Reservoir Formation Damage Episode 2 Part 9 pot

432 Reservoir Formation Damage The rate of deposition in the nonplugging tubes can be expressed by (Civan, 1994, 1995, 1996): subject to the initial condition Here, (14-76) (14-77) * 0 , T W > k = 0 , i < T (14-78) (14-79) For simplification purposes, Civan (1995) assumed that organic deposits are sticky and, therefore, once deposited they cannot be removed. Con- sequently, the second term in Eq. 14-76 can be dropped. Mansoori (1997) tends to support this argument. Although Leontaritis (1998) considered the possibility of erosion of deposits, it is not apparent if he actually implemented this possibility in his calculational steps. k d and k e are the surface deposition and mobilization rate constants. r\ e is the fraction of the uncovered deposits estimated by: (14-80) i cr is the minimum shear stress necessary to mobilize the surface deposits. i w is the wall shear-stress given by the Rabinowitsch-Mooney equation (Metzner and Reed, 1995): T w =*'(8v/D)"' (14-81) in which the interstitial velocity, v, is related to the superficial velocity, u, by: and the mean pore diameter is given by: 1/2 (14-82) (14-83) Formation Damage by Organic Deposition 433 The permeabilities of the plugging and nonplugging pathways are given by the following empirical relationships (Civan, 1994): K = exp [-a(<|>, o - exp (-aej) (14-84) and (14-85) Then, the average permeability of the porous medium is given by: K = f p K p +f np K np (14-86) The superficial flows in the plugging and nonplugging pathways are given respectively, by: u p =uK p /K (14-87) » v ="Knp/K (14-88) The total superficial flow is given by (Gruesbeck and Collins, 1982): u = f p u p +f»p» H p (14-89) Considering the simultaneous deposition of paraffins and asphaltenes, e p and e np in Eqs. 14-62 through 69 denote the sum of the paraffins and asphaltenes, that is, = F + F ° ~ ^ np, par ' ^np, asp (14-90) (14-91) Description of Fluid and Species Transport The preceding treatment of the porous media impairment phenomena implies that the suspended particle and dissolved species concentrations may be different in the plugging and nonplugging pathways. Then, separate sets of balance equations are required for the plugging and 434 Reservoir Formation Damage nonplugging pathways. Consequently, the numerical solution would require highly intensive computational effort. However, this problem can be conveniently circumvented by assuming that there is hydraulic inter- action between these pathways (i.e., they are not isolated from each other). 1. The mass balances are considered for the following four pseudo- components a. Gas b. Oil c. Suspended paraffins and asphaltenes d. Dissolved paraffins and asphaltenes 2. Total thermal equilibrium energy balance is considered 3. Non-Newtonian fluid description using the Rabinowitsch-Mooney equation is resorted 4. The Forhheimer equation for the non-Darcy flow description is used 5. The average flow is defined as a volume fraction weighted linear sum of the flow through the plugging and nonplugging paths accord- ing to Gruesbeck and Collins (1982) 6. The average permeability is defined as a volume weighted linear sum of the permeabilities of the plugging and nonplugging paths according to Gruesbeck and Collins (1982) a. In the plugging paths, a snowball deposition effect is represented by an exponential decay function b. In the nonplugging paths, a gradual pore size reduction, repre- sented by the power law function, is considered 7. The precipitation of the asphaltene and paraffin is predicted, apply- ing Chung's (1992) thermodynamic model for non-ideal solutions to determine the cloud point and the quantity of the precipitates to be formed. The total mass balance of the gas component is given by: — v + p L u L w g L (14-92) The first, second, and third terms respectively represent the accumu- lation, transport, and well production. Assuming that the oil component exists only in the liquid phase and does not vaporize into the gas phase, the oil component mass balance is given by: — Formation Damage by Organic Deposition 435 (14-93) for which Ring et al. (1994) assumed w oL = 1.0. Considering that organic precipitates only exist in the liquid phase, because it is the wetting phase for these particles, the suspended paraffin and asphaltene particle mass balances are expressed by: — ; p = asphaltene, paraffin (14-94) Note that (14-95) If the particle density, p p , is assumed constant, and the suspended particle content is expressed by the volume fraction of organic substance (paraffin or asphaltene), C7 pL , according to Eq. 14-95, then Eq. 14-94 can be simplified as: 3 I da_ L ~| — | §S L D pL —^— I; p = asphaltene or paraffin (14-96) Note that both Ring et al. (1994) and Civan (1996) neglected the term on the right, representing the dispersion of particles. The mass balances of the paraffin and asphaltene dissolved in oil is given by: d/dx(p L u L x iL ) = 9/9jt[(J)S L £> L 3/3jt(p L Jt lL )] : i = asphaltene or paraffin (14-97) S is the saturation, p is the density, t is the time, x is distance, u is the volume flux, G p L is the volume fraction of the organic precipitates in 436 Reservoir Formation Damage the liquid phase, w p<L denotes the mass fraction, x iL is the mole fraction of organic dissolved in the oil, M i is the molecular weight and D iL is the dispersion coefficient. 