusion equations with small time steps, the hybrid MFE method can produce solutions with small unphysical over- and undershoots (Hoteit et al., 2002a, b; Mazzia, 2008). A lumped formulation of the hybrid MFE method was developed by Younes et al. (2006) to improve its monotonicity and reduce nonphysical oscillations. The lumped formulation ensures that the maximum principle is respected for parabolic diffusion equations on acute triangulations (Younes et al., 2006). For more general 2D and 3D element shapes, the lumping procedure allows to significantly improve the monotonous character of the hybrid MFE solution (Younes et al., 2006; Koohbor et al., 2020). As an illustration, the lumped formulation was shown to be more efficient and more robust than the standard hybrid formulation for the simulation of the challenging nonlinear problem of water infiltration into an initially dry soil (Belfort et al., 2009). The lumped formulation has recently been used for flow discretization in the case of density-driven flow in saturated–unsaturated porous media (Younes et al., 2022a).

However, the MFE method remains little used for the discretization of the full transport equation. When employed to the advection–dispersion equation, the MFE method can generate solutions with strong numerical instabilities in the case of advection-dominated transport because of the hyperbolic nature of the advection operator. To avoid these instabilities, one of the most popular and easiest ways is to use an upwind scheme. Indeed, although upwind schemes introduce some numerical diffusion leading to an artificial smearing of the numerical solution, they avoid unphysical oscillations and remain useful, especially for large domains and regional field simulations. In the literature, some upwind mixed finite element schemes have been employed to improve the robustness of the MFE method for advection-dominated problems (Dawson, 1998; Dawson and Aizinger, 1999; Radu et al., 2011; Vohralík, 2007; Brunner et al., 2014).

The main idea of an upwind scheme for an element

In this work, a new upwind MFE method is proposed for solving the full transport equation without requiring any approximation of the upwind concentration. The new scheme is a combination of the upwind edge/face centered finite volume (FV) scheme with the lumped formulation of the MFE method. It guarantees continuity of both advective and dispersive fluxes at element interfaces. Further, the new upwind MFE scheme maintains the time derivative continuous, and thus allows to employ high-order time integration methods via the method of lines (MOL), which was shown to be very efficient for solving nonlinear problems (see, for instance, Fahs et al., 2009 and Younes et al., 2009).

This article is structured as follows. In Sect. 2, we recall the hybrid MFE method for the discretization of the transport equation. In Sect. 3, we introduce the new upwind MFE method based on the combination of the upwind edge/face FV scheme with the lumped formulation of the MFE method. In Sect. 4, numerical experiments are performed for transport in saturated and unsaturated porous media to investigate the robustness of the new developed upwind MFE scheme. Some conclusions are given in the last section of the article.

The mass conservation of the contaminant in variably saturated porous media
is

Vectorial basis functions for the MFE method.

The water content

To apply the hybrid MFE method to the transport Eq. (5), we approximate the
dispersive flux

The variational formulation of Eq. (2) using the test function

Denoting the local matrix

Substituting Eq. (2) into Eq. (12) yields

The hybridization of the MFE method is performed in the following two steps.

The flux Eq. (10) is substituted into the mass conservation Eq. (13), which
is then discretized in time using the first-order implicit Euler scheme,

Hence, the mean concentration at the new time level

The mean concentration given by Eq. (15) is then substituted into the flux
Eq. (10), which allows expressing the dispersive flux

Continuity of concentration and total flux between adjacent elements with the hybrid MFE method.

Note that the advective flux

Note that the hybrid MFE Eq. (18), obtained by approximating the dispersive flux with RT0 basis functions, is equivalent to the new MFE method proposed in Radu et al. (2011).

In this section, we recall the main principles of two existing approaches, developed to improve the stability of the MFE solution of the transport equation. The first approach is the upwind hybrid MFE scheme of Radu et al. (2011), developed for advection dominated transport. The second approach is the lumped hybrid MFE method of Younes et al. (2006), developed for dispersive transport.

