OPTIMIZING THE SOLUTION OF THE BUCKLEY-LEVERETT NONLINEAR STOCHASTIC EQUATION FOR THE SEPARATION OF OIL AND WATER IN THE RESERVOIR WITH NUMERICAL EXPANSION METHODS
Keywords:
Buckley-Leverett equation, Colocation method, Jacobi polynomial, Airfoil polynomial, Bernstein polynomial, Genocci polynomial, Boubaker polynomial, Petroleum Reservoir and gas.Abstract
There are many numerical simulation methods for reservoir modeling, The Buckley-Leverett equation provides important insight into the physical operation of oil-water separation.These equations are known as two-phase equations.In this article, considering the importance of observing environmental factors during oil and gas extraction and the importance of separating water from oil products,By introducing the Buckley-Leverett equation, it is investigated and presented an optimal method to solve it, using the colocalization method with orthogonal bases.Then, in a comparison, the optimality of the accuracy of the answer with Genocci's base We specified Airfoil, Jacopi, Bernstein, Bubaker bases. Differential equation with partial derivatives is one of the most important and widely used equations in maintaining oil and gas reservoir. This nonlinear equation is used in porous media. Solving these equations is very difficult due to their non-linearity and often involves discontinuities. According to the equation considered in this research, a level is randomly selected and water deficit flow is investigated. The fractional flow curve is obtained instead of the relative permeability curves, and the reservoir conditions are entered into the Buckley-Leverett model through the fractional flow function. With the equation, the relative permeability and displacement of two fluids, such as oil and water, which are flowing together, are investigated, Separation occurs in such a way that the initial saturation corresponds to the co-saturation. In this article, by implementing numerical methods and using orthogonal bases on the desired equation, non-linear devices are obtained, which can be solved by non-linear device solving methods, such as Newton's iterative method. In this method, the stop criterion ,can be reached with few repetitions. This issue shows the correctness of the desired method. Also, the uniqueness and convergence of the method have been investigated.
