The Flow through saturated-unsaturated porous media is extremely important in various natural and industrial based applications. While the Darcy's law with various modifications are used to model the flow through a porous media, the flow through unsaturated porous media is largely based on conservation of mass and modified Darcy's law where non-linear relationship exists between the pressure head and the fluid saturation coupled with fluid density variations. This paper represents mathematical modelling of flow through unsaturated porous media using constant and variable fluid density. The variable density model is further split into thermal and Isothermal models. The mathematical model is applied to an unsaturated porous media filled with water and oil having immiscible flow. The variables describing the models like permeability, capillary pressure, fluid saturation and their constituent relations are considered. The models are extremely important for different industrial applications like enhancing oil recovery, sea water filtration, nuclear waste disposal, chemical clean-up of soil, underground hydrology, soil physics etc.

1. H. Darcy, "Les fontaines Publiques de la Ville de Dijon," Vols. Dalmont, Paris, 1856.
2. J. Dupuit, "Estudes Théoriques et Pratiques sur le mouvement des Eaux dans les canaux découverts et à travers les terrains perméables," (Second ed. ). , Paris, Dalmont, 1863.
3. H. A, "Some physical properties of sand and grazels with special reference to their use in filtration," Massachusetts State Broad of Health, Twenty four annual report, Massachusetts State, 1893.
4. M. M. , The flow of homogeneous fluid through porous media, United State of America: Mcgraw Hill, 1937.
5. B. H. C. , "A calculation of viscous force exerted by a flowing fluid on a dense swarm of paricles," Applied Scientific Research, vol. 1, pp. 27-34, December 1949.
6. Irmay, "Extention of Darcy law to unsteady unsaturated flow through porous media," Division of hydraulic Engg. , Israel Institute of Technology, Haifa, pp. 57-66, 1958.
7. L. A. Richards, "Capillary Conduction of Liquids Through Porous Mediums," American Institute of Physics, Vols. Volume 1, , p, no. Issue 5, pp. 318-333, 1931.
8. G. W. and J. Widtsoe, "Infiltration," Trans. Am. Geophys. Union, vol. 27, pp. 126-128, 1946.
9. V. Genuchten, "A closed form equation for predicting the hydraulic conductivity of unsaturated soils," Soil Science Society of America Journal, vol. 44, pp. 892-898, 1980.
10. E. N. E. and A. A. B. C. , "Thermodynamic of soil moisture," Hilgardia, vol. 15, pp. 31-298, 1943.
11. J. L. Nieber, R. Z. Dautov, A. G. Egorov and A. Y. Sheshukov, "Dynamic Capillary Pressure Mechanism for Instability in Gravity-Driven Flows; Review and Extension to Very Dry Conditions," Transport in Porous Media, vol. 58, no. 1-2, 2005.
12. H. K. Dahle, M. A. Celia and S. M. Hassanizadeh, "Bundle-of-Tubes Model for Calculating Dynamic Effects in the Capillary- Pressure-Saturation Relationship," Transport in Porous Media, vol. 58, pp. 1-2, 2005.
13. D. Yang, R. P. Currier and Duan Z. Zhang, "Ensemble phase averaged equations for multiphase flows in porous media. Part 1: The bundle-of-tubes model," International Journal of Multiphase Flow, vol. 35, p. 628–639, 2009.
14. F. E. Teruel and Rizwan-uddin, "Characterization of a porous medium employing numerical tools: Permeability and pressure-drop from Darcy to turbulence," International Journal of Heat and Mass Transfer, vol. 52, p. 5878–5888, 2009.
15. Z. Chaia, B. Shia, J. Lub and Z. Guoc, "Non-Darcy flow in disordered porous media: A lattice Boltzmann study," Computers & Fluids, vol. 39, no. 10, p. 2069–2077, December 2010.
16. M. T. Pamuk and M. Özdemir, "Friction factor, permeability and inertial coefficient of oscillating flow through porous media of packed balls," Experimental Thermal and Fluid Science, vol. 38, pp. 134-139, 2012.
17. X. Wang, F. Thauvin and K. Mohanty, "Non-Darcy flow through anisotropic porous media," Chemical Engineering Science, vol. 54, no. 12, p. 1859–1869, June 1999.
18. C. Hsu and P. Cheng, "Thermal dispersion in a porous medium," Int. J. of heat and mass transfer, vol. 33, pp. 1587-1597, 1990.
19. Z. Zeng and R. Grigg, "A Criterion for Non-Darcy Flow in Porous Media," Petroleum Recovery Research Center, New Mexico Institute of Mining and Technology, Socorro, NM 87801, U. S. A. , 2006.
20. Y. Su, J. H. Davidson and F. A. Kulacki, "A geometry factor for natural convection inopen cell metal foam," International Journal of Heat and Mass Transfer, vol. 62, pp. 697-710, 2013.
