A new dual-permeability model for naturally fractured reservoirs

Naturally fractured reservoirs (NFRs) are the reservoirs with two distinct types of porous media called the fracture and matrix. The pressure behavior of naturally fractured reservoirs is usually studied by using the Warren and Root model [Warren, J.E. and Root, P.J., The Behavior of Naturally Fractured Reservoirs, SPEJ, Soc. Pet. Eng. J., vol. 3, no. 3, pp. 245–255, 1963]. The Warren and Root model assumes that production from the naturally fractured system goes from the matrix to the fracture and thence to the wellbore. However, this assumption is oversimplified if the contrast between the permeability of the matrix system and that of the fracture system is not significant. In order to estimate the limits of validity of solutions based on the Warren-Root model and to study the pressure behavior of a naturally fractured reservoir when the contrast between the two permeabilities are not significant, it is necessary to solve the original model proposed by Barenblatt and Zheltov [Barenblatt, G.I. and Zheltov, Y.P., Fundamental Equations of Filtration of Homogeneous Liquids in Fissured Rocks, Soviet Phys., vol. 5, p. 522, 1960]. But the analytical solutions to this model which were obtained by numerical analysis or numerical inversion are very complex and inconvenient to use. Assuming that both the matrix system and fracture system produce directly into the wellbore, a new mathematical model for dual-permeability naturally fractured reservoirs is presented in this paper. Based on our proposed model, it is concluded that there are four stages for the pressure behavior of NFRs. The dual-permeability system behaves like a reservoir with a constant pressure boundary when the dimensionless time approaches infinite. The solution procedure proposed in this paper is a fast tool to evaluate a vertical well performance in a dual-permeability naturally fractured reservoir.