# A Deforming Mixed-Hybrid Finite Element Model for Robust Groundwater Flow Simulation in 3D Unconfined Aquifers with Unstructured Layered Grids

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{−2}–10

^{−3}). New developments presented the numerical solution of 3D unconfined seepage problems with the smoothed finite element method borrowed from solid mechanics [16]. A limitation of the fixed mesh approach is that the exact position and shape of the water table could only be determined by post-processing computed groundwater heads. Hence, it is expected that a moving mesh technique will always lead to a more accurate position and a smooth shape of the phreatic surface when compared with a fixed mesh approach using the same resolution.

## 2. Theory

#### 2.1. Governing Equations

^{−1}] is the specific storage coefficient, $h$ [L] is the hydraulic head, $t$ [T] is the time, ${\mathit{q}}_{s}$ [T

^{−1}] denotes the source/sink strength term positive for sources and negative for sinks, $\nabla $ [L

^{−1}] is the differential operator, and $\mathit{q}$ [LT

^{−1}] is the specific groundwater discharge, following Darcy’s law:

^{−1}] is the hydraulic conductivity tensor. In the presented model, we use a full tensor for each soil type in the stratigraphic domain:

#### 2.2. Boundary Conditions

#### 2.2.1. Phreatic Surface

^{−1}] is the specific yield and $N\left(x,y,t\right)=-N\nabla z$ [LT

^{−1}] is the flux associated with net recharge (i.e., inflows such as recharge minus outflows such as evapotranspiration) across the surface. The kinematic condition is nonlinear, owing, first, to the occurrence of the quadratic terms in $\nabla h$ and, second, because the conductivity coefficients depend upon the location of the water table inside the stratigraphic domain. It indicates that the flux vector across the water table is not entirely vertical. It is a combination of linear and nonlinear flux terms. The nonlinearity derives from the changing shape of the deforming computational domain volume. A direct numerical discretization of Equation (4) subject to boundary conditions (5) and (6), in particular, increases the model’s nonlinearity and impacts its efficiency. Alternatively, we use a linearized version of Equation (6) borrowed from Dagan [43] that was used also by Neuman [44]:

#### 2.2.2. Seepage Face

#### 2.2.3. Recharge

^{−1}], is prescribed directly on the moving water table boundary, as presented in Equations (8) and (10), for transient and steady-state regimes, respectively. Under transient conditions, the recharge may take a non-negligible time to reach the saturated zone from the land surface. This is known as the unsaturated zone time lag, ${t}_{u}\left(x,y\right)$, which is site-specific and depends on many parameters, such the water table depth, layered heterogeneity in the unsaturated zone, rainfall intensity, crops water demand, etc. Many approaches have been developed in the literature [45,46,47] to provide first-order estimates of this parameter which can be provided as a model input to delay the recharge flux prescribed at the horizontal coordinates $\left(x,y\right)$ and time $t$ as $RE\left(x,y,t-{t}_{u}\left(x,y\right)\right)$. Obviously, a delayed recharge flux is not needed under steady-state conditions. Based on observed rainfall and water levels time series in six chalk aquifers, ref. [48] concluded that for shallower water tables, a linear relationship exists between the lag time and the depth to the water table. In [49], simple analytical methods based on a steady-state assumption and rigorous one-dimensional transient simulations were compared to assess the travel time in unsaturated zone profiles. The authors found that none of the results obtained with the simplified methods matched the transient simulations.

#### 2.2.4. Evapotranspiration

^{−1}], is prescribed directly on the moving water table boundary, such that the net recharge in Equations (8) and (10) is $N=RE-ET$. This condition is characterized by two conceptual parameters. The extinction elevation, ${z}_{EXT}$, below which evapotranspiration from the water table ceases. Next, the ET surface elevation, ${z}_{ET}>{z}_{EXT}$, above which the evapotranspiration loss from the water table occurs at a fixed rate, $E{T}_{max}$. In between, it is assumed that the $ET$ flux varies linearly with the water table elevation. Hence,

