I am trying to understand the Darcy law of flow of fluid in porous media.
The law in differential form is simple equation: $q = K \nabla p$, where $q$ is a fluid flux, $K$ is some permeability coeff, and $p$ is pressure. The idea is that flux increases where the pressure is greater and where permeability is greater.
I want to use this law for the problem drawn in the picture below.
I have some media with vertical gradient of pressure. I have some inclined zone with high permeability representing some fracture inside a rock. My intuition tells me that fluid would lift straight upward until it meets fracture, and then, it will go upward and slightly to the side because it's easier for the fluid to move inside a zone with higher permeability (show above 2).
But according to the law (if I I understand math in the law correctly), fluid should go straight upward and when it meets the zone with higher permeability it will only increase its speed but will not change its direction at all.
Where I'm wrong?
This form of Darcy's law where the permeability $K$ is a scalar applies to isotropic porous media.
Let the vertical direction be associated with Cartesian coordinate $x$. If the pressure gradient has a non-zero component only in the $x$-direction then the components of the flux vector are $(q_x,0,0)$ with no motion in the $y-$ and $z-$ directions and where
$$q_x = K \frac{\partial p}{\partial x}$$
Even if $K$ varies with spatial position the flux is unidirectional (although non-constant) as illustrated in Figure 3.
More generally for anistotropic porous media, the permeability is represented by a tensor $\mathbf{K}$ and Darcy's law is written as
$$\mathbf{q} = \mathbf{K}\cdot \nabla p$$
In that case, a pressure gradient imposed in the $x-$direction could give rise to flux components in the other directions, e.g.
$$q_x = K_{xx}\frac{\partial p}{\partial x}, \quad q_y = K_{yx}\frac{\partial p}{\partial x}$$