3e,/3f represents the volume rate of retention of organic deposits in porous media determined according to Eqs. 14-68, 69, and 76. Assuming that the various phases are at thermal equilibrium at a temperature of T v = T L = T s = T , the total porous media energy balance is given by: par (p yUyHy + P L u L H L ) + <$>S v q v + $S u v dP v /dx + u (14-98) + B par k par + £ asp k asp where U and H are the internal energy and enthalpy, respectively, q is the energy loss, p is pressure, k denotes the thermal conductivity, and T is temperature. Ring et al. (1994) simplified Eq. 14-98 as: (14-99) The first, second, and third terms represent the accumulation, con- vection, and conduction heat transfer. The last terms represent the heat carried away by production at wells, heat losses into formation sur- rounding the reservoir and the external heat losses. The deposition of organic precipitates in porous media reduces the flow passages causing the fluids to accelerate. Therefore, Darcy's law is modified as following, considering the inertial effects, according to the Forchheimer equation (Civan, 1996): Formation Damage by Organic Deposition 437 '3jc : J = VorL (14-100) where K is the permeability, PJ is the fluid pressure, and the non-Darcy number is given by: N^ =(l + Re y )" 1 (14-101) in which the porous media Reynolds number is given by: Re y = p y UjK$/\ij (14-102) where P is the inertial flow coefficient, and p y and |ii y denote the density and viscosity of a fluid phase J. Note that the formulations presented here are applicable for jnulti- dimensional cases encountered in the field if 3/3x is replaced by V • and a vector-tensor notation is applied. Phase Transition The source terms appearing on the right of Eqs. 14-92 through 99 are considered a sum of the internal (rock-fluid and fluid-fluid interactions) and external (wells) sources. When the oil is supersaturated, the internal contribution to the source terms in Eq. 14-94 is determined as the excess quantity of organic content of oil above the organic solubility at saturation conditions determined by Chung's (1992) thermodynamic model: P ,L P,L m pL = Q,x p L <x s p L ; p = asphaltene or paraffin (14-103) (14-104) where jc* L represents the mole-fraction of dissolved organic at saturation. Civan (1995) carried out case studies similar to Ring et al. (1994) using the Sutton and Roberts data (1974). Figure 14-51 shows a comparison of the predicted and measured permeability impairments by paraffin deposition for below and above bubble point pressure cases. Note that, above the bubble point pressure, only the liquid phase exists and there is more severe formation damage. Whereas, below the bubble point pressure, both the liquid and vapor phases exist and there is less severe formation damage. 438 Reservoir Formation Damage 2 4 Pore Volume Throughput, PV > Below bubble point-Sutton and Roberts data i Above bubble point-Sutton and Roberts data — Below bubble point- Simulation - -Above bubble point- Simulation Figure 14-51. Comparison of the Sutton and Roberts (1974) experimental data and simulation results for permeability reduction by organic deposition below and above bubble point pressure. Single-Porosity and Two-Phase Model for Organic Deposition Ring et al. (1994) developed a two-phase model considering only the paraffin precipitation. They assumed that (1) oil is always saturated with the paraffin, (2) the solution is ideal, (3) paraffin deposition obeys a first order kinetics, (4) pores undergo an irreversible continuous plugging, and (5) permeability reduction obeys a power law: (14-105) Ring et al. (1994) determined that m = 8 for paraffin deposition. Wang et al. (1999) developed an improved model considering the simultaneous deposition of asphaltenes and paraffins. Wang et al. (1999) Formation Damage by Organic Deposition 439 model incorporates the features of Civan's (1995) dual-porosity model for a single-porosity treatment. The formulation of the Wang et al. (1999) model and its experimental verification are described in the following. Formulation* Wang et al. (1999) used the ideal-solution theory to predict the solubility and precipitation of paraffin and asphaltene in crude oil; an improved one- dimensional, three-phase, and four-pseudo-component model to