# Permeability Relationships

The permeability relationships can be classified in two groups: static and dynamic. The static correlations have been derived using the properties of various porous materials that have not been subjected to formation damage processes. The dynamic correlations or models consider porous media undergoing alteration due to rock-fluid interactions during formation damage and, therefore, are preferred for formation damage prediction. In the following, selected models pertaining to formation damage are reviewed and presented with some modifications for consistency and applications in the formation damage prediction.

## Contents

## The Carman-Kozeny Hydraulic T\ibes Model

The hydraulic tubes model was derived based on the analogy between the flow of fluid through porous media and parallel flow through a bundle of tortuous capillary tubes (Carman-Kozeny, 1938). The number, diameter, and the tortuous length of the hydraulic tubes are denoted by n, Dh, and Lh, respectively. The porosity, specific pore or grain surface, and length of the porous media are (|>, Z, and L. Vp and Vb denote the pore and bulk volumes, respectively. The tortuosity of porous media is expressed as the ratio of the actual tortuous tube length to the length of porous media: where Dg is the grain diameter. Next consider that the laminar flows through porous media and the bundle of tortuous tubes can be described by the

Darcy and the Hagen- Poiseuille laws given, respectively, as:

A" is the intrinsic permeability of porous media. The cross-sectional area of porous media open for flow can be expressed by:

Therefore, equating Eqs. 5-9 and 10, and substituting Eqs. 5-1 and 10 results in the following relationship for the mean hydraulic tube diameter:

Bourbie et al. (1986) determined that n = 1 for ()><0.05 and n = 3 for 0.08 <(j)<0.25. In view of this evidence and Eq. 5-14, the Carman-Kozeny equation appears to be valid for the 0.08 < <)) < 0.25 fractional porosity range. Reis and Acock (1994) warn that these exponents may be low "because the permeabilities were not corrected for the Klinkenberg effect."

## The Modified Carman-Kozeny Equation Incorporating the Flow Units Concept

Based on the Carman-Kozeny model, Eq. 5-14, Adin's (1978) correlation of experimental data leads to a permeability-porosity model as:

where oc and n are some empirical parameters. Arshad's equation accounts for the formation of the dead-end pores during deposition, which do not conduct fluids.

## The Flow Efficiency Concept

Rajani (1988) concluded that permeability function can be separated into and expressed as a product of a function incorporating the pore geometry and a function of porosity as:

This approach is particularly useful in porous media undergoing alteration during formation damage. Frequently, the Carman-Kozeny equation fails to represent the cases where the pore throats are plugged without significant porosity reduction. This problem can be alleviated by introducing a flow efficiency factor, y, in view of Eq. 5-19 (Ohen and Civan, 1993; Chang and Civan, 1991, 1992, 1997). Hence, the permeability variation can be expressed by (Chang and Civan, 1997):

where a, b, and c are some empirically determined parameters and K0 and §0 denote the permeability and porosity at some initial or reference state. The flow efficiency factor, y, can be interpreted as a measure of the fraction of the open pore throats allowing fluid flow. Thus, when the pore throats are plugged, then y = 0, and therefore K = 0, even if <|) * 0. This phenomenon is referred to as the "gate or valve effect" of the pore throats (Chang and Civan, 1997; Ochi and Vernoux, 1998). In order to estimate the flow efficiency factor, Ohen and Civan (1993) assumed that, although the pore throat sizes vary with time, they always remain log-normally distributed:

in the range of dl<y<dh, where sd is the standard deviation and dt is the mean pore throat diameter. Then, assuming that the pore throats smaller than the size, dp, of the suspended particles will be plugged, the flow efficiency factor is estimated by the fraction of pores remaining open at a given time:

where Ep is the plugging efficiency factor. Particles that are sticky and deformable can mold into the shape of pore throats and seal them. Then, the plugging is highly efficient and Ep is close to unity. Particles that are rigid and nonsticky cannot seal the pore throats effectively and still allow for some fluid flow. Thus, E < 1 for such plugs.

The lower and upper bounds of the pore throat size range are estimated by a simultaneous solution of the non-linear integral equations given by:

for which the mean pore throat size is estimated by solving the following equation which relates the pore throat size variation to the rate of eposition:

where k6 is a rate constant and ep is the volume of deposition per unit bulk volume, subject to the initial mean pore throat diameter, either etermined from the initial pore throat size distribution using Eq. 5-28, or estimated as a fraction of the mean pore diameter using:

Note that r\ is not a fraction because it is a lumped coefficient including the mentioned fraction, some unit conversion factors, and the shape factor. Chang and Civan (1991, 1992, 1997) considered that the pore throat and particle diameters can be better represented by bimodal distribution functions over finite diameter ranges, given by Popplewell et al. (1989) as:

where w is an adjustable weighting factor in the range of 0 < w < 1, and ./i(y) and/2(j) denote the distribution functions for the fine and coarse fractions, each of which are described by:

with different values of the parameters a, ra, dt, and dh. Chang and Civan (1991, 1992, 1997) used the critical particle diameter, \dp] , necessary for pore throat jamming, determined according to the criteria. For applications with multiphase flow systems, Liu and Civan (1993, 1994, 1995, 1996) used a simplified empirical equation for permeability reduction in porous media as:

## References

1 Adin, A., "Prediction of Granular Water Filter Performance for Optimum Design," Filtration and Separation, Vol. 15, No. 1, 1978, p. 55-60.

