Reservoir Formation Damage Episode 1 Part 5 potx

82 Reservoir Formation Damage n (5-8) (5-9) A" is the intrinsic permeability of porous media. The cross-sectional area of porous media open for flow can be expressed by: A P = (5-10) Therefore, equating Eqs. 5-9 and 10, and substituting Eqs. 5-1 and 10 results in the following relationship for the mean hydraulic tube diameter: D h = (5-11) Equating Eqs. 5-6 and 11 leads to the following Carman-Kozeny equa- tion (1938): (5-12) Two alternative forms can also be derived. Substitution of Eq. 5-7 into 12 yields: 6V2rll-(|) (5-13) or substituting Eq. 5-5 into Eq. 5-12 and then rearranging yields a power law type relationship given as (Kozeny, 1927): K = 1 2TZ2 (5-14) Based on the analysis of data by Jacquin (1964), Adler et al. (1990) suggested a power low correlation of permeability with respect to poros- ity as: (5-15) Permeability Relationships 83 Bourbie et al. (1986) determined that n = 1 for ()><0.05 and n = 3 for 0.08 <(j)<0.25. In view of this evidence and Eq. 5-14, the Carman-Kozeny equation appears to be valid for the 0.08 < <)) < 0.25 fractional porosity range. Reis and Acock (1994) warn that these exponents may be low "because the permeabilities were not corrected for the Klinkenberg effect." The Modified Carman-Kozeny Equation Incorporating the Flow Units Concept The derivation of the Carman-Kozeny equation presented in the pre- ceding section inherently assumed uniform diameter cyclindrical flow tubes analogy. Therefore, for applications to nonuniform diameter flow tubes, the Carman-Kozeny equation has been modified by inserting a geometric shape factor, F s (Amaefule et al., 1993), as: (5-16) Hearn et al. (1984, 1986) introduced the "flow units" concept and Amaefule et al. (1993) defined a lumped parameter as following, called the "flow zone indicator" to combine the three unknown parameters, F s , i and Z g , into one unknown parameter: FZI = 1 (5-17) Therefore, a plot of experimental data based on the logarithmic form of Eq. 5-16 (Amaefule et al., 1993) 4> (5-18) should yield a straightline with a slope of two. Hence, the FZI 2 value can be obtained as the value of K/§ at the (|) = 0.5 value. Implicit in Eq. 5-18 is the assumption that formations with similar flow characteristics can be represented by the same characteristic flow zone indicator parameter values. Consequently, formations having distinct flow zone parameters can be identified as different flow units. 84 Reservoir Formation Damage The Modified Carman-Kozeny Equation for Porous Media Altered by Deposition Based on the Carman-Kozeny model, Eq. 5-14, Adin's (1978) cor- relation of experimental data leads to a permeability-porosity model as: 1/2' (5-19) where the volumetric fraction of bulk porous media occupied by the deposits is given by: e = § o - <|) (5-20) This equation facilitated the following expression for the porosity reduc- tion by deposition: -*- (5-21) The alternative empirical expression proposed by Arshad (1991) can be modified as (Civan, 1998): 1 — = 1 - a — (5-22) where oc and n are some empirical parameters. Arshad's equation accounts for the formation of the dead-end pores during deposition, which do not conduct fluids. The Flow Efficiency Concept Rajani (1988) concluded that permeability function can be separated into and expressed as a product of a function incorporating the pore geometry and a function of porosity as: (5-23) Permeability Relationships 85 This approach is particularly useful in porous media undergoing alteration during formation damage. Frequently, the Carman-Kozeny equation fails to represent the cases where the pore throats are plugged without sig- nificant porosity reduction. This problem can be alleviated by introducing a flow efficiency factor, y, in view of Eq. 5-19 (Ohen and Civan, 1993; Chang and Civan, 1991, 1992, 1997). Hence, the permeability variation can be expressed by (Chang and Civan, 1997): K (5-24) where a, b, and c are some empirically determined parameters and K 0 and § 0 denote the permeability and porosity at some initial or reference state. The flow efficiency factor, y, can be interpreted as a measure of the fraction of the open pore throats allowing fluid flow. Thus, when the pore throats are plugged, then y = 0, and therefore K = 0, even if <|) * 0. This phenomenon is referred to as the "gate or valve effect" of the pore throats (Chang and Civan, 1997; Ochi and Vernoux, 1998). In order to estimate the flow efficiency factor, Ohen and Civan (1993) assumed that, although the pore throat sizes