The classical problem used to model the response of an unconfined aquifer to a sudden change in boundary head is considered in this paper. This problem is usually modeled using the nonlinear groundwater Boussinesq equation. Due to the nonlinearity of this problem, no exact solutions exist. Solutions to the Boussinesq equation are therefore obtained using numerical techniques or approximate analytical methods. In this paper, we present a novel closed-form approximate analytical solution to this problem. The Boussinesq equation is converted to a first-order ordinary differential equation (ODE) by means of the Boltzmann transformation and by introducing a new variable related to the water flow. The first-order ODE is then solved analytically after introducing an intermediate approximation involving two fitting parameters. To avoid any numerical treatment, closed-form polynomial expressions of the fitting parameters are proposed. The final-form solution is simple to use and is obtained in terms of the incomplete gamma function, which is valid for both recharge and discharge. The derived solution is tested and compared to efficient numerical solutions, as well as to two types of analytical solutions: an accurate series expansion solution and an equivalent closed-form solution. The results show excellent agreement between the proposed solution and the numerical and series solutions. The proposed solution offers a key advantage in terms of both accuracy and simplicity; notably, it can be implemented using a simple spreadsheet.

本文考虑了用于模拟无承压含水层对边界水头突然变化的响应的经典问题。该问题通常使用非线性地下水 Boussinesq 方程进行建模。由于该问题的非线性，不存在精确解。因此，可以使用数值技术或近似分析方法获得 Boussinesq 方程的解。在本文中，我们针对该问题提出了一种新颖的封闭式近似解析解。通过玻尔兹曼变换并引入与水流相关的新变量，将 Boussinesq 方程转换为一阶常微分方程 (ODE)。然后在引入涉及两个拟合参数的中间近似后，对一阶 ODE 进行解析求解。为了避免任何数值处理，提出了拟合参数的闭式多项式表达式。最终形式的解决方案使用简单，并且是根据不完全伽玛函数获得的，对于充电和放电都有效。对导出的解进行测试，并将其与有效的数值解以及两种类型的解析解进行比较：精确的级数展开解和等效的封闭式解。结果表明，所提出的解与数值解和级数解之间非常吻合。所提出的解决方案在准确性和简单性方面具有关键优势；值得注意的是，它可以使用简单的电子表格来实现。