Reservoir Sand Migration and Gravel-Pack Damage 657 where v pv is the induced velocity of particles and f sw is the shear wave frequency. In case of the lack of information for Eq. 20-4, Hayatdavoudi (1999) recommends estimating a, by: a,. =0.19* (20-5) where g denotes the gravitational acceleration. Hayatdavoudi (1999) points out the importance of the buoyant unit weight of the in-situ particles when determining the particle mass, and estimates the in-situ particle mass by: m = (y ls]z (20-6) V | (2Vg / O / ~ ^ ' where z represents the depth or height measured from a reference datum (ft) and j ave is the average specific weight of the formation sand (lb/ ft 3 ). The latter is expressed as: • avg ri\ I in the I in the ^ V °U zone water zone (20-7) in which the specific weights of the sand grains in the water and oil zones are given, respectively, by: I in the water zone l + e (20-8) i in the oil zone (20-9) where the void ratio, e, in terms of the fractional porosity, (j>, is given by: (20-10) and j w and J 0 are the specific weights of the water and oil phases, and G is the specific gravity of the sand grains. As a fluid flows over the face of a cohesionless bed of particles, such as sand or gravel, the particles can be detached and lifted-off when the fluid shear-stress exceeds the minimum, critical shear-stress. Yalin and 658 Reservoir Formation Damage Karahan (1979) developed a dimensionless correlation to predict the critical conditions for onset of particle mobilization (or scouring) by fluid shear. Following their approach, Tremblay et al. (1998) developed: M cr =0.122Re^ 206 (20-11) in which Re cr is the critical particle Reynolds number given by r» cr R& cr=~ (20-12) where v cr is the critical shear velocity, d is the mean particle diameter, and p and (I are the density and viscosity of the fluid flowing over the particle bed. M cr is the critical mobility number given by pv 2 (20-13) where y s denotes the specific weight of the particles suspended in the fluid. Applying Eq. 20-11, Tremblay et al. (1998) correlated their experi- mental data of laminar flow of various liquids over a loose bed of sand particles linearly on a full logarithmic scale and obtained the following expression for the critical shear velocity: v cr = 0.385 (^/p) a0934 Y?- 453 J°- 36 p^ 453 (20-14) Then, they predict the critical shear-stress on the scouring face by: T cr =pv 2 r (20-15) Massive Sand Production Model 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 Reservoir Sand Migration and Gravel-Pack Damage 659 cases, such as horizontal and deviated wells, and different formations may require further developments. The decline of pressure during production causes flow and stress- induced 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): (20-16) where <5 r and a e denote the radial and tangential stresses, respectively, (Pa), c is the cohesive strength (Pa), y is a friction coefficient (dim- ensionless), and p is the fluid pressure (Pa). intact zone sand flow flowing, yielded zone ^ of yielding front intact zone fry? Figure 20-3. Growing yielded zone and the intact zone around a producing well (reprinted from Journal of Petroleum Science and Engineering, Vol. 17, Geilikman, M. B., and Dusseault, M. B, "Fluid Rate Enhancement from Massive Sand Production in Heavy-Oil Reservoirs," pp. 5-18, ©1997, with permission from Elsevier Science). 660 Reservoir Formation Damage The stress equilibrium condition for the solid skeleton is given by: (20-17) in which b is the coefficient of the body force, approximated by: " = T (20-18) where <(),. is the porosity of the intact zone; K is permeability; \i is viscosity; v f and v s denote the fluid and solid phase velocities, respectively; and r is the radial distance. The Darcy law is applied for the mobile fluid phase dp (2(M9) 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: (20-20) (20-21) In the intact zone, the porosity, ((),, is assumed constant. Thus, Eqs. 20-20 and 21 simplify as: (20-22) (20-23) Thus, the fluid velocities in the yielded and intact zones can be expressed, respectively, by: < 2 °- 24 > Reservoir Sand Migration and Gravel-Pack Damage 661 v, = 2n^r Similarly, the solid velocity is given by: (20-25) (20-26) Therefore, substituting Eqs. 20-23 through 26 into Eq. 20-19 and inte- grating, yields the following fluid pressure profiles in the yielded and intact zones, respectively: (cr) = — , r<r<# (20-27) and -^ R<r<r < r (20-28) The consistency and compatibility conditions for the fluid flow at the moving front are given, respectively, by: -V, r = R(t) p=Pi, r = R(t) (20-29) (20-30) r w and r e denote the wellbore and reservoir radii, respectively, and p w and p e are the fluid pressures at these locations. The consistency and compatibility conditions for the solid flow at the mov- ing front between the yielded and intact zones are given, respectively, by: (20-31) (20-32) where V is the front velocity, given by: V = dR/dt 662 Reservoir Formation Damage in which R=R(t) denotes the radial distance to the front. Substitut- ing Eqs. 20-26 and 32 into Eq. 20-31, and solving the resulting expres- sion for the cumulative volume of solids production, S c , yields: (20-33) (20-34) The volumetric rate of solid production is given by q,=dSJdt In the yielded zone, eliminating the tangential stress, a e , between Eqs. 20-16 and 17 leads to: dr subject to the conditions at the wellbore and at the moving front a r = p y ,r = *(r) (20-35) (20-36) (20-37) 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: r,., - = 0 (20-38) where A, = 2y/(l-y). Substituting Eqs. 20-27 through 30 into Eq. 20-38, q s can be calculated. Incorporating some simplifying approximations, Geilikman and Dusseault (1997) obtain the following expression for sand production rate: Reservoir Sand Migration and Gravel-Pack Damage 663 dS r r_. _ / i \-\-b (1-<>,) dt + MP./P) (20-39) ~(l-y)*np which can be numerically integrated assuming a wellbore fluid pressure history, represented by the following decay function: P w (0 = Ac + (p c - P~) exp (-t/t p ) (20^0) where t p is a characteristic time scale, p c is some critical fluid pressure at which the yield criterion is met, and p^ is the limit value of the wellbore pressure for t~»t p . The volumetric rates of fluid production is given by: loft ^ln(±\ + *M-ln\*®. <lf(t) = in + In (20-41) in which q 0 (t} is the rate of fluid production without any sand pro- duction, given by: (20-42) Geilikman and Dusseault (1997) defined dimensionless sand production rate, time, characteristic time, and fluid production enhancement ratio, respectively as: (20-43) 664 Reservoir Formation Damage (20^4) _ =t P P (20-45) = q f (t)/q 0 (t) (20-46) Figures 20-4 and 20-5 by 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 <b 10 20 30 40 30 60 70 I/0.01 Figure 20-4. Dimensionless volumetric sand production rate vs. dimensionless time: Curves 1, 2, and, 3 are for T= 0.1, 0.5, and, 1.0, respectively (reprinted from Journal of Petroleum Science and Engineering, Vol. 17, Geilikman, M. B., and Dusseault, M. B, "Fluid Rate Enhancement from Massive Sand Production in Heavy-Oil Reservoirs," pp. 5-18, ©1997, with permission from Elsevier Science). Reservoir Sand Migration and Gravel-Pack Damage 665 0.5 10 20 30 40 50 60 70 .t/0.01 Figure 20-5. Short-term fluid production improvement vs. dimensionless time: Curves 1, 2, and, 3 are for T= 0.1, 0.5, and, 1.0, respectively (reprinted from Journal of Petroleum Science and Engineering, Vol. 17, Geilikman, M. B., and Dusseault, M. B, "Fluid Rate Enhancement from Massive Sand Production in Heavy-Oil Reservoirs," pp. 5-18, ©1997, with permission from Elsevier Science). 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 Chapter 10 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 gravel- pack, and (b) the clay swelling effects are not considered. As attested by the results given in Figures 20-6 and 20-7, 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 Exhibi- tion, September 25-28, 1994, New Orleans, Louisiana, pp. 179-191. 666 Reservoir Formation Damage Simulation a Low Flow Rate M High Flow Rate 5 10 15 20 Distance from Sand-Gravel Interface, cm Figure 20-6. Simulation of experimental data for low and high flow rate profiles of migrated sand particles in a 7.5 gravel to sand ration gravel-pack (after Bouhroum et al., ©1994 SPE; reprinted by permission of the Society of Petroleum Engineers). Simulation • Low Flow Rate m High Flow Rate 5 10 15 20 Distance from Sand-Gravel Interface, cm Figure 20-7. Simulation of experimental data for low and high flow rate profiles of migrated sand particles in a 6.3 gravel to sand ration gravel-pack (after Bouhroum et al., ©1994 SPE; reprinted by permission of the Society of Petroleum Engineers). [...]