On the use of the Boussinesq equation for interpreting recession hydrographs from sloping aquifers

The method of recession analysis proposed by Brutsaert and Nieber (1977) remains one of the few analytical tools for estimating aquifer hydraulic parameters at the field scale and beyond. In the method, the recession hydrograph is examined as − dQ / dt = f ( Q ), where Q is aquifer discharge and f is an arbitrary function. The observed function f is parameterized through analytical solutions to the one‐dimensional Boussinesq equation for unconfined flow in a homogeneous and horizontal aquifer. While attractive in its simplicity, as originally presented it is not applicable to settings where slope is an important driver of flow, or where hydraulic parameters vary greatly with depth. We compare analytical solutions to the linearized one‐dimensional Boussinesq equation for a sloping aquifer to numerical solutions of the full nonlinear equation. The behavior of the nonlinear Boussinesq equation is also assessed when the aquifer is heterogeneous wherein the lateral saturated hydraulic conductivity k varies as a power law with height z above the impermeable layer ( k ∼ z n , n constant ≥ 0). All of the analytical solutions differ in key aspects from the nonlinear solution when plotted as − dQ / dt = f ( Q ) and thus are inappropriate for a Brutsaert and Nieber‐type analysis. However, new analytical solutions for a sloping aquifer are derived “empirically” from the numerical simulations that are applicable during the late period of recession when the recession curve converges to − dQ / dt = aQ b , where b = (2 n + 1)/( n + 1) and a is a function of the dimensions and hydraulic properties of the aquifer.