# Ordinary Differential Equations

In this section, several examples are given to illustrate the numerical solution of ordinary differential equation models. Specifically, the simplified formation damage and filtration models, developed in previous articles, are solved.

## Contents

## Example 1: Wojtanowicz et al. Fines Migration Model

a. Derive a numerical solution for the following modified Wojtanowicz etal. (1987, 1988) fines migration model

b. Plot c and o versus t using the following data until ɸ = 0 :

A = 1cm2, L = 1cm, ɸ0 =0.20, q = 0.5cm3/min, Cin =0.85gr/cm3, p = 1.00 grl cm3, P= 1.00grlcm3, kr= 0.7mm-1, ke=0.2mm-1

Expanding Eq. 16-1 and then substituting Eqs. 16-2 and 3 and rearranging yields

A simultaneous solution of Eqs. 16-2 and 5 as a function of time, subject to the initial conditions given by Eq. 16-4, can be readily obtained using
an appropriate method, such as by the Runge-Kutta-Fehlberg four (five) method available in many ordinary differential equation solving software
(IMSL, 1987, for example). Then, the porosity variation is calculated by Eq. 16-3. A typical numerical solution is presented in Figure 16-1.

## Example 2: Ceriiansky and Siroky Fines Migration Model

The numerical solution is carried out for T^. =0. Here, the numerical solution approach presented by Cernansky and Siroky (1985) is described. Define the dimensionless time and distance, respectively, by:

Thus, invoking Eqs. 16-6 and 7, Eqs. 10-84 and 91, respectively, become:

Eqs. 16-8 and 9 are a system of hyperbolic partial differential equations, which can be transformed into a system of ordinary differential equations
by means of the method of characteristics as:

Applying the condition given by Eq. 16-15, Eq. 16-10 becomes:

The system of ordinary differential equations given by Eqs. 16-10 and 11 are solved by means of the fourth-order Runge-Kutta method, subject
to the conditions given by Eqs. 16-15 and 16 along the characteristic represented by Eq. 16-14. Figure 16-2 shows the dimensionless effluent particles concentration as a function of the cumulative volume injected per unit area. Figures 16-3 and 16-4 show typical suspended particle concentration and the particles retained in porous media as a function of distance along the porous media at different times.

## Example 3: Civan's Incompressive Cake Filtration without Fines Invasion Model

The equations of Civan's (1998, 1999) incompressive cake filtration model are given in this article. The ordinary differential equations of this model have been solved by the Runge-Kutta- Fehlberg four (five) numerical scheme (Fehlberg, 1969), subject to the initial condition given by Eq. 12-14.

## Example 4: Civan's Compressive Cake Filtration Including Fines Invasion Model

The equations of Civan's (1998, 1999) compressive cake filtration including fines invasion model are given in this article. As described in

The ordinary differential equations of this model have been solved by the Runge-Kutta-Fehlberg four (five) numerical scheme (Fehlberg, 1969), subject to the initial condition given by Eq. 12-14.

## References

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.

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.

Burnett, D. S., Finite Element Analysis, Addison-Wesley Publishing Company, Massachusetts, 1987, 844 p.

Cernansky, A., & Siroky, R., "Hlbkova Filtracia Polydisperznych Castic z Kvapalin na Vrstvach z Vlakien," Chemicky Prumysl, Vol. 32 (57), No. 8, 1982, pp. 397-405.

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.

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.

Civan, F., A Generalized Model for Formation Damage by Rock-Fluid Interactions and Particulate Processes, SPE 21183, Proceedings of SPE 1990 Latin American Petroleum Engineering Conference, Rio de Janeiro, Brazil, October 14-19, 1990, 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, "Numerical Simulation by the Quadrature and Cubature Methods," SPE 28703 paper, Proceedings of the SPE International Petroleum Conference and Exhibition of Mexico, October 10-13, 1994, Veracruz, Mexico, pp. 353-363.

Civan, F.,"Solving Multivariable Mathematical Models by the Quadrature and Cubature Methods," Journal of Numerical Methods for Partial Differential Equations, Vol. 10, 1994, pp. 545-567.

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

Civan, F. /'Practical Implementation of the Finite Analytic Method," Applied Mathematical Modeling, Vol. 19, No. 5, 1995, pp. 298-306.

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

Civan, F. "A Time-Space Solution Approach for Simulation of Flow in Subsurface Reservoirs," Turkish Oil and Gas Journal, Vol. 2, No. 2, June 1996, pp. 13-19.

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

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

Civan, F., "Quadrature Solution for Waterflooding of Naturally Fractured Reservoirs," SPE Reservoir Evaluation & Engineering J., April 1998, pp. 141-147.

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.

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

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

Escobar, F. H., Jongkittinarukorn, K., & Civan, F, "Cubature Solution of the Poisson Equation," Communications in Numerical Methods in Engineering, Vol. 13, 1997, pp. 453-465.

Fehlberg, E., "Low-Order Classical Runge-Kutta Formulas with Stepsize Control and their Application to Some Heat Transfer Problems," NASA TR R-315, Huntsville, Alabama, July 1969.

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

IMSL—FORTRAN Subroutines for Mathematical Applications IMSL Inc., Houston, Texas, Version 1.0, April 1987.

Malik, M., & Civan, F., "A Comparative Study of Differential Quadrature and Cubature Methods Vis-A-Vis Some Conventional Techniques in Context of Convection-Diffusion-Reaction Problems," Chemical Engineering Science, Vol. 50, No. 3, 1995, pp. 531-547.

Ohen, H. A., & Civan, R, Simulation of Formation Damage in Petroleum Reservoirs, SPE 19420 paper, Proceedings of the SPE 1990 Symposium on Formation Damage Control, February 22-23, 1990, Lafayette, Louisiana.

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

Thomas, G. W., Principles of Hydrocarbon Reservoir Simulation, International Human Resources Development Corporation, Publishers, Boston, 1982, 207 p.

Wojtanowicz, A. K., Krilov, Z., & Langlinais, J. P., "Experimental Determination of Formation Damage Pore Blocking Mechanisms," Trans. oftheASME, Journal of Energy Resources Technology, Vol. 110, 1988, pp. 34-42.

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