Numerical solution of instability phenomenon arising in double phase flow through inclined homogeneous porous media

In the petroleum reservoir at an early stage the oil is recovered due to existing natural pressure and such type of oil recovery is referred as primary oil recovery. It ends when pressure equilibrium occurs and still large amount of oil remains in the reservoir. Consequently, secondary oil recovery process is employed by injection water into some injection wells to push oil towards the production well. The instability phenomenon arises during secondary oil recovery process. When water is injected into the oil filled region, due to the force of injecting water and difference in viscosities of water and native oil, protuberances occur at the common interface. It gives rise to the shape of fingers (protuberances) at common interface. The injected water shoots through inter connected capillaries at very high speed. It appears in the form of irregular trembling fingers, filled with injected water in the native oil field; this is due to the immiscibility of water and oil. The homogeneous porous medium is considered with a small inclination with the horizontal, the basic parameters porosity and permeability remain uniform throughout the porous medium. Based on the mass conservation principle and important Darcy's law under the specific standard relationships and basic assumptions considered, the governing equation yields a non-linear partial differential equation. The Crank–Nicolson finite difference scheme is developed and on implementing the boundary conditions the resulting finite difference scheme is implemented to obtain the numerical results. The numerical results are obtained by generating a MATLAB code for the saturation of water which decreases with the space variable and increases with time. The obtained numerical solution is efficient, accurate, and reliable, matches well with the physical phenomenon.