Applying a new method to solve diffusivity equation for gas reservoirs

Abstract

Diffusivity equation is the general partial differential equation used to describe the flow of any fluid flowing in a radial direction in porous media. The main objective of this study is to present a new numerical scheme based on orthogonal collocation (OC) method to solve the diffusivity equation for heterogeneous and homogeneous gas reservoirs. OC is an approximate analytical technique which categorizes in the weighted residuals methods. The advantage and priority of the OC method over exact analytical solution (i.e., Laplace transform) is in the cases that the heterogeneity and variation of reservoir properties such as porosity and permeability with position or pressure could not be neglected; in these cases, the exact analytical solution is very tedious and may be impossible. The diffusivity equation has been solved by both OC and exact analytical solutions. To demonstrate the reliability of the proposed method, the results of OC method have been compared with those achieved using exact analytical solution. Average absolute deviation percent (AAD%) 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 numbers of collocation points result in significant improvement of OC method accuracy and its capability on dynamic pseudo pressure prediction. The minimum AAD% of 0.113 from the exact analytical predictions has been obtained with 25 collocation points. The results indicate that the proposed approximate method with these numbers of collocation points can predict the reservoir pseudo pressure trend with an acceptable accuracy in heterogeneous and homogeneous gas reservoirs.