The Analytical Solution of an Unsteady State Heat Transfer Model for the Confined Aquifer under the Influence of Water Temperature Variation in the River Channel

Abstract

The effect of water temperature variation in a river channel on groundwater temperature in the confined aquifer it cuts can be generalized to a one-dimensional thermal convection-conduction problem in which the boundary water temperature rises instantaneously and then remains constant. The basic equation of thermal transport for such a problem is the viscous Burgers equation, which is difficult to solve analytically. To solve this problem, the Cole–Hopf transform was used to convert the second-order nonlinear thermal convection-conduction equation into a heat conduction equation with exponential function-type boundary conditions. Considering the difficulty of calculating the inverse of the image function of the boundary function, the characteristics and properties of the Laplace transform were used to derive the theoretical solution of the model without relying on the transformation of the boundary function, and the analytical solution was obtained by substituting the boundary condition into the theoretical solution. The analytical solution was used to analyze the temperature response laws of aquifers to parameter variation. Subsequently, a 40-day numerical simulation was conducted to analyze the boundary influence range and the results from the analytical method were compared to those from the numerical method. The study shows that: (1) the greater the distance from the river canal and the lower the aquifer flow velocity, the slower the aquifer temperature changes; (2) the influence range of the river canal boundary increases from 18.19 m to 23.19 m at the end of simulation period as the groundwater seepage velocity v increases from 0.08 m/d to 0.12 m/d; (3) the relative errors of the analytical and numerical methods are mostly less than 5%, confirming the rationality of the analytical solution.