Computation Of Dimensionless Pressure In A Vertical Well Using Gauss-Chebyshev Quadrature, Gauss-Kronrod Quadrature And Runge-Kutta Fourth Order

Abstract The determination of pressure distribution in a vertical well using analytical methods have posed a problem to petroleum engineers due to its cumbersome equations and difficulty in programming in computer system. In this paper three numerical methods are used to determine dimensionless pressure, which are (Gauss-Chebyshev quadrature, Gauss-Kronrod quadrature and Runge-Kutta fourth order). There results were compared with the analytical solutions based on the following assumptions: (1) An infinite acting reservoir (2) the well is producing at constant flow rate (3) the reservoir is at a uniform pressure when production begins (4) the well is centered in a cylindrical reservoir at radius re and (5)No flow across the outer boundary. The result shows that as rD increases, PD decreases, this indicate that when the radius of the wellbore is increasing, productivity decreases for a vertical well. This work also show that the accuracy of the quadrature methods depends on the nth terms used and also increases with rD. That is at higher rD the quadrature methods approximate to the exact solution. Runge-Kutta fourth method was found to give exact solution at lower step time but required high computation time.