# Simplified Analytic Model for Asphaltene-Induced Formation Damage in Single-Phase

Leontaritis (1998) developed a simplified model for prediction of formation damage and productivity decline by asphaltene deposition in under saturated (above bubble-point pressure) asphaltenic oil reservoirs. This model consists of a set of algebraic equations. In this section, the Leontaritis model is presented with some modifications for consistency with the rest of the presentation of this artcle. As schematically shown in this article, for analysis, Leontaritis (1998) considers the portion of the reservoir defined by the radius of drainage of a production well. In this region, the flow is assumed radial. Schematically depicts the variation of the flowing bottom hole pressure during constant rate production, while the external reservoir pressure and the onset of the asphaltene flocculation pressure remain constant. The calculational steps of this model are described briefly in the following.

Step 1. The initiation time for asphaltene precipitation is referred to as zero (i.e., t = 0). Given the well productivity index, PI, the flowing bottom hole pressure, pw [=Q, prior to asphaltene damage is calculated from the definition of the productivity index:

Then, the steady-state radial pressure profile prior to damage is calculated by:

The asphaltene deposition is assumed to occur within the near wellbore region, rw<r<rAF, where the pressure is below the asphaltene flocculation pressure, pAF. The radius of this region, rAF, is determined by Eq. 14-16 for p = pAF. Leontaritis (1998) assumes that the pressure beyond this region (i.e., rAF ≤r≤re) is not influenced by asphaltene deposition in the near wellbore region. The region rw ≤ r ≤ rAF is divided into a number of sections of finite width Δr. Steps 2 and 3 calculations are carried out over each Δr segment for a time increment by Δt, consecutively, as described in the following.

Step 2. Similar to Wojtanowicz et al. (1987, 1988), Leontaritis considers the porous media as a bundle of tortuous flow tubes. Thus, the mean hydraulic diameter is estimated by the ratio of the total pore volume to the total pore surface area of the flow channels according to:

where A, L, and ɸ denote the cross-sectional area, length, and porosity of a core plug, and Ag and Vg are the mean surface area and the mean volume of the porous media grains. If the mean, spherical grain diameter is denoted by d , then Eq. 14-17 can be expressed as:

Next, the tube size distribution function, f(DA), the mole fraction, XA, and molar volume, VA , of the flocculated asphaltenes, and the moles of reservoir fluid, mRF, at the prevailing pressure and temperature conditions within the near wellbore region are determined according to Leontaritis
(1997).

Leontaritis (1998) assumes that permeability impairment primarily occurs by pore throat plugging and generalizes the one-third rule-of-thumb of filtration as the particles larger than a certain fraction of the pore size cannot penetrate a filter, and determines the fraction of the particles, ftrap,
which are captured and deposited at the pore throats. Thus, the rule-ofthumb for trapment of particles at the pore throats is generalized to estimate the critical particle diameter for plugging as a fraction of the hydraulic tube diameter as:

in which a is as an empirical factor accounting for the smallest particle that can be filtered. Its value is determined to match the model predictions
to measured data, instead of setting it to a prescribed value, like 1/3. Then, the fraction of the asphaltene particles that cannot pass through the pore throats and, therefore, are captured at the pore throats is determined by:

Step 3. The incremental moles of asphaltene particles trapped and the incremental flow area closed within the A? time interval are estimated, respectively, by:

where aA is the specific surface of the asphaltene particles retainedin porous media, estimated as:

where DA is the mean diameter of the asphaltene particles retained, p is an empirical factor accounting for the plugging by asphaltene particles. Therefore, combining Eqs. 14-20 through 23 over a number of N consecutive, discrete time steps, Af, the cumulative flow area closed to flow by pore throat plugging is estimated by:

Hence, the area open to flow during damage is given by:

According to Wojtanowicz et al. (1987, 1988), the area open to flow during formation damage by pore throat plugging is linearly related to permeability:

Therefore, the instantaneous permeability is given by:

in which the productivity index is defined by:

Hence, combining Eqs. 14-25 and 30-32 yields the following expression for the productivity ratio:

Note that Eq. 14-33 is different than the equation given by Leontaritis (1998) because Eq. 14-33 is squared. The pressure loss by damage is calculated by Eqs. 14-31 through 33 as:

Step 4. When Steps 2 and 3 over all the Δr segments are completed, the pressure loss by skin and the skin factor are calculated as following. Note that the drawdown pressure is given during damage as:

where s is the van Everdingen-Hurst skin factor. Thus, the loss of the pressure by the skin effect is given by:

Consequently, comparing Eqs. 14-16 and 36 in view of Eq. 14-37 yields:

Once the pressure loss by skin is calculated by Eq. 14-38, the skin factor can then be calculated by Eq. 14-37.

