This paper presents a method for constructing approximate semi-analytical solutions for the radially symmetric generalized Boussinesq equation, also known as the porous medium equation (PME). This work is a further extension of analysis done by Telyakovskiy et al. (2016), that studied water injection governed by the Boussinesq equation into an unconfined initially dry aquifer. The newly derived approximate solution also has two parts; one to approximate singular behavior near the well and a polynomial part to model far-field behavior. With the prescribed initial and boundary conditions, the analyzed problem can be rewritten in terms of similarity variables resulting in a boundary value problem for a nonlinear ordinary differential equation. Using higher-order physically significant moments of the solution, a closed-form expression for the position of the wetting front and the shape of the phreatic surface is obtained. The introduced solution is valid over a wide range of injection regimes: the time-independent, the power-law injection function with positive exponents, and exponential-law of injection. A highly accurate numerical solver similar to Telyakovskiy et al. (2016) is used to validate the presented approximate solutions. Plus, comparison is made with the experimental data on prediction of the wetting front position for the special case of the Boussinesq equation.

本文提出了一种构造径向对称广义 Boussinesq 方程的近似半解析解的方法，也称为多孔介质方程 (PME)。这项工作是 Telyakovskiy 等人所做分析的进一步扩展。（2016 年），研究了由 Boussinesq 方程控制的注水到无侧限的初始干燥含水层中。新导出的近似解也有两部分；一个近似于井附近的奇异行为和一个多项式部分来模拟远场行为。在规定的初始条件和边界条件下，可以根据相似变量重写分析的问题，从而导致非线性常微分方程的边界值问题。使用解的高阶物理重要时刻，得到了润湿前沿位置和潜水面形状的闭式表达式。引入的解决方案适用于广泛的注入方案：与时间无关的、具有正指数的幂律注入函数和注入的指数律。类似于 Telyakovskiy 等人的高精度数值求解器。（2016）用于验证提出的近似解。此外，还与预测 Boussinesq 方程特殊情况的润湿前沿位置的实验数据进行了比较。类似于 Telyakovskiy 等人的高精度数值求解器。（2016）用于验证提出的近似解。此外，还与预测 Boussinesq 方程特殊情况的润湿前沿位置的实验数据进行了比较。类似于 Telyakovskiy 等人的高精度数值求解器。（2016）用于验证提出的近似解。此外，还与预测 Boussinesq 方程特殊情况的润湿前沿位置的实验数据进行了比较。