APPENDIX A

 Darcian Flow:

 

In the fluid mechanics of porous media, the place of momentum equations of force balances is occupied by the numerous experimental observations summarized mathematically as the “Darcy Law”.  The observations were first reported by Darcy who, based on measurement alone, discovered that the area averaged fluid velocity through a column of porous material is proportional to the pressure gradient established along the column. Subsequent experiments proved that the area-averaged velocity is, in addition, inversely proportional to the viscosity () of the fluid seeping through the porous material.

 

So one can write:

                                     (A-1)

 

where K is an empirical constant called permeability. The dimensions of K must be

 

                     (A-2) 

 

Darcy flow is the macroscopic manifestation of a highly viscous flow through the pores of the permeable structure, and  is a length scale representative of the effective pore diameter. Ergun (1952) proposed

                                 (A-3)

 

as a correlation for the measured permeabilities of columns of packed spheres of diameter d and porosity .

 

In the presence of a body force per unit volume  the Darcy Law (A-1) becomes

 

                           (A-4)

 

acknowledging the fact that the flow through the porous column stops when the externally controlled pressure gradient dP/dx matches the hydrostatic gradient .