تعداد نشریات | 206 |

تعداد شمارهها | 5,156 |

تعداد مقالات | 56,829 |

تعداد مشاهده مقاله | 92,961,097 |

تعداد دریافت فایل اصل مقاله | 75,101,384 |

## Application of Homotopy Perturbation Method to Nonlinear Equations Describing Cocurrent and Countercurrent Imbibition in Fractured Porous Media | ||

Journal of Chemical and Petroleum Engineering | ||

مقاله 2، دوره 46، شماره 1، شهریور 2012، صفحه 13-29
اصل مقاله (334.2 K)
| ||

نوع مقاله: Original Paper | ||

شناسه دیجیتال (DOI): 10.22059/jchpe.2012.1890 | ||

نویسندگان | ||

Hossein Fazeli ^{} ^{1}؛ Reza Fathi^{1}؛ Abbas Atashdehghan^{2}
| ||

^{1}Institute of Petroleum Engineering, Faculty of Engineering, Tehran University, Tehran, Iran | ||

^{2}Gachsaran Oil and Gas Production Company, Gachsaran, Iran | ||

چکیده | ||

In oil industry, spontaneous imbibition is an important phenomenon in recovery from fractured reservoirs which can be defined as spontaneous uptake of a wetting fluid into a porous solid. Spontaneous imbibition involves both cocurrent and countercurrent flows. When a matrix block is partially covered by water, oil recovery is dominated by cocurrent imbibition i.e. the production of non wetting phase has the same direction of flow as the wetting phase. However if the matrix block is completely covered by water then countercurrent flow takes place, and the production of non wetting phase has an opposite direction of flow to that of the imbibing wetting phase. Each of these processes can be described by a nonlinear partial differential equation (PDE). In this paper, the homotopy perturbation method (HPM) which is a powerful series-based analytical tool, is used to approximate the solutions of cocurrent and countercurrent equations. HPM decomposes a complex partial differential equation under study to a series of simple ordinary differential equations that are easy to be solved. The solutions obtained by HPM are compared with that found using a common numerical method applied by MATLAB software. The difference between the two is seemed to be virtually negligible. A good agreement is also achieved from the comparison of the solutions obtained by HPM with those of a numerical method (NM). | ||

کلیدواژهها | ||

Fracture porous media؛ Homotopy perturbation method (HPM)؛ Cocurrent imbibition؛ Countercurrent imbibition؛ Spontaneous Imbibition | ||

