The common methods to model hydraulic fractures are based on the assumption of constant fracture conductivity. However, in some cases the hydraulic fracture conductivity may be a function of stress/pressure, and thus significantly change during production. It is reported that the hydraulic fracture conductivity may be reduced from a few to hundreds of folds in the literature.

This paper presents a semi-analytical model to facilitate transient pressure analysis for hydraulically fractured wells with stress-dependent hydraulic fracture conductivities. This model is developed for both hydraulically fractured vertical wells and multi-stage fractured horizontal wells. Considering the stress-dependent hydraulic fracture conductivities leads to the mathematical model being strongly non-linear. In order to solve the problem, hydraulic fractures are discretized into several slab source segments. The fluid flow from formation directly to fractures and flow inside fractures are calculated at each time step. Pressure distribution can be computed with complete fluid flow distribution. Then, the conductivities of hydraulic fractures are updated based on the pressure distribution. In each time step, an iteration process is used to deal with the relationship between fracture conductivities and the pressure.

The effect of stress-sensitive conductivities on transient pressure behavior is studied and type curves are documented. As the fracture conductivity decrease, the pressure and corresponding pressure derivative curves rise quickly and when the conductivity declines to the minimum value, the increasing pressure drop slows down. Therefore, a hump is formed on the pressure derivative curves. The slope of the hump is close to unit in log-log plot. The time of the hump's appearance and its size are determined by characteristics of hydraulic fractures, reservoir properties and production rate. Field examples from fractured vertical/horizontal wells are analyzed and reliable results are obtained.

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