Near wellbore mud filtrate and fines invasion during drilling operations and the resulting formation damage and filtercake formation are amongst the most important problems involving the petroleum reservoir exploitation. This chapter reviews the fundamental processes and their mathematical formulation necessary to develop models that can be used for assessment of the damaged zone, filtrate and fines concentrations, fluid saturations, and the filtercake thickness and permeability alteration during drilling. The effects of under and over balance drilling on near wellbore formation damage are discussed. The models for simulation of the single and two-phase flow situations in the formation with water or oil based drilling mud cases are described. External particle invasion prior to filtercake buildup and its effect on the formation damage by particle invasion and retention and filtercake formation are described. These models are demonstrated by various applications. The models presented here can be used for accurate estimation of the near wellbore fluid saturations and resistivity profiles, which are necessary for accurate welllog interpretation.

## Simplified Single Phase Mud Filtrate Invasion Model

Similar to Donaldson and Chernoglazov (1987), Civan and Engler (1994) assumed that the mud filtrate mixes with the reservoir fluid and the salt concentration varies. This model implicitly assumes a piston type immiscible displacement of oil similar to the formulations by Collins (1961) and Olarewaju (1990).

Thus, the fluid zone can be viewed in two parts: the water phase and oil phase zones behind and ahead of the displacement front, located at a distance, re(t). In this case, the front moves with time. The formulation is also applicable when the mud filtrate can mix with the reservoir fluid (i.e., of the same wetting type). The filtrate mixture is considered as a pseudo-component. The filtrate mass balance is given by:

The boundary conditions at the wellbore and the moving front are given, respectively, by:

in which the filtrate invasion rate is assumed to follow an empirically determined exponential decay law according to Donaldson and Chernoglazov (1987):

The dispersion coefficient is expressed as power-law function of the volume flux (Donaldson and Chernoglazov, 1987):

Next, three dimensionless groups are defined for computational convenience and scaling purposes. The dimensionless concentration is defined as:

The dimensionless time can be defined based on the dispersion or convection time scales, respectively, as (Civan, 1994):

Because the process is mainly convection dominated, we use Eq. 18-11. The porous media peclet number is expressed as:

where «0 and D0 are some characteristic values that are the maximum values of u and D, determined as following. Note that the filtration rate varies in a range of

where it can be shown by means of Eqs. 18-5, 6, and 11 that:

Therefore, Eqs. 18-1 through 4 can be transformed into a set of dimensionless equations, respectively, as:

Finally, substituting Eqs. 18-18 and 22 into Eqs. 18-23 through 26 and dropping the subscript "D" for dimensionless quantities, Eqs. 18-23 through 26, respectively, become:

In Eq. 18-27, the parameters a and (3 are given by:

Notice that Eq. 18-27 is linear, because a and (3 do not depend on the concentration, c.The exterior radius, re(t), of the invaded region can be determined from the following volumetric balance:

Civan and Engler (1994) considered the mixing of the mud filtrate with the resident fluid within a fixed, but sufficiently long, radial distance (re = constant] and obtained a numerical solution of the model using the Crank-Nicholson finite difference scheme as described in this article.

## References

Bennion, D. B., Thomas, F. B., Bennion, D. W., & Bietz, R. E, "Underbalanced Drilling and Formation Damage—Is It a total solution?" J. Canadian Petroleum Technology, Vol. 34, No. 9, November 1995, pp. 34-41.

Bilardo, U., Alimonti, C., Chiarabelli, A., & Caetani, F. C., "Formation Water Saturation from Drilling Fluid Filtrate Invasion: Comparison of Displacement Modelling and Induction Well Log Response,"J. Petroleum Science and Engineering, Vol. 15, Nos. 2/4, August 1996, pp. 251-259.

Briscoe, B. J., Luckham, P. F., & Ren, S. R., "The Properties of Drilling Muds at High Pressures and Temperatures," Phil. Trans. R. Soc. London. A., Vol. 348, 1994, pp. 179-207.