vary with time, they always remain log-normally distributed: f(y) = (5-25) in the range of d l <y<d h , where s d is the standard deviation and d t is the mean pore throat diameter. Then, assuming that the pore throats smaller than the size, d p , of the suspended particles will be plugged, the flow efficiency factor is estimated by the fraction of pores remaining open at a given time: "I l d \ = l-E p jf(y)dynf(y)dy (5-26) where E p is the plugging efficiency factor. Particles that are sticky and deformable can mold into the shape of pore throats and seal them. Then, the plugging is highly efficient and E p is close to unity. Particles that are rigid and nonsticky cannot seal the pore throats effectively and still allow for some fluid flow. Thus, E < 1 for such plugs. 86 Reservoir Formation Damage The lower and upper bounds of the pore throat size range are estimated by a simultaneous solution of the non-linear integral equations given by: (5-27) a h \yf(y)dy = (5-28) for which the mean pore throat size is estimated by solving the follow- ing equation which relates the pore throat size variation to the rate of deposition: , t>0 (5-29) where k 6 is a rate constant and e p is the volume of deposition per unit bulk volume, subject to the initial mean pore throat diameter, either determined from the initial pore throat size distribution using Eq. 5-28, or estimated as a fraction of the mean pore diameter using: (5-30) Note that r\ is not a fraction because it is a lumped coefficient including the mentioned fraction, some unit conversion factors, and the shape factor. Chang and Civan (1991, 1992, 1997) considered that the pore throat and particle diameters can be better represented by bimodal distribution functions over finite diameter ranges, given by Popplewell et al. (1989) as: (5-31) where w is an adjustable weighting factor in the range of 0 < w < 1, and ./i(y) and/ 2 (j) denote the distribution functions for the fine and coarse fractions, each of which are described by: (5-32) Permeability Relationships 87 with different values of the parameters a, ra, d t , and d h . Chang and Civan (1991, 1992, 1997) used the critical particle diameter, \d p ] , necessary for pore throat jamming, determined according to the criteria described in Chapter 8. For applications with multiphase flow systems, Liu and Civan (1993, 1994, 1995, 1996) used a simplified empirical equation for permeability reduction in porous media as: (5-33) where K 0 and <|> 0 are the reference permeability and porosity, K f , is the residual permeability of plugged formation, and / is a flow efficiency factor given by: (5-34) where i and / denote the species and phases, k f are some rate constants and e u are the quantity of the pore throat deposits. The instantaneous porosity is given by: (5-35) where (e, /) is amount of surface deposits. The Plugging-Nonplugging Parallel Pathways Model The porous media realization is based on the plugging and nonplugging pathways concept according to Gruesbeck and Collins (1982). Relatively smooth and large diameter flowpaths mainly involve surface deposition and are considered nonplugging. Flowpaths that are highly tortuous and having significant variations in diameter are considered plugging. In the plugging pathways, retainment of deposits is assumed to occur by jam- ming and blocking of pore throats when several particles approach narrow flow constrictions. Deposits that are sticky and deformable usually seal the flow constrictions (Civan, 1990, 1994, 1996). Therefore, conductivity of a flow path may diminish without filling the pore space completely. Fluid seeks alternative flow paths until all the flow paths are eliminated. 88 Reservoir Formation Damage Then the permeability diminishes even though the porosity may be nonzero. Another important issue is the criteria for jamming of pore throats. As demonstrated by Gruesbeck and Collins (1982) experimentally for perforations, the probability of jamming of flow constrictions depends strongly on the particle concentration of the flowing suspension and the flow constriction-to-particle diameter ratio. The pore plugging mechanisms are analyzed considering an infini- tesimally small width slice of the porous core. The total cross-sectional area, A, of the porous slide can be separated into two parts: (1) the area A p , containing pluggable paths in which plug-type deposition and pore filling occurs, and (2) the area, A np , containing nonplugging paths in which nonplugging surface deposition occurs. Thus, the total area of porous media