I'm going to provide background on the parameters in the title's question. Then I'm going to explain the origin of my question. Hopefully, then, my title question (and additional questions) will be more clear and the focus can be on the mathematical proof. (Note I have only recently became aware about tensors and their use for physical problems).
Background: Fluid flow through porous media is commonly described using Darcy's Law, $$\tag{1} v=\frac{k_d}{\mu} \frac{\Delta p}{\Delta x}$$
where $k_d$ is known as the medium's absolute coefficient of permeability, its a proportionality constant that reflects the porous medium's void space configuration. Permeability is one of the parameters of focus for this question/post. It was originally only considered to be a scalar quantity (Darcy's original experiments from which his equation was determined only considered isotropic media).
However, for anisotropic porous media, quoting wiki:
To model permeability in anisotropic media, a permeability tensor is needed. Pressure can be applied in three directions, and for each direction, permeability can be measured (via Darcy's law in 3D) in three directions, thus leading to a 3 by 3 tensor. The tensor is realized using a 3 by 3 matrix being both symmetric and positive definite.
The other parameter of focus is the so-called Klinkenberg factor, $b$. It is a correction factor required when flowing gas at low mean pressure. Like permeability, the Klinkenberg factor is a function of the porous medium's void space structure, namely, how large (or small) the radii ($r$) of the interconnected void spaces are within the medium.
The origin of my question: Paraphrasing a paper I am reading,
Using one-dimensional flow, Kinkenberg came to the following conclusions and verified them experimentally.
- As the constant $b$ (Klinkenberg factor) is inversely proportional to $\bar r$ (radius of capillaries), the value of $b$ may be expected to be small for highly permeable samples, and to be large for less permeable samples.
- The apparent permeability, $k_{app}$, extrapolated to infinite mean pressure ($1/p_m=0$) should give the absolute permeability. Klinkenberg’s proposed equation for isotropic permeability is: $$\tag{2} k_{app}=k_d(1+b/p_m)$$ Given the consistency of Klinkenberg's conclusions, one may generalize Eqn (2) for three dimensional flow. From conclusion 2, the apparent permeability tensor $k_{app}^{ij}$ for anisotropic rock extrapolated to infinite mean pressure should give the absolute permeability tensor. Therefore, it should be a contravariant tensor of order two since absolute permeability tensor is a contravariant tensor of order two. From conclusion 1, the Klinkenberg factor for anisotropic rocks also should be a tensor. Since the Klinkenber factor is related to the absolute permeability, it is convenient to use the same kind of tensor as the absolute permeability. Thus, one may by analogy write the following equation: $$\tag{3} k_{app}^{ij}=k_d^{ij} \left(\delta_\alpha^j+g_{\alpha \beta} \frac{b^{\beta j}}{p_m}\right)$$ where $\delta_\alpha^j$ = Kronecker delta, and $g_{\alpha \beta}$ = metric tensor.
Restating the Title's question: Can a physical parameter originally defined as a scalar (the Klinkenberg factor $b$) be shown to be a tensor through its relationship to another parameter that is a tensor (the absolute permeability $k_d^{ij}$)?
Additional questions:
- In line with my proof-explanation tag, does the arguments given by the paper I am reading make sense as a proof to conclusively say that the Kinkenberg factor is a symmetric positive-definite tensor of rank 2 like the absolute permeability tensor?
- In line with my alternative-proof tag, regardless of the answer to the previous question, can an alternative proof be given to show that the Kinkenberg factor is (or is not) a symmetric positive-definite tensor of rank 2 like the absolute permeability tensor?