Chin, W. C., Formation Invasion, Gulf Publishing Co., Houston, TX, 1995, 240 p.

Civan, F., & Engler, T., "Drilling Mud Filtrate Invasion—Improved Model and Solution," /. of Petroleum Science and Engineering, Vol. 11, pp.183-193, 1994.

Civan, F., "Rapid and Accurate Solution of Reactor Models by the Quadrature Method," Computers & Chemical Engineering, Vol. 18. No. 10, 1994, pp. 1005-1009.

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., "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., "Practical Model for Compressive Cake Filtration Including Fine Particle Invasion," AlChE J., Vol. 44, No. 11, November 1998, pp. 2388-2398.

Civan, F., "Incompressive Cake Filtration: Mechanism, Parameters, and Modeling,", AIChE J., Vol. 44, No. 11, November 1998, pp. 2379-2387.

Civan, F., "Phenomenological Filtration Model for Highly Compressible Filter Cakes Involving Non-Darcy Flow," SPE 52147 paper, Proceedings of the 1999 SPE Mid-Continent Operations Symposium, March 28-31, 1999, Oklahoma City, Oklahoma, pp. 195-201.

Civan, F., "Predictive Model for Filter Cake Buildup and Filtrate Invasion with Non-Darcy Effects," SPE 52149 paper, Proceedings of the 1999 SPE Mid-Continent Operations Symposium, March 28-31, 1999, Oklahoma City, Oklahoma, pp. 203-210.

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

Donaldson, E. C., & V. Chernoglazov, "Drilling Mud Fluid Invasion Model", J. Pet. Sci. Eng., Vol. 1, No. 1, 1987, pp. 3-13.

Liu, X., & Civan, F., "A Multi-Phase Mud Fluid Infiltration and Filter Cake Formation Model," SPE 25215 paper, Proceedings, SPE International Symposium on Oilfield Chemistry, February 28-March 3, 1993, New Orleans, Louisiana, pp. 607-621.

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, Feb. 9-10, 1994, Lafayette, Louisiana, pp. 259-274.

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., "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.

Olarewaju, J. S., "A Mathematical Model of Permeability Alteration Around Wells," Intl. J. for Numerical and Analytical Methods in Geomechanics, Vol. 14, 1990, pp. 191-207.

Phelps, G. D., "Computation of Mud Filtrate Invasion Profiles," J. Canadian Petroleum Technology, Vol. 34, No. 1, January 1995, pp. 18-27.

Ramakrishnan, T. S., & Wason, D. T., "Effect of Capillary Number on the Relative Permeability Function for Two-Phase Flow in Porous Media," Powder Technology Journal, Vol. 48, 1986, pp. 99-124.

Ramakrishnan, T. S., & Wilkinson, D. J., "Formation Producibility and Fractional Flow Curves from Radial Resistivity Variation Caused by Drilling Fluid Invasion," Phys. Fluids, Vol. 9, No. 4, April 1997, pp. 833-844.

Ramakrishnan, T. S., & Wilkinson, D. J., "Water-Cut and Fractional-Flow Logs from Array-Induction Measurements," SPE Reservoir Evaluation and Engineering Journal, Vol. 2, No. 1, February 1999, pp. 85-94.

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

Simpson, J. P., "Drilling Fluid Filtration Under Simulated Downhole Conditions," SPE Paper 4779, 1974.

Yao, C. Y, & Holditch, S. A., "Reservoir Permeability Estimation from Time-Lapse Log Data," SPE Paper 25513, Proceedings of the Production Operations Sym. Held in Oklahoma City, OK, March 21-23, 1993, pp. 963-975.

Yao, C. Y, & Holditch, S. A., "Reservoir Permeability Estimation from Time-Lapse Log Data," SPE Formation Evaluation, June 1996, pp. 69-74.