In spite of many experimental studies of the formation damage of oil and gas bearing formations, there have been only a few reported attempts to mathematically model the relevant processes and develop formation damage simulators (Civan, 1990, 1992, 1994, 1996). The use of these models in actual reservoir analysis and management has been rather limited because of the difficulties in understanding and implementing these models, as well as due to the limitations in the applicability of these models. Most present formation damage models consider a single fluid phase and the dominant formation damage mechanism is assumed to be the mobilization, migration, and retention of fine particles in porous matrix. Although, these models have been validated using experimental data obtained from reservoir core samples under controlled laboratory conditions, their applicability is rather limited in the field conditions. Most formation damage cases encountered in actual reservoirs are associated with multiphase flow and other factors which are not considered in the present single phase formation damage models. In addition, determination of the model parameters have not been well addressed.

Formation damage refers to permeability impairment by alteration of porous media due to rock-fluid and fluid-fluid interactions in geological porous formations. The phenomena leading to formation damage is a rather complicated process involving mechanical, physical, thermal, biological, and chemical factors. A formation damage model is a mathematical expression of the permeability impairment due to the alteration of the porous media texture and surface characteristics. This must be a dynamic model, which is coupled with a porous media fluid flow model to predict the mutual effects of formation damage and flow conditions in oil and gas reservoirs. Therefore, although the main emphasis and objective are to develop a formation damage model, we must also address the modeling of fluid flow in porous media. Thus, the basic constituents of the overall modeling effort involve:

1 porous media realization,

2 formation damage model,

3 fluid and species transport model,

4 numerical solution,

5 parameter estimation, and

6 model validation and application.

In reality, porous matrix and the fluids contained within pore volume display a discrete structure. For convenience, however, a continuum approach using average properties over representative elemental porous media volume is preferred. Porous media is considered in two parts:

1 the flowing phase, denoted by the subscript/, consisting of a suspension of fine particles flowing through and

2 the stationary phase, denoted by the subscript s, consisting of the porous matrix and the particles retained

Although it would be more rigorous to proceed through these steps, we resort to a continuum modeling approach using the average properties over representative elemental porous media for simplification purposes. The loss of information on the process details are then compensated by empirical formulations. Empiricism cannot be avoided because of the irregular structure of geological porous media and the disposition of various fluid phases and particulate matter.

Description of Fundamental Model Equations

The basic model equations are the mathematical expressions for the following (Civan, 1994):

1. Mass balance

a. Fluid phases (convection) Gas/Liquid 1/Liquid 2

b. Species (convection/dispersion)

- Solid (indigenous/external, water wet/oil wet/intermediately wet, swelling/non-swelling)

- Ionic (anions/cations)

- Molecular

2. Momentum balance

- Fluid phases (Gas/Liquid 1/Liquid 2) (Forchheimer/Darcy)

3. Porosity-Permeability-Texture relationship

4. Particle transport efficiency factor

5. Swelling rate

a. Formation

b. Particle

6. Pore throat plugging criteria and rate

7. Particle mobilization rate

8. Internal and external filter cake formation

9. Plug-type deposition rate

10. Pore surface deposition rate

11. External fluid infiltration rate

a. Oil-based b. Water-based c. Emulsion

12. Effective porosity

13. Interphase particle and species exchange rates

14. Critical salt concentration

15. Critical velocity

16. Wettability index (water wet, oil wet, intermediately wet)

17. Phase equilibrium conditions

18. Non-Darcy coefficient correlation

19. Dynamic pore size distribution

20. Dynamic pore throat distribution

21. Chemical equilibrium

22. Particle size growth

The simulation input file requires the following information (Gruesbeck and Collins, 1982; Amaefule et al., 1988; Baghdikian et al., 1989; Civan et al., 1989; Chang and Civan, 1992; Civan, 1994):

1. Phenomena considered:

  • Dissolution/precipitation of mineral salts
  • Mobilization/retention of particles
  • Cation exchange between rock and fluids
  • Liquid absorption by porous matrix
  • Particle size variation
  • Crystallization processes

