Numerical Modeling for Enhancement of Oil Recovery via Direct Current

Abstract

Differential Quadrature (DQ) method is a numerical discretization technique which achieves accurate numerical solutions compared with the other lower order numerical method such as the Finite Difference (FD) method and Finite Element (FE) method by using a small number of grid points. Although DQ method can achieves an acceptable accuracy, but it is sensitive to the number of grid points. The Localized Differential Quadrature (LDQ) is developed from the DQ method which used to reduce the complexity of the system. Comparing with DQ method, LDQ method can yield very accurate numerical results by using large number of grid points. In this paper, the LDQ method is used in combination with Multiscale scheme and applied in two-dimensional wave equation, together with the fourth-order Runge-Kutta (RK) method. The Multiscale method is applied in critical area where the oscillations appeared. In the Multiscale process, additional layer of grid point is added which involved real and fictitious points. Thus, in other words, Multiscale method will produce localized course grids and fine grids which will be solved using RK method coupling with Lagrange interpolation. The Multiscale scheme with LDQ method is characterized by approximating the derivatives at the certain grid point which selected using the weighted sum of points in the neighbourhood. The proposed method is applicable in one- and two-dimensional wave equation problems. LDQ method with Multiscale scheme enables us to solve more complicated problems. Detailed proposed method is described, and numerical examples are used to demonstrate the present method which produces acceptable accuracy results.