Many models with varying degrees of predictive capabilities are available for sand production. Here, the radial continuum model for massive sand production, coupling fluid, and granular matrix flows by Geilikman and Dusseault (1994, 1997) is presented for instructional purposes. This is a physics-based approach that includes the essential ingredients of a sand production model. However, applications to other cases, such as horizontal and deviated wells, and different formations may require further developments.

The decline of pressure during production causes flow and stressinduced damage in the near-wellbore region. The increase in the deviatoric stress above the yield condition in unconsolidated sandstone formations cause instabilities and plastic flow leading to sand production. As depicted in Figure 20-3, Geilikman and Dusseault (1997) considers two regions for modeling purposes:

1 a yielded-zone, initiating from the wellbore and extending to a propagating front radius, R=R(t), and

2 an intact-zone, beyond the propagating front of the yielded-zone.

They consider a two-phase continuum medium:

1 a viscoplastic solid skeleton, and

2 an incompressible and viscous saturated fluid.

The modeling is carried out per unit formation thickness. The yield function, F, for granular matrix is defined as (Jackson, 1983; Collins, 1990; Pitman, 1990; Drescher, 1991):

where <5r and ae denote the radial and tangential stresses, respectively, (Pa), c is the cohesive strength (Pa), y is a friction coefficient (dimensionless), and p is the fluid pressure (Pa).

The stress equilibrium condition for the solid skeleton is given by:

where <(),. is the porosity of the intact zone; K is permeability; \i is viscosity; vf and vs denote the fluid and solid phase velocities, respectively; and r is the radial distance. The Darcy law is applied for the mobile fluid phase

Assuming that the fluid and solid phases are incompressible, the volumetric balance equations (equation of continuity) of the fluid and solid phases are given by:

Therefore, substituting Eqs. 20-23 through 26 into Eq. 20-19 and integrating, yields the following fluid pressure profiles in the yielded and intact zones, respectively:

The consistency and compatibility conditions for the fluid flow at the moving front are given, respectively, by:

rw and re denote the wellbore and reservoir radii, respectively, and pw and pe are the fluid pressures at these locations. The consistency and compatibility conditions for the solid flow at the moving front between the yielded and intact zones are given, respectively, by:

in which R=R(t) denotes the radial distance to the front. Substituting Eqs. 20-26 and 32 into Eq. 20-31, and solving the resulting expression for the cumulative volume of solids production, Sc, yields:

Thus, substituting Eqs. 20-25 and 26 into Eq. 20-35 and solving, leads to the following expression for the radial stress in the yielded zone:

Substituting Eqs. 20-27 through 30 into Eq. 20-38, qs can be calculated. Incorporating some simplifying approximations, Geilikman and Dusseault (1997) obtain the following expression for sand production rate:

which can be numerically integrated assuming a wellbore fluid pressure history, represented by the following decay function:

where tp is a characteristic time scale, pc is some critical fluid pressure at which the yield criterion is met, and p^ is the limit value of the wellbore pressure for t~»tp.The volumetric rates of fluid production is given by:

in which q0(t} is the rate of fluid production without any sand production, given by:

Geilikman and Dusseault (1997) defined dimensionless sand production rate, time, characteristic time, and fluid production enhancement ratio, respectively as:

Geilikman and Dusseault (1997) present typical solutions for the rate of sand production and enhancement of fluid production.

## Sand Retention in Gravel-Packs

As stated by Bouhroum et al. (1994): Sand production poses serious problems to tubular material, surface equipment and the stability of the well. . . . A popular method of

combating sand production is using gravel-packs. Gravel-packs have a protective function to inhibit the flow of sand particulates into the well.

Bouhroum et al. (1994) essentially applied the Ohen and Civan (1993) model, given in article with several simplifications for prediction of the gravel-pack permeability impairment by sand deposition. The important simplifying assumptions of this model are:

a. the sand particles are generated in the near-wellbore formation and deposited in the gravelpack, and

b. the clay swelling effects are not considered. As attested by the results given in this article their predictions accurately match the experimental values.

## References

Bouhroum, A., Liu, X., & Civan, F., "Predictive Model and Verification for Sand Particulates Migration in Gravel-Packs," SPE 28534 paper, Proceedings of the SPE 69th Annual Technical Conference and Exhibition, September 25-28, 1994, New Orleans, Louisiana, pp. 179-191.

