Development of new analytical model for series and parallel triple porosity models and providing transient shape factor between different regions

Fluid flow architecture in fractured reservoirs is mainly based on the dual porosity model. In this model, the matrix block is considered as a fluid storage source and fracture as a flow path. Nevertheless, in naturally fractured reservoirs, different properties of matrix blocks and fracture systems have caused the triple porosity model to be more applicable than the traditional dual porosity model. The existed triple porosity models presented so far are mainly based on pseudo steady state flow in the matrix block and vug and via a semi-analytic form. In this paper, two analytical models of triple porosity model are presented with consideration of transient flow; the first model being: the series flow of the matrix block to the vug and the vug to the fracture, and the second model: transient parallel flow of the matrix block and vug to the fracture. In this paper, for the first time, an analytical and transient shape factor between different regions is put forward. Using the developed new solution, various flow regimes in triple porosity model have been identified. In the series triple porosity model, changing the parameters of vug and matrix did not affect the early production while it influenced the rate of the middle and lately times. The shape factor between the matrix and vug was independent of the physical properties of vug while the shape factor between vug and fracture was influenced by the physical properties of vug and matrix. In the parallel triple porosity model, the shape factor of matrix-fracture and vug-fracture were not dependent on the properties of the other regions of rock. Moreover, a decrease in inter-porosity coefficient and an increase in storativity ratio increased the shape factor in transient time. Additionally, in order to validate the analytical model, the results of the model are compared with the numerical simulations. This study presents a new analytical and transient shape factor which finds application in simulation of triple porosity formations.