Novel Mathematical Model for Transient Pressure Analysis of Multifractured Horizontal Wells in Naturally Fractured Oil Reservoirs

Multifractured horizontal wells have gained significant attention within the petroleum industry for commercial development. Despite considerable developments of transient pressure analysis or flow rate behaviors for horizontal wells in naturally fractured reservoirs, some significant problems are yet to be resolved, including high heterogeneity of reservoirs, pressure sensitivity of hydraulic fractures, and non-Darcy flow effect, which may occur during the production life. This paper presents a more pragmatic mathematical model for multifractured horizontal wells in naturally fractured reservoirs based on the fractal system, the theory of permeability modulus, and the time-fractional calculus correspondingly as an extension of the classic trilinear flow model. This new model comprises three modules: high heterogeneity of the reservoir based on the fractal system, the permeability modulus typically showing the pressure sensitivity of hydraulic fractures, and the anomalous diffusion describing non-Darcy flow turbulence. This investigation evaluates a trilinear dual-permeability dual-porosity flow model, with the dual-porosity model for the unstimulated outer reservoir flow region, the dual-permeability model for the stimulated inner reservoir flow region, and the permeability modulus for the flow region of hydraulic fractures. The comprehensive sensitivity analysis conducted indicates how the key parameters, such as fractal dimension, hydraulic fracture permeability modulus and conductivity, interporosity flow coefficient, storativity ratio, etc., affect the transient pressure behaviors, along with their reasons for the change in behavior. Application to a field case study further demonstrates the validity of the mathematical model, and the results presented may play a guiding role in well test interpretation.