Model Validation, Refinement, and Parameter Estimation
As stated by Civan (1994), confidence in the model cannot be established without validating it by experimental data. However, the microscopic phenomena is too complex to study each detail individually. Thus, a practical method is to test the system for various conditions to generate its input-output response data. Then, determine the model parameters such that model predictions match the actual measurements within an acceptable tolerance. However, some parameters may be directly measurable. A general block diagram for parameter identification and model development and verification is given in this article (Civan, 1994).
Contents
Experimental System
The experimental system is a reservoir core sample subjected to fluid flow. The input variables are injection flow rate or pressure differential and its particles concentration, temperature, pressure, pH, etc. The output
variables are the measured pressure differential, pH, and species concentration of the effluent. Frequently, data filtering and smoothing are required to remove noise from the data as indicated in this article. However, some important information may be lost in the process. Millan-Arcia and Civan (1990) reported that frequent breakage of particle bridges at the pore throat may cause temporary permeability improvements, which are real and not just a noise. Baghdikian et al. (1989) reported that accumulation and flushing of particle floccules can cause an oscillatory behavior during permeability damage.
Parity Equations
An integration of the model equations over the length of core yields the equations of a macroscopic model called the "parity equations." However, for a complicated model of rock-fluid-particle interactions in geological porous formations it is impractical to carry out such an integration analytically. Hence, an appropriate numerical method, such as described in this article, is facilitated to generate the model response (pressure differential across the core or sectional pressure differentials, and the effluent conditions) for a range of input conditions (i.e., the conditions of the influent, confining stress, temperature, pressure, pH, etc.).
Parameter Estimation with Linearized Models
Luckert (1994) points out that estimating parameters using linearized model equations obtained by transformation is subject to uncertainties and errors because of the errors introduced by numerical transformation of the experimental data. Especially, numerical differentiation is prone to larger errors than numerical integration. Luckert (1994) explains this problem on the determination of the parameters K and q of the following filtration model:
Thus, a straightline plot of Eq. 17-41 using a least-squares fit provides the values of log K and q as the intercept and slope of this line, respectively.
Luckert (1994) points out, however, this approach leads to highly uncertain results because numerical differentiation of the experimental data involves some errors, second differentiation involves more errors than the first derivative, and numerical calculation of logarithms of the second
numerical derivatives introduce further errors.
Therefore, Luckert (1994) recommends linearization only for preliminary parameter estimation, when the linearization requires numerical processing of experimental data for differentiation. Luckert (1994) states that "From a statistical point of view, experimental values should not be transformed in order that the error distribution remains unchanged." Using the parameter values determined this way, Civan (1998) has shown that the model predictions compared well with the measured filtrate volumes and cake thicknesses.
History Matching for Parameter Identification
The model equations contain various parameters dealing with the rate equations. They are determined by a procedure similar to history matching commonly used in reservoir simulation. In this method, an objective function is defined as
where Ym is the measured values, Yc is the calculated values, W is the weighting matrix, W = V"1, V is the variance-covariance matrix of the
measurement error, and n is the number of data points. Szucs and Civan (1996) facilitated an alternative formulation of the objective function,
called the p-form, which lessens the effect of the outliers in the measured data points. Then, a suitable method is used for minimizing the objective function to obtain the best estimates of the unknown model parameters (Ohen and Civan, 1990, 1993).
Note that the number of measurements should be equal or greater than the number of unknown parameters. When there are less measurements than the unknown parameters, additional data can be generated by interpolation between the existing data points. However, for meaningful estimates of the model parameters the range of the data points should cover a sufficiently long test period to reflect the effect of the governing formation damage mechanisms. The above described method has difficulties. First, it may require a lot of effort to converge on the best estimates of the parameter values. Second, there is no guarantee concerning the uniqueness of the parameter values determined with nonlinear models.
However, some parameters can be eliminated for less important mechanisms for a given formation and fluid system. The remaining parameters are determined by a history matching procedure. In this method, the best estimates of the unknown parameters are determined in such a way that the model predictions match the measurements obtained by laboratory testing of cores within a reasonable accuracy (Civan, 1994).
Frequently used optimization methods are:
1 trial-and-error [tedious and time consuming];
2 Levenberg-Marquardt (Marquardt, 1963) method [requires derivative evaluation, computationally intensive];
3 simulated annealing [algebraic and practical (Szucs and Civan, 1996; Ucan et al., 1997)].
Detailed examples of the history matching method by Ohen and Civan (1990, 1993) have been presented in this article.
Sensitivity Analysis
Once the model is developed and verified with experimental data, studies of the sensitivity of the model predictions with respect to the assumptions, considerations, and parameters of the model can be conducted. Consequently, the factors having negligible effects can be determined and the model can be simplified accordingly. Models simplified this way are preferred for routine, specific applications. The solid line represents the best fit of the experimental data using the best estimates of the model parameters obtained by history matching. When the parameter values were
perturbed in a random manner, a significantly different trend, represented by the dashed line, was obtained.
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 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.
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.
