Based on the Dupuit approximation some modifications can be made for a sloping aquifer, as first proposed by Boussinesq. Boussinesq proposed a modified version of Darcys law:
where is the groundwater flow to the stream, is the hydraulic conductivity of the aquifer along the slope, is the groundwater level above the aquifer base, is the slope of the aquifer. For the meaning of variables see the sketch below:
When combined with the continuity equation, the so-called 'Boussinesq-Equation' is obtained:
Brutsaert (1994) linearized the equation:
where is a coefficient that equals more or less 0.5 and adjust for the use of instead of .
The Boussinesq equation can be simplified to:
There is a shape parameter that describes the ratio between advection and diffusion terms:
A large means steep slopw and shallow aquifer and a predomincance of advection, a small means shallow slope and thick aquifer and a dominance of diffusion, for a slope of the solution approaches that of the Dupuit equation.
For a shallow aquifer with or the Dupuit conditions, the solution of the Boussinesq equation simplifies to
This equation represents a series of terms, the importance of terms for higher diminishes rapidly. If only the first term is retained, we get
this is the equation for the drainage of a Dupuit aquifer.
A general solution of the equation is given by Brutsaert(1994). For a steep aquifer and large , some simplifications can be introduced:
where is the gravity drainable water. In addition the outflow can be described by Darcy's law as:
and hence (if no vertical recharge occurs:
After substitution this leads to
The equation can be used to estimate the outflow from a sloping (steep and shallow) aquifer as a function of hydraulic conductivity , thickness of the aquifer , slope of the aquifer , storativity and length of the aquifer as a function of time . This equation can be simplified to
where .