Step 5. Another time increment, Δt, is taken and Steps 2-4 are repeated until either the final time considered for the calculation is reached or the flow rate of production can no longer be kept constant, which is the condition imposed for the above described model. Leontaritis considers that a steady-state is attained when the deposition and erosion rates equal. Then, the asphaltene deposition stops and the area open to flow attains a certain minimum limit value. Because of the lack of a better asphaltene erosion theory, Leontaritis assumes that the area of flow can be empirically expressed as some fraction of the initial area. His equation can be expressed in terms of Eq. 14-31 as:

Based on Eqs. 14-40 and 41, it can be concluded that 0≤a≤l and 0≤b≤l. However, there is no clear evidence of the use of Eqs. 14-39 through 41 in his calculational procedure.

## References

Acevedo, S., Ranaudo, M. A., Escobar, G., Gutierrez, L., & Ortega, P., "Adsorption of Asphaltenes and Resins on Organic and Inorganic Substrates and Their Correlation with Precipitation Problems in Production Well Tubing," Fuel, Vol. 74, No. 4, 1995, pp. 595-598.

Ali, M. A., & Islam, M. R., "The Effect of Asphaltene Precipitation on Carbonate-Rock Permeability: An Experimental and Numerical Approach," SPE Paper No. 38856, Proceedings of the 1997 SPE Annual Conference and Exhibition, held in San Antonio, Texas, October 1997, pp. 139-146.

All, M. A., & Islam, M. R., "The Effect of Asphaltene Precipitation on Carbonate-Rock Permeability: An Experimental and Numerical Approach," SPE Production and Facilities Journal, August 1998, pp. 178-183.

Amaefule, J. O., Kersey, D. G., Norman, D. L., & Shannon, P. M., "Advances in Formation Damage Assessment and Control Strategies," CIM Paper No. 88-39-65, Proceedings of the 39th Annual Technical Meeting of Petroleum Society of CIM and Canadian Gas Processors Association, June 12-16, 1988, Calgary, Alberta, 16 p.

Andersen, S. I., Keul, A., & Stenby, E., "Variation in Composition of Subfractions of Petroleum Asphaltenes," Petroleum Science and Technology, Vol. 15, No. 7 & 8, 1997, pp. 611-645.

Chang, F. F., & Civan, F., "Practical Model for Chemically Induced Formation Damage," J. of Petroleum Science and Engineering, Vol. 17, 1997, pp. 123-137.

Chang, C.-L., & Fogler, H. S., "Peptization and Coagulation of Asphaltenes in Apolar Media Using Oil-Soluble Polymers," Fuel Science and Technology International, Vol. 14, No. 1 & 2, 1996, pp. 75-100.

Chang, C.-L., & Fogler, H. S., Langmuir, Vol. 10, 1994, pp. 1749-1758.

Chung, T.-H., "Thermodynamic Modeling for Organic Solid Precipitation," SPE 24851, Proceedings of the 67th Annual technical Conference and Exhibition of the SPE held in Washington, D.C., October 4-7, 1992, pp. 869-878,

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.

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

Civan, F., "A Multi-Phase Mud Filtrate Invasion and Well Bore Filter Cake Formation Model," SPE 28709 paper, Proceedings of the SPE International Petroleum Conference & Exhibition of Mexico, October 10-13, 1994, Veracruz, Mexico, pp. 399-412.

Civan, F., "Modeling and Simulation of Formation Damage by Organic Deposition," Proceedings of the First International Symposium on Colloid Chemistry in Oil Production: Asphaltene 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 paper, the SPE Formation Damage Control Symposium, Lafayette, Louisiana, February 14-15, 1996, pp. 311-326.

