خطی سازی ترم نفوذپذیری نسبی در حل عددی معادلات جریان دوفازی در محیط‌های متخلخل

نوع مقاله : مقاله پژوهشی

نویسندگان

1 گروه مکانیک سنگ دانشگاه تربیت مدرس، تهران، ایران

2 Department of Civil and Environmental Engineering, Norwegian University of Science and Technology (NTNU),Trondheim, Norway

چکیده

This paper presents a control volume finite element model (CVFEM) to simulate simultaneous flow of two immiscible fluids in non-deformable porous media. The method is fully conservative at the local and global level. It keeps the data structure of the common finite element method (FEM). A pressure-based formulation is presented in this paper. The proper choice of primary unknown variables is a critical step in developing an efficient solution of the multiphase subsurface flow problems. Pressure-based models are one of the common choices to this end. This type of models consists of strong nonlinear terms and encounters convergence difficulties when the Jacobian matrix are poorly approximated. The most severe problem is related to the relative permeability term that appears as a function of volume fraction (or degree of saturation) of the wetting phase. Since water saturation is not a primary unknown variable, the relative permeability terms become a function of two primary unknowns, i.e. wetting and non-wetting pressures, together. A fully implicit first order accurate finite difference scheme is employed for temporal discretization of the equations. A full Newton method with exact Jacobian is considered in this work, and a rapid convergence has been achieved. The model is used to simulating a five-spot problem in a block heterogenous porous medium.

کلیدواژه‌ها


عنوان مقاله [English]

Treatment of relative permeability term in two-phase flow problems in porous media

نویسندگان [English]

  • Ehsan Taheri 1
  • Ali Ghoreishian 2
1 استادیار
2 Department of Civil and Environmental Engineering, NTNU
چکیده [English]

This paper presents a control volume finite element model (CVFEM) to simulate simultaneous flow of two immiscible fluids in non-deformable porous media. The method is fully conservative at the local and global level. It keeps the data structure of the common finite element method (FEM). A pressure-based formulation is presented in this paper. The proper choice of primary unknown variables is a critical step in developing an efficient solution of the multiphase subsurface flow problems. Pressure-based models are one of the common choices to this end. This type of models consists of strong nonlinear terms and encounters convergence difficulties when the Jacobian matrix are poorly approximated. The most severe problem is related to the relative permeability term that appears as a function of volume fraction (or degree of saturation) of the wetting phase. Since water saturation is not a primary unknown variable, the relative permeability terms become a function of two primary unknowns, i.e. wetting and non-wetting pressures, together. A fully implicit first order accurate finite difference scheme is employed for temporal discretization of the equations. A full Newton method with exact Jacobian is considered in this work, and a rapid convergence has been achieved. The model is used to simulating a five-spot problem in a block heterogenous porous medium.

کلیدواژه‌ها [English]

  • Two-phase flow
  • immiscible fluids
  • Porous media
  • Pressure-based solution
  • Relative permeability
[1] Settari A, Aziz K (1975) Treatment of nonlinear terms in the numerical solution of partial differential equations for multiphase flow in porous media. International Journal of Multiphase Flow 1 (6):817-844.
[2] Chen Z, Huan G, Wang H (2006) Computer Simulation of Compositional Flow Using Unstructured Control Volume Finite Element Methods. Computing 78 (1):31-53.
[3] de Carvalho DKE, Willmersdorf RB, Lyra PRM (2007) A node-centred finite volume formulation for the solution of two-phase flows in non-homogeneous porous media. International Journal for Numerical Methods in Fluids 53 (8):1197-1219.
[4] Ghoreishian Amiri SA, Sadrnejad SA, Ghasemzadeh H, Montazeri GH (2013) Application of control volume based finite element method for solving the black-oil fluid equations. Petroleum Science 10 (3):361-372.
[5] Wu YS, Forsyth PA (2001) On the selection of primary variables in numerical formulation for modeling multiphase flow in porous media. Journal of contaminant hydrology 48 (3-4):277-304.
[6] Ataie-Ashtiani B, Raeesi-Ardekani D (2010) Comparison of Numerical Formulations for Two-phase Flow in Porous Media. Geotechnical and Geological Engineering 28 (4):373-389.
[7] Taheri E, Sadrnejad SA, Ghasemzadeh H (2017) Application of M3GM in a Petroleum Reservoir Simulation. Journal of Petroleum Science and Technology 7 (3):33-46.
[8] Sadrnejad SA, Ghasemzadeh H, Taheri E Multiscale advanced features in modeling oil transport in porous media. In: 21st annual international conferences on mechanical engineering, Tehran, 2013.
[9]  Pinder GF, Abriola LM (1986) On the simulation of nonaqueous phase organic compounds in the subsurface. Water Resources Research 22 (9S):109S-119S.
[10] Yin S, Dusseault MB, Rothenburg L (2009) Thermal reservoir modeling in petroleum geomechanics. International Journal for Numerical and Analytical Methods in Geomechanics 33 (4):449-485. [11] Sadrnejad SA, Ghasemzadeh H, Ghoreishian Amiri SA, Montazeri GH (2012) A control volume based finite element method for simulating incompressible two-phase flow in heterogeneous porous media and its application to reservoir engineering. Petroleum Science 9 (4):485-497.
[12] Ghoreishian Amiri SA, Taheri E, Lavasan AA (2021) A hybrid finite element model for non-isothermal two-phase flow in deformable porous media. Computers & Geotechnics 135:104199.