I was reading a paper (equation 1 in linked paper) where they used 'linearized Boussinesq' equation to model groundwater in unconfined aquifer. The linearized Boussinesq equation described in the paper was $$\frac{\partial h}{\partial t} = \frac{kh^*}{\phi} \frac{\partial^2 h}{\partial x^2}$$ where $k$ is hydraulic conductivity $h^*$ is average height of water table and $\phi$ is specific yield. This equation is very much similar to diffusion equation used normally to model groundwater flow in confined aquifer as defined below.$$\frac{\partial h}{\partial t} = \frac{kd}{S} \frac{\partial^2 h}{\partial x^2}$$ where $d$ is thickness and $s$ is storativity of confined aqufier. When I googled about Boussinest equation, it is noway similar to what I have presented above. My question is is 'linearized form of Boussinesq equation similar to diffusion equation in this case? If yes then how?

  • $\begingroup$ can you edit your question to show the reference of the paper you are reading ? $\endgroup$
    – gansub
    May 9, 2017 at 13:05
  • 1
    $\begingroup$ mathworld.wolfram.com/BoussinesqEquation.html the linear boussinesq equation as presented here is similar to what you have in the paper. You take the total material derivative. Set the advective non linear terms to be zero and then you have that equal to the RHS $\endgroup$
    – gansub
    May 11, 2017 at 2:21
  • $\begingroup$ @gansub how? The partial derivative w.r.t time is second derivative in the given link? Also the term on right hand side of equation 1 in link as no corresponding term in the equation that I have given here. $\endgroup$ May 12, 2017 at 21:05
  • $\begingroup$ you write out the total material derivative - you have two terms - one the partial deriviative of h with respect to t and the advection term. The advection term is nonlinear and you set it to zero. Then you have a linear form of the material derivative. Regarding the right hand side you will have to explain from hydrology what that is. $\endgroup$
    – gansub
    May 14, 2017 at 2:06


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