Quasi-steady flow in sloping aquifers

Mass conservation links the storage S and the outflow Q of an aquifer. A relation between them (an S-Q relation) provides then a model governing the evolution of these quantities. In this work we construct an analytical quasi-steady state model which exploits the properties of the exact S-Q relation associated with steady state solutions of the Boussinesq equation for the sloping aquifer (that is, the Henderson and Wooding [1964] solutions). The model is derived by matching the asymptotic forms of the exact S-Q relation which arise for small and large values of the Henderson and Wooding parameter λ. These asymptotic forms provide a novel rederivation of well-known semiempirical S-Q relations of the form Q ∝ S and Q ∝ S², and they lead to soluble quasi-steady state models. The quadratic asymptotic relation turns out to hold for surprisingly low values of λ. This characteristic and its formal properties allow smooth matching with the linear relation at λ=π²/4=2.47. The obtained model holds over the entire parameter space. An important characteristic of the model, stemming from its derivation, is that it involves only the geometric and hydraulic quantities present in the exact Boussinesq equation. The model is tested by best fitting four data sets from experiments simulating aquifer drainage. The derived curves for the drained volume are in excellent agreement with the data. The estimated values for k and n are also in overall very good agreement with their reference values.