Reservoir Formation Damage Episode 1 Part 9 docx

Part Formation Damage by Particulate Fines Mobilization, Migration, and Deposition III Proccesses Single-Phase Formation Damage by Fines Migration and Clay Swelling* Summary A review of the primary considerations and formulations of the various single-phase models for formation damage by fines migration and clay swelling effects is presented. The applicability and parameters of these models are discussed. Introduction The majority of the formation damage models were developed for single phase fluid systems. This assumption is valid only for very specific cases such as the production of particles with oil flow and for special core tests. Nevertheless, it is instructive to understand these models before looking into the multi-phase effects. Therefore, the various processes involving single-phase formation damage are discussed and the selected models available are presented along with some modifications and critical evaluation as to their practical applicability and limitations. The method- ology for determination of the model parameters are presented. The parameters that can be measured directly are identified. The rest of the parameters are determined by means of a history matching technique. The applications of the models and the parameter estimation method are demonstrated using several examples. Parts reprinted by permission of the Society of Petroleum Engineers from Civan, ©1992 SPE, SPE 23787 paper, and by permission of the U.S. Department of Energy from Civan, 1994. 183 Chapter 10 184 Reservoir Formation Damage An evaluation and comparison of six selected models bearing direct relevance to formation damage prediction for petroleum reservoirs are carried out. The modeling approaches and assumptions are identified, interpreted, and compared. These models are applicable for special cases involving single-phase fluid systems in laboratory core tests. Porous media is considered in two parts: (1) the flowing phase, denoted by the subscript /, consists of a suspension of fine particles flowing through and (2) the stationary phase, denoted by the subscript s, consists of the porous matrix and the particles retrained. The Thin Slice Algebraic Model Model Formulation Wojtanowicz et al. (1987, 1988) considered a thin slice of a porous material and analyzed the various formation damage mechanisms assum- ing one distinct mechanism dominates at a certain condition. Porous medium is visualized as having tortuous pathways represented by N h tubes of the same mean hydraulic equivalent diameter, D h , located between the inlet and outlet ports of the core as depicted in Figure 10-1. The cross- sectional area of the core is A and the length is L. The tortuosity factor for the tubes is defined as the ratio of the actual tube length to the length of the core. -c = L h /L (10-1) The cross-sectional area of the hydraulic tubes are approximated by (10-2) in which C l is an empirical shape factor that incorporates the effect of deviation of the actual perimeter from a circular perimeter. As a suspension of fine particles flows through the porous media, tubes having narrow constrictions are plugged and put out of service. If the number of nonplugged tubes at any given time is denoted by N np and the plugged tubes by N p , then the total number of tubes is given by: N, = N +N ^ h ly p T "n The area open for flow is given by (10-3) (10-4) oo 5' •n o 3 to OQ CD Of Figure 10-1. Hydraulic tubes realization of flow paths in a core (after Civan, 1994; reprinted by permission of the U.S. Department of Energy; after Civan 1992 SPE, reprinted by permission of the Society of Petroleum Engineers). O to d. n 3 00 00 186 Reservoir Formation Damage The Darcy and Hagen-Poiseuille equations given respectively by (10-5) and (10-6) are considered as two alternative forms of the porous media momentum equations, q is the flowrate of the flowing phase and Ap is the pressure differential across the thin core slice. Thus, equating Eqs. 10-5 and 10-6 and using Eqs. 10-1 and 10-2 the relationship between permeability, K, and open flow area, A is obtained as: K = A f A h I C\ in which the new constant is defined by C\ = (10-7) (10-8) The permeability damage in porous media is assumed to occur by three basic mechanisms: (1) gradual pore reduction (pore narrowing, pore lining) by surface deposition, (2) single pore blocking by screening (pore throat plugging) and (3) pore volume filling by straining (internal filter cake formation by the snowball effect). Gradual pore reduction is assumed to occur by deposition of particles smaller than pore throats on the pore surface to reduce the cross-sectional area, A, of the flow tubes gradually as depicted in Figure 10-2. Thus, the number of tubes open for flow, N np , at any time remains the same as the total number of tubes, N h , available. Hence, N h =N np ,N p =Q (10-9) Then, using Eq. 10-9 and eliminating A between Eqs. 10-4 and 10-7 leads to the following equation for the permeability to open flow area relationship during the surface deposition of particles: (10-10) (10-11) in which the new constant is defined by c _ c N 1/2 ^—"i ~~~ *—"> * * fc