Applying orthogonal collocation for rapid and reliable solutions of transient flow in naturally fractured reservoirs

Naturally fractured reservoirs (NFR) contain a significant portion of the worldwide oil and gas reserves. Pressure transient analysis is the most well-established methodology for characterizing these heterogeneous reservoirs based upon well-test data. Pressure transient responses (PTR) of oil and gas reservoirs to various production/injection scenarios is widely used for forecasting and optimizing their production performance and ultimate resource recoveries. In this study, orthogonal collocation (OC) which is a straightforward and very efficient approximate approach for solving differential equations is employed as a novel approach to simulate the PTR of NFRs. The governing equations describe fluid flow in NFR for pseudo-steady state conditions. Constant production rate from a wellbore and circular no-flow are employed as an inner and outer boundary conditions, respectively. The results of OC are verified with exact analytical solutions using the Laplace transform. Thereafter, the effects of three parameters (storativity ratio, interporosity flow coefficient, and radial distance of the outer boundary from the wellbore) are studied using pressure-derivative plots. Employing the superposition theorem, the pressure response of NFR to the multi-rate production from the wellbore is calculated. The results indicate that this approximate approach can predict the pressure transient response of NFRs with an acceptable accuracy, i.e., absolute average relative deviation (AARD) of 0.16%. The equations can be adapted to deal with other flow conditions, boundary conditions and fracture stimulated reservoirs. The proposed method provides a straightforward, faster and more accurate solution in comparison with other methods commonly used. As the OC method is simpler to execute than analytical solutions (e.g. Laplace Transform), it is easier to modify and expand for addressing more complex reservoir scenarios. Likewise, the numerical methods (e.g. finite difference and finite elements) commonly applied to model fractured reservoirs involve significantly more time-consuming calculations, making them more cumbersome to use in reservoir simulations than the proposed orthogonal collocation method.