Wojtanowicz et al. (1987, 1988) have analyzed and developed diagnostic equations for two special cases of practical importance:

1 . Deposition of the externally introduced particles during the injection of a suspension of particles

2. Mobilization and subsequent deposition of the indigeneous particles of porous medium during the injection of a particle free solution

Deposition of Externally Introduced Particles

Three distinct permeability damage mechanisms are analyzed for a given injection fluid rate and particle concentration. Particles are retained mainly in the thin core section near the inlet face. In this region the concentration of the flowing phase is assumed the same as the injected fluid (i.e., pp/ - ( P p f ) . ). Gradual pore reduction by surface deposition occurs when the particles of the injected suspension are smaller than the pore constrictions. Assume that the surface deposition is the dominant mechanism compared to the entrainment, that is, kr»ke.

Then, the solution of Eqs. 10-35 and 36 yields:

A substitution of Eq. 10-37 into Eq. 10-29 leads to the following expression for the area occupied by the surface deposits

Substitution of Eqs. 10-10 and 38 into Eq. 10-28 yields the following diagnostic equation:

in which the empirical constant is given by

Single pore blocking occurs when the size of the particles in the injected fluid are comparable or bigger than the size of the pore constrictions. A substitution of Eqs. 10-12, 13, and 14 into Eq. 10-28 yields the following diagnostic equation:

Cake formation near the inlet face of the porous media occurs when the particles in the injected solutions are large relative to the pore size and at high a concentration. Combining Eqs. 10-22 and 17 yields the following diagnostic equation:



Mobilization and Subsequent Deposition of Indigeneous Particles

This case deals with the injection of a clear (particle free) solution into a porous media. A core is visualized as having two sections designated as the inlet and outlet sides. The particles of the porous media entrained by the flowing phase in the inlet part are recaptured and deposited at the outlet side of the core. Near the inlet port, the mobilization and entrainment of particles by the flowing phase is assumed to be the dominant mechanism compared to the particles retention (i.e., ke » kr}. Thus, dropping the particle retention term, Eqs. 10-35 and 36 yield the following solution for the mass of particles remaining on the pore surface

Substituting Eq. 10-45 and (pp/). =0, Eq. 10-32 yields the following expression for the particle concentration of the flowing phase passing from the inlet to the outlet side of the core as

Depending on the particle concentration and size of the flowing phase entering the core, the outlet side diagnostic equations for three permeability damage mechanisms mentioned previously are derived next.


Gradual Pore Reduction by Surface Deposition and Sweeping

Assume that the mass of the indigeneous or previously deposited particles on the pore surface is m*. Then, the area occupied by these particles is given by Eq. 10-29 as

Afg denotes the open flow area when all the deposits are removed. If simultaneous, gradual pore surface deposition and sweeping ar occurring near the outlet region, then both the entrainment and retention terms are considered equally important. Thus, substituting Eq. 10-46, Eq. 10-35 yields the following ordinary differential equation:

The solution of Eq. 10-49, subject to the initial condition mp=m*p (previously deposited particles), is obtained by the integration factor method as:

Particle.png

Then, the area occupied by the remaining particles is given by Eq. 10-29 as:

Eliminating Afo between Eqs. 10-48 and 53, substituting Eqs. 10-47 and 52 for A* and Ap, and then applying Eq. 10-10 for Af and A*f yields the following diagnostic equation:

Normally, mpg =m*p. Wojtanowicz et al. (1987) simplified Eq. 10-54 by substituting C7 = 0 when the mass of the particles initially available on the pore surface is small compared to the mass of particles deposited later (i.e., mJsO). If only pore sweeping occurs, then kr « ke. Thus, substitute kr = 0 in Eq. 10-56 to obtain C8 =0 and Eq. 10-54 becomes:


If only gradual surface deposition is taking place in the outlet region, then kr » ke . Therefore, dropping the particle retention term and substituting Eq. 10-46, Eqs. 10-35 and 36 for mpg = 0 are solved to obtain the amount of particles retained as:


Single Pore Blocking

If the permeability damage is solely due to single pore blocking, then substituting Eqs. 10-46, 12, 13 and 14 into Eq. 10-28 yields the following diagnostic equation:

Cake Formation

If the permeability damage is by cake formation, then substituting Eqs. 10-46 and 22 into Eq. 10-17 yields the following diagnostic equation

A list of the diagnostic equations derived in this section are summarized in this article for convenience.


