Steady state analytical solutions for pumping in a fully bounded rectangular aquifer

Abstract

AbstractUsing the Schwartz‐Christoffel conformal mapping method together with the complex variable techniques, we derive steady state analytical solutions for pumping in a rectangular aquifer with four different combinations of impermeable and constant‐head boundaries. These four scenarios include: (1) one constant‐head boundary and three impermeable boundaries, (2) two pairs of orthogonal impermeable and constant‐head boundaries, (3) three constant‐head boundaries and one impermeable boundary, and (4) four constant‐head boundaries. For these scenarios, the impermeable and constant‐head boundaries can be combined after applying the mapping functions, and hence only three image wells exist in the transformed plane, despite an infinite number of image wells in the real plane. The closed‐form solutions reflect the advantage of the conformal mapping method, though the method is applicable for the aspect ratio of the rectangle between 1/10.9 and 10.9/1 due to the limitation in the numerical computation of the conformal transformation from a half plane onto an elongated region (i.e., so‐called “crowding” phenomenon). By contrast, for an additional scenario with two parallel constant‐head boundaries and two parallel impermeable boundaries, an infinite series of image wells is necessary to express the solution, since it is impossible to combine these two kinds of boundaries through the conformal transformation. The usefulness of the results derived is demonstrated by an application to pumping in a finite coastal aquifer.