# Fluid and Species Transport

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.

## References

Civan, F., "A Generalized Model for Formation Damage by Rock-Fluid Interactions and Particulate Processes," SPE Paper 21183, Proceedings of the SPE 1990 Latin American Petroleum Engineering Conference, October 14-19, 1990, Rio de Janeiro, Brazil, 11 p.

Civan, F., Predictability of Formation Damage: An Assessment Study and Generalized Models, Final Report, U.S. DOE Contract No. DE-AC22- 90-BC14658, April 1994.

Civan, F., "A Multi-Phase Mud Filtrate Invasion and Well Bore Filter Cake Formation Model," SPE Paper No. 28709, Proceedings of the SPE International Petroleum Conference & Exhibition of Mexico, October 10-13, 1994, Veracruz, Mexico, pp. 399-412.

Civan, F., "A Multi-Purpose Formation Damage Model," SPE 31101 paper, Proceedings of the SPE Formation Damage Symposium, Lafayette, Louisiana, February 14-15, 1996, pp. 311-326.

Civan, F., "Convenient Formulations for Immiscible Displacement in Porous Media," SPE Paper 36701, Proceedings of the 71st SPE Annual Tech. Conf. and Exhibition, Denver, Colorado, October 6-9, 1996, pp. 223-236.

Collins, E. R., Flow of Fluids Through Porous Materials, Penn Well Publishing Co., Tulsa, Oklahoma, 1961, 270 p.

Craig, F. F., Jr., The Reservoir Engineering Aspects of Waterflooding, Third Printing, November 1980, Society of Petroleum Engineers of AIME, New York, 1971, 134 p.

Dake, L. P., Fundamentals of Reservoir Engineering, Elsevier Scientific Publ. Co., New York, 1978, 443 p.

Eleri, O. O., & Ursin, J-R., "Physical Aspects of Formation Damage in Linear Flooding Experiments," SPE 23784 paper, presented at the SPE Intl. Symposium on Formation Damage Control, Lafayette, Louisiana, February 26-27, 1992.

Gruesbeck, C., & Collins, R. E., "Particle Transport Through Perforations," SPEJ, December 1982, pp. 857-865.

Gruesbeck, C., & Collins, R. E., "Entrainment and Deposition of Fine Particles in Porous Media," SPEJ, December 1982, pp. 847-856.

Jiao, D., & Sharma, M. M., "Formation Damage Due to Static and Dynamic Filtration of Water-Based Muds," SPE 23823 paper, presented at the SPE Intl. Symposium on Formation Damage Control, Lafayette, Louisiana, February 26-27, 1992.

Khilar, K. C., & Fogler, H. S., "Water Sensitivity of Sandstones," SPEJ, February 1983, pp. 55-64.

Ku, C-A., & Henry, Jr., J. D., "Mechanisms of Particle Transfer from a Continuous Oil to a Dispersed Water Phase, J. Colloid and Interface ScL, 1987, Vol. 116, No. 2, pp. 414-422.

Liu, X., & Civan, F., "Characterization and Prediction of Formation Damage in Two-Phase Flow Systems, SPE 25429 paper, Proceedings of the SPE Production Operations Symposium, March 21-23, 1993, Oklahoma City, Oklahoma, March 21-23, 1993, pp. 231-248.

Liu, X., & Civan, F, "Formation Damage and Skin Factors Due to Filter Cake Formation and Fines Migration in the Near-Wellbore Region," SPE 27364 paper, Proceedings of the 1994 SPE Formation Damage Control Symposium, February 9-10, 1994, Lafayette, Louisiana, pp. 259-274.

Liu, X., & Civan, E, "Formation Damage by Fines Migration Including Effects of Filter Cake, Pore Compressibility and Non-Darcy Flow—A Modeling Approach to Scaling from Core to Field," SPE Paper #28980, SPE International Symposium on Oilfield Chemistry, February 14-17, 1995, San Antonio, TX.

Liu, X., & Civan, F., "Formation Damage and Filter Cake Buildup in Laboratory Core Tests: Modeling and Model-Assisted Analysis," SPE Formation Evaluation J., Vol. 11, No. 1, March 1996, pp. 26-30.

Liu, X., Civan, F, & Evans, R. D., "Correlation of the Non-Darcy Flow Coefficient, J. of Canadian Petroleum Technology, Vol. 34, No. 10, 1995, pp. 50-54.

Luan, Z., "Splitting Pseudospectral Algorithm for Parallel Simulation of Naturally Fractured Reservoirs," SPE Paper 30723, Proceedings of the Annual Tech. Conf. & Exhibition held in Dallas, TX, October 22-25.

Muecke, T. W., "Formation Fines and Factors Controlling their Movement in Porous Media," JPT, pp. 147-150, Feb. 1979.

Peng, S. J., & Peden, J. M., "Prediction of Filtration Under Dynamic Conditions," paper SPE 23824 presented at the SPE Intl. Symposium on Formation Damage Control held in Lafayette, LA, February 26-27, 1992, pp. 503-510.

Rahman, S. S., & Marx, C., "Laboratory Evaluation of Formation Damage Caused by Drilling Fluids and Cement Slurry," J. Can. Pet. Tech., November-December, 1991, pp. 40-46.

Richardson, J. G., "Flow Through Porous Media," In: V. L. Streeter (Editor), Handbook of Fluid Dynamics, Section 16, McGraw-Hill, New York, 1961, pp. 68-69.

Sarkar, A. K., "An Experimental Investigation of Fines Migration in Two-Phase Flow," MS Thesis, U. of Texas, Austin, 1988.

Sarkar, A. K., & Sharma, M. M., "Fines Migration in Two-Phase Flow," JPT, May 1990, pp. 646-652.

Sutton, G. D., & Roberts, L. D., "Paraffin Precipitation During Fracture Stimulation," JPT, September 1974, pp. 997-1004.

Ucan, S., & Civan, R, "Simultaneous Estimation of Relative Permeability and Capillary Pressure for Non-Darcy Flow-Steady-State," SPE Paper 35271, Proceedings of the 1996 SPE Mid-Continent Gas Symposium, Amarillo, TX, April 29-30, 1996, pp. 155-163.

Yokoyama, Y, & Lake, L. W., "The Effects of Capillary Pressure on Immiscible Displacements in Stratified Porous Media," SPE 10109 paper, presented at the 56th Annual Fall Technical Conference and Exhibition of the Society of Petroleum Engineers of AIME, San Antonio, TX, October 5-7, 1981.