References

Cernansky, A., & Siroky, R. "Deep-bed Filtration on Filament Layers on Particle Polydispersed in Liquids," Int. Chem. Eng., Vol. 25, No. 2, 1985, pp. 364-375.

Cernansky, A., & Siroky, R., "Hlbkova Filtracia Polydisperznych Castic z Kvapalin na Vrstvach z Vlakien," Chemicky Prumysl, Vol. 32 (57), No. 8, 1982, pp. 397-405.

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. "Evaluation and Comparison of the Formation Damage Models," SPE 23787 paper, Proceedings of the SPE International Symposium on Formation Damage Control, February 26-27, 1992, Lafayette, Louisiana, pp. 219-236.

Civan, F., & Knapp, R. M. "Effect of Clay Swelling and Fines Migration on Formation Permeability," SPE Paper No. 16235, Proceedings of the SPE Production Operations Symposium, Oklahoma City, Oklahoma, 1987, pp. 475-483.


Civan, F. "A Multi-Phase Mud Filtrate Invasion and WellBore 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., Knapp, R. M., & Ohen, H. A. "Alteration of Permeability by Fine Particle Processes," J. Petroleum Science and Engineering, Vol. 3, Nos. 1/2, October 1989, pp. 65-79.

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

Civan, F. "Modeling and Simulation of Formation Damage by Organic Deposition," Proceedings of the First International Symposium on Colloid Chemistry in Oil Production: Asphaltenes and Wax Deposition, ISCOP'95, Rio de Janeiro, Brazil, November 26-29, 1995, pp. 102-107.

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

Civan, F. "Interactions of the Horizontal Wellbore Hydraulics and Formation Damage," SPE 35213, Proceedings of the SPE Permian Basin Oil & Gas Recovery Conf., Midland, Texas, March 27-29, 1996, pp. 561-569.

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

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

Khilar, K. C., & Fogler, H. S. "Colloidally Induced Fines Migration in Porous Media," in Amundson, N. R. & Luss, D. (Eds.), Reviews in Chemical Engineering, Freund Publishing House LTD., London, England, January-June 1987, Vol. 4, Nos. 1 and 2, pp. 41-108.

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

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.

Metzner, A. B., & Reed, J. C. "Flow of Non-Newtonian Fluids—Correlation of the Laminar, Transition, and Turbulent Flow Regions," AIChE J., Vol. 1, No. 4, 1955, pp. 434-440.

Nayak, N. V, & Christensen, R. W. "Swelling Characteristics of Compacted Expansive Soils," Clay and Clay Mineral, Vol. 19, No. 4, December 1970, pp. 251-261.

Ohen, H. A., & Civan, F. "Predicting Fines Generation, Migration and Deposition Near Injection and Production Wells," Proceedings of the First Regional Meeting, American Filtration Society, Houston, Texas, October 30-November 1, 1989, pp. 161-164.

Ohen, H. A., & Civan, F. "Simulation of Formation Damage in Petroleum Reservoirs," SPE Advanced Technology Series, Vol. 1, No. 1, April 1993, pp. 27-35.

Ohen, H. A., & Civan, F. "Simulation of Formation Damage in Petroleum Reservoirs," SPE 19420 paper, Proceedings of the 1990 SPE Symposium on Formation Damage Control, Lafayette, Louisiana, February 22-23, 1990, pp. 185-200.

Schechter, R. S., Oil Well Stimulation, Prentice Hall, Englewood Cliffs, New Jersey, 1992, 602 p.

Seed, H. B., Woodward, Jr., R. J., & Lundgren, R. "Prediction of Swelling Potential for Compacted Clays," /. Soil Mech. Found. Div., Proc. Am. Soc. Civ. Eng., 88(SM3), June 1962, pp. 53-87.

Wojtanowicz, A. K., Krilov, Z., & Langlinais, J. P. "Study on the Effect of Pore Blocking Mechanisms on Formation Damage," SPE 16233 paper, presented at Society of Petroleum Engineers Production Operations Symposium, Oklahoma City, Oklahoma, March 8-10, 1987, pp. 449-463.

Wojtanowicz, A. K., Krilov, Z., & Langlinais, J. P. "Experimental Determination of Formation Damage Pore Blocking Mechanisms," Trans, of the ASME, Journal of Energy Resources Technology, Vol. 110, 1988, pp. 34-42.