# Iterative Numerical Scheme for Non-Isothermal Two-Phase Flow in Heterogeneous Porous Media

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Modeling and Mathematical Formulation

^{−3}–10

^{−4}. For a temperature variation of moderate amount, we have

^{3}] is density of the solid phase; ${c}_{p,s}$ [kg/m

^{3}] is the heat capacity of the solid phase; ${c}_{p,\alpha}$ [J/Kg·K] is the heat capacity of the fluid phase $\alpha $; T [K] is the temperature; ${Q}_{T}$ [m

^{3}/s] is the heat source term; $\tau $ is the tortuosity of the water phase; and ${h}_{s}$ [J/K·m·s] is the thermal conductivity of the solid phase. The thermal conductivity of the phase $\alpha $ is expressed as

## 3. Iterative Method

## 4. Spatial Discretization

## 5. Numerical Investigations

## 6. Conclusions

## Funding

## Conflicts of Interest

## References

- Aziz, K.; Settari, A. Petroleum Reservoir Simulation; Applied Science Publishers: London, UK, 1979. [Google Scholar]
- Collins, D.A.; Nghiem, L.X.; Li, Y.K.; Grabenstetter, J.E. An efficient approach to adaptive implicit compositional simulation with an equation of state. SPE Res. Eng.
**1992**, 7, 259–264. [Google Scholar] [CrossRef] - Dawson, C.N.; Kloe, H.; Wheeler, M.F.; Woodward, C.S. A parallel, implicit, cell-centered method for two-phase flow with a preconditioned Newton- Krylov solver. Comput. Geosci.
**1997**, 1, 215–249. [Google Scholar] [CrossRef] - Ascher, U.; Ruuth, S.J.; Wetton, B.R. Implicit-Explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal.
**1995**, 32, 797–823. [Google Scholar] [CrossRef] - Boscarino, S. Error analysis of IMEX Runge-Kutta methods derived from differential-algebraic systems. SIAM J. Numer. Anal.
**2007**, 45, 1600–1621. [Google Scholar] [CrossRef] - Coats, K.H. IMPES stability: Selection of stable time steps. SPE J.
**2003**, 8, 181–187. [Google Scholar] [CrossRef] - Jamei, M.; Ghafouri, H. A novel discontinuous Galerkin model for two-phase flow in porous media using an improved IMPES method. Int. J. Num. Meth. Heat Fluid Flow
**2016**, 26, 284–306. [Google Scholar] [CrossRef] - Chen, Z.; Huan, G.; Ma, Y. Computational Methods for Multiphase Flows in Porous Media; SIAM Computational Science and Engineering: Philadelphia, PA, USA, 2006. [Google Scholar]
- Watts, J.W. A compositional formulation of the pressure and saturation equations. SPE Reserv. Eng.
**1986**, 1, 243–252. [Google Scholar] [CrossRef] - Young, L.C.; Stephenson, R.E. A generalized compositional approach for reservoir simulation. SPE J.
**1983**, 23, 727–742. [Google Scholar] [CrossRef] - Lacroix, S.; Vassilevski, Y.V.; Wheeler, J.A.; Wheeler, M.F. Iterative solution methods for modeling multiphase flow in porous media fully implicitly. SIAM J. Sci. Comput.
**2003**, 25, 905–926. [Google Scholar] [CrossRef] - Lu, B.; Wheeler, M.F. Iterative coupling reservoir simulation on high performance computers. Pet. Sci.
**2009**, 6, 43–50. [Google Scholar] [CrossRef] [Green Version] - Kou, J.; Sun, S. A new treatment of capillarity to improve the stability of IMPES two-phase flow formulation. Comput. Fluids
**2010**, 39, 1923–1931. [Google Scholar] [CrossRef] - El-Amin, M.F.; Kou, J.; Salama, A.; Sun, S. An iterative implicit scheme for nanoparticles transport with two-Phase flow in porous media. Procedia Comput. Sci.
**2016**, 80, 1344–1353. [Google Scholar] [CrossRef] - El-Amin, M.F.; Kou, J.; Sun, S. Convergence analysis of the nonlinear iterative method for two-phase flow in porous media associated with nanoparticle injection. Int. J. Num. Meth. Heat Fluid Flow
**2017**, 27, 2289–2317. [Google Scholar] [CrossRef] - Falta, R.W.; Pruess, K.; Javandel, I.; Witherspoon, P.A. Numerical Modelling of Steam Injection for the Removal of Nonaqueous Phase Liquids from the Subsurface. 2. Code Validation and Application. Adv. Water Res.
**1992**, 28, 451–465. [Google Scholar] [CrossRef] - Forsyth, P.A. A positivity preserving method for simulation of steam injection for NAPL site remediation. Adv. Water Res.
**1993**, 16, 351–370. [Google Scholar] [CrossRef] - Cortellessa, G.; Arpino, F.; Fraia, S.D.; Scungio, M. Two-phase explicit CBS procedure for compressible viscous flow transport in porous materials. Int. J. Num. Meth. Heat Fluid Flow
**2018**, 28, 336–360. [Google Scholar] [CrossRef] - Nelson, P.H. Pore-throat sizes in sandstones, tight sandstones, and shales. AAPG Bull.
**2009**, 93, 329–340. [Google Scholar] [CrossRef] - Bear, J. Dynamics of Fluids in Porous Media; American Elsevier: New York, NY, USA, 1972. [Google Scholar]
- Hassanizadeh, S.M.; Gray, W.G. Toward an improved description of the physics of two-phase flow. Adv. Water Resour.
**1993**, 16, 53–67. [Google Scholar] [CrossRef] - Helmig, R. Multiphase Flow and Transport Processes in the Subsurface, 1st ed.; Series of Environmental Science and Engineering; Springer: Berlin/Heidelberg, Germany, 1997. [Google Scholar]
- Olivella, S.; Gens, A. Vapor transport in low permeability unsaturated soil with capillary effects. Trans. Porous Med.
**2000**, 40, 219–241. [Google Scholar] [CrossRef] - Ortega, J.M.; Rheinboldt, W.C. Iterative Solution of Nonlinear Equations in Several Variables; SIAM: Philadelphia, PA, USA, 2000. [Google Scholar]
- Wheeler, M.F.; Yotov, I. A Multipoint Flux Mixed Finite Element Method. SIAM J. Numer. Anal.
**2006**, 44, 2082–2106. [Google Scholar] [CrossRef] [Green Version] - Arbogast, T.; Wheeler, M.F.; Yotov, I. Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal.
**1997**, 34, 828–852. [Google Scholar] [CrossRef] - Al-Dhafeeri, A.M.; Nasr-El-Din, H.A. Characteristics of high-permeability zones using core analysis, and production logging data. J. Pet. Sci. Eng.
**2007**, 55, 18–36. [Google Scholar] [CrossRef]

