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### Table of Contents

# Boussinesq equation

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:

$$q_{GW}=-K_h*h*{\delta h}/{\delta x}*cos(\beta)-K_h*h*sin(\beta)$$

where $q_{GW}$ is the groundwater flow to the stream, $K_h$ is the hydraulic conductivity of the aquifer along the slope, $h$ is the groundwater level above the aquifer base, $\beta$ 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:

$$\frac{\delta}{\delta x}* \left ( h*\frac{\delta h}{\delta x} \right ) * cos(\beta) + \frac{\delta h}{\delta x}*sin(\beta) = \frac{S_y}{K_h}*\frac{\delta h}{\delta t}$$

## Brutsaerts simplification

Brutsaert (1994) linearized the equation:

$$q_{GW}=-\alpha*K_h*h_0*\frac{\delta h}{\delta x}*cos(\beta)-K_h*h*sin(\beta)$$

where $\alpha$ is a coefficient that equals more or less 0.5 and adjust for the use of $h_0$ instead of $h$.

The Boussinesq equation can be simplified to:

$$\left ( \frac{\alpha*K_h*h_0*cos(\beta) }{S_y} \right ) * \left ( \frac{\delta^2(h_0-h)}{\delta x^2} \right ) + \left ( \frac{K_h*sin(\beta)}{S_y} \right ) * \left ( \frac{\delta*(h_0-h)}{\delta x} \right ) = \frac{\delta h}{\delta t}$$

## The shape parameter

There is a shape parameter that describes the ratio between advection and diffusion terms:

$$ J = \frac{\frac{K_h*sin \beta * X}{S_y}}{\frac{\alpha*K_h*h_0*cos \beta}{S_y}}=\frac{tan \beta * X}{\alpha * h_0}$$

A large $J$ means steep slopw and shallow aquifer and a predomincance of advection, a small $J$ means shallow slope and thick aquifer and a dominance of diffusion, for a slope of $\beta=0$ the solution approaches that of the Dupuit equation.

## Solution for shallow slopes

For a shallow aquifer with $\beta = 0$ or the Dupuit conditions, the solution of the Boussinesq equation simplifies to

$$q_{GW}= \left ( \frac{h^2_0*K_h}{X} \right ) * \sum_{i=1}^{n}*exp \left [ - \frac{(2*i-1)^2*\pi^2*K_h*h_0}{8*X^2*S_y}*t \right ]$$

This equation represents a series of terms, the importance of terms for higher $i$ diminishes rapidly. If only the first term is retained, we get

$$q_{GW}= \left ( \frac{h^2_0*K_h}{X} \right ) * exp \left [ \frac{1.23*K_h*h_0}{X^2*S_y}*t \right ]$$

this is the equation for the drainage of a Dupuit aquifer.

## Solution for steep slopes

A general solution of the equation is given by Brutsaert(1994). For a steep aquifer and large $J$, some simplifications can be introduced:

$$B = h*S_y*X$$

where $B$ is the gravity drainable water. In addition the outflow can be described by Darcy's law as:

$$q_{GW}=h*k_h*tan \beta$$

and hence (if no vertical recharge occurs:

$$q_{GW}=-\frac{dB}{dt}$$

After substitution this leads to

$$q_{GW}=K_h*h_0*tan \beta * exp \left ( \frac{-K_h*tan \beta}{S_y * X}*t \right ) $$

The equation can be used to estimate the outflow from a sloping (steep and shallow) aquifer as a function of hydraulic conductivity $K_h$, thickness of the aquifer $h_0$, slope of the aquifer $tan \beta$, storativity $S_y$ and length of the aquifer $X$ as a function of time $t$. This equation can be simplified to

$$q_{GW}=a * h_0 * exp \left ( \frac{-a}{S_y * X}*t \right ) $$

where $a=K_h*tan \beta$.