Diffusion Problem for Multiscale Porous Media

Bentonite is a composite material with micron sized grains, composed mainly of quartz, and nanometer sized clay minerals as described previously. We present here a multiscale HA theory for diffusion in such a composite porous medium. 

We consider a material that has macro-/meso-/mlcro-structures as shown in Fig. 8.21, where Q/ and are the domains of the fluid phase and solid phase, respectively, Fj/ is the solid/fluid interface, F/ is the periodic boundary of the fluid phase, and is the mixture domain of smectite stacks and fluid in the meso-domain. 

Let a , and be coordinate systems introduced in the macro-/meso-/micro-domains, respectively. We consider a triaxial test on the typical bentonite of Japan, Kunigel VI , using a specimen of about 10 m in diameter, which is the macrodomain, where the size of the quartz grains are about e = which is the meso-domain, and the size of clay stacks is about 10 m. Then we can set e=10 , and introduce the following relations  

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