2. Injection fluid conditions:

  • Rate or pressure specified
  • Particle free or particle containing

3. Rock properties:

  • Core length
  • Core diameter
  • Initial porosity
  • Dispersion coefficient in porous media
  • Diffusion coefficient for liquid in porous matrix

4. Properties of the fluid and suspended particles:

  • Viscosity of liquid
  • Density of porous matrix material
  • Density of mineral salts
  • Density of clay particles
  • Density of externally injected particles
  • Critical velocity for particle mobilization
  • Critical salt concentration

5. Stoichiometric coefficients for chemical reactions in liquid media

  • Aqueous Phase

6. Conditions of the problem:

  • Initial conditions in porous media

— Concentration of ionic, molecular, and particulate species

  • Boundary conditions

— Injection end

— Constant pressure or flux

— Species concentration or flux

— Outlet end

— Constant pressure

— Species flux

7. Model parameters:

  • Delete the parameters of the mechanisms neglected for a specific problem
  • Assign the measured values for the parameters that are directly measurable by laboratory procedures
  • Identify the parameters for which the best estimates will be obtained by history matching

8. Laboratory core flow test data that will be used for history matching:

  • Input-Output pressure differential or input volume flux versus pore volume injected
  • Effluent species concentrations versus pore volume injected

9. Output that can be requested:

  • Best estimates of the unknown parameters
  • Predicted versus measured data
  • Simulation of pressure; various species concentrations in the flowing fluid and the pore surface; porosity and permeability as functions of pore volume injected or time

Numerical Solution of Formation Damage Models

Depending on the level of sophistication of the considerations, theoretical approaches, mathematical formulations, and due applications, formation damage models may be formed from algebraic and ordinary and partial differential equations, or a combination of such equations. Numerical solutions are sought under certain conditions, defined by specific applications. The conditions of solution can be grouped into two classes:

1 initial conditions, defining the state of the system prior to any or further formation damage, and

2 boundary conditions, expressing the interactions of the system with its surrounding during formation damage.

Typically, boundary conditions are required at the surfaces of the system, through which fluids enter or leave, such as the injection and production wells or ports, or that undergo surface processes, such as exchange or reaction processes. Algebraic formation damage models are either empirical correlations and/or obtained by analytical solution of differential equation models for certain simplified cases. Numerical solution methods for linear and nonlinear algebraic equations are well developed. Ordinary differential equation models describe processes in a single variable, such as either time or one space variable. However, as demonstrated in the following sections, in some special cases, special mathematical techniques can be used to transform multi-variable partial differential equations into single-variable ordinary differential equations. Amongst these special techniques are the methods of combination of variables and separation of variables, and the method of characteristics.

The numerical solution methods for ordinary differential equations are well developed. Partial differential equation models contain two or more independent variables. There are many numerical methods available for solution of partial differential equations, such as the finite difference method (Thomas, 1982), finite element method (Burnett, 1987), finite analytic method (Civan, 1995), and the method of weighted sums (the quadrature and cubature methods) (Civan, 1994, 1994, 1995, 1996, 1998; Malik and Civan, 1995; Escobar et al., 1997). In general, implementation of numerical methods for solution of partial differential equations is a challenging task.

In the following sections, several representative examples are presented for instructional purposes. They are intended to provide some insight into the numerical solution process. Interested readers can resort to many excellent references available in the literature for details and sophisticated methods. For most applications, however, the information presented in this chapter is sufficient and a good start for those interested in specializing in the development of formation damage simulators. Although numerical simulators can be developed from scratch as demonstrated by the examples given in the following sections, we can save a lot of time and effort by taking advantage of ready-made softwares available from various sources. For this purpose, the spreadsheet programs are particularly convenient and popular. Various softwares for solving algebraic, ordinary, and partial differential equations are available. Commercially available reservoir simulators can be manipulated to simulate formation damage, such as by paraffin deposition as demonstrated by Ring et al. (1994).


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.