Prediction of transient pressure response in the petroleum reservoirs using orthogonal collocation

Abstract

Diffusivity equation is one of the most widely used equations in petroleum engineering. It is basically derived to predict the dynamic pressure response and behavior of fluids flow in porous media with respect to time and position. The main objective of this study is the feasibility of applying of Orthogonal Collocation (OC) method to solve diffusivity equation in the radial transient flow system. Orthogonal collocation is an approximate analytical approach which categorizes in the weighted residuals methods. The advantage and priority of this method over numerical or exact analytical solution (i.e., Laplace transform) is in the cases which the heterogeneity and variation of reservoir properties such as porosity and permeability with position or pressure could not be neglected, in these situations, the numerical or exact analytical solution is very tedious and may be impossible. The diffusivity equation for an oil reservoir which obtained from literature has been solved by both orthogonal collocation and exact analytical solutions. To demonstrate the reliability of the proposed method, the results of this method have been compared with those achieved using exact analytical solution. MADP% (Mean Absolute Deviation Percent) has been used for determining the suitable number of collocation points to give acceptable error and best matching between approximate and analytical results. Sensitivity analysis indicates that increasing the number of collocation points result in significant improvement in its accuracy and capability on dynamic pressure prediction. The minimum MADP% of 0.1047 from the exact analytical predictions has been obtained by 20 collocation points. The results indicate that the proposed approximate method with these numbers of collocation points can predict the reservoir pressure trend with an acceptable accuracy.