2 Adler, P. M., Jacquin, C. G., & Quiblier, J. A., "Flow in Simulated Porous Media," Int. J. Multiphase Flow, Vol. 16, No. 4, 1990, pp. 691-712.

3 Amaefule, J. O., Altunbay, M., Tiab, D., Kersey, D. G., & Keelan, D. K., "Enhanced Reservoir Description: Using Core and Log Data to Identify Hydraulic (Flow) Units and Predict Permeability in Uncored Intervals/Wells," SPE 26436, Proceedings of the 68th Annual Technical Conference and Exhibition of the SPE held in Houston, TX, October 3-6, 1993, pp. 205-220.

4 Arshad, S. A., "A Study of Surfactant Precipitation in Porous Media with Applications in Surfactant-Assisted Enhanced Oil Recovery Processes," Ph.D. Dissertation, University of Oklahoma, 1991, 285 p.

5. Bear, J., Dynamics of Fluids in Porous Media, American Elsevier Publ. Co., Inc., New York, New York, 1972, 764 p.

6. Bhat, S. K., & Kovscek, A. R., "Statistical Network Theory of Silica Deposition and Dissolution in Diatomite," In-Situ, Vol. 23, No. 1, 1999, pp. 21-53.

7. Bourbie, T., Coussy, O., & Zinszner, B., Acoustique des Milieux Poreux, Technip, Paris, 1986.

8. Bustin, R. M., "Importance of Fabric and Composition on the Stress Sensitivity of Permeability in Some Coals, Northern Sydney Basin, Australia: Relevance to Coalbed Methane Exploitation, AAPG Bulletin (November 1997) Vol. 81, No. 11, 1894-1908.

9. Carman, P. C, "The Determination of the Specific Surfaces of Powders. I," July 1938, pp. 225-234.

10. Carman, P. C., Flow of Gases Through Porous Media, Butterworths, London, 1956.

11. Cernansky, A., & Siroky, R., "Deep-bed Filtration on Filament Layers on Particle Polydispersed in Liquids," Chemicky Prumysl, Vol. 32 (57), No. 8, pp. 397-405, 1982.

12. 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.

13. Chang, F. F., & Civan, F., "Predictability of Formation Damage by Modeling Chemical and Mechanical Processes," SPE 23793 paper, Proceedings of the SPE International Symposium on Formation Damage Control, February 26-27, 1992, Lafayette, Louisiana, pp. 293-312.

14. Chang, F. F., & Civan, F., "Modeling of Formation Damage due to Physical and Chemical Interactions between Fluids and Reservoir Rocks," SPE 22856 paper, Proceedings of the 66th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers, October 6-9, 1991, Dallas, Texas.

15. Chang, F. F., & Civan, F., "Practical Model for Chemically Induced Formation Damage," /. of Petroleum Science and Engineering, Vol. 17, No. 1/2, February 1997, pp. 123-137.

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

17. 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.

18. 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.

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

20. Civan, R, "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.

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

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

23. Civan, R, "Practical Model for Compressive Cake Piltration Including Fine Particle Invasion," AIChE J. (November 1998) 44, No. 11, 2388-2398.

24. Civan, R, "Predictability of Porosity and Permeability Alterations by Geochemical and Geomechanical Rock and Pluid Interactions," Paper SPE 58746, Proceedings of the SPE International Symposium on Formation Damage held in Lafayette, Louisiana, 23-24 February 2000.

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

26. Gdanski, R. D., & Shuchart, C. E., "Advanced Sandstone-Acidizing Designs with Improved Radial Models," SPE Production & Facilities Journal, November 1998, pp. 272-278.

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

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

29. Gupta, A., and Civan, F, "Temperature Sensitivity of Formation Damage in Petroleum Reservoirs," paper SPE 27368, Proceedings of the 1994 SPE Formation Damage Control Symposium, Lafayette, Louisiana, February 9-10, 1994, 301-328.

30. Hearn, C. L., Ebanks Jr., W. J., Tye, R. S., & Ranganathan, V., "Geological Factors Influencing Reservoir Performance of the Hartzog Draw Field, Wyoming," Journal of Petroleum Technology, Vol. 36, No. 9, August 1984, pp. 1335-1344.