In the case of advection-dominated transport, solving the hybrid MFE Eq. (18) can yield solutions with strong instabilities. A common way to avoid such instabilities is to use an upwind scheme for the advective flux. Thus, for an element

The advective flux of Eq. (19) is rewritten in the following condensed form:

However, this expression is incompatible with the hybridization procedure.
Indeed, if we replace, in the Eq. (14), the advective term

To avoid this difficulty, Radu et al. (2011) suggested replacing

Plugging Eq. (21) into Eq. (20), the advective flux

The assumption given by Eq. (21) can be a rough approximation, especially in
the case of a heterogeneous domain where dispersion can vary with several
orders of magnitudes from element to element. For such a situation, the edge
concentration can be significantly different from the average of the mean
concentrations of adjacent elements. Furthermore, the advective flux is not
uniquely defined at the interface and can be different for the two adjacent
elements

In this section, we recall the main principles of the lumped hybrid MFE method of Younes et al. (2006), developed to improve the stability of the MFE solution in the case of dispersive transport.

Considering only dispersion, Eq. (5) simplifies to

Therefore, the mean concentration of Eq. (15) becomes

The lumping region

Associating the edge concentration

Besides, applying the steady-state dispersive transport Eq. (26) on the
simplex region

In the lumped formulation Eq. (32), the term of mass (with time derivative) has a contribution only on the diagonal term of the final system matrix. This improves the monotonous character of the solution (see Younes et al., 2006). For instance, in the case of an acute triangulation, the maximum principle is respected by the lumped formulation Eq. (32) whatever the heterogeneity of the porous medium (Younes et al., 2006).

Contrarily to the standard hybrid MFE scheme, where the discretization of the temporal derivative performed in Eq. (14) was necessary to obtain the final system given by Eq. (18), the lumped scheme given by Eq. (32) keeps the time derivative continuous which allows the use of efficient high-order temporal discretization methods via the MOL.

In the case of 2D triangular elements, the lumped formulation Eq. (32) is algebraically equivalent to the nonconforming Crouzeix–Raviart (Crouzeix and Raviart, 1973) finite element method (see Younes et al., 2008). The nonconforming Crouzeix–Raviart method uses the chapeau functions as basis functions to approximate the concentration, like the standard finite element method, but seed nodes are the midpoints of the edges.

To avoid the rough approximation (Eq. 21), we develop hereafter a new upwind MFE scheme where the advection term is calculated using upwind edge
concentration instead of upwind mean concentration of the element

The integration of the whole mass conservation Eq. (5) over the lumping
region

Thus, the final system to solve becomes

Description of the problem of the contamination of a 2D saturated porous medium.

In the case of a first-order Euler implicit time discretization, Eq. (37)
becomes

It is easy to see that, due to upwinding, the system matrix corresponding to
Eq. (28) is always an

Further, Eq. (37) expresses the total exchange between

In this section, a first test case dealing with transport in saturated porous media is simulated with the standard hybrid MFE and the new upwind MFE schemes. The results are compared against an analytical solution in order to validate the new developed scheme and to show its robustness for solving advection-dominated transport problems compared to the standard one. The second test case deals with transport in the unsaturated zone and aims to investigate the robustness of the new scheme when combined with the MOL for solving highly nonlinear problems.

The hybrid and upwind MFE formulations are compared against the analytical
solution developed by Leij and Dane (1990) for a simplified 2D transport problem (Fig. 4). The test case has been employed by Putti et al. (1990) and Siegel et al. (1997) for the verification of transport codes. It deals with the contamination from the left boundary of a 2D rectangular domain of dimension (0, 100)

The boundary conditions for the transport are of Dirichlet type at the inflow (left vertical boundary), with

The analytical solution of this test case for an infinite domain is given by
Leij and Dane (1990),

The final distributions of the concentration with both hybrid MFE and
upwind MFE schemes are depicted in Fig. 5. Although we have used an unstructured mesh, the two schemes yield almost symmetrical results. The
hybrid MFE scheme (Fig. 5a) yields a solution with unphysical oscillations. Indeed, around 1.2 % of the contaminated region (i.e., the region with

Concentration distribution with the hybrid MFE and the upwind MFE methods for the 2D saturated transport problem (only the region

The results of Fig. 6 show that the solution of both hybrid MFE and upwind MFE methods are in very good agreement with the analytical solution, which validates the new upwind MFE numerical model. Note, however, that a small numerical diffusion is observed with the upwind MFE solution, which is especially visible in Fig. 6b. Indeed, for the simulated problem, the transverse dispersivity is much smaller than the longitudinal one, and, as a consequence, the concentration front is sharper in the vertical section than in the horizontal one. This explains why the numerical diffusion generated by the upwind MFE method is more pronounced in Fig. 6b than in Fig. 6a.

