Full and boundary-layer solutions of the steady air sparging problem

The one previous analytical solution to the steady sparging problem in a homogeneous soil was in the boundary-layer approximation that neglects the vertical capillary pressure gradient. The present study gives an analytical solution to the full air-flow equation, with the corresponding boundary-layer solution for comparison. The full solution predicts sparging below the air-injection point, whereas the boundary-layer solution does not. Solutions are expressed both in terms of the dimensionless Kirchhoff potential θ (increasing monotonically with capillary pressure and air saturation also) and the dimensionless Stokes stream function F (mapping the pattern of air flow). Both θ and F show the full solution predicting more effective sparging near the injection point than does the boundary-layer solution. The overall boundary-layer relative error, however, decreases as dimensionless air-injection depth below the undisturbed water-table increases.