Single pore blocking is assumed to occur by elimination of flow tubes from service by plugging of a pore throat or constriction, that may exist Single-Phase Formation Damage by Fines Migration and Clay Swelling 187 Surface deposits Figure 10-2. Pore surface deposition in a core (after Civan, 1994; reprinted by permission of the U.S. Department of Energy; after Civan 1992 SPE; reprinted by permission of the Society of Petroleum Engineers). somewhere along the tube, by a single particle to stop the flow through that particular tube. Therefore, the cross-sectional areas of the individual tubes, A h , do not change. But, the number of tubes, N np , open for the flow is reduced as depicted in Figure 10-3. The area of the tubes eliminated from service is given by: A p =N p A h (10-12) The number of tubes plugged is estimated by the ratio of the total volume of pore throat blocking particles to the volume of a single particle of the critical size. (10-13) The critical particle size, d, is defined as the average size of the critical pore constrictions in the core. f d is the volume fraction of particles in the flowing phase, having sizes comparable or greater than d. p p is the particle grain density. p p/ is the mass concentration of particles in the flowing suspension of particles. Because A h is a constant, Eq. 10-7 leads to the following permeability to open flow area relationship: 188 Reservoir Formation Damage Plugged tube Nonplugged tube ^ ^ S~A Figure 10-3. Pore throat plugging in a core (after Civan, 1994; reprinted by permission of the U.S. Department of Energy; after Civan 1992 SPE; reprinted by permission of the Society of Petroleum Engineers). A f = C 4 K in which the new constant is given by: C — C 2 I A <- — <- ' ™- (10-14) (10-15) Pore filling occurs near the inlet face of the core when a suspension of high concentration of particles in sizes larger than the size of the pore throats is injected into the core as depicted in Figure 10-4. The per- meability, K c , of the particle invaded region decreases by accumulation of particles. But, in the uninvaded core region near the outlet, the permeability of the matrix, K m , remains unchanged. The harmonic mean permeability, K, of a core section (neglecting the cake at the inlet face) can be expressed in terms of the permeability, K c , of the L c long pore filling region and the permeability, K m , of the L m long uninvaded region as (10-16) (10-17) which can be written as: K(t) = L/[L c R c (t) Single-Phase Formation Damage by Fines Migration and Clay Swelling 189 + \ ^ L r L c *^ i L m H Filter Cake Region Figure 10-4. Pore filling and internal filter cake formation in a core (after Civan, 1994; reprinted by permission of the U.S. Department of Energy; after Civan 1992 SPE; reprinted by permission of the Society of Petroleum Engineers). R c (t) and R m are the resistances of the pore filling and uninvaded regions defined by R c (t) = \lK c (t) (10-18) R m = l/K m (10-19) The rate of increase of the filtration resistance of the pore filling particles is assumed proportional to the particle mass flux of the flowing phase according to: dR c /dt = (k c /L c )(q/A)p pf subject to the initial condition Then, solving Eqs. 10-20 and 21 yields: (10-20) (10-21) (10-22) k c is the pore filling particles resistance rate constant. 190 Reservoir Formation Damage The instantaneous porosity of a given cross-sectional area is given by: ty = ty 0 -£ p (10-23) (|) 0 and (() denote the initial and instantaneous porosity values, e is the fractional bulk volume of porous media occupied by the deposited particles, given by e p =m p /p p (10-24) m p is the mass of particles retained per unit volume of porous media and p p is the particle grain density. For convenience, these quantities can be expressed in terms of initial and instantaneous open flow areas, A fo and Ay> and the area covered by the particle deposits, A , as ty = A f /A (10-25) $ 0 =A fo /A (10-26) e p =A p /A (10-27) Substituting Eqs. 10-25 through 10-27, Eqs. 10-23 and 24 become, respectively A /0 =A /+ A p (10-28) A p =Am p /p p (10-29) The particle mass balance for a thin core slice is given by: + m )} = 0 (10-30) subject to the initial conditions: (10-31) and (Ppf) are the particle mass concentrations in the flowing phase at themlet and outlet of the core. Wojtanowicz et al. (1987, 1988) omitted the accumulation of particles in the thin core slice and simplified Eq. 10-30 to express the concentration of particles leaving a thin section by: Single-Phase Formation Damage by Fines Migration and Clay Swelling 191 The rate of particle retention on the pore surface is assumed proportional to the particle mass concentrations in the flowing phase according to: r r =(dm p /dt) r =k r p pf (10-33) The rate of entrainment of the surface deposited particles by the flowing phase is assumed proportional to the mass of particles available on the pore surface according to: k e m p (10-34) Then, the net rate of deposition is given as the difference between the retention and entrainment rates as: dm p /dt = k r p pf -k e m p (10-35) subject to the initial condition given by: 0 (10-36) Diagnostic Equations for