TY - JOUR

T1 - Infiltration-induced phreatic surface flow to periodic drains

T2 - Vedernikov–Engelund–Vasil'ev's legacy revisited

AU - Kacimov, A. R.

AU - Obnosov, Yu V.

N1 - Funding Information:
This work was supported by Sultan Qaboos University via the grant “Rise of Water-table and Its Mitigation at SQU Campus” and it was carried out as part of the development program of the Scientific and Educational Mathematical Center of the Volga Federal District, agreement No. 075-02-2020-1478.
Publisher Copyright:
© 2020

PY - 2021/3

Y1 - 2021/3

N2 - 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).

AB - 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).

KW - 2-D saturated and unsaturated seepage flows

KW - Complex potential and hodograph

KW - Conformal mappings

KW - HYDRUS2D simulations

KW - Isobars, isotachs, streamlines and phreatic surfaces

KW - Laplace's and Richards’ equations

KW - Zhukovsky's slit drain and circular drains

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U2 - 10.1016/j.apm.2020.10.029

DO - 10.1016/j.apm.2020.10.029

M3 - Article

AN - SCOPUS:85096199344

SN - 0307-904X

VL - 91

SP - 989

EP - 1003

JO - Applied Mathematical Modelling

JF - Applied Mathematical Modelling

ER -