# Sensitivity Analysis—Stability and Conditionality

Sensitivity analysis is an important tool for systematic evaluation of mathematical models (Lehr et al., 1994). Sensitivity analysis can be use for various purposes, including model validation, evaluating model behavior, estimating model uncertainties, decision making using uncertain models, and determining potential areas of research (Lehr et al., 1994). Sensitivity analysis provides information about the effect of the errors and/or variations in the variables and/or parameters and models on the predicted behavior. Sensitivity of a model to changes in its input data determines the condition of the model (Chapra and Canale, 1998). The sensitivity of a system's outcome or response to changes in a variable is defined by the partial derivative (Lehr et al., 1994):

Relative sensitivity (Lehr et al., 1994) or the condition number (Chapra and Canale, 1998) is defined as the ratio of the relative change or error in the function to the relative change or error in the variable or parameter value. Thus, for a single parameter function, the relative sensitivity can be expressed by means of Lehr et al., 1994; (Chapra and Canale, 1998):

Thus, the condition number or relative sensitivity can be used as a criteria to evaluate the effect of an uncertainty in the x variable on the condition of a system as (Chapra and Canale, 1998):

Given the differential equations of a model, the sensitivity equations can be formulated for determining the sensitivity trajectory. The following
example by Lehr et al. (1994) illustrates the process. Consider a mathematical model given by an ordinary differential equation as:

A differentiation of the g(f,x,t) function with respect to the variable (or parameters) x leads to:

Substituting Eqs. 17-34 and 37 into Eq. 17-38 and rearranging yield the following sensitivity trajectory equation:

One of the practical applications of the sensitivity analysis is to determine the critical parameters, which strongly effect the predictions of models (Lehr et al., 1994). Lehr et al. (1994) studied the sensitivity of an oil spill evaporation model.

Lehr et al. (1994) depict the sensitivity of the fractional oil evaporation,f, from an oil spill with respect to the initial bubble point, TB, and the rate of bubble point variation by the fraction of oil evaporated (the slope of the evaporation curve), TG =d2TB/dtdf, respectively.

## References

Amaefule, J. O., Kersey, D. G., Norman, D. L., & Shannon, P. M., "Advances in Formation Damage Assessment and Control Strategies", CIM 88-39-65 paper, 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.

Baghdikian, S. Y., Sharma, M. M., & Handy, L. L., Flow of Clay Suspensions Through Porous Media, SPE Reservoir Engineering, Vol. 4., No. 2. , May 1989, pp. 213-220.

Bethke, C. M., Geochemical Reaction Modeling, Concepts and Application, Oxford University Press, New York, 1996, 397 p.

Bu, T., & Damsleth, E., "Errors and Uncertainties in Reservoir Performance Predictions," SPE Formation Evaluation, September 1996, pp. 194-200.

Chang, F. F., & Civan, F., "Modeling of Formation Damage due to Physical and Chemical Interactions between Fluid 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.

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.

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

Chapra, S. C., & Canale, R. P., Numerical Methods for Engineers, 3rd ed., McGraw-Hill, Inc., 1998, Boston, 924 p.

Civan, F., "Review of Methods for Measurement of Natural Gas Specific Gravity," SPE 19073 paper, Proceedings of the SPE Gas Technology Symposium, June 7-9, 1989, Dallas, Texas, pp. 173-186.

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

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

Cook, A. R., Introduction to Engineering, ENGR 1113 Class Notes, Civil Engineering and Environmental Science, University of Oklahoma, 1980.

Demir, I., "Formation Water Chemistry and Modeling of Fluid-Rock Interaction for Improved Oil Recovery in Aux Vases and Cypress Formations," Department of Natural Resources, Illinois State Geological Survey, Illinois Petroleum Series 148, 1995, 60 p.

Demir, I. and Seyler, B., "Chemical Composition and Geological History of Saline Waters in Aux Vases and Cypress Formations, Illinois Basin," Aquatic Geochemistry, Vol. 5, pp. 281-311, 1999

Duda, J. L., "A Random Walk in Porous Media," Chemical Engineering Education Journal, Summer 1990, pp. 136-144.

Frenklach, M., & Miller, D. L., "Statistically Rigorous Parameter Estimation in Dynamic Modeling Using Approximate Empirical Model," AIChE Journal, Vol. 31, No. 3, March 1985, pp. 498-500.

Gadiyar, B., & Civan, R, "Acidization Induced Formation Damage—Experimental and Modeling Studies," SPE 27400 paper, Proceedings of the 1994 SPE Formation Damage Control Symposium, February 9-10, 1994, Lafayette, Louisiana, pp. 549-560.

