Assuming incompressible species, the volumetric balance of species j transported via phase J through porous media is given by:

where Ԑy indicates the volume fraction of phase J in porous media, αjj is the volume fraction of species j in phase J, uj is the volumetric flux of phase J through porous media and qjJL represents the volume rate of transfer of species j from phase J to phase L. Djj denotes the coefficient of dispersion of species j in phase Pj and pj is the density of phase J, which varies by its composition even if the individual constituent species may be considered incompressible, jc and t denote the distance along the flow direction and time. The dispersion term for particles is usually neglected. The volumetric rate of particle lost per unit bulk media by various processes is given by:

in which qju denotes the volume rate of transformation of species j type to species ʅ type in phase J expressed per unit bulk volume. Summing Eq. 11-1 over all species j in phase J and considering that the dispersion terms of various species j cancel each other out in a given phase, the volumetric equation of continuity for phase J is obtained as:

in which the volumetric loss of all types of particles from phase J is given by:

Finally, by summing Eq. 11-3 for all phases Ј, the total equation of continuity for the multi-phase fluid system is obtained as:

where the total volumetric flux and all types of particles lost by the multiphase fluid system are given, respectively, by:

Considering the possibility of the generation of inertial effects by rapid flow due to the narrowing of pores during formation damage, the volumetric flux of phase J is represented by the non-Darcy flow equation given by:

where θ is the angle of inclination of the flow path and PJ and |iy are the pressure and viscosity of phase J. krj is the relative permeability of phase J and K is the permeability of porous media. NndJ is the phase J non-Darcy number given according to the Forchheimer equation as:

in which Rej is the phase J Reynolds number given by (Ucan and Civan, 1996):

where β is the inertial flow coefficient given by a suitable correlation, such as by Liu et al. (1995).

Determination of Fluid Saturations and Pressures

Two alternative convenient formulations can be taken for solution of the equations of continuity and motion given by Eqs. 11-3 and 8 for pressures and saturations of the various phases flowing through porous media. In the first approach, Eq. 11-8 is substituted into Eq. 11-3 to obtain:

The capillary pressure is defined as the difference between the nonwetting and wetting phase pressures according to:

Thus, substituting Eqs. 11-12 and 11-13 into Eq. 11-11 yields the following equations for the wetting and nonwetting phases, respectively:

Note that the saturations add up to one:

Therefore, adding Eqs. 11-14 and 11-15 yields the following equation:

where the total volumetric loss of particles from the two-phase system is given by:

Eqs. 11-14 and 17 can be solved simultaneously to determine the wetting phase pressure and saturation, pw and Sw. A second and more convenient approach facilitates the fractional flow formulation. This is especially suitable for incompressible systems described by the equation of continuity given by Eq. 11-3. Civan's (1996) formulation based on Richardson's (1961) formulation can be modified as the following to include the loss terms and inertia flow affect in the saturation equation:

for which the capillary and gravity dispersion coefficients are given, respectively, by:

The zero capillary pressure and zero gravity fractional flow term is given by:

In the fractional flow formulation, the saturation of the wetting phase is calculated by solving Eq. 11-19. But the pressure of the wetting phase is still determined by solving Eq. 11-17. As explained by Civan (1996), the solution of equations presented above requires the capillary pressure and relative permeability data for the two-phase system. These data continuously vary during formation damage and empirical models, such as those given in this article, are required to incorporate these affects in the solution. This problem can be alleviated in a practical manner by resorting to an end-point mobility ratio formulation similar to Civan (1996) and Luan (1995), by extending and generalizing the unit mobility ratio formulations given by Craig (1971), Collins (1961) and Dake (1978). In view of the uncertainties in determining the exact nature of the variations of these data, it is reasonable to make the following assumptions. First, similar to Liu and Civan (1996), the capillary pressure affect can be neglected. Second, the relative permeabilities can be approximated by linear relationships with respect to the phase saturations as (Yokoyama and Lake, 1981):

Consequently, Eq. 11-19 can be simplified significantly by substituting Eqs. 11-25 and 11-26. In addition, the non-Darcy effect can be neglected by substituting

The end-point relative permeabilities and fluid densities may be replaced by average values as:

As a result of substituting Eqs. 11-27 and 11-28, Eq. 11-5 can be simplified as (Civan, 1996):

Determination of Species Concentrations in Various Phases

Once the phase saturations are determined, then the species concentrations can be determined by solving the following equation obtained by combining Eqs. 11-1 and 11-3:

The dispersion term is considered for dissolved species, such as those contained in the aqueous phase, but it is usually neglected for the particles. In accordance with the experimental observations by Muecke (1979), Liu and Civan (1993, 1995, 1996) have assumed that wettable particles remain in the wetting phase and nonwettable particles remain in the nonwetting phase and the intermediately wet particles are situated along the interface. They did not consider the possibility of wettability alteration of the particles and the pore surface in porous media and they assumed that the dispersion terms are negligible for the particles.

They considered that the porous media has uniform wetting properties. Under these circumstances, Eq. 11-31 simplifies significantly because qju = 0 and the particle loss only occurs from the fluid phases to the solid matrix Liu and Civan (1996) considered a water/oil system flowing through a homogeneous (i.e., one type—either water-wet or oil-wet—porous media). They assumed that the wettability of the porous medium does not change during the short period of time involving the typical laboratory core tests.


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