... Sulfur 669 670 Reservoir Formation Damage deposition can occur both in the well and reservoir formation (Hyne, 19 68, 1 980 , 19 83 ; Kuo, 1972; Roberts, 1997) In this section, a brief discussion of the simplified analytical modeling effort by Roberts (1997) for prediction of the formation damage by sulfur deposition is presented Hyne (19 68, 1 980 , 19 83 ) has considered the possibility of formation of hydrogen... 1962, pp 130 1- 133 1 Woll, W., "The Effect of Sour Gases on the Pressure-Melting Point Curve of Sulfur," Erdoel, Erdgas Z., Vol 9, 19 83 , p 297 Part VII Diagnosis and Mitigation of Formation Damage Measurement, Control, and Remediation Chapter 22 Field Diagnosis and Measurement of Formation Damage Summary Methods for inferring formation damage in petroleum reservoirs and expressing formation damage and... of Oklahoma, Norman, Oklahoma, March 3- 5 Kuo, C H., "On the Production of Hydrogen Sulfide-Sulfur Mixtures from Deep Formations," JPT, September, 1972, p 1142 Leontaritis, K J., "Asphaltene Near-Wellbore Formation Damage Modeling," SPE 39 446 paper, Proceedings of the 19 98 SPE Formation Damage Control Conference, February 18- 19, 19 98, Lafayette, Louisiana, pp 277- 288 Roberts, B E., "The Effect of Sulfur... No 86 , 1 982 , p 30 16 Hyne, J B., "Study Aids Prediction of Sulfur Deposition in Sour Gas Wells," Oil & Gas J., November 25, 19 68, p 107 Hyne, J B., "Controlling Sulfur Deposition in Sour Gas Wells," World Oil, pp 35 , August 19 83 6 78 Reservoir Formation Damage Hyne, J B., & Derdall, G., "Sulfur Deposition in Reservoirs and Production Equipment: Sources and Solutions," Paper presented at the 1 980 Annual... by formation damage is expressed by alternative measures, and the various tests available for measurement and diagnosis of formation damage problems in the field are described 680 Field Diagnosis and Measurement of Formation Damage 681 Diagnosis and Evaluation of Formation Damage in the Field As stated by Yeager et al (1997), "No individual test or tool can provide the only information needed for damage. .. SPE 37 506 paper, SPE Production Operations Symposium, March 9-11, 1997, Oklahoma City, Oklahoma, pp 87 1 -87 9 Spangler, M G., & Handy, R L., Soil Engineering, Harper & Row, New York City, New York, 1 982 Tiffin, D L., King, G E., Larese, G E., & Britt, R E., "New Criteria for Gravel and Screen Selection for Sand Control," SPE 39 437 paper, SPE Formation Damage Control Conference, February 18- 19, 19 98, Lafayette,... (21 -3) In Eq 21 -3, cr represents the concentration of the solid sulfur dissolved in the gas expressed as mass per unit reservoir gas volume Formation Damage by Scale Deposition 671 [g /reservoir ra3), p is the density of the gas (&g/m 3 ), T is the reservoir gas temperature, and k, A, and B are some empirically determined parameters As shown by Roberts (1997), using the data by Brunner and Woll (1 980 ,... 1 980 , p 37 7 Brunner, E., Place, M C Jr., & Woll, W H., "Sulfur Solubility in Sour Gas," JPT, December 1 988 , p 1 587 Chernik, P S., & Williams, P J., "Extended Production Testing of the Bearberry Ultra-Sour Gas Resource," Paper SPE 26190, presented at the 19 93 SPE Gas Technology Symposium, Calgary, June 28 -30 Chrastil, J., "Solubility of Solids and Liquids in Supercritical Gases," / Phys Chem., No 86 ,... to the flashing point from the wellbore, with 0.119 and 0.0642 m3/s brine flow rates, in 50 m thick production zones of permeabilities of 4. 935 x10~ 13 and 9 .86 9x10~ 13 m2, respectively (after Satman et al., ©1999; reprinted by permission of the Int Institute for Geothermal Research, Italy) Formation Damage by Scale Deposition 677 h»60m 0. 08 \> Skin « -2 —V 0.06 Skin = 0 \ \ 0.04 0.02 1 10 1000 1C 0 1... 1996, pp 39 1 -39 8 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 6 68 Reservoir Formation Damage Saucier, R J., "Successful Sand Control Design for High Rate Oil and Water Wells," J of Petroleum Technology, Vol 21, 1969, p 11 93 Saucier, . well performance. Sulfur 669 670 Reservoir Formation Damage deposition can occur both in the well and reservoir formation (Hyne, 19 68, 1 980 , 19 83 ; Kuo, 1972; Roberts, 1997). In . Roberts (1997) for prediction of the formation damage by sulfur deposition is presented. Hyne (19 68, 1 980 , 19 83 ) has considered the possibility of formation of hydrogen polysulfides . expressed as mass per unit reservoir gas volume Formation Damage by Scale Deposition 671 [g /reservoir ra 3 ), p is the density of the gas (&g/m 3 ), T is the reservoir gas temperature,