مراجع | ||

[1] Saidi, A.M. (1983). “Simulation of Naturally Fractured Reservoirs.” SPE paper 12270 presented at Seventh SPE Symposium on Reservoir Simulation, San Francisco, CA.
[2] German, E.R. (2002). “Water Infiltration in Fractured Porous Media: In-situ Imaging, Analytical Model, and Numerical Study.” PhD Dissertation, Stanford University.
[3] Bourblaux, B.J. and Kalaydjian, F.J. (1990). “Experimental study of cocurrent and countercurrent flows in natural porous media.” SPE Reservoir Engineering, Vol. 5, PP.361–368.
[4] Chimienti, M.E, Illiano, S.N and Najurieta, H.L. (1999). “Influence of temperature and interfacial tension on spontaneous imbibition process.” SPE paper 53668 presented as Latin American and Caribbean Petroleum Engineering Conference, Caracas, Venezuela.
[5] Pooladi-Darvish, M. and Firoozabadi, A. (2000). “Cocurrent and countercurrent imbibition in a water-wet matrix block.” SPE Journal, Vol. 5, PP. 23–11.
[6] Najurieta, H.L., Galacho, N., Chimienti, M.E. and S. N. Illiano, S.N. (2001). “Effects of temperature and interfacial tension in different production mechanisms.” SPE paper 69398 presented at Latin American and Caribbean Petroleum Engineering, Buenos Aires, Argentina.
[7] Tang, G.Q. and Firoozabadi, A. (2001). “Effect of pressure gradient and initial water saturation on water injection in water-wet and mixed-wet fractured porous media.” SPE Reservoir Evaluation and Engineering, Vol. 4, PP.516–524.
[8] Parsons, R.W. and Chaney, P.R. (1966). “Imbibition model studies on water-wet carbonate rocks.” SPE Journal, Vol. 6, PP.26–34.
[9] Iffly, R., Rousselet, D.C. and Vermeulen, J.L. (1972). “Fundamental study of imbibition in fissured oil fields.” SPE paper 4102 presented at Annual Fall Meeting, San Antonio
[10] Hamon, G. and Vidal, J. (1986). “Scaling-up the capillary imbibition process from laboratory experiments on homogeneous samples.” SPE paper 15852 presented at SPE European Petroleum Conference, London, England.
[11] Al-Lawati, S. and Saleh, S. (1996). “Oil recovery in fractured oil reservoirs by low IFT imbibition process.” SPE paper 36688 presented at SPE Annual Technical Conference and Exhibition, Denver, Colorado.
[12] Morrow, N.R. and Mason, G. (2006). “Recovery of oil by spontaneous imbibitions.” Current Opinion in Colloid and Interface Science, Vol. 6, PP.321-337
[13] Kashchiev, D. and Firoozabadi, A. (2003). “Analytical Solutions for 1D Countercurrent Imbibition in Water-Wet Media.” SPE Journal, Vol. 8, PP.401-408.
[14] Tavassoli, Z., Zimmerman, R.W. and Blunt, M.J. (2005). “Analysis of counter-current imbibition with gravity in weakly water-wet systems.” Journal of Petroleum Science and Engineering, Vol. 48, PP.94– 104.
[15] Silin, D. and Patzek, T. (2004). “On Barenblatt’s Model of Spontaneous Countercurrent Imbibition.” Transport in Porous Media, Vol. 54, PP.297–322.
[16] Behbahani, H.Sh., Donato G.D. and Blunt, M.J. (2006).“Simulation of counter-current imbibition in water-wet fractured reservoirs.” Journal of Petroleum Science and Engineering, Vol. 50, PP.21– 39.
[17] Cai, X.C., Wu, W.Y. and Li, M.S. (2006). “Approximate period solution for a kind of nonlinear oscillator by He’s homotopy perturbation method.” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 7, PP.109–117.
[18] Cveticanin, L. (2006). “ Homotopy perturbation method for pure nonlinear differential equation.” Chaos, Solitons& Fractals, Vol. 30, PP.1221–1230.
[19] El-Shahed, M. (2005). “Application of He’s homotopy perturbation method to Volterra’s integro-differential equation.” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 6, PP.163–168.
[20] Abbasbany, S. (2006). “Application of He’s homotopy perturbation method for Laplace transform.” Chaos, Solitons& Fractals, Vol. 30, PP.1206–1212.
[21] Belendez, A., Hernandez, A., Belendez, T., Fernández, E., Álvarez, M.L. and Neipp, C. (2007). “Application of He’s homotopy perturbation method to the Duffing-harmonic oscillator.” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 8, PP.79–88.
[22] He, J.H. (2006). “New interpretation of homotopy perturbation method.” International Journal of Modern Physics B, Vol. 20, PP.2561–2568.
[23] Rafei, M. (2006). “Solitary wave solutions for a generalized Hirota-Satsuma coupled KdV equation by homotopy perturbation method.” Physics Letter A, Vol. 356, PP.131-137.
[24] Ganji, D.D. and Rajabi, A. (2006). “Assessment of Homotopy-Perturbation and Perturbation Methods in Heat Radiation Equations.” International Communications in Heat and Mass Transfer, Vol. 33, PP.391-400.
[25] Ganji, D.D. (2006). “The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer.” Physics Letter A, Vol. 355, PP.337-341.
[26] He, J. H. (1999). “Homotopy perturbation technique.” Computer Methods in Applied Mechanics and Engineering, Vol. 178, PP.257–262.
[27] He, J. H. (2005). “Application of homotopy perturbation method to nonlinear wave equations.” Chaos Solitons Fractals, Vol. 26, PP.695–700.
[28] He, J. H. (2005). “Homotopy perturbation method for bifurcation of nonlinear problems.” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 6, PP.207–208.
[29] He, J. H. (2006). “Homotopy perturbation method for solving boundary value problems.” Physics Letter A, Vol. 350, PP.87–88.
[30] He, J. H. (2006). “Some asymptotic methods for strongly nonlinear equations.” International Journal of Modern Physics B, Vol. 20, PP.1141–1199.
[31] He, J. H. (2008). “Recent development of the homotopy perturbation method”. Topological Methods in Nonlinear Analysis, Vol. 31, PP.205– 209.
[32] He, J. H. (2008). “An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering.” International Journal of Modern Physics B, Vol. 22, PP.3487–3578.
[33] Yildirim, A. and Sezer, S.A. (2012). “Analytical solution of MHD stagnation point flow in porous media by means of the homotopy perturbation method.” Journal of Porous Media, Vol. 15, No. 1, PP.83-94
[34] He, J.H, (2005). “Periodic solutions and bifurcations of delay-differential equations.” Physics Letter A, Vol. 347, PP.228–230.
[35] He, J.H. (2005). “Limit cycle and bifurcation of non-linear problems.” Chaos, Solitons & Fractals, Vol. 26, PP.827–833.
[36] Siddiqui, A.M., Mahmood, R. and Ghori, Q.K. (2008). “Homotopy perturbation method for thin film flow of a third grade fluid down an inclined plane.” Chaos, Solitons& Fractals, Vol. 35, PP.140–147.
[37] Biazar, J. and Ghazvini, H. (2009). “He’s homotopy perturbation method for solving systems of Volterra integral equations of the second kind.” Chaos, Solitons& Fractals, Vol. 39, PP.770–777.
[38] Cveticanin, L. (2009).“ Application of homotopy-perturbation to non-linear partial differential equations.” Chaos, Solitons& Fractals, Vol. 40, PP.221–228.
[39] Ravi Kanth, A.S.V. and Aruna, K. (2009). “ He’s homotopy perturbation method for solving higher-order boundary value problems.” Chaos, Solitons& Fractals, Vol. 41, PP.1905–1909.
[40-] Cai, X.C. and Wu, W.Y. (2009).“ Homotopy perturbation method for nonlinear oscillator equations.” Chaos, Solitons& Fractals, Vol. 41, PP.2581-2583
[41] Fathizadeh, M. and Rashidi, F. (2009).“ Boundary layer convective heat transfer with pressure gradient using Homotopy Perturbation Method (HPM) over a flat plate.” Chaos, Solitons& Fractals, Vol. 42, PP.2413-2419
[42] Chen, Z., Huan, G. and Ma, Y. (2006). “Computational Methods for Multiphase Flows in Porous Media.” Philadelphia: Society for Industrial and Applied Mathematics, PP. 260-276.
[43] Scheidegger, A.E. and Johnson, E.F. (1961). “The statistical behavior of instabilities in displacement process in porous media.” Canadian Journal of Physics, Vol. 39, No. 2, PP.326-33. | ||

آمار تعداد مشاهده مقاله: 9,156 تعداد دریافت فایل اصل مقاله: 6,636 |