TY - JOUR

T1 - Analytical solution and shape optimization for groundwater flow through a leaky porous trough subjacent to an aquifer

AU - Kacimov, A. R.

PY - 2006

Y1 - 2006

N2 - Steady two-dimensional groundwater flow in a porous low-permeable trough is studied by the method of boundary-value problems for holomorphic functions. In the overlying highly permeable aquifer the hydraulic head varies linearly, i.e. flow is unidirectional. The exchange of groundwater between aquifer and trough does not affect the flow in the aquifer. It is assumed that through a horizontal aquifer-trough interface the head is transmitted into the trough, where the bounding effect of the bed causes circulatory seepage. Triangular troughs are studied and an isosceles form with a base angle of 38° is proved to have the highest circulation rate at a given cross-sectional area. In the class of arbitrary forms, solution to this optimal shape design problem is obtained by tackling the Schwartz and Signorini singular integrals and a unique and global maximum of the rate is found. The extreme curve coincides with an optimal soil channel of minimal seepage losses having depth to width ratio of 0.371. The global maximum differs from that one for the triangular class in 3% only that corroborates stability and robustness of the optimization criterion and makes possible isoperimetric estimates of the seepage intensity through arbitrary troughs.

AB - Steady two-dimensional groundwater flow in a porous low-permeable trough is studied by the method of boundary-value problems for holomorphic functions. In the overlying highly permeable aquifer the hydraulic head varies linearly, i.e. flow is unidirectional. The exchange of groundwater between aquifer and trough does not affect the flow in the aquifer. It is assumed that through a horizontal aquifer-trough interface the head is transmitted into the trough, where the bounding effect of the bed causes circulatory seepage. Triangular troughs are studied and an isosceles form with a base angle of 38° is proved to have the highest circulation rate at a given cross-sectional area. In the class of arbitrary forms, solution to this optimal shape design problem is obtained by tackling the Schwartz and Signorini singular integrals and a unique and global maximum of the rate is found. The extreme curve coincides with an optimal soil channel of minimal seepage losses having depth to width ratio of 0.371. The global maximum differs from that one for the triangular class in 3% only that corroborates stability and robustness of the optimization criterion and makes possible isoperimetric estimates of the seepage intensity through arbitrary troughs.

KW - Aquifer

KW - Holomorphic functions

KW - Optimal shape design

KW - Seepage

KW - Signorini formula

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U2 - 10.1098/rspa.2005.1617

DO - 10.1098/rspa.2005.1617

M3 - Article

AN - SCOPUS:33748286336

SN - 1364-5021

VL - 462

SP - 1409

EP - 1423

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

IS - 2069

ER -