Orthogonal Collocation Method for Solving the Diffusivity Equation: Application on Dual Porosity Reservoirs With Constant Pressure Outer Boundary

Abstract

Fluid flow inside heterogeneous structure of dual porosity reservoirs is presented by two coupled partial differential equations (PDE). Finding an analytical solution for the diffusivity equations is tedious or even impossible in some circumstances due to the heterogeneity of dual porosity reservoirs. Therefore, in this study, orthogonal collocation method (OCM) is proposed for solving the governing equations in dual porosity reservoirs with constant pressure outer boundary. Since no analytical solution has been proposed for this system, validation is carried out by comparing the OCM-obtained results for “dual porosity reservoirs with circular no-flow outer boundary” with both exact analytical solution and real field data. Sensitivity analyses reveal that the OCM with 13 collocation points is a good candidate for prediction of pressure transient response (PTR) in dual porosity reservoirs. OCM predicts the PTR of a real field draw-down test with an absolute average relative deviation (AARD) of 0.9%. Moreover, OCM shows a good agreement with the analytical solution obtained by Laplace transform (AARD = 0.16%). It is worth noting that OCM requires a smaller computational effort. Thereafter, PTR of dual porosity reservoirs with a constant production rate in the wellbore and constant pressure outer boundary is simulated by OCM for wide ranges of operating conditions. Accuracy of OCM and its low required computational time justifies that this approximate method can be considered as a practical candidate for pressure transient analysis in dual porosity reservoirs.