represent the transport of paraffin and asphaltene precipitates; and a deposition model including the static and dynamic pore surface depositions and pore throat plugging to describe the deposition of the paraffin and asphaltene. The model was developed for analysis of the laboratory core flow tests. The gravity effects and the capillary pressure effects between vapor and liquid phases have been neglected. The oil, gas, and solid phases were assumed at thermal equilibrium. The oil, gas, paraffin, and asphaltene pseduo-components are denoted by O, G, P, and A, respectively. The vapor and the liquid phases are denoted by V and L, respectively. Considering both the free and dissolved gases, the gas component mass balance equation is given by: ~\ — -\ — (14-106) where $ represents the porosity of the porous media, S v ,p v ,u v are the saturation, density and flux of the vapor phase, respectively, and S L ,p L ,u L are the saturation, density and flux of the liquid phase, respectively. W GL represents the mass fraction of the dissolved gas in the liquid phase. Considering that the oil component exists only in the liquid phase, its mass balance is given by: — ($S L p L w OL ) + — (p L u L w OL ) = 0 (14-107) where W OL is the mass fraction of the oil component in the liquid phase. The paraffin mass balance equation is expressed by considering that it may be partly dissolved and/or suspended as particles in the liquid phase and deposited in porous media: Reprinted by permission of the Society of Petroleum Engineers from Wang et al., ©1999 SPE, SPE 50746 paper. 440 Reservoir Formation Damage -\ — (PL U L W SPL+PL U L W PL) = (14-108) where S P is the saturation of the suspended paraffin in the oil phase and p p is the density of the paraffin. W PL represents the mass fraction of the dissolved paraffin in the liquid phase. W SPL is the mass ratio of the paraffin precipitates suspended in the liquid phase to the liquid phase. £ p is the volume fraction of the deposited paraffin in the bulk porous media. The asphaltene mass balance equation is written similarly as: + PL U L W AL) = -PA (14-109) dt in which S A is the saturation of the suspended asphaltene and p A is the density of asphaltene. W AL represents the mass fraction of the dissolved asphaltene in the liquid phase. W SAL is the mass ratio of the asphaltene precipitates suspended in the liquid phase to the liquid phase. £ A is the volume fraction of the deposited asphaltene in the bulk porous media. The vapor and liquid phase volumetric fluxes are given by Darcy's equation, respectively, as: Kk RV dx (14-110) u, =-• Kk, (14-111) where K is the absolute permeability of the porous media, and k RV and |LL V are the relative permeability and viscosity of the vapor phase, respec- tively. k RL and (J, L are the relative permeability and viscosity of the liquid phase, respectively. Neglecting the capillary pressure between the vapor and liquid phases, p represents the pressure of the pore fluids. The total thermal equilibrium energy balance equation is expressed as: Formation Damage by Organic Deposition 441 (pv u v H v + PL U L H L) = (14-112) where H v ,H L ,H p ,H A and H F are the enthalpies of the vapor, liquid, paraffin, asphaltene, and porous media, respectively. K V ,K L ,K P ,K A and K F are the thermal conductivities of the vapor, liquid, paraffin, asphaltene, and porous media, respectively. T is the equilibrium tempera- ture of the system. The saturations of the vapor and liquid phases, and the paraffin and asphaltene precipitates suspended in the liquid phase, add up to 1: S v + S L + S P + S A = (14-113) The ideal-solution theory (Weingarten and Euchner, 1988; Chung, 1992) is applied for the paraffin and asphaltene solubility predictions, respectively, as: X PL = ex P 1 1 /Mr r, Pm (14-114) X AL = ex P A/Ml 1 R [T T t Am (14-115) where X PL and X AL indicate the mole fractions of the paraffin and asphaltene dissolved in the oil, respectively, and X ps and X As are the mole fractions of the paraffin and asphaltene at saturation. A// P and A// A are the latent heats of fusion of the paraffin and asphaltene, respectively. [...]... 0.0 AA, l/sec 0.0008 . Engineers. Formation Damage by Organic Deposition 445 CASE 3 4 5 6 Oil Properties T, °C (i (at 20 °C),cp API ^(%) M A , g/gmol 50 13 29 5.3 6,000 50 13 29 5.3 6,000 - 5 .27 29 . 29 53® 6,000 - 5 .27 29 . 29 53® 6,000 Core . g/gmol 50 13 29 5.3 6,000 50 13 29 5.3 6,000 - 5 .27 29 . 29 53® 6,000 - 5 .27 29 . 29 53® 6,000 Core Properties L, cm D, cm k ( , darcy A 5.08 2. 3 0.107 0.131 5.08 2. 3 0.0 122 0 .24 3 26 .5 2. 56 0.011 0.35 26 .5 2. 56 0.011 0.35 Data for Simulation Ax, . SPE International Symposium on Formation Damage Control, February 26 -27 , 19 92, Lafayette, Louisiana, pp. 21 9 -23 6. Civan, F, Predictability of Formation Damage: An Assessment Study