# Modeling Groundwater Flow in Heterogeneous Porous Media with YAGMod

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## Abstract

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`YAGMod`(yet another groundwater flow model), a model based on a finite-difference conservative scheme and implemented in a computer code developed in Fortran90.

`YAGMod`simulates also the presence of partially-saturated or dry cells. The proposed algorithm and other alternative methods developed to manage dry cells in the case of depleted aquifers are analyzed and compared to a simple test. Different approaches yield different solutions, among which, it is not possible to select the best one on the basis of physical arguments. A possible advantage of

`YAGMod`is that no additional non-physical parameter is needed to overcome the numerical difficulties arising to handle drained cells.

`YAGMod`also includes a module that allows one to identify the conductivity field for a phreatic aquifer by solving an inverse problem with the comparison model method.

## 1. Introduction

`YAGMod`(yet another groundwater flow model), developed in Fortran90, for the simulation of constant-density, groundwater flow under stationary conditions, which is the extension of the codes developed by our research team over the years [7,8,9,10,11,12,13,14,15,16,17].

`YAGMod`is based on a conservative finite difference scheme for stationary conditions and is oriented to the simulation of flow in saturated media. It takes into account the possible drying of shallow blocks of the domain with an original approach, which limits the number of additional parameters that have to be assigned by the user and that have a weak physical significance. Notice that

`YAGMod`considers both prescribed distributed sources and variable point sources. While the former can be used to simulate aquifer recharge, the latter can be used to simulate draining systems or the effects of the water head drawdown on a water-well discharge.

`YAGMod`to handle desaturated cells with the above-mentioned algorithms proposed by Doherty [3], Keating and Zyvoloski [4] and Niswonger et al. [6]. This is performed by means of a simple, but significant test case.

`YAGMod`includes a module for the model calibration, namely the identification of transmissivity from the knowledge of the reference head and source fields all over the domain. The calibration is performed for 2D flow conditions implementing the comparison model method (CMM). This method was originally proposed by Scarascia and Ponzini [18], successively developed by Ponzini and Lozej [19] and cast in a more formal mathematical framework by Ponzini and Crosta [20]. The CMM has been applied and implemented with success to study 2D hydraulic flow in regional aquifers [14,16,17,21,22,23], and therefore, for the moment, its implementation within

`YAGMod`covers these kinds of systems only.

`YAGMod`takes into account the variation of the saturated thickness of each cell for the solution of the forward problem.

## 2. Forward Model

`YAGMod`, and particular emphasis will be given to its innovative features.

#### 2.1. Mathematical Model and Discretization

`I`identifies the internal cells, for which the hydraulic head can vary freely;

`D`identifies the cells where Dirichlet conditions are assigned, i.e., the hydraulic head is prescribed;

`E`identifies the cells external to the domain or where no flow takes place.

**Figure 1.**(

**a**) Plan view of a domain’s layer; (

**b**) vertical section view. Red arrows are examples of groundwater fluxes through first-neighborhood cells and considered in the discrete model.

**Figure 2.**Domain code example in a 2D domain:

`I`,

`E`and

`D`codes correspond, respectively, to internal, external and prescribed-head (Dirichlet boundary conditions) cells; the blue line denotes the border of the domain.

#### 2.2. Boundary Conditions and Source Terms

`D`label for the domain code: in that case, the hydraulic head does not change during the computation of the solution. Neumann and Robin boundary conditions are implemented as specific types of source terms, as specified hereinafter.

#### 2.2.1. Distributed Source/Sink Terms

#### 2.2.2. Local Source/Sink Terms

`YAGMod`code, local sources or sinks, i.e., those that are concentrated in a single cell, are modeled with the paradigmatic equation:

