From Boussinesq Equation to Porous Medium Equation: Analytical Extensions and Structural Reformulations in Nonlinear Groundwater Flow
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Authors
Pratt, Phillip Alexander
Issue Date
2025
Type
Dissertation
Language
en_US
Keywords
Boussinesq Equation , Nonlinear Diffusion , Partial Differential Equations , Porous Medium Equation , Self-Similar Solution
Alternative Title
Abstract
This dissertation develops analytical and semi-analytical methods for the porous medium equation (PME), with emphasis on recharge problems, boundary discontinuities, and extensions of classical groundwater flow models. Beginning with the cylindrical formulation of recharge into an initially dry aquifer, a semi-analytical solution is constructed that clarifies the roles of geometry, initial conditions, and nonlinear diffusion, extending the injection framework of Telyakovskiy et al. (2016) from the Boussinesq setting to the full PME.
The analysis then addresses one-dimensional boundary problems, where step discontinuities are resolved through families of power-law and exponential–polynomial solutions, reformulating and extending the closure approach of Tolikas, Sidiropoulos, and Tzimopoulos (1984) to the PME context.
A further contribution develops a quadratic non-selfsimilar solution as an extension of the framework introduced by Parlange et al. (2000), revealing new analytic solutions to regimes of drainage and infiltration. In each case, the PME-based formulations not only yield new solution forms and boundary treatments but also recover the earlier Barenblatt-based results when the corresponding parameter limits are applied. This ensures that the present work complements and enhances prior models, embedding them within a broader and more flexible analytical structure.
Taken together, these developments broaden the analytical toolkit for nonlinear diffusion in porous media, generalizing existing results across wider parameter ranges and geometries while recasting governing structures to sharpen tractability and physical interpretation. The outcome is a more comprehensive mathematical framework for porous medium flows, bridging classical groundwater theory with contemporary nonlinear diffusion analysis.