Solution to the Unsteady Seepage Model of Phreatic Water with Linear Variation in the Channel Water Level and Its Application

For a semi-infinite aquifer controlled by a river channel boundary, when the Laplace transform is used to solve a one-dimensional unsteady seepage model of phreatic water while considering the influence of the vertical water exchange intensity ε with the change in the river channel water level f(t), a complicated and tedious integral transformation process is required. By replacing f(t) with an operator, this study first derived the analytic formula of the ε term based on the properties of the Laplace transform without the direct participation of f(t) in the transformation. By using f(t) in the form of several types of linear functions, the Laplace transform and inverse transform laws were summarized. The analytical solution to the problem was easily obtained by applying the “integral property” of the transformation to the linear function term with time t. The relative error between the numerical solution and the analytical solution of the example was less than 0.2%, which verified the rationality of the model linearization method and the reliability of the analytical solution. For different boundary conditions, the process of establishing and applying the inflection point method and the curve-fitting method for calculating the model parameters by using dynamic monitoring data for phreatic water is presented with examples.