- Drain:${F}_{(+)}={F}_{(-)}=0$, ${\mathcal{K}}_{(-)}=0$; ${\mathcal{K}}_{(+)}$ represents the drain conductance; ${\mathcal{H}}^{\left(\mathrm{act}\right)}={\mathcal{H}}^{\left(\mathrm{cal}\right)}$ represents the drain elevation.
- Robin boundary conditions:These conditions can be used if the aquifer interacts with another water body and water exchange is controlled by the difference of the water head in the aquifer and in the external water body. They are introduced through Equation (6), by assigning the following parameters: ${F}_{(+)}={F}_{(-)}=0$; ${\mathcal{H}}^{\left(\mathrm{act}\right)}={\mathcal{H}}^{\left(\mathrm{cal}\right)}$ are the reference hydraulic heads; ${\mathcal{K}}_{(+)}$ and ${\mathcal{K}}_{(-)}$ represent the conductances for flux out or into the cell. ${\mathcal{K}}_{(+)}$ and ${\mathcal{K}}_{(-)}$ depend on the conductivity of the materials that separate the aquifer from the water body at the reference water head and on the distance from this water body. Notice that for the simulation of limited domains of aquifers with a large extension, it is usually impossible to prescribe physically-based boundary conditions. In those cases, Robin boundary conditions are very useful to introduce fictitious boundary conditions, which are more flexible than prescribed head (Dirichlet) or flux (Neumann) boundary conditions. In these situations, ${\mathcal{H}}^{\left(\mathrm{act}\right)}={\mathcal{H}}^{\left(\mathrm{cal}\right)}$ should be close to the estimated water head far from the aquifer system, and the conductances ${\mathcal{K}}_{(+)}$ and ${\mathcal{K}}_{(-)}$ can assume different values to take into account the geometrical, geological and hydrological characteristics of the aquifer.
- River/aquifer interaction:${\mathcal{H}}^{\left(\mathrm{act}\right)}$ is the height of the bottom of the river: therefore, if $h\ge {\mathcal{H}}^{\left(\mathrm{act}\right)}$, the river and groundwater are in contact, whereas, if $h<{\mathcal{H}}^{\left(\mathrm{act}\right)}$, then they are separated by a vadose zone, i.e., partly-saturated sediments or rocks. In the first situation, ${\mathcal{H}}^{\left(\mathrm{cal}\right)}$ is the river stage; then, the river drains the aquifer if $h>{\mathcal{H}}^{\left(\mathrm{cal}\right)}$ and recharges the aquifer if $h<{\mathcal{H}}^{\left(\mathrm{cal}\right)}$. It is quite common to assume ${F}_{(+)}=0$ and to consider ${\mathcal{K}}_{(+)}$ as a function of the conductivity of the river bed sediments, their thickness and the area of the contact surface between the river bed and the aquifer in the considered cell. In the second situation, namely if $h<{\mathcal{H}}^{\left(act\right)}$, the river bed is assumed to be composed of fine-grained materials, which could be almost saturated, but poorly permeable, whereas the vadose zone between the river bed and the water table could be more permeable than the river bed sediments and approximated as dry. Therefore, the water flows through the river bed under a gravity-controlled, unit hydraulic gradient and freely flows through the relatively permeable vadose zone: then, ${\mathcal{K}}_{(-)}=0$, whereas ${F}_{(-)}$ depends on the conductivity, thickness and extension of the river bed sediments in the considered cell and on the river stage.

#### 2.2.3. Screened Wells

`YAGMod`considers a new kind of source term that takes into account the dependence of the wells’ extraction rate on the aquifer water head. In particular, this kind of source term allows one to turn off the pumping if a cell becomes dry. Sources in this category are denoted as “screened wells”, as the user has to give as input data not only the $(x,y)$ coordinates of the well, i.e., the node indices ${i}_{W}$ and ${j}_{W}$, but also the top and bottom elevation of the screened interval ($to{p}_{\mathrm{W}}$ and $bo{t}_{\mathrm{W}}$) and the maximum well extraction rate, ${q}_{w}$.

#### 2.3. Solution of the Balance Equations

`D`nodes. $\mathbf{A}$ is a sparse, symmetric, diagonally-dominant matrix, which is strictly diagonally dominant if at least one

`D`node is present in the domain; its elements are built with transmittances and, therefore, depend on $\mathbf{x}$, as shown by Equations (3) and (4).

`YAGMod`, a simple approach, based on a generalization of the relaxation methods for the solution of systems of algebraic linear equations, is proposed. This choice is optimal from the point of view of the memory requirement. Other approaches, e.g., Newton’s or conjugate-gradient methods, could be more efficient in terms of elapsed running time, if the code is properly modified to profit from parallel computers. However, it should be noted that the specific problem addressed in this paper includes non-differentiable terms in the system of equations, like those introduced by Equation (6) and by the sequence of equations from Equations (7) to (11). Several tests showed that the generalization of relaxation methods is in general quite robust, in particular for complex physical situations.

#### 2.4. Check of the Physical Consistency of the Solution

`YAGMod`prints a warning message to the standard output and in the summary output file.

#### 2.5. An Example

**Figure 3.**Contour lines of the hydraulic head (2-meter contour interval) and map of the saturation field (fully-saturated cells are drawn in blue; partially-saturated cells are drawn in green; dry cells are drawn in orange) along a vertical section for a 3D problem solved with

`YAGMod`(yet another groundwater flow model). (

**top**) A single deep well, whose fixed extraction rate is about $0.06\phantom{\rule{4pt}{0ex}}{\mathrm{m}}^{3}/s$, is located at $x=125\phantom{\rule{4pt}{0ex}}\mathrm{m}$ and $z=20\phantom{\rule{4pt}{0ex}}\mathrm{m}$, beneath a low conductivity lens (dashed area). (

**bottom**) The single deep well is replaced by a screened well, from $20\phantom{\rule{4pt}{0ex}}\mathrm{m}$ to $45\phantom{\rule{4pt}{0ex}}\mathrm{m}$ (yellow line).

