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

Abstract

[1] 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 ÀdQ/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 ÀdQ/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 ÀdQ/dt = aQ b , where b = (2n + 1)/(n + 1) and a is a function of the dimensions and hydraulic properties of the aquifer.

Are streamflow recession characteristics really characteristic?

“…Brutsaert and Nieber (1977) stated that values of b from 1 for late recession segments (long-time behavior) to 3 for early stages of recession (short-time behavior) are in a physically reasonable range. They identified decreasing slope b in power-law storage-outflow relationships as a function of continuous drawdown, whereas Rupp and Selker (2006b) summarized that the range of slope b may be used for aquifer characterization since values ranging from 1 to 2 indicated sloping aquifers and from 1.5 to 2 horizontal aquifers. If we consider the upper and lower 10 % of calculated values to be outliers, in this study slope b ranged from 1.1 to 2.1 with an average of 1.55.…”

Investigating storage‐discharge relations in a lowland catchment using hydrograph fitting, recession analysis, and soil moisture data

“…However, local storage computed with a groundwater model has been used in direct storage-discharge fitting [Rupp et al, 2009]. Examples of studies in which solutions to the Boussinesq equation for sloping aquifers have been employed in order to investigate storage-discharge relations are Troch et al [1993], Brutsaert [1994], and Rupp and Selker [2006b]. We refer to the review article by Troch et al [2013] and references therein for a more complete overview.…”

On the effectiveness of recession analysis methods for capturing the characteristic storage-discharge relation: An intercomparison study

“…Using linear least-squares regression weighted by the reciprocal of the square of the standard errors of that bin,  and  parameters were obtained through fitting ln(−dQ/dt) vs. ln(Q) data. This approach weighs high Q values more and 25 limits the influence of low Q values on the regression, as low Q values are more likely to have measurement errors caused by instruments' precision and stage-discharge relations [Rupp and Selker, 2006a].…”

“…The relation 15 between S and Q was then obtained by fitting a least-squares regression line through the lower envelope of the ln(−dQ/dt) vs. ln(Q) scatter. When the magnitude of streamflow change is smaller than the precision of the stream gauge, the derived lower envelope of ln(−dQ/dt) vs. ln(Q) could be an artifact [Rupp and Selker, 2006a]. Therefore, we followed Palmroth et al [2010] and adopted a "scaled-t " method to obtain −dQ/dt , where instead of using a constant t, a varying t was used.…”