Open Access Establishment of Mathematical Model and Sensitivity Analysis of Plugging Capacity of Multi-Component Foam Flooding

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1 Send Orders or Reprints to 84 The Open Chemical Engineering Journal, 2015, 9, Open Access Establishment o Mathematical Model and Sensitivity Analysis o Plugging Capacity o Multi-Component Foam Flooding Chengli Zhang 1,*, Dezhi Liang 1, Haiqing Cui 1, Daiyin Yin 1 and Guoliang Song 2 1 Key Laboratory o Enhanced Oil and Gas Recovery o Ministry o Education, Northeast Petroleum University 1, Daqing, , China 2 College o Mathematics and Statistics, Northeast Petroleum University 2, Daqing, , China Abstract: Based on mass conservation, oam total amount balance thought and oam properties characterization model, a oam looding mathematical model with three-phase and ive components is established and numerically solved. The explicit is used to solve the saturation equation and the implicit is used to solve the pressure equation in this model. By itting the results o oam plugging capacity experiments and numerical calculation, the correctness o the model is veriied. On this basis, the conceptual model is established to simulate and evaluate the sensitivity o inluencing parameters o oam looding plugging ability. The results show that: the plugging ability o oam system increases with the increase o oaming agent concentration. When the concentration o oaming agent is more than 0.3%, the resistance coeicient tends to steady; when gas-liquid ratio is 2:1, the oam plugging ability is the strongest. The plugging ability o oam system increases with the increase o permeability showing good permeability selectivity. I the permeability in the reservoir is higher than the proile, control capability will also be stronger. Keywords: Numerical calculation, oam composite system looding, mathematical model, plugging capacity, sensitivity analysis. 1. INTRODUCTION Because water looding recovery can only reach about 40%, so we can enhance oil recovery by the tertiary oil recovery. A disadvantage o the conventional suractant looding is that it can not achieve mobility control and easily orm a cross low phenomenon. To solve this problem, the displacing luid resistance must be increased. I the oaming properties o the suractant which is injected is good, then the oam looding under the reservoir conditions can be ormed [1]. It can be applied to high salinity, high temperature reservoirs and can avoid the cross low phenomenon. The oam has a high viscosity at the same time and it can control mobility and expand swept volume. It can also reduce the oil-water interacial tension and increase the displacement eiciency, which to be a new research direction o chemical looding. Foam system has a unique seepage characteristic and oam or high permeability layer has strong plugging capacity [2-12], but or low permeability layer it has good swept eect. Foam has a higher apparent viscosity which can signiicantly improve water and oil mobility ratio. This paper established a oam looding mathematical model with three-phase and ive components, veriied its correctness, simulated and evaluated the sensitivity o inluencing parameters o oam looding plugging ability. *Address correspondence to this author at the College o Petroleum Engineering, Northeast Petroleum University, Daqing, , China; Tel: ; Fax: ; zhangchengli0303@163.com 2. MATHEMATICAL MODELS 2.1. Assumed Conditions (1) Considering three-phase: the oil, gas, water and ivecomponents: oil, nitrogen, water, oam, suractant. Nitrogen and oam are included in the gas phase.(2) Suractant is only in the aqueous phase, ignoring its eect on the density and the viscosity o the aqueous phase because o its small concentrations, and ignoring the dissolution o nitrogen in the water. (3) The seepage o the luids in the reservoir and the injected luids satisy the generalized Darcys law. (4) Using a modiied gas phase relative permeability model to characterize the relative permeability o the oam Mass Conservation Equations Oil, gas and water phase mass conservation equations Oil phase: ( ) % kk (! " ro o!( P µ o # $ Z o )* & o * + q o = +," o S o +t ) Water phase: % kk (! " rw w!( P µ w # $ Z w )* & w * + q w = +," w S w +t ) Gas phase: ( ) (1) (2) / Bentham Open