## 3. A Simple Test Case to Compare Different Approaches to Handle Dry Cells

`YAGMod`is compared to those approaches by means of a test case, which is very simple, but permits one to emphasize some significant properties of the different methods. In particular, the basic characteristics of the analyzed algorithms are briefly recalled using a simplified notation based on this example. A simple 2D domain has been constructed with a grid of $3\times 1\times 2$ cells whose size is $100\phantom{\rule{4pt}{0ex}}\mathrm{m}\times 100\phantom{\rule{4pt}{0ex}}\mathrm{m}\times 20\phantom{\rule{4pt}{0ex}}\mathrm{m}$. This 2D domain is illustrated in Figure 4, together with the cell numbering used in the following for the sake of simplicity.

`YAGMod`; g and f, which are numerical parameters, and ${\vartheta}_{r}$, the residual saturated thickness, are parameters supplied by the user. To ensure that the function defined in Equation (18) is continuous and continuously differentiable, the following relationship must be satisfied:

**Figure 5.**Map of the balance error given by Equation (17), for the approach of Doherty [3], as a function of the hydraulic heads ${h}_{\left(2\right)}$ (x axis) and ${h}_{\left(5\right)}$ (y axis), for different values of m ((

**left**) $m=1$; (

**center**) $m=10$; (

**right**) $m=100$). At the bottom right corner of each graph, the zone containing the minimum error value is enlarged.

`YAGMod`), for three cases corresponding to an extraction rate Q varying from $0.1\phantom{\rule{4pt}{0ex}}{\text{m}}^{3}\phantom{\rule{0.166667em}{0ex}}{\text{s}}^{-1}$ to $0.2\phantom{\rule{4pt}{0ex}}{\text{m}}^{3}\phantom{\rule{0.166667em}{0ex}}{\text{s}}^{-1}$ and $0.3\phantom{\rule{4pt}{0ex}}{\text{m}}^{3}\phantom{\rule{0.166667em}{0ex}}{\text{s}}^{-1}$, are represented in Figure 6. Notice that with the approach by Keating and Zyvoloski [4], ${h}_{\left(5\right)}$ cannot drop below the cell bottom; therefore, the results obtained with this method for extraction rates of $0.2\phantom{\rule{4pt}{0ex}}{\text{m}}^{3}\phantom{\rule{0.166667em}{0ex}}{\text{s}}^{-1}$ and $0.3\phantom{\rule{4pt}{0ex}}{\text{m}}^{3}\phantom{\rule{0.166667em}{0ex}}{\text{s}}^{-1}$ could not be significantly compared to those from other algorithms. Besides this remark, all of the methods give realistic results, even if the values that yield the least total balance error for alternative algorithms differ from each other.

**Figure 6.**Map of the total quadratic balance error given by Equation (17), as a function of the hydraulic heads ${h}_{\left(2\right)}$ (x axis) and ${h}_{\left(5\right)}$ (y axis). From left to right, respectively, the results obtained with

`YAGMod`and the approaches by Doherty [3], Keating and Zyvoloski [4] and Niswonger et al. [6]. From top down, the results obtained with extraction rates of $0.1\phantom{\rule{4pt}{0ex}}{\text{m}}^{3}\phantom{\rule{0.166667em}{0ex}}{\text{s}}^{-1}$, $0.2\phantom{\rule{4pt}{0ex}}{\text{m}}^{3}\phantom{\rule{0.166667em}{0ex}}{\text{s}}^{-1}$ and $0.3\phantom{\rule{4pt}{0ex}}{\text{m}}^{3}\phantom{\rule{0.166667em}{0ex}}{\text{s}}^{-1}$. The blue circles point out the zone where the least value of the total quadratic balance error is located.

`YAGMod`for an extraction rate of $0.1\phantom{\rule{4pt}{0ex}}{\text{m}}^{3}\phantom{\rule{0.166667em}{0ex}}{\text{s}}^{-1}$ are smaller than those obtained with the other approaches; on the other hand, for higher extraction rates, the behavior is more complex, and no systematic difference is shown.

`YAGMod`, ${\u03f5}_{\mathrm{tot}}^{2}$ increases from the least value more rapidly than for other methods.

`YAGMod`, is based on some auxiliary parameters. Such parameters are not related to physical processes or quantities, but are necessary to apply artifices to face the numerical problems arising from the simulation of drying cells. These auxiliary parameters, which are listed in Table 1, have to be assigned by the user. On the other hand,

`YAGMod`does not require any additional parameter. This is useful to limit the arbitrariness associated with the assignment of non-physical, auxiliary quantities.

