Page 152 - Reservoir Formation Damage

134 Reservoir Formation Damage

The species i mass balance equations for the water, oil, gas and solid
phases are given by:

+ V • + V • = (7-34)

in whch w tj is the mass fraction of species i in the j th phase, jy denotes
the spontaneous or dispersive mass flux of species i in the j th phase given
by modifying the equation by Olson and Litton (1992):

A,' Vw, + -2- A; • VO. + Y —^- D s, • Vw . (7-35)
JtT •^ w

where D i} is the coefficient of dispersion of species i in the j th phase, k
is the Boltzmann constant, and T is temperature. The first term represents
the ordinary dispersive transport by concentration gradient. For particulate
species of relatively large sizes the first term may be neglected. The
second term represents the dispersion induced by the gradient of the
potential interaction energy, <E> (y. When the particles are subjected to
uniform interaction potential field then the second term drops out. The
third term represents the induced dispersion of bacterial species by
substrate or nutrient, 5, concentration gradient due to the chemotaxis
phenomena (Chang et al., 1992). D sj is the substrate dispersion coefficient.
Incorporating Eq. 7-33 into Eq. 7-34 leads to the following alternative
form:

+ V • = (7-36)

Adding Eq. 7-34 over all the phases gives the total species / mass
balance equation as:

:
y (7-37)

Considering the possibility of the inertial flow effects due to the
narrowing of pores by formation damage, the Forchheimer (1901) equation
is used for the momentum balance. Although more elaborate forms of the
macroscopic equation of motion are available, Blick and Civan (1988)
have shown that Forchheimer's equation is satisfactory for all practical