Drainage of a horizontal Boussinesq aquifer with a power law hydraulic conductivity profile

Solutions to the Boussinesq equation describing drainage into a fully penetrating channel have been used for aquifer characterization. Two analytical solutions exist for early- and late-time drainage from a saturated, homogeneous, and horizontal aquifer following instantaneous drawdown. The solutions for discharge Q can be expressed as dQ/dt = -aQ(b), where a is constant and b takes on the value 3 and 3/2 for early and late time, respectively. Though many factors can contribute to departures from the two predictions, we explore the effect of having permeability decrease with depth, as it is known that many natural soils exhibit this characteristic. We derive a new set of analytical solutions to the Boussinesq equation for k proportional to z(n), where k is the saturated hydraulic conductivity, z is the height above an impermeable base, and n is a constant. The solutions reveal that in early time, b retains the value of 3 regardless of the value of n, while in late time, b ranges from 3/2 to 2 as n varies from 0 to infinity. Similar to discharge, water table height h in late time can be expressed as dh/dt = -ch(d), where d = 2 for constant k and d --> infinity as n --> infinity. In theory, inclusion of a power law k profile does not complicate aquifer parameter estimation because n can be solved for when fitting b to the late-time data, whereas previously b was assumed to be 3/2. However, if either early- or late-time data are missing, there is an additional unknown. Under appropriate conditions, water table height measurements can be used to solve for an unknown parameter.