INTERDYSCYPLINARY CENTRE |
AND COMPUTATIONAL MODELLING
The problem of finding the effective permeability of a granular medium is common to many branches of numerical engineering, including some in which the quantity of interest would not be called "permeability". Let me therefore restrict to those two I know where this word is the right one: porous media and magnetic materials. Perhaps the former is more germane to everyone's intuition, so let's rather discuss porous media here. (The associated document (.ps file, 77 kb, or .pdf file, 18 kb) uses the terminology and notation of magnetostatics instead.)
As a typical problem, let's consider a porous wall, or perhaps a layer of terrain, etc., which separates two fluid reservoirs at non-equal pressures. Liquid leaks through. At which rate? At a rate which is proportional to the pressure drop (that's Darcy's law), the multiplicative factor involved being the wall's permeability. This macroscopic parameter is what we are usually after.
If the wall was made of a single homogeneous, isotropic, material,
there would be no problem: just measure, in laboratory conditions, the
permeability of a sample, hence the result by simple scaling, or at worst,
by solving a simple boundary-value problem. But real walls are not homogeneous,
if considered at a small enough scale, nor are they isotropic, as a rule.
On the other hand, we may measure and know the permeability of each of
the wall's constituents, and know how these are arranged. In many cases
of interest, the structure of the macroscopic porous medium can faithfully
be described as a periodic arrangement, crystal like, of different
well-characterized homogeneous materials, and we can describe the structure
of the "periodicity cell" the way numerical analysts do, as a set union
of domains, each of them with its own, uniform, permeability.
Homogenization theory then takes over. Each of the fields of interest (pressure, velocity, etc.) is the superposition of a large-scale, slowly varying one, plus a quasi-periodic meso-scale fluctuation, whose spatial periods are those of the underlying lattice. The goal of homogenization is to "filter out" these fluctuations. Hence a homogenized Darcy's law, that can be used in large-scale numerical simulations at a reasonable cost, whereas a detailed computation, which would have to take account of the fine structure of each cell, would be overly expensive.
As theory shows, one can achieve this goal by solving a particular system of PDEs on the mesoscale cell, called the "cell problem". The document gives heuristic justifications about the form of this problem: The idea of "filtering out" is realized as the minimization of some energy-like functional under constraints which model both the periodicity of the cell-scale fluctuations and the trends in the large-scale components of the fields, in the spirit of [TZB]. The result is a 3 x 3 matrix of permeabilities, symmetric, which can be diagonalized by a suitable change of axes. Hence three "homogenized permeabilities", not equal as a rule, one for each principal direction.
The procedure, of course, implies a systematic error, in addition to the error inherent in any numerical simulation. Now, suppose it's critical not to underestimate, at any cost, the flow through the wall. Can we make certain to err on the side of security?
The document contributes to this goal by proving a conjecture
earlier formulated by A. Trykozko and W. Zijl: There is a way, based on
the time-honored "hypercircle" theory by Synge [Sy],
to give bilateral bounds for the above principal permeabilities.
[Sy] J. L. Synge: The Hypercircle in Mathematical
Physics, Cambridge U.P. (Cambridge), 1957. Return