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

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

DeBoer, R. B., Leerlooyer, K., Eigner, M. R. P., & van Bergen, A. R. D., "Screening of Crude Oils for Asphalt Precipitation: Theory, Practice, and the Selection of Inhibitors," SPEPF, February 1995, pp. 55-61.

Dubey, S. T, & Waxman, M. H., "Asphaltene Adsorption and Desorption From Mineral Surfaces," SPE Reservoir Engineering, August 1991, pp. 389-395.

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

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

Gupta, S. P., & Greenkorn, R. A., "Dispersion During Flow in Porous Media with Bilinear Adsorption," Water Resources Research, Vol. 9, 1973, pp. 1357-1368.

Haskett, C. E., & Tartera, M., "A Practical Solution to the Problem of Asphaltene Deposits-Hassi Messaoud Field, Algeria," JPT, April 1965, pp. 387-391.

Houchin, L. R., & Hudson, L. M., "The Prediction, Evaluation and Treatment of Organic Damage Caused by Organic Deposition," SPE 14818 paper, Proceedings of the Seventh SPE Symposium on Formation Damage Control, February 26-27, 1986, Lafayette, Louisiana, pp. 83-90.

Khalil, C. N., Rocha, N. O., & Silva, E. B., "Detection of Formation Damage Associated to Paraffin in Reservoirs of the Reconcavo Baiano, Brazil," Proceedings of the 1997 SPE International Symposium on Oil Field Chemistry held in Houston, Texas, February 18-21, pp. 277-281.

Leontaritis, K. J., "Application of a Thermodynamic-Colloidal Model of Asphaltene Flocculation," presented at the Symposium of Solids Deposition, Area 16C of Fuels and Petrochemical Division, AIChE Spring National Meeting and Petroleum Exposition, March 28-April 1, 1993, Houston, Texas.

Leontaritis, K. J., "The Asphaltene and Wax Deposition Envelopes," Fuel Science and Technology International, Vol. 14, No. 1 & 2, Marcel Dekker, Inc., New York, 1996, pp. 13-39.

Leontaritis, K. J., "PARA-Based (Paraffin-Aromatic-Resin-Asphaltene) Reservoir Oil Characterizations," SPE Paper 37252, Proceedings of the 1997 SPE International Symposium on Oilfield Chemistry held in Houston, Texas, February 18-21, 1997, pp. 421-440.

Leontaritis, K. J., "Asphaltene Near-Wellbore Formation Damage Modeling," SPE Paper 39446, Proceedings of the 1998 SPE Formation Damage Control Conference held in Lafayette, Louisiana, February 18-19, 1998, pp. 277-288.

Leontaritis, K. J., & Mansoori, G. A., "Asphaltene Flocculation During Oil Production and Processing: A Thermodynamic-Colloidal Model," SPE Paper 16258, Proceedings of the SPE International Symposium on Oil Field Chemistry, San Antonio, Texas, January 1987, pp. 149-158.

Leontaritis, K. J., Amaefule, J. O., and Charles, R. E., "A Systematic Approach for the Prevention and Treatment of Formation Damage Caused by Asphaltene Deposition," SPE Paper 23810, Proceedings of the SPE International Symposium on Formation Damage Control, Lafayette, LA, February 26-27, 1992, pp. 383-395.

Lichaa, P. M., "Asphaltene Deposition Problem in Venezuela Crudes, Usage of Asphaltenes in Emulsion Stability," Oil Sands, June 1997, pp. 609-624.

Lira-Galeana, C., & Firoozabadi, A., "Thermodynamics of Wax Precipitation in Petroleum Mixtures," AIChE Journal, Vol. 42, No. 1, January 1996, pp. 239-248.

Manoranjan, V. S., & Stauffer, T. B., "Exact Solution for Contaminant Transport with Kinetic Langmuir Sorption," Water Resources Research, Vol. 32, No. 3, 1996, pp. 749-752.

Mansoori, G. A., "Modeling of Asphaltene and Other Heavy Organic Depositions," Journal of Petroleum Science and Engineering, Vol. 17, 1997, pp. 101-111.

