Reservoir Formation Damage Episode 1 Part 7 potx

132 Reservoir Formation Damage (7-21) Several other relationships, which may be convenient to use in the for- mulation of the transport phenomena in porous media, are given in the following: The volume flux , My, and the velocity , Vy, of a phase j are related by: (7-22) where e jr is the volume fraction of the irreducible phase j in porous media. When an irreducible residual fluid saturation, S jr exists in porous media, Eq. 7-22 should be substituted into Eq. 7-15 for the flowing phase volume flux as: (7-23) In deforming porous media, the volumetric flux of the solid phase can be expressed in terms of the velocity according to the following equation: «,=6,v, (7-24) where e 5 and v s denote the solid phase volume fraction and velocity, respectively. Substituting Eq. 7-14, Eq. 7-24 becomes: ",=(l-4>)v, (7-25) Accounting for the immobile fluid fraction, e jr , in deforming porous media, the volumetric flux of the fluid relative to the deforming solid phase is given by Civan (1994, 1996): (7-26) The volume fraction of species i of phase j in the bulk system is given by: (7-27) Multi-Phase and Multi-Species Transport in Porous Media 133 or by P, e y =e *y %• (7-28) The mass concentration of species / in phase j is given by: c ( ,=p,-o ( , (7-29) The molar concentration of species / in phase j is given by: C^Cy/M; (7-30) The volume flux of species / in phase j is given by: «</=<Vry (7-31) where u r - is the volume flux of phase j. The mass flux of species i in phase j is given by: ~ C ij U rj ~ C ikj U rkj (7-32) Multi-Species and Multi-Phase Macroscopic Transport Equations The macroscopic description of transport in porous media is obtained by elemental volume averaging (Slattery, 1972). The formulations of the macroscopic equations of conservations in porous media have been carried out by many researchers. A detailed review of these efforts is presented by Whitaker (1999). The mass balances of various phases are given by (Civan, 1996, 1998): , p,) + V-( 9j uj) = V• (e, Dj • Vp y )+ (7-33) where u rj is the fluid flux relative to the solid phase, t is the time and V • is the divergence operator. p ; is the phase density, m • is the net mass rate of the phase j added per unit volume of phase j. Dj is the hydraulic dispersion coefficient which has been omitted in the petroleum engineer- ing literature. 134 Reservoir Formation Damage The species i mass balance equations for the water, oil, gas and solid phases are given by: + V • + V • = (7-34) in whch w tj is the mass fraction of species i in the j th phase, jy denotes the spontaneous or dispersive mass flux of species i in the j th phase given by modifying the equation by Olson and Litton (1992): A,' Vw, + -2- A; • VO. + Y —^- D s , • Vw . JtT •^ w (7-35) where D i} is the coefficient of dispersion of species i in the j th phase, k is the Boltzmann constant, and T is temperature. The first term represents the ordinary dispersive transport by concentration gradient. For particulate species of relatively large sizes the first term may be neglected. The second term represents the dispersion induced by the gradient of the potential interaction energy, <E> ( y. When the particles are subjected to uniform interaction potential field then the second term drops out. The third term represents the induced dispersion of bacterial species by substrate or nutrient, 5, concentration gradient due to the chemotaxis phenomena (Chang et al., 1992). D sj is the substrate dispersion coefficient. Incorporating Eq. 7-33 into Eq. 7-34 leads to the following alternative form: + V • = (7-36) Adding Eq. 7-34 over all the phases gives the total species / mass balance equation as: : y (7-37) Considering the possibility of the inertial flow effects due to the narrowing of pores by formation damage, the Forchheimer (1901) equation is used for the momentum balance. Although more elaborate forms of the macroscopic equation of motion are available, Blick and Civan (1988) have shown that Forchheimer's equation is satisfactory for all practical Multi-Phase and Multi-Species Transport in Porous Media 135 purposes. The Forchheimer equation for multi-dimensional and multi- phase fluids flow can be written for the j th phase as (Civan, 1994; Tutu et al., 1983; Schulenberg and Miiller, 1987): -V4*. = • u + Tl-'rr'.