An exact solution for a constant-strength line-sink satisfying the modified Helmholtz equation for groundwater flow

An exact solution for the discharge potential of a linear sink feature in a leaky rectangular aquifer

“…Furthermore, since the leakage induced by a well vanishes remotely from that well, it is generally agreed that the leakage may be ignored beyond a distance of order 8λ from any point on the linesinks [1]. Such a condition we shall call remoteness condition in the sequel is widely verified in all numerical evaluations [6]. This condition defines a limited influence area of the leaky feature allowing one to consider the result (3) as equivalent to the discharge potential for a leaky aquifer occupying a finite region with no-flow boundary conditions (homogeneous Neumann boundary conditions).…”

“…Often, such integrations in closed-form are intricate so that in practice polynomial or truncated series representations of K 0 (among other methods, see e.g. [1,6]) are used for obtaining a good approximation of the solution. Furthermore, since the leakage induced by a well vanishes remotely from that well, it is generally agreed that the leakage may be ignored beyond a distance of order 8λ from any point on the linesinks [1].…”

“…Such flow system is governed by the modified Helmholtz equation δ(x − ξ l (t))δ(y − η l (t))dt, (2) δ(z) being the Dirac delta function at z = 0. In previous attempts (e.g., [1]), the aquifer is assumed to be of infinite extent for simplicity and the potential is computed on the whole Euclidean plane as the double convolution product (called in that context, integral line elements [5,6]), * Tel. : +33 262938233.…”