Homogenization Analysis and Seepage Problem of Porous Media

1 Homogenization Analysis and Seepage Problem of Porous Media 

Let us introduce a perturbation in terms of both the and coordinate systems for the velocity vf and the pressure p  

X gives the size of a unit cell for each direction. The perturbation of the velocity V implies that vf consists of the sum of the global change in terms of the macrodomain and the local change in terms of the micro-domain, and the conditions are the same for the pressure p. The reason why the initial order of perturbation is different between v (starting with the term e ) and (starting with the term e°) is that the order of differentiation for v is different from the order of p in the governing equation (8.1). 

By substituting (8.6) and (8.9) into the mass conservation equation (8.2), we have 

By subsituting (8.18) and (8.19) into (8.14) and (8.16), we obtain the following incompressible flow equations in the micro-domain, referred to as the microscale equations for Stokes flow. 

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