Helmholtz evolution of a semi-infinite aquifer drained by a circular tunnel

The evolution equation of a drained aquifer during the consolidation process when time is transformed into the Laplace variable is the modified Helmholtz equation. The governing equation of the steady state of a heterogeneous aquifer which hydraulic conductivity when plotted against depth in a semi-log graph has a constant slope is also the modified Helmhotlz equation. The same equation comes out when the slopes of the hydraulic conductivity plotted against depth and against the hydraulic potential in a semi-log graph are constants. The modified Helmholtz equation will be solved exactly considering a semi-infinite aquifer drained by a circular tunnel. A unique state function, which according to the case considered has different interpretations, is obtained in closed form as an infinite sum involving modified Bessel functions. The amount of water that flows into the tunnel contrarily to the state function may change from case to case and will be calculated exactly and in closed form for the different cited cases. The analytic solution has a wide range of application, is valid for different cases, and within every case needs being adapted to the particular problem to be solved. An illustrative application will show an adaptation of the solution to rock masses when the hydraulic conductivity plotted against the effective stress in a semi-log graph has a constant slope. This will allow estimating the relative precision of approximated formulae for the water inflow in fissured rock masses such as the Zhang and Franklin equation and the first order approximation.