#### 2.2.5. Pumping and Injection Wells

^{3}T

^{−1}], depends on the position of the water table relative to the well screen. When the water table position is higher than ${z}_{top}^{w}$, the flow rate is distributed among the formations perforated by the well screen; otherwise, when it intersects, the well screen length the flow rate is distributed among the formations lying between the water table position and the bottom of the well screen. A third case corresponds to the situation when the water table is below the bottom of the well screen, leading to shutting-down the well (i.e., ${Q}_{w}=0$). Additionally, we have implemented an option for a gradual decrease in the prescribed flow rate of a pumping well when the water table position starts to decrease below a threshold elevation, ${z}_{thr}^{w},({z}_{\mathit{b}ot}^{w}{z}_{thr}^{w}{z}_{top}^{w})$, as was introduced in [9]. To summarize, the well boundary condition is formulated as follows:

#### 2.2.6. River

#### 2.2.7. Drain

## 3. The Discrete Moving Mesh Model

#### 3.1. Mixed-Hybrid Finite Element Approximation (MHFEM)

_{i}, of element e. This implies that the basis function ${\mathit{b}}_{i}$ contributes a unit flux to face number i, and a null flux to all others.

_{e}is the cell-centered groundwater head.

_{e}. From Equation (21), it is clear that this shape factor depends only on the geometry and the hydraulic conductivity components of the hexahedral element e.

_{i}, shared by two neighbor elements, e

_{1}and e

_{2}, such that:

_{1}and e

_{2}) into Equation (24) yields:

^{3}] is the volume of element e, $n$ is a time index, $\Delta {t}^{n}={t}^{n+1}-{t}^{n}$ [T] is the time step at time level $n$, and ${Q}_{s}^{n+1}={\mathit{q}}_{s}^{n+1}{V}_{e}$ [L

^{3}T

^{−}

^{1}] is a volumetric source/sink term.

_{e}. Equation (35) is rewritten in globally assembled matrix form as:

#### 3.2. Mesh Adaption to Water Table Movement

#### 3.3. Numerical Implementation of Boundary Conditions

#### 3.3.1. Phreatic Surface

#### 3.3.2. Seepage Face

#### 3.3.3. Constant Head

#### 3.3.4. Recharge and Evapotranspiration

#### 3.3.5. Pumping/Injection Well

#### 3.3.6. River

#### 3.3.7. Drain

#### 3.4. Computer Code

^{©}is the computer code implementing the adaptive MHFEM method presented in this paper. It is entirely programmed using the object-oriented (OO) C++ language. Although there was an increasing support of OO methodologies in other widely used computer languages by the groundwater modeling community, such as the Fortran 2003 standard, C++ was preferred because it is tightly integrated to high-level scripting languages (i.e., such as Python), which are planned to be used in the future to accelerate the addition of new features. Furthermore, the extended support of OO features and template-based metaprogramming were other features needed in this project. The core computational framework includes classes for sparse linear algebra, structured/unstructured mesh management, iterative sparse linear solvers, and boundary condition packages. Similar to MODFLOW, all supported types of boundary conditions could be loaded on-the-fly from their input files to the discrete sparse system of the equations. This flexibility supports the flexible and modular testing of groundwater flow models using different sets of hypothetical boundary conditions when using several conceptual models.

## 4. Results

^{©}with the same solution obtained by MODFLOW. The last example is an adapted three-dimensional model of an alluvial aquifer which is used as a test bed to compare the two models’ performance when simulating an unconfined groundwater flow.

#### 4.1. Pumping in a Two-Dimensional Anisotropic Aquifer

^{3}s. A constant time step of 2 s is selected. The hydraulic head drawdown is monitored at two piezometers, P

_{1}and P

_{2}, as shown in Figure 2b. Figure 3a,b show the evolution of the numerically computed drawdowns with those analytically evaluated using Equation (47) at piezometers P

_{1}and P

_{2}for the diagonal and fully symmetric transmissivity tensors, respectively. In all cases, the MHFEM and analytical solutions are in good agreement. For the second case, it is clear that the hydraulic head drawdown is higher in P

_{2}and lower in P

_{1}than in the first case as expected, at least qualitatively, from the change in the transmissivity values provided in Table 1.