Mansoori, G. A., "Modeling and Prevention of Asphaltene and Other Heavy Organic Deposition in Oil Wells," Paper SPE 27070, proceedings of the International Symposium on Formation Damage Control held in Lafayette, Louisiana, 9-10 February 1994, pp. 9-18.

Mansoori, G. A., "Modeling and Prevention of Asphaltene and Other Heavy Organic Deposition in Oil Wells," SPE 27070 paper, presented at the Third Latin American/Caribbean Petroleum Engineering Conference, April 27-29, 1994, Beunos Aires, Argentina.

Marquardt, D. W, "An Algorithm for Least-squares Estimation of Nonlinear Parameters," J. Soc. Indust. Appl. Math., Vol. 11, 1963, p. 431.

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.

Minssieux, L., "Core Damage From Crude Asphaltene Deposition" SPE 37250 paper, 1997 SPE International Symposium on Oilfield Chemistry, February 18-21, 1997, Houston, Texas.

Nghiem, L. X., & Coombe, D. A., "Modeling Asphaltene Precipitation During Primary Depletion," SPE Journal, Vol. 2, June 1997, pp. 170-176.

Philp, R. P., Bishop, A. N., Del Rio, J.-C, and Allen, J., "Characterization of High Molecular Weight Hydrocarbons (>C40) in Oils and Reservoir Rocks," in The Geochemistry of Reservoirs, Cubitt, J. M. and England, W. A. (Eds.), Geological Society Special Publication No. 86, 1995, pp. 71-85.

Ring, J. N., Wattenbarger, R. A., Keating, J. F., and Peddibhotla, S., "Simulation of Paraffin Deposition in Reservoirs," SPE Production & Facilities, February 1994, pp. 36-42.

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

Sircar, S., Novosad, J., & Myers, A. L., "Adsorption from Liquid Mixtures on Solids, Thermodynamics of Excess Properties and Their Temperature Coefficients," I&EC Fund., Vol. 11, 1972, p. 249.

Speight, J. G., "Asphaltenes in Crude Oil and Bitumen: Structure and Dispersion," Chapter 8, pp. 377-401, in Suspensions: Fundamentals and Applications in the Petroleum Industry, Schramm, L. L. (Ed.), Advances in Chemistry Series 251, American Chemical Society, 1996, Washington, DC, 800 p.

Speight, J. G., "The Chemical and Physical Structure of Petroleum: Effects on Recovery Operations," J. of Petroleum Science and Engineering, Vol. 22, Nos. 1-3, 1999, pp. 3-15.

Speight, J. G., "Solvent Effects in the Molecular Weights of Petroleum Asphaltenes," Preprints ACS, Div. Pet. Chem., pp. 825-832.

Speight, J. G., & Long, R. B., "The Concept of Asphaltenes Revisited," Fuel Science and Technology International, Vol. 14, No. 1 & 2, 1996, pp. 1-12.

Srivastava, R. K., & Huang, S. S., "Asphaltene Deposition During CO2 Flooding: A Laboratory Assessment," Paper SPE 37468, Proceedings of the 1997 SPE Productions Operations Symposium, held in Oklahoma City, Oklahoma, March 9-11, 1997, pp. 617-635.

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

Wang, S., Civan, F., & Strycker, A. R., "Simulation of Paraffin and Asphaltene Deposition in Porous Media," SPE 50746 paper, SPE International Symposium on Oilfield Chemistry, February 16-16, 1999, Houston, Texas, pp. 449-458.

Weingarten, J. S., & Euchner, J. A., "Methods for Predicting Wax Precipitation and Deposition," SPERE, February 1988, pp. 121-132.

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.

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

Yarranton, H. W., & Masliyah, J. H., "Molar Mass Distribution and Solubility Modeling of Asphaltenes," AIChE Journal, Vol. 42, No. 12, December 1996, pp. 3533-3543.

Zhou, X., Thomas, F. B., & Moore, R. G., "Modeling of Solid Precipitation from Reservoir Fluid," Journal of Canadian Petroleum Technology, Vol. 35, No. 10, December 1996, pp. 37-45.