Burton, R. C., "Perforation-Tunnel Permeability Can Assess Cased-Hole Gravel-Pack Performance," J. of Petroleum Technology, March 1998, pp. 86-87.

Collins, I. F., "The Mathematical Structure of the Equations for Quasi-Static Plane Strain Deformations of Granular Material," in D. D. Joseph & D. G. Schaeffer (Eds.), Two Phase Flow and Waves, The IMA Volumes in Mathematics and its Applications Series 26, Springer, New York, p. 30, 1990.

Drescher, A., "Analytical Methods in Bin-Load Analysis," Developments in Civil Engineering Series 36, Elsevier, Amsterdam, 255 p., 1991.

Dusseault, M. B., & Santarelli, F. J., "A Conceptual Model for Massive Solids Production in Poorly-Consolidated Sandstones," Rock at Great Depth, Maury & Fourmantraux (Eds.), Balkema, Rotterdam, 1989, pp. 789-797.

Geilikman, M. B., Dusseault, M. B., & Dullien, F. A. L., "Sand Production as a Viscoplastic Granular Flow," SPE 27343 paper, SPE International Symposium on Formation Damage Control, February 9-10, 1994, Lafayette, Louisiana, pp. 41-50.

Geilikman, M. B., & Dusseault, M. B, "Fluid Rate Enhancement from Massive Sand Production in Heavy-Oil Reservoirs," /. of Petroleum Science and Engineering, Vol. 17, No. 1/2, 1997, pp. 5-18.

Hayatdavoudi, A., "Formation Sand Liquefaction: A New Mechanism for Explaining Fines Migration and Well Sanding," SPE 52137 paper, SPE Mid-Continent Operations Symposium, March 28-31, 1999, Oklahoma City, Oklahoma, pp. 177-180.

Jackson, R., "Some Mathematical and Physical Aspects of Continuum Models for the Motion of Granular Materials," in R.E. Meyer (Ed.), Theory of Dispersed Multiphase Flow, Academic Press, New York, New York, 1983.

JPT, "Specialists Share Knowledge of Sand-Control Methods," J. of Petroleum Technology, July 1995, pp. 550-609.

Kanj, M., Zaman, M., & Roegiers, J-C., "Highlights from the Knowledge-Base of a Sanding Advisor Called SITEX," Computer Methods and Water Resources III, Abousleiman, Y, Brebbia, C. A., Cheng, A.H.-D., & Ouazar, D. (Eds.), Computational Mechanics Publications, Southampton, Boston, 1996, pp. 391-398.

Pitman, E. B., "Computations of Granular Flow in Hopper," in D. D. Joseph & D. G. Schaeffer (Eds.), Two Phase Flow and Waves, The IMA Volumes in Mathematics and its Applications Series 26, Springer, New York, 1990, p. 30.

Saucier, R. J., "Successful Sand Control Design for High Rate Oil and Water Wells," J. of Petroleum Technology, Vol. 21, 1969, p. 1193.

Saucier, R. J., "Considerations in Gravel-Pack Design," J. of Petroleum Technology, Vol. 26, 1974, p. 205.

Skjaerstein, A., & Tronvoll, J., "Gravel Packing: A Method of Wellbore Re-enforcement or Sand Filtering?," SPE 37506 paper, SPE Production Operations Symposium, March 9-11, 1997, Oklahoma City, Oklahoma, pp. 871-879.

Spangler, M. G., & Handy, R. L., Soil Engineering, Harper & Row, New York City, New York, 1982.

Tiffin, D. L., King, G. E., Larese, G. E., & Britt, R. E., "New Criteria for Gravel and Screen Selection for Sand Control," SPE 39437 paper, SPE Formation Damage Control Conference, February 18-19, 1998, Lafayette, Louisiana, pp. 201-214.

Tremblay, B., Sedgwick, G., & Forshner, K., "Modeling of Sand Production from Wells on Primary Recovery," Journal of Canadian Petroleum Technology, Vol. 37, No. 3, March 1998, pp. 41-50.

Yalin, M. S., & Karahan, E., "Inception of Sediment Transport," Journal of the Hydraulics Division, Vol. 105 (HY11), American Society of Civil Engineers, November 1979, pp. 1433-1443.