21. D. A. Neid and A. Bejan, Convection in Porous Media, New York, USA: Springer, 2006.
22. S. Manthey, S. M. Hassanizadeh and Rainer Helmig, "Macro-Scale Dynamic Effects in Homogeneous and Heterogeneous Porous Media," Transport in Porous Media, vol. 58, no. 1-2, 2005.
23. P. H. Valvatne, M. Piri, X. Lopez and M. J. Blunt, "Predictive Pore-Scale Modeling of Single and Multiphase Flow," Transport in Porous Media, vol. 58, no. 1-2, 2005.
24. H. Eichel, R. Heling, I. Neuweiler and O. A. Cripka, "Upscaling of Two-Phase Flow Processes in Porous Media," Transport in Porous Media, vol. 58, no. 1-2, 2005.
25. R. B. Yedder and F. Erchiqui, "Convective flow and heat transfer in a tall porous cavity side-cooled with temperature profile," International Journal of Heat and Mass Transfer, vol. 52, p. 5712–5718, 2009.
26. . R Cortez, B. Cummins, K. Leiderman and D. Varela, "Computation of three-dimensional Brinkman flows using regularized methods," Journal of Computational Physics , vol. 229, p. 7609–7624, 2010.
27. K. Vafai and C. Tien, "Boundary and inertia effects on flow and heat transfer in porous media," Int. J. Heat Mass Transfer, vol. 24, no. printed in Great Britain, pp. 195-203, 1981.
28. Y. Pachepskya, D. Timlinb and W. Rawls, "Generalized Richards' equation to simulate water transport in unsaturated soil," Journal of Hydrology, Elsevier, vol. 272, pp. 3-13, 2003.
29. J. Auriault, C. Boutin and C. Geindreau, Homogenization of Coupled phenomena in Hetrogeneous media, Hoboken: Wiley, 2009.
30. G. F. Pinder and W. G. Gray, Essentials of Multiphase flow and transport in porous media, New Jersey: John Wiley & Sons, Inc. , 2008.
31. J. Bear, Dynamics of fluids in porous media, Amsterdam: Elseview, 1972.
32. Y. Mualem, "A new model for predicting the hydraulic onductivity of unsaturated porous media," Wafer Resour. Res. , vol. 12, pp. 513-522, 1976 a.
33. J. L. Lage and B. V. Antohe, "Darcy's Experiments and the Deviation to Nonlinear Flow Regime," Journal of Fluids Engineering , ASME, vol. 122, no. September, pp. 619-625, 2000.
34. Petrowiki, "Flow equations for gas and multiphase flow," Society of Petroleum Engineers, 08 June 2015. [Online]. Available: http://petrowiki. org/Flow_equations_for_gas_and_multiphase_flow. [Accessed 01 Feb 2016].
35. M. K and S. K, "Regenerator models for Stirling cycles," Thermal Sci Eng. , vol. 5, pp. 31-40, 1997.
36. Q. Zhu, M. Xie, J. Yang and Y. Li, "A fractal model for the coupled heat and mass transfer in porous fibrous media," vol. xxx, 2010.
37. N. S. •. B. F. •. R. H. •. B. I. Wohlmuth, "Dimensionally reduced flow models in fractured porous media: crossings and boundaries," Funded by German Research Foundation (DFG) within the International Research Training Group, University of Stuttgart, 2015.
38. Springer, Springer, [Online]. Available: http://spring. delta-h. de/download/SPRING4_Webhilfe_E/Stat_Stroemung13. htm. [Accessed 2015].
39. S. Irmay, "Sur le mouvement des eaux dans le sol," Rev. Universelle des Mines, vol. 3, no. 4, 1947.
40. J. Kozeny, "Ü ber die kapillare Leitung des Wassers im Boden," Sitzungsberichte der Akademieder Wissenschaften in Wien, vol. 136a, pp. 271-306, 1927.
41. P. C. Carman, "Fluid flow through granular beds," Trans. Institute of Chem. Engineers, London, no. 15, pp. 150-157, 1937.
42. M. Y, "A new model for predicting the hydraylic conductivity of unsaturated porous media," Water Resource Reserver, vol. 12, pp. 513-522, 1976.
43. J. Delleur, The Handbook of Groundwater Engineering, Boca Raton: CRC Press, 1999.
44. "Flow through porous media," [Online]. Available: http://echo2. epfl. ch/VICAIRE/mod_3/chapt_6/main. htm. [Accessed 10 Dec 2014].
45. C. Fetter, Contaminent Hydrology, Upper Saddle River, NJ: Prentice Hall, 1999.

Unsaturated Porous Media Flow Mathematical Modeling Porous Media Variable Density Local Thermal Equilibrium Immiscible Flow