Does groundwater flow depend on the geometry of the aquifer?

Question

I consider two cases of a confined aquifer and want to calculate the pressure / hydraulic head in a quasi 1d case. These two cases are shown in my drawing.
I assume homogenious, isotropic darcy flow, and use the groundwater flow equation, without time dependency (steady state flow) and no sources or sinks. So the equation simplifies to a simple Laplace equation
$$ 0 = alpha nabla^2 h $$,
where $h$ is the hydraulic head. In one dimension this can be written as
$$ 0 = alpha frac{partial^2 h}{partial x^2} $$.
If the boundary conditions are known, $h_1$ at the left boundary, $h_2$ at the right boundary, $h_2 > h_1$, I conclude, that $h$ probably looks like I have drawn it in bottom left, because if the second derivative of $h$ is zero, then the first derivative has to be a straight line, and therefor $h$ needs to be a parabola.

So my first question is, given that I have not made a mistake until here, how do I know if the parabola has positive or negative curvature? (line or dashed line in the drawing).

And the second question: What does change in the case, that I have drawn on the right? Is the hydraulic head the same? Is the pressure different?

Chet Miller · Answer

If the formation height is constant and you use the 1D form of the equation, then x must be measured along the channel and h is a linear function of x. In particular, $$v=-Kfrac{dh}{dx}$$ where v is the superficial velocity (constant) in the x direction and K is the hydraulic conductivity. If $h_2>h_1$, the flow is in the negative x direction. In the second figure, v will be lower in magnitude because the path is longer.

Answered by Chet Miller on November 26, 2020

Hassan Saleem · Answer

to answer this question

the right figure has variable aquifer bottom, unlike the left one which has constant aquifer bottom

from groundwater model prospective on real case, it has been proven that the variation in the aquifer bottom dose affects the groundwater flow in some way.In fact this is the actual case for most groundwater flow systems that you don't have uniform aquifer bottom
hence different aquifer geometry

for more detail on this problem chick Never Krisk Book title: Hydrogeology and Groundwater Modeling 2007

Hassan Saleem · Answer

you can't have constant velocity unless you have a uniform hydraulic conductivity (k). In other words, constant velocity that means homogeneous-isotropic aquifer

Does groundwater flow depend on the geometry of the aquifer?

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