^+pj^F,. (7-38) in which ¥• is the interfacial drag force, r\ rj =k rj (Liu et al., 1995), T| = 1/P and \|/ is the flow potential given by: (7-39) where the first term is the fluid-content-dependent potential or simply the negative of the "effective stress" due to the interactions of the fluid with the pore surface, g is the gravitational acceleration, g(z-z 0 ) is the potential of fluid due to gravity, z is the positive upward distance measured from a reference at z 0 , and Q is the overburden potential, which is the work of a vertical displacement due to the addition of fluid into porous media (Smiles and Kirby, 1993). K and |3 denote the Darcy or laminar permeability and the non-Darcy or inertial flow coefficient tensors, respectively. K rj and p r; are the relative permeability and relative inertial flow coefficient, respectively. Eq. 7-38 can be written as, for convenience in which v is the kinematic viscosity (or momentum diffusivity) given by v j = (7-41) and N nd is the non-Darcy number for anisotropic porous media given by, neglecting the interfacial drag force Fji N.* - (7-42) where / denotes a unit tensor and Re^ is the tensor Reynolds number for flow of phase j in an anisotropic porous media given by (7-43) 136 Reservoir Formation Damage = («£ (7-44) The permeability and inertial flow coefficient for porous materials are determined by means of laboratory core flow data and thus correlated empirically (Civan and Evans, 1998). Liu et al. (1995) give: 8.91xl0 8 T (7-45) where (3 is in ft \ k is in mD, and 0 is in fraction. The energy balance equations for the water, oil, gas, and solid phases are given by: <x=l (7-46) q j and q ja denote the external and interface heat transfer to the phase j per unit volume of phase y; & y is the thermal conductivity of phase j, Note that the enthalpy Hj and internal energy Uj per unit mass of phase j are related according to: H J= U J + Pj/?J Thus, Eq. 7-46 can also be written as: (7-47) oc=l (7-48) When the system is at thermal equilibrium (i.e. T w = T 0 = T g = T s = T} then Eq. 7-48 can be written for each phase and then added to obtain the total energy balance equation as: Multi-Phase and Multi-Species Transport in Porous Media 137 = v- (7-49) Invoking Eq. 7-33, Eq. 7-46 can be written in an alternative form as: p,.[e y dHj V • (EJ k • V7})+ EJ + L«J.a a=l a*; (7-50) The equation of motion given by Chase and Willis (1992), for deforming porous matrix can be written as following: (7-51) where T 5 is the shear stress tensor for the solid matrix. The jump mass balance equations, given by Slattery (1972) can be simplified to express the boundary conditions as: (7-52) (7-53) (7-54) The superscript a denotes a quantity associated with the dividing surface, which is moving at a macroscopic velocity of w°, and n a is the unit vector normal to the dividing surface. r°, rf and r? are the rates of addition of mass of the porous matrix, the th phase, and the species i 138 Reservoir Formation Damage in the j th phase, respectively. [I I] denotes a jump in a quantity across a dividing surface defined by: f-H-r -(•••)• (7-55) where the signs + and - indicate the post and fore sides, respectively, of the dividing surface. Exercises 1. Show that the balance of species / in phase j can also be expressed in the following forms: jD 9 -V(c iy /p,. (7 _ 56) (7-57) 2. Show that, for incompressible flow and incompressible species, Eq. 7-56 simplifies as (7-58) References Blick, E. F., & Civan, F., "Porous Media Momentum Equation for Highly Accelerated Flow," SPE Reservoir Engineering, Vol. 3, No. 3, 1988, pp. 1048-1052. Chang, M M., Bryant, R. S., Stepp, A. K., & Bertus, K. M. "Modeling and Laboratory Investigations of Microbial Oil Recovery Mechanisms in Porous Media," Topical Report No. NIPER-629, FC22-83FE60149, U.S. Department of Energy, Bartlesville, Oklahoma, 1992, p. 27. Chase, G. G., & Willis, M. S., "Compressive Cake Filtration," Chem. Eng. Sci., Vol. 47, 1992, pp. 1373-1381. Multi-Phase and Multi-Species Transport in Porous Media 139 Civan, F., "Waterflooding of Naturally Fractured Reservoirs—An Efficient Simulation Approach," SPE Production Operations Sympsoium, March 21-23, 1993, Oklahoma City, Oklahoma, pp. 395-407. 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. "A Multi-Phase Mud Filtrate Invasion and Well Bore Filter Cake Formation Model," SPE Paper No. 28709, Proceedings of the SPE International Petroleum Conference & Exhibition of Mexico, October 10-13, 1994, Veracruz, Mexico, pp. 399-412. Civan, F. "A Multi-Purpose Formation Damage Model," SPE 31101 paper, Proceedings of the SPE Formation Damage Symposium, Lafayette, LA, February 14-15, 1996, pp. 311-326. Civan, F., "Quadrature Solution for Waterflooding of Naturally Fractured Reservoirs," SPE Reservoir Evaluation and Engineering, April 1998, pp. 141-147. Civan, F, & Evans, R. D., "Determining the Parameters of the Forchheimer Equation from Pressure-Squared vs. Pseudopressure Formulations," SPE Reservoir Evaluation and Engineering, February 1998, pp. 43-46. Forchheimer, P., "Wasserbewegung durch Boden," Zeitz. ver. Deutsch Ing. Vol. 45, 1901, pp. 