Identification of Characteristic Spatial Scales to Improve the Performance of Analytical Spectral Solutions to the Groundwater Flow Equation

Analytical solutions for the three-dimensional groundwater flow equation have been widely used to gain insight about subsurface flow structure and as an alternative to computationally expensive numerical models. Of particular interest are solutions that decompose prescribed hydraulic head boundaries (e.g., Dirichlet boundary condition) into a collection of harmonic functions. Previous studies estimate the frequencies and amplitudes of these harmonics with a least-square approach where the amplitudes are fitted given a pre-assigned set of frequencies. In these studies, an ad hoc and structured discretization of the frequency domain is typically used, excluding dominant frequencies while assigning importance to spurious frequencies, with significant consequences for estimating the fluxes and residence times. This study demonstrates the advantages of using a pre-assigned frequency spectrum that targets the dominant frequencies based on rigorous statistical analysis with predefined significance levels. The new approach is tested for three hydrologic conceptualizations: (a) a synthetic periodic basin, (b) synthetic bedforms, and (c) a natural mountainous watershed. The performance of the frequency spectrum selection is compared with exact analytical or approximate numerical solutions. We found that the new approach better describes the fluxes and residence times for Dirichlet boundaries with well-defined characteristics spatial scales (e.g., periodic basins and bedforms). For more complex scenarios, such as natural mountainous watersheds, both pre-assigned frequency spectrums present similar performance. The spectral solutions presented here can play a central role in developing reduced-complexity models for assessing regional water and solute fluxes within mountain watersheds and hyporheic zones.