2 Establishment o Mathematical Model and Sensitivity Analysis The Open Chemical Engineering Journal, 2015, Volume 9 85 ( ) % kk ( +," rg! " g! P µ g # $ Z * g g ( )* + q g = g S g +t & ) (3) Where! o,! w,! g - the density o oil phase, water phase and gas phase respectively; φ - porosity; K -the absolute permeability o the rock; K ro, K rw, K rg - the relative permeability o oil phase, water phase and gas phase respectively; µ o,µ w,µ g - the viscosity o the oil phase, water phase and gas phase respectively ; P o, P w, P g - the pressure o oil phase, water phase and gas phase respectively ;! o,! w,! g - the weight o the oil phase, water phase and gas phase respectively ; S o,s w,s g the saturation o oil phase, water phase - and gas phase respectively ; Z - the vertical depth o the reservoir; q -source sink term The Foam Total Amount Balance Equation Foam will regenerate, burst continually when it transports in porous media, since it is a dynamic equilibrium process. The basic thought o oam total amount balance: Foam increment = inlow -outlow + production-burst + source and sink term. Establish the oam total amount balance equation [5-12]:!!t ( )% & = (4) ( ) +"S g ( r g r c ) + q #" S X n + S t X t n $ t ( ) n u Where n, n - is the average density o the lowing t oam and trapped oam respectively; X, X t - represents the shunt volume o the lowing oam and trapped oam respectively ; u - gas Darcy velocity; rg -oam generating speed, c r - oam burst speed; q -the amount o injection oam per unit volume and per unit time. r (1) Foam generating speed Foam generation rate expression [13] n = k v v 1 n 0 1/3 g 1 w * ω Where k oam generating rate constant; (5) vw -water phase velocity; v - lowing oam velocity; n * - the upper limit value o oam density;! - the constant is determined by experiment. (2) Foam burst speed Foam burst speed expression: " 0 P % r c = k c!1 $ P * # c! P c & 2 v n (6) P * * 0 c = P c,max tanh( C S / C S ) (7) 0 Where k!1 - oam burst rate constant; P c - capillary orce; P * * c - critical capillary orce; P c,max -critical capillary orce maximum; C S - suractant concentration; C S 0 - reerence value o suractant concentration Foam Properties Parameters (1) Foam viscosity calculation Foam system presents the characteristics o non- Newtonian luid in porous medium. The oam eective viscosity can be characterized as [14]: µ = µ g +!n v 1/3 (8) Where µ -oam eective viscosity; µ g - gas phase viscosity without oam;! - constant related to suractant properties. (2) Gas phase relative permeability The increase o the gas phase low resistance ater generating oam is relected by the decrease o the gas phase relative permeability. The oam molar concentration o the gas phase is a macro relection o the oam stability. The expression o gas phase relative permeability is: k rg 0 k rg = (9) 1+ (R!1)X g where k rg, k 0 rg represents the gas phase relative permeability with and without oam; R - decreasing coeicient o the gas phase relative permeability; X g - oam molar concentration o the gas phase, which characterizes the oam stability. (3) Score calculation o the trapped oam The trapped oam raction can be characterized as: "!n X t = X t % t,max $ # 1+!n t & (10) Where X t,max - maximum trapped oam raction;! - empirical constants.

3 86 The Open Chemical Engineering Journal, 2015, Volume 9 Zhang et al. Table 1. Pressure dierence itting results o oam plugging experiment. Injection Pore Volume /PV Experimental ΔP /MPa Numerical Calculation ΔP /MPa Relative Error/% average error /% Auxiliary Equation (1) Saturation constraint equation So + Sw + S = 1 (11) (2) Capillary pressure equation P cow = P o! P w (12) P cgo = P g! P o (13) 3. MODEL VALIDATION IMPES method (the implicit pressure-explicit-its saturation method) is used to calculate the three-phase and ivecomponent oam looding mathematical model. It is implicit pressure explicit method, in which implicit is used to solve the pressure equation and explicit is used to solve saturation equation. The results o numerical calculation o established model are used in the oam plugging experiment, and then the correctness o the model can be veriied. An orthogonal grid is established or simulation o nitrogen oam looding plugging experiment, which is 45m 1m 1m and grid spacing is 1m 5cm 5cm. The simulation parameters are as ollows:! = 0.28, K = 1200!10 "3 µm 2, µ w = 0.6mPa!s, µ g = 2.15!10 "2 mpa #s, k 0 1 = 2.03!10 15 s 1/3 m "13/3, k 0!1 = 9.5m!1, * P c,max = 27kPa, X t,max = 0.81,! = 8.2 "10 #18 Pa $s 2/3 m 10/3, C S 0 = 0.075wt%, C S = 0.48wt%, n * = 1!10 12 mm "3,! = 2.9. The itting results are shown in Table 1. It can be inerred that simulation results are basically consistent with the experimental results, and the itting eect o experiment is good. Hence, the rationality o established model and the correctness o the numerical calculation method are veriied. 