#### 4.2. Flow in a Heterogeneous Porous Medium

^{2}m.d

^{−1}to 10

^{−3}m.d

^{−1}. Groundwater flows from north to south between two fixed head boundaries at 100 m and 99 m, respectively. A total of 19 particles were placed at an equal distance of 5 m at the top boundary tracing their flow paths. MODPATH simulation uses a semi-analytical particle tracking method [54]. Meanwhile, particle tracking simulation using the continuous specific discharge fields obtained by the MHFEM approximation is based on previously published algorithms [34].

^{©}and MODFLOW/MODPATH computer codes, respectively. In the first column of Figure 5, the simulated path lines using the MHFEM approximation with increasing grid resolutions from top to bottom are shown for the 10 × 10, 20 × 20, and 30 × 30 grid resolutions. Similarly, the second column shows the path lines simulated with MODPATH with identical grid resolutions. Notably, in all subfigures, the reference path lines are plotted with a red color for a direct comparison of both MHFEM and FDF approximations. These reference path lines are identical to the reference solution presented by the authors in the original publication [28].

#### 4.3. Two-Dimensional Free Surface in a Homogeneous Earth Dam

#### 4.4. Two-Dimensional Free Surface in a Pumped Phreatic Aquifer

^{3}/d. The well is screened in the lower half of the aquifer thickness. Owing to the aquifer homogeneity, the flow rate is equally divided among the cells lying along the well perforation.

^{©}solves a problem with 2530 unknowns using the deforming mesh algorithm developed in this paper.

^{−12}). In the FDM- based simulation, a Picard iteration is adopted with a convergence closure equal to 10

^{−3}m. The same parameter was used as a convergence closure for the heads in the mesh adaption loop. Likewise, a maximum number of 100 outer iterations was similarly taken for both algorithms. All 10 upper horizontal lines of the 2D mesh were allowed to move during the iterative adaption process.

#### 4.5. Three-Dimensional Free Surface in a Pumped Phreatic Aquifer

^{3}/d. The well is screened in the lower 10 m of the aquifer.

^{©}solves a problem with 53,435 unknowns.

^{−2}m. This is because the MODFLOW simulation does not converge when the last parameter was increased up to 10

^{−3}m. Contrarily, a higher outer convergence criterion does not pose any challenge to the adaptive mesh model as only three more mesh iterations were required. All four upper horizontal slices of the 3D mesh were allowed to move during the iterative adaption process.

## 5. Discussion

^{−3}m, while this impacts even the simplest models with a fixed mesh approach.

## 6. Conclusions

- The mixed-hybrid finite element approximation is suitable for highly heterogeneous and anisotropic aquifers. It provides accurate solutions for the discharge fields even for coarser grids than the FDMs. Therefore, it is a suitable approximation technique prior to advective particle tracking or solute/heat transport simulations.
- The moving mesh technique based on the MHFEM approximations provides an accurate solution for the water table interface, groundwater heads, and specific discharge fields. The water table is accurate because it is free from cellwise interpolation and independent of the vertical grid size, as used in fixed mesh methods.
- The robustness of the moving mesh method cannot be surpassed by a fixed mesh alternative. The nonlinearity associated with the latter is replaced by a series of saturated flow problems which are always guaranteed to converge for some preconditioners. Moreover, the moving mesh outer iteration loop accepts higher convergence closure parameters.
- Owing to the simplicity of the mesh adaption scheme, the moving mesh model proves to be equally efficient. For all of the studied problems, it was faster than the Picard loop-based FDM solutions when using the same resolution.
- The efficiency of the moving mesh MHFEM model is supported by an almost quadratic rate of convergence.
- The presented samples prove that it is possible to confidently develop practical groundwater applications using HydroTec
^{©}code. Nevertheless, additional benchmarks and testing will strengthen users’ confidence.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Expressions of the Vector Basis Functions

## Appendix B. MHFEM Approximation on a Rectangular Element with a Diagonal Hydraulic Conductivity Tensor

## Appendix C. Equivalence with the FDM Approximation for a Diagonal Hydraulic Conductivity Tensor

## Appendix D. Assembly of the Local Mixed-Hybrid Finite Element Matrix

**K**

_{e}, for the soil material filling the same element. Outputs are the six-by-six inverse of the local mixed-hybrid matrix, ${A}_{e}^{-1}$, the six-by-one array of its row-sums (i.e., the shape factors ${\alpha}_{e,i}i=1,\dots ,6$), the sum of all entries in ${A}_{e}^{-1}$: ${\alpha}_{e}$, and the hexahedral element volume V

_{e}.

