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 -