Typical Cases Wojtanowicz et al. (1987, 1988) have analyzed and developed diagnostic equations for two special cases of practical importance: 1 . Deposition of the externally introduced particles during the injection of a suspension of particles 2. Mobilization and subsequent deposition of the indigeneous particles of porous medium during the injection of a particle free solution Deposition of Externally Introduced Particles Three distinct permeability damage mechanisms are analyzed for a given injection fluid rate and particle concentration. As depicted in Figure 10-4, particles are retained mainly in the thin core section near the inlet face. In this region the concentration of the flowing phase is assumed the same as the injected fluid (i.e., p p / -(P p f). ). Gradual pore reduction by surface deposition occurs when the particles of the injected suspension are smaller than the pore constrictions. Assume that the surface deposition is the dominant mechanism compared to the entrainment, that is, k r »k e . [...]... liquid are constant, the volumetric balance of particles in porous media is given by: 9/ 9?((|>(j + e) + 9/ 9jt(a«) = 0 (10 -11 9) Substituting Eq 10 -10 0 into Eq 10 -11 9, and then rearranging, an alternative convenient form of Eq 10 -11 9 can be obtained as: ($0 - e) 9a /9? + 3/9x(ou) + (1 - a) de/9r = 0 (10 -12 0) Following Gruesbeck and Collins ( 19 82), Eq 10 -12 0 can be simplified for cases where e and a are... fpKp + fnpKnp (10 -11 6) Note that Eq 10 -11 6 was derived independently by Civan ( 19 92 ) and Schechter ( 19 92 ) and is different than the expression given by Gruesbeck and Collins ( 19 82) The superficial flows in the plugging and nonplugging pathways are given, respectively, by: up=uKp/K (10 -11 7) 206 Reservoir Formation Damage unp=uKnp/K (10 -11 8) Considering that the physical properties of the particles and the... Eqs 10 -46, 12 , 13 and 14 into Eq 10 -28 yields the following diagnostic equation: K/K0 = l-C 7 [l-exp(-M)] (10 - 61) in which C7 = 6fdALmpJ(nd3ppC4) (10 -62) Cake Formation If the permeability damage is by cake formation, then substituting Eqs 10 -46 and 22 into Eq 10 -17 yields the following diagnostic equation KJK = l + C10[l - exp(-*ef)] (10 -63) in which Clo=kcmpJRm (10 -64) 19 6 Reservoir Formation Damage. .. Single-Phase Formation Damage by Fines Migration and Clay Swelling 205 and the mean pore diameter is given by: (10 -11 0) where C is an empirical shape factor It can be shown that Eqs 10 -10 1 and 10 5 simplify to the deposition rate equations given by Gruesbeck and Collins ( 19 82): (10 -11 1) (10 -11 2) where c1? c 2 , c 3 ,and c4 are some empirically determined coefficients This requires that the effects of the permeability.. . 19 2 Reservoir Formation Damage Then, the solution of Eqs 10 -35 and 36 yields: (10 -37) A substitution of Eq 10 -37 into Eq 10 - 29 leads to the following expression for the area occupied by the surface deposits (10 -38) Substitution of Eqs 10 -10 and 38 into Eq 10 -28 yields the following diagnostic equation: (K/K0)l/2=\-C5t (10 - 39) in which the empirical constant is given by (10 -40) Single... , , r te « 1 c ta " « Pore throat blocking T1 (K/K )vs t Simplified pore surface deposition and sweeping for negligibfe initial particle content Pore surface deposition Eq # T\ 1 11 / vs./ f r/ , \ i/ ) Tl-8 vs.t * After Wojtanowicz et al., 19 87, 19 88; Civan, 19 92 SPE; reprinted by permission of the Society of Petroleum Engineers, and Civan, 19 94 ; reprinted by permission of the U.S Department of... plugging and nonplugging pathways of the porous media (Civan, 19 94 , 19 95 ) PULHJU, F=t> on on •==> Figure 10 -6 Non-plugging and plugging paths realization in a core (after Civan, 19 94 ; reprinted by permission of the U.S Department of Energy; after Civan 19 92 SPE; reprinted by permission of the Society of Petroleum Engineers) Single-Phase Formation Damage by Fines Migration and Clay Swelling 203 These values... kpUpGpp ep=e/v t = 0 (10 -10 1) (10 -10 2) where \ ~\ 1 ) \l-e~ket) K /K -I C 01 10 et (l-e ) Tl-4a ' / > \n \K/K0) =1 + C7 ,,/., n K K =1 L I 0 - 1 fit e Pore surface sweeping Pore filling ~1 IK /K\vs.t 1 / , \i/2 \K/K0... substituting Eq 10 -46, Eq 10 -35 yields the following ordinary differential equation: 19 4 Reservoir Formation Damage dmp/dt + kemp = [krkeALmpg /q]exp(-ket) (10 - 49) The solution of Eq 10 - 49, subject to the initial condition mp=m*p (previously deposited particles), is obtained by the integration factor method as: (10 -50) in which Cu=krkeALmp/q (10 - 51) Then, the area occupied by the remaining particles is . from Civan, 19 94. 18 3 Chapter 10 18 4 Reservoir Formation Damage An evaluation and comparison of six selected models bearing direct relevance to formation damage prediction for petroleum reservoirs . condition Then, solving Eqs. 10 -20 and 21 yields: (10 -20) (10 - 21) (10 -22) k c is the pore filling particles resistance rate constant. 19 0 Reservoir Formation Damage The instantaneous porosity . Wojtanowicz et al., 19 87, 19 88; Civan, 19 92 SPE; reprinted by permission of the Society of Petroleum Engineers, and Civan, 19 94; reprinted by permission of the U.S. Department of