Infiltration-induced phreatic surface flow to periodic drains: Vedernikov–Engelund–Vasil'ev's legacy revisited

A. R. Kacimov*, Yu V. Obnosov

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


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).

Original languageEnglish
Pages (from-to)989-1003
Number of pages15
JournalApplied Mathematical Modelling
Publication statusPublished - Mar 2021


  • 2-D saturated and unsaturated seepage flows
  • Complex potential and hodograph
  • Conformal mappings
  • HYDRUS2D simulations
  • Isobars, isotachs, streamlines and phreatic surfaces
  • Laplace's and Richards’ equations
  • Zhukovsky's slit drain and circular drains

ASJC Scopus subject areas

  • Modelling and Simulation
  • Applied Mathematics


Dive into the research topics of 'Infiltration-induced phreatic surface flow to periodic drains: Vedernikov–Engelund–Vasil'ev's legacy revisited'. Together they form a unique fingerprint.

Cite this