TY - JOUR

T1 - Analytical solutions for seepage near material boundaries in dam cores

T2 - The Davison-Kalinin problems revisited

AU - Kacimov, Anvar

AU - Obnosov, Yurii

N1 - Funding Information:
This work was supported by DOPSAR project “Effects of siltation behind Al-Khod dam on the soil properties and recharge efficiency” (Oman) and RFBR Grant No. 09-01-97008-r_povolgh’e_a (Russia). Helpful comments by two anonymous referees are appreciated.

PY - 2012/3

Y1 - 2012/3

N2 - Steady Darcian seepage through a dam core and adjacent shells is analytically studied. By conformal mappings of the pentagon in the hodograph plane and triangle in the physical plane flow through a low-permeable dam core is analyzed. Mass-balance conjugation of flow in the core and downstream highly-permeable shell of the embankment is carried out by matching the seepage flow rates in the two zones assuming that all water is intercepted by a toe-drain. Seepage refraction is studied for a wedge-shaped domain where pressure and normal components of the Darcian velocities coincide on the interface between the core and shell. Mathematically, the problem of R-linear conjugation (the Riemann-Hilbert problem) is solved in an explicit form. As an illustration, flow to a semi-circular drain (filter) centered at the triple point (contact between the core, shell and impermeable base) is studied. A piece-wise constant hydraulic gradient in two adjacent angles making a two-layered wedge (the dam base at infinity) is examined. Essentially 2-D seepage in a domain bounded by an inlet constant head segment, an outlet seepage-face curve, a horizontal base and with a straight tilted interface between two zones (core and shell) is investigated. The flow net, isobars, and isotachs in the core and shell are reconstructed by computer algebra routines as functions of hydraulic conductivities of two media, the angle of tilt and the hydraulic head value at a specified point.

AB - Steady Darcian seepage through a dam core and adjacent shells is analytically studied. By conformal mappings of the pentagon in the hodograph plane and triangle in the physical plane flow through a low-permeable dam core is analyzed. Mass-balance conjugation of flow in the core and downstream highly-permeable shell of the embankment is carried out by matching the seepage flow rates in the two zones assuming that all water is intercepted by a toe-drain. Seepage refraction is studied for a wedge-shaped domain where pressure and normal components of the Darcian velocities coincide on the interface between the core and shell. Mathematically, the problem of R-linear conjugation (the Riemann-Hilbert problem) is solved in an explicit form. As an illustration, flow to a semi-circular drain (filter) centered at the triple point (contact between the core, shell and impermeable base) is studied. A piece-wise constant hydraulic gradient in two adjacent angles making a two-layered wedge (the dam base at infinity) is examined. Essentially 2-D seepage in a domain bounded by an inlet constant head segment, an outlet seepage-face curve, a horizontal base and with a straight tilted interface between two zones (core and shell) is investigated. The flow net, isobars, and isotachs in the core and shell are reconstructed by computer algebra routines as functions of hydraulic conductivities of two media, the angle of tilt and the hydraulic head value at a specified point.

KW - Analytic functions

KW - Free boundary problems

KW - Hydraulic gradient

KW - Lapalce's equation

KW - Refraction

KW - Seepage

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U2 - 10.1016/j.apm.2011.07.088

DO - 10.1016/j.apm.2011.07.088

M3 - Article

AN - SCOPUS:83655201361

VL - 36

SP - 1286

EP - 1301

JO - Applied Mathematical Modelling

JF - Applied Mathematical Modelling

SN - 0307-904X

IS - 3

ER -