Advection–diffusion–reaction modeling of contaminant transport in groundwater: Analysis and simulation

Owolabi K. M., Alagöz S.

Nonlinear Science, cilt.5, ss.1-36, 2025 (Hakemli Dergi)

Özet

We develop a unified mathematical and numerical framework for modeling reactive contaminant transport in saturated porous media. The governing advection–dispersion–reaction (ADR) equation describes solute concentration evolution driven by Darcy velocity advection, Fickian dispersion, and general nonlinear reaction kinetics, subject to suitable initial and boundary conditions. Using semigroup theory and energy estimates, we establish well-posedness of the ADR system, including existence, uniqueness, and positivity of solutions. A nondimensional formulation identifies the Péclet and Damköhler numbers as the key dimensionless parameters governing the relative roles of transport and reaction. For dissipative kinetics, the longtime dynamics converge to global attractors, reflecting stabilization of plume evolution. From a computational perspective, we design two complementary time-integration strategies: an implicit–explicit (IMEX) Euler method, which stabilizes stiff diffusion–reaction terms, and a split-step Fourier method (SSFM), which achieves higher-order accuracy via Strang splitting of transport and reaction operators. Rigorous stability analyses confirm the conditional stability of IMEX under a CFL constraint and the unconditional stability of the linear SSFM component. Both methods preserve positivity and capture plume dynamics faithfully. Numerical experiments in one and two dimensions illustrate plume advection, dispersion-driven spreading, and nonlinear reactive decay, together with emergent instabilities such as Kelvin–Helmholtz roll-up under shear-driven flows. Connections to related nonlinear PDE models—including Gray–Scott pattern formation and Darcy–Korteweg roll-up phenomena—highlight the broader relevance of the ADR framework as a bridge between contaminant transport and instability-driven mixing processes. The results demonstrate that combining rigorous PDE analysis, nondimensional characterization, and structure-preserving numerics provides a robust foundation for predictive modeling of reactive transport in groundwater systems.