4. THE SENSITIVITY ANALYSIS OF MODEL PA- RAMETERS The plugging ability o oam is an important indicator to evaluate its perormance [15-17], and there are many actors inluencing the plugging ability o oam. Besides the stability o oam [18], the concentration o oam, gas-liquid ratio o oam system, and rock permeability are all important actors. Foam resistance coeicient is used to describe the ability o oam plugging. It is the ratio o pressure dierence at both ends o core and straight water looding. Its value is the key index to evaluate the quality o oam system. The sensitivity analysis o parameters, which aects the plugging ability o oam, is conducted through the research on the numerical

4 Establishment o Mathematical Model and Sensitivity Analysis The Open Chemical Engineering Journal, 2015, Volume 9 87 Fig. (1). Relationship between the concentration o suractant and the resistance coeicient / residual resistance actor. Fig. (2). Resistance coeicient / residual resistance actor under dierent gas-liquid ratio. simulation. The basic model parameters are mentioned beore Concentration o Foaming Agent Through establishing a conceptual model, numerical simulation o oam looding is studied by changing the concentration o suractant in the injecting model to calculate the resistance coeicient and residual resistance coeicient, as shown in Fig. (1). As can be seen rom the calculation, both resistance coeicient and residual resistance coeicient increase with the increase o concentration o oaming agent. But resistance coeicient and residual resistance coeicient keep constant, and tend to be gentle, when the concentration o oaming agent is more than 0.3% Gas-Liquid Ratio (GLR) Gas-liquid ratio o oam system is an important indicator o the oam looding. The best gas-liquid ratio is optimized by the studying o oam plugging ability through gas-liquid ratio, so the plugging ability o oam system can be stronger. Through establishing a conceptual model, numerical simulation o oam looding is studied by changing the gas-liquid ratio o oam system in the injecting model to calculate the resistance coeicient and residual resistance coeicient, as shown in Fig. (2). It is evident rom the igure that the relationship between the gas-liquid ratio and the resistance coeicient / residual resistance coeicient exhibits the oam that is low at two ends and high in the middle. Foam plugging ability enhances with the increase o the gas-liquid ratio when the gas-liquid ratio is low. The resistance coeicient / residual resistance coeicient reaches the maximum o 45.52, when the gas liquid ratio is 2:1. The resistance coeicient / residual resistance coeicient o the oam system decreases with the increase o the gas-liquid ratio, when the gas-liquid ratio is more than 2:1. So the oam plugging ability is strongest when the gas-liquid ratio is 2: The Reservoir Permeability Through establishing conceptual model o dierent permeability, numerical simulation o oam looding is studied

5 88 The Open Chemical Engineering Journal, 2015, Volume 9 Zhang et al. Fig. (3). Relationship between permeability and resistance coeicient/residual resistance actor. by setting oaming agent concentration to 0.3% and gasliquid ratio to 2:1 to calculate the relationship between the resistance coeicient, residual resistance coeicient and permeability as shown in Fig. (3). The relationship between the oam resistance coeicient and permeability is the distinctive eature o oam system. Plugging ability o oam system increases with the increase o permeability and presents good linear relationship showing excellent selective permeability. The characteristic o oam is the result o the oam apparent viscosity which is relected by additional pressure dierence caused by elastic deormation o oam in the pores, when oam system transports in the porous media. When the larger volume o oam migrates in the larger pore, there is stronger ability o proile control, as well as the larger deormation amount. In terms o reservoir, the higher the permeability is, the stronger the ability o proile control is. CONCLUSION (1) Based on mass conservation, oam total amount balance thought and oam properties characterization model, a oam looding mathematical model with three-phase and ive components is established and numerically solved. (2) The numerical calculation results show that the resistance coeicient / residual resistance coeicient increases with the increase o the oaming agent