Porous Medium Homogeneity and Representative Elemental Volume
We have defined earlier what is a porous medium [1]. What constitutes a homogeneous porous medium?
We are aware of the concept of homogeneity [2]. A homogeneous material has its properties invariant with location and size of the material portion from which those properties are to be measured or determined.
According to this definition, interestingly, even heterogeneous material are often classified as homogeneous because they are rendered incapable of sizing (or seeing) of their individual heterogeneity in a particular visual resolution level. The location or size of the individual heterogeneity could be small and hence negligible when compared with the size of the material sample.
The example in the figure illustrates this situation. The region in the figure can be treated as homogeneous (material) in a visual resolution level provided its size is considerably bigger than the size of the single black dot.
On the other hand, uniformly spread heterogeneity in a particular material sample may allows the heterogeneous material to be treated as a homogeneous material. A porous medium comprising of at least two homogeneous material constituents, presenting identifiable interfaces between them in a resolution level, is considered homogeneous in this sense. The following figures explain the concept.
The LHS figure is a geometric structure that satisfies the definition of a porous medium. Based on certain properties that we could measure, this porous medium can be considered homogeneous. Let us measure the volume fraction of the porous medium defined as the ratio of total volume of the pores and total volume occupied by the porous medium. For the two dimensional example porous medium the volume fraction reduces to a surface area fraction. The surface fraction can be defined as the ratio between the total area of one constituent of the porous medium (say, the white region in the figure) and the total area of the sample considered. Figure 1b illustrates a method to measure this surface fraction.
Considering the area of the sample size enclosed by the rectangle marked as 1 in the figure, we can see the surface fraction of the white region would be unity. On the other hand, if we do use region marked 2, the measurement would result in a surface fraction between one and zero. Proceeding in this manner, We could perform subsequent experiments by increasing the measurement window. The result could look something as shown in the figure.
The Φ symbol on the ordinate represents the surface or volume fraction of the sample and the abscissa represents the increasing size of the measurement window. After the initial wiggles for small values of the measurement window, as the size of the sampling area (or volume) increases, the surface (or volume) fraction settles down to a certain fixed value, within certain percentage of error. For the example porous medium considered in the earlier figure, this happens beyond a sample area size represented as 4 (arbitrarily, not to scale). Beyond this size, any further increase in the sampling size should result in a fixed value for the surface fraction. In other words, beyond the visual resolution of this sampling area, the porous medium would exhibit homogeneity in terms of the property, surface (or volume) fraction.
Before finalizing this sampling area we need to do one more check. That the surface fraction determined using a particular measurement window yields identical surface fraction (within agreed error), even when the experiment is performed at several different locations in the porous medium sample. This is illustrated in the next figure that shows a metal foam type porous medium.
In the figure, the surface (or volume) fraction measured in the sampling size marked 1 should remain invariant when tested at different locations of the larger sample. If it doesn’t satisfy the check, a larger sampling area (marked 2) should be tested as a suitable candidate.
A homogeneous porous medium is a concept based on a property of the porous medium. In the examples discussed here, the surface fraction is the utilized property. Such a definition of homogeneity is not new. It is how, for instance, the concept of continuum is defined.
The accompanying figure is reproduced from page 5 of [3]. It defines continuum using the density variation over a volume sampling for a ”continuous” medium. The analogy with what we discussed so far for a porous medium is obvious.
Observe in our example that once a critical sample size is determined based on the invariance of the surface fraction, the actual porous medium can be constructed by repeating the critical sample size endlessly. This is possible because, by virtue of the procedure we followed, the determined critical sample size and the total porous medium both should yield identical surface fraction. The sampling fraction (volume, in 3D) beyond which homogeneity can be claimed for a porous medium based on the volume fraction, is called the Representative Elemental Volume for that porous medium. Representative, because, the structure contained inside that volume represents completely the entire porous medium; Elemental because, with the help of such volume regions, we can construct the entire porous medium.
Analogous to the continuum concept defined for single constituents – as done using density in the earlier example – a homogeneous porous medium is defined over a porous continuum. In a porous continuum, every point represents not the individual constituents but the porous medium itself. A point in a porous continuum represents a finite volume – the REV – of the porous medium. The picture of the metal foam above, when viewed at a distance, would blur the interfaces between the metal matrix and the voids, resulting in a continuous indistinguishable haze. That visual resolution (or equivalently, the sample size) defines a porous continuum.
References
- http://arunn.info/porous-medium-definition.html
- http://en.wikipedia.org/wiki/Homogenous
- G. K. Batchelor, An Introduction to Fluid Dynamics, Indian reprint Amazon Link
Related posts:
- Porous Medium Modeling: Homogeneity and Representative Elemental Volume
- Notes on the Volume averaged Energy Equation for Porous Medium Flows
- What is a Porous Medium
- Flow Through Porous Media Summary
- Predicting Flow Transition in Porous Media