## 4. Inverse Modeling with the Comparison Model Method

`YAGMod`permits the application of the comparison model method (CMM) to estimate the conductivity field for a 2D flow field by solving an inverse problem. The CMM was proposed to identify T at every node of a discretization grid [18] and successively developed to directly compute internode transmissivities [19]. Further modifications were proposed by Ponzini and Crosta [20], Ponzini et al. [26], Pasquier and Marcotte [27], Ponzini et al. [28]. The CMM was applied to alluvial aquifers in Italy [14,16,21], Switzerland [22] and Canada [23,29,30] and to a carbonatic aquifer in southern Italy [17,31].

#### 4.1. Fundamentals of the CMM Algorithm

#### 4.2. A Test

**Figure 7.**Simple test of the comparison model method (CMM): hydraulic conductivity field; the color scale refers to ${log}_{10}K$, with K expressed in $\text{m}\phantom{\rule{0.166667em}{0ex}}{\text{s}}^{-1}$. (

**left**) Reference field; (

**central**) field estimated from the CMM with the integral approach; (

**right**) field estimated from the CMM with the differential approach.

**Figure 8.**Simple test of the CMM: image plot of the reference hydraulic head and contour lines of the reference h field (purple) and of the h fields obtained for the K fields estimated with the CMM: yellow lines refer to the K field obtained with the integral approach (central map of Figure 7) and green lines to the K field obtained with the differential approach (right map of Figure 7).

## 5. Conclusions

`YAGMod`, has been described in this paper. The basic characteristics of this model are briefly summarized below.

- When the water head in a cell is lower than the cell’s top coordinate, the cell is considered as partially saturated, and its saturated thickness is used for the water balance. If the water head is lower than the cell bottom, then the cell is considered as dry, but it is not excluded from the water balance calculations. In fact, the water head in this cell is used to compute a vertical water balance that permits one to transfer source terms (namely, aquifer recharge) from the shallow layers down to the deep ones.
- A large number of different sources or boundary conditions, which depend on the water head in the cell, are simulated with a single prototype equation.
- If the position of the screened intervals of a water abstraction well is known, the extraction rate can be limited when the water head falls within the screened interval or turned off when the water head falls below the bottom of the screens.

`YAGMod`and those proposed by Doherty [3], Keating and Zyvoloski [4] and Niswonger et al. [6] shows that the different approaches yield different solutions, among which, it is not possible to select the best one on the basis of physical arguments. Nevertheless, the advantage of

`YAGMod`is that it does not introduce any additional parameter whose values should be assigned by the user and which mostly have a limited physical significance. Differently, the other approaches require that the user defines one or more additional parameters. These parameters mostly have a limited physical significance, making the parameterization of the model more complex.

`YAGMod`can be used also to solve the inverse problem with the CMM for a phreatic aquifer, with either an integral or a differential approach and taking into account the variability of saturation. This further strengthens the features of

`YAGMod`, because the CMM is embedded in the source code and can be applied to estimate the hydraulic conductivity field, directly at the model scale. The present version of

`YAGMod`is developed to apply the CMM with 2D flow fields, but it can be easily extended to 3D flows, if data with high quality and high density are available.

`YAGMod`is sufficiently flexible to be adapted to other situations, as was done to model groundwater flow in multi-layered coastal aquifers by De Filippis et al. [31], who modified

`YAGMod`to cope with the variable thickness of the aquifer saturated with fresh water and to identify the spatial variability of a fractured and karst carbonatic aquifer with the application of the CMM to a single layer, while the parameters of other layers are fixed with estimates based on prior information.

`YAGMod`to cope with draining cells can be easily applied to integrated finite differences [34], but can be adapted also to other numerical techniques, which are based on the discretization of the integral balance equation over a given subdomain.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Cattaneo, L.; Comunian, A.; De Filippis, G.; Giudici, M.; Vassena, C.
Modeling Groundwater Flow in Heterogeneous Porous Media with YAGMod. *Computation* **2016**, *4*, 2.
https://doi.org/10.3390/computation4010002

**AMA Style**

Cattaneo L, Comunian A, De Filippis G, Giudici M, Vassena C.
Modeling Groundwater Flow in Heterogeneous Porous Media with YAGMod. *Computation*. 2016; 4(1):2.
https://doi.org/10.3390/computation4010002

**Chicago/Turabian Style**

Cattaneo, Laura, Alessandro Comunian, Giovanna De Filippis, Mauro Giudici, and Chiara Vassena.
2016. "Modeling Groundwater Flow in Heterogeneous Porous Media with YAGMod" *Computation* 4, no. 1: 2.
https://doi.org/10.3390/computation4010002