A novel analytical solution for constant-head test in a patchy aquifer

A mathematical model describing the hydraulic head distribution for a constant-head test performed in a well situated at the centre of a patchy aquifer is presented. The analytical solution for the mathematical model is derived by the Laplace transforms and the Bromwich integral method. The solution for the hydraulic head has been shown to satisfy the governing equations, related boundary conditions, and continuity requirements for the hydraulic head and flow rate at the interface of the patch and outer regions. An efficient numerical approach is proposed to evaluate the solution, which has an integral covering an integration range from zero to infinity and an integrand consisting the product and square of the Bessel functions. This solution can be used to produce the curves of dimensionless hydraulic head against dimensionless time for investigating the effect of the contrast of formation properties on the dimensionless hydraulic head distribution. Define the ratio of outer-region transmissivity to patch-region transmissivity as α. The dimensionless hydraulic head for α = 0.1 case is about 2.72 times to that for α = 10 case at dimensionless large time (e.g.t≥10⁶) when the dimensionless distance (ρ) equals 10. The results indicate that the hydraulic head distribution highly depends on the hydraulic properties of two-zone formations.