**Figure 1.**Distribution of the saturation on heterogenous permeability map after 0.35, 0.5, 0.75, and 0.85 pore volume (PV).

**Figure 2.**Distribution of the temperature on heterogenous permeability map after 0.35, 0.5, 0.75, and 0.85 PV.

**Figure 5.**Saturation profiles at y = 0.01 m against x-axis for different number of iterations (NIT). The lower graph is scaled up.

**Figure 6.**Temperature profiles at y = 0.04 m against x-axis for different number of iterations. The lower graph is scaled up.

**Figure 7.**Distribution of saturation on heterogeneous permeability for different number of iterations.

**Figure 8.**Distribution of saturation on heterogeneous permeability for different number of iterations.

Parameter | Value | Units |
---|---|---|

${T}_{in}$ | 360 | K |

${T}_{0},{T}_{r}$ | 300 | K |

${h}_{s}$ | 0.718 | W/(m/K) |

${h}_{w}$ | 0.6 | W/(m/K) |

${h}_{n}$ | 0.2 | W/(m/K) |

${\rho}_{s}$ | 2500 | kg/m^{3} |

${\rho}_{w}$ | 1000 | kg/m^{3} |

${\rho}_{n}$ | 660 | kg/m^{3} |

${c}_{p,s}$ | 800 | J/kg·K |

${c}_{p,w}$ | 4000 | J/kg·K |

${c}_{p,n}$ | 2300 | J/kg·K |

${\beta}_{w}$ | 0.005 | K^{−1} |

${\beta}_{n}$ | 0.003 | K^{−1} |

${S}_{wr},{S}_{nr}$ | 0.001 | – |

$\varphi $ | 0.3 | – |

${\mu}_{w}$ | 1 | cP = 1.0 × 10^{−3} Pa·s |

${\mu}_{n}$ | 0.45 | cP |

${k}_{rw0},{k}_{rn0}$ | 1 | – |

${P}_{e}$ | 50 | bar = 1.0 × 10^{5} Pa |

**Table 2.**Error estimates for different values of the time steps number (TSN) and iterations number (ITN).

TSN | ITN | $\parallel {\mathit{s}}_{\mathit{w}}^{\mathit{n}+1,\mathit{k}+1}-{\mathit{s}}_{\mathit{w}}^{\mathit{n}+1}\parallel $ | $\parallel {\mathit{T}}^{\mathit{n}+1,\mathit{k}+1}-{\mathit{T}}^{\mathit{n}+1}\parallel $ |
---|---|---|---|

2000 | 1 | 1.6205 × 10^{−3} | 6.9198 × 10^{−4} |

1000 | 1 | 3.2422 × 10^{−3} | 1.3857 × 10^{−3} |

500 | 1 | 6.4895 × 10^{−3} | 2.7781 × 10^{−3} |

100 | 1 | 3.2657 × 10^{−2} | 1.4169 × 10^{−2} |

100 | 10 | 1.3387 × 10^{−2} | 5.8768 × 10^{−6} |

100 | 30 | 1.2699 × 10^{−2} | 8.7596 × 10^{−8} |

100 | 50 | 1.2699 × 10^{−2} | 8.7596 × 10^{−8} |

**Table 3.**Comparison of error estimates of saturation $\parallel {s}_{w}^{n+1,k+1}-{s}_{w}^{n+1}\parallel $ with a previous work.

ITN | Present Work | Previous Work [15] |
---|---|---|

1 | 0.0093 | 0.0096 |

10 | 0.0094 | 0.0096 |

30 | 0.0093 | 0.0094 |

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**MDPI and ACS Style**

El-Amin, M.F.
Iterative Numerical Scheme for Non-Isothermal Two-Phase Flow in Heterogeneous Porous Media. *Algorithms* **2019**, *12*, 117.
https://doi.org/10.3390/a12060117

**AMA Style**

El-Amin MF.
Iterative Numerical Scheme for Non-Isothermal Two-Phase Flow in Heterogeneous Porous Media. *Algorithms*. 2019; 12(6):117.
https://doi.org/10.3390/a12060117

**Chicago/Turabian Style**

El-Amin, Mohamed F.
2019. "Iterative Numerical Scheme for Non-Isothermal Two-Phase Flow in Heterogeneous Porous Media" *Algorithms* 12, no. 6: 117.
https://doi.org/10.3390/a12060117