Non-linear Singular Integral Equations for Multiphase Flows in Petroleum Reservoir Engineering

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A very important problem of petroleum reservoir engineering is investigated by the present research. Consequently, an innovative and groundbreaking mathematical approach is introduced for the determination of the properties of the earth subsurface, when oil reserves together with water are moving through porous media. So, the multiphase flow problem of petroleum together with water is reduced to the solution of a non-linear singular integral equation , which is numerically solved by using the Singular Integral Operators Method (SIOM). The porous medium equation of the multiphase flows is defined as a Helmholtz differential equation and several properties are further analyzed and investigated for the above equation. Then by using the proposed new method, the estimation of the future oil production from a reservoir can be determined. An application is finally proposed for a well testing to be checked when a heterogeneous oil reservoir together with water in a multiphase flow is moving through a porous medium in the earth's subsurface. Hence, by using the SIOM, then the pressure response from the well test conducted in the above heterogeneous oil and water reservoir, is numerically calculated and investigated. Keyword and Phrases: Multiphase Flows of Oil with Water, Non-linear Singular Integral Equations, Singular Integral Operators Method (SIOM), Porous Medium Analysis, Helmholtz Differential Equation, Petroleum Reservoir Engineering