Page 153 - Reservoir Formation Damage

Multi-Phase and Multi-Species Transport in Porous Media 135

purposes. The Forchheimer equation for multi-dimensional and multi-
phase fluids flow can be written for the j th phase as (Civan, 1994; Tutu
et al., 1983; Schulenberg and Miiller, 1987):

-V4*. = • u + Tl-'rr'.^+pj^F,. (7-38)

in which ¥• is the interfacial drag force, r\ rj=k rj (Liu et al., 1995),
T| = 1/P and \|/ is the flow potential given by:

(7-39)

where the first term is the fluid-content-dependent potential or simply
the negative of the "effective stress" due to the interactions of the
fluid with the pore surface, g is the gravitational acceleration, g(z-z 0)
is the potential of fluid due to gravity, z is the positive upward distance
measured from a reference at z 0 , and Q is the overburden potential, which
is the work of a vertical displacement due to the addition of fluid into
porous media (Smiles and Kirby, 1993).
K and |3 denote the Darcy or laminar permeability and the non-Darcy
or inertial flow coefficient tensors, respectively. K rj and p r; are the
relative permeability and relative inertial flow coefficient, respectively.
Eq. 7-38 can be written as, for convenience

in which v is the kinematic viscosity (or momentum diffusivity) given by

v (7-41)
j =
and N nd is the non-Darcy number for anisotropic porous media given by,
neglecting the interfacial drag force Fji

N.* - (7-42)

where / denotes a unit tensor and Re^ is the tensor Reynolds number
for flow of phase j in an anisotropic porous media given by

(7-43)