Concentration profiles at

The test problem is then simulated using different mesh refinements to
investigate the order of convergence of the new method. We start with a
uniform mesh formed by 1000 triangles and a time step

The runs are performed on a single computer with an Intel Xeon E-2246G processor and 32 GB memory. The results of the computations, summarized in Table 1, clearly show optimal first-order convergence in space and time for the developed upwind hybrid MFE method.

Numerical results for the new upwind hybrid MFE method.

In this test case, the developed upwind MFE method is combined with the MOL for solving contaminant transport in a variably-saturated porous medium. The advection–dispersion equation is transformed to an ordinary differential equation (ODE) using the new upwind MFE formulation for the spatial discretization, whereas the time derivative is maintained continuously. Therefore, high-order time integration methods included in efficient ODE solvers can be employed. With these solvers, both the time step size and the order of the time integration can vary during the simulation to deliver accurate results in an acceptable computational time.

To investigate the robustness and efficiency of the combination of the developed upwind MFE method with the MOL, we simulate in this section the problem of contaminant infiltration into a variably-saturated porous medium.

The domain (Fig. 7) is a rectangular box of 3 m

Description of the problem of contaminant infiltration into a 2D variably-saturated porous medium.

In this problem, the flow and transport are coupled by the velocity, which
is obtained by solving the following pressure-head form of the nonlinear
Richards' equation:

We use the standard van Genuchten (1980) model for the relationship between
water content and pressure head,

Parameters for the problem of infiltration into a 2D variably-saturated porous medium.

Concentration distribution, with the hybrid MFE

The simulation is performed for 80 h using a triangular mesh formed by
4273 triangular elements. Two test cases are investigated. In the first test
case, the longitudinal and transverse dispersivities are

The coupled nonlinear flow–transport system is solved using the MOL, which allows the use of efficient high-order time integration methods, for both the hybrid MFE and the upwind MFE schemes. To this aim, a hybrid MFE formulation with continuous time derivative was developed by extending the lumping procedure, developed in Younes et al. (2006) for the flow equation, to the advection–dispersion transport Eq. (5).

The time integration is performed with the DASPK time solver which uses an efficient automatic time-stepping scheme based on the fixed leading coefficient backward difference formulas (FLCBDF). The linear systems arising at each time step are solved with the preconditioned Krylov iterative method. The nonlinear problem is linearized using the Newton method with a numerical approximation of the Jacobian matrix.

The results of the hybrid MFE and the upwind MFE methods are depicted in Fig. 8 for the first test case involving high dispersion. Good agreement can be observed between the results of the hybrid MFE (Fig. 8a) and upwind MFE (Fig. 8b) schemes when combined with the MOL. In these figures, the contaminant progresses essentially vertically through the unsaturated zone of the soil. When the saturated zone is reached, the contaminant progresses horizontally and remains close to the water table. Note that the results of both schemes are stable and free from unphysical oscillations (Fig. 8a and b).

For the second test case with lower dispersion (

Concentration distribution with the hybrid MFE

MFE is a robust numerical method well adapted for diffusion problems on heterogeneous domains and unstructured meshes. When applied to transport equations, the MFE solution can exhibit strong unphysical oscillations due to the hyperbolic nature of advection. Upwind schemes can be used to avoid such oscillations, although they introduce some numerical diffusion. In this work, we developed an upwind scheme that does not require any approximation for the upwind concentration. The method can be seen as a combination of an upwind edge/face centered FV method with the lumped formulation of the hybrid MFE method. It ensures continuity of both advective and dispersive fluxes between adjacent elements and allows to maintain the time derivative continuous, which facilitates employment of high-order time integration methods via the method of lines (MOL) for nonlinear problems.

Numerical simulations for the transport in a saturated porous medium show that the standard hybrid MFE method can generate unphysical oscillations due to the hyperbolic nature of advection. These unphysical oscillations are completely avoided with the new upwind MFE scheme. The simulation of the problem of contaminant transport in a variably-saturated porous medium shows that only the upwind MFE scheme provides a stable solution. The results point out the robustness of the developed upwind MFE scheme when combined with the MOL for solving nonlinear transport problems.

All data presented in this paper as well as the hybrid MFE and the upwind MFE Fortran transport codes are available under request from the first author.

AY defined the aim and the conception of study, developed the numerical model, performed first numerical simulations and drafted the manuscript. HH verified the numerical model. RH revised the manuscript. MF worked on the validation of the numerical model and literature review.

The contact author has declared that none of the authors has any competing interests.

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This paper was edited by Mauro Giudici and reviewed by Thomas Graf and two anonymous referees.