Abstract An explicit analytical solution is obtained to an old problem of a potential steady-state 2-D saturated Darcian flow in a homogeneous isotropic soil towards systematic drains modeled as line sinks (submerged drains under an overhanging of a phreatic surface), placed on a horizontal impervious substratum, with a constant-rate infiltration from the vadose zone. The corresponding boundary-value problem brings about a quarter-plane with a circular cut. A mathematical clue to solving the Hilbert problem for a two-dimensional holomorphic vector-function is found by engaging a hexagon, which has been earlier used in analytical solution to the problem of phreatic flow towards Zhukovsky’s drains (slits) on a horizontal bedrock. A hodograph domain is mapped on this hexagon, which is mapped onto a reference plane where derivatives of two holomorphic functions are interrelated via a Polubarinova-Kochina type analysis. HYDRUS2D numerical simulations, based on solution of initial and boundary value problems to the Richards equation involving capillarity of the soil, concur with the analytical results. The position of the water table, isobars, isotachs, and streamlines are analyzed for various infiltration rates, sizes of the drains, boundary conditions imposed on them (empty drains are seepage face boundaries; full drains are constant piezometric head contours with various backpressures).

中文翻译：

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渗透诱导的潜水表流到周期性排水沟：重新审视 Vedernikov-Engelund-Vasil'ev 的遗产

摘要 对于在均质各向同性土壤中朝向系统排水管的潜在稳态 2-D 饱和达西流的老问题，获得了明确的解析解，该系统排水管被建模为线汇（潜水面悬垂下的淹没排水管），放置在一个水平不透水基质，从包气带以恒定速率渗透。相应的边值问题产生一个带有圆形切口的四分之一平面。解决二维全纯矢量函数的希尔伯特问题的数学线索是通过使用六边形找到的，六边形早先已用于解析水平基岩上朝向茹科夫斯基排水沟（狭缝）的潜水流问题。一个hodograph域映射在这个六边形上，它被映射到一个参考平面，其中两个全纯函数的导数通过 Polubarinova-Kochina 类型分析相互关联。HYDRUS2D 数值模拟基于对涉及土壤毛细作用的理查兹方程的初值和边值问题的求解，与分析结果一致。分析了地下水位、等压线、等压线和流线的位置，以了解各种渗透率、排水管尺寸、施加在它们上的边界条件（空排水管是渗流面边界；全排水管是具有各种背压的恒定测压水头等高线）。与分析结果一致。分析了地下水位、等压线、等压线和流线的位置，以了解各种渗透率、排水管尺寸、施加在它们上的边界条件（空排水管是渗流面边界；全排水管是具有各种背压的恒定测压水头轮廓）。与分析结果一致。分析了地下水位、等压线、等压线和流线的位置，以了解各种渗透率、排水管尺寸、施加在它们上的边界条件（空排水管是渗流面边界；全排水管是具有各种背压的恒定测压水头轮廓）。