1782-1788. Liu, X., Civan, F., & Evans, R. D. "Correlation of the Non-Darcy Flow Coefficient, J. of Canadian Petroleum Technology, Vol. 34, No. 10, 1995, pp. 50-54. Olson, T. M., & Litton, G. M. "Colloid Deposition in Porous Media and an Evaluation of Bed-Media Cleaning Techniques," Chapter 2, pp. 14-25, in Transport and Remediation of Subsurface Contaminants, Colloidal, Interfacial, and Surfactant Phenomena, Sabatini, D. A. and R. C. Knox (Eds.), ACS Symposium Series 491, American Chemical Society, Washington, DC (1992). Schulenberg, T., & Miiller, U., "An Improved Model for Two-Phase Flow Through Beds of Coarse Particles," Int. J. Multiphase Flow, Vol. 13, No. 1, 1987, pp. 87-97. Slattery, J. C. Momentum, Energy and Mass Transfer in Continua, McGraw- Hill Book Co., New York, 1972, pp. 191-197. Smiles, D. E., & Kirby, J. M., "Compressive Cake Filtration—A Com- ment," Chem. Eng. ScL, Vol. 48, No. 19, 1993, pp. 3431-3434. Tutu, N. K., Ginsberg, T., & Chen, J. C., "Interfacial Drag for Two-Phase Flow Through High Permeability Porous Beds," Interfacial Transport Phenomena, Chen, J. C. & Bankoff, S. G., (eds.), ASME, New York, pp. 37-44. Whitaker, S., The Method of Volume Averaging, Kluwer Academic Publishers, Boston, 1999, 219 p. Chapter 8 Particulate Processes in Porous Media Summary Physico-chemical, chemical, hydrodynamic, and mechanical processes frequently lead to the mobilization, generation, migration and deposition of fine particles, which in turn cause formation damage in petroleum bearing formations. This chapter is devoted to the review of the various types of internal particulate processes that occur in porous media, and the factors and forces affecting these processes. Introduction There are three primary sources of fine particles in petroleum bearing formations: 1. Invasion of foreign particles carried with the fluids injected for completion, workover, and improved recovery purposes, 2. Mobilization of in-situ formation particles due to the incompatibility of the fluids injected into porous media and by various rock-fluid interactions, and 3. Production of particulates by chemical reactions, and inorganic and organic precipitation. Fluids injected into petroleum reservoirs usually contain iron colloids produced by oxidation and corrosion of surface equipment, pumps, steel casing, and drill string (Wojtanowicz et al., 1987). Brine injected for waterflooding may contain some fine sand and clay particles. Mud fines can invade the formation during overbalanced drilling. These are some examples of the externally introduced particles. Petroleum bearing formation usually contains various types of clay and other mineral species attached to the pore surface. These species can be 140 Paniculate Processes in Porous Media 141 released by colloidal forces or mobilized by hydrodynamic shear of the fluid flowing through porous media. Fine particles can also be generated by deformation of rock during compression and dilatation. This is due to variation of the net overburden stress and loss of the integrity of rock grains. Fine particles are unleashed and liberated because of the integrity loss of rock grains by chemical dissolution of the cementing materials in porous rock, such as by acidizing or caustic flooding. These are the typical internal sources of indigenous fine particles. Paniculate matter can be produced by various chemical reactions such as the salt formation reactions that occur when the seawater injected for waterflooding mixes with the reservoir brine, and formation of elemental sulfur during corrosion. Paniculate matter can also be produced by precipitation due to the change of the thermodynamic conditions and of the composition of the fluids by dissolution or liberation of light gases (Amaefule et al., 1988). These are typical" mechanisms of particle pro- duction in porous media. Once entrained by the fluids flowing through porous media, the various particles migrate by four primary mechanisms (Wojtanowicz et al., 1987): 1. Diffusion 2. Adsorption 3. Sedimentation 4. Hydrodynamics The transport of the fine particles are affected by six factors (Wojtanowicz et al., 1987): 1. Molecular forces 2. Electrokinetic interactions 3. Surface tension 4. Fluid pressure 5. Friction 6. Gravity As the fine particles move along the tortuous flow pathways existing in porous media, they are captured, retained, and deposited within the porous matrix. Consequently, the texture of the matrix is adversely altered to reduce its porosity and permeability. Frequently, this phenomena is referred to as formation damage measured as the permeability impairment. Particulate Processes The various particulate processes, schematically depicted in Figure 8-1, can be classified in two groups