concentration, but when the oaming agent concentration is greater than 0.3%, resistance coeicient/residual resistance coeicient remains the same, and tends to be steady. (3) The relationship between the gas-liquid ratio and the resistance coeicient / residual resistance coeicient exhibits the oam that is low at two ends and high in the middle. Foam plugging ability enhances with the increase o the gasliquid ratio when the gas-liquid ratio is low. The resistance coeicient / residual resistance coeicient reaches the maximum, when the gas liquid ratio is 2:1; The resistance coeicient / residual resistance coeicient o the oam system decreases with the increase o the gas-liquid ratio, i.e. when the gas-liquid ratio is more than 2:1. So the oam plugging ability is strongest when the gas-liquid ratio is 2:1. (4) Plugging ability o oam system increases with the increase o permeability and presents good linear relationship. In terms o reservoir, the higher the permeability, the stronger the ability o proile control. CONFLICT OF INTEREST The authors conirm that this article content has no conlict o interest. ACKNOWLEDGEMENTS This work is inancially supported by the Natural Science Foundation o China under Grant No REFERENCES [1] Q. Wang, P. Guo, Z. Wang, and X. Li, Foam luid low characteristics in porous media, Oil Drilling & Production Technology, vol.30, no.2, pp , [2] B. Li Z. Li, and Z. Liu, Experiment on proile control and looding by multiphase oam system, Journal o China University o Petroleum (Edition o Natural Science), vol. 34, no. 4, pp , [3] S. Li Z. Li, and R. Lin, Mathematical model building or oam diversion acidizing and its application, Journal o China University o Petroleum (Edition o Natural Science) vol. 32, no. 5, pp , [4] Z. Li M. Sun and R. Lin, Laboratory study on oam plugging and selective divided-low, Acta Petrolei Sinica vol.28, no. 4, pp , [5] A. R. Kovscek, C. J. Radke, Fundamentals o Foam Transport in Porous Media, Foams: Fundamentals and Applications in the Petroleum Industry, Washington, DC: American Chemical Society, pp , [6] G. Chen, G. Liao, J. Niu, Equivalent numerical simulation model or oam lowing in porous medium, Petroleum Geology & Oilield Development in Daqing, pp , [7] H. Cheng, and Z. Lang, Capillary Crosslow in Foam Flood and Its Numerical Simulation, Journal o Chongqing University, vol. 23, pp , 2003.

6 Establishment o Mathematical Model and Sensitivity Analysis The Open Chemical Engineering Journal, 2015, Volume 9 89 [8] A. R. Kovscek, T. W. Patzek, and C. J. Radke, Simulation o Foam Transport in Porous Media, SPE26402, pp , [9] W. R. Roseen, S. C. Zeilinger, J. Shi, and M. T. Lim, Mechanistic Simulation o Foam Processes in Porous Media, SPE, 28940, pp , [10] A. H. Palls, G. J. Hirasaki, T. W. Patzek, D. A. Gaualltz, D. D. Miller, T. Ratulowsk, Development o a Mechanistic Foam Simulator: The Population Balance and Generation by Snap-O, SPE14961, pp , [11] T. W. Patzek, Description o Foam Flow in Porous Media by the Population Balance Method, Society o Petroleum Engineers, pp. 1-9, [12] T. Lu, Z. Li, J. Li, R. Li, A Mathematical Model o Foam Flooding Based on Foam Microscopic Seepage Characteristics, Chinese Journal o Computational Physics, vol. 29, no. 4, pp , [13] A. H. Falls, and P. A. Gauglitz, and G. J. Hirasaki, Development o a mechanistic oam simulator: The population balance and generation by snap-o, SPE 14961, [14] Z. I. Khatlb, G. J. Hirasaki, and A. H. Falls, Eects o Capillary Pressure on Coalescence and Phase Mobilities in Foams Flowing Through Porous Media, SPE15442, pp , [15] Q. Wang, G. Zhou, P. Guo, and X. Li, Experimental study o oam sealing ability, Southwest Petroleum Technology, vol. 25, no.6, pp , [16] A. R. Kovscek, and H. J. Bertin, Foam Mobility in Heterogeneous Porous Media (I: Scaling Concepts), Transport in Porous Media, vol. 52, no.1, pp.17-35, [17] Y. Liu, H. Liu, and Z. Pang, Numerical simulation o nitrogen oam injection to control water cresting by bilateral horizontal well in bottom water reservoir, J. Open Fuels and Energy Science Journal, vol. 5, no.1, pp , [18] H. Mei, H. Dong, H. Gu, M. Ren, and L. Chen, Frother stable perormance evaluation experiments, Xinjiang Petroleum Geology, vol. 25, no.6, pp , Received: June 16, 2015 Revised: July 23, 2015 Accepted: August 16, 2015 Zhang et al.; Licensee Bentham Open This is an open access article licensed under the terms o the ( which permits unrestricted, non-commercial use, distribution and reproduction in any medium, provided the work is properly cited.