The Solution of Instability Phenomenon Arising in Homogeneous Porous Media by Crank-Nicolson Finite Difference Method
Ravi N. Borana1, V. H. Pradhan2, and M. N. Mehta3
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During secondary oil recovery process, when water is injected in oil formatted area to recover remaining oil, the important phenomenon Instability (fingering) occurs due to the injecting force, difference of viscosity and wettability of the water and oil gives rise to the protuberances (instability) at common interface. The injected water will shoot through the porous medium at relatively high speed through the inter connected capillaries and its shape is as good as fingers, therefore it is also called fingering phenomenon and fingers are unstable due to the injecting force, so it is also known as instability phenomenon. The mathematical formulation yields to a non-linear partial differential equation known as Boussinesq equation. Its solution has been obtained by using unconditionally stable Crank-Nicolson finite difference scheme with appropriate initial and boundary conditions. The solution has been compared with the solution of this phenomenon. It is concluded that the solution obtained is very close to the solution obtained by finite element method and analytical method. The solution represents saturation of injected water which is increasing as length of fingers x -increases for given time t Ã¯ÂÂ¾ 0. The solution represents saturation of injected water occupied by average cross-sectional area of schematic fingers for instability phenomenon. The Matlab coding is used for numerical values and graphical presentation of the solution.