Development of a nonlinear coupled model for identifying groundwater-level and salinity-change patterns under third-type boundary conditions

The object of this research is the coupled dynamics of groundwater level and mineralization in a two-layer hydrodynamic medium under third-type, or Robin-type, boundary conditions. Unlike earlier single-layer models or formulations in which groundwater flow and salt transport are considered separately, the present research integrates the phreatic groundwater level h(x,y,t), the confined-water level H(x,y,t), and the salt concentrations θ1(x,y,t) and θ2(x,y,t) within a unified nonlinear coupled framework that accounts for interlayer hydrodynamic interaction and mass exchange. The paper provides a mathematical description of the combined influence of atmospheric precipitation, evaporation, irrigation infiltration, groundwater abstraction and artificial recharge, interlayer exchange, filtration properties, porosity, layer thickness and discharge intensity on groundwater-level variations and mineralization processes. To solve the nonlinear coupled model, a numerical algorithm based on the finite-difference method, an implicit approximation scheme, iterative refinement and the sweep method was employed. The numerical experiments produced specific quantitative estimates of the hydrodynamic and hydrochemical response of the two-layer aquifer system. In the first computational case, the normalized phreatic groundwater level varied from 0.74 to 0.97, while the discharge field changed from 0.45 to 0.70. The salt concentration in the phreatic layer θ1 varied from 0.66 to 0.90, whereas the salt concentration in the confined layer θ2 changed within a narrower interval of 0.50–0.63. In the second computational case, the confined-water level H varied from 0.59 to 0.81, the discharge field Q from 0.27 to 0.37, θ1 from 0.68 to 0.90, and θ2 from 0.58 to 0.65. These values show that the phreatic layer forms stronger hydrodynamic and hydrochemical contrasts, while the confined layer exhibits a more inertial and smoothed response. The scenario-based sensitivity analysis showed that a 25% increase in infiltration raises the phreatic-head maximum by approximately 4–7% and reduces the maximum θ1 value by about 5–8% due to dilution. Increased evaporation, by contrast, raises the maximum θ1 value by approximately 6–10%. A 30% increase in well discharge increases the local Q maximum by about 10–15%, while a 50% increase in the interlayer exchange coefficient reduces the h–H contrast by approximately 8–12% and increases the maximum θ2 value by about 3–5%. These results confirm that salinization risk is controlled not only by groundwater-level variations but also by filtration gradients, interlayer exchange and boundary interactions. The numerical verification confirmed the reliability of the proposed implicit finite-difference algorithm. The manufactured-solution test demonstrated second-order convergence: the maximum error decreased from 2.23 × 10⁻4 on a 20 × 20 grid to 3.08 × 10⁻6 on a 160 × 160 grid. The obtained results demonstrate that the developed model can be used for forecasting groundwater regimes, assessing salinization risk, supporting the rational use of water resources and providing a scientific basis for hydroecological monitoring in irrigated and environmentally sensitive areas

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