31. Hearn, C. L., Hobson, J. P., & Fowler, M. L., "Reservoir Characterization for Simulation, Hartzog Draw Field, Wyoming," In: Reservoir haracterization, pp. 341-372, L. W. Lake & H. B. Carroll, Jr. (Eds.), Academic Press, Inc., Orlando, Florida, 1986, 659 p.

32. Jacquin, C. G., "Correlation Entre la Permeabilite et les Caracteristiques Geometriques du Gres de Fontainebleau," Revue Inst. Fr Petrole, Vol. 19, 1964, pp. 921-937.

33. Koh, C. J., Dagenais, P. C., Larson, D. C., and Murer, A. S., "Permeability Damage in Diatomite Due to In-Situ Silica Dissolution/Precipitation," paper SPE/DOE 35394, proceedings of the 1996 SPE/DOE Tenth Symposium on Improved Oil Recovery held in Tulsa, Oklahoma, April 21-24, 1996, 511-517.

34. Kozeny, J., "Uber Kapillare Leitung des Wasser im Boden," Sitzungsber, Akad. Wiss. Wien, No. 136, 1927, pp. 271-106.

35. 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.

36. Liu, X., & Civan, K, "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.

37. Liu, X., & Civan, F., "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 No. 28980, SPE International Symposium on Oilfield Chemistry, February 14-17, 1995, San Antonio, TX.

38. 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.

39. 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.

40. Liu, X., Ormond, A., Bartko, K., Li, Y., & Ortoleva, P., "A Geochemical Reaction-Transport Simulator for Matrix Acidizing Analysis and Design," Jour, of Petroleum Science and Engineering, Vol. 17, No. 1/2, February 1997, pp. 181-196.

41. Nieto, J. A., Yale, D. P., and Evans, R. J., "Improved Methods for Correcting Core Porosity to Reservoir Conditions," The Log Analyst (May-June 1994) 21-30.

42. Nolen, G., Amaefule, J. O., Kersey, D. G., Ross, R., & Rubio, R., "Problems Associated with Permeability and Vclay Models from Textural Properties of Unconsolidated Reservoir Rocks," SCA 9225 paper, 33rd Annual Symposium of SPWLA Society of Core Analysts, Oklahoma City, Oklahoma, June 15-17, 1992.

43. Ochi, J., and Vernoux, J.-F, "Permeability Decrease in Sandstone Reservoirs by Fluid Injection-Hydrodynamic and Chemical Effects," J. of ydrology (1998) 208, 237-248.

44. Okoye, C. U., Onuba, N. L., Ghalambor, A., and Hayatdavoudi, A., "Characterization of Formation Damage in Heavy Oil Formation During Steam Injection," paper SPE 19417, presented at the 1990 SPE Formation Damage Control Symposium, Lafayette, Louisiana, February 22-23, 1990.

45. 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.

46. Popplewell, L. M., Campanella, O. H., & Peleg, M., "Simulation of Bimodal Size Distributions in Aggregation and Disintegration Processes," Chem. Eng. Progr., August 1989, pp. 56-62.

47. Rajani, B. B., "A Simple Model for Describing Variation of Permeability with Porosity for Unconsolidated Sands," In Situ, Vol. 12, No. 3, 1988, pp. 209-226.

48. Rege, S. D., & Fogler, H. S., "Network Model for Straining Dominated Particle Entrapment in Porous Media," Chemical Engineering Science, Vol. 42, No. 7, 1987, pp. 1553-1564.

49. Rege, S. D., & Fogler, H. S., "A Network Model for Deep Bed Filtration of Solid Particles and Emulsion Drops," AIChE J., Vol. 34, No. 11, 1988, pp. 1761-1772.

50. Reis, J. C., & Acock, A. M., "Permeability Reduction Models for the Precipitation of Inorganic Solids in Berea Sandstone," In Situ, Vol. 18, No. 3, 1994, pp. 347-368.

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

52. Sharma, M. M., & Y. C. Yortsos, "A Network Model for Deep Bed Filtration Processes", AIChE J., Vol. 33, No. 10, 1987, pp. 1644-1653.

53. Verlaan, M. L., Dijkgraaf, H. K., and van Kruijsdijk, C.P.J.W., "Effect of a Wetting Immobile Phase on Diffusion and Macroscopic Dispersion in Unconsolidated Porous Media," paper SPE 56417, presented at the 1999 SPE Annual Technical Conference and Exhibition held in Houston, Texas, October 3-6, 1999, 13 p.

54. Tien, C., Bai, R., & Ramarao, B. V, "Analysis of Cake Growth in Cake Filtration: Effect of Fine Particle Retention," AIChE J., Vol. 43, No. 1, January 1997 pp. 33-44.