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

Haggerty, D. J., & Seyler, B., "Investigation of Formation Damage from Mud Cleanout Acids and Injection Waters in Aux Vases Sandstone Reservoirs," Department of Natural Resources, Illinois State Geological Survey, Illinois Petroleum Series 152, 1997, 40 p.

ISGS Oil and Gas Section, "Improved and Enhanced Oil Recovery Through Reservoir Characterization: Standard Operating and QA/QC Procedures," Illinois State Geological Survey, Open File Series 1993-13.

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

Ku, H. K., "Precision Measurement and Calibration," National Bureau of Standards, Special Publication 300, Vol. 1, 1969, Washington, pp. 331-341.

Leetaru, H. E., "Application of Old Electrical Logs in the Analysis of Aux Vases Sandstone (Mississippian) Reservoirs in Illinois," Illinois State Geological Survey, Illinois Petroleum Series 134, 1990, 21 p.

Lehr, W., Calhoun, D., Jones, R., Lewandowski, A., & Overstreet, R., "Model Sensitivity Analysis in Environmental Emergency Management: A Case Study in Oil Spill Modeling," Proceedings of the 1994 Winter Simulation Conference, J. D. Tew, S. Manivannan, D. A. Sadowski, and A. F. Seila (eds.), 1994, pp. 1198-1205.

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 28980 paper, 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.

Luckert, K., "Model Selection Based on Analysis of Residue Dispersion, Using Solid-Liquid Filtration as an Example," International Chemical Engineering, Vol. 34, No. 2, April 1994, pp. 213-224.

Marquardt, D. W., "An Algorithm for Least Squares Estimation of Nonlinear Parameters," SIAM J. Appl. Math., Vol. 11, 1963, pp. 431-441.

Mickley, H. S., Sherwood, T. K., & Reed, C. E., Applied Mathematics in Chemical Engineering, 1957, McGraw-Hill, New York, pp. 49-52.

Millan-Arcia, E., & Civan, F. "Characterization of Formation Damage by Paniculate Processes," J. Canadian Petroleum Technology, Vol. 31, No. 3, March 1992, pp. 27-33.

Miller, D., & Frenklach, M., "Sensitivity Analysis and Parameter Estimation in Dynamic Modeling of Chemical Kinetics," International Journal of Chemical Kinetics, Vol. 15, 1983, pp. 677-696.

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

Reilly, P. M., "A Statistical Look at Significant Figures," Chemical Engineering Education, Summer 1992, pp. 152-155.

Schenck, H., Jr., Theories of Engineering Experimentation, 1961, McGraw-Hill, New York.

Schlumberger Log Interpretation Charts for 1989, Schlumberger Education Services, 1989, Houston, Texas, 150 p.

Seyler, B., "Geologic and Engineering Controls on Aux Vases Sandstone Reservoirs in Zeigler Field, Illinois—A Comprehensive Study of a Well-Managed Oil Field," Illinois Petroleum Series 153, 1998, Department of Natural Resources, Illinois State Geological Survey, 79 p.

Sharma, M. M., & Yortsos, Y. C., "Fines Migration in Porous Media," AIChE J., Vol. 33, No. 10, 1987, pp. 1654-1662.

Spiegel, M. R., Theory and Problems of Statistics, 1961, Schaum Publ. Co., New York, p. 71.

Szucs, P., & Civan, F., "Multi-Layer Well Log Interpretation Using the Simulated Annealing Method," J. Petroleum Science and Engineering, Vol. 14, Nos. 3/4, May 1996.

Ucan, S., Civan, F., & Evans, R. D., "Uniqueness and Simultaneous Predictability of Relative Permeability and Capillary Pressure by Discrete and Continuous Means," J. of Canadian Petroleum Technology, Vol. 36, No. 4, pp. 52-61, 1997.

Vitthal, S., Sharma, M. M., & Sepehrnoori, K., "A One-Dimensional Formation Damage Simulator for Damage Due to Fines Migration," SPE 17146 paper, Proceedings of the SPE Formation Damage Control Symposium, Bakersfield, California, February 8-9, 1988, pp. 29-42.

Willhite, G. P., Green, D. W., Thiele, J. L., McCool, C. S., & Mertes, K. B., "Gelled Polymer Systems for Permeability Modification in Petroleum Reservoirs, Final Report," Contract No. DE-FG07-89 ID 12846, U.S. Department of Energy, Bartlesville, Oklahoma, September 1991.

Ziauddin, M., Berndt, O., & Robert, J., "An Improved Sandstone Acidizing Model: The Importance of Secondary and Tertiary Reactions," SPE 54728 paper, Proceedings of the 1999 SPE European Formation Damage Conference, May 31-June 1, 1999, The Hague, The Netherlands, pp. 225-237.