Richards equation (RE) of Brooks–Corey soil is invariant under some scaling transformations and can be reduced to ordinary differential equations (ODEs) with scale symmetry analysis. From the reduced ODE, it is convenient to get the solution of RE (mostly numerically). The admissible scaling transformations of RE can transform a given solution to other solutions, by which the explicit relations between wetting front (or its velocity) and boundary moisture and between cumulative infiltration and boundary moisture for horizontal infiltration are deduced. For horizontal infiltration into dry soil with constant boundary moisture, the contour of water distribution curve can be represented by a shape factor R, which can be regarded as a constant for a given soil. Based on the reduced ODE and the parameter R, an explicit approximate solution for horizontal soil water infiltration is proposed, which just relies on R. The shape factor R is related to Brooks–Corey power exponent n and independent of the boundary moisture. With the obtained R–n relations, the relative deviation of the approximate solution can be less than 0.001. The approximate solution keeps high precision in any time range of the infiltration process.

中文翻译：

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水平入渗Richards方程的标度分析及其近似解

Brooks-Corey 土壤的理查兹方程 (RE) 在一些尺度变换下是不变的，并且可以通过尺度对称分析简化为常微分方程 (ODE)。从简化的 ODE 中可以方便地得到 RE 的解（主要是数值上的）。RE 的可容许标度变换可以将给定的解转换为其他解，由此推导出润湿锋（或其速度）与边界水分之间以及累积入渗与水平入渗边界水分之间的显式关系。对于边界水分恒定的干燥土壤的水平入渗，水分分布曲线的等高线可以用形状因子R 表示，对于给定的土壤，可以将其视为常数。基于简化的 ODE 和参数R，提出了水平土壤水分入渗的显式近似解，它仅依赖于R。形状因子R与 Brooks-Corey 幂指数n相关，与边界湿度无关。有了得到的R - n关系，近似解的相对偏差可以小于0.001。近似解在渗透过程的任何时间范围内都保持高精度。