1 Begin Function AssembleLocalMatrix (Input e, K_{e}) Returns ${A}_{e}^{-1}$, ${\alpha}_{e,i}$, ${\alpha}_{e}$, V_{e}2 Obtain K^{−1} the inverse of K_{e}3 Set V _{e} to 04 Set A _{e} to 05 Begin Cycle for Gauss in [1,6] 7 Obtain the transpose of the Jacobian J^{T} and its determinant |J|8 Obtain the mixed basis function b_{Gauss} at Gauss point 9 Add W _{Gauss} x |J| to V_{e} 10 Begin Cycle for face i in [1,6] 11 Obtain W _{i} = K^{−1} x J^{T} x b_{Gauss}(i)12 Begin Cycle for face j in [1,6] 13 If j is lower than or equals i Add ((W _{i} x b_{Gauss}(i)) . (J^{T} x b_{Gauss}(j)))/|J| to A_{e}(i,j)14 End Cycle 15 End Cycle 16 End Cycle 17 Begin Cycle for face i in [1,6] 18 Begin Cycle for face j in [1,6] 19 If j is greater than i Set A _{e}(j,i) to A_{e}(i,j)20 End Cycle 21 End Cycle 22 Obtain ${A}_{e}^{-1}$ the inverse of A _{e}23 Set ${\alpha}_{e,i}$ to $\sum}_{j=1}^{6}{A}_{e}^{-1}\left(i,j\right)$ for i in [1,6] 24 Set ${\alpha}_{e}$ to $\sum}_{i=1}^{6}{\alpha}_{e,i$ 25 End Function |

**J**

^{T}, from the real to the reference element and its determinant |J|, and (iii) the vector basis function at the Gauss point. This loop also calculates the element volume by adding each Gauss point contribution, as shown in line nine.

## Appendix E. Assembly of the Global Mixed-Hybrid Finite Element Matrix

**Q**at the current time step. For steady-state conditions, the CellHead array may take any initial set of head values. The number of elements ne in line 2 may be deduced from the shape of the CellHead array.

1 Begin Function AssembleGlobalMatrix (Input $\Delta t$, FixedHead, CellHead, Q) Returns Matrix, Rhs 2 Begin Cycle for e in [1,ne] 3 Obtain the element hydraulic conductivity tensor K_{e}4 AssembleLocalMatrix (e, K_{e}) Returns ${A}_{e}^{-1}$, ${\alpha}_{e,i}$, ${\alpha}_{e}$, V_{e}5 Obtain ${S}_{e}$ the specific storage coefficient in element e 6 If the flow is steady-state Set ${\beta}_{e}$ to 1 7 If the flow in transient Set ${\beta}_{e}$ to $\left({\alpha}_{e}\Delta t/{S}_{e}\mathrm{Ve}+{\alpha}_{e}\Delta t\right)$ 8 Begin Cycle for face i in [1,6] 9 Obtain fi the global number of face i 10 If fi is a constant head face Cycle 11 Begin Cycle for face j in [1,6] 12 Obtain fj the global number of face j 13 If fj is not a constant head face 14 If fj is lower than or equals fi Add ${A}_{e}^{-1}$(i,j) − ${\beta}_{e}\frac{{\alpha}_{e,i}{\alpha}_{e,j}}{{\alpha}_{e}}$ to Matrix(fi,fj)15 Else 16 Add -(${A}_{e}^{-1}$(i,j)-${\beta}_{e}\frac{{\alpha}_{e,i}{\alpha}_{e,j}}{{\alpha}_{e}}$) FixedHead(fj) to Rhs(fi)17 End If 18 End Cycle 19 Add Q(e)$\frac{{\beta}_{e}{\alpha}_{e,i}}{{\alpha}_{e}}$ + ${\alpha}_{e,i}\left(1-{\beta}_{e}\right)$ CellHead(e) to Rhs(fi)20 End Cycle 21 End Cycle 22 End Function |

