My intuition tells me that permeability $K_{ij}$ of Darcy's Law is symmetric, $K_{ij}=K_{ji}$, and I am looking for an answer to show me why this is not the case.
When measuring the permeability of a cylindrical sample in a conventional permeameter in the conventional manner, flowing in one direction, it is quite inconceivable to me that the permeability would vary upon reversal of the direction of flow. Such an effect would be equivalent to a valve-action of the porous medium which, although not a priori impossible, I am not considering this as a possibility -- here I will consider a fixed, rigid, unchanging porous medium.
Contrary to my intuition, I read the following from Krizek (1968) (emphasis mine):
Although $K_{ij}$ is usually assumed to be a symmetric tensor, it is not intuitively obvious why this must be the case. In fact, Collins (17) states that it cannot be expected that all anisotropic porous media can be associated with a symmetric permeability tensor. For example, the soil formation described by Dapples and Rominger (3) where elongated mineral grains are larger at one end than the other and oriented with their larger ends pointing in one direction would probably not produce a symmetric permeability tensor. Since a nonsymmetric tensor cannot be diagonalized, results cannot be expressed in terms of principal permeabilites lying along orthogonal axes.
Krizek holds a view that is contrary to my intuition, saying, "it is not intuitively obvious why permeability symmetry must be the case." I'm not sure why one would have this intuition. If you hold the same intuition as Krizek's, could you explain to me your thoughts with regards to this?
Krizek also mentions an example that Dapples and Rominger describes, where elongated mineral grains are larger at one end than the other and oriented with their larger ends pointing in one direction. I have my own visualization of what a porous medium composed of grains as described, but I’m not sure if what I am envisioning is correct. Could you provide me with an image of this (your own depiction or Dapples and Rominger’s)?
Are there any other examples of non-symmetric permeability porous mediums that you know of? Images/illustrations of these would be most helpful.
As a point of note, Bear says the following:
In thermodynamics, the rate of entropy produciton, denoted by $\dot S$ is related to the thermodynamic driving force, $X_i$, and to the thermodynamic flux, $Y_i$, by De Groot and Mazur (1962, p.65): $$\tag{3.4.38} \dot S=Y_i X_i.$$ Furthermore, the rate of entropy production must be positive, i.e., $\dot S \ge 0$.
And when considering Darcy's Law ($q_i=-K_{ij} \partial h / \partial x_j$) he says,
We make use of (3.4.38), with $X_i \equiv q_i$, and $Y_i \equiv \partial h/\partial x_i$. In this case, the, $\dot S$ is related to $X_i$ and $Y_i$, by: $$\dot S \equiv Y_i X_i=(-K_{ij} \partial h / > \partial x_j)(-\partial h / \partial x_i) \ge 0.$$ Hence, the matrix $K_{ij}$ is symmetric and definite positive. (emphasis mine)
So, would this mean that any non-symmetric medium would violate this entropy generation relation?
