y = IL 2-3. Flow in Porous Media: Darcy's Law A relationship used to model flow in isotropic porous media is Darcy's law, (VP Pg) ==VP where k is the Darcy permeability (units of m3, g is the gravitational acceleration, and P is the dynamic pressure (Chapter 6). The velocity and pressure in Darcy's lax are each averaged over a length scale that is large compared with the pore dimensions, but small compared with the dimen- sions of the system of interest. That is, microstructural details are ignored, but variations in v and P at a more macroscopic level are considered. The pore structure enters only through its effect on K, which is usually determined by experiment. (a) For a material of average porosity e (volume fraction of pores), use Darcy's law to obtain expressions for conservation of mass in the fluid phase, in integral form and then in differ- ential form. (b) Show that, for an incompressible fluid, the differential equation from part (a) reduces to v?P=0. (c) Consider an idealized porous material consisting of straight, cylindrical pores of diameter d. Assume that all pores are parallel to the x, y, or z axes and that they intersect at points described by a simple cubic lattice of dimension l d. The overall dimensions of a sam- ple of this material are of the order of L, where L. The volume flow rate Q in a single pore segment is described by Poiseuille's law, Q T|AP|d4 128ul where |AP| is the pressure drop per segment. Evaluate x for this material. =