## Appendix F. Retrieving Cell-Centered Heads and Normal Face Fluxes from Head Traces

1 Begin Function Hybridization (Input $\Delta t$, FaceHead, CellHead, Q) Returns CellHead, FaceFlux 2 Begin Cycle for e in [1,ne] 3 Obtain the element hydraulic conductivity tensor K_{e}4 AssembleLocalMatrix (e, K_{e}) Returns ${A}_{e}^{-1}$, ${\alpha}_{e,i}$, ${\alpha}_{e}$, V_{e}5 Obtain ${S}_{e}$ the specific storage coefficient in element e 6 If the flow is steady-state Set ${\beta}_{e}$ to 1 7 If the flow in transient Set ${\beta}_{e}$ to $\left({\alpha}_{e}\Delta t/{S}_{e}\mathrm{Ve}+{\alpha}_{e}\Delta t\right)$ 8 Set sum to 0 9 Begin Cycle for face i in [1,6] 10 Obtain fi the global number of face i 11 Add ${\alpha}_{e,i}$ FaceHead(fi) to sum 12 End Cycle 13 Set CellHead(e) to $\left(1-{\beta}_{e}\right)$ CellHead(e) + (Q(e)+sum)$\frac{{\beta}_{e}}{{\alpha}_{e}}$ 14 Begin Cycle for face i in [1,6] 15 Obtain fi the global number of face i 16 Set FaceFlux(i,e) to ${\alpha}_{e,i}$ CellHead(e)17 Begin Cycle for face j in [1,6] 18 Obtain fj the global number of face j 19 Add -${A}_{e}^{-1}$(i,j) FaceHead(fj) to FaceFlux(i,e)20 End Cycle 21 End Cycle 22 End Cycle 23 End Function |

**Q**at the current time step.

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**Figure 1.**Illustration of different approaches for modeling unconfined groundwater flow in subsurface aquifers. (

**A**) An example of flow processes occurring in an unconfined aquifer such as recharge, abstraction, and aquifer-river interactions. (

**B**) A discrete variably saturated groundwater flow model seeks to determine the saturation state in all cells; the water table shape and position are post-processed from simulated pressure heads. (

**C**) A discrete unconfined flow model neglecting processes in the unsaturated zone using a fixed mesh. The numerical model identifies wet, partially wet, and dry cells. The water table is not represented explicitly and is post-processed from simulated heads. (

**D**) A discrete unconfined flow model neglecting processes in the unsaturated zone using a moving mesh. The numerical model works exclusively with fully saturated cells. The water table is explicitly represented by the upper mesh slice and is, therefore, not post-processed.

**Figure 2.**(

**a**) Plan view of the unstructured mesh used for simulating the hydraulic head drawdown in an anisotropic aquifer. (

**b**) Enlarged view around the first level of mesh refinement within a 10 m radius from the domain center. The two piezometers are shown. (

**c**) Enlarged view around the second level of mesh refinement within a 1 m radius from the domain center. The well is approximated with a regular pentagon and has a radius of 0.1 m.

**Figure 3.**The hydraulic head drawdown in an anisotropic aquifer: comparison of the Papadopulos analytical solution with results of the MHFEM model at piezometers P

_{1}and P

_{2}with (

**a**) a diagonal transmissivity tensor, and (

**b**) a full symmetric transmissivity tensor.

**Figure 4.**Conceptual model of the second test problem. Groundwater is flowing between two fixed head boundaries, through a heterogeneous porous medium. The hydraulic conductivity distribution spans four orders of magnitude.

**Figure 5.**Simulated path lines (black lines) with both the MHFEM code HydroTec

^{©}and the finite-difference-based MODFLOW/MODPATH codes are presented in the first (

**a**–

**c**) and second columns (

**d**–

**f**), respectively, and for the 10 × 10, 20 × 20, and 30 × 30 grid resolutions starting from the top. The reference path lines (red lines) are computed using a mixed approximation with a 320 × 320 grid resolution. The MHFEM approximation yields a more accurate solution at a 20 × 20 resolution (

**b**) than a finite difference approximation at a higher resolution (

**f**).

**Figure 6.**Calculated relative errors in simulated residence times of two particles released at (25, 100) (

**a**) and (95, 100) (

**b**) using the MHFEM and FDM approximations.

**Figure 7.**Simulated steady-state water table position using the adaptive mesh MHFEM technique for the third test problem. All horizontal lines of the two-dimensional mesh were allowed to move during the iterative procedure. The water table is favorably compared with numerical solutions provided by other authors [55,56].

