Analytical solution and shape optimization for groundwater flow through a leaky porous trough subjacent to an aquifer
Abstract
Steady twodimensional groundwater flow in a porous lowpermeable trough is studied by the method of boundaryvalue 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 aquifertrough 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 crosssectional 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.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 May 2006
 DOI:
 10.1098/rspa.2005.1617
 Bibcode:
 2006RSPSA.462.1409K
 Keywords:

 seepage;
 aquifer;
 holomorphic functions;
 Signorini formula;
 optimal shape design