Abstract
In many low-permeability gas reservoirs, producing a well at constant rate is very difficult or, in many cases, impossible. Constant-pressure production is much easier to attain and more realistic in practice. This is seen when production occurs into a constant-pressure separator or during the reservoir depletion phase, when the rate-decline period occurs. Geothermal reservoirs, which produce fluids that drive backpressure turbines, and open-well production both incorporate the constant-pressure behavior.
For finite-conductivity vertically fractured systems, solutions for the constant-pressure case have been presented in the literature. In many high-flow-rate wells, however, these solutions may not be useful since high velocities are attained in the fracture, which results in non-Darcy effects within the fracture.
In this study, the effects of non-Darcy flow within the fracture are investigated. Unlike the constant-rate case, it was found that the fracture conductivity does not have a constant apparent conductivity but rather an apparent conductivity that varies with time. Semianalytical solutions as well as graphical solutions in the form of type curves are presented to illustrate this effect.
An example is presented for analyzing rate data by using both solutions for Darcy and non-Darcy flow within the fracture. This example relies on good reservoir permeability from prefracture data to predict the non-Darcy effect accurately.
Introduction
To fully analyze the effects of constant-bottomhole-pressure production of hydraulically fractured wells, it is necessary that we understand the pressure behavior of finite-conductivity fracture systems producing at constant rate as well as the effects of non-Darcy flow on gas flow in porous media.
Probably one of the most significant contributions in the transient pressure analysis theory for fractured wells was made by Gringarten et al.1,2 In the 1974 paper,2 general solutions were made for infinite-conductivity fractures. Cinco et al.3 found a more general solution for the case of finite-conductivity fractures and further extended this analysis in 1978 to present a graphical technique to estimate fracture conductivity.4
For the case of constant pressure at the wellbore, solutions were presented in graphical form by Agarwal et al.5 In his paper, a graph of log (1/qD) vs. log (tDxf) can be used to determine the conductivity of the fracture by using type-curve matching. Although such a contribution is of great interest, unique solutions are difficult to obtain.
More recently, Guppy et al.6 showed that the Agarwal et al. solutions may be in error and presented new type curves for the solution to the constant-pressure case assuming Darcy flow in the fracture. That paper developed analytical solutions which can be applied directly to field data so as to calculate the fracture permeability-width (kfbf) product.
