Backward similarity solution of the Boussinesq groundwater equation

TL;DR

This work analyzes groundwater flow in an unconfined aquifer under a rapidly growing boundary water level, modeled by the Boussinesq equation with a backward power-law head condition. A Boltzmann-type similarity transformation reduces the PDE to a nonlinear ODE, allowing a quadratic approximate solution and accurate numerical solutions via Shampine's method, with validation against a finite element discretization. The approach recovers known limits: Kalashnikov's solution as $\alpha \to -1$ and the exponential/forward limit as $\alpha \to -\infty$, while providing a flexible framework for front propagation under finite-time blow-up conditions. Together, these results furnish practical tools for predicting wetting-front dynamics and establish robust benchmarks for numerical schemes handling backward boundary conditions in groundwater flow.

Abstract

Groundwater flow in an unconfined aquifer resting on a horizontal impermeable layer with a boundary condition of a rapid increase in the source water level is considered in this work. The newly introduced condition, referred to as the backward power-law head condition, represents a situation where the water level in the adjacent water body increases more rapidly than do conventional problems, which can only represent a situation akin to a traveling wave under rising water level conditions, given its consideration of infinite time. This problem admits the similarity transformation which allows the nonlinear partial differential equation to be converted into a nonlinear ordinary differential equation via the Boltzmann transformation. The reduced boundary value problem is interpreted as the initial value problem for a system of ordinary differential equations (ODE), which can be numerically solved via Shampine's method. The numerical solutions are in good agreement with Kalashinikov's special solution, which is also introduced into the Boussinesq equation. We find that the solution is consistent with the limit of the forward power-law head condition. The new approximate analytical solution and the associated wetting front position are derived by assuming that the solution has the form of quadratic polynomials, which enables the description of the time progression of a real front position. The obtained approximation is compared to Shampine's solution to check the accuracy. Furthermore, the finite element method is applied to the original partial differential equation (PDE), which validates Shampine's solution for use as a benchmark.

Figures (8)

• Figure 1: Schematic illustration of groundwater flow into an unconfined aquifer from an adjacent water source with a rapidly increasing boundary water level.
• Figure 2: Time variation of the water front position $x_0$ for various $\alpha$ when $K=U =1$, $S=1/2$, and $T=3$ in the inlet condition (\ref{['InletCondition']}).
• Figure 3: Numerical solutions of (\ref{['eq:BoussinesqODE']})-(\ref{['FrontCondition']}) for various $\alpha$ (solid line). Kalashnikov's solution for the lower limit and a numerical result of the exponential condition for the upper limit (dotted line) are also presented.
• Figure 4: Comparison between the numerical solution and the approximate analytical solution.
• Figure 5: Errors defined by Shampine's numerical solution minus the quadratic approximate solution for various $\alpha$.