## 2 Materials and Methods
### Modeling Setup
#### 2.2.2 The Site Mathematical Computer Modeling Structure  
The conceptual approach to model development for this study is shown in Figure 4.  
The mathematical model structure is determined with consideration of the following factors: the modeled object's nominal hydrogeological conditions, which are modeled so that the model content corresponds to the natural object; the possibility of constructing a model that allows solving the tasks with the required accuracy; maintaining option of adjusting the model and introducing various amendments to it during the modeling process; performing modeling while maintaining the specified accuracy of the solution in the most straightforward ways and within adequate timeframes is considered. The groundwater flow structure simplification is carried out by reducing spatial filtration to a simpler one, for example, by reducing the flow dimensions. The Zhem Artesian  
Figure 4: The Zhem Artesian Basin conceptual model development working process. Basin model area significant dimensions in terms of the aquifer thickness, almost always make it possible to switch from spatial filtration to a planned filtration. The basin's aquifers are hydraulically interconnected and form a single geofiltration flow, which has a common piezometric surface. The inconsistent clayey layer between the Albian sandy formations and Cenomanian strata suggests a single aquifer complex. (Figure 2). It should be noted that there is insufficient factual material that would allow the modeling of the filtration heterogeneity of the sub-aquifers and the dynamics of the changes in the groundwater heads along the vertical section of the aquifers. This unique definition of these predetermined factors enabled a single-layer creation for the two-dimensional model of the Zhem Artesian Basin Middle-Upper Albian, Cenomanian, and Quaternary aquifer complex. The mathematical model represents groundwater flow through porous media using a two-dimensional partial differential equation for a one-layer geofiltration to an unconfined or confined aquifer. The undifferentiated Middle-Upper Albian and partly Cenomanian Cretaceous sediments and Quaternary sediments are unconfined conditions represented by flow Equation (1) [32], as follows:  
\[\partial/\partial x[Kx(Hx-\eta x)(\partial H/\partial x)]+\partial y[Ky(Hy- \eta y)(\partial H/\partial y)]\pm W=Sy(\partial H/\partial t), \tag{1}\]  
where \(Kx\) and \(Ky\) are the hydraulic conductivity along the principal permeability directions of the \(x\)- and \(y\)-coordinate axes, respectively; \(H\) is the potentiometric head; \(x\) and \(y\) are the unconfined aquifer floor elevations along the \(x\)- and \(y\)-coordinate axes; \(W\) indicates the fluxes representing recharge, evaporation, and water pumping; \(Sy\) is the porous material specific yield; and \(t\) is time. For confined conditions, \((Hx-\eta x)\) and \((Hy-\eta y)\) are replaced by \(mx\) and \(my\), and \(Sy\) is replaced by \(Ss\), where \(m\) is the thickness of the aquifer and \(Ss\) is the specific storage. The visual MODFLOW version 4.12 software package [33] ([[https://www.pdfdrive.com/visual-modflow-20111-users-manual-d22572407.html](https://www.pdfdrive.com/visual-modflow-20111-users-manual-d22572407.html)]([https://www.pdfdrive.com/visual-modflow-20111-users-manual-d22572407.html](https://www.pdfdrive.com/visual-modflow-20111-users-manual-d22572407.html)), accessed on 20 April 2025) [27] was selected for the study because of its ability to calculate flow fluxes across cell boundaries and predict groundwater spatial and temporal head-level distribution [34]. The transformation from unconfined flow to confined conditions and vice versa is automatic with differential calculations of the surface elevation and the calculated groundwater level at each time step. The model boundary conditions are represented as head and flow and their combination. Equation (1) mathematically represents the study area's groundwater flow model.