**Figure 8.**Simulated steady-state water table position and groundwater heads using the adaptive mesh MHFEM technique for the fourth test problem. All horizontal lines of the two-dimensional mesh were allowed to move during the iterative procedure (vertical aspect ratio = 80).

**Figure 9.**Performance of the adaptive mesh iteration for the fourth test problem. The water table error decreases with a rate greater than a linear trend. The bars show the number of linear iterations at each outer iteration. Eight mesh iteration steps were sufficient to achieve accuracy with a 10

^{−3}m closure criterion.

**Figure 10.**Comparison between the water table simulated with the adaptive MHFEM approximation and the post-processed FDM approximation. There is a good match for the well drawdown. The observed discrepancies far from the well are mainly attributed to post-processing interpolation errors.

**Figure 11.**Simulated steady-state water table position and groundwater heads using the adaptive mesh MHFEM technique for the three-dimensional problem. The cutaway view shows the pumping well’s depression cone and the head contours at the aquifer base (vertical aspect ratio = 50).

**Figure 12.**Performance of the adaptive mesh iteration for the three-dimensional problem. The water table error decreases with a rate greater than a linear trend. The bars show the number of linear iterations at each outer iteration. Thirteen mesh iterations were sufficient to achieve accuracy with a 10

^{−2}m closure criterion.

**Figure 13.**Comparison between the water table profile (parallel to X-axis and crossing the well position) simulated with the adaptive MHFEM approximation and the post-processed FDM approximation for the three-dimensional problem. The maximum difference occurs at the well due to the coarse vertical discretization of the FDM fixed mesh.

**Table 1.**Physical parameters used in the pumping test problem. Case 1 and 2 correspond to diagonal and fully symmetric transmissivity tensors, respectively.

Parameter | Value—Case 1 | Value—Case 2 |
---|---|---|

$\mathrm{Transmissivity}\mathrm{along}\mathrm{X},{T}_{xx}$ m^{2}.s^{−1} | 2.93 × 10^{−5} | 2.57 × 10^{−5} |

$\mathrm{Transmissivity}\mathrm{along}\mathrm{Y},{T}_{yy}$ m^{2}.s^{−1} | 1.47 × 10^{−5} | 1.84 × 10^{−5} |

$\mathrm{Transmissivity}\mathrm{along}\mathrm{XY},{T}_{xy}={T}_{yx}$ m^{2}.s^{−1} | 0 | 6.36 × 10^{−6} |

$\mathrm{Specific}\mathrm{storage},{S}_{p}$ m^{−1} | 1.957 × 10^{−3} | 1.957 × 10^{−3} |

$\mathrm{Pumping}\mathrm{flow}\mathrm{rate},{Q}_{w}$ m^{3}.s^{−1} | 5 × 10^{−3} | 5 × 10^{−3} |

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## Share and Cite

**MDPI and ACS Style**

Sbai, M.A.; Larabi, A.
A Deforming Mixed-Hybrid Finite Element Model for Robust Groundwater Flow Simulation in 3D Unconfined Aquifers with Unstructured Layered Grids. *Water* **2023**, *15*, 1177.
https://doi.org/10.3390/w15061177

**AMA Style**

Sbai MA, Larabi A.
A Deforming Mixed-Hybrid Finite Element Model for Robust Groundwater Flow Simulation in 3D Unconfined Aquifers with Unstructured Layered Grids. *Water*. 2023; 15(6):1177.
https://doi.org/10.3390/w15061177

**Chicago/Turabian Style**

Sbai, Mohammed Adil, and Abdelkader Larabi.
2023. "A Deforming Mixed-Hybrid Finite Element Model for Robust Groundwater Flow Simulation in 3D Unconfined Aquifers with Unstructured Layered Grids" *Water* 15, no. 6: 1177.
https://doi.org/10.3390/w15061177