as the internal and external processes. [...].. .14 2 Reservoir Formation Damage Hydrodynamlc mobilization Colloidal expulsion Liberation of particles by cement dissolution Surface deposition Pore throat plugging Internal cake formation by small particles Internal and External cake formation by small particles External cake formation by large particles Figure 8 -1 Various particulate processes Paniculate Processes in Porous Media 14 3 The external... according to (Ohen and Civan, 19 90, 19 93): (8 -19 ) D,=fD Then, the ration of the particle to pore throat diameters can be approximated by: F = 5 1= DP = p D, o (8-20) King and Adegbesan (19 97) state that the ratio of the median particle diameter to pore throat diameter is given by (Dullien, 19 79 ): (8- 21) A comparison of Eqs 8-20 and 21 implies that, even if / = 1. 0, Eq 8- 21 is applicable for tight porous... leads to two dimensionless groups (Civan, 19 96) The first is an aspect ratio representing the critical pore throat to particle diameter ratio necessary for plugging given by: 15 2 Reservoir Formation Damage MAXIMUM PARTICLE CONCENTRATION - VOLUME/VOLUME 10 0.08 0 .15 0. 21 0. 27 0. 31 0.58 • TAP WATER • 10 0 cp HYDROXYETHYL CELLULOSE SOLUTION BRIDGING REGION 2 4 6 8 10 MAXIMUM GRAVEL CONTENT - LB/GAL 30 Figure... cake formation at a rate proportional to the particle flux, uap, and the pore volume, (|), available (8 -13 ) subject to e?=ero,? = 0 (8 -14 ) kt is a pore filling rate constant given by: kt * 0 for t> tcr when p < pcr (8 -15 ) 15 0 Reservoir Formation Damage Figure 8-4 Particles approaching a pore throat and k, = 0 otherwise (8 -16 ) tcr represents the critical time when the pore throats are first jammed by particles... (Ives, 19 85; Khilar and Fogler, 19 87) Forces Related to the Detachment Mechanisms Shearing Force This is the friction or drag force When the shear stress of the liquid flowing over the deposited particles creates a shearing force greater than those attaching the particles to the grain surface, then the particles can be detached and mobilized (Ives, 19 85): dv dr (8-9) 14 8 Reservoir Formation Damage. .. Media 14 7 and its other forms such as those "relating to the shear gradient, the relative velocity between particle and liquid, the angular velocity of the rotating particle, and the frequency of pulsation liquid have been suggested." Khilar and Fogler (19 87) expressed the hydrodynamic lift force pulling a spherical particle off the pore surface by the following equation given by Hallow (19 73 ): 1/ 2 (8 -7) ... crushing, and deformation 4 Chemical and physico-chemical formation Unswollen Particles in the Porous Matrix Swollen Particles Extending from Pore Surface Deposited Particles Particles Suspension T DEPOSITION FLOWING PHASE Plague ENTRAPMENT & Figure 8-3 Particulate processes in porous media (after Civan, 19 94; reprinted by permission of the U.S Department of Energy, modified after Civan et al 19 89, from... rock deformation and crushing, liberation of fine particles by chemical dissolution of cement, coagulation/disintegration) e Interphase Transport or Exchange (c) Figure 8-2 Mechanism of pore throat blocking: a) plugging and sealing, b) flow restriction, c) bridging (after Civan, 19 94; reprinted by permission of the U.S Department of Energy) 14 4 Reservoir Formation Damage The net amount of particles... Fine Particle Processes," pp 65 -79 , 19 89, with permission from Elsevier Science) Paniculate Processes in Porous Media 14 5 The fundamental particle retention mechanisms are: 1 2 3 4 Surface deposition (physico-chemical) Pore throat blocking (physical jamming) Pore filling and internal filter cake formation (physical) Screening and external filter cake formation (physical) Forces Acting Upon Particles... that the flow of a particulate suspension into porous media may lead to one of the following phenomena (King and Adegbesan, 19 97) : a P < 3 , external filtercake formation b 3 < ( 3 < 7 , internal filtercake formation c p > 7 , negligible filtercake involvement Pautz et al (19 89) point out that these rules-of-thumb have been derived based on experimental observations The values 3 and 7 denote the critical . Multiphase Flow, Vol. 13 , No. 1, 19 87, pp. 87- 97. Slattery, J. C. Momentum, Energy and Mass Transfer in Continua, McGraw- Hill Book Co., New York, 19 72 , pp. 19 1 -19 7. Smiles, D. E.,. Mexico, October 10 -13 , 19 94, Veracruz, Mexico, pp. 399- 412 . Civan, F. "A Multi-Purpose Formation Damage Model," SPE 311 01 paper, Proceedings of the SPE Formation Damage Symposium,. LA, February 14 -15 , 19 96, pp. 311 -326. Civan, F., "Quadrature Solution for Waterflooding of Naturally Fractured Reservoirs," SPE Reservoir Evaluation and Engineering, April 19 98, pp.