facing the flow is given by: A - A p + A np (5-36) The fractions of the bulk volume containing the plugging and nonplugging pathways can be estimated by (Civan, 1995, 1996): f = (5-37) fnp = A npI A ~ §np/$ (5-38) where <|> p , § np , and <() denote the porosities of the plugging, nonplugging, and overall flow pathways. Thus, by definition: /„+/«, = ! (5-39) The fraction of the plugging pathways is a characteristic property of porous media and the particles of the critical size, comparable or larger than the pore throat size (Gruesbeck and Collins, 1982; Schechter, 1992). As explained in Chapter 8, the pore size distribution of the porous medium and the size distribution of the particles determine its value. However, its value varies because the nonplugging pathways undergo a transition to become plugging during formation damage. The volumetric flow rate, q, can also be expressed as a sum of the flow rates, q p and q np , through the pluggable and nonpluggable paths as: q = q p + q np (5-40) . Permeability Relationships 89 The volumetric flows and volumetric fluxes are related by the fol- lowing expressions: q = Au (5-41) Vnp = A np U ip~'np (5-42) (5-43) Thus, by means of Eqs. 5-36 through 43 the total superficial flow is expressed as (Gruesbeck and Collins, 1982): 'p p Jnp np (5-44) Applying the Darcy law, the volumetric fluxes through the porous media and the pluggable and nonpluggable paths can be expressed as: (5-45) (5-46) (5-47) K p and K np represent the permeabilities of the pluggable and non- pluggable fractions of the core. Assuming that the plugging and non- plugging paths are interconnective and hydraulically communicating, the pressure gradients are taken equal: (5-48) Then, it can be shown that, the average permeability of the porous medium is given by (Civan, 1992; Schechter, 1992): K = f K + f K (*)—4Q^ J p p J np no \ ' and the superficial flows in the plugging and nonplugging pathways are given respectively, by: 90 Reservoir Formation Damage u p =uK p /K (5-50) u np =uK np /K (5-51) Let <|> and <j) K denote the initial pore volume fractions of the plugging and nonplugging pathways of the porous media (Civan, 1995) and £ p and £ np represent the fractions of the bulk volume occupied by the deposits in the respective pathways. Thus, the instantaneous porosities in the plugging and nonplugging pathways are given by: Po (5-52) $ np =$n Po -z n p (5-53) Total instantaneous and initial porosities are given, respectively, by: (b = (b + (b (5—54) (|) = (j> + cj) (5—55) The total deposit volume fraction and the instantaneous available porosity are given by: e = e p + z np (5-56) (J) = (|> o — e (5—57) The permeabilities of the plugging and nonplugging pathways are given by the following empirical relationships by Civan (1994) by generalizing the expressions given by Gruesbeck and Collins (1982): K = (5-58) and (5-59) Permeability Relationships 91 where n { and n 2 are the permeability reduction indices, a is a coefficient and K and K npg are the permeabilities at the reference porosities § PO and § np of the plugging and nonplugging pathways, respectively. Eq. 5-58 represents the snow-ball effect of plugging on permeability, while Eq. 5-59 expresses the power-law effect of surface deposition on perme- ability. Eqs. 5-58 and 5-59 have been also verified by Gdanski and Shuchart (1998) and Bhat and Kovscek (1999), respectively, using experi- mental data. Bhat and Kovscek (1999) have shown that the power-law exponent in Eq. 5-59 can be correlated as a function of the coordination number and the pore body to throat aspect ratio, applying the statistical network theory for silica deposition in silicaous diatomite formation. Note that, for n 2 <0 and £ np /$ npg «1, Eq. 5-59 simplifies to the expression given by Gruesbeck and Collins (1982): (5-60) where b = n 2 /§ npg . Eq. 5-60 is a result of a truncated series approxi- mation of Eq. 5-59. Thus, substitution of Eqs. 5-58 and 59 into Eq. 5-49 results in the following expression for the permeability of the porous media (Civan, 1994, 1996): K = /,*, o exp(-ae J) + f np K npg (l - z np /<$> npo )" 2 (5-61) Cernansky and Siroky (1985) proposed an empirical relationship as: l ~ E expGl-—M-l expG-1 11 K 0 " (5-62) where E and G are some empirical constants. It can be shown that K = K 0 for £ = 0. Civan (2000) pointed out that, when E= 1, Eq. 5-62 yields a linear model as: (5-63) When E = 0, Eq. 5-62 yields a nonlinear model as: [...]