A Permeability Theory for Multiscale Porous Media

We can extend the two-scale HA theory to a multiscale porous medium, which is schematically shown in Fig. 8.21 where 2/ is the fluid phase, 2 is the solid phase, sf is the mixture phase of fluid and solid (i.e., a mixture of stacks of beidellite and fluid) in the meso-domain, F / is the solid/fluid interface, and F/ is the periodic boundary for the fluid phase. 

The three scales are as follows the clay mineral is the microscale medium, which has a thickness of 10 m and a width of about 10 m the quartz grain, which constitutes the mesoscale, has dimensions of 10 -10 m the triaxial specimen used in the experiments has dimensions 10 m. Considering these scales, the parameter s for a smectitic bentonite is about 10 . Then we can introduce the following relations between the coordinate systems  

These relations are approximate however, if the size of each domain is widely different, we need to introduce another form. The size of a unit cell of the meso-domain is jA = and the size of a unit cell of the micro-domain is AT.  

Since we introduce the multiscale systems jc , x and x, the original coordinates should be changed to 

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