Mathematical model to study The spread of spilled oil in the soil

A mathematical model describing the spread of spilled oil through the soil is discussed. The spread of spilled oil in soil is controlled by the flow of water and is described by multiphase equations. In this context, the two-phase flow characteristics of oil-water flow with varying viscosity in the subsurface coupled to an advective-diffusion equation are examined to study the transport of oil. The terms that model the interaction between the multiple phases are introduced at the boundary, such as the slip condition at the porous-fluid interface, shear stress condition at the fluid-fluid interface, and the continuity of velocity at both the interfaces. The effect of various physical parameters such as Schmidt number, retardation factor, viscosity ratio, porous and slip parameter on the velocity and concentration profiles are discussed in detail with the help of graphs. The surface plots of velocity and concentration of oil against axial distance at different time are also analyzed. The obtained results show that the velocity of oil accelerates linearly with axial length and there is a decrease in the concentration of the spilled oil through the media. The validity of the results obtained is verified by comparison with available experimental result, and good agreement is found