Production from a fractured well with finite fracture conductivity in a closed reservoir: An exact analytical solution for pseudosteady-state flow

Pseudosteady-state (PSS) flow is a dominant time-dependent flow regime during constant-rate production from a finite closed reservoir. For a vertical well with a fully penetrated vertical fracture in a circular drainage area modeled as bounded by a slightly elliptical boundary, Prats et al. (1962) obtained an exact analytical solution for such a flow for the case of infinite fracture conductivity. For finite fracture conductivity, the current method to achieve a PSS solution is to run numerical simulations to long times. This paper extends the work of Prats et al. (1962), and presents the first exact analytical solution for PSS flow for a fully penetrated fractured vertical well with finite fracture conductivity. The exact analytical solution is expressed in terms of elementary functions, and it provides simple expressions for the PSS constant, dimensionless productivity index (PI), and effective wellbore radius. The present work eliminates the need of performing time-consuming numerical simulation for obtaining PSS solution for fractured wells in such a reservoir, and the new analytical solution can be used to generate approximate solutions for reservoirs of other geometrical shapes. The new solution can also be used for fracturedesign optimization and production-rate decline analysis.