... Oklahoma, April 21- 24, 19 96, 51 1 - 51 7 Kozeny, J., "Uber Kapillare Leitung des Wasser im Boden," Sitzungsber, Akad Wiss Wien, No 13 6, 19 27, pp 2 71- 106 Liu, X., & Civan, F., "Characterization and Prediction of Formation Damage in Two-Phase Flow Systems, SPE 254 29 paper, Proceedings of the SPE Production Operations Symposium, March 21- 23, 19 93, Oklahoma City, Oklahoma, March 21- 23, 19 93, pp 2 31- 248 Liu, X.,... Formation Damage by Organic Deposition," Proceedings of the First International Symposium on Colloid Chemistry in Oil Production: Asphaltenes and Wax Deposition, ISCOP' 95, Rio de Janeiro, Brazil, November 26-29, 19 95, pp 10 2 -10 7 Civan, R, "A Multi-Purpose Pormation Damage Model," SPE 311 01, Proceedings of the SPE Formation Damage Symposium, Lafayette, Louisiana, February 14 - 15 , 19 96, pp 311 -326 Civan,... Chemistry, February 14 -17 , 19 95, San Antonio, TX Liu, X., & Civan, F., "Formation Damage and Filter Cake Buildup in Laboratory Core Tests: Modeling and Model-Assisted Analysis," SPE Formation Evaluation J., Vol 11 , No 1, March 19 96, pp 26-30 Liu, X., Civan, F., & Evans, R D., "Correlation of the Non-Darcy Flow Coefficient, J of Canadian Petroleum Technology, Vol 34, No 10 , 19 95, pp 50 -54 Liu, X., Ormond,... Unconsolidated Sands," In Situ, Vol 12 , No 3, 19 88, pp 209-226 Rege, S D., & Fogler, H S., "Network Model for Straining Dominated Particle Entrapment in Porous Media," Chemical Engineering Science, Vol 42, No 7, 19 87, pp 15 53 - 15 64 Rege, S D., & Fogler, H S., "A Network Model for Deep Bed Filtration of Solid Particles and Emulsion Drops," AIChE J., Vol 34, No 11 , 19 88, pp 17 61- 1772 Reis, J C., & Acock, A M.,... Interactions and Particulate Processes," SPE 211 83 paper, Proceedings of the SPE 19 90 Latin American Petroleum Engineering Conference, October 14 -19 , 19 90, Rio de Janeiro, Brazil, 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, 19 92, Lafayette, Louisiana, pp 219 -236 Civan,... AAPG Bulletin (November 19 97) Vol 81, No 11 , 18 94 -19 08 Carman, P C, "The Determination of the Specific Surfaces of Powders I," July 19 38, pp 2 25- 234 Carman, P C., Flow of Gases Through Porous Media, Butterworths, London, 19 56 Cernansky, A., & Siroky, R., "Deep-bed Filtration on Filament Layers on Particle Polydispersed in Liquids," Chemicky Prumysl, Vol 32 (57 ), No 8, pp 397-4 05, 19 82 Cernansky, A., &... Eq 5- 49, which considers Nnd = 1 for Darcy flow Gdanski and Shuchart (19 98) have correlated their permeability vs porosity measurements obtained during sandstone-acidizing by: 1/ 3 .5 -2 (5- 89) They pointed out that Eq 5- 89 is remarkably similar to Civan's (19 94) equation (Eq 5- 58) Note that, for permeability enhancement by acid stimulation, Eq 5- 58 can be written as: (5- 90) By comparison of Eqs 5- 89... June 15 -17 , 19 92 Ochi, J., and Vernoux, J.-F, "Permeability Decrease in Sandstone Reservoirs by Fluid Injection-Hydrodynamic and Chemical Effects," J of Hydrology (19 98) 208, 237-248 Okoye, C U., Onuba, N L., Ghalambor, A., and Hayatdavoudi, A., "Characterization of Formation Damage in Heavy Oil Formation During Steam Injection," paper SPE 19 417 , presented at the 19 90 SPE Permeability Relationships 10 1... comparison of Eqs 5- 89 and 5- 90, show that the parameter values are Kpg = 0. 01 md, (^ = 0, a = 15 and n{ = 0.29 (Civan, 2000) 3 Show that Eq 5- 60 can be derived by a truncated series approximation of Eq 5- 59 for n2 < 0 and £np/$npo«l (Civan, 19 94) References Adin, A., "Prediction of Granular Water Filter Performance for Optimum Design," Filtration and Separation, Vol 15 , No 1, 19 78, p 55 -60 Adler, P M., Jacquin,... Fluids and Reservoir Rocks," SPE 22 856 paper, Proceedings of the 66th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers, October 6-9, 19 91, Dallas, Texas Chang, F F., & Civan, F., "Practical Model for Chemically Induced Formation Damage, " / of Petroleum Science and Engineering, Vol 17 , No 1/ 2, February 19 97, pp 12 3 -13 7 Civan, F., "A Generalized Model for Formation Damage . 19 95, pp. 10 2 -10 7. Civan, R, "A Multi-Purpose Pormation Damage Model," SPE 311 01, Proceedings of the SPE Formation Damage Symposium, Lafayette, Louisiana, February 14 - 15 , . Dissolution/Precipitation," paper SPE/DOE 353 94, proceedings of the 19 96 SPE/DOE Tenth 10 0 Reservoir Formation Damage Symposium on Improved Oil Recovery held in Tulsa, Oklahoma, April 21- 24, 19 96, 51 1 - 51 7 . Kozeny, . by: A P = (5 -10 ) Therefore, equating Eqs. 5- 9 and 10 , and substituting Eqs. 5 -1 and 10 results in the following